criticality in power networks · jori, marta, clara, alessandro g (and chiara), murtuza, and my...

Criticality in power networks

Citation for published version (APA):Sloothaak, F. (2020). Criticality in power networks: a probabilistic approach. Technische Universiteit Eindhoven.

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Criticality in power networksA probabilistic approach

This work was financially supported by The Netherlands Organization for Sci-entific Research (NWO) through the Gravitation Networks grant 024.002.003.

c© Fiona Sloothaak, 2020

Criticality in power network: A probabilistic approach

A catalogue record is available from the Eindhoven University of Technology libraryISBN: 978-94-6380-660-2

Printed by ProefschriftMaken || www.proefschriftmaken.nlCover by Bregje Jaspers || www.proefschriftontwerp.nl

Criticality in power networksA probabilistic approach

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de TechnischeUniversiteit Eindhoven, op gezag van de rector magnificus

prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door hetCollege voor Promoties, in het openbaar te verdedigen op

donderdag 16 januari 2020 om 16:00 uur

door

Foekje Sloothaak

geboren te Lemsterland

Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van depromotiecommissie is als volgt:

voorzitter: prof.dr.ir. B. Koren1e promotor: prof.dr. A.P. Zwart2e promotor: prof.dr.ir. S.C. Borstleden: prof.dr. N.V. Litvak (Universiteit Twente)

prof.dr. J.S.H. van Leeuwaardenprof.dr. J.L. Hurink (Universiteit Twente)dr. M. Vlasioudr. V. Shneer (Heriot-Watt University)

Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven is uitgevoerd in overeen-stemming met de TU/e Gedragscode Wetenschapsbeoefening.

Acknowledgments

It is very difficult to point out the most important part of the thesis. Yet, if onewould simply measure it by the number of times that a section is read, it is likelythat the acknowledgments will be the absolute winner. I would like to takethis opportunity to express my gratitude towards the many people who havesupported and encouraged me during my PhD.

Without a doubt, the two persons I am mostly indebted to, are my supervisors BertZwart and Sem Borst. Bert, you have an inspiring drive and passion where yougive your all to whatever is on your mind. Using the approach of total honesty,you allowed me to get the most out of my PhD: I was able to work on a widerange of different topics and pursue my own ideas. You have been a tremendoussupport throughout these years, whether it was when you had to convince meto work on a conference paper I believed was hopeless (indeed, I was wrong),providing a creative perspective on a problem, or advising me on academic life.Sem, you were definitely the calming note that I needed from my supervisors.Your thoroughness, deep insights and writing skills are really exceptional. Youtaught me that to occasionally slow down and focus on some critical details first.Together with Bert, I could not have wished for better academic parents.

I would also like to thank all the members of my defense committee: Nelly Litvak,Johan van Leeuwaarden, Johann Hurink, Maria Vlasiou and Seva Shneer. Thankyou for taking the time to read this body of work. A special thanks goes to Johanand Maria, who were of pivotal importance in my choice to pursue a PhD. Johan,you introduced me to the world of academics and I think it is safe to say thatwithout your infectious enthusiasm, I would not be at this point of my life. Maria,I am not sure if you even remember, but it was a particular conversation we hadthat convinced me to choose this specific PhD project. Thank you for this, andalso for the many endless conversations we had throughout the years.

I was lucky to have had the opportunity to work with several researchers. I wouldlike to mention Vitali Wachtel, who opened up my eyes to a whole new world of(new) mathematical techniques, which led to the results in Chapter 3. I want tothank Lorenzo Federico for our collaboration that resulted in Chapter 5. TommasoNesti, without your knowledge, insights, devotion and hard work, the results

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in Chapter 6 would probably not have seen the light of day. James Cruise andSeva Shneer, thank you for introducing me to topic of battery swapping systems,resulting in Chapter 7 and 8 of my thesis. I would also like to thank AlessandroZocca for our discussions (even though they never actually led to a project), andfor our joint effort to organize the winter school on energy systems.

The STO section has been an amazing place to work in. We owe that to a largeextend to our incredible secretariat and workshop officer at EURANDOM. Petra,Chantal and Patty, your doors were always open to talk about all parts of life thatdo not involve mathematics. You created a safe and friendly environment wherewe can share all the good and the bad. I would like to also acknowledge the otherstaff that contribute to the great atmosphere: Ellen, Onno, Jacques, Marco, Stella,Koo, Remco and Edwin (when present).

I really feel like I lived through two generations of PhD students. I would liketo thank the first generation for making me feel like home: Jaron, Gianmarco,Jori, Marta, Clara, Alessandro G (and Chiara), Murtuza, and my first office mates.Thomas, I really appreciated your sense of humor, that seemed to become dryerand dryer as the end of your PhD life came closer. Britt, thank you for yourfriendship and offering a listening ear whenever I needed it. Fabio, thank you foryour positivity, sharing your and Anna’s special day with us, and the many eventsyou suggested during your time in Eindhoven (we only started appreciating themafter you left the Netherlands).

And before you know it, you become the office senior. I would like to thank Youri,Collin and Mona for keeping the office just as "gezellig"; it was certainly noisierafter you arrived. I think I refrain from dishing out any more compliments untilafter the defense party. I would also like to thank Richard and Kay for making thecommute back home less boring, Angelos for being my default conference buddy,and also Ellen, Joost (the desk-thief), Liron, Mark, Mayank and Rik. Marieke,Jan-Pieter, Mariska, Birgit, Margriet, Pieter, David, Jaap, Tom and Brendan: thanksfor making the NETWORKS training weeks so enjoyable.

I was fortunate to have the support of my friends and family. Thank you, Nienkeand Peter, Sida and Hans, Kirsten, Marlies, Willeke and Mark, JC capae, JC capi,Marijn and Anneke, Guus, Vincent and Ruby. Ik wil ook mijn moeder bedankenvoor haar steun door de jaren heen, alsmede mijn zussen en aanhang: Gerda, Anja,Marjolein, Andries, Jesse, Milan, Rinske en Jan-Douwe. Ik kan ook zeker mijn(schoon)familie niet vergeten: Ed, Jeanine, Job en Anne, Jelle en Annet, Frank,Monique, Tim en Daphne. Bedankt voor het bieden van all gezellige momentennaast het "schoolwerk".

I would also like to address the ones on the cover of this thesis, Markov and Bart.I was happy for Markov to bring some relief of stress in the final stages of myPhD. But my final words are for you, Bart. You bring laughter, happiness and lovein my life. I feel lucky that I can share my life with you, and look forward to themany more joyful moments together in the future.

Contents

1 Introduction 11.1 Power grid operations . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Blackout characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Cascading failure models . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Sandpile models . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 CASCADE model . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Branching models . . . . . . . . . . . . . . . . . . . . . . . . 71.3.4 Hidden failure model . . . . . . . . . . . . . . . . . . . . . . 71.3.5 OPA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Contributions to cascading failure models . . . . . . . . . . . . . . . 91.4.1 Robustness of power-law behavior . . . . . . . . . . . . . . . 91.4.2 Effect of network disintegration . . . . . . . . . . . . . . . . 101.4.3 Causal effect of city population sizes . . . . . . . . . . . . . . 11

1.5 Electric vehicle battery swapping systems . . . . . . . . . . . . . . . 12

2 Robustness of power-law behavior in cascading failure models 152.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Affine case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Proof of Theorem 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Identifying thresholds where power-law behavior prevails . . . . . 39

3 First-passage asymptotics for random walk bridges 433.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Model description and preliminaries . . . . . . . . . . . . . . . . . . 463.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5 Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6 Threshold close to return point . . . . . . . . . . . . . . . . . . . . . 50

3.6.1 Density of random walk . . . . . . . . . . . . . . . . . . . . . 503.6.2 Proof of Theorem 3.4.2 . . . . . . . . . . . . . . . . . . . . . . 63

4 Impact of a network disconnection 694.1 Model description and preliminaries . . . . . . . . . . . . . . . . . . 70

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iv Contents

4.2 Main results and road maps of the proofs . . . . . . . . . . . . . . . 724.2.1 Balanced component sizes . . . . . . . . . . . . . . . . . . . . 734.2.2 Disparate component sizes . . . . . . . . . . . . . . . . . . . 78

4.3 Proofs of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3.1 Very few or many failures in one component . . . . . . . . . 804.3.2 Asymptotic behavior of some summation terms . . . . . . . 854.3.3 Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . . . . 874.3.4 Proof of Theorem 4.2.2 . . . . . . . . . . . . . . . . . . . . . . 894.3.5 Proof of Theorem 4.2.3 . . . . . . . . . . . . . . . . . . . . . . 93

4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.4.1 Our approximation scheme . . . . . . . . . . . . . . . . . . . 974.4.2 Balanced component sizes . . . . . . . . . . . . . . . . . . . . 984.4.3 Disparate component sizes . . . . . . . . . . . . . . . . . . . 99

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.5.1 Load surge function . . . . . . . . . . . . . . . . . . . . . . . 1004.5.2 General network topologies . . . . . . . . . . . . . . . . . . . 1014.5.3 Heterogeneous edge capacities . . . . . . . . . . . . . . . . . 102

5 Cascading failures on complex networks 1055.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . 1055.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.1.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2 Proof strategy for Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . 1115.2.1 Relation of failure process and sequential removal process . 1115.2.2 Disintegration of the network through sequential removal . 1135.2.3 Impact of disintegration on the failure process . . . . . . . . 114

5.3 Disintegration of the network . . . . . . . . . . . . . . . . . . . . . . 1165.3.1 Percolation on the connected configuration model . . . . . . 1165.3.2 Explosion algorithm . . . . . . . . . . . . . . . . . . . . . . . 1185.3.3 Typical structure of the percolated configuration model . . . 1205.3.4 Probabilistic bounds on component sizes outside the giant . 1305.3.5 First disconnections . . . . . . . . . . . . . . . . . . . . . . . 1325.3.6 Number of edges outside the giant component . . . . . . . . 1385.3.7 Linear number of edge removals . . . . . . . . . . . . . . . . 143

5.4 Cascading failure process . . . . . . . . . . . . . . . . . . . . . . . . 1435.4.1 No edge disconnections . . . . . . . . . . . . . . . . . . . . . 1445.4.2 Random walk formulation . . . . . . . . . . . . . . . . . . . 1475.4.3 Behavior of the number of edge failures in the giant . . . . . 1505.4.4 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . 162

5.5 Universality principle . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6 Relation between city population sizes and blackout sizes 1716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.2.1 Preliminaries: DC approximation . . . . . . . . . . . . . . . . 1736.2.2 Our framework . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Contents v

6.3 Main result and road map of the proof . . . . . . . . . . . . . . . . . 1786.4 Principle of a single city with large demand . . . . . . . . . . . . . . 1806.5 Closed-form solution for the operational OPF in a special case . . . 1816.6 Convergence of the cascade sequence . . . . . . . . . . . . . . . . . 1826.7 Asymptotic behavior of power imbalance . . . . . . . . . . . . . . . 1886.8 Cascade analysis for 6-node topology . . . . . . . . . . . . . . . . . 191

6.8.1 Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.8.2 Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.8.3 Tail behavior of blackout size . . . . . . . . . . . . . . . . . . 193

6.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7 Battery swapping dynamics within a single facility 1997.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.3 QED regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2027.4 Steady-state distribution . . . . . . . . . . . . . . . . . . . . . . . . . 2037.5 Limiting queue length behavior . . . . . . . . . . . . . . . . . . . . . 2047.6 Interchange of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.7 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . . . 2127.8 Unlimited number of charging points for single station system . . . 214

7.8.1 Fluid and diffusion limits . . . . . . . . . . . . . . . . . . . . 2157.8.2 Steady-state limits . . . . . . . . . . . . . . . . . . . . . . . . 2177.8.3 Performance measures . . . . . . . . . . . . . . . . . . . . . . 221

7.9 Unlimited number of swapping servers for single station system . . 2227.9.1 Diffusion limits . . . . . . . . . . . . . . . . . . . . . . . . . . 2227.9.2 Steady-state limits . . . . . . . . . . . . . . . . . . . . . . . . 2247.9.3 Performance measures . . . . . . . . . . . . . . . . . . . . . . 227

7.10 Provisioning scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8 Battery swapping dynamics in a network setting 2318.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2318.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.3 System dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2358.4 Fluid limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2378.5 Diffusion limit and system performance . . . . . . . . . . . . . . . . 2428.6 Proof of Theorem 8.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 245

8.6.1 Hydrodynamic scaling and its limiting process . . . . . . . . 2468.6.2 The SSC function . . . . . . . . . . . . . . . . . . . . . . . . . 2548.6.3 Multiplicative state space collapse . . . . . . . . . . . . . . . 2568.6.4 Strong state space collapse . . . . . . . . . . . . . . . . . . . . 260

8.7 Simulation experiments . . . . . . . . . . . . . . . . . . . . . . . . . 2658.7.1 State space collapse for exponential charging times . . . . . 2668.7.2 Non-Markovian charging time distribution . . . . . . . . . . 270

8.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

References 273

Summary 283

1Chapter 1

Introduction

Modern-day society relies heavily on a steady and secure supply of electricity,making power grids critical and indispensable infrastructures. Fortunately, gridshave proven to be capable of providing an extremely high level of reliability [14].Still, in the last two decades, several highly visible blackouts have taken place, suchas the Northeast blackout of 2003, the India blackout in 2012, or the recent blackoutin Argentina, Paraguay and Uruguay in June, 2019. Each blackout affected tens tohundreds of millions of people, causing major societal and economical disruptions.The increase in the frequency at which these major blackouts occur is extremelyalarming [14], and the prevention of these catastrophic events is perceived as oneof the most important and challenging problems we face in the 21st century [85].

In order to prevent power outages in the future, a better understanding is neededof how and why these events can take place, and what has changed over theyears to cause them to occur more often. Naturally, power grids have grownconsiderably in size and complexity throughout the last century. Moreover, tech-nological advances, such as renewables, are creating significant structural changesto existing power grids and the way they operate. Traditionally, the delivery ofelectricity was a top-down system where energy producers had to meet demand,and transmit the production to their consumers [126]. Control of power gridoperations has therefore been designed and developed from this point of view.The introduction of more sustainable energy sources, such as solar, wind andgeothermal power, caused power generation to become more decentralized andvolatile.

Nowadays, a major tool for grid operators to ensure reliability is to apply theN−1rule [14], which dictates that at all times the failure of any single component doesnot cause the grid to destabilize. The complexity of this problem is significant: anyfailure of one of the N components gives rise to different large sets of reliabilityconstraints, and for each case there needs to be a feasible solution. In addition,there is a time-varying element that plays a role. Existing frameworks used incurrent grid operations typically rely on (deterministic) optimization techniquestogether with extensive scenario testing. The complexity of this N − 1 reliab-ility problem is significant, especially since N is usually large. The increase of

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2 Chapter 1. Introduction

variability in power generation, as well as demand, makes this problem truly com-putationally infeasible. Existing frameworks need to be enriched with stochasticfeatures to deal with increasing volatility, as well as to obtain a more fundamentalunderstanding why severe blackouts occur.

In this thesis, we introduce macroscopic stochastic models to obtain a betterunderstanding of the interaction between different elements in power networks.Using a probabilistic approach, we consider how small disturbances can cascadethrough a network and possibly lead to severe blackouts. To prevent these poweroutages, it is also crucial to reduce variability in power demand, and a batteryswapping infrastructure for electric vehicles may contribute to this purpose. Forsuch systems, we establish capacity levels that guarantee good quality-of-serviceand near-optimal resource utilization under a dynamic control policy.

Throughout this thesis, we use the following conventions in our notation. For ran-

dom variables (Xn)n∈N, we write Xnd→ X and Xn

P→ X to denote convergence indistribution and in probability as n→∞, respectively. For real-valued sequences(an)n∈N and (bn)n∈N, we write an = o(bn) and an = O(bn) if limn→∞ an/bn tendsto zero or is bounded, respectively. Similarly, we write an = ω(bn) and an = Ω(bn)if limn→∞ bn/an tends to zero or is bounded, respectively. We write an = Θ(bn) ifboth an = O(bn) and an = Ω(bn) hold. We say that an ∼ bn if limn→∞ an/bn = 1.We adopt an analogous notation in case we consider functions instead of sequences.Finally, Poi(λ) denotes a Poisson distributed random variable with mean λ, Exp(λ)an exponentially distributed random variable with parameter λ, and Bin(n, p) abinomially distributed random variable with parameters n and p.

The remainder of this introduction is organized as follows. First, we provide ahigh-level description of power transmission systems and the central elementsin Section 1.1. In Section 1.2, we describe the main characteristics of poweroutages that are of critical importance in this thesis. These include the notionsthat blackouts typically occur through a cascade of failures, and that analysis ofhistorical blackouts suggest that the "blackout size" exhibits scale-free behavior. InSection 1.3 we review several cascade models in related literature, and we discussthe contributions of this thesis to cascading failure models in Section 1.4. Finally,we move to electric vehicle battery swapping systems in Section 1.5.

1.1 Power grid operations

A power grid can be seen as an interconnection system that conveys power fromgenerators to demand locations, also called loads [14]. Generally, the grid consistsof two subsystems: the high-voltage transmission system that typically carriespower over long distances, and a collection of low-voltage distribution systems.In Chapters 2-6, we focus on high-voltage transmission systems and only considerstatic models where power injections are not time-dependent. On the other hand,a distribution system carries power through a narrow geographical area, such ascities, and is connected to the transmission system at a single point. Chapters 7and 8 relate to the charging of electric vehicles, which has a direct impact on thedynamics of the distribution systems. Inherently, it also affects the transmissionsystem, but to a much smaller extend.

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1.1. Power grid operations 3

A transmission system can be represented through an undirected graph. Nodes(buses) represent either (a collection of) generators or points where a distributionsystem is attached (a load), while edges (branches) represent transmission linesthat convey power between nodes. In order to transmit power over long distanceswith minimal loss, grids are typically built to deal with alternating-current (AC)power. We provide a high-level description of the operations of a (static) ACpower grid next. For a more detailed account, we refer to [14, 126].

Every node has a complex-valued power injection, where the real part is calledthe active power and the imaginary part is called the reactive power. The voltageat a node is also a complex-valued quantity that can be described through its(voltage) magnitude and phase angle. The voltage drop on a line is the differenceof the voltages between the two connected nodes. Ohm’s law states that thecurrent on a line is proportional to the voltage drop of its (end) nodes, where thefactor of proportionality is called the admittance. The admittance of a line can bewritten as the inverse of a complex-valued quantity, where the real part is calledresistance and the imaginary part the reactance. Next to Ohm’s law, the systemobeys Kirchoff’s (current) law stating that the sum of currents flowing into a nodeequals the sum of currents flowing out of that node, implying that the currentinjected in a node equals the total sum of currents at the lines attached to thatnode. Moreover, the power at a node is defined as the multiplication of the nodevoltage and the conjugate of the nodal current. Using the above three relations,one can derive the non-linear AC power flow equations, which directly relate thepower injections to the power flows through the lines.

A central question in power grid operations is how the generation of power shouldbe distributed among the locations. Typically, the objective adopted in powerengineering practice is minimizing a separable convex (cost) function in terms ofpower generation per location, subject to a set of constraints. These constraintsare set in place to prevent damage to any equipment, or power lines to loose theirtensile strength. The constraints typically describe bounds for the temperature(or current/power), voltages and reactive power. Due to the non-convexity ofthe underlying AC power flow equations, the generation distribution problem isextremely complex. An often-used tool is the Direct Current (DC) approximationmodel, and also plays a central role in Chapter 6.

The DC approximation entails three assumptions: i) the voltage magnitudes areequal at every node and set to one, ii) the phase angles of the currents are smallfor all lines, such that the currents at the line can be approximated by a linearequation (using polar coordinates), iii) at any line the resistance is much smallerthan its reactance, implying that the admittance can be approximated by thecomplex inverse of the reactance. These operations allow for a linearization ofthe AC power flow equations, and the result is also known as the DC power flowequations. More specifically, suppose the grid is represented by a network with nnodes and m edges. Using the above three assumptions, the topology of the gridand the line reactances can be encoded by a matrix V ∈ Rm×n. The active linepower flows f = (f1, ..., fm) can then be written as a linear mapping of the activepower injections p = (p1, ..., pn) through the simple relation

f = Vp.

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To find an optimal way to distribute power generation among the nodes, oneminimizes a convex separable cost function subject to the constraints that the sumof the power injection equals zero (generation equals demand), the power flowsare between certain bounds at all lines, and the power generations at the nodesare bounded. The constraints are all linear, simplifying the optimization problemsignificantly compared to the AC power framework.

It may be apparent that the DC power framework involves a static deterministicoptimization problem. Most tools used in the power engineering community relyon these types of techniques. With the increase of volatility in power generationand demand, this may cause problems in securing a steady and reliable deliveryof electricity [14, 85]. To prevent major blackouts in the future, these frameworksneed to be upgraded with stochastic elements to deal with the variability in thesystems. In Chapters 2-6, we therefore focus on stochastic models that contributeto obtaining a fundamental understanding of the mechanisms that lead to severepower outages.

1.2 Blackout characteristics

Blackouts typically occur through a cascade of failures that accelerate and outstripcontrol capabilities [14, 22, 57, 105, 130]. The understanding of the failure mech-anism in cascading blackouts is extremely challenging due to various aspects,including the huge size of the system (curse of dimensionality [54]), limited com-putational power (N −2 contingency analysis is typically already computationallyinfeasible [66]), mechanisms in an evolving system (thermal dynamics, voltagecollapse, etc. [10]) and poorly-understood interactions between components [65,98]. Moreover, although power outages occur increasingly often, such eventsremain relatively rare. Consequently, validation of models remains difficult sinceonly a limited number of blackouts have occurred up to this day [116].

Many models and tools have been developed over the years to obtain a betterunderstanding of cascading blackouts. A very active field of research in the powerengineering community involves the development of detailed simulation toolsthat localize fragile parts of the power grid through (offline) scenario testing [110].It may be apparent that the computational power required becomes prohibitive asmore details are taken into account. Rare-event simulation, such as importancesampling and splitting [70, 106], can be used to speed up the computations andanalyze larger and more involved power networks. Also, microscopic models thatmake simplifying assumptions [44, 86, 105, 108], such as the DC approximationframework, are used extensively in practice. Yet, the propagation of cascadingfailures remains hard to analyze, and provides little insight in the mechanism thatleads to a severe blackout on a fundamental level.

A macroscopic characteristic that appears in data analysis of historical blackoutdata, is that both the total number of affected customers as well as the amountof lost power have a power-law tail [24, 26, 38, 57]. More specifically, a randomvariable X has a power-law tail with exponent α > 0 if for some c > 0, as x→∞,

P(X > x) ∼ cx−α. (1.2.1)

1

1.3. Cascading failure models 5

A power-law distribution is a statistical law that arises in many applications ofscience and engineering [11, 12, 31, 107, 112]. It is often also called scale-free, asrealizations of X take a wide range of values and exhibit an extreme amount ofvariability [12]. A central question in this thesis is why scale-free behavior (1.2.1)emerges in blackout sizes.

Various literature streams provide an explanation for the emergence of scale-freebehavior in other applications. In many of these applications, e.g. the internet,the scale-free phenomena relate to the nodal degree distribution [12]. There is amassive amount of literature in complex network theory that derives propertiesfor these systems. Yet, making a direct analogy to power networks is difficult [50]as power grid topologies are not scale-free [127].

Another explanation of how large-scale system-wide failures can happen involvesself-organized criticality (SOC), a theory that comes from the field of statisticalphysics [6, 8, 9, 52]. A system that exhibits SOC is one that is in a critically stablestate, and when it is perturbed, the system naturally moves to another criticallystable state. Typically, the evolution of these dynamical systems can be describedthrough simple local relations, and leads to a point where power-law distributedevent sizes appear. Power grids are designed to operate efficiently subject to alarge set of reliability constraints. This causes power grids to behave as dynamicalsystems that naturally evolve to critical states that are barely stable, which is verymuch in the spirit of SOC. However, it is not straightforward to determine whetherpower system blackouts are governed by SOC since there are no strictly definingcriteria [116]. Nevertheless, the ideas of SOC have initiated a stream of literature ofcascading failure models for power systems, which we discuss in the next section.

1.3 Cascading failure models

Cascading failures can happen in many different systems, such as electric powersystems [24, 38, 96], communication networks [73, 83, 84], road systems [29, 134],material science [95], epidemiology theory [82, 128] and more. The idea behindcascades is simple: there is an interconnected system consisting of componentsthat have a certain capacity. An initial disruption/failure puts an additional strainon the system, which may lead to capacity limits of components to be exceededand in turn cause component failures. Every failure leads to a redistribution ofload, possibly causing more components to fail and triggering knock-on effects.Next, we review five (macroscopic) models that have served as inspiration for thecascading failure models that we consider in this thesis. For a more comprehensivelist of models, we refer the reader to [116].

1.3.1 Sandpile models

Although our interest is in the application of power systems, it may be goodto realize that the general study of cascading failures has been of interest fora long time in a broad range of contexts. A classical setting involves the Bak-Tang-Wiesenfeld (BTW) sandpile model [8, 9] that was developed to understandavalanche behavior [7]. With respect to graphs, this model can be explained asfollows. Consider a (fixed) network with n nodes, where di denotes the degree of

1


node i. Every node can hold a capacity of Ci grains of sand, and usually, thesecapacities are set as Ci = di − 1. Every node has a random initial load that doesnot exceed its capacity, and the cascade is triggered by dropping one additionalgrain at a randomly chosen node. Whenever a node exceeds its capacity, it topplesover a grain of sand to all of its neighbors such that its own stock is lowered by di,and all of its neighbors have (possibly) received one additional grain. A grain thatmoves from one node to another can be lost with a small probability ε.

This model was originally introduced on lattices, and later also considered ondifferent types of networks such as the Erdös-Rényi random graph [17] and scale-free networks [52], i.e. networks where the degree distribution has a power-lawdistribution. A main feature in these models is the emergence of a power-law tailin the number of toppling events (the avalanche size). In particular, the avalanchesize distribution in the Erdös-Rényi random graph follows a power law withexponent 3/2 (and hence the tail is a power law with exponent 1/2). For scale-freenetworks whose degree distribution follows a power law with exponent γ, theavalanche size distribution also follows a power law, yet, the exponent of thepower law depends on the value of γ. If 2 < γ < 3, then the exponent is γ/(γ − 1),and if γ > 3, then the exponent is 3/2 as well.

The theory of SOC was instigated by the Bak-Tang-Wiesenfeld sandpile model [9],which may explain why this model has received attention in the application ofpower systems as well. In [28], the blackout size distribution from empirical datais compared with results from a one-dimensional idealized sandpile model, andthey show that the blackout data seems consistent with SOC.

1.3.2 CASCADE model

The CASCADE model was introduced in [40], and involves a stochastic load-dependent cascading failure model. In this section, we adopt their frameworkwith an altered notation for consistency with the notation used in Chapter 2.

Consider a system with n identical components, each carrying an initial load thatis uniformly distributed. Component failures occur whenever the load exceedsa certain threshold, which in turn transfers a fixed amount of load to all othercomponents. The cascade is initialized by an additional loading of all componentsby another fixed amount. We note that this framework can be normalized suchthat the initial loads are uniformly distributed in [0, 1] and the failing threshold isone. The load surges are given by the function

ln(i) =θ + (i− 1)λ

n, i = 1, ..., n,

where θ/n > 0 denotes the (normalized) initial disturbance at every line, andλ/n > 0 the (normalized) fixed load surge after every failure. The number offailed components An follows a (modified) quasi-binomial distribution, givenby [40]

P (An = k) =

(nk

)θn

(θ+λkn

)k−1 (1− θ+λk

n

)n−kif 0 ≤ k ≤ min

n−θλ , n− 1

,

0 if n−θλ < k ≤ n− 1,

1−∑n−1j=0 P (An = j) if k = n.

1

1.3. Cascading failure models 7

For large networks, this distribution can be approximated by letting n→∞, forwhich it converges to a generalized Poisson distribution:

P (An = k) ≈ θ (θ + λk)k−1

k!e−(θ+λk).

This distribution is closely related to a branching process [71], as it correspondsto the offspring size in a branching process where the first generation produces aPoisson distributed offspring with mean θ, and every subsequent child producesan offspring with mean λ. In the critical case where λ = 1, it is well-known [89]that the offspring distribution obeys a power-law distribution with exponent 3/2.

1.3.3 Branching models

The relation to branching processes that emerged in the CASCADE model promp-ted a line of research in branching-type models aimed at attaining a better un-derstanding of blackout propagation [39, 71, 72, 96]. A branching process canbe described as follows [4, 56]. The cascade is initiated by the outage of oneor several components. In every generation, each “parent” outage produces arandom number of “children” outages in the next generation. That is, the childrenof one generation become parents to the next generation. If the number of outagesin a generation is zero, then the cascade stops. The total number of outages is thefailure/blackout size.

We stress that in this general description of branching processes we make noassumptions on the offspring distribution. It is well-known that the mean of theoffspring prescribes the behavior of the branching process in a meaningful way: ifthe mean is less than one, then with high probability the failure size is finite, whilea mean strictly larger than one corresponds to a supercritical branching processwith a strictly positive probability that the cascade does not end in finite time(i.e. the failure size is infinite). The critical window where the mean equals onecorresponds to the case where power-law behavior appears for the failure size,corresponding to the critical point where the model seems to be governed by SOC.

There is a strand of literature that uses branching models together with statisticalanalysis of historical and simulated data to derive more insight in cascadingblackouts. A challenging aspect in this line of research concerns the processingof data: much effort is put in discretization of continuous data [96], validatingthe parameters and deriving meaningful conclusions. For example, the objectof interest is the total amount of load that is shed (i.e. amount of lost power),a continuous value that cannot be captured solely by a discrete process. Thereare generalizations of the framework to continuous [72] or multi-type branchingprocesses [97].

1.3.4 Hidden failure model

A hidden failure of a protection system in power grids is a permanent defectthat is typically difficult to detect during normal operations. Whenever otherdisturbances occur in the system, this hidden failure can be exposed. Yet, it maybe too late to adequately deal with the hidden failure and restore the system to

1


stability. Although we do not consider this aspect in this thesis, the hidden failuremodel [30, 94] is worth mentioning as it has some elements in common with themodels in this thesis.

The model considers a network where an initial disturbance is caused by tripping arandomly selected line. Power flows are redistributed using the DC approximation.All lines that violate the line flow constraints are tripped. If there is no violation,all lines connected to the last tripped line(s) are tripped with a small probability.If the network disintegrates into multiple islands, a redispatch is performed usinga load shedding minimization problem. The hidden failure model reveals animportant aspect of cascading blackouts: the cascade of failures occurs non-locallyand depends on the global network structure, which continually changes as thefailure process continues.

1.3.5 OPA model

The OPA model [25, 27, 37, 100] is a simulation tool that considers the patterns ofcascading blackouts of a power system under the complex dynamics of a growingdemand and the engineering responses to failure [116]. The model representstransmission lines, loads, and generators and computes the network power flowswith a DC load flow. It has a two-loop structure including fast and slow dynamics.The fast dynamics in the inner loop simulates power flow and cascading failuresthat occur in a fast time scale of minutes to hours. The slow dynamics in the outerloop simulates the evolution of a power grid considered on a daily scale. After thesimulation of the current day the load is increased and the system is upgraded,based on which the simulation of the next day is performed.

Each simulation run starts from a solved base case solution for the power flows,generation, and loads that satisfy circuit laws and constraints. To obtain diversityin the runs, the system loads at the start of each run are varied randomly aroundtheir mean values by multiplying by a factor uniformly distributed in [2− γ, γ]with γ ∈ [0, 1). In the fast dynamics, initial line outages are generated randomly byassuming that each line can fail independently with a fixed probability. Whenevera line fails, the generation and load is redispatched to satisfy the transmissionline and generation constraints using the DC framework. The optimization costfunction is weighted to ensure that load shedding is avoided wherever possible.If any lines were overloaded during the optimization, then these lines are likely tohave experienced high stress, and each of these lines fails independently with aspecified probability. The process of redispatching and testing for line outages isiterated until there are no more outages. The slow dynamics update the load andgeneration limits by increasing these quantities by a fixed factor. In addition, thetransmission line limits are increased by a factor at all lines that are overloaded inthe slow dynamics.

The improved OPA model [77] considers the unwanted operation of protectiverelays and the failure of the energy management system or communications. Thisis modeled by including a probability of failure at lines that are not loaded, whichdepends on how close the power flow at the line is to its limit. The simulation toolhas also been extended to the AC power flow framework [78, 79].

1

1.4. Contributions to cascading failure models 9

1.4 Contributions to cascading failure models

In this thesis, we make the following contributions to the field of cascading failuremodels.

1.4.1 Robustness of power-law behavior

To obtain a better understanding why scale-free behavior appears in the failuresizes of cascading failure models, we start this thesis by considering the CASCADEmodel. To the best of our knowledge, this model was one of the first that usesan analytic approach in order to provide a possible explanation why power-lawbehavior appears in the blackout size. The model does not account for any of thepower flow dynamics, but does capture some salient features such as the large sizeof the system, the initial disturbance stressing the network, the component failurewhen its capacity is exceeded, and the additional strain on other componentswhen a component failure has occurred. This results in a tractable model thatallows for a rigorous derivation of the number of component failures.

In this thesis, we consider the number of line (edge) failures in a power grid,which is modeled by a graph. We translate the CASCADE model to the settingof this thesis by saying that components correspond to power lines, where everyline failure increases the load of the other lines. This can be modeled through agraph that is shaped as a star: every line failure disconnects a single node fromthe graph, which remains star-shaped.

The scale-free behavior is identified by first letting the number of lines grow large,and then considering large values of the failure size An, i.e.

limk→∞

limn→∞

k3/2P(An = k) =θ√2π.

Naturally, this provides a rough estimation of the failure size. We extend thisresult in two ways. First, we identify the asymptotic behavior if k := kn dependson n. More specifically, for every k := kn for which k → ∞ and n − k → ∞ asn→∞,

P(An = k) ∼ θ√2π

1√1− k/n

k−3/2,

or equivalently,

P(An ≥ k) ∼ 2θ√2π

√n− kn

k−1/2.

In other words, we obtain a more accurate approximation of the failure size forvalues of k for which limn→∞ k/n ∈ (0, 1].

Secondly, the assumptions in the CASCADE model are rather restrictive, e.g. theuniformly distributed surplus capacities. We develop a framework that allows forany continuous distribution with a strictly positive density in zero. We write

F (ln(i)) =θ + i− 1 + ∆(i, n)

n, (1.4.1)

1


where F (·) is the surplus capacity distribution function, and ∆(·, ·) can be con-sidered as the perturbation function with respect to the original setting. The centralquestion is under which conditions on the perturbation function the scale-freebehavior prevails. In Chapter 2, we provide sufficient conditions such that

P(An ≥ k) ∼ V (θ,∆)

√n− kn

k−1/2, (1.4.2)

for some constant V (θ,∆) ∈ (0,∞).

The proof techniques used in Chapter 2 give rise to a range of perturbationfunctions that lead to (1.4.2). Naturally, we are interested in the complete range ofperturbations function for which scale-free behavior prevails. In Chapter 3, werelate this problem to a random walk bridge, where the failure size corresponds tothe first-passage time over a moving boundary. The conditions for the movingboundary directly imply a rather general description under which power-lawbehavior appears in the failure size tail.

More specifically, for i.i.d. incrementsXj , j ≥ 1 with E(Xj) = 0 and finite variance,we consider the random walk Si =

∑ij=1Xj . Moreover, we assume that the

perturbations can be written as a sequence ∆(i) := ∆(i, n), i ≥ 1. It turns out thatwe can translate the failure size problem to

P(An ≥ k) ∼ P(Si ≥ 1− θ + ∆(i), i = 1, ..., k

∣∣Sm = 0).

We show that if ∆(i) = o(√i) as i→∞, then for every k such that limn→∞ k =∞

and limn→∞ k/n < 1,

P(An ≥ k) ∼√

2

πLθ,∆(k)

√n− kn

k−1/2,

where Lθ,∆(·) is a slowly varying function. Moreover, if k = n−o(n), an intriguingphase-transition takes place.

1.4.2 Effect of network disintegration

Although the CASCADE model and its generalization captures some typicalelements of the cascading mechanism that can lead to power outages, it neglectsthe following important aspect. Successive line failures can cause the powergrid to disintegrate in disjoint (graph) components. After every disconnection,the failure propagation in one component no longer affects the dynamics of therest of the network. This phenomenon is also known as islanding in the powerengineering community [14]. It can be a powerful tool to prevent large-scaleblackouts by isolating the most severe discrepancies in a small part of the network.

In the (generalization of the) CASCADE model, it is assumed that every failurecontinues to affect all other remaining lines during the cascade. In Chapter 4,we make a first step to understanding the effect of network disconnections byassuming that initially the network disconnects in two components, after which nomore disconnections take place. Assuming that the load surges continue similarlyas in the CASCADE model in both components, we rigorously derive the tail of the

1

1.4. Contributions to cascading failure models 11

failure size. More specifically, we derive the asymptotic behavior of the exceedanceprobability P(An ≥ k) for various thresholds k := kn, satisfying both k →∞ andn− k →∞ as n→∞. If the two component sizes are of the same order, and thethreshold k is (sufficiently) smaller than the size of the largest component, thenthe dominant contribution comes from the failure size in a single component. Thisimplies that the power-law behavior prevails with an exponent of −1/2. If insteadk is such that limn→∞ k/n < 1 and sufficiently larger than the larger component,it is most likely that both components have a proportional number of failures.This causes the failure size to still exhibit power-law behavior, but reduces theexponent to become −1. Our analysis allows us to rigorously determine the waythis phase-transition takes place, i.e. we also provide the asymptotic behavior ofthe exceedance probability if k is close to the larger component size. In addition,we consider the case that k = n − o(n) and/or the component sizes are not ofequal order.

In Chapter 5, we use the intuition obtained in Chapter 4 to establish the heavy-tailed characteristics of (edge) failure sizes for more general network topologies. Inparticular, we focus on a connected configuration model satisfying some regularitycondition on the degree sequence, and determine the behavior of the number ofedge failures. It turns out that the scale-free behavior prevails under criticalload conditions for sublinear thresholds. In contrast to more commonly studiedprocesses in the literature, an intrinsic feature in this work is that the propagationof failures occurs non-locally and depends on the global network structure, whichcontinually changes as the cascade advances.

In the framework of Chapter 5, two elements of randomness are involved in thecascading failure process: the surplus capacities of the edges and the way thenetwork disintegrates as edge failures occur. Exploiting the assumption that thesurplus capacities are i.i.d., we explain that these two elements of randomnesscan be decomposed. That is, the load surges depend on the way the networkdisintegrates as the cascade continues. We can relate this process to a percolationprocess with an adequately chosen removal probability. It is well-known thatas long as the removal probability is smaller than some critical parameter, the(unique) largest component of the percolated connected configuration modelcontains a non-vanishing proportion of the vertices and edges [60, 81], and allother components are likely to be much smaller. This observation leads to thenotion that the only dominant contribution to the failure size comes from thenumber of failures in this largest component. The failure size in the largestcomponent can in turn be translated to a first-passage time problem of a randomwalk bridge over a moving boundary, much in the same spirit as in Chapter 3.We derive that the tail of the total failure size has a power-law distribution forsublinear thresholds.

1.4.3 Causal effect of city population sizes

The models considered in Chapters 2-5 rely on the notion that power grids operatein a critically stable state and power outages are governed by SOC. Although theobtained insights are interesting by themselves from a theoretical point of view,it is important to realize that these type of models, as well as critical branching

1


process models, always lead to failure sizes that have infinite mean. However,recent findings based on advanced statistical analysis of actual blackout datasuggest the contrary [26, 57]. Moreover, analogies between power systems andthese types of models do not clearly translate to effective preventive measures.

To explain how scale-free behavior in power grids arises, we take a perspective inChapter 6 that contrasts the traditional views. We adopt a model similar to thefast dynamics of the OPA framework, and explore the causal mechanism betweencity populations and blackout sizes while accounting for the physical laws. Thatis, it is documented that city population sizes exhibit scale-free behavior [13, 103],as well as the power demand [121]. In our model, we therefore assume i.i.d.heavy-tailed demands at the cities for any general network topology. We adopt theDC approximation to model the power flow dynamics, where the line limits arechosen relative to the city sizes. The cascade is initiated by the failure of a singleline, and failures occur subsequently at the lines that are relatively overloadedmost (if any). Using large deviations, we show that the dominant scenario inwhich a large-scale power outage occurs is when there is a single city whosepower demand is so excessive that the power demands of other cities are relativelynegligible. In other words, we show that the scale-free nature of city populationsizes (and power demands) drives the scale-free nature of the blackout sizes.

Currently, the power engineering community is mainly focused on network up-grades to enhance the resilience of power grids, i.e. increase the capacity of thosepower lines that are most likely to fail and lead to large-scale blackouts. A funda-mental insight that can be deduced from our results in Chapter 6 is that, typically,network upgrades do not affect the scale-free behavior, but only have an effect onthe prefactor. Instead, it may be more effective to focus on enhancing the resilienceof cities. Temporarily covering a large surge in power demand by a cities’ ownreserves could provide an immense relief to the strain on the power grid, anddecrease the overall volatility. In other words, storage should become an integralpart of a secure and stable power system. On the other hand, it is also important tomanage increasing demand fluctuations in power demand on a local level, causedby e.g. electric vehicles.

1.5 Electric vehicle battery swapping systems

To reduce carbon emission in the transportation sector, there is a steady shifttaking place towards an electrified transportation system. Despite the apparentneed of alternative energy sources in the transportation sector, the adoption ofelectric vehicles (EVs) has been slow due to various practical challenges, suchas high purchase costs of an EV, battery life problems and long battery chargingtimes [115]. Several of these issues are addressed by a battery swapping infra-structure: whenever EV users are faced with a battery (collection) that is close todepletion, they move to a facility to swap their battery system (collection) for onethat is fully charged. Using a battery swapping infrastructure, the upfront costs ofpurchase of an EV can be significantly reduced when battery swapping stationoperators own and lease batteries to customers, the batteries can be charged moreappropriately to prolong batteries’ lifetime [118], and EV users can experience a

1

1.5. Electric vehicle battery swapping systems 13

fast exchange of batteries in contrast to long charging times. Furthermore, theaggregation of a large number of batteries at charging stations can provide a com-prehensive range of flexibility services to transmission and distribution networkservice operators. That is, battery swapping facilities can relieve the strain on thegrid in times of high demand, especially if the control of EV charging can be cent-ralized. As such, a battery swapping infrastructure can significantly contribute todecreasing the volatility in power demand.

Although a battery swapping system is a promising solution for many of theseissues, it is not immediate how to build an efficient design. More specifically, EVusers move to a battery swapping facility whenever their battery (collection) isalmost depleted to exchange it for a fully-charged one. Every battery swappingfacility has three types of assets: (spare) batteries that can be exchanged for thosein the EVs that arrive at the facility, charging points that can charge (incoming)batteries, and the swapping servers where the exchange of batteries takes place.How many of these assets does the facility need in order to have an efficient andadequate operation?

Naturally, within a battery swapping facility, there is an immediate trade-offbetween service performance and resource utilization. Adequately balancing thetwo metrics is very much in the spirit of the Quality-and-Efficiency-Driven (QED)regime known from asymptotic many-server queueing theory [55]. Typically, thisgives rise to a square-root slack provisioning policy for the capacity levels, whichhas been successfully implemented in many applications such as call centers [19,69, 132, 133], healthcare systems [74, 123, 124, 131], and more. This policy leadsto favorable performance for large systems: as the number of users r growslarge, the waiting probability tends to a value strictly between zero and one, thewaiting time vanishes with a rate O(1/

√r), and near-optimal resource utilization

of 1−O(1/√r) is achieved. To inherit such properties for the battery swapping

framework, we adopt a similar capacity level design policy for both the numberof charging servers and the number of spare batteries.

r +B batteries

∞

Exp(λ)

F

B

Exp(µ)

Figure 1.1: Visual representation of the battery swapping facility.

Inspired by the above-mentioned arguments, we take a queueing perspective inChapter 7. We suppose there are r EV users that return a to battery swappingfacility after an exponentially distributed time with mean λ. The battery swapping

1


facility has B spare batteries, F charging points and G swapping servers. Weassume that the charging of a battery takes an exponential amount of time withrate µ. Since swapping time is relatively short with respect to charging, we assumethis to be negligible. Whenever there is an EV arrival at a station, a swappingserver takes out the almost depleted battery and exchanges it for a fully-chargedone if available. As long as the exchange has not finished, the EV cannot movefrom the swapping server until a fully-charged battery is available. Consequently,having more charging points creates no additional charging capacity, and hencewe can assume that F ≤ B + G. Typically, due to the high purchase costs, thenumber of swapping servers G equals one. Under the condition F ≤ B +G, thenumber of swapping servers therefore does not play an actual role. An illustrationof this model is given in Figure 1.1.

In view of the QED regime, the square-root slack provisioning policy prescribesthat the number of charging points and spare batteries should be given by

B =λr

µ+ β

√λr

µ, β ∈ R,

F =λr

µ+ γ

√λr

µ, γ ≤ β.

(1.5.1)

Indeed, we show that under policy (1.5.1) we obtain the typical QED properties asthe number EVs grows large.

This is not the first work on the operation and control of a single battery swappingstation, see for example [113–115, 117, 118]. However, there is a clear gap in theliterature when extending this to the operation a wider network of stations. To de-velop an adequate model, one should account for the proliferation of smartphonesand online technologies as well, since a range of service providers may utilizethese tools to provide occupancy level information to customers to improve delayperformance. In a battery swapping system, such information can motivate EVusers to visit the location in their direct vicinity that is least busy. This leads to aload-balancing effect in the arrival streams, and resource capacity levels should beset accordingly. In Chapter 8, we introduce a novel stochastic model describing anetwork of battery swapping stations within the context of a fixed region. Withinthe region, there is a fixed number of swapping facilities and vehicles. In general,the vehicles do not leave the region, leading to the conservation of batteries. Themodel incorporates a load-balancing policy for which we establish suitable capa-city levels in the spirit of the QED regime that account for the inherent tradeoffbetween EV users’ quality-of-service and operational costs.

2Chapter 2

Robustness of power-law behaviorin cascading failure models

Based on:Robustness of power-law behavior in cascading line failure models

F. Sloothaak, S.C. Borst, B. ZwartStochastic Models, 34(1), 2018, pp. 45–72.

Inspired by reliability issues in electric transmission networks, we use a probabil-istic approach to study the occurrence of large failures in a stylized cascading linefailure model. In this chapter, we generalize the CASCADE model as introducedin Section 1.3.2. Under certain critical conditions, the probability that the numberof line failures exceeds a large threshold obeys a power-law distribution, a dis-tinctive property observed in empiric blackout data. In this chapter, we examinethe robustness of the power-law behavior by exploring under which conditionsthis behavior prevails. That is, we consider under what range of perturbationfunctions in (1.4.1) the failure size exhibits scale-free behavior.

This chapter is organized as follows. In Section 2.1, we provide a detailed modeldescription. In Section 2.2, we analyze the special case where the perturbationfunction in (1.4.1) is zero. We state our main result in Section 2.3, classifying arange of perturbation functions under which power-law behavior prevails. Weprovide a proof of our main result in Section 2.4, and illustrate how to apply ourmain result through examples in Section 2.5.

2.1 Model description

We consider a stochastic load-dependent cascading line failure model which canbe seen as a generalization of the CASCADE model. We already provided a high-level description of this model in the introduction of this thesis, but in this section,we also provide all details as well as justifications for our modeling choices.

15

2

16 Chapter 2. Robustness of power-law behavior in cascading failure models

Suppose we consider a graph that is shaped as a star where load demands, im-posed on the lines, are initially exceeded by the line capacities. A possible cas-cading failure effect is initiated by a disturbance that additionally loads all lines.When the load demand outstrips the capacity on a particular line, it fails. Eachline failure induces changes in the load distribution in the surviving network,possibly causing further lines to trip in succession and triggering knock-on ef-fects. The cascading failure propagation continues until each surviving line inthe network has enough capacity to meet its load demand. Since the evolution ofpower system operation, upgrade, maintenance, and design is much slower than ablackout cascade, it is reasonable to assume a fixed system during the progressionof any particular cascade [38]. That is, the capacities remain fixed throughout thecascading failure process.

More specifically, suppose the star-shaped network consists of n lines, see Fig-ure 2.1. Each line has a limited capacity for the amount of load it can carry beforeit trips. To simplify the model, we consider a setting where all lines are statisticallyindistinguishable. We assume that the network is initially stable, i.e. all lineshave capacities that exceed their initial load. We observe that what matters is thedifference between load demands and line capacities, which we refer to as thesurplus capacity. We assume that the differences between the initial load demandsand capacities are random variables that are independent and identically distrib-uted, and denote these by Cni for line i. Moreover, we assume that its distributionfunction F (·) is continuous with a strictly positive density in zero. We will seethat the continuity assumption ensures we can use the theory of order statisticsin our analysis. The reason for assuming a strictly positive density is merely atechnical one, and will become apparent in Example 2.5.2.

In order to trigger a possible cascading failure effect, we include an initial dis-turbance that causes all lines to be additionally loaded. When the capacity on aline is exceeded by its load demand, that line fails. Every line failure causes anadditional loading of the surviving lines in the network, which we refer to as theload surge. We assume that the load surge can be described through a deterministicnon-decreasing function. We write ln(1) for the initial load surge on all lines,and ln(i) for the total load surge on the surviving lines when i − 1 lines havefailed. That is, all randomness is captured by the differences between the initialload demands and the line capacities, while the load surges are deterministic.This is a modeling choice which serves to create a setting with a single source ofrandomness.

Our model does not explicitly account for many complexities that exist in realelectric power transmission systems, such as the length of time between failureoccurrences or the network topology that can lead to non-identical line capacitydistributions or non-equal load distributions. Yet, this model does capture twoimportant features of large blackouts: the initial disturbance loading the systemand the cascading line failure mechanism.

The main objective in this chapter involves the probability that An, the numberof failed lines in the network, exceeds a certain threshold k as n grows large. Toexpress this in mathematical terms, we take a closer look at the cascading failureprocess. After the dummy line has tripped, a next line will fail when the smallest

2

2.1. Model description 17

surplus capacity is exceeded by the load surge ln(1). If so, another line will fail ifand only if ln(2) exceeds the second smallest surplus capacity and so forth. Denoteby Cn(1) ≤ Cn(2) ≤ ... ≤ Cn(n) the ordered surplus capacities. The above observationyields that the failure size is given by

An = maxk ≤ n : Cn(i) ≤ ln(i), i = 1, ..., k (2.1.1)

and An = 0 if Cn(1) > ln(1). The probability that the failure size exceeds an integerk is thus given by

P(An ≥ k) = P(Cn(i) ≤ ln(i), i = 1, ..., k

). (2.1.2)

CN(1)

CN(2)

CN(3)

CN(4)

CN(5)

CN(6)

CN(N)

= lN(1)

(a) Initial disturbance.

CN(1)

CN(2)

CN(3)

CN(4)

CN(5)

CN(6)

CN(N)

= lN(1)

= lN(2)− lN(1)

(b) Load surge after one line failure.

CN(1)

CN(2)

CN(3)

CN(4)

CN(5)

CN(6)

CN(N)

= lN(1)

= lN(2)− lN(1)

= lN(3)− lN(2)

(c) Load surge after two line failures.

Figure 2.1: Illustration of cascading line failure process with An = 2.

Equation (2.1.2) can be rewritten into an expression that is easier to analyze. LetUn(i) denote the standard uniformly distributed ordered statistics for i = 1, ..., n.Since F (·) is continuous, it follows from (2.4.1) in [2] that (F (Cn(i)))i=1,...,n and(Un(i))i=1,...,n are equal in distribution. Therefore, (2.1.2) is equivalent to

P(An ≥ k) = P(Un(i) ≤ F (ln(i)), i = 1, ..., k

), (2.1.3)

where F ln is a non-decreasing function in the number of failed lines withvalues in [0, 1]. Much is known on uniformly distributed order statistics (see e.g.

2


Section 4.7 in [2]), e.g. the density of every order statistic. In turn this property canbe exploited to derive the asymptotic behavior of (2.1.3) in our framework.

Our framework can be seen as an extension of the CASCADE model presentedby Dobson et al. [40]. In the original setting as described in [40], we note that theprobability that the number of failed lines ADn exceeds a threshold k is given by

P(ADn ≥ k) = P(Un(n−i+1) +

θ + (i− 1)λ

n≥ 1, i = 1, ..., k

)

= P(

1− Un(n−i+1) ≤θ + (i− 1)λ

n, i = 1, ..., k

)

= P(Un(i) ≤

θ + (i− 1)λ

n, i = 1, ..., k

).

Comparing this to (2.1.3), we observe that this is equivalent to the case in ourframework with

F (ln(i)) =θ + (i− 1)λ

n, i ≥ 1 (2.1.4)

with constants θ > 0 and λ > 0.

Next, we briefly recall some notions that we discussed in the introduction ofthis thesis. The main result in [40] is that the number of failures follows a quasi-binomial distribution. Moreover, it is indicated that as n→∞, the quasi-binomialdistribution converges to a generalized Poisson distribution. This latter distri-bution also appears in the setting of branching processes, where it correspondsto the number of offspring. In [39], the branching process relation is used as anapproximation for the failure size, and note that λ = 1 corresponds to a criticalwindow where a power-law dependence manifests itself. In fact, it yields theapproximation

P(An = k) ≈ θ√2πk−3/2 (2.1.5)

for all large k independent of n.

In view of (2.1.4), we refer to the particular setting in [40] with λ = 1 as the affinecase. In this chapter, we aim to find a broader set of scenarios for which thispower-law behavior prevails. Firstly, we will extend to thresholds that allow for adependency on the network size. Secondly, we will explore how the assumptionson the load surge function and the surplus capacity distribution can be relaxedwhile preserving the power-law behavior for the line failure distribution.

2.2 Affine case

The main object of interest is the probability that the number of failed lines Anexceeds network-dependent thresholds k := k(n) that satisfy both k → ∞ andn− k →∞ as n→∞. We refer to such thresholds as "growing". For compactness,we suppress the dependence of the threshold k on n in the remainder of thischapter.

2

2.2. Affine case 19

In this section, we first examine the robustness of the power-law behavior of theaffine case. As indicated in Section 2.1, this covers all cases with

F (ln(i)) =θ + i− 1

n(2.2.1)

for some constant θ > 0. That is, the composition F ln needs to be linearlyincreasing with step increments 1/n. For example, Equation (2.2.1) holds forstandard exponentially distributed surplus capacities, i.e. F (x) = e−x for all x ≥ 0,and load surge function ln(i) = − log ((θ + i− 1)/n) for all 1 ≤ i ≤ k.

In [39], it is shown that a branching process approach yields the approxima-tion (2.1.5). This method essentially uses a double limit approach: first the asymp-totic behavior is derived as n → ∞ for fixed k, followed by considering thebehavior as k →∞. Specifically, (2.1.5) originates from the result

limk→∞

k3/2 limn→∞

P(An = k) =θ√2π. (2.2.2)

Proof of (2.2.2). The probability distribution of the number of failed lines is givenby a quasi-binomial distribution [40], i.e.

P(An = k) =

(nk

)θn

(θ+kn

)k−1 (1− θ+k

n

)n−k, if k ≤ n− θ,

0, if n− θ < k < n,n∑

i=bn−θc+1

(ni

)θn

(θ+in

)i−1 (1− i+θ

n

)n−i, if k = n.

(2.2.3)

This distribution converges to a generalized Poisson distribution [40],

limn→∞

P(An = k) = θ(θ + k)k−1

k!e−(θ+k).

Applying Stirling’s approximation, we obtain

limk→∞

k3/2 limn→∞

P(An = k) = limk→∞

k3/2θ(θ + k)k−1

k!e−(θ+k)

= limk→∞

θ√2π

(1 +

θ

k

)k−1

e−θ =θ√2π.

However, this approach does not allow thresholds that grow large with the net-work size, such as k = αn with α ∈ (0, 1). Our result accounts for the dependencybetween k and n, and justifies an approximation for such thresholds as well.

Proposition 2.2.1. Let F ln be as in (2.2.1) with constant θ > 0 for each n ∈ N. Letk? := k?(n) and k? := k?(n) be functions of n with k? ≥ k?, k? →∞ and n− k? →∞as n→∞. Then,

limn→∞

supk∈[k?,k?]

∣∣∣∣k3/2√

1− k/nP(An = k)− θ√2π

∣∣∣∣ = 0. (2.2.4)

2


For network-size dependent thresholds k, we thus obtain the approximation

P(An = k) ≈ θ√2π

1√1− k/n

k−3/2. (2.2.5)

Particularly, if k = αn for some fixed coefficient α ∈ (0, 1), our approximationleads to a different prefactor than the branching process approximation (2.1.5).

Yet, when accounting for the dependency of k on n, the analytic expression for theprobability distribution function of the number of failed lines can be exploited toderive the correct prefactor.

Proof of Proposition 2.2.1. Recall that the probability distribution of the number offailed lines is given by (2.2.3). The idea of the proof is to use Stirling’s approxima-tion to derive an upper and lower bound for (2.2.3) and show they asymptoticallycoincide.

Since we consider k ∈ [k?, k?], we are only concerned with the probability for

k ≤ n− θ. With Stirling’s approximation (formula (6.1.38) of [1]) we have that forevery integer m > 0

m! =√

2πmm+1/2e−m+y(m)12m ,

for some 0 < y(m) < 1. So the binomial term is bounded by(n

k

)≥ 1√

2π

nn

kk(n− k)n−k

√n

k(n− k)e−

112k e−

112(n−k) ,

(n

k

)≤ 1√

2π

nn

kk(n− k)n−k

√n

k(n− k)e

112n .

Using these bounds for the binomial term in (2.2.3) yields

k3/2√

1− k/nP(An = k) ≥ θ√2π

(1 +

θ

k

)k−1(1− θ

n− k

)n−ke−

112k e−

112(n−k)

and

k3/2√

1− k/nP(An = k) ≤ θ√2π

(1 +

θ

k

)k−1(1− θ

n− k

)n−ke

112n

for any k? ≤ k ≤ k?. Note that for every constant θ > 0 the functions (1 + θ/x)x−1

and (1 − θ/x)x are both monotone increasing in x > 0. Moreover, the functione−1/(12x) is monotone increasing in x > 0. Therefore, we obtain the lower bound

supk∈[k?,k?]

k3/2√

1− k/nP(An = k)

≥ θ√2π

supk∈[k?,k?]

(1 +

θ

k

)k−1

e−1

12k supk∈[k?,k?]

(1− θ

n− k

)n−ke−

112(n−k)

=θ√2π

(1 +

θ

k?

)k?−1(1− θ

n− k?

)n−k?e−

112k? e−

112(n−k?) .

2

2.2. Affine case 21

Moreover, since (1 + θ/x)x−1 ≤ eθ and (1− θ/x)x ≤ e−θ for all x > 0, we have theupper bound

supk∈[k?,k?]

k3/2√

1− k/nP(An = k) ≤ θ√2π

e1

12n .

We observe that both the upper bound and the lower bound converge to θ/√

2π asn→∞ under the given assumptions on k? and k?, implying that (2.2.4) holds.

Dobson et al. show in [40] that the failure size for the affine case follows a quasi-binomial distribution, and the proof of Proposition 2.2.1 relies heavily on theexplicit form of that distribution function. We note that the same technique can beused for fixed k, which yields the generalized Poisson distribution, or for k = n− lwith l > θ a fixed integer. This gives rise to the results summarized in Table 2.1.

k fixed k growing k = n− l, l fixed

P(An = k) θ(θ+k)k−1

k! e−(θ+k) θ√2π

k−3/2√1−k/n

θ(l−θ)ll! e−(l−θ)n−1

Table 2.1: Asymptotic behavior affine case.

Proposition 2.2.1 can be used to derive the asymptotic behavior of the tail of thefailure size distribution.

Theorem 2.2.1. Let F ln be of the form (2.2.1) with constant θ > 0 for each n ∈ N,and k := k(n) < n a positive function of n such that k →∞ and n−k →∞ as n→∞.Then,

limn→∞

√kn

n− kP(An ≥ k) =2θ√2π.

Theorem 2.2.1 yields the approximation

P(An ≥ k) ≈ 2θ√2π

√1− k/nk−1/2. (2.2.6)

We conclude from Theorem 2.2.1 that the power-law behavior for the affine modelextends to thresholds k that are appropriately growing functions of the networksize.

To prove Theorem 2.2.1, we bound the discrete density function of the failuresize by two continuous functions that grow arbitrarily close to one another for alli ∈ [k, n−log(n−k)], see Figure 2.2. We conclude the proof by deriving the integralcounterparts of the continuous functions and showing that P(An ≥ n− log(n− k))is asymptotically negligible.

2


i = 0.75N i

Figure 2.2: Continuous bounds for P(An = i).

Proof of Theorem 2.2.1. Set k? = n − log(n − k). Observe that k? ≥ k and notethat this choice ensures k? = n − o(n) and n − k? → ∞ as n → ∞. Due toProposition 2.2.1, it follows that for every ε > 0 there is a Nε > 0 such that for alln ≥ Nε and i ∈ [k, k?]

θ√2πi−3/2(1− i/n)−1/2(1− ε) ≤ P(An = i) ≤ θ√

2πi−3/2(1− i/n)−1/2(1 + ε).

Next, we use this observation to bound the exceedance probability from aboveand below and show that these bounds coincide as ε ↓ 0.

An upper bound for the exceedance probability is given by

P(An ≥ k) ≤ P(An = k) +

k?∑

i=k+1

θ(1 + ε)√2π

i−3/2

(1− i

n

)−1/2

+

n∑

i=k?+1

P(An = i).

We consider the three terms separately. For the first term, note that for everyinteger m, Stirling’s bound [1] yields

√2πmm+1/2e−m ≤ m! ≤ e

√2πmm+1/2e−m.

Therefore, as n→∞,√

kn

n− kP(An = k) ≤ e

√n

n− k

√n

n− kθ

θ + k

(1 +

θ

k

)k (1− θ

n− k

)n−k

≤ en

k(n− k)

θ

θ/k + 1→ 0

For the second term, we consider the integral∫ n

k

x−3/2(

1− x

n

)−1/2

dx

=

∫ π/2

arcsin(√k/n)

n−3/2 sin(u)−3(1− sin(u)2)−1/22n sin(u) cos(u) du

= 2n−1/2

∫ π/2

arcsin(√k/n)

sin(u)−2 du = 2n−1/2

√1− k/n√k/n

= 2

√n− kkn

,

2

2.2. Affine case 23

where we applied the variable substitution x = n sin(u)2. Then, we obtain thebound

k?∑

i=k+1

θ√2πi−3/2(1− i/n)−1/2 ≤

∫ n

k

θ√2πi−3/2(1− i/n)−1/2 di =

2θ√2π

√n− kkn

.

For the third term, observe that

n∑

i=k?+1

P(An = i) ≤ (n− k?) supi∈(k?,n]

P(An = i).

To determine the supremum, we take a closer look at (2.2.3). For all i ∈ (n− θ, n),if any, P(An = i) = 0. Moreover, for all integers i ∈ (k?, n − θ], Stirling’s boundyields

(n

i

)θ

n

(θ + i

n

)i−1(1− θ + i

n

)n−i

≤ e

√n

i(n− i)θ

θ + i

(1 +

θ

i

)i(1− θ

n− i

)n−i≤ e

√n

n− θθ

θ + k?.

Therefore, supi∈(k?,n−θ] P(An = i) ≤ c1/k? for some constant c1 > 0, and

P(An = n) =θ

n

(1 +

θ

n

)n−1

+

n−1∑

i=bn−θc+1

(n

i

)θ

n

(θ + i

n

)i−1(1− θ + i

n

)n−i

≤ θeθ

n+ θe

√n

n− 1

θ

θ + k?≤ c2k?

for some constant c2 > 0. Recall k ≤ k? = n− log(n− k), and set c = maxc1, c2.This yields

√kn

n− k (n− k?) supi∈(k?,n]

P(An = i) ≤ c√kn

k?n− k?√n− k

= c

√k/n

1− log(n− k)/n︸︷︷︸=O(1)

log(n− k)√n− k︸︷︷︸

=o(1)

as n→∞. We conclude that

lim supn→∞

√kn

n− kP(An ≥ k) ≤ (1 + ε)2θ√2π.

2


A lower bound is given by

P(An ≥ k) ≥ (1− ε)k?∑

i=k

θ√2πi−3/2(1− i/n)−1/2

≥ (1− ε)∫ k?

k

θ√2πi−3/2(1− i/n)−1/2 di

= (1− ε) 2θ√2π

(√n− kkn

−√n− k?k?n

).

It follows that

lim infn→∞

√kn

n− kP(An ≥ k)

≥ lim infn→∞

(1− ε) 2θ√2π

1−

√k

k?︸︷︷︸=O(1)

√n− k?n− k︸︷︷︸

=o(1)

= (1− ε) 2θ√

2π.

As ε ↓ 0, the limsup and liminf coincide.

2.3 Main result

Theorem 2.2.1 shows that the probability to exceed a network-dependent thresholdhas a power-law distribution when the composition F ln is of the specificform (2.2.1). To understand why this leads to power-law behavior, recall (2.1.3).In expectation, the difference between two consecutive uniformly distributedorder statistics is 1/(n + 1) ≈ 1/n. When F ln is of the form as in (2.2.1), thestep increments are 1/n and thus (nearly) equal to the expected difference of twoconsecutive uniform order statistics. It is this critical correspondence that leads toheavy-tailed behavior for the exceedance probability.

What other forms of the composition F ln lead to power-law behavior for thenumber of failed lines? To answer this question, we include additive perturbationswith respect to (2.2.1). Specifically, we consider compositions F ln of the form

F (ln(i)) =θ + i− 1 + ∆(i, n)

n, (2.3.1)

where ∆(·, ·) represents the perturbations with respect to the corresponding affinecase. We impose suitable conditions on the magnitude of the perturbations suchthat the power-law behavior prevails as n→∞.

Intuitively, these conditions can be understood as follows. The limiting behaviorof the perturbations ∆(i, n) can be of any (finite) size as long as i is fixed. Yet,for larger values of i ≤ k that grow to infinity with n, the perturbations need tobecome close to zero in this domain. These conditions ensure that for most valuesup to k, the step increment of F ln still equals the expected difference of twoconsecutive uniform order statistics, and consequently remains in the frameworkwhere power-law behavior appears.

2

2.3. Main result 25

Theorem 2.3.1. Let k := k(n) < n be a positive function of n such that k → ∞ andn− k →∞ as n→∞. Let F ln be as in (2.3.1) with ∆(·, ·) satisfying the followingproperties:

(A) ∆(i, n)→ ∆(i) pointwise as n→∞ for some well-defined function ∆(·) : N→ Rwith limi→∞∆(i) = 0;

(B) For all i(n) ≤ k that are growing with n, i.e. limn→∞ i(n) = ∞, it holds thatlimn→∞∆(i(n), n) = 0.

Then, there exists a constant V (θ,∆) ∈ (0,∞) such that

limn→∞

√kn

n− kP(An ≥ k) = V (θ,∆).

That is, if ∆(·, ·) satisfies (A) and (B) specified in Theorem 2.3.1, then there exists afinite, strictly positive constant V (θ,∆) (not depending on k) such that

P(An ≥ k) ≈ V (θ,∆)

√n− kkn

for large n. The main point to take from this result is that the perturbations onlychange the prefactor, while the decay rate remains the same. For instance, for allk = αn with α ∈ (0, 1), the power of the exponent of the power-law remains 1/2.

Algorithm 1: Approximation scheme for V (θ,∆).Input: Target error δ > 0, constant θ > 0 and perturbations ∆(·, ·)

satisfying (A) and (B) in Theorem 2.3.1.Output: Approximation VMε

(θ,∆) such that |V (θ,∆)− VMε(θ,∆)| < δ.

1. Determine ε > 0 such that 8ε(1+ε)√2π≤ δ.

2. Determine pair (Mε, Nε) such that |∆(i, n)| < ε for all n ≥ Nε and allMε ≤ i ≤ k(n).

3. Return VMε(θ,∆) defined in (2.3.4).

The constant V (θ,∆) is generally difficult to compute explicitly, but we can ap-proximate its value with arbitrary precision through Algorithm 1. The idea is toaccount for only the first M <∞ perturbations, where M is chosen large enough,and take the corresponding prefactor VM (θ,∆) as an approximation of the truevalue. The definition of the approximation VM (θ,∆) requires some notation. Writeci,n = nF (ln(i)) = θ + i− 1 + ∆(i, n) and for every fixed i ∈ N,

ci = limn→∞

nF (ln(i)) = θ + i− 1 + ∆(i). (2.3.2)

2


Recall that F (ln(i)) is non-decreasing in i, and hence ci is a non-decreasing se-quence. Let βi, i ∈ N, be defined as

βi =∑ij=1

(−1)j+1

j! βi−j(ci−j+1)j , β0 = 1. (2.3.3)

Finally, let γ(·, ·) denote the lower incomplete gamma distribution:

γ(s, x) =

∫ x

0

ts−1e−t dt.

Then, the value of VM (θ,∆) can be expressed as

VM (θ,∆) =2√2π

(θ

(M − 1)!γ(M, cM ) +

(cM )M

(M − 1)!e−cM

+

M−1∑

j=1

βj−1e−cj∆(j)

(M − j)!γ(M − j + 1, cM − cj)

−M−1∑

j=1

βj−1(cM − cj)M−j+1

(M − j)! e−cM

.

(2.3.4)

The constant V (θ,∆) is thus defined as

V (θ,∆) = limM→∞

VM (θ,∆). (2.3.5)

2.4 Proof of Theorem 2.3.1

This section is devoted to proving Theorem 2.3.1. Whether we obtain power-law behavior for the black-out size distribution depends on the surplus capacitydistribution, the load surge function and the threshold k. Due to relation (2.1.3),we observe that the relation between the surplus capacity distribution and theload surge function is captured by the composition F ln, see Figure 2.3. Inthis section we prove that if F ln has a form as in (2.3.1) with perturbations∆(·, ·) satisfying (A) and (B) as in Theorem 2.3.1, the power-law behavior for theexceedance probability prevails.

Recall (2.3.2) and note that by conditions (A) and (B), ci, i ∈ N, is a well-definednon-decreasing sequence that satisfies

limi→∞

ci − (θ + i− 1) = limi→∞

δ(i) = 0.

We will show that these conditions result in power-law behavior for the exceedanceprobability.

The proof of Theorem 2.3.1 uses a simple idea. In view of (2.1.1), (2.1.3) and (2.3.1),we observe that

An = min

i ∈ N : Un(i) >

θ + i− 1 + ∆(i, n)

n

− 1.

2

2.4. Proof of Theorem 2.3.1 27

F (x)

xlN (k)

k

∆(i,N)

i

F (lN (i))

Figure 2.3: Relation surplus capacity distribution function and loadsurge function.

In other words, the probability that the failure size exceeds a threshold k corres-ponds to a first-passage time problem of uniformly distributed order statisticsover a function. For every fixed ε > 0, we bound this function from above andbelow by appropriately chosen functions, which in turn give rise to an upperand lower bound for the exceedance probability. We show that the asymptoticbehavior of the bounds converge to the same expressions as ε ↓ 0, from which wecan conclude Theorem 2.3.1.

To prove Theorem 2.3.1, we will leverage two basic asymptotic properties asformulated in the following two lemmas.

Lemma 2.4.1. Let k be a function of n such that both k →∞ and n−k →∞ as n→∞.Then for every fixed M1,M2 ∈ N,

limn→∞

√kn

n− kP(Un−M1

(i) ≤ θ + i− 1

n−M1, ∀i ≤ k −M2

)=

2θ√2π.

Proof. Recall (2.1.3) and (2.2.1), and observe that the case with M1 = M2 = 0is implied by Theorem 2.2.1. The general case follows from a straightforward

2


calculation:√

kn

n− kP(Un−M1

(i) ≤ θ + i− 1

n−M1, i = 1, ..., k −M2

)

=

√k

k −M2︸︷︷︸→1

√n

n−M1︸︷︷︸→1

√n− k +M2 −M1

n− k︸︷︷︸→1

·√

(k −M2)(n−M1)

n− k +M2 −M1P(Un−M1

(i) ≤ θ + i− 1

n−M1, i = 1, ..., k −M2

)n→∞−→ 2θ√

2π.

The last convergence follows from noting (2.1.3) and applying Theorem 2.2.1 to anetwork with n−M1 lines and threshold k −M2.

Lemma 2.4.2. Let k be a function of n such that both k →∞ and n−k →∞ as n→∞.Then for every fixed M ∈ N

limn→∞

√kn

n− kP(Un−M(i) ≤ θ + i− 1

n, i = 1, ..., k

)=

2θ√2π. (2.4.1)

Proof. Note that

lim supn→∞

√kn

n− kP(Un−M(i) ≤ θ + i− 1

n, i = 1, ..., k

)

≤ lim supn→∞

√kn

n− kP(Un(i) ≤

θ + i− 1

n, i = 1, ..., k

)=

2θ√2π.

To obtain a lower bound, we first consider the case M = 1. Consider a Poissonprocess with unit rate where the epoch of the i’th event is denoted by Si =

∑ij=1Ej

with Ej standard independent exponential random variables for all j ≥ 1. Notethat, given Si = t, the joint distribution of (S1, ..., Si−1) is the same as the jointdistribution of i − 1 ordered independent uniform random variables on (0, t).Therefore, Equation (2.4.1) with M = 1 is equivalent to

limn→∞

√kn

n− kP(SiSn≤ θ + i− 1

n, i = 1, ..., k

)=

2θ√2π.

We observe that for every ε > 0,

P(

SiSn+1

≤ θ + i− 1

n, ∀i ≤ k

)

≤ P(Si ≤

(θ + i− 1)Sn+1

n, ∀i ≤ k;En+1 ≤ εSn

)+ P (En+1 > εSn)

≤ P(Si ≤ (θ + ε+ i− 1)

Snn, ∀i ≤ k

)+ P (En+1 > εSn) .

2


Since

P (En+1 > εSn) = E(e−εSn

)= E

(e−εS1

)n=

(1

1 + ε

)n,

it follows that

lim infn→∞

√kn

n− kP(SiSn≤ θ + i− 1

n, ∀i ≤ k

)

≥ lim infn→∞

√kn

n− k

(P(

SiSn+1

≤ θ − ε+ i− 1

n, ∀i ≤ k

)−(

1

1 + ε

)n)

=2(θ − ε)√

2π

for every ε > 0. The result for M = 1 follows by letting ε ↓ 0. Using inductionyields the result for any fixed M > 0.

In view of (2.1.1), (2.1.3) and (2.3.1), it is convenient to introduce the stoppingtimes

τnθ,∆ = min

i ∈ N : Un(i) >

θ + i− 1 + ∆(i, n)

n

− 1 (2.4.2)

for all constants θ ∈ R and functions ∆ : N× N→ R. In particular, if the constantθ and function ∆(·, ·) are chosen as in (2.3.1), then An = τnθ,∆. Yet, the advantageof the notation as in (2.4.2) appears when we compare the asymptotic exceedanceprobability for different constants θ and functions ∆(·, ·).

Our derivation of the asymptotic behavior of the exceedance probability makesuse of similar arguments multiple times in the proof. We present these argumentsseparately by means of the next two lemmas.

Lemma 2.4.3. Let k := k(n) ≤ n be a positive function of n such that k → ∞ andn− k →∞ as n→∞. Let ci, i ∈ N, be as in (2.3.2) and for some fixed M ∈ N, suppose∆(i, n) = 0 for all i ≥ M and n ≥ N0 for some N0 ∈ N. For all constants a, b ∈ R≥0

with a ≤ b,

limn→∞

√kn

n− kP(τnθ,∆ ≥ k;Un(M) ∈

[a

n,b

n

])

=2√2π

∫ b

a

P(UM−1

(i) ≤ ciy, ∀i ≤M − 1

)(θ +M − y)

yM−1

(M − 1)!e−y dy. (2.4.3)

Proof. The density of the M ’th order statistic of a sample of n standard uniformlydistributed random variables is given by a beta distribution [2, pp. 78–79]

fUn(M)

(x) =n!

(M − 1)!(n−M)!xM−1(1− x)n−M .

2


Conditioning on the M ’th order statistic yields

P(τnθ,∆ ≥ k) =

∫ bn

an

P(UN(i) ≤

ci,nn, ∀i ≤ k

∣∣Un(M) = x)fUn

(M)(x) dx

=

∫ b

a

P(UN(i) ≤

ci,nn, ∀i ≤ k

∣∣Un(M) =y

n

) fUn(M)

(yn

)

ndy

=

∫ b

a

P(UM−1

(i) ≤ ci,ny, ∀i ≤M − 1

)

· P(Un−M(i) ≤ θ +M − y + i− 1

n(1− yn )

, ∀i ≤ k −M) fUn

(M)

(yn

)

ndy.

The latter equality follows from the Markov property: Given that Un(M) = y/n,the first M − 1 order statistics are independent of the other order statistics anddistributed as M − 1 uniformly distributed random variables on the interval[0, y/n]. Similarly, the other order statistics are independent of the first M orderstatistics, and have the same law as n−M uniformly distributed random variableson the interval [y/n, 1]. Rescaling the intervals results in the above expression.

Next, we show that an interchange of limit and integration is justified by boundingall three terms within the integral form above. First, observe that for all y ∈ [a, b],

fUn(M)

(yn

)

n=

(n− 1)!

(M − 1)!(n−M)!

( yn

)M−1 (1− y

n

)n−M

≤ nM−1

(M − 1)!

( yn

)M−1

≤ bM−1

(M − 1)!<∞.

Second, we show that the second term multiplied with√kn/(n− k) is also

bounded. Let M? = dbe, and hence for all y ∈ [a, b], n−M? ≤ n− y ≤ n,

P(Un−M(i) ≤ θ +M − y + i− 1

n(1− yn )

, ∀i ≤ k −M)

≥ P(Un−M(i) ≤ θ +M − y + i− 1

n, ∀i ≤ k −M

)

and

P(Un−M(i) ≤ θ +M − y + i− 1

n(1− yn )

, ∀i ≤ k −M)

≤ P(Un−M(i) ≤ θ +M − y + i− 1

n−M?, ∀i ≤ k −M

).

2


Applying Lemmas 2.4.1 and 2.4.2 and subsequently the squeeze theorem yields

limn→∞

√kn

n− kP(Un−M(i) ≤ θ +M − y + i− 1

n(1− yn )

, ∀i ≤ k −M)

= lim infn→∞

√kn

n− kP(Un−M(i) ≤ θ +M − y + i− 1

n, ∀i ≤ k −M

)

= lim supn→∞

√kn

n− kP(Un−M(i) ≤ θ +M − y + i− 1

n−M?, ∀i ≤ k −M

)

=2(θ +M − y)√

2π.

We find that the second term multiplied with√kn/(n− k) is indeed bounded,

since every converging sequence is bounded. Finally, the first term is triviallybounded by one, and therefore the dominated convergence theorem justifies aninterchange of limit and integration. Since UM−1

(i) , i = 1, ...,M − 1, have a densitynot depending on n, it holds that

limn→∞

P(UM−1

(i) ≤ ci,ny, ∀i ≤M − 1

)= P

(UM−1

(i) ≤ ciy, ∀i ≤M − 1

),

and moreover,

limn→∞

1

nfUn

(M)

( yn

)=

yM−1

(M − 1)!e−y.

We conclude that (2.4.3) holds.

To obtain a more quantitative handle on the integral expression in (2.4.3), we needto have a deeper understanding of the probability term within the integral. Thesecond lemma expresses this probability by means of a recursive formula.

Lemma 2.4.4. Let M ∈ N be fixed, and suppose ∆(i, n) = 0 for all i ≥M and n ≥ N0

for some N0 ∈ N. Let ci, i ∈ N be as in (2.3.2) and for every y ∈ R≥0, define σM (y) = 0if c1 > y and otherwise

σM (y) = maxi ∈ N : i ≤M, ci < y.

Then,

P(UM(i) ≤

ciy, ∀i ≤M

)= 1− M !

yM

σM (y)∑

j=1

βj−1(y − cj)M−j+1

(M − j + 1)!,

where βi are defined as in (2.3.3).

The proof of Lemma 2.4.4 uses the two following identities.

2


Lemma 2.4.5. For βk, k ≥ 1 defined as in (2.3.3),

βk =

∫ 0

−c1

∫ y1

−c2· · ·∫ yk−1

−ckdyk · · · dy1.

Proof. The proof is by induction. For k = 1 we indeed have∫ 0

−c1 dy1 = c1 = β1.Suppose the lemma holds for all integers strictly smaller than k. Then,∫ 0

−c1

∫ y1

−c2· · ·∫ yk−1

−ckdyk · · · dy1 =

∫ 0

−c1

∫ y1

−c2· · ·∫ yk−2

−ck−1

yk−1 dyk−1 · · · dy1 + ckβk−1

=

∫ 0

−c1

∫ y1

−c2· · ·∫ yk−3

−ck−2

(yk−2)2

2dyk−2 · · · dy1 −

(ck−1)2

2βk−2 + ckβk−1

=

∫ 0

−c1

1

(k − 1)!yk−1

1 dy1 +

k−1∑

j=1

(−1)j+1

j!βk−j(ck−j+1)j

=

k∑

j=1

(−1)j+1

j!βk−j(ck−j+1)j = βk.

Lemma 2.4.6. Let (ci)i∈N be a non-negative, non-decreasing sequence and βk, k ≥ 0defined as in (2.3.3). If x ≥ ck, then

βk +

k∑

j=1

βj−1

k+1−j∑

l=0

(x− cj)ll!

=

k∑

l=0

xk

k!.

Proof. Particularly, we note that the identity is true for k = 0. For k ≥ 1, we findthat due to the binomial formula,

k∑

j=1

βj−1

k+1−j∑

l=0

(x− cj)ll!

=

k∑

j=1

k+1−j∑

l=0

l∑

m=0

βj−1

(l

m

)xm(−cj)l−m

l!

=

k∑

j=1

k+1−j∑

m=0

k+1−j∑

l=m

βj−1xm(−cj)l−mm!(l −m)!

=

k∑

j=1

k+1−j∑

l=0

βj−1(−cj)ll!

+

k∑

m=1

xm

m!

k+1−m∑

j=1

k+1−j∑

l=m

βj−1(−cj)l−m(l −m)!

=

k∑

j=1

βj−1 +

k∑

j=1

k+1−j∑

l=1

βj−1(−cj)ll!

+

k∑

m=1

xm

m!

k+1−m∑

j=1

βj−1

+

k∑

m=1

xm

m!

k−m∑

j=1

k+1−m−j∑

l=1

βj−1(−cj)ll!

.

2


In the second term and the fourth term, we observe a double summation for allpairs of integers in a triangle. We apply the variable substitution u = l+ j − 1 andv = l to sum over all pairs in the triangle via the diagonal lines. For the secondterm, this yields

k∑

j=1

k+1−j∑

l=1

βj−1(−cj)ll!

=

k∑

u=1

u∑

v=1

βu−v(−cu−v+1)v

v!= −

k∑

u=1

βu.

Similarly, this argument can be applied to the fourth term. In the end, we obtain

k∑

j=1

βj−1

k+1−j∑

l=0

(x− cj)ll!

=

k∑

j=1

βj−1 −k∑

u=1

βu +

k∑

m=1

xm

m!

k+1−m∑

j=1

βj−1 −k−m∑

u=1

βu

= β0 − βk + β0

k∑

m=1

xm

m!= −βk +

k∑

m=0

xm

m!.

Proof of Lemma 2.4.4. First, note that if σM (y) = 0, then y ≤ ci for all i = 1, ...,Mand the probability equals one, and hence the identity holds in this case.

To show the result for σM (y) ∈ (0,M), we need the joint density of M orderstatistics. This is given by the constant M ! [2, p. 11], yielding

P(UM(i) ≤

ciy, ∀i ≤M

)

=

∫ c1/y

0

∫ c2/y

u1

· · ·∫ cσM (y)/y

uσM (y)−1

∫ 1

uσM (y)

· · ·∫ 1

uM−1

M ! duM · · · du2du1

=M !

yM

∫ c1

0

∫ c2

v1

· · ·∫ cσM (y)

vσM (y)−1

∫ y

vσM (y)

· · ·∫ y

vM−1

1 dvM · · · dv2dv1,

=M !

yM

∫ c1

0

∫ c2

v1

· · ·∫ cσM (y)

vσM (y)−1

(y − vσM (y))M−σM (y)

(M − σM (y))!dvσM (y) · · · dv2dv1

= −M !

yMβσM (y)−1

(y − cσM (y))M−σM (y)+1

(M − σM (y) + 1)!

+M !

yM

∫ c1

0

∫ c2

v1

· · ·∫ cσM (y)−1

vσM (y)−2

(y − vσM (y))M−σM (y)+1

(M − σM (y) + 1)!dvσM (y)−1 · · · dv2dv1

= −M !

yM

σM (y)∑

j=2


(M − j + 1)!+M !

yM

∫ c1

0

(y − v1)M−1

(M − 1)!dv1

= 1− M !

yM

σM (y)∑

j=1


(M − j + 1)!.

where we used the change of variable ui = vi/y for i = 1, ...,M , and appliedLemma 2.4.5 multiple times.

2


For σM (y) = M , we observe that y > ci for all i ≤M . Therefore, this case requiresa separate analysis. Note that

yM

M !P(UM(i) ≤

ciy, ∀i ≤M

)=

∫ c1

0

∫ c2

v1

· · ·∫ cM

vM−1

dvM · · · dv2dv1

= βM =

M∑

j=0

yj

j!−

M∑

j=1

βj−1

M+1−j∑

l=0

(y − cj)ll!

=yM

M !−

M∑

j=1

βj−1(y − cj)M+1−j

(M + 1− j)! +M−1∑

j=0

yj

j!−M−1∑

j=1

βj−1

M−j∑

l=0

(y − cj)ll!

− βM−1

=yM

M !−

M∑

j=1

βj−1(y − cj)M+1−j

(M + 1− j)! ,

where we applied Lemma 2.4.6 twice.

Next, we use these results to prove Theorem 2.3.1. As a first step, we prove apartial result that only includes the scenarios with finitely many perturbations,see Figure 2.4.

∆(i,N)

M k

Figure 2.4: Effect of perturbations for the truncated case.

Proposition 2.4.1. Let k := k(n) ≤ n be a positive function of n such that k →∞ andn− k → ∞ as n → ∞. Let F ln be as in (2.3.1) with ∆(i, n) = 0 for all i ≥ M andn ≥ N0 for some fixed M ∈ N and N0 ∈ N, and let ci, i ∈ N, be as in (2.3.2). Then, thereexists a constant VM (θ,∆) ∈ (0,∞) such that

limn→∞

√kn

n− kP(An ≥ k) = VM (θ,∆).

Let βi, i ∈ N, be as in (2.3.3) and let γ(·, ·) denote the lower incomplete gamma distribu-tion. The value of VM (θ,∆) can be expressed as in (2.3.4).

2


Proof. Noting (2.1.3) and (2.4.2), applying Lemma 2.4.3 with a = 0 and b = cMand subsequently invoking Lemma 2.4.4 yields

limn→∞

√kn

n− kP(An ≥ k)

=2√2π

∫ cM

0

P(UM−1

(i) ≤ ciy, ∀i ≤M − 1

)(θ +M − y)

yM−1

(M − 1)!e−y dy

=2√2π

∫ cM

0

(θ +M − y)yM−1

(M − 1)!e−y dy

− 2√2π

∫ cM

0

(θ +M − y)

σM−1(y)∑

j=1

βj−1(y − cj)M−j

(M − j)! e−y dy.

The first term can also be expressed as∫ cM

0

(θ +M − y)yM−1

(M − 1)!e−y dy =

(θ +M)γ(M, cM )−Mγ(M, cM ) + (cM )Me−cM

(M − 1)!

=θ

(M − 1)!γ(M, cM ) +

(cM )M

(M − 1)!e−cM .

The second term yields

∫ cM

0

(θ +M − y)

σM−1(y)∑

j=1

βj−1(y − cj)M−j

(M − j)! e−y dy

=

M−1∑

m=1

m∑

j=1

∫ cm+1

cm

(θ +M − y)βj−1(y − cj)M−j

(M − j)! e−y dy

=

M−1∑

j=1

∫ cM

cj

βj−1(θ +M − y)(y − cj)M−j

(M − j)! e−y dy

=

M−1∑

j=1

βj−1e−cj∫ cM−cj

0

(θ +M − cj − u)uM−j

(M − j)! e−u du,

=

M−1∑

j=1

βj−1e−cj(θ +M − cj(M − j)! γ(M − j + 1, cM − cj)

−M − j + 1

(M − j)! γ(M − j + 1, cM − cj) +(cM − cj)M+j−1

(M − j)! e−(cM−cj))

=

M−1∑

j=1

βj−1e−cj−∆(j)

(M − j)!γ(M − j + 1, cM − cj)

+

M−1∑

j=1

βj−1(cM − cj)M+j−1

(M − j)! e−cM .

Subtracting the second term from the first concludes the proof.

2


Next, we allow for all perturbations that satisfy conditions (A) and (B) as inTheorem 2.3.1. It turns out that for the proof it is more convenient to use thefollowing equivalent condition: for every ε > 0 there exists a pair (Mε, Nε) ∈ N×Nsuch that |∆(i, n)| < ε for all n ≥ Nε and all Mε ≤ i ≤ k(n). Conditions (A) and(B) as stated in Theorem 2.3.1 are more intuitive and tractable when consideringexamples as opposed to its more technical description.

Lemma 2.4.7. Conditions (A) and (B) for perturbations ∆(·, ·) defined in Theorem 2.3.1are equivalent to the following: For every ε > 0 there exists a pair (Mε, Nε) ∈ N × Nsuch that |∆(i, n)| < ε for all n ≥ Nε and all Mε ≤ i ≤ k(n).

Proof. (⇒) The boundedness of ∆(·) is an immediate consequence of the bounded-ness of ∆(·, ·). By definition of ∆(·), we can pick a Ni,ε ≥ Nε/2 for every ε > 0 andfor all i ≥Mε/2 such that |∆(i, n)−∆(i)| < ε/2 for all n ≥ Ni,ε. Then,

|∆(i)| ≤ |∆(i, Ni,ε)|+ ε/2 < ε

for all i ≥Mε/2, showing that limi→∞∆(i) = 0. For condition (B) to hold, supposeε > 0 and let Nε ∈ N be such that i(n) ≥Mε for all n ≥ Nε and Nε ≥ Nε. Then, byassumption we obtain |∆(i(n), n)| < ε for all n ≥ Nε.(⇐) If not, then ∃ε > 0 such that for every (Mε, Nε) ∈ N× N there exists an i > Mε

and n ≥ Nε such that |∆(i, n)| ≥ ε. In particular, if we choose Mε = k(Nε)/2, thenthere exists an ε > 0 such that for every Nε ∈ N there are a k(Nε)/2 ≤ i ≤ k(Nε)and n ≥ Nε such that |∆(i, n)| ≥ ε, contradicting condition (B).

Mε

ε

∆

Figure 2.5: Illustration of infinitely many perturbations setting.

Next, we prove Theorem 2.3.1 by considering the bounds illustrated by the dashedlines in Figure 2.5 for every fixed ε > 0. That is, for an upper bound, we considerthe exceedance probability in case of an initial disturbance (θ + ε)/n and allowingfor the first Mε − 1 perturbations. Indeed, this yields an upper bound for alln ≥ Nε: the values are the same for all pairs (i, n) with i ≤Mε − 1, and for i ≥Mε,we have θ + i − 1 + ∆(i, n) ≤ θ + ε + i − 1 for all n ≥ Nε. Similarly, for a lowerbound we consider the case with initial disturbance (θ − ε)/n where we allow for

2


the first Mε − 1 perturbations. By applying Proposition 2.4.1, we can determinethe asymptotic behavior of the bounds explicitly. We show that as ε ↓ 0, the upperand lower bound converges to the same constant V (θ,∆) ∈ (0,∞) as defined in(2.3.5).

Proof of Theorem 2.3.1. By assumption and Lemma 2.4.7, we know that ∀ε > 0there exists a pair (Mε, Nε) ∈ N× N such that |∆(i, n)| < ε for every n ≥ Nε andMε ≤ i ≤ k(n). Fix ε > 0, and define for all (i, n) ∈ N× N,

∆1(i, n) =

∆(i, n)− ε if i < Mε,0 if i ≥Mε,

and

∆2(i, n) =

∆(i, n) + ε if i < Mε,0 if i ≥Mε.

Recall the definition of the stopping times defined in (2.4.2) and particularly,τnθ,∆ = An. Observe that the case of the upper and lower bound described abovethus correspond to stopping times τnθ+ε,∆1

and τnθ−ε,∆2respectively. Applying

Proposition 2.4.1 to these cases with M = Mε yields

limn→∞

√kn

n− kP(τnθ+ε,∆1≥ k) = VMε(θ + ε,∆1),

limn→∞

√kn

n− kP(τnθ−ε,∆2≥ k) = VMε(θ − ε,∆2).

Couple τnθ+ε,∆1, τnθ,∆ = An and τnθ−ε,∆2

. The inequalities τnθ−ε,∆2≤ An ≤ τnθ+ε,∆1

hold, and hence we obtain

limn→∞

√kn

n− kP(An ≥ k) ∈ [VMε(θ − ε,∆2), VMε

(θ + ε,∆1)] .

Next, we show that the limits of the upper and lower bound coincide as ε ↓ 0, i.e.

limε↓0

[VMε(θ + ε,∆1)− VMε

(θ − ε,∆2)] = 0.

For this, we condition on the value of Un(Mε):

VMε(θ + ε,∆1)− VMε(θ − ε,∆2)

= limn→∞

√kn

n− k(P(τnθ+ε,∆1

≥ k)− P(τnθ−ε,∆2≥ k)

)≤ z1(ε) + z2(ε),

where

z1(ε) = limn→∞

√kn

n− kP(τnθ+ε,∆1

≥ k; τnθ−ε,∆2< k;Un(Mε)

∈ I1),

and

z2(ε) = limn→∞

√kn

n− kP(τnθ+ε,∆1

≥ k;Un(Mε)∈ I2

)

2


with I1 =[0, θ−ε+Mε−1

n

]and I2 =

[θ−ε+Mε−1

n , θ+ε+Mε−1n

].

Note that for all i < Mε and n ∈ N, ci,n are the same for τnθ+ε,∆1and τnθ+ε,∆1

bydefinition of ∆1 and ∆2. Applying Lemma 2.4.3 to z1(ε), we obtain

z1(ε) =2√2π

∫ θ−ε+Mε−1

0

P(UMε−1

(i) ≤ ciy, ∀i ≤Mε − 1

)

·(

(θ + ε+Mε − y)yMε−1

(Mε − 1)!e−y − (θ − ε+Mε − y)

yMε−1

(Mε − 1)!e−y)dy

≤ 4ε√2π

∫ θ+ε+Mε−1

0

yMε−1

(Mε − 1)!e−y dy ≤ 4ε√

2π.

For every fixed M ∈ N, differentiating yMe−y with respect to y and determiningits roots shows that this function has one maximum attained at y = M and hence,yMe−y ≤ MMe−M . Using Lemma 2.4.3, the previous argument and Stirling’sbound yields

z2(ε) =2√2π

∫ θ+ε+Mε−1

θ−ε+Mε−1

(θ + ε+Mε − y)yM−1e−y

(M − 1)!dy

≤ 2√2π

∫ θ+ε+Mε−1

θ−ε+Mε−1

(1 + 2ε)(M − 1)M−1

(M − 1)!e−(M−1) dy ≤ 4ε(1 + 2ε)√

2π.

Consequently,

VMε(θ + ε,∆1)− VMε(θ − ε,∆2) ≤ 8ε(1 + ε)√2π

,

and since the difference is non-negative, it must converge to zero as ε ↓ 0.

What remains to be shown is that the limit of VMε(θ + ε,∆1) exists, and thus alsoVMε(θ − ε,∆2), and is the same as V (θ,∆) defined in (2.3.5). The existence of thelimit follows from monotonicity. That is, VMε

(θ + ε,∆1) is non-decreasing andbounded from below by a strictly positive constant, for example VMε

(θ − ε,∆2)with ε = 1. Since every monotone bounded function in a complete metric spaceconverges, it follows that the limit exists as ε ↓ 0. Moreover, since it holds thatV (θ,∆) ∈ [V (θ − ε,∆2), V (θ + ε,∆1)] for every ε > 0, the value of the limit mustin fact be V (θ,∆).

We would like to conclude this section by remarking that the proof of Theorem 2.3.1shows why Algorithm 1 constitutes an approximation for V (θ,∆) that is withina preset distance from its true value. That is, suppose that for some fixed ε > 0we determined the pair (Mε, Nε) such that |∆(i, n)| < ε for all n ≥ Nε and allMε ≤ i ≤ k(n). Since V (θ,∆) lies between V (θ−ε,∆2) and V (θ+ε,∆1), it followsfrom the proof of Theorem 2.3.1 that

|V (θ,∆)− VMε(θ,∆)| ≤ 8ε(1 + ε)√2π

.

2

2.5. Identifying thresholds where power-law behavior prevails 39

2.5 Identifying thresholds where power-law behavior prevails

The purpose of this section is to illustrate the use of Theorem 2.3.1. When thesurplus capacity distribution and/or load surge function are given, we would liketo know what (growing) thresholds k := k(n), if any, yield power-law behaviorfor the exceedance probability. A sufficient condition is provided by Theorem 2.3.1and accordingly, we need to determine the thresholds k such that (A) and (B)are satisfied. A particularly compelling example involves the case where theloads of the failed lines are equally redistributed over the remaining lines, seee.g. [106]. Every time a line fails, the total load is redistributed over all survivinglines. Theorem 2.3.1 implies that in this case for power-law behavior to prevail, thestep increments of the composition F ln should become approximately 1/n upto the k’th failure from a certain point on. The key approach to identify thresholdswhere scale-free behavior prevails involves a Taylor expansion. To conclude thissection, we also consider an approach that can be used to numerically explore theasymptotic behavior for settings that do not fall in the framework of Theorem 2.3.1.

Example 2.5.1. First, we illustrate the impact of a different surplus capacity distribution.That is, suppose

ln(i) =i

n, i ≥ 1,

and let the surplus capacities be exponentially distributed with mean one. Using a Taylorexpansion we obtain

F (ln(i)) = 1− e−in =

i

n+O

((i

n

)2),

where O(·) denotes the big-O notation in relation with n→∞. Hence, ∆(i, n) = i2/nfor all (i, n) ∈ N× N and condition (A) is satisfied. We note that for (B) to hold, we needk = o(

√n). All thresholds that satisfy k = o(

√n) thus result in power-law behavior for

the exceedance probability.

The final claim holds generally for any surplus capacity distribution with a positive densityin zero. That is, in general, Taylor expansion yields

F (ln(i)) = F ′(0)ln(i) +O((ln(i))2). (2.5.1)

We observe that (2.5.1) leads to an approximation of the composition that only requiresinformation on the value F ′(0) and the load surge function. That is, the only propertyof the surplus capacity distribution we need for checking whether power-law behaviorprevails, is its behavior near its minimum. In particular, the mean of the surplus capacitydoes not play any role.

If the load surge function is given by

ln(i) =i

F ′(0)n, i ≥ 1,

and k = o(√n), we thus remain in the setting described in Theorem 2.3.1.

2


Example 2.5.2. Next, we verify and formalize the claims for the model in [106], wherethe total load is redistributed over all surviving lines. Specifically, the load surge functionis given by

ln(i) =an

n− i − a =ai

n− i ,

and suppose that a = 1/F ′(0). Then, applying the Taylor expansion (2.5.1), we obtain

∆(i, n) = O

((i

n

)2)

+O

(n

(i

n− i

)2)

for all (i, n) ∈ N× N. Again, we have pointwise convergence ∆(i) = 0 for all i ∈ N. Inaddition, we require that k = o(

√n) for condition (B) to hold for all i ≤ k.

We close this section by setting the threshold k to a certain fixed integer, whichallows us to analyze cases where the perturbations do not satisfy conditions (A)and (B). We suggest a method to explore the asymptotic behavior numerically forthese cases.

Observe that if the value ∆(1, n) tends too close to its lower bound as n→∞, thesystem does not perceive an initial disturbance and no line will fail. On the otherhand, if ∆(k, n) becomes too large as n→∞, the system cannot deal with such astrong increase of load and the threshold k will certainly be exceeded. If ∆(1, n) isnot too small and ∆(k, n) is not too large as n→∞, we obtain a non-degeneratelimit for the exceedance probability.

Proposition 2.5.1. Let ci,n := n · F (ln(i)) for (i, n) ∈ N × N and ci = limn→∞ ci,nfor i ∈ N, which is a non-decreasing sequence. If c1 > 0 and ck = O(1), then

limn→∞

P(An ≥ k) = 1−k∑

j=1

βj−1e−j ,

where βi, i ∈ N, are defined as in (2.3.3).

This result can be proven by applying results from extreme value theory. Pro-position 2.5.1 thus provides a method to determine the asymptotic exceedanceprobability for every fixed k, and can be used to numerically explore how theasymptotic tail of the number of failed lines decays as k grows large.

Proof of Proposition 2.5.1. It is known that the distribution function of a standarduniformly distributed random variable is contained in the maximum domain ofattraction of a Weibull distribution:

P(n(Un(n) − 1) ≤ x

)= P

(Un(n) ≤ 1 +

x

n

)−→

ex x ≤ 0,1 x > 0,

as n→∞.

2

2.5. Identifying thresholds where power-law behavior prevails 41

Then for every fixed k ∈ N, the first k order statistics converge in distribution to [2,Chapter 8]

(n(Un(n−i+1) − 1)

)i=1,...,k

d−→(Y (i)

)i=1,...,k

as n→∞, where the joint density of(Y (1), Y (2), ..., Y (k)

)is given by

ψ1(x1, ..., xk) = exk , xk < ... < x1 < 0.

This observation is essential to determine the asymptotic exceedance probability,which we derive next.

First suppose that ci,n does not depend on n, i.e. ci,n = ci for all n ∈ N. Then, theproof follows by induction. For k = 1 the statement holds, since

limn→∞

P(Un(1) ≤

c1n

)= limn→∞

1−(

1− c1n

)n= 1− e−c1 .

Suppose the statement holds for all integers strictly smaller than k. Then,

limn→∞

P(Un(i) ≤

cin, ∀i ≤ k

)= limn→∞

P(Un(n−i+1) > 1− ci

n, ∀i ≤ k

)

= P(Y (i) > −ci, ∀i ≤ k

)

=

∫ 0

−c1

∫ y1

−c2· · ·∫ yk−1

−ckeyk dyk · · · dy1

=

∫ 0

−c1

∫ y1

−c2· · ·∫ yk−1

−ck−1

eyk−1 dyk−1 · · · dy1 − e−ck∫ 0

−c1

∫ y1

−c2· · ·∫ yk−2

−ck−1

dyk−1 · · · dy1

= 1−k−1∑

j=1

βj−1e−j − eckβk−1 = 1−k∑

j=1

βj−1e−j .

By induction, the statement thus holds for all k ≥ 1.

Next, suppose ci,n does depend on n, i.e. there is at least one n ∈ N such thatci,n 6= ci. Then,

P(An ≥ k) = limn→∞

P(Un(i) ≤

ci,nn, ∀i ≤ k

)

n→∞−→ P(n(Un(n−i+1) − 1) ≥ −ci(1 + o(1)), ∀i ≤ k

).

Note that for every ε > 0 (small enough) there exists a N0 ∈ N such that for alln ≥ N0 and 1 ≤ i ≤ k,

P(An ≥ k) ≥ P(n(Un(n−i+1) − 1) ≥ −ci + ε,∀i ≤ k

)

and

P(An ≥ k) ≤ P(n(Un(n−i+1) − 1) ≥ −ci − ε,∀i ≤ k

).

2


Write V1, V2 for the integration area of the upper bound and the lower boundrespectively, and V for the integration area corresponding to c1, ..., ck. Since ex < 1for all x < 0, it follows that

lim supn→∞

P(An ≥ k) =

∫

V1

ey dy ≤∫

V

ey dy +

∫

V1\V1 dy

≤ 1−k∑

j=1

βj−1e−j + k(ck + ε)k−1ε.

Similarly, for the lower bound we have

lim infn→∞

P(An ≥ k) ≥∫

V

ey dy −∫

V \V2

1 dy ≥ 1−k∑

j=1

βj−1e−j − k(ck)k−1ε.

Letting ε ↓ 0 we obtain that both the upper bound and the lower bound convergeto 1−∑k

j=1 βj−1e−j .

3

Chapter 3

First-passage asymptoticsfor random walk bridges

Based on:First-passage time asymptotics over moving boundaries for random walk bridges

F. Sloothaak, V. Wachtel, B. Zwart.Journal of Applied Probability, 55(2), 2018, pp. 627–651.

In this chapter, we study the asymptotic tail behavior of the first-passage time overa moving boundary for a random walk conditioned to return to zero, where theincrements of the random walk have zero mean and finite variance. Typically, theasymptotic tail behavior may be described through a regularly varying functionwith exponent −1/2, where the impact of the boundary is captured by the slowlyvarying function. Yet, the moving boundary may have a stronger effect when thetail is considered at a time close to the return point of the random-walk bridge,leading to a possible phase transition depending on the order of distance betweenzero and the moving boundary.

This section is outlined as follows. In Section 3.1, we illustrate why this particularfirst-passage problem is of interest in the setting of cascading failure models, andwe provide a short overview of known results on first-passage times over movingboundaries in Section 3.2. In Section 3.3, we describe the model framework andcomment on the necessary assumptions. We establish the tail behavior of thefirst-passage time of the random-walk bridge in Section 3.4, where we distinguishbetween two cases. In the first case, we consider the tail behavior at a time thatis significantly smaller than the return time of the random-walk bridge, and weprove the result in Section 3.5. In the second case, the time is close to the returnpoint of the random-walk bridge, which may lead to a phase transition. We provethis result in Section 3.6.

43

3

44 Chapter 3. First-passage asymptotics for random walk bridges

3.1 Motivation

To understand our interest in this setting, we translate the framework described inChapter 2 to an equivalent formulation. Recall the model description in 2.1, wherewe adopt the notation in this chapter. Recall that a measure of system reliabilityis the number of component failures at the end of the cascading failure process,denoted by An. Since F (·) is continuous, the identity

P (An ≥ k) = P(Un(i) ≤ F (ln(i)) , i = 1, ..., k

)

is satisfied, where Un(i) denotes the i’th order statistic of n uniformly distributedrandom variables with support [0, 1]. We are interested in which choices of F (·)and ln(·) exhibits power-law behavior for large values of k. In this chapter, weconsider a setting with

F (ln(i)) =θ + i− 1 + ∆(i)

n, (3.1.1)

where ∆(·) correspond to the perturbations with respect to the affine case, i.e. thecase where ∆(i) = 0 for all i ≥ 1. Therefore, the central question is under whichrange of perturbation functions ∆(·) the failure size exhibits scale-free behavior.

This problem can be related to a random-walk bridge framework where theincrements have zero mean and finite variance. Consider Sn =

∑ni=1(1−Ei) where

(Ei)i∈N are independent identically exponentially distributed random variableswith mean one. It is well-known that

(Un(1), U

n(2), ..., U

n(n)

)d=

(E1

n,

∑2i=1Ein

, ...,

∑ni=1Ein

∣∣∣∣n+1∑

i=1

Ei = n

).

Then the probability that the number of component failures exceeds k can bewritten as

P(An ≥ k) = P(Un(i) ≤

θ + i− 1 + ∆(i)

n, i = 1, ..., k

)

= P (Si ≥ 1− θ −∆(i), i = 1, ..., k|Sn+1 = 1)

∼ P (Si ≥ 1− θ −∆(i), i = 1, ..., k|Sn = 0) .

In other words, we are interested in the probability that the first-passage time of arandom-walk bridge over a boundary sequence gi = 1− θ −∆(i), i ≥ 1, exceedsk.

In this chapter, we consider exactly this setting for a broad class of moving bound-aries. More precisely, we consider a random walk Si, i ≥ 0, conditioned to returnto zero at time n with increments that have zero mean and finite variance. Thepurpose is to derive the asymptotic tail of the first-passage time τg over a movingboundary (gi)i∈R for this random-walk bridge. We stress that we are only con-sidering the tail of τg for all times k := kn that are well before the time that therandom-walk bridge returns to zero. In other words, we extend a random-walkbridge in the Brownian setting to stay above a moving boundary over part of itsinterval, see Figure 3.1.

3

3.2. Literature review 45

0i

(Si)i∈R

(gi)i∈R τg

nk

Figure 3.1: Illustration of the random-walk bridge.

3.2 Literature review

The asymptotic behavior of random walks has long been an extremely populartopic in probability. In 1951, Donsker [43] showed that a suitably rescaled randomwalk converges to a Brownian motion. Many extensions have been studied overthe years, such as generalizations to random walks in the domain of attractionof a stable law [109] and additional conditioning properties. For example, aninvariance principle was shown for random-walk bridges in [75], whereas [16,41, 59] developed invariance principles for random walks conditioned to staypositive. Recently, these two types of conditioning have been combined in [23] toan invariance principle for random-walk bridges conditioned to stay positive overthe entire interval.

A natural question that arises is whether and how these results extend to movingboundaries. That is, how does the random walk behave asymptotically, condi-tioned it stays above a boundary sequence that is not necessarily zero or evenconstant? This topic, as well as the closely related first-passage asymptotics, areaddressed in [5, 34, 35, 53, 87, 88, 90] and many more papers. Our contributionlies in extending the results to random-walk bridges.

That is, the main object of interest in this chapter is the asymptotic behavior ofP(τg > k|Sn = 0). Clearly, the asymptotic behavior of this first-passage timedepends on the boundary sequence. We consider all boundary sequences thatare within square-root order from zero, and hence relatively not too far fromzero with respect to time. Our results distinguish between two regimes. The firstconcerns values of k that are significantly smaller than the time that the randomwalk returns to zero, whereas the second considers k close to the return point, i.e.k = n − o(n). In the first case, the asymptotic tail of the first-passage time canbe described through a regularly varying function with exponent −1/2. Whenthe boundary sequence satisfies certain additional conditions, as explained ine.g. [34] and in Section 3.3, the probability of the event τg > k|Sn = 0 to occurhas a power-law decay with a prefactor that can be interpreted in a probabilisticway. However, for the second regime, a phase transition possibly occurs when k

3


is close to the return point of the random-walk bridge, depending on how closethe boundary remains to zero. This intriguing phenomenon reflects the strongdependence on the boundary and constitutes a distinctive feature of random-walkbridges: the effect may not solely be captured in a slowly varying function, butcan affect the behavior much more drastically.

3.3 Model description and preliminaries

Let Xi, i ≥ 1, be independent, identically distributed random variables withE(Xi) = 0 and E(X2

i ) = 1 for all i ≥ 1. Define the random walk

Sm :=∑mi=1Xi, m ≥ 1.

We refer to gii∈N as the boundary sequence. Define the stopping time

τg := mini ≥ 1 : Si ≤ gi,i.e. the first-passage time of the random walk below the (moving) boundary. Incase gi = x for all i ≥ 1, we write Tx := τg for the stopping time to emphasize thefact that the boundary is constant.

Our framework makes three assumptions, where the first concerns the increments.

Assumption 3.3.1. The increments of the random walk are independent and identicallydistributed with mean zero and variance one. Additionally, we assume that the law of theincrements has a density f(·) (almost everywhere) and that there exists an n0 ∈ N suchthat fn0

(·), the density corresponding to Sn0, is bounded (almost everywhere).

We point out that the boundedness requirement on the density function of therandom walk for some n0 in Assumption 3.3.1 is a necessary and sufficient con-dition for uniform convergence between the scaled density of the position of therandom walk towards the standard normal density [93, p. 198]. Specifically, let φ(·)and Φ(·) denote the density function and the distribution function of a standardnormal random variable, respectively. Then,

limn→∞

supx∈R

∣∣∣∣P(Sn√n≤ x

)− Φ(x)

∣∣∣∣ = 0, limn→∞

supx∈R

∣∣√nfn(√nx)− φ(x)

∣∣ = 0.

(3.3.1)

Next, we assume that the boundary sequence gii∈N does not move too far fromzero.

Assumption 3.3.2. The boundary sequence gii∈N satisfies

|gi| = o(√i), (3.3.2)

and

P (τg > n) > 0, ∀n ≥ 1. (3.3.3)

3

3.3. Model description and preliminaries 47

Under Assumption 3.3.2, it is known that the position of the rescaled random walk,conditioned to stay above the boundary, converges to a Rayleigh distribution [34].To be precise, as n→∞,

P(Sn > gn + v

√n|τg > n

)∼ e−v2/2, ∀v ≥ 0. (3.3.4)

The first-passage time itself has a regularly varying tail,

P(τg > n) ∼√

2

π

Lg(n)√n, (3.3.5)

where Lg(·) is a positive, slowly varying function.

The slowly varying function in (3.3.5) has a probabilistic interpretation:

Lg(n) = E(Sn − gn; τg > n) ∼ E(−Sτg ; τg ≤ n) ∈ (0,∞). (3.3.6)

The literature offers many discussions and results for which this slowly varyingfunction converges to a finite constant Lg(∞) := limn→∞ Lg(n). We note thatin case that Lg(∞) < ∞ exists, the slowly varying term can be replaced by theconstant E(−Sτg ).

To the best of our knowledge [34] provide the weakest conditions that allow forthe existence of a finite Lg(∞). These conditions are a bit cumbersome, and as analternative, we mention two easily checked cases here that are known from theliterature.

In [53], it is shown that if the boundary sequence gn, n ≥ 1, is non-increasing andconcave, then Lg(∞) exists and

∞∑

n=1

−gnn3/2

<∞⇐⇒ Lg(∞) = E(−Sτg ) ∈ (0,∞).

In particular, this holds for all finite constant boundaries. In Theorem 5 of [125], theconcavity condition is relaxed, but a stronger summability condition is required.Specifically, it is shown that if gn, n ≥ 1 is non-increasing,

∞∑

n=1

log1/2 n

n3/2(−gn) <∞ =⇒ Lg(∞) = E(−Sτg ) ∈ (0,∞).

In this chapter, we consider a random walk that is conditioned to return to zero attime n. The objective is to derive the asymptotic behavior of the probability thatthis random walk stays above a moving boundary over part of its interval. Thatis, given that the random walk returns to zero at time n, what is the probabilityof the random walk staying above the moving boundary up to time k := kn? Weassume that this time k is at least ω(1) distance from both zero and n.

Assumption 3.3.3. Time k := kn satisfies both k →∞ and n− k →∞ as n→∞.

3


3.4 Main results

We distinguish between two cases: one where k is not too close to the point ofreturn of the random-walk bridge, and one where it is.

Theorem 3.4.1. Suppose lim supn→∞ k/n < 1. Then, uniformly in k/n,

P (τg > k|Sn = 0) ∼√

2

πLg(k)

√n− kn

k−1/2 (3.4.1)

as n→∞.

Note that when k is not too close to the boundary, the impact of the boundary iscompletely captured by the slowly varying function. When k moves closer to n,the behavior of the boundary becomes more relevant and possibly results in achange in asymptotics.

Theorem 3.4.2. Suppose k = n− o(n). In addition to Assumption 3.3.2, suppose thereexists an ε ∈ (0, 1) such that

supj∈[(1−ε)k,k]

|gj − gk| ≤ α(ε)|gk|, (3.4.2)

for every large enough k, with α(ε)→ 0 as ε ↓ 0. Then, as n→∞,

P (τg > k|Sn = 0) ∼

√2πLg(k)

√n−kk , if |gk| = o(

√n− k),√

2πLg(k)γ

(|gk|√n−k

) √n−kk , if |gk| = Θ(

√n− k),

2Lg(k) |gk|k , if |gk| = ω(√n− k), gk < 0,

(3.4.3)

where

γ(y) := e−y2

2 − y∫∞x=y

e−x2

2 dx. (3.4.4)

A typical example that is covered by this framework is when gi = −iα, i ∈ N,with α < 1/2. The additional assumption (3.4.2) is merely technical: it ensuresthat the boundary does not fluctuate too much as it moves closer to k. That is, forevery ε > 0 there is a value α(ε) <∞ such that the boundary does not fluctuatemore than 2α(ε)gk in the interval [(1− ε)k, k] for large enough n. Particularly, thisimplies that ε ∈ (0, 1) can be chosen small enough such that α(ε) < 1, and hencethe boundary sequence at time [(1 − ε)k, k] has the same sign (either positive,negative or zero). Cases where the boundary sequence strongly oscillates close totime k are thus excluded from our framework.

The phase transition that appears in Theorem 3.4.2 reflects the strong influence ofthe boundary sequence in this case. It might not be captured solely by the slowlyvarying function, but can have a much stronger effect. Furthermore, this effect isonly influenced by the behavior of the boundary sequence close to time k. This

3


observation is best explained by our approach. We track the position of a randomwalk at time k, conditioned that it stays above the moving boundary till that point.Then we evaluate how likely a reversed random walk moving back from time ncan reach that point. Due to a local limit theorem, the random walk is likely tostay within

√n− k = o(

√k) distance from zero. When gk < 0, those values are

thus likely of order max√n− k, |gk| distance from the boundary. The phase

transition is then a consequence of how likely the random walk staying above theboundary sequence can move to such values.

If we return to our original question on the failure size of the cascading fauiluremodel, we observe that Theorems 3.4.1 and 3.4.2 yield the result immediately.That is, as n→∞, we obtain

P(An ≥ k) ∼ P (τg > k|Sn = 0) ,

where the behavior of the latter term is given in Theorems 3.4.1 and 3.4.2. In theaffine case where ∆(i) = 0 for all i ≥ 1, we observe that

P(An ≥ k) ∼√

2

πL1−θ(k)

√n− kkn

with

L1−θ(k) ∼ E(−ST1−θ

)= −(1− θ) + 1 = θ,

where the equality is due to the memoryless property of exponentials. Moreover,the results of Theorems 3.4.1 and 3.4.2 allows for a much broader range of possibleperturbations, and quantify their effect on the prefactor in a probabilistic way.


We first consider the case where lim supn→∞ k/n < 1 as in Theorem 3.4.1. Definethe reversed random walk as

Sm =

m∑

i=1

Xm, 1 ≤ m ≤ n, (3.5.1)

where

Xm = −Xn+1−m, 1 ≤ m ≤ n.

Therefore, Sm obeys the same law as −Sm for all 1 ≤ m ≤ n.

Proof of Theorem 3.4.1. Note that

P (τg > k|Sn = 0) =

∫ ∞

u=gk

P (τg > k;Sk ∈ du|Sn = 0)

=1

fn(0)

∫ ∞

u=gk

P (Sk ∈ du; τg > k) fn−k(u)

=P(τg > k)

fn(0)

∫ ∞

u=gk

P (Sk ∈ du|τg > k) fn−k(u),

3


where fn−k(·) is the density of the reversed random walk at time n− k. Since thereversed random walk has i.i.d. increments with zero mean and finite variance,it also satisfies (3.3.1) and hence there is a uniform convergence to the normaldensity. Note that since limn→∞ k/n < 1, it holds that limn→∞ k/(n − k) < ∞.Therefore,

P (τg > k|Sn = 0) = (1 + o(1))P(τg > k)

1/√

2πn

∫ ∞

u=gk

P (Sk ∈ du|τg > k)e−

u2

2(n−k)

√2π(n− k)

= (1 + o(1))P(τg > k)

√n

n− kE(e−

S2k

2(n−k)

∣∣∣∣τg > k

).

It follows from (3.3.4) that

E(e−

S2k

2(n−k)

∣∣∣∣τg > k

)∼∫ ∞

0

e−v2

2k

n−k ve−v2/2 dv =

n− kn

.

Using (3.3.5), we conclude that

P (τg > k|Sn = 0) = (1 + o(1))

√n− kn

P(τg > k) = (1 + o(1))

√2

πLg(k)

√n− kkn

.

3.6 Threshold close to return point

Unfortunately, the analysis in the previous section does not follow through whenk = n−o(n). In particular, in our analysis we consider all the ways for the randomwalk to stay above the moving boundary until time k, and in time n−k to get backto its return point. In view of (3.3.4), the random walk conditioned on τg > kis likely to be at a position of order Θ(

√k) at time k. However, for u = Θ(

√k),

we can no longer replace fn−k(u) by an appropriately scaled normal density ifn− k = o(k). We therefore need to refine our approach, which we elaborate on inthis section.

3.6.1 Density of random walk

For the evaluation of the first-passage time of the random-walk bridge, it makessense to consider the position of a random walk at time k itself. A uniformconvergence result is given by Proposition 18 of [42] in case of constant boundaries.As this result is crucial in our analysis, we pose it here for our setting.

Proposition 3.6.1 (Proposition 18 of [42]). Let x := xn ≥ 0 denote the starting pointof a random walk (depending on n) and let y := yn be a sequence of non-negative numbers.Let U(·) denote the renewal function in the (strict) increasing ladder height process, andV (·) the renewal function corresponding to the decreasing ladder height process. LetE(−ST0

) be the expected position of a random walk at stopping time T0, and E(−ST0)

the expected position of a random walk with increments −Xi, i ≥ 0 at stopping time T0.Then the following results hold uniformly for every ∆ ∈ (0,∞) as n→∞.

3

3.6. Threshold close to return point 51

(i) For maxx/√n, y/√n → 0,

P (Sn ∈ [y, y + ∆), T0 > n|S0 = x) ∼V (x)

∫ y+∆

yU(w) dw

√2πn3/2

. (3.6.1)

(ii) For any (fixed) D > 1 with x/√n→ 0 and y/

√n ∈ [D−1, D],

P (Sn ∈ [y, y + ∆), T0 > n|S0 = x) ∼√

2

π

E(−ST0)V (x)∆√n

y

ne−

y2

2n , (3.6.2)

and uniformly for y/√n→ 0 and x/

√n ∈ [D−1, D],

P (Sn ∈ [y, y + ∆), T0 > n|S0 = x) ∼√

2

π

E(−ST0)U(y)∆√n

x

ne−

x2

2n . (3.6.3)

(iii) For any (fixed) D > 1 with x/√n ∈ [D−1, D] and y/

√n ∈ [D−1, D],

P (Sn ∈ [y, y + ∆), T0 > n|S0 = x) ∼ ∆q(x/√n, y/

√n)√

n, (3.6.4)

where q(x, y) is the density of P(W (1) ∈ dy, inf0≤t≤1W (t) > 0|W (0) = x) withW (t), t ≥ 0 the standard Wiener process. This has the explicit form [45]

q(u, v) =1√2π

(e−(u−v)2

2 − e−(u+v)2

2

), (3.6.5)

for every u, v > 0.

The asymptotic behaviors of V (·) andU(·) are quite well-understood: the functionsare both non-decreasing functions and regularly varying with exponent one. Inparticular, as t→∞,

U(t) ∼ t/E(−ST0), V (t) ∼ t/E(−ST0

). (3.6.6)

Moreover, for all random walks with finite variance σ2 = 1 it holds that

E(−ST0)E(−ST0

) =σ2

2=

1

2. (3.6.7)

The goal is to exploit Proposition 3.6.1 to derive the asymptotic behavior of therandom walk at time k, while staying above the moving boundary. Intuitively,we derive this by looking at the position of the random walk at time (1 − ε)k,where ε ∈ (0, 1) satisfies (3.4.2). Due to the additional assumption (3.4.2), one canreplace the boundary between (1− ε)k and k by a constant boundary with value(approximately) gk. The density is then derived using (3.3.4), (3.3.5) and the resultof Doney [42] for constant boundaries. This yields the following result.

3


Proposition 3.6.2. Let t ≥ gk with t − gk = Θ(|gk|) and (t − gk) → ∞ as k → ∞.Then, uniformly as k →∞,

P (Sk ∈ dt; τg > k)

dt∼√

2

π

Lg(k)

k3/2·

E(−ST0

)U(t− gk) if t = o(

√k),

te−t2

2k if t = Θ(√k).

(3.6.8)

For the proof of Proposition 3.6.2, we separate three cases depending on theposition of the random walk at time (1 − ε)k: a position close to the boundary(Lemmas 3.6.1 and 3.6.2), a position very far from the boundary (Lemmas 3.6.3and 3.6.4), or a position that lies in a typical distance from the boundary (proof ofProposition 3.6.2). We will show that the first two cases are very unlikely to occurcompared to the final case.

The two following lemmas show that it is unlikely for the random walk to be closeto its boundary at time (1− ε)k.

Lemma 3.6.1. Suppose t = o(√k) such that t − gk = Ω(|gk|) and (t − gk) → ∞ as

k →∞. Let ε ∈ (0, 1) be such that (3.4.2) is satisfied, and choose xε > 0 small enoughsuch that

1− e−−xε22(1−ε) < ε3/2 (3.6.9)

holds. Let vε,k = g(1−ε)k + xε√k. Then, there exists a constant C1 ∈ (0,∞) such that

for all ε ∈ (0, 1),

lim supk→∞

k3/2

Lg(k)U(t− gk)

P(Sk ∈ dt;S(1−ε)k < vε,k; τg > k

)

dt≤ C1

xε√(1− ε)

.

Proof. Define

g+k,ε = gk − α(ε)|gk|, (3.6.10)

and note that


)≤ P

(S(1−ε)k ≤ vε,k; τg > (1− ε)k

)

· supv∈[g(1−ε)k,vε,k]

P(Sεk ∈ dt;Tg+k,ε > εk|S0 = v

).

For v = o(√k), Equation (3.6.1) yields


)

dt∼U(t− g+

k,ε)√2π(εk)3/2

V (v − g+k,ε),

3


uniformly in v = o(√k) and t = o(

√k). On the other hand, if v = Θ(

√k),

then (3.6.3) yields


)

dt∼√

2

πE(−ST0

)U(t− g+

k,ε)

(εk)3/2(v − g+

k,ε)e−

(v−g+k,ε

)2

2εk ,

uniformly in v = Θ(√k) and t = o(

√k). We observe that e−x ≤ 1 for all x ≥ 0, and

E(−ST0) ∈ (0,∞) since the increments of the random walk have finite variance.

Moreover, due to (3.6.6) and (3.6.7), there exists a constant c1 ∈ (0,∞) such that

supv∈[g(1−ε)k,vε,k]


)≤ c1(vε,k − g+

k,ε)U(t− g+

k,ε)

(εk)3/2dt.

Due to assumption (3.4.2), we have that (g(1−ε)k− g+k,ε) < 2α(ε)|gk| = o(

√k). Also,

as U(·) is non-decreasing and (3.6.6) holds, there exists a constant c2 ∈ (0, 1) suchthat

U(t− g+k,ε) ≤ c2(1 + α(ε))U(t− gk).

Consequently, there exists a c3 ∈ (0,∞),



)

dt≤ c3xε

U(t− gk)

ε3/2k.

Finally, since (3.3.4) and (3.3.5) hold with Lg(·) a slowly varying function,

P(S(1−ε)k ≤ vε,k; τg > (1− ε)k

)

= P(S(1−ε)k ≤ vε,k|τg > (1− ε)k

)P (τg > (1− ε)k)

∼(

1− e−−xε22(1−ε)

)√2

π

Lg(k)√(1− ε)k

<

√2

π

ε3/2√1− ε

Lg(k)√k.

Multiplying the final two expressions yields the result.

Next we prove a similar result as Lemma 3.6.1, but where t = Θ(√k).

Lemma 3.6.2. Suppose t = Θ(√k) such that t ≥ gk. Let ε ∈ (0, 1) be such that (3.4.2)

holds, and choose xε small enough such that (3.6.9) is satisfied. Define g+k,ε as in (3.6.10)

and let vε,k = g(1−ε)k + xε√k. Then, there exists a constant C2 ∈ (0,∞) such that for

all ε ∈ (0, 1),

lim supk→∞

k3/2

Lg(k)te−t2/(2k)


)

dt≤ C2

xε√1− ε .

3


Proof. The proof is similar to the proof of Lemma 3.6.1, but in this case we have toconsider the asymptotics for t = Θ(

√k). Note that


)≤ P

(S(1−ε)k ≤ vε,k; τg > (1− ε)k

)

· supv∈[g(1−ε)k,vε,k]


).

For v = o(√k), Equation (3.6.2) yields


)

dt∼√

2

πE(−ST0

)V (v − g+

k,ε)

(εk)3/2(t− g+

k,ε)e−

(t−g+k,ε

)2

2εk ,

uniformly in t = Θ(√k) and v = o(

√k). Note e−x ≤ 1 for all x ≥ 0 and g+

k,ε =

o(√k). Since V (·) is non-decreasing and satisfies (3.6.6), we find that there exists a

c1 ∈ (0,∞) such that


√2

πE(−ST0)

V (v − g+k,ε)

(εk)3/2(t− g+

k,ε)e−

(t−g+k,ε

)2

2εk

≤ c1xε√k

(εk)3/2te−

t2

2k e−(1−ε)t2

2εk ≤ c1xεε3/2k

te−t2

2k .

On the other hand, if v = Θ(√k), then (3.6.3) yields


)

dt∼ 1√

2πεk

(e−

(v−t)22εk − e−

(v+t−2g+k,ε

)2

2εk

)

∼ 1√2πεk

(e−

(v−t)22εk − e− (v+t)2

2εk

),

uniformly in t = Θ(√k) and v = Θ(

√k). Using a Taylor expansion, we obtain

(e−

(v−t)22εk − e− (v+t)2

2εk

)= e−

v2

2εk− t2

2εk

(evtεk − e− vt

2εk

)≤ e− t

2

2k

(2vt

εk+ o

(vt

εk

)).

Therefore there exists a c2 ∈ (0,∞) such that


1√2πεk

(e−

(v−t)22εk − e− (v+t)2

2εk

)≤ c2

xεε3/2k

te−t2

2k .

We can conclude that there must exist a c3 ∈ (0,∞) such that



)

dt≤ c3

xεε3/2k

te−t2

2k .

3


Again, since (3.3.4) and (3.3.5) hold with Lg(·) a slowly varying function,

P(S(1−ε)k ≤ vε,k; τg > (1− ε)k

)∼(

1− e−−xε22(1−ε)

)√2

π

Lg(k)√(1− ε)k

≤√

2

π

ε3/2√1− ε

Lg(k)√k.

Multiplying this with the previous expression then concludes the proof.

The next two lemmas imply that it is unlikely for the random walk to be very farabove the boundary at time (1− ε)k.

Lemma 3.6.3. Suppose t = o(√k) such that t − gk = Ω(|gk|) and (t − gk) → ∞

as k → ∞. Let ε ∈ (0, 1) be such that (3.4.2) is satisfied. Then there exist constantsC3, C4 ∈ (0,∞) such that for all ε ∈ (0, 1),

lim supk→∞

k3/2

Lg(k)U(t− gk)

P(Sk ∈ dt;S(1−ε)k >

√k/ε; τg > k

)

dt

≤ C3 (1 + C4α(ε))1

ε√

1− εe− 1ε(1−ε) .

Proof. Let gk,ε be defined as in (3.6.10). Since t = o(√k), it follows from (3.6.3) that

there exists a c1 <∞ such that for every v = Ω(√k),


)

dt≤ c1

U(t− g+k,ε)√

εk.

Then it follows that


√k/ε; τg > k

)

dt

≤∫ ∞

v=√k/ε

P(S(1−ε)k ∈ dv; τg > (1− ε)k

) P(Sεk ∈ dt;Tg+k,ε > εk|S0 = v

)

dt

≤ c1U(t− g+

k,ε)

εkP(S(1−ε)k >

√k/ε; τg > (1− ε)k

).

In view of (3.3.4),

P(S(1−ε)k >

√k/ε; τg > (1− ε)k

)∼ P (τg > (1− ε)k) e−

1ε(1−ε)

∼√

2

π

Lg((1− ε)k)√(1− ε)k

e−1

ε(1−ε) ,(3.6.11)

3


where Lg(·) is slowly varying, and hence

lim supk→∞

Lg((1− ε)k)

Lg(k)= 1.

Due to (3.6.6) and (3.4.2),

lim supk→∞

U(t− g+k,ε)

U(t− gk)= 1 + α(ε) lim sup

k→∞

|gk|t− gk

≤ 1 + c2α(ε)

for some constant c2 ∈ (0,∞), since we assume that t− gk = Ω(|gk|). Therefore,

lim supk→∞

k3/2

Lg(k)U(t− gk)P(Sk ∈ dt;S(1−ε)k >

√k/ε; τg > k

)

≤√

2

πc1 (1 + c2α(ε))

1

ε√

1− εe− 1ε(1−ε) .

Next we prove a similar result as Lemma 3.6.3, but where t = Θ(√k).

Lemma 3.6.4. Suppose t = Θ(√k) such that t ≥ gk. Let ε ∈ (0, 1) be such that (3.4.2)

holds. Then there exists a constant C5 ∈ (0,∞) such that for all ε ∈ (0, 1),

lim supk→∞

k3/2

Lg(k)te−t2/(2k)P(Sk ∈ dt;S(1−ε)k >

√k/ε; τg > k

)

≤ C5√ε(1− ε)

e−1

ε(1−ε) dt.

Proof. The proof proceeds along the same lines as the proof of Lemma 3.6.3. Again,let gk,ε be defined as in (3.6.10). Since t = Ω(

√k) and (3.6.4), there exists a constant

c1 <∞ such that for every v = Ω(√k),


)

dt≤ c1√

εk,

and hence


√k/ε; τg > k

)

dt≤ c1√

εkP(S(1−ε)k >

√k/ε; τg > (1− ε)k

).

In view of (3.3.4), again (3.6.11) holds with Lg(·) a slowly varying function. Thatis, there exists a constant c2 ∈ (0,∞) such that

P(S(1−ε)k >

√k/ε; τg > (1− ε)k

)≤ c2

Lg(k)√(1− ε)k

e−1

ε(1−ε) .

3


The result follows by combining these two expressions and noting that

lim supk→∞

k1/2

te−t2/(2k)<∞.

Before proving Proposition 3.6.2, we provide a useful identity that is used.

Lemma 3.6.5. For every c, d > 0,

∫ ∞

y=0

ye−y2

2

(e−

(y−c)22d − e− (y+c)2

2d

)dy =

√2πc

√d

(1 + d)3e−

c2

21

1+d .

Proof. First note that for every a, b ∈ R,

∫ye−

y2

2a+ yb dy = −ae− y

2

2a+ yb − a3/2

bea

2b2

∫ a−by√ab

s=0

e−s2

2 ds.

Therefore,∫ ∞

y=0

ye−y2

2

(e−

(y−c)22d − e− (y+c)2

2d

)dy = e−

c2

2d

∫ ∞

y=0

ye−y2

21+dd

(eycd − e− ycd

)dy

= e−c2

2d

(d

1 + d

)3/2c

de

12

d1+d

c2

d2√

2π =√

2πc

√d

(1 + d)3e−

c2

21

1+d .

Next, we prove Proposition 3.6.2.

Proof of Proposition 3.6.2. We consider the position of the random walk at time(1 − ε)k with ε ∈ (0, 1). Specifically, fix ε ∈ (0, 1) such that (3.4.2) holds and,additionally,

α(ε) < lim infk→∞

t− gk|gk|

(3.6.12)

is satisfied. Note that the right-hand side is of order Ω(1) due to our assumptions,and hence such a ε ∈ (0, 1) satisfying (3.6.12) exists.

We partition the event Sk ∈ dt; τg > k in three disjoint ones depending on theposition of the random walk at time (1 − ε)k. Let vε,k = g(1−ε)k + xε

√k, where

xε > 0 is chosen small enough such that (3.6.9) is satisfied. Note that this choice ofxε implies that xε/

√ε(1− ε)→ 0 as ε ↓ 0, since

limε↓0

xε√ε(1− ε)

< limε↓0

√−2 log(1− ε3/2)

ε= 0.

3


Then,

P (Sk ∈ dt; τg > k) = P(Sk ∈ dt;S(1−ε)k < vε,k; τg > k

)

+ P(Sk ∈ dt;S(1−ε)k ∈ [vε,k,

√k/ε]; τg > k

)

+ P(Sk ∈ dt;S(1−ε)k >

√k/ε; τg > k

).

(3.6.13)

We consider an upper and lower limiting bound for P (Sk ∈ dt; τg > k) as k →∞,and show they coincide as ε ↓ 0. For readability of the proof, we consider the casest = o(

√k) and t = Θ(

√k) separately.

First, suppose t = o(√k), and we will derive an upper bound. Lemma 3.6.1

provides an upper bound for the first term in (3.6.13), and Lemma 3.6.3 yields anupper bound for the third term in (3.6.13). Recalling definition (3.6.10), we seethat the second term in (3.6.13) can be bounded by

P(Sk ∈ dt;S(1−ε)k ∈ [vε,k,

√k/ε]; τg > k

)

=

∫ √k/ε

v=vε,k

P(S(1−ε)k ∈ dv; τg > (1− ε)k

)

· P(Sk ∈ dt; τg > k|S(1−ε)k = v; τg > (1− ε)k

)

≤∫ √k/ε

v=vε,k

P(S(1−ε)k ∈ dv; τg > (1− ε)k

)P(Sεk ∈ dt;Tg+k,ε > εk|S0 = v

).

Due to Proposition 3.6.1, it holds uniformly in t = o(√k) and v = Θ(

√k) as

k →∞,


)

dt

= (1 + o(1))

√2

πE(−ST0

)U(t− g+

k,ε)√εk

v − g+k,ε

εke−

(v−g+k,ε

)2

2εk .

This yields

∫ √k/ε

v=vε,k

P(S(1−ε)k ∈ dv; τg > (1− ε)k

) P(Sεk ∈ dt;Tg+k,ε > εk|S0 = v

)

dt

= (1 + o(1))P (τg > (1− ε)k)

√2

πE(−ST0

)U(t− g+

k,ε)√εk

·∫ √k/ε

v=vε,k

v − g+k,ε

εke−

(v−g+k,ε

)2

2εk P(S(1−ε)k ∈ dv

∣∣τg > (1− ε)k).

First, due to (3.3.5) and the fact that Lg(·) is slowly varying,

P (τg > (1− ε)k) = (1 + o(1))

√2

π

Lg((1− ε)k)√(1− ε)k

= (1 + o(1))

√2

π

Lg(k)√(1− ε)k

.

(3.6.14)

3


Second, note that due to (3.6.6) and (3.4.2),

lim supk→∞

U(t− g+k,ε)

U(t− gk)= 1 + α(ε) lim sup

k→∞

|gk|t− gk

≤ 1 + c1α(ε)

for some c1 ∈ (0,∞) (since t− gk = Ω(|gk|)). Third, invoking (3.3.4) yields

∫ √k/ε

v=vε,k

v − g+k,ε

εke−

(v−g+k,ε

)2


∣∣τg > (1− ε)k)

= (1 + o(1))

∫ 1√ε(1−ε)

z=xε/√

1−ε

√1− εε√kze−

z2

21−εε ze−z

2/2 dz

≤ (1 + o(1))

√1− εε√k

∫ ∞

z=0

z2e−z2

2ε dz =

√π

2

√ε(1− ε)

k.

We conclude that for every ε ∈ (0, 1),

lim supk→∞

k3/2

Lg(k)U(t− gk)P(Sk ∈ dt;S(1−ε)k ∈ [vε,k,

√k/ε]; τg > k

)

≤√

2

πE(−ST0) (1 + c1α(ε)) .

Then, this expression together with Lemma 3.6.1 for the first term in (3.6.13) andLemma 3.6.3 for the third term in (3.6.13) yields the upper bound

lim supk→∞

k3/2

Lg(k)U(t− gk)P (Sk ∈ dt; τg > k) ≤ C1

xε√1− ε

√2

πE(−ST0)

+

√2

πE(−ST0) (1 + c1α(ε)) + C3 (1 + C4α(ε))

1

ε√

1− εe− 1ε(1−ε)

for every ε > 0. Letting ε ↓ 0, we conclude

lim supk→∞

k3/2

Lg(k)U(t− gk)P (Sk ∈ dt; τg > k) ≤

√2

πE(−ST0

). (3.6.15)

The proof of the lower bound follows along similar lines. Define

g−k,ε = gk + α(ε)|gk|. (3.6.16)

In view of (3.6.13), we bound it from below by the second term only. That is,

P (Sk ∈ dt; τg > k) ≥ P(Sk ∈ dt;S(1−ε)k ∈ [vε,k,

√k/ε]; τg > k

)

≥∫ √k/ε

v=vε,k

P(S(1−ε)k ∈ dv; τg > (1− ε)k

)P(Sεk ∈ dt;Tg−k,ε > εk|S0 = v

)

= (1 + o(1))P (τg > (1− ε)k)

√2

πE(−ST0

)U(t− g−k,ε)√

εkdt

·∫ √k/ε

v=vε,k

v − g−k,εεk

e−(v−g−

k,ε)2


∣∣τg > (1− ε)k).

3


First, we observe that (3.6.14) remains true for the lower bound. Second, note thatdue to (3.6.6) and (3.6.12),

lim infk→∞

U(t− g−k,ε)U(t− gk)

= 1− α(ε) lim infk→∞

|gk|t− gk

∈ (0,∞).

Third, invoking (3.3.4) and using partial integration we obtain∫ √k/ε

v=vε,k

v − g−k,εεk

e−(v−g−

k,ε)2


∣∣τg > (1− ε)k)

= (1 + o(1))

√1− εε√k

∫ 1√ε(1−ε)

z=xε/√

1−εz2e−

z2

2ε dz

= (1 + o(1))

xε√

ke−

x2ε2ε(1−ε) −

√ε

ke− 1

2ε2(1−ε) +

√ε(1− ε)

k

∫ 1ε√

1−ε

y= xε√ε(1−ε)

e−y2

2 dy

.


lim infk→∞

k3/2

Lg(k)U(t− gk)


dt

≥√

2

πE(−ST0)

(1− α(ε) lim inf

k→∞|gk|t− gk

)

·√

2

π

(xε√

ε(1− ε)e−

x2ε2ε(1−ε) − 1√

1− εe− 1

2ε2(1−ε) +

∫ 1ε√

1−ε

y=xε/√ε(1−ε)

e−y2

2 dy

).

We note that

limε↓0

(xε√

ε(1− ε)e−

x2ε2ε(1−ε) − 1√

1− εe− 1

2ε2(1−ε) +

∫ 1ε√

1−ε

y=xε/√ε(1−ε)

e−y2

2 dy

)=

√π

2,

and

limε↓0

(1− α(ε) lim inf

k→∞|gk|t− gk

)= 1.

In conclusion,

lim infk→∞

k3/2

Lg(k)U(t− gk)


dt≥√

2

πE(−ST0).

Note that this coincides with the upper bound (3.6.15), proving the result in caseof t = o(

√k).

Next, we consider the case t = Θ(√k). An upper bound for the first and third

term in (3.6.13) are given by Lemmas 3.6.2 and 3.6.4. The second term in (3.6.13)can again be bounded by


√k/ε]; τg > k

)

≤∫ √k/ε

v=vε,k

P(S(1−ε)k ∈ dv; τg > (1− ε)k

)P(Sεk ∈ dt;Tg+k,ε > εk|S0 = v

).

3


Due to Proposition 3.6.1, it holds uniformly in t = Θ(√k) and v = Θ(

√k) as

k →∞,


)

dt∼ 1√

2πεk

(e−

(v−t)22εk − e−

(v+t−2g+k,ε

)2

2εk

),

and hence


√k/ε]; τg > k

)

dt≤ (1 + o(1))

P (τg > (1− ε)k)√2πεk

·∫ √k/ε

v=vε,k

(e−

(v−t)22εk − e−

(v+t−2g+k,ε

)2

2εk

)P(S(1−ε)k ∈ dv

∣∣τg > (1− ε)k).

Again, we find that (3.6.14) holds. Moreover, due to (3.3.4),

∫ √k/ε

v=vε,k

(e−

(v−t)22εk − e−

(v+t−2g+k,ε

)2

2εk


∣∣τg > (1− ε)k)

= (1 + o(1))

∫ 1√ε(1−ε)

z=xε/√

1−ε

(e−

(z−t/√

(1−ε)k)2

2ε/(1−ε) − e−(z+t/

√(1−ε)k)2

2ε/(1−ε)

)ze−

z2

2 dz.

Applying Lemma 3.6.5,

∫ √k/ε

v=vε,k

(e−

(v−t)22εk − e−

(v+t−2g+k,ε

)2

2εk


∣∣τg > (1− ε)k)

≤ (1 + o(1))

∫ ∞

z=0

(e−

(z−t/√

(1−ε)k)2

2ε/(1−ε) − e−(z+t/

√(1−ε)k)2

2ε/(1−ε)

)ze−

z2

2 dz

= (1 + o(1))√

2π√ε(1− ε) t√

ke−

t2

2k .


lim supk→∞

k3/2

Lg(k)te−t2/(2k)


√k/ε]; τg > k

)

dt≤√

2

π.

Recalling (3.6.13) and invoking Lemmas 3.6.2 and 3.6.4, we derive for every ε ∈(0, 1)

lim supk→∞

k3/2

Lg(k)te−t2/(2k)


dt

≤ C2xε√

(1− ε)+

√2

π+

C5√ε(1− ε)

e−1

ε(1−ε) .

Letting ε ↓ 0 concludes

lim supk→∞

k3/2

Lg(k)te−t2/(2k)


dt≤√

2

π. (3.6.17)

3


For the lower bound of the probability that Sk ∈ dt; τg > k occurs, we observethat

P (Sk ∈ dt; τg > k) ≥ P(Sk ∈ dt;S(1−ε)k ∈ [vε,k,

√k/ε]; τg > k

)

≥ (1 + o(1))P (τg > (1− ε)k)√

2πεk

·∫ √k/ε

v=vε,k

(e−

(v−t)22εk − e−

(v+t−2g−k,ε

)2

2εk


∣∣τg > (1− ε)k).

Recall (3.6.14) and moreover, due to (3.3.4),∫ √k/ε

v=vε,k

(e−

(v−t)22εk − e−

(v+t−2g−k,ε

)2

2εk


∣∣τg > (1− ε)k)

= (1 + o(1))

∫ 1√ε(1−ε)

z=xε/√

1−ε

(e−

(z−t/√

(1−ε)k)2

2ε/(1−ε) − e−(z+t/

√(1−ε)k)2

2ε/(1−ε)

)ze−

z2

2 dz.

Note that [xε/√

1− ε, 1/√ε(1− ε)] = [0,∞]\[0, xε/

√1− ε]\[1/

√ε(1− ε),∞], and

also that it always holds that(e−

(z−t/√

(1−ε)k)2

2ε/(1−ε) − e−(z+t/

√(1−ε)k)2

2ε/(1−ε)

)≤ e−

(z−t/√

(1−ε)k)2

2ε/(1−ε) ≤ 1

for any ε ∈ (0, 1). Then, invoking Lemma 3.6.5 yields∫ √k/ε

v=vε,k

(e−

(v−t)22εk − e−

(v+t−2g−k,ε

)2

2εk


∣∣τg > (1− ε)k)

≥ (1 + o(1))

√2πε(1− ε) t√

ke−

t2

2k −∫ xε√

1−ε

z=0

ze−z2

2 dz −∫ ∞

z= 1√ε(1−ε)

ze−z2

2 dz

= (1 + o(1))

(√2π√ε(1− ε) t√

ke−

t2

2k −(

1− e−x2ε

2(1−ε)

)− e− 1

ε(1−ε )

).

Therefore, for every ε ∈ (0, 1),

lim infk→∞

k3/2

Lg(k)te−t2/(2k)P (Sk ∈ dt; τg > k)

≥ lim infk→∞

k3/2

Lg(k)te−t2/(2k)P(Sk ∈ dt;S(1−ε)k ∈ [vε,k,

√k/ε]; τg > k

)

≥√

2

π− 1

πlim infk→∞

(t√ke−

t2

2k

)−1

(1− e−

x2ε2(1−ε)

)

√ε(1− ε)

+1√

ε(1− ε)e−

1ε(1−ε)

.

We observe that as ε ↓ 0, this expression tends to√

2/π due to our choice of xε,and hence

lim infk→∞

k3/2

Lg(k)te−t2/(2k)P (Sk ∈ dt; τg > k) ≥

√2

π.

3


Since this coincides with the upper bound in (3.6.17), the proposition followsuniformly for t = Θ(

√k).

3.6.2 Proof of Theorem 3.4.2

Recall that Sm, m ≥ 1, denotes the reversed random walk defined in (3.5.1), andfm(·) the corresponding density function at time m. Then,

P (τg > k|Sn = 0) =

∫ ∞

u=gk

P (τg > k;Sk ∈ du|Sn = 0)

=1

fn(0)

∫ ∞

u=gk

P (Sk ∈ du; τg > k) fn−k(u).

(3.6.18)

It may be clear from the above identity that one may use Proposition 3.6.2 to derivethe main result. Yet, Proposition 3.6.2 does not provide the asymptotic behaviorfor the complete interval [−gk,∞], and we need to account for the behavior at itsextremes, i.e. for values within o(|gk|) distance from the boundary and values thatare beyond distance Θ(

√k). The next two lemmas turn out to be useful to bound

the behavior at these values.

Lemma 3.6.6. For every stopping time τg , irrespective of boundary (gi)i∈N, there exist ak0 ∈ N and constant c1 <∞ such that uniformly for all k ≥ k0 and for all x ≥ gk,

P (Sk ∈ dx; τg > k)

dx≤ c1

P (τg > bk/2c)√k

.

In particular, if the boundary satisfies gi = o(√i), there exists a constant c2 <∞ such

that


dx≤ c2

Lg(k)

k.

Proof. Set m = bk/2c. Then,


dx≤ P (Sk ∈ dx; τg > m)

dx

=

∫ ∞

y=gm

P (Sm ∈ dy; τg > m)P(Sk−m ∈ dx

∣∣S0 = y)

dx

=

∫ ∞

y=gm

P (Sm ∈ dy; τg > m) fk−m(x− y).

Assumption 3.3.1 implies that there exists a constant c3 <∞ such that

supz∈R

fk−m(z) ≤ c3√k −m

.

3


Therefore,


dx≤ c3√

k −m

∫ ∞

y=gm

P (Sm ∈ dy; τg > m) ≤√

2c3√kP (τg > m) .

For the second assertion, note that due to (3.3.5) with Lg(·) slowly varying,

P (τg > m) ∼√

2

π

Lg(m)√m∼ 2√

π

Lg(k)√k

=√

2P (τg > k) .

The next lemma quantifies how likely the random walk staying above the movingboundary is to have a position relatively close to the boundary.

Lemma 3.6.7. Suppose xk = o(√k) is such that xk = Ω(|gk|) and xk →∞ as k →∞.

There exists a constant c1 <∞ such that

P (Sk ≤ gk + xk; τg > k) ≤ c1x2k

Lg(k)

k3/2.

Proof. Recall (3.6.10) for some fixed ε ∈ (0, 1), and let m = bk/2c. Note that

P (Sk ≤ gk + xk; τg > k)

=

∫ ∞

u=gm

P (Sm ∈ du; τg > m)

∫ gk+xk

v=gk

P(Sk−m ∈ dv;Tg+k,ε

> k −m∣∣S0 = u

)

≤ P (τg > m)xk supv∈[gk,gk+xk],u≥gm


> k −m∣∣S0 = u

)

dv.

Applying Lemma 3.6.6,


> k −m∣∣S0 = u

)

dv≤ c2√

k −mP(Tg+k,ε−v

> b(k −m)/2c)

for some constant c2 <∞. Taking its supremum over v ∈ [gk, gk + xk] yields

P (Sk ≤ gk + xk; τg > k)

≤ c2√k −m

P (τg > m)xkP(T−(α(ε)|gk|+xk) > b(k −m)/2c

).

In view of (3.3.5) with Lg(·) slowly varying, we have that there exists a c3 < ∞such that

P (τg > m) ≤ c3Lg(k)√

k.

3


Moreover, for constant boundaries it holds that for some constants c4 ≤ c5 <∞,

P(T−(α(ε)|gk|+xk) > b(k −m)/2c

)≤ c4

U(α(ε)|gk|+ xk)√b(k −m)/2c

≤ c5xk√k,

where the latter inequality follows from (3.6.6) and since xk = Ω(|gk|). This yieldsthat there exists a c1 <∞ such that

P (Sk ≤ gk + xk; τg > k) ≤ c2√k −m

c3Lg(k)√

kxkc5

xk√k≤ c1x2

k

Lg(k)

k3/2.

Next, we prove our main result.

Proof of Theorem 3.4.2. As in the proof of Proposition 3.6.2, we provide an appro-priate upper and lower bound for P (τg > k|Sn = 0), and show that these behaveidentically in the limit. Fix δ ∈ (0, 1), and recall (3.6.18), i.e.

P (τg > k|Sn = 0) =1

fn(0)

∫ ∞

u=gk

P (Sk ∈ du; τg > k) fn−k(u).

Let xk = |gk| if |gk| → ∞ as k → ∞, and xk = (n − k)1/4 if gk = O(1). For theupper bound, we will partition the integration area in three intervals, namely[gk, gk + δxk], [gk + δxk, δ

√k] and [δ

√k,∞].

For values close to the boundary, we observe the following. Assumption 3.3.1implies that there exists a constant c1 ∈ [0,∞) such that

supx∈R

fn−k(x) ≤ c1√n− k

,

and hence, applying Lemma 3.6.7

∫ gk+δxk

u=gk

P (Sk ∈ du; τg > k) fn−k(u) ≤ c1√n− k

P (Sk ∈ [gk, gk + δxk]; τg > k)

≤ c2√n− k

δ2x2kLg(k)

k3/2

for some c2 <∞. Due to (3.3.1),

1

fn(0)∼√

2πn ∼√

2πk, (3.6.19)

and hence

P(τg > k;Sk ≤ gk + δxk

∣∣Sn = 0)

= δ2 ·O(x2kLg(k)

k√n− k

).

Note that this is at most of an order stated in (3.4.3) in all three cases. As δ ↓ 0, thisterm is therefore negligible with respect to (3.4.3).

3


For values relatively far from the boundary, we observe that due to Lemma 3.6.6,there exists a constant c3 <∞,

∫ ∞

u=δ√k

P (Sk ∈ du; τg > k) fn−k(u) ≤ c3Lg(k)

kP(Sn−k > δ

√k).

Applying Chebyshev’s inequality on Sn−k with Var(Sn−k) = n− k,

P(Sn−k > δ

√k)≤ 1

δ2

n− kk

.

Recalling (3.6.19), we observe that

P(τg > k;Sk ≥ δ

√k∣∣Sn = 0

)=

1

δ2O

(Lg(k)

k

n− k√k

)

In all three cases this term is of strictly smaller order as k →∞ than stated in (3.4.3),and hence is also negligible with respect to (3.4.3).

For values in [gk + δxk, δ√k], we will use Proposition 3.6.2. Using (3.6.6) and the

fact that e−x2 ≤ 1 for all x ∈ R, Proposition 3.6.2 implies that uniformly for all

u ∈ [gk + δxk, δ√k],

P (Sk ∈ du; τg > k)

du≤ (1 + o(1))

√2

π

Lg(k)

k3/2(u− gk).

Therefore,

∫ δ√k

u=gk+δxk


≤ (1 + o(1))

√2

π

Lg(k)

k3/2

∫ δ√k

u=gk+δxk

(u− gk)fn−k(u)

= (1 + o(1))

√2

π

Lg(k)

k3/2

√n− k E

(Sn−k − gk√

n− k;Sn−k√n− k

∈[gk + δxk√n− k

,δ√k√

n− k

]).

Using the Central Limit Theorem, we observe that as n→∞,

E

(Sn−k − gk√

n− k;Sn−k√n− k

∈[gk + δxk√n− k

,δ√k√

n− k

])

∼

∫∞v=0

1√2πve−v

2/2 dv = 1√2π

if gk = o(√n− k),

∫∞v=

gk+δ|gk|√n−k

1√2π

(v − gk√

n−k

)e−v

2/2 dv if gk = Θ(√n− k),

−gk√n−k if gk = ω(

√n− k), gk < 0.

3


Adding the three terms corresponding to the three disjoint intervals shows thatfor every 0 < δ < 1 we have the upper bound,

∫ ∞

u=gk

P (Sk ∈ du; τg > k) fn−k(u) ≤ (1 + o(1))

√2

π

Lg(k)

k3/2(n− k)

·

1√2π


∫∞v=

gk+δ|gk|√n−k

1√2π

(v − gk√

n−k

)e−v

2/2 dv if gk = Θ(√n− k),


√n− k), gk < 0.

For the lower bound, Proposition 3.6.2 and (3.6.6) imply that it holds uniformly inu ∈ [gk + δxk, δ

√k],

P (Sk ∈ du; τg > k)

du≥ (1 + o(1))

√2

π

Lg(k)

k3/2(u− gk)e−

δ2

2 .

Therefore, we obtain the lower bound

∫ ∞

u=gk

P (Sk ∈ du; τg > k) fn−k(u) ≥∫ δ√k

u=gk+δx


≥ (1− o(1))

√2

π

Lg(k)

k3/2

√n− ke−δ2/2

·

∫∞v=0

1√2πve−v

2/2 dv = 1√2π


∫∞v=

gk+δ|gk|√n−k

1√2π

(v − gk√

n−k

)e−v

2/2 dv if gk = Θ(√n− k),


√n− k), gk < 0.

We observe that as δ ↓ 0 the lower and upper bound coincide. Since we haveidentity (3.6.18) with (3.6.19), we conclude that (3.4.3) holds.

4

Chapter 4

Impact of a network disconnection

Based on:Impact of network splitting on cascading failure blackouts

F. Sloothaak, S.C. Borst, and B. ZwartIEEE Power and Energy Society General Meeting (PESGM), 2017.

andThe impact of a network split on cascading failure processes

F. Sloothaak, S.C. Borst, and B. ZwartStochastic Systems, to appear.

Cascading failure models are typically used to capture the phenomenon wherefailures possibly trigger further failures in succession, causing knock-on effects.In the previous two chapters, our goal was to identify the range of load surgefunctions and surplus capacity distributions leading to scale-free behavior for thefailure size. In this framework, it is assumed that every failure continues to affectall other remaining lines during the cascade.

Yet, in reality, successive line failures may cause the underlying network to dis-integrate in disjoint components. Once a network split has occurred, the failurepropagation continues independently among the various components. Networksplitting is also known as islanding, and is sometimes used as a tool in powersystems to prevent blackouts from cascading to large-scale proportions [14]. In thischapter, we are interested in the impact of islanding on the power-law exponent.To allow for a rigorous derivation, we assume that a single line failure immediatelysplits the network in two components and causes no consecutive disconnectionsanymore. The results provide valuable qualitative insights that are crucial firststeps towards understanding more complex network splitting scenarios.

This chapter is structured as follows. In Section 4.1, we describe in detail themodel that we consider in this chapter. We describe the tail behavior of the failuresize in Section 4.2, as well as an intuitive explanation why this type of behaviorappears. In our results, we distinguish between several different cases, dependingon the sizes of the two disconnected components and the value of the threshold.

69

4

70 Chapter 4. Impact of a network disconnection

We prove our results in Section 4.3. The asymptotic behavior gives rise to anapproximation scheme for the actual failure size, which we describe and validatethrough simulation experiments in Section 4.4. Finally, we discuss a few possibleextensions of our framework in Section 4.5.

4.1 Model description and preliminaries

We consider a network consisting of two star-shaped components connected bya single line, see Figure 4.1. More specifically, we consider a network with n+ 2nodes, where n is large. The network consists of two components connected bya single line. The smaller component consists of l := ln ≤ n/2 lines, whereas theother component has n − l lines, and hence l ≤ n − l. Each line has a limitedcapacity for the amount of load it can carry before it fails. We assume that thenetwork is initially stable in the sense that every line has enough capacity to carryits load. The difference between the initial load and capacity is called the surpluscapacity, and we assume it to be independent and standard uniformly distributedat each of the n lines.

1

2

3

l-1

l

1

2

3

4

n-l-2

n-l-1

n-l

Figure 4.1: Visual representation of the network.

The cascading failure process is initiated by the failure of the single line connectingthe two components. This event creates two disjoint components, and causes theload at all other lines to increase by θ/n for a certain constant θ > 0. If this loadincrease exceeds the capacity of one or more lines, those lines will fail. Everysubsequent failure again results in a load increase at the surviving lines, and wecall such an increase the load surge. This cascading failure process continues untilthe surplus capacity for every surviving line exceeds its load. We assume that theload surge caused by each consecutive line failure in the smaller component is 1/l,and in the larger component 1/(n−l), and that both components remain connectedafter every consecutive line failure. In other words, the cascading failure processesbehave independently between the two components and no further disconnectionoccurs.

The objective is to examine the impact of a single immediate network split onthe exceedance probability. It turns out that the power-law property, which

4

4.1. Model description and preliminaries 71

appeared in case of a star topology, mostly prevails. However, the splittingfeature may possibly change the prefactor and the exponent depending on thethreshold and component sizes. The results can intuitively be interpreted asfollows. When the threshold is sufficiently smaller than the size of the smallercomponent, the threshold is most likely exceeded in just one of the componentsalone. If the threshold is approximately between the size of the smaller and thelarger component, the threshold is most likely exceeded in the bigger componentalone. In both cases, this property will imply that the power-law exponent is−1/2 as is the case in a single star network. For larger threshold values, bothcomponents need to have a significant number of line failures. Consequently, it ismuch less likely for the threshold to be exceeded, which causes a phase transition:the power-law exponent is reduced to −1.

Our methodology uses an asymptotic analysis for the sum of two independentquasi-binomially distributed random variables. We distinguish between differentcases: the balanced case where the sizes of both components are of the sameorder of magnitude, and the disparate case where one is of a smaller order. Inthe analysis many subtleties need to be accounted for, which are most apparentwhen the threshold is close to the size of the larger component. These obstaclescannot be handled with existing techniques from the area of heavy-tailed distri-butions [48, 102]. Our analysis in Section 4.2 aims to provide physical insights inthese subtleties.

The case of a single star network, where each line failures causes a single node tobecome isolated, has been studied rigorously in Chapter 2. Specifically, this caseinvolves a star network consisting of n+1 nodes, n lines with uniformly distributedsurplus capacities, an initial load surge of θ/n at all lines, and subsequent loadsurges of 1/n at all surviving lines. In that case, the following result holds.

Theorem 4.1.1. Let k? := kn? and k? := kn? be growing sequences of n with k? ≤ k?,

i.e. both k? →∞ and n− k? →∞ as n→∞. Then,

limn→∞

supk∈[k?,k?]

∣∣k3/2

√n− kn

P (An = k)− θ√2π

∣∣ = 0, (4.1.1)

and

limn→∞

supk∈[k?,k?]

∣∣k1/2

√n

n− kP (An ≥ k)− 2θ√2π

∣∣ = 0. (4.1.2)

Proof of Theorem 4.1.1. Equation (4.1.1) is Proposition 2.2.1. The second statementfollows the lines of the proof of Theorem 2.2.1, but it is adapted here to show itholds uniformly in k ∈ [k?, k

?].

Choose k = n− log(n− k?) and fix ε > 0. Following the proof of Theorem 2.2.1,

4


we observe that for every k ∈ [k?, k?] that

√kn

n− kP (An ≥ k) ≤ e θ

θ/k + 1

n

k(n− k)

+ (1 + ε)2θ√2π

+ c ·√k/n

1− log(n− k)/n

log(n− k)√n− k

for some positive constant c, and

√kn

n− kP (An ≥ k) ≥ (1− ε) 2θ√2π

1−

√k

k

√n− kn− k

for large enough n. Therefore,

supk∈[k?,k?]

∣∣∣∣√

kn

n− kP(An ≥ k)− 2θ√2π

∣∣∣∣

≤ supk∈[k?,k?]

max

ε

2θ√2π

1−

√k

k

√n− kn− k

+

2θ√2π

√k

k

√n− kn− k ,

ε2θ√2π

+ eθ

θ/k + 1

n

k(n− k)+

c ·√k/n

1− log(n− k)/n

log(n− k)√n− k

.

We see that as n→∞, this gives

limn→∞

supk∈[k?,k?]

∣∣∣∣√

kn

n− kP(An ≥ k)− 2θ√2π

∣∣∣∣ ≤ ε2θ√2π.

Letting ε ↓ 0 concludes the proof.

4.2 Main results and road maps of the proofs

The exceedance probability naturally depends on the threshold and the componentsizes. In essence, we derive the tail distribution of An = Al,n + An−l,n, whereAl,n and An−l,n are independent random variables. Note that Al,n correspondsto the number of line failures in a single star network with initial load surgevalue θ/n = θ/l · l/n and consecutive load surges 1/l. Therefore, Al,n obeysa quasi-binomial distribution [40], where the asymptotic behavior is given byTheorem 4.1.1 (with θ replaced by θ · l/n). We point out that Al,n is thus heavy-tailed for all values that are not too close to l. We derive the asymptotic behavior ofthe probability that the sum of two quasi-binomial distributed random variablesexceeds a network-size dependent threshold k.

As mentioned earlier, we distinguish between two cases: the balanced case whereβ = limn→∞ l/n > 0, and the disparate case where l = o(n). Henceforth, letα = limn→∞ k/n. Our results apply to thresholds k that are "growing with n",i.e. thresholds for which both k →∞ and n− k →∞ hold as n→∞.

4

4.2. Main results and road maps of the proofs 73

4.2.1 Balanced component sizes

In this section, we consider the case where the two component sizes are of thesame order, and derive the tail of An. This tail behavior reflects the most likelyscenarios for the number of line failures to exceed threshold k. Recall that Anis essentially the sum of two heavy-tailed random variables (since l → ∞ asn→∞). Moreover, the tail of both random variables typically obeys a power-lawdistribution with exponent −1/2 in the balanced case.

l

n− l n An−l,n

Al,n

k

k

P(An ≥ k)

(a) Case 0 ≤ α < β.

l

n− l n An−l,n

Al,n

k

k

P(An ≥ k)

(b) Case β ≤ α < 1− β.

l

n− l n An−l,n

Al,n

k

k

P(An ≥ k)

(c) Case 1− β ≤ α < 1.

Figure 4.2: Asymptotic contributions to the exceedance probability inTheorem 4.2.1.

This observation yields intuition for the asymptotic behavior of the exceedanceprobability. Figure 4.2 illustrates this intuition, where the bolder areas reflectwhich scenarios asymptotically contribute to the exceedance probability. Whenthe threshold is significantly smaller than both component sizes (α < β), the mostlikely scenario to exceed k is when it is exceeded in one of the components alone.In other words, the event where both Al,n and An−l,n attain large values is muchless likely to occur. Similarly, if the threshold is only significantly smaller than thelarger component size (β ≤ α < 1− β), the most likely scenario for An to exceed kis when it is exceeded in the larger component, while the smaller component onlyhas very few line failures. We observe that in both cases the tail of An thereforeobeys a power-law distribution with exponent −1/2.

If 1− β < α < 1, both components must have many line failures. The threshold isthen most likely to be exceeded if in both components a non-negligible fractionof the lines have failed. This causes the power-law exponent to decrease to−1/2− 1/2 = −1, i.e. a phase transition appears at α = 1− β. These notions leadto the following theorem.

Theorem 4.2.1. Suppose β ∈ (0, 1/2] and α 6= 1− β. As n → ∞, An asymptoticallybehaves as follows. If 0 ≤ α < β, then

P (An ≥ k) ∼ 2βθ√2π

√1− k

lk−1/2 +

2(1− β)θ√2π

√1− k

n− l k−1/2. (4.2.1)

If β ≤ α < 1− β, then

P (An ≥ k) ∼ 2(1− β)θ√2π

√1− k

n− l k−1/2. (4.2.2)

4


Lastly, if 1− β < α < 1, then

P (An ≥ k) ∼ α√β(1− β)θ2

πc(α, β)k−1,

where

c(α, β) =

∫ 1

x=α−(1−β)

β

x−3/2

√1− x

√s(x)

1− s(x)dx,

s(x) =βx− (α− (1− β))

1− β .

To prove Theorem 4.2.1, we partition the event of exceeding the threshold in threeterms:

An ≥ k = An ≥ k;Al,n ≤ s? ∪ An ≥ k; s? < Al,n < s?∪ An ≥ k;Al,n ≥ s?,

(4.2.3)

where s? is chosen appropriately small, and s? appropriately large. Table 4.1illustrates which term will yield the dominant behavior in each of the cases inTheorem 4.2.1.

P (An ≥ k;Al,n ≤ s?) P (An ≥ k;Al,n ∈ (s?, s?)) P (An ≥ k;Al,n ≥ s?)

α ∈ [0, β) P (An−l,n ≥ k) negligible P (Al,n ≥ k)α ∈ [β, 1− β) P (An−l,n ≥ k) negligible 0 or negligibleα ∈ (1− β, 1) 0 dominant 0

Table 4.1: Road map for proof of Theorem 4.2.1.

The reasoning turns more subtle at the boundary where the threshold is eitherclose to the larger component size, or when it is close to n itself. In view ofTheorem 4.2.1, the first case (α = 1− β) corresponds to the interval of thresholdvalues where we move from a power-law distribution with exponent −1/2 toone with exponent −1. When the larger component remains significantly largerthan the smaller one (0 < β < 1− β < 1), this phase transition occurs as follows.As long as threshold k is sufficiently smaller than l, the most likely scenario toexceed k remains when it is already exceeded in the larger component alone.However, the closer k is to l, the smaller this probability is and it is in fact zerowhen l > k. From some specific point, the scenario where the number of linefailures in the larger component is close to k, yet not exceeding it, becomes themost likely one. If α = β = 1− β = 1/2, a similar likely event can also occur forthe smaller component. Figure 4.3 reflects this intuition of Theorem 4.2.2. Again,the bold areas indicate which scenarios possibly asymptotically contribute to theexceedance probability.

Finally, if the threshold is close to the network size itself (α = 1), almost all linesin both of the components need to have failed. Visually, this case is comparable tothe one in Figure 4.2(c), where the triangle is minuscule.

4


l

n− l n An−l,n

Al,n

k

k

P(An ≥ k)

(a) Case β ∈ (0, 1/2).

l

n− l n An−l,n

Al,n

k

k

P(An ≥ k)

(b) Case β = 1− β = 1/2.

Figure 4.3: Asymptotic contributions to P(An ≥ k) in Theorem 4.2.2 ifα = 1− β.

Theorem 4.2.2. Suppose α = 1− β and β ∈ (0, 1/2], and write r := rn = k − (n− l)and t := tn = k − l. Then as n → ∞, An asymptotically behaves as follows. Ifβ ∈ (0, 1/2), then

P (An ≥ k) ∼

2(1−β)θ√2π

√−rk if r < 0,−r = ω((log k)2),

2(1−β)θ√2π

√−rk + β(1−β)θ2

πlog kk if r < 0,−r ∝ (log k)2,

β(1−β)θ2

πlog kk if r < 0,−r = o(log k)2,

β(1−β)θ2

πlog(k/r)

k otherwise.

If β = 1/2, write η := limn→∞ t/r. Then,

P (An ≥ k)

∼

θ√2π

√−r+√−tk if − r = ω((log k)2), η > 0,

θ√2π

√−rk if − r = ω((log k)2), η ≤ 0,

θ√2π

√−r+√−tk + θ2

2πlog kk if r ∝ −(log k)2, η > 0,

θ√2π

√−rk + θ2

4πlog k+log(k/(|t|+1))

k if r ∝ −(log k)2, η ≤ 0, |t| 6= k1−o(1),θ√2π

√−rk + θ2

4πlog kk if r ∝ −(log k)2, t = k1−o(1),

θ2

4πlog(k/(|r|+1))+log(k/(|t|+1))

k if r else, k|t|+1 6=

(k|r|+1

)o(1)

,

θ2

4πlog(k/(|r|+1))

k otherwise.

If α = 1, then as n→∞,

P (An ≥ k) ∼ θ2

2(n− k)k−2. (4.2.4)

Since there is a sharp transition from a power-law with exponent−1/2 to one withexponent −1 when α = 1− β, it is natural to consider the number of failures inthe bigger component in more detail. In the proof of Theorem 4.2.2, we partitionthe event of exceeding the threshold with respect to the number of line failures inthe bigger component.

4


When α = 1− β with β ∈ (0, 1/2), we use the identity

P (An ≥ k) = P (An−l,n ≥ k) +P (An ≥ k;An−l,n ∈ [k − s?, k))

+ P (An ≥ k;An−l,n ∈ (k − s?, k − s?))+ P (An ≥ k;An−l,n ∈ [k − l, k − s?]) ,

(4.2.5)

where s? and s? are chosen in a specific way. Labeling the probability terms onthe right-hand side I, II, III, IV respectively, the asymptotic behavior of each termcan be evaluated separately, which yields the result as in Table 4.2.

Term Probability Asymptotic behaviorI P (An−l,n ≥ k) 2(1−β)θ√

2π

√−rk 1−r>0 growing

+O(k−1) · 1r≤0 fixedII P (An ≥ k;An−l,n ∈ [k − s?, k)) o

(log(l/(|r|+1))

k

)

III P (An ≥ k;An−l,n ∈ (k − s?, k − s?)) β(1−β)θ2

πlog(k/(|r|+1)

k

IV P (An ≥ k;An−l,n ∈ [k − l, k − s?]) o(

log(l/(|r|+1))k

)

Table 4.2: Asymptotic behavior of terms in (4.2.5).

The result then follows by determining the dominant terms of (4.2.5) for thevarious cases of the threshold. It turns out there is a transition in dominantbehavior when −r ∝ (log k)2. For a smaller threshold, the threshold remains mostlikely to be exceeded in the larger component alone. Otherwise, it is most likelyexceeded due to almost all lines failing in the larger component in conjunction witha growing number of line failures in the smaller component. This is summarizedby Table 4.3.

I II III IV−r = ω((log k)2) 2(1−β)θ√

2π

√−rk negligible negligible negligible

−r ∝ (log k)2 2(1−β)θ√2π

√−rk negligible β(1−β)θ2

πlog kk negligible

|r| = o(log k)2 0 or negligible negligible β(1−β)θ2

πlog kk negligible

r > 0 growing 0 negligible β(1−β)θ2

πlog(k/r)

k negligible

Table 4.3: Road map for proof of Theorem 4.2.2 with α = 1 − β withβ ∈ (0, 1/2).

The situation turns even more subtle when α = β = 1/2. In the most extremecase, we may have l = n− l = n/2 and one cannot distinguish between a smallerand larger component. The cascading process in the component of size l cantherefore become more significant, leading to more possible scenarios likely tohave occurred if the threshold is exceeded. To capture these scenarios, we need to

4


refine the partitioning of events in (4.2.5) to

P (An ≥ k) = P (An−l,n ≥ k) + P (An ≥ k;An−l,n ∈ [k − s?, k))

+ P (An ≥ k;An−l,n ∈ (k − s?, k − s?))+ P (An ≥ k;An−l,n ∈ (k − q?, k − s?])

+ P (An ≥ k;An−l,n ∈ [k − q?, k − q?])+ P (An ≥ k;An−l,n ∈ [k − l, k − q?)) .

(4.2.6)

In other words, we partition the event An ≥ k;An−l,n ∈ [k − l, k − s?] in (4.2.5)in three disjoint events in this case. In the proof, we determine the asymptoticbehavior of the various terms in the identity (4.2.6), which leads to the result givenin Table 4.4. Table 4.5 illustrates which terms contribute to the asymptotic tailbehavior of An.

Term Probability Asymptotic behaviorI P (An−l,n ≥ k) 2(1−β)θ√

2π

√−rk 1−r>0 growing+O(k−1) · 1r≤0 fixed

II P (An ≥ k;An−l,n ∈ [k − s?, k)) o(

log(l/(|r|+1))k

)

III P (An ≥ k;An−l,n ∈ (k − s?, k − s?)) θ2

4πlog kk

IV P (An ≥ k;An−l,n ∈ (k − q?, k − s?]) o(

log(k/(|r|+1)k + log(k/(|t|+1)

k

)

V P (An ≥ k;An−l,n ∈ [k − q?, k − q?]) θ2

4πlog(k/(|t|+1)

k

VI P (An ≥ k;An−l,n ∈ [k − l, k − q?))

θ√2π

√−tk 1−t>0 growing

+o(

log(l/(|t|+1))k

)

Table 4.4: Asymptotic behavior of terms in (4.2.6).

I II III IV V VI−r = ω((log k)2),

limn→∞ t/r > 0 θ√2π

√−rk negl. negl. negl. negl. θ√

2π

√−tk

−r = ω((log k)2),limn→∞ t/r ≤ 0 θ√

2π

√−rk negl. negl. negl. negl. negl.

−r ∝ (log k)2,limn→∞ t/r > 0 θ√

2π

√−rk negl. θ2

4πlog kk negl. θ2

4πlog kk

θ√2π

√−tk

−r ∝ (log k)2,|t| 6= k1−o(1),limn→∞ t/r ≤ 0 θ√

2π

√−rk negl. θ2

4πlog kk negl. θ2

4π

log( k|t|+1

)

k negl.−r ∝ (log k)2,t = k1−o(1) θ√

2π

√−rk negl. θ2

4πlog kk negl. negl. negl.

r otherwise,k|t|+1 6=

(k|r|+1

)o(1)

negl. negl. θ2

4π

log( k|r|+1

)

k negl. θ2

4π

log( k|t|+1

)

k negl.

otherwise negl. negl. θ2

4π

log( k|r|+1

)

k negl. negl. negl.

Table 4.5: Road map for proof of Theorem 4.2.2 with α = 1− β = 1/2.

4


The final case of Theorem 4.2.2 involves the case where the threshold is close to thenetwork size, i.e. α = 1. Both component sizes are therefore significantly smallerthan the threshold. In this case, we partition the event of exceeding the thresholdin only three disjoint events:

P (An ≥ k) = P (An ≥ k;An−l,n ∈ (k − s?, n− l])+ P (An ≥ k;An−l,n ∈ [k − s?, k − s?])

+ P (An ≥ k;An−l,n ∈ (k − l, k − s?]) . (4.2.7)

For appropriate choices of s? and s?, we show that the second term is dominantand yields the result in Theorem 4.2.2.

4.2.2 Disparate component sizes

Next, we turn to the case l = o(n). The smaller component is hence of a size thatis (almost) negligible compared to the larger component. Essentially, this resultsin a framework that for most thresholds (0 < α < 1), no matter what occurs in thesmaller component, the only likely manner to exceed the threshold is when it isexceeded in the larger component alone. This intuition remains true for α = 0: theinitial disturbance θ/n = θ/l · l/n is relatively minor in the smaller componentand unlikely to cause the cascading failure process to propagate further.

When α = 1 − β = 1, other scenarios to exceed k may become relevant. Inparticular, when k > n − l the number of line failures in the larger componentalone cannot exceed k. The partitioning of the event of exceeding threshold kneeds to be done carefully, resulting in many phase transitions.

Theorem 4.2.3. Suppose β = 0 and r = k−(n− l). If α < 1, or α = 1 with−r = Ω(l),then as n→∞,

P (An ≥ k) ∼ 2θ√2π

√1− k

n− l k−1/2. (4.2.8)

If k ≤ n− l, −r = o(l) is growing with l, then as n→∞,

P (An ≥ k) ∼

2θ√2π

√−rk , if l = o

(n√−r

logn

),

2θ√2π

√−rk + θ2l log l

π k−2, if l ∝ n√−r

logn ,θ2l log l

π k−2, if l = ω(n√−r

logn

).

(4.2.9)

If r ≤ 0 is fixed, then as n→∞,

P (An ≥ k) ∼

∑max−r,dθe−1m=0

θ(m−θ)mm! e−(m−θ)k−1, if l = o

(n

logn

),

θ2l log lπ k−2

+∑max−r,bθcm=0

θ(m−θ)mm! e−(m−θ)k−1, if l ∝ n

logn ,θ2l log l

π k−2, if l = ω(

nlogn

).

(4.2.10)

4


If k > n− l and r = o(l) is growing with l, then as n→∞,

P (An ≥ k) ∼ θ2l log(l/r)

πk−2. (4.2.11)

If k > n− l and γ := limn→∞ r/l ∈ (0, 1), then as n→∞,

P (An ≥ k) ∼ θ2

πc(γ)

√l

k2, (4.2.12)

where

c(γ) =

∫ 1

y=γ

√1− y

(y − γ)ydy.

Finally, if k > n− l and r = l − o(l), then as n→∞,

P (An ≥ k) ∼ θ2

2(n− k)k−2. (4.2.13)

The proof of Theorem 4.2.3 is analogous to the proofs of Theorems 4.2.1 and 4.2.2(excluding the case α = β = 1/2). Yet, we need to account for the disparity incomponent sizes, changing the points where the phase transitions occur.

Remark 4.2.1. We would like to stress that the main results in this chapter can beextended immediately to more general surplus capacity distributions. The reasonwhy is described in Chapter 2 in case of the star topology. That is, suppose thatthe surplus capacities are i.i.d. with a distribution function F (·) that is continuousand has a strictly positive density in zero. Write Cn(i), i = 1, ..., n, for the orderstatistics corresponding to the surplus capacities, and similarly, Un(i), i = 1, ..., n,

for the standard uniformly distributed order statistics. Write ln(·) for the loadsurge function. It holds that

P (An ≥ k) = P(Cn(i) ≤ ln(i), i = 1, ..., k

)

= P(F (Cn(i)) ≤ F (ln(i)), i = 1, ..., k

)

= P(Un(i) ≤ F (ln(i)), i = 1, ..., k

).

In other words, we require that

F (ln(i)) =θ + i− 1 + ∆(i)

n(4.2.14)

for the power-law behavior to prevail, where ∆(i), i ≥ 1, corresponds to theallowed perturbation functions as specified in Chapter 3. In particular, it shouldhold that ∆(i) = o(

√i). Using this observation, it is immediate to extend the

results in this chapter to more general surplus capacities: we require F (ll(i)) ≈(θ+ i−1)/l for the component with l edges, and F (ln−l(i)) ≈ (θ+ i−1)/(n− l) forthe other component. We refrained from doing so for the purpose of readability.

4


4.3 Proofs of main results

In order to prove Theorems 4.2.1-4.2.3, we use an asymptotic analysis for the sumof two independent heavy-tailed random variables. Specifically, we determine theasymptotic tail of An = Al,n +An−l,n, where Al,n and An−l,n are independent. Infact, the distribution of Al,n and An−l,n is well-understood due to the followingobservation. The first line failure disconnects the network in two separate starnetworks of sizes l and n − l. For the first component, this causes an initialdisturbance of θ/n = (θ/l) · (l/n), and every consecutive line failure causes anadditional load surge of 1/l. Hence, it falls in the framework of the model studiedin Chapters 2 and 3, where l is the network size and θ · l/n is the initial disturbanceconstant. It implies that for every growing k,

P (Al,n ≥ k)

= 0 if k > l,∼ χ(l) · l−1 if l − k ≥ 0 fixed,

∼ 2θ·l/n√2π

√l−kkl otherwise,

(4.3.1)

where the latter holds uniformly in accordance with Theorem 4.1.1 with

χ(l) =

maxl−k,dθ·l/ne−1∑

m=0

θ · l/n(m− θ · l/n)m

m!e−(m−θ ln ).

Similarly, (4.3.1) holds for An−l,n with l replaced by n− l. We see that it implies acertain order of magnitude, i.e.

P (Al,n ≥ k) =

0 if k > l,O( ln l

−1) if l − k ≥ 0 fixed,O(ln

√l−kl

)if l − k = o(l) growing,

O

(ln

√l−kkl

)if limn→∞ k/l ∈ (0, 1),

O(lnk−1/2

)if k = o(l),

and again, similarly for An−l,n with l replaced by n− l. This will be used extens-ively throughout the proofs.

To derive the main results stated in Section 4.2, we determine which scenarios aremost likely to cause An = Al,n +An−l,n to exceed the threshold.

4.3.1 Very few or many failures in one component

The strategy in all our proofs involves an appropriate partition of the eventof exceeding the threshold. In this section, we state results on the asymptoticbehavior of such joint events where there are very few or many line failures in onecomponent.

Smaller component

The proof of Theorem 4.2.1 partitions the event of exceeding threshold k in jointevents where the number of failures in the smaller component is in a certain

4

4.3. Proofs of main results 81

interval. The next two lemmas quantify the probability of An ≥ k;Al,n ≤ s?with s? very small, An ≥ k;Al,n ≥ s?with s? very large.

Lemma 4.3.1. Let s? = o(mink, l) be growing. Then, as n→∞,

P (An ≥ k;Al,n ≤ s?)∼ P (An−l,n ≥ k) if α < 1− β,= 0 if 1− β < α < 1.

Proof. Note that if 1− β < α < 1, we must be in the balanced case. Therefore, forlarge enough n, s? ≤ k − (n− l), which proves the second assertion.

Next, suppose α < 1 − β (this can be either the disparate or the balanced case).We then have to prove that the joint event that the threshold is exceeded andthe smaller component has few line failures is dominated by the event that k isexceeded in the larger component. Note that An ≥ k;Al,n ≤ s? implies that atleast An−l,n ≥ k − s?. Moreover, An−l,n ≥ k implies An ≥ k. Then,

(1− P(Al,n > s?))︸︷︷︸=1−o(1)

P (An−l,n ≥ k)︸︷︷︸∼ 2(1−β)θ√

2π

≤ P (An ≥ k;Al,n ≤ s?) ≤ P (An−l,n ≥ k − s?)︸︷︷︸∼ 2(1−β)θ√

2π

.

Therefore,

P (An ≥ k;Al,n ≤ s?) ∼2(1− β)θ√

2π

√n− l − kk(n− l) ∼ P (An−l,n ≥ k) .

Lemma 4.3.2. Let s? be growing defined by s? = k − o(k) if k < l and s? = l − o(l)otherwise. Then, as n→∞,

P (An ≥ k;Al,n ≥ s?)

∼ P (Al,n ≥ k) if α < β,= o(k−1/2) if β ≤ α < 1− β,= o(k−1) if 1− β < α < 1.

Proof. First, suppose α < β. Then we must be in the balanced case, and k < l ands? = k − o(k) for n large enough. Basically, we want to show in this case that it ismost likely that Al,n already exceeds k given that it exceeds s?. Note

P (Al,n ≥ k) ≤ P (An ≥ k;Al,n ≥ s?) ≤ P (Al,n ≥ s?) .

Equation (4.3.1) yields P (Al,n ≥ s?) ∼ P (Al,n ≥ k) as n → ∞ in case that l isbalanced and α < β. This coincides with the lower bound, and hence

P (An ≥ k;Al,n ≥ s?) ∼ P (Al,n ≥ k) .

4


If β ≤ α < 1− β, we can have both the disparate and the balanced case. When thecomponent sizes are disparate,

P (An ≥ k;Al,n ≥ s?) ≤ P (Al,n ≥ s?) = O

(l

n

√l − s?l

s?−1/2

)= o(k−1/2).

When the component sizes are balanced, note that the condition β ≤ α < 1 − βimplies that (l − s?)/l = o(1), and hence

P (An ≥ k;Al,n ≥ s?) ≤ P (Al,n ≥ s?) = O

(√l − s?l

s?−1/2

)= o(k−1/2).

Finally, if 1 − β < α < 1, we have a balanced case and k > l for n large enough.Then,

P (An ≥ k;Al,n ≥ s?) ≤ P (Al,n ≥ s?)︸︷︷︸=o(k−1/2)

P (An−l,n ≥ k − l)︸︷︷︸=O(k−1/2)

= o(k−1).

Larger component: balanced case

The proof of Theorem 4.2.2 partitions the event of exceeding threshold k in jointevents where the number of failures in the larger component is in a certain interval.The next lemma shows the asymptotic behavior where almost all lines in the largercomponent have failed. Henceforth, recall that r = k − (n− l).

Lemma 4.3.3. Suppose α = 1− β with β 6= 0. If r < 0 and −r is growing with n or |r|fixed, let s? = o(l) be growing. Then,

P (An ≥ k,An−l,n ∈ [k − s?,mink − 1, n− l]) = O

(s?n− l

). (4.3.2)

If α = 1 − β and r > 0 is growing, let s? = r + o(l) be growing such that s? − r isgrowing. Then,

P (An ≥ k,An−l,n ≥ k − s?) = o

(√s? − rn− l

). (4.3.3)

Finally, if α = 1 and β 6= 0, let s? = k − (n− l) + o(n− k) = r + o(n− k) such thats? − r is growing. Then,

P (An ≥ k,An−l,n ≥ k − s?) = o

(n− kk2

). (4.3.4)

Proof. For the first claim, note that s? − r > 0 is growing, and

P (An ≥ k,An−l,n ∈ [k − s?,mink − 1, n− l])

≤ s? supi∈[0,s?−r]

P (An−l,n = n− l − i) = O

(s?n− l

).

4


Next, in the second case,

P (An ≥ k,An−l,n ≥ k − s?)

≤ P (Al,n ≥ r)︸︷︷︸=O(r−1/2)

P (An−l,n ≥ k − s?)︸︷︷︸=O(

√s?−r/(n−l))

= o

(√s? − rn− l

).

For the final case, observe that r = k − (n− l) = l − (n− k) = l − o(l), and hence

P (An ≥ k,An−l,n ≥ k − s?)

≤ P (Al,n ≥ r)︸︷︷︸=O(√n−k/l)

P (An−l,n ≥ k − s?)︸︷︷︸=o(√n−k/(n−l))

= o

(n− kk2

).

The second lemma in this section yields the asymptotic behavior where at least asignificant number of lines in the smaller component have failed for α = 1− β orα = 1.

Lemma 4.3.4. If β ∈ (0, 1/2) and α = 1− β, suppose s? = o(l) such that s? − r > 0is growing. Then,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) = O(s?−1/2k−1/2

). (4.3.5)

If α = β = 1− β = 1/2 and l − k is not growing, set s? = l − o(l) such that l − s? isgrowing. Then,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) = O

(√l − s?k

). (4.3.6)

If α = β = 1−β = 1/2 and l−k > 0 is growing, set s? = k− o(l−k) such that k− s?is growing. Then as n→∞,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) ∼ θ√2π

√l − kk

. (4.3.7)

Finally, if α = 1 and β ∈ (0, 1/2], set s? = l − o(n− k) such that l − s? = ω(1). Then,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) = o

(n− kk2

). (4.3.8)

Proof. For (4.3.5), note that k − l is of order n and hence,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) ≤ P (Al,n ≥ s?)P (An−l,n ≥ k − l)= O

(s?−1/2k−1/2

).

4


For (4.3.6), observe l ∼ k as n→∞ and hence,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) ≤ P (Al,n ≥ s?) = O

(√l − s?k

).

For (4.3.7), we thus want to show that it is most likely that if An−l,n is at mostk − s? = o(l − k), the threshold is exceeded in the smaller component itself. Asn→∞,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) ≤ P (Al,n ≥ s?)

∼ 2θ · 1/2√2π

√l − s?ls?

∼ θ√2π

√l − kk

.

For the lower bound,

P (An ≥ k,An−l,n ∈ [k − l, k − s?]) ≥ P (Al,n ≥ k)P (An−l,n ≤ k − s?)

∼ θ√2π

√l − kk

.

Since the lower and upper bounds coincide, we observe that (4.3.7) holds. Finally,if α = 1 and β ∈ (0, 1/2], then

P (An ≥ k,An−l,n ∈ [k − l, k − s?])≤ P (Al,n ≥ s?)P (An−l,n ≥ n− l − (n− k))

= O

(√l − s?l

√n− kn− l

)= o

(n− kk2

).

Larger component: disparate case

In this section, we consider only the case α = 1−β = 1 with β = 0, where r = o(l).

Lemma 4.3.5. Suppose |r| = o(l). If −r > 0 is growing, let s? = o(l) be growing withl such that it satisfies s? = o(log l). Otherwise, let s? = r + v where v is growing andsatisfying v = o(log(l/(|r|+ 1))). Then,

P (An ≥ k;An−l,n ∈ [k − s?, k − 1]) = o

(1r≤0

l log l

k2+ 1r>0

l log(l/r)

k2

).

Proof. When r < 0 is growing, we have (for n large enough)

P (An ≥ k;An−l,n ∈ [k − s?, k − 1]) =

s?∑

j=1

P (Al,n ≥ j)P (An−l,n = k − j)

≤ P (Al,n ≥ 1) s? supj∈[1,s?]

P (An−l,n = k − j) .

4


Due to our choice of s?, we obtain the inequality,

P (An ≥ k;An−l,n ∈ [k − s?, k − 1]) = O

(l

ks?

1√−rk

)= o

(l log l

k2

).

When r is fixed,

P (An ≥ k;An−l,n ∈ [k − s?, k − 1])

≤ P (Al,n ≥ 1) s? supj∈[1,s?]

P (An−l,n = k − j) = O

(l

ks?

1

k

)= o

(l log l

k2

).

When r > 0 is growing,

P (An ≥ k;An−l,n ∈ [k − s?, k − 1])

≤ P (Al,n ≥ r)P (An−l,n ≥ n− l − v) = O

(l

kr−1/2

√v

k

)= o

(l log(l/r)

k2

).

Lemma 4.3.6. Suppose |r| = o(l), and let s? = o(l) be growing with l such that itsatisfies s? = ω(l/(log l)2) when r ≤ 0, and s? = ω(l/(log(l/r))2) when r > 0. Then,

P (An ≥ k;An−l,n ∈ [k − l, k − s?])

= o

(1r≤0

(l log l

k2

)+ 1r>0

l log(l/r)

k2

).

Proof. Observe that

P (An ≥ k;An−l,n ∈ [k − l, k − s?]) ≤ P (Al,n ≥ s?) · P (An−l,n ≥ k − l)

= O

(l

ks?−1/2

√l

k

)· o(1r≤0

(l log l

k2

)+ 1r>0

l log(l/r)

k2

).

4.3.2 Asymptotic behavior of some summation terms

In our analysis determining the asymptotic behavior often boils down to derivingthe asymptotics of some summation terms. In this section we provide two ofsuch results that are used. Henceforth, for sequences (an)n∈N and (bn)n∈N, writean . bn if limn→∞ an/bn ≤ 1, and similarly, write an & bn if limn→∞ an/bn ≥ 1.

Lemma 4.3.7. Suppose |r| = o(l). Let s? be such that s? = o(log(l/(|r|+1))) is growingif −r > 0 is growing or |r| fixed, and s? = r + o(r) be such that s? − r is growing ifr > 0 is growing. Let s? = o(l) be growing such that s? = ω (l/ log(l/(|r|+ 1))). Then,as l→∞, s? . s? and

s?∑

j=s?

j−1/2(j − r)−1/2 ∼ log

(l

|r|+ 1

). (4.3.9)

4


Proof. First, we have that s? ≤ s? as l → ∞. That is, if −r > 0 is growing or |r|fixed,

s? . log

(l

|r|+ 1

)≤ log(l) .

l

log l≤ l

log(

l|r|+1

) . s?,

and if r > 0 is growing,

s? ∼ r = lr

l.

l

log(l/r). s?.

Next, observe that the expression in the summation is a decreasing function, andtherefore

s?∑

j=s?

j−1/2(j − r)−1/2 ≤∫ s?

j=s?−1

j−1/2(j − r)−1/2

= 2 log

( √s? +

√s? − r√

s? − 1 +√s? − r − 1

),

ands?∑

j=s?

j−1/2(j − r)−1/2 ≥∫ s?

j=s?

j−1/2(j − r)−1/2 = 2 log

(√s? +

√s? − r√

s? +√s? − r

).

It is apparent that the asymptotic behavior of the upper and lower bounds is thesame. It remains to derive this behavior in terms of l and r.

For an asymptotic upper bound, we observe that due to our choice of s?, it holdsthat√s? +

√s? − r .

√l and

√s? +

√s? − r &

√|r|+ 1. Therefore,

s?∑

j=s?

j−1/2(j − r)−1/2 . 2 log

(√l

|r|+ 1

)∼ log

(l

|r|

).

For an asymptotic lower bound, recall that |r| . l/ log(l/(|r| + 1)) and thus√s? +

√s? − r & 2

√l/ log(l/(|r|+ 1)). Since log log x = o(log x), we derive that

as l→∞,s?∑

j=s?

j−1/2(j − r)−1/2 & log

(4 · l/ log(l/(|r|+ 1))

max|r|, log(l/(|r|+ 1))

)∼ log

(l

|r|+ 1

).

Lemma 4.3.8. Suppose |r| = o(l). Let s? be such that s? = o(log(l/(|r| + 1))) isgrowing if −r > 0 is growing or |r| fixed, and if r > 0 is growing let s? = r + o(r)be such that s? − r = ω(r/ log(l/r)) is growing. Let s? = o(l) be growing such thats? = ω (l/ log(l/(|r|+ 1))). Then, there exists a s? satisfying the assumptions, as l→∞,s? . s? and

s?∑

j=s?

j−1/2(j − r)−3/2 ∼ log

(l

|r|+ 1

). (4.3.10)

4


Proof. It is not immediate that if r > 0 is growing, there exists a s? that satisfiesboth s? = r + o(r) and s? − r = ω(r/ log(l/r)). Yet, we observe that log(l/r)→∞as l → ∞ and hence r/ log(l/r) = o(r). Therefore, there exists a s? that satisfiesthe stated conditions.

The claim that s? . s? as l→∞ is already proven in Lemma 4.3.7.

Finally we have to show (4.3.10). Note that the expression in the summation is adecreasing function, and therefore

s?∑

j=s?

j1/2(j − r)−3/2 ≤∫ s?

j=s?−1

j1/2(j − r)−3/2

= 2

√s? − 1

s? − 1− r − 2

√s?

s? − r+ 2 log

( √s? +

√s? − r√

s? − 1 +√s? − 1− r

).

Similarly,

s?∑

j=s?

j−1/2(j − r)−1/2 ≥∫ s?

j=s?

j−1/2(j − r)−1/2

= 2

√s?

s? − r− 2

√s?

s? − r+ 2 log

(√s? +

√s? − r√

s? +√s? − r

).

It is apparent that the bounds asymptotically coincide, and it remains to express theasymptotics in terms of l and r. First, as we have seen in the proof of Lemma 4.3.7,r = O(s?), and hence

2

√s?

s? − r = O(1) = o

(log

(l

|r|+ 1

)).

Next, if r ≤ 0 or |r| fixed, then clearly,

2

√s?

s? − r= O(1) = o

(log

(l

|r|+ 1

)).

If r > 0, then

2

√s?

s? − r∼ 2

r

s? − r= o

(log

(l

r

)).

Finally, it follows from the proof of Lemma 4.3.7 that

2 log

(√s? +

√s? − r√

s? +√s? − r

)∼ log

(l

|r|+ 1

),

as l→∞. Adding the above expressions yields the result.


Next, we prove Theorem 4.2.1 using the approach outlined in Table 4.1.

4


Proof of Theorem 4.2.1. If α < β, set s? = o(k) and s? = k − ω(k/s?) such that bothare growing large with n. Then,

P (An ≥ k; s? < Al,n < s?) ≤ P (Al,n ≥ s?)︸︷︷︸=O(s

−1/2? )

P (An−l,n ≥ k − s?)︸︷︷︸=o((k/s?)−1/2)

= o(k−1/2).

Applying Lemmas 4.3.1 and 4.3.2, together with (4.3.1), yields

P (An ≥ k) ∼ P (An−l,n ≥ k) + P (Al,n ≥ k)

∼ 2βθ√2π

√1− k

lk−1/2 +

2(1− β)θ√2π

√1− k

n− l k−1/2

as n→∞.

If β ≤ α < 1− β, set s? = o(mink, l) and s? = mink, l− o(mink, l) such thatboth are growing with n. Lemmas 4.3.1 and 4.3.2 imply that as n→∞,

P (Al,n ≥ k) + P (An−l,n ≥ k) ∼ P (An−l,n ≥ k)

∼ 2(1− β)θ√2π

√1− k

n− l k−1/2.

The result follows in this case as well since

P (An ≥ k; s? < Al,n < s?) ≤ P (Al,n ≥ s?)︸︷︷︸=o(1)

P (An−l,n ≥ k − l)︸︷︷︸=O(k−1/2)

= o(k−1/2).

Finally, we consider the case 1 − β < α < 1. For every s? = o(l), it holds thatP(An ≥ k;Al,n < s?) = 0. Lemma 4.3.2 implies for s? = l − o(l),

P(An ≥ k;Al,n ≥ s?) = o(k−1).

In addition, we have for q? = n− l − o(n− l)P (An ≥ k;An−l,n ≥ q?)≤ P (Al,n ≥ k − (n− l))︸︷︷︸

=O(k−1/2)

P (An−l,n ≥ q?)︸︷︷︸=O(q?−1/2))

= o(k−1).

Therefore, it remains to be shown that for these choices of q? and s?,

P(An ≥ k;Al,n < s?;An−l,n < q?) ∼ α√β(1− β)θ2

πc(α, β)k−1 (4.3.11)

as n→∞, where c(α, β) is defined as in the theorem.

Remark 4.3.1. We note that c(α, β) is a positive finite constant. That is, the functionwithin the integral is non-negative and has a positive mass over the interval weintegrate, and hence it is positive. Moreover, since x/(1 − x) is a non-negativeincreasing function for all x ∈ [0, 1) and s(·) is a linearly increasing function,∫ 1

α−(1−β)β

(1− x)−1/2

x−3/2

√s(x)

1− s(x)dx ≤ 2

√1− α

α− (1− β)

√s(1)

1− s(1)<∞.

Indeed, the integral expression is a positive finite constant.

4


Theorem 4.1.1 yields as n→∞,

P(An ≥ k;Al,n < s?;An−l,n < q?) ∼ β(1− β)θ2

2π

·s?−1∑

j=k−q?+1

j−(k−(n−l))∑

m=(n−l)−q?+1

√l

(l − j)j3

√n− l

m(n− l −m)3.

An upper bound for the summation term is given by

s?−1∑

j=k−q?+1

j−(k−(n−l))∑

m=(n−l)−q?+1

√l

(l − j)j3

√n− l

m(n− l −m)3

≤∫ s?

j=k−q?

∫ j−(k−(n−l))

m=(n−l)−q?

√l · (n− l)

(l − j)j3m(n− l −m)3dmdj

≤ 1√l(n− l)

∫ 1

x=k−(n−l)

l

∫ lx−(k−(n−l))n−l

y=0

(1− x)−1/2

x−3/2 · y−1/2(1− y)−3/2 dy dx

=2√

l(n− l)

∫ 1

x=k−(n−l)

l

(1− x)−1/2

x−3/2

·√

(lx− (k − (n− l)))/(n− l)1− (lx− (k − (n− l)))/(n− l) dx.

Similarly, a lower bound is given by

s?−1∑

j=k−q?+1

j−(k−(n−l))∑

m=(n−l)−q?+1

√l

(l − j)j3

√n− l

m(n− l −m)3

≥ 1√l(n− l)

∫ s?−2l

x= k−q?+2l

∫ lx−(k−(n−l))n−l

y=n−l−q?+2n−l

(1− x)−1/2

x−3/2

· y−1/2(1− y)−3/2 dy dx.

Due to our choices of q? and s?, then as n → ∞, the two integral expressionsconverge to the same constant. That is,

P(An ≥ k;Al,n < s?;An−l,n < q?) ∼ β(1− β)θ2

2π

· 2√l(n− l)

∫ 1

x=α−(1−β)

β

x−3/2

√1− x

√s(x)

1− s(x)dx

as n→∞, which asymptotically coincides with (4.3.11).


Next, we prove Theorem 4.2.2 using the approach outlined in Table 4.2.

4


Proof of Theorem 4.2.2. First consider the case that β ∈ (0, 1/2). Using (4.2.5), weobserve that it suffices to show that the asymptotic behavior provided in Table 4.2holds. That is, the result is immediate from Table 4.3, which in turn only highlightsthe dominant terms of Table 4.2.

Let s? = o(log(l/(|r| + 1))) be growing if −r > 0 is growing or |r| fixed, ands? = r + o(minr, log(l/r)) such that s? − r is growing if r > 0 is growing. Lets? = o(l) be growing such that s? = ω (l/ log(l/(|r|+ 1))). Note that due to thischoice, s? > s? for all n large enough. Then Lemmas 4.3.3 and 4.3.4 yield

s?∑

j=1

P (Al,n ≥ j)P (An−l,n = k − j) = o

log(

l|r|+1

)

k

,

l∑

j=s?

P (Al,n ≥ j)P (An−l,n = k − j) = o

log(

l|r|+1

)

k

.

Moreover, uniformly as n→∞,

s?∑

j=s?

P (Al,n ≥ j)P (An−l,n = k − j) ∼ β(1− β)θ2

π

s?∑

j=s?

j−1/2 (j − r)−1/2

n− l .

By assumption, k ∼ n− l and log l ∼ log k. Invoking Lemma 4.3.7 hence yields

s?∑

j=s?

P (Al,n ≥ j)P (An−l,n = k − j) ∼ β(1− β)θ2

π

log(k/(|r|+ 1)

k

as n→∞. Using (4.3.1), we obtain as n→∞,

P (An ≥ k) ∼ P (An−l,n ≥ k) +β(1− β)θ2

π

log(

k|r|+1

)

k,

where the asymptotic behavior of P (An−l,n ≥ k) is given by equation (4.3.1). Theresult follows by observing that phase transitions occur when −r ∝ log2 k. Inwords, the threshold is most likely exceeded in the larger component alone, orboth components have a significant number of line failures. The latter turnsdominant as soon as the difference between the threshold and larger componentsize becomes small enough.

Next, we prove the second case of the theorem with β = 1/2. That is, the twocomponent sizes are approximately the same, making the analysis more delicate.Effectively, we follow the same strategy as before, but make some modifications asthe smaller component is approximately of the same size as the bigger component.Equation (4.3.1) provides the asymptotic behavior of P (An−l,n ≥ k). Again, let s?and s? be as above. Using the analysis above shows that

s?∑

j=max1,rP (Al,n ≥ j)P (An−l,n = k − j)

∼ β(1− β)θ2

π

log(k/(|r|+ 1)

k=θ2

4π

log(k/(|r|+ 1)

k

4


remains valid in this case, covering the asymptotic behavior of terms II and III.

Let q? = l − o(l) satisfy l − q? = ω (l/ log(l/(|t|+ 1))) and let it be growing. Letq? = k− o(l− k) = l− (|t|+ o(|t|)) be growing such that k− q? = ω(|t|/ log(l/|t|))is growing if −t = l − k > 0 is growing, and q? = l − o(l) such that l − q? =o(log(l/(|t|+ 1))) is growing otherwise. We observe that for this choice of q?, termIV yields

q?∑

j=s?

P (Al,n ≥ j)P (An−l,n = k − j) ≤ P (Al,n ≥ s?)P (An−l,n ≥ k − q?)

= O

(1√

s?(k − q?)

)= o

(√log(l/(|r|+ 1)) log(l/(|t|+ 1))

k

)

= o

(log(k/(|r|+ 1) + log(k/(|t|+ 1)

k

).

It follows from Theorem 4.1.1 that uniformly as n→∞,

q?∑

j=q?

P (Al,n ≥ j)P (An−l,n = k − j) ∼ θ2

4π

q?∑

j=q?

(l − j)1/2

l(k − j)−3/2.

Applying Lemma 4.3.8 results into q? < q? for all n large enough, and

q?∑

j=q?

P (Al,n ≥ j)P (An−l,n = k − j) ∼ θ2

4π

log(k/(|t|+ 1)

k.

That is, it describes the asymptotic behavior of all events where almost all lineshave failed in the smaller component while the number of failures in the largercomponent is substantial, yet relatively small. Finally, Lemma 4.3.4 implies asn→∞,

l∑

j=q?

P (Al,n ≥ j)P (An−l,n = k − j) ∼ θ√2π

√−tk

if −t > 0 is growing, and

l∑

j=q?

P (Al,n ≥ j)P (An−l,n = k − j) = o

log(

l|t|+1)

)

k

otherwise. In other words, the event that the threshold is exceeded in the smallercomponent alone contributes to the dominant behavior only if k is significantlysmaller than l. Combining the above results then concludes the result for β = 1/2.

Finally, we have to show the result for α = 1. The threshold is close to n itself, andhence both components can only have a few surviving lines after the cascadingfailure process. Recall (4.2.7) and the results of Lemmas 4.3.3 and 4.3.4. Observe

4


that r > 0 is of order n, and hence for any s? = r + o(n− k) and s? = l − o(n− k)satisfying the conditions in Lemmas4.3.4 and 4.3.3 yield

s?∑

j=r

P (Al,n ≥ j)P (An−l,n = k − j) = o(k−1),

l∑

j=s?

P (Al,n ≥ j)P (An−l,n = k − j) = o

(n− kk2

).

To finalize the proof, we hence have to show that for suitable s? and s? satisfyingthe conditions above,

s?∑

j=s?

P (Al,n ≥ j)P (An−l,n = k − j) ∼ θ2

2(n− k) k−2 (4.3.12)

as n→∞. Fix ε > 0, then for large enough n,

s?∑

j=s?


=

s?−r∑

j=s?−rP (Al,n ≥ j + r)P (An−l,n = n− l − j)

≤ (1 + ε)2

∫ n−k

x=0

2βθ√2π

√l − r − x

l

(1− β)θ√2π

1√x(n− l) dx

= (1 + ε)2 β(1− β)θ2

πl(n− l)

∫ n−k

x=0

(n− k − x)1/2x−1/2 dx

= (1 + ε)2 β(1− β)θ2

2

n− kl(n− l) .

For the lower bound, note that we can set s?− r = l− s?, which is done to simplifythe integration term in the lower bound. That is,

s?∑

j=s?


≥ (1− ε)2 β(1− β)θ2

πl(n− l)

∫ s?−r−1

x=s?−r+1

(n− k − x)1/2x−1/2 dx

≥ (1− ε)2 β(1− β)θ2(n− k)

πl(n− l) arctan

(n− k − 2(s? − r)

2√

(n− k − (s? − r))(s? − r)

).

We note that

limn→∞

arctan

(n− k − 2(s? − r)

2√

(n− k − (s? − r))(s? − r)

)=π

2.

Letting ε ↓ 0 shows that the bounds coincide and hence (4.3.12) holds.

4



Lastly, we prove Theorem 4.2.3.

Proof of Theorem 4.2.3. In order to see that (4.2.8) holds for α < 1, we chooses? = o(mink, l and s? = mink, l − o(mink, l). In addition, if k = O(l), lets? = mink, l − ω(mink, l/s?). Lemma 4.3.1 and (4.3.1) yield

P (An ≥ k;Al,n ≤ s?) ∼ P (An−l,n ≥ k) ∼ 2θ√2π

√1− k

n− l k−1/2,

and

P (An ≥ k;Al,n ∈ (s?, s?)) ≤ P (Al,n ≥ s?)︸︷︷︸

=O(l/ns−1/2? )

P (An−l,n ≥ k − s?)︸︷︷︸=O((k−s?)−1/2)

= o(k−1/2).

Moreover, if k < l,

P (An ≥ k;Al,n ≥ s?) ≤ P (Al,n ≥ s?) = O

(l

n

√l − s?s?l

)

= O

(√l(l − s?)n

k−1/2

)= o(k−1/2),

and if k ≥ l,

P (An ≥ k;Al,n ≥ s?) ≤ P (Al,n ≥ s?) = O

(l

n

√l − s?s?l

)

= O

(√l − s?n

)= o(k−1/2).

Due to (4.2.5), we can therefore conclude that (4.2.8) holds when α < 1.

Next, suppose α = 1. Equation (4.3.1) then translates to

P (An−l,n ≥ k)

∼ 2θ√

2π

√−rk if − r > 0 growing,

∼ χ(r)k−1 if r fixed,= 0 if r > 0,

where

χ(r) =

max−r,dθe−1∑

m=0

θ(m− θ)mm!

e−(m−θ).

If −r = Ω(l), we have the bound

P (An ≥ k;An−l,n < k) ≤ P (Al,n ≥ 1)P (An−l,n ≥ k − l) .

4


Since Al,n obeys a quasi-binomial distribution [40],

P (Al,n ≥ 1) = 1−(

1− θ

n

)l= o(1),

and the second term is bounded by

P (An−l,n ≥ k − l) = O

(√n− kn− l

)= O

(√−rk

).

Again, due to identity (4.2.5), we observe that (4.2.8) holds in this case as well.

Next, suppose α = 1 with |r| = o(l). Choose s? small enough and s? large enoughsuch that the conditions in Lemmas 4.3.7 and 4.3.5 are satisfied. Then, uniformly,

s?−1∑

j=s?+1


∼s?−1∑

j=s?+1

θ2

π

l

k2j−1/2(j − r)−1/2 ∼ θ2

π

l

k2log

(l

|r|+ 1

).

Recalling (4.2.5), (4.3.1), and applying Lemmas 4.3.5 and 4.3.6 then yields that asn→∞,

P (An ≥ k) ∼ P (An−l,n ≥ k) +θ2

π

l log(l/(|r|+ 1))

k2. (4.3.13)

It follows immediately that (4.2.11) holds if r > 0. Moreover, if r ≤ 0 and fixed,the exceedance of the threshold in the larger component alone already yieldsa term of order k−1. Since log(l/(|r| + 1)) < log(n) and k ∼ n, it is necessarythat l = Ω(n/ log n) for the second term in (4.3.13) to be non-negligible. It is alsosufficient since l = n/ log n yields

l log(l/(|r|+ 1))

k2∼ n log(n/ log n)

k2 log n∼ n log n

k n log n=

1

k.

That is, (4.2.10) holds as well. When −r > 0 is growing, the same reasoning asbefore shows that it is necessary that l = Ω(n

√−r/ log n) for the second termin (4.3.13) to be non-negligible. Note this implies

√−r = O (l/n log n) = o(log n).This condition is also sufficient: when l = n

√−r/ log n,

l log(l/(|r|+ 1))

k2=n√−r log(n/(

√−r log n))

k2 log n∼√−r log n

k log n=

√−rk

.

In conclusion, also the case (4.2.9) holds.

Next, suppose α = 1 with γ = limn→∞ r/l ∈ (0, 1). Choose s? = r + o(l) such thats? − r is growing, and s? = l − o(l) such that l − s? is growing. Then,

P (An ≥ k;An−l,n ≥ k − s?) ≤ P (Al,n ≥ r)P (An−l,n ≥ k − s?)

= O

(l

nr−1/2 (k − s?)1/2

n− l

)= o

(l

k2

),

4


and

P (An ≥ k;An−l,n ≤ k − s?) ≤ P (Al,n ≥ s?)P (An−l,n ≥ k − l)

= O

(l

n

(l − s?)1/2

l

√n− kn− l

)= o

(l

k2

).

Using (4.3.1), we obtain that as n→∞,

P (An ≥ k;An−l,n ∈ (k − s?, k − s?))

=

s?−1∑

j=s?+1

P (An−l,n = k − j)P (Al,n ≥ j)

∼s?−1∑

j=s?+1

θ√2π

1

(n− l)√n− l − k + j· 2θ√

2π

l

n

√l − jl · j

∼ θ2

π

√l

k2

s?−1∑

j=s?+1

√l − j

(j − r)j .

Note that the function within the summation is (strictly) decreasing on (r, l]. Hence,an upper bound for the summation term is given by

s?−1∑

j=s?+1

√l − j

(j − r)j ≤∫ l

x=s?

√l − j

(j − r)j dx =√l

∫ 1

y=s?/l

√1− y

(y − r/l)y dy

∼√l

∫ 1

y=γ

√1− y

(y − γ)ydy,

and a lower bound is given by

s?−1∑

j=s?+1

√l − j

(j − r)j ≥∫ s?

x=s?+1

√l − j

(j − r)j dx =

∫ s?/l

y=(s?+1)/l

√l(1− y)

(y − r/l)y dy

∼√l

∫ 1

y=γ

√1− y

(y − γ)ydy.

As the asymptotic behavior of the upper and lower bound coincides, we obtain

P (An ≥ k;An−l,n ∈ (k − s?, k − s?)) ∼θ2

π

∫ 1

y=γ

√1− y

(y − γ)ydy

√l

k2.

We observe that∫ 1

y=γ

√(1− y)/((y − γ)y) dy is a constant, since

∫ 1

y=γ

√1− y

(y − γ)ydy ≤

∫ 1

y=γ

√1

(y − γ)γdy <∞.

Recalling (4.2.7) yields the result in this case.

4


Finally, we consider α = 1 with r = l − o(l), and hence both components canonly have a few surviving lines after the cascading failure process. The proofis analogous to the case where β 6= 0, and is merely adapted below to accountfor the disparity between the component sizes. Choose s? = r + o(n − k) ands? = l − o(n− k). Then,

P (An ≥ k;An−l,n ≥ k − s?) ≤ P (Al,n ≥ r)P (An−l,n ≥ k − s?)

= O

(l

n

√l − rl

√k − s?n− l

)= o

(n− kk2

),

and

P (An ≥ k;An−l,n ≤ k − s?) ≤ P (Al,n ≥ s?)P (An−l,n ≥ k − l)

= O

(l

n

√l − s?l

√n− kn− l

)= o

(n− kk2

).

Fix ε > 0. Using (4.3.1) yields

s?−1∑

j=s?+1


=

s?−1−r∑

j=s?+1−rP (Al,n ≥ j + r)P (An−l,n = n− l − j)

≤ (1 + ε)2

∫ n−k

x=0

l

n

2θ√2π

√l − r − x

l

θ√2π

1√x(n− l) dx

= (1 + ε)2 θ2

πn(n− l)

∫ n−k

x=0

(n− k − x)1/2x−1/2 dx = (1 + ε)2 θ2

2

n− kn(n− l) .

For the lower bound, note that we can set s? − r = l − s? without violating theassumptions on s? and s?. This is done to simplify the integration term in thelower bound, i.e.,

s?−1∑

j=s?+1


≥ (1− ε)2 θ2

πn(n− l)

∫ s?−r−2

x=s?−r+2

(n− k − x)1/2x−1/2 dx

≥ (1− ε)2 θ2

πn(n− l) (n− k) arctan

(n− k − 2(s? − r)

2√

(n− k − (s? − r))(s? − r)

).

We note that

limn→∞

arctan

(n− k − 2(s? − r)

2√

(n− k − (s? − r))(s? − r)

)=π

2.

4

4.4. Simulation results 97

Letting ε ↓ 0 shows that the bounds coincide and hence

s?−1∑

j=s?+1

P (Al,n ≥ j)P (An−l,n = k − j) ∼ θ2

2(n− k) k−2.

Combining the results shows that the theorem also holds in this final case.

4.4 Simulation results

In this section, we use our asymptotic results to devise an approximation schemefor the exceedance probability, and compare this with simulation results. Welook at all thresholds k ∈ [k?, k

?], where k? = dlog ne and k? = n − blog nc. Oursimulation results are obtained by averaging over 5n2 samples.

4.4.1 Our approximation scheme

Note that the balanced and disparate regimes only have strict meaning in anasymptotic sense. For finite values of n and l, it is not immediate which asymptoticregime provides the most natural fit. To illustrate this point, suppose n = 1000and l = 100, then both l = 0.1n and l = n2/3 hold. In our approximation schemewe treat l < r := 0.05n as a disparate case, and adopt the balanced case otherwise.

Write

(a) :=2θl/n√

2π

√l − kl

k−1/2,

(b) :=2(1− l/n)θ√

2π

√n− l − kn− l k−1/2,

(c) :=l/n(1− l/n)θ2

π

log k

k,

(d) :=l/n(1− l/n)θ2

π

log(k/(k − n+ l))

k,

(e) :=θ2

4π

log(k/(k − n+ l))

k,

(f) :=θ2

4π

log(k/(k − l))k

,

(g) :=θ2

4π

log k

k.

We provide the approximation An,l,θ,k for the exceedance probability as follows.If l < 0.05n, then

An,l,θ,k =

(b) if k < n− l − r,∑max−r,dθe−1m=0

θ(m−θ)mm! e−(m−θ)k−1 if k ∈ [n− l − r, n− l],

θ2

π c((k − n+ l)/l)√l

k2 , if k > n− l.

4


where c(·) is as in Theorem 4.2.3. If 0.05n ≤ l ≤ 0.45n, then

An,l,θ,k =

(a) + (b) if k < l,max (b), (c) if k ∈ [l, n− l],(d) if k ∈ (n− l, n− l + r],k/n√l/n(1−l/n)θ2

π c(k/n, l/n)k−1, if k ∈ (n− l + r, n− r],θ2

2 (n− k)k−2 if k > n− r,

where c(·, ·) is as in Theorem 4.2.1. On the other hand, if 0.45n < l ≤ 0.5n,

An,l,θ,k =

max (a) + (b), 2(g) , if k ≤ l,max(b), (f) + (g) if k ∈ (l, n− l],(e) + (f) if k ∈ (n− l, n− l + r],k/n√l/n(1−l/n)θ2

π c(k/n, l/n)k−1, if k ∈ (n− l + r, n− r],θ2

2 (n− k)k−2 if k > n− r.

We briefly motivate a few salient features of our approximation scheme. Since wethink of n to be in the range of several hundreds to many thousands, we take alll ≥ r as balanced cases and all l < r as disparate cases.

For l disparate, and k > n − l − r, a very large l is needed for an asymptoticphase transition between the cases to occur. For example, for l > n/ log n, whilel ≤ 0.05n, one needs n > e20 = 4.8 · 108. Therefore, we choose only those formulasin Theorem 4.2.3 with relatively widest range.

For l balanced, we consider all β > 1/2 − r = 0.45 as if β = 1/2. The reasonbehind the maximum operator is that (a) and (b) are decreasing to zero in k, andhence eventually move below (g). It provides a small improvement in accuracyfor thresholds k ≤ n− l that are close to the component size.

4.4.2 Balanced component sizes

To illustrate how close the approximation are to the actual exceedance probability,we compare our approximation to the simulation result. We consider the casewhere n = 1000, l = 300 and θ = 1, see Figure 4.4. From our approximationscheme, we observe that for thresholds k? ≤ k ≤ n− l, the exceedance probabilityis of order (close to) k−1/2, and it is of order close to k−1 for k ∈ [n − l, n − r].Therefore, we scale the exceedance probability accordingly to obtain a betterview of how close the approximation is to its true value. From Figures 4.4(a)and 4.4(b), it appears that our approximation is quite close, not only with respectto power-law exponent, but also the prefactor is quite close.

We have different approximations for 0.05n ≤ l ≤ 0.45n and if l > 0.45n. Thedifference between the two approximations manifests itself at thresholds close ton − l. Figure 4.5 demonstrates that incorporating this distinction improves theaccuracy, which is most apparent for the case l = 0.5n.

The key take-away message from the above observations is that the power-lawexponent is well captured by our approximation for most scenarios. The power-law exponent of −1/2 for threshold values sufficiently smaller than n− l is due to

4

4.4. Simulation results 99

200 400 600 800 10000

0.2

0.4

0.6

0.8

k

k1/2P(A

n≥k)

SimulationApproximation

(a) Complete interval.

700 750 800 850 900 950 10000

0.5

1

1.5

k

kP(A

n≥k)


(b) Large thresholds.

Figure 4.4: Results for n = 1000, l = 300 and θ = 1.

the most likely scenario that the blackout size exceeds k in one of the componentsalone. The power-law exponent exhibits a sudden change from −1/2 to −1 forthreshold values close to the bigger component size. This change reflects that forlarger threshold values a substantial number of line failures needs to occur in bothcomponent. In conclusion, our qualitative insights into the most likely occurrenceof large blackouts are pertinent to large finite network sizes.

460 480 500 520 5400

0.005

0.01

0.015

k

P(A

n≥k)


Approximation β 6= 1/2

Figure 4.5: Results for n = 1000, l = 500 and θ = 1.

4.4.3 Disparate component sizes

Since l is already relatively small, most threshold values will be sufficiently smallerthan n− l. For such thresholds, the exceedance probability reflects the power-lawexponent −1/2, which is also implied by our approximation. We point out thatthe exponent becomes more difficult to capture as the thresholds becomes closerto n. In particular, our approximation makes a transition when k = n− l − r andwhen k = n− l. As is apparent from Figure 4.6, this point can even cause a sudden

4


increase in the approximation. Nevertheless, the approximation remains relativelyclose to its true value.

880 900 920 940 960 980 10000

0.002

0.004

0.006

0.008

k

P(A

n≥k)


Figure 4.6: Results for n = 1000, l = d√ne = 32 and θ = 1.

4.5 Discussion

The results of Theorems 4.2.1-4.2.3 identify how the power-law exponent and itsprefactor are affected when a single immediate split occurs. A highly relevantand interesting problem is to move to other network structures that also accountfor more general properties. In this section, we discuss a few possible generaliz-ations/extensions, and offer some ideas how to deal with the analysis in thesesettings.

4.5.1 Load surge function

In general, we are interested in the question why and how the number of failuresexhibits power-law behavior in cascading failure models. In this chapter, weconsider the impact of a single immediate disconnection that leads to no furtheredge disconnections (i.e. every possible consecutive edge failure only causes anisolated node to be disconnected). The reason that power-law behavior appearsin this setting is due to the way the load surge behaves. More specifically, thesurplus capacities are uniformly distributed and the corresponding order statisticsU1

(1), ..., U1(l) and U2

(1), ..., U2(n−l) describe their values from smallest to largest in the

first and second component, respectively. We note that in expectation, E(U1(i)) =

i/l and E(U2(i)) = i/(n − l). That is, the expected spacings after an edge failure

between two order statistics (i.e. 1/l and 1/(n − l)) are exactly equal to theadditional load surges in both components. This causes the evolution of the failureto exhibit some form of criticality, leading to the heavy-tailed behavior of thenumber of failed edges.

In Chapter 2 and 3, the robustness of the power-law behavior is studied for thesingle star topology. This setting allows small perturbations on the total loadsurge function. More specifically, recall (4.2.14). The approximation operator can

4

4.5. Discussion 101

be specified as

ln(i) =θ + i− 1 + ∆(i)

n.

The results in Chapters 2 and 3 suggest that if ∆(i) = o(√i) as i→∞, then for all

k := kn such that k/n ∈ [0, 1)

P(An ≥ k) ∼ c(θ,∆)k−1/2

as n→∞, where c(θ,∆) ∈ (0,∞) is a constant that depends on the values θ andthe perturbations ∆(·). In a similar way, we can extend the results in this chapterto allow for perturbations of this size.

We point out that if the load surges move beyond this critical window, the behaviorof the number of failed edges becomes significantly different. If the load surgesare much smaller, the distribution of the failure size becomes light-tailed: theprobability distribution of the number of failed edges decays exponentially in thetail. On the other hand, if the load surge is much larger, then for both componentsthere is a strictly positive (non-vanishing) probability for the entire component tofail (for both components). In fact, given that the number of failed edges is ω(1)in one component, then with high probability all edges in that component havefailed.

4.5.2 General network topologies

Our focus in this chapter is on the impact of a single immediate disconnectionthat leads to no further edge disconnections. For the power-law behavior to appearin more general topologies, we need to understand how the load surges need tobehave such that this criticality property prevails when more disconnections canoccur as the cascade continues. This topic will be discussed in more detail in thenext chapter.

500 1000 1500 20000

0.2

0.4

0.6

0.8

k

k1/2P(A

n≥k)

Simulation l = n− l = 1000Simulation l = 500 n− l = 1500

2θ/√2π√

(n− k)/n

Figure 4.7: Scaled tail probability for network as in Figure 4.1 withrandom first failure over 100 000 runs.

A question that we are briefly considering in this section is what happens forthe network in Figure 4.1 if the initial disturbance does not necessarily cause

4


the bridging edge to fail immediately, but is due to the failure of one edge (orseveral) chosen uniformly at random. The failure of the bridging edge is a randomvariable itself in this setting, and (only) likely to occur at the σ’th step withlimn→∞ σ/n ∈ (0, 1). Using the results for the star topology and the insightsobtained in this chapter, one can show that for every k such that k = ω(1) andα = limn→∞ k/n ∈ [0, 1),

P (An ≥ k) ∼ c(θ, α, β)k−1/2 (4.5.1)

as n → ∞, where c(θ, α, β) ∈ (0,∞) is some constant depending on the initialdisturbance θ, α ∈ [0, 1) and β = limn→∞ l/n ∈ [0, 1/2]. In particular, if β = 0and/or α = 0, then c(θ, α, β) = 2θ/

√2π. Indeed, as is illustrated in Figure 4.7,

simulation experiments seem to confirm the asymptotic behavior as in (4.5.1).Summarizing, the scale-free behavior prevails if the splitting does not occurimmediately.

4.5.3 Heterogeneous edge capacities

Another interesting question is what happens if the capacity distribution is notthe same at every edge. One possible way to include heterogeneity between edgesis introducing edge classes. That is, suppose there are I classes of edges. Anedge in class i has a surplus capacity that is uniformly distributed between [0, ai]with a1 ≤ ... ≤ aK . For power-law behavior to appear in the tail distribution, werequire that some form of criticality is induced by the load surge function. Tounderstand what kinds of load surge functions result in power-law behavior forthe tail, one needs to define the correlation between the edge classes, i.e. the effectof edge failure to edges of other classes. We consider the two extreme scenarios inthis section: one with no correlation and one with full correlation. For simplicity,we consider these scenarios only for the star topology where class i has li edgesand l1 + ...+ lI = n.

Firstly, suppose that an edge failure of class i does not yield an additional loadsurge to other edges of different classes. As we have observed in the previoussection with general network topologies, the load surges need to be approximatelyequal to the expected spacings between two uniformly distributed order statistics.More specifically, suppose there is an initial disturbance of θi/n at every edge ofclass i = 1, ..., I . Every edge failure of class i causes an additional load surge ofai/li at every surviving edge of the same class, and none at edges of other classes.In this case, the tail of the number of edge failures obeys a power-law distributionwith an exponent (and prefactor) that depends on a1, ..., aI and threshold k. Infact, we note that the special case where I = 2 and θ1 = θ2 = θ reduces to themain setting of this chapter. A similar proof strategy can be used to rigorouslyderive the asymptotic behavior of this slightly more general setting.

Secondly, suppose that an edge failure of class i yields an additional load surgeto all other edges of different classes. We suppose that this occurs in a fair way,i.e. the load surge is proportional to the surplus capacity distributions. Morespecifically, suppose there is an initial disturbance of θai/n at every edge in class iand every class i = 1, ..., I . Every edge failure (regardless of the class) causes anadditional load surge of ai/n to every surviving edge of class i. Write U l(i) for the

4

4.5. Discussion 103

i’th uniformly distributed order statistic on [0, 1] with 1 ≤ i ≤ l. We observe that

P (An ≥ k) = P

⋃

j1+...+jI=j

I⋂

i=1

U li(ji) ≤θ + (ji − 1)

n, j = 1, ..., k

= P(Un(j) ≤

θ + (j − 1)

n, j = 1, ..., k

).

In other words, this case reduces to the setting with a single class of edges, andhence

P (An ≥ k) ∼ 2θ√2π

√n− kkn

.

To bridge the two extreme scenarios discussed in this section in a general frame-work is an extremely interesting problem. A key element is how to define thedependence between edge classes. The challenging part of the problem is todetermine the way the load surge function needs to behave such that power-lawbehavior prevails.

5

Chapter 5

Cascading failureson complex networks

Based on:Non-local cascading failures in a connected configuration model

L. Federico and F. Sloothaak

In this chapter, we introduce a cascading mechanism on a complex networkwhose nodal distribution does not exhibit scale-free behavior. The cascade isinitialized by a small additional loading of the network, and failures occur onedges whenever its load capacity is exceeded. In Chapter 4, we provided a criticalloading mechanism that leads to scale-free behavior for the failure size after asingle immediate disconnection. We generalize this notion in this chapter byintroducing a critical load function for a general network topology. An intrinsicfeature of this setup is that the propagation of failures occurs non-locally anddepends on the global network structure which continuously changes as thefailure process advances.

5.1 Introduction

In this section, we introduce the model that we consider in this chapter, we providean overview of the notation, and we state our main result.

5.1.1 Model description

In the previous chapters, we focused on (variants of) star topologies, where eachedge failure causes a single vertex/node to be detached. Consequently, it wasunnecessary to distinguish between vertex and edge sizes. Since we considermore general network topologies in this chapter, we slightly adapt our notation.Let G = (V,E) denote a graph, where V denotes the vertex set with |V | = n, andE denotes the edge set with |E| = m. Typically, we consider graphs that can bescaled in the number of vertices/edges.

105

5

106 Chapter 5. Cascading failures on complex networks

Suppose that each edge in the network is subjected to a load demand that isinitially exceeded by its edge capacity. The difference between the initial loadand the edge capacity is called the surplus capacity of the edge, and we assumethe surplus capacities to be independently and uniformly distributed on [0, 1]at the various edges. The failure process is triggered by an initial disturbance:all edges are additionally loaded with θ/m for some constant θ > 0. If the totalload increase surpasses the surplus capacity of one or more edges, these edgesfail and in turn, cause an additional load increase on all surviving edges thatare in the same component upon failure. We call these load increments the loadsurges. We make two assumptions: edge failures occur subsequently with thesmallest surplus capacity that is exceeded by its load (if any), and all edges inthe same component upon failure experience the same load surge. We continuewith the failure of edges that have insufficient surplus capacity to handle the loadsurges till there are no more. We are interested in the number of edge failures, alsoreferred to as the failure size.

We are interested in the setting where the failure size exhibits scale-free beha-vior. To this purpose, we define the load surge function lmj (i) at edge j afterexperiencing i load surges to be

lmj (1) = θ/m,

lmj (i+ 1) = lmj (i) +1−lmj (i)

|Emj (i−1)| ,(5.1.1)

where |Emj (i)| is the number of edges in the component that contains edge j afterperceiving i edge failures in that component. We observe that as long as twoedges remain in the same component during the cascade, this recursive relationimplies that the load surges are the same at both edges. Moreover, as long as alledges remain to be in a single component, it holds that |Emj (i)| = m− i at everysurviving edge, and hence (5.1.1) is solved by

lmj (i) =θ

m+ (i− 1) · 1− θ/m

m≈ θ + i− 1

m. (5.1.2)

In other words, applying (5.1.1) to a star topology with n + 1 nodes and m = nedges reduces to the setting of Chapter 2.

We point out that the load surges defined in (5.1.1) are typically non-deterministic,and are only well-defined as long as edge j has not yet failed throughout thecascade. Edge failures may cause the network to disintegrate in multiple compon-ents, which affects |Emj (i)| in a probabilistic way that depends on the structureof the graph. In contrast to processes in epidemiology or percolation models,the propagation of failures thus occurs non-locally and depends on the globalstructure of the graph throughout the cascade.

To provide an intuitive understanding why (5.1.1) gives rise to scale-free behaviorfor the failure size, recall the discussion in Section 4.5.1. That is, for a scale-freefailure size to appear, the cascade propagation should occur in some form ofcriticality. This happens when the load surges are close to the expected values ofthe ordered surplus capacities. More specifically, since the surplus capacities areindependently and uniformly distributed on [0, 1], the mean of the i-th smallest

5

5.1. Introduction 107

surplus capacity is i/(m+ 1). If no disconnections occurred during the cascade,this would imply that the additional load surges at every edge should be closeto 1/(m + 1) after every edge failure. Yet, whenever the network disintegratesin different components, in order for the cascade to remain in the window ofcriticality, the consecutive load surges need to be of a size such that the loadsurge is close the smallest expected surplus capacity of the remaining edges. Inparticular, suppose that the first disconnection occurs after k edge failures andsplits the graph in two components of edge size l and m − k − l, respectively.Due to the properties of uniformly distributed random variables, we note thatthe expectation of the smallest surplus capacity in the first component equalsk/(m+ 1) + (1− k/(m+ 1))/l. In other words, the additional load surges shouldbe close to (1 − k/(m + 1))/l after every edge failure until the cascade stops inthat component, or another disconnection occurs. In particular, this reduces to thesetting of Chapter 4 if k = 0. This process can be iterated, and gives rise to loadsurge function (5.1.1).

In this chapter, our main focus is to apply failure mechanism (5.1.1) to the (connec-ted) configuration model CMn(d) on n vertices with a prescribed degree sequenced = (d1, ..., dn). The configuration model is constructed by assigning dv half-edgesto each vertex v ∈ [n] := 1, ..., n, after which the half-edges are paired randomly:first we pick two half-edges at random and create an edge out of them, then wepick two half-edges at random from the set of remaining half-edges and pair theminto an edge, and we continue this process until all half-edges have been paired.For consistency, we therefore assume that the total degree

∑ni=1 di is even. The

construction can give rise to self-loops and multiple edges between vertices, butthese events are relatively rare when n is large [58, 62, 63]. We point out that thenumber of edges is thus a function of n and d, i.e. m := mn(d) :=

∑ni=1 di/2,

where we suppress the dependency on n and d for the sake of exposition.

Define ni as the number of vertices of degree i, and let Dn denote the degree of avertex chosen uniformly at random from [n]. We assume the following conditionon the degree sequence.

Condition 5.1.1 (Regularity conditions). Unless stated otherwise, we assume thatCMn(d) satisfies the following conditions:

• There exists a limiting degree variable D such that Dn converges in distribution toD as n→∞;

• n0, n1 = 0;

• p2 := limn→∞ n2/n ∈ (0, 1);

• nj = 0 for all j ≥ n1/4−ε for some ε > 0;

• d := limn→∞ E[Dn] <∞.

Under these conditions, we can write pi := limn→∞ ni/n as the limiting fraction ofdegree i vertices in the network. Moreover, under these conditions it is known thatthere is a positive probability for CMn(d) to be connected [47, 76]. Our startingpoint will be such a configuration model conditioned to be connected, denoted by

5


CMn(d), on which we apply the edge failure mechanism (5.1.1). We are interestedin quantifying the resilience of the connected configuration model under (5.1.1),measured by the number of edge failures An,d.

5.1.2 Notation

For convenience, we introduce some additional notation in this chapter. Unlessstated otherwise, all limits are taken as n→∞, or equivalently by Condition 5.1.1,asm→∞. A sequence of events (An)n∈N happens with high probability (w.h.p.) ifP(An)→ 1. For sequences of random variables (Xn)n∈N and (Yn)n∈N, we denoteXn = oP(Yn) if Xn/Yn

P→ 0. For real-valued non-negative sequences (an)n∈N,(bn)n∈N, we write an bn if an = o(bn). We will introduce an extensive numberof symbols in this chapter, and without going into detail, we provide an overviewin Tables 5.1-5.4.

5.1.3 Main result

We are interested in the probability that the failure size exceeds a threshold k. Inthis chapter, we mainly focus on thresholds satisfying 1 k m1−δ for someδ ∈ (0, 1), which we refer to as the sublinear case. Our main result shows that thefailure size has a power-law distribution.

Theorem 5.1.1. Suppose k := km such that 1 k m1−δ for some δ ∈ (0, 1). Then,

P (An,d ≥ k) ∼ 2θ√2πk−1/2. (5.1.3)

To see why (5.1.3) holds, we need to understand the typical behavior of the failureprocess as the cascade continues. A first result we show is that it is likely that thenumber of edge failures that need to occur for the network to become disconnectedis of order Θ(

√m). This suggests that as long as the threshold satisfies k = o(

√m),

the tail is the same as in the case of a star topology, i.e. given by (5.1.3).

As long as the cascade continues, we show that it typically disconnects smallcomponents. This suggests that up to a certain point, the cascading failure mech-anism creates a network with a single large component that contains almost allvertices and edges, referred to as the giant component, and some small disconnectedcomponents with few edges. It turns out that the total number of edges that areoutside the giant component is sufficiently small that the dominant contributionto the failure size comes from the number of edges that are contained in the giantcomponent upon failure. Moreover, due to small sizes of the components outsidethe giant component, the load surge function for the edges in the giant as pre-scribed by (5.1.1) is relatively close to (5.1.2). We show that as long as thresholdk m1−δ for some δ ∈ (0, 1), the perturbations are sufficiently small such thatthe failure size behavior in the giant component is similar to the setting of a startopology, and hence (5.1.3) prevails.

5

5.1. Introduction 109

Table 5.1: List of variables commonly used throughout this chapter.

n # verticesd (Fixed) degree sequence

m = mn(d) # edges in a configuration model with degree sequence d

CMn(d) Configuration model with degree sequence d

CMn(d) Configuration model with degree sequence d conditioned to beconnected

ni # vertices with degree ipi Fraction of vertices with degree iDn Degree of a vertex chosen uniformly at random from [n]

D Limiting degree variable of Dn

d Average degree of d

θ Disturbance constantlmj (i) Total load surge at edge j after experiencing i load surges in a

graph with m edges|Emj (i)| # edges in the component containing edge j after experiencing i

load surgesAn,d # edge failures after the cascading failure process

Table 5.2: List of variables commonly used in percolation/sequentialedge-removal process.

q Removal probability in the percolation processCMn(d, q) Percolated configuration model with removal probability q

C Component of a graph, sometimes also denoted as C(x) whenreferring to the component that contains vertex or edge x

Cmax Largest component of a graphTn,d # edges that are sequentially removed uniformly at random for

the first disconnection to occur in the connected configurationmodel

T Limiting variable of m1/2Tn,d

|Em(i)| # edges in the giant (largest) component after i edges have beenremoved uniformly at random

|Em(i)| # edges outside the giant (largest) component after i edges havebeen removed uniformly at random

5


Table 5.3: List of variables commonly used in the explosion algorithm.

Rn # removed half-edges in the first step of the explosion algorithmd′ Degree sequence of configuration graph in step three of the

explosion method (d′ ∈ Nn+Rn )CMN (d′) Configuration model in step three of the explosion method with

degree sequence d′

CMn(d, q) Resulting configuration model in step four of the explosionmethod, indistinguishable from the percolated configurationmodel with removal probability q

n′i # vertices of degree i in CMN (d′)

p′i limn→∞ n′i/n

nl,j # vertices of degree l in d that have degree j in d′

pl,j Probability for a vertex of degree l to retain j half-edges afterthe first step of the explosion algorithm

L′k(n) # components that are lines of length k ≥ 2 in CMN (d′)

C ′k(n) # components that are cycles of length k ≥ 1 in CMN (d′)

Table 5.4: List of variables commonly used in the failure process.

An,d # edges that were contained in the largest component upon failureduring the cascade

An,d # edges that were contained outside the largest component upon failureduring the cascade

A∗n+1 # edge failures in the cascading failure process on a star topology withn+ 1 nodes and m = n edges

κ(i) # edges that are contained in the largest component upon removalwhen i edges are sequentially removed uniformly at random

υ(i) Minimum # edges that need to be removed uniformly at random forthe sum of υ(i) and # edges outside the giant to exceed i

%(i) # edges that need to be removed uniformly at random such that i edgeswere contained in the giant component upon failure

Si∑ij=1

(1− Expj(1)

)

Li,m Scaled perturbed load surge, formally defined as in (5.4.15)Si,m Random walk defined as

∑ij=1 Lj,m − Expj,m(1)

Yi,m Random walk∑ij=1(1− Lj,m)

τm First-passage time of random walk Si,m to be less than 1− θTg First-passage time of random walk Si to move below a boundary

sequence (gi)i∈N

5

5.2. Proof strategy for Theorem 5.1.1 111

We will illustrate in Section 5.5 that the power-law tail prevails beyond the sublin-ear case up to a certain critical point. However, we heuristically explain why andhow the prefactor is affected in this case, i.e. the failure size tail is different fromthe star topology in this case.

We would like to remark that the disintegration of the network by the cascadingfailure mechanism is closely related to percolation, a process where each edge isfailed/removed with a corresponding removal probability. In fact, percolationresults will be crucial in our analysis to prove our main result. Before we lay out aroad map to prove Theorem 5.1.1, we provide an outline of the remainder of thischapter.

5.1.4 Outline

The proof of Theorem 5.1.1 requires many different steps, and therefore we providea road map of the proof in Section 5.2. We explain that we need to derive novelpercolation results for the sublinear case, which we show in Section 5.3. Werigorously identify the impact of the disintegration of the network on the cascadingfailure process in Section 5.4, which we use to prove our main result. Finally, westudy the failure size behavior beyond the sublinear case in Section 5.5 throughextensive simulation experiments.

5.2 Proof strategy for Theorem 5.1.1

Our proof of Theorem 5.1.1 requires several steps. In this section, we provide ahigh-level road map of the proof.

5.2.1 Relation of failure process and sequential removal process

There are two elements of randomness involved in the cascading failure process:the initial surplus capacities of the edges and the way the network disintegratesas edge failures occur (since CMn(d) is a non-deterministic graph). The secondaspect determines the values of the load surges, and only when the surpluscapacity of an edge is insufficient to handle the load surge, the cascading failureprocess continues. Recall that as long as edges remain in the same component,they experience the same load surge. Since the surplus capacities are i.i.d., itfollows that every edge in the same component is equally likely to be the nextedge to fail as long as the failure process continues in that component. This is acrucial observation, as this provides a relation with the sequential edge-removalprocess. That is, suppose that we sequentially remove edges uniformly at randomfrom a graph. Given that a new edge removal occurs in a certain component,each edge in that component is equally likely to be removed next, just as in thecascading failure process. Consequently, this observation gives rise to a couplingof the disintegration of the network through the cascading failure process to onethat is caused by sequentially removing edges uniformly at random.

More specifically, suppose that sequentially removing edges uniformly at randomyields the permutation e(1), ..., e(m). For the cascading failure process, sample muniformly distributed random variables and order them so that Um(1) ≤ ... ≤ Um(m)

5


and assign to edge e(j) surplus capacity Um(j). In particular, this implies that if thecascading failure process would always continue until all edges have failed, thenthe edges would fail in the same order as the sequential edge-removal process.Moreover, it is possible to exactly compute the load surge function from theorder e(1), ..., e(m) over the edge set. We illustrate this claim by the examplein Figure 5.1. In this example, we observe that m = 11, and we consider thesequential removal process up to five edge removals. We see that the first threeedge failures do not cause the graph to disconnect. It holds for every dottededge ej ∈ [m],

|Emj (i)| =

11− i if i ∈ 0, 1, 2, 34 if i = 4,3 if i = 5,

and for every dashed edge ej ∈ [m],

|Emj (i)| =

11− i if i ∈ 0, 1, 2, 33 if i = 4.

Recursion (5.1.1) yields the corresponding load surge values.

(5) (1)

(4)

(2)

(3)

Figure 5.1: Sample of sequentially removing five edges uniformlyat random from a connected graph. We remove the red edges sub-sequently in successive order, leaving two disconnected component.The first component is connected by the dotted lines, whereas thesecond component is connected by the dashed lines.

This example illustrates that using our coupling, the sequential removal processgives rise to the load surge values as prescribed in (5.1.1). We point out that dueto our coupling, if after step j − 1 an edge e(j) for some j ∈ [m] has sufficientsurplus capacity to deal with the load surge, then so do all the other edges in thatcomponent. In other words, the cascade stops in that component. To determinethe failure size of the cascade, one needs to subsequently compare the load surgevalues to the surplus capacities in all components until the surplus capacities atethe surviving edges are sufficient to deal with the load surges.

In summary, sequentially removing edges uniformly at random gives rise to theload surge values in every component. This idea decouples the two sources of ran-domness: first one needs to understand the disintegration of the network through

5


the sequential removal process, leading to the load surge values through (5.1.1).Then, we can determine the failure size by comparing the surplus capacities to theload surges up to the point where the cascade stop in every component.

5.2.2 Disintegration of the network through sequential removal

The next step is to determine the typical behavior of the sequential removal processon the connected configuration model CMn(d). A first question that needs to beanswered is how many edges need to fail, or equivalently, need to be removeduniformly at random from the network CMn(d) to become disconnected. It turnsout that this is likely to occur when Θ(

√m) edges are removed.

Theorem 5.2.1. Suppose that we subsequently remove edges uniformly at random fromCMn(d). Define Tn,d as the smallest number of edges that need to be removed for thenetwork to be disconnected. Then,

m−1/2Tn,dd→ T (D),

where T (D) has a Rayleigh distribution with density function

fT (D)(x) =4p2x

d− 2p2e−

2x2p2d−2p2 . (5.2.1)

After this point, more disconnections start to occur as more edges are removeduniformly at random from the graph. In Section 5.3.3, we focus on the typicalnetwork structure if

√m i m1−δ edges have been removed uniformly

at random for some δ ∈ (0, 1/2). Typically, there is a giant component thatcontains almost all edges and vertices. The components that detach from the giantcomponent are isolated nodes, line components, and possibly isolated cycles. Weshow that the total number of edges contained in these components are likelyto be of order Θ(i2/m), while the number of edges in more complex componentstructures are negligible in comparison. This leads to the following result, whichis proved in Section 5.3.6.

Theorem 5.2.2. Suppose we remove i := im edges uniformly at random from the con-nected configuration model CMn(d) with

√m i m1−δ for some δ > 0. Write

Em(i) for the set of edges in the largest component of this graph, and let |Em(i)| denoteits cardinality. Then,

(m− i− |Em(i)|)mi2

P→ 4p22

(d− 2p2)2. (5.2.2)

We stress that determining the typical behavior of the network disintegration isnot enough to prove our main result. In addition, for each of our results, we needto show that it is extremely unlikely to be far from its typical behavior, which weconsider in Section 5.3.4. Moreover, we combine these large deviations results toshow the following result in Section 5.3.6.

5


Theorem 5.2.3. Suppose i := im edges have been removed uniformly at random from theconnected configuration model CMn(d), where

√m i mα for some α ∈ (1/2, 1).

Then,

P(m− i− |Em(i)| > iα

)= O(m−3). (5.2.3)

Moreover, for every k := km for which k = o(mα) for some α ∈ (0, 1),

P(m− i− |Em(i)| > iα for some i ≤ k

)= o(m−1/2). (5.2.4)

5.2.3 Impact of disintegration on the failure process

The sequential removal process gives rise to the load surge function at every edge,and we need to compare these values to the surplus capacities in every componentuntil the cascade stops. To keep track of the cascading failure process in everycomponent may seem cumbersome at first glance. However, due to the way theconnected configuration model is likely to disintegrate, it turns out that it onlymatters what happens in the largest component, i.e. the component that containsthe largest number of edges (and vertices). Henceforth, we refer to this componentas the giant (component).

Intuitively, this can be understood as follows. By Theorem 5.2.1 and 5.2.2, remov-ing any sublinear number im of edges always leaves o(im) edges outside the giant.Therefore, if k := km m edges are sequentially removed, then only o(k) ofthese edges were contained outside the giant upon removal. Moreover, even if thecascading failure process struck every edge of the components outside the giant,the contribution to the failure size would be at most o(k). This is negligible withrespect to the k − o(k) failures that occur in the giant. The failure size An,d shouldtherefore be well-approximated by the failure size in the giant. We formalize theseideas in Section 5.4.

More specifically, write

An,d = # edges contained in the giant upon failure during the cascade,

An,d = # edges contained outside the giant upon failure during the cascade,

and hence An,d = An,d + An,d. Moreover, define

|Em(i)| =# remaining edges outside the giant when i edges are removed ,uniformly at random,

κ(i) =# edges removed from giant when i edges are removed uniformly atrandom.

The main idea is that since the number of edges outside the giant is likely tobe o(i) when a sublinear number of i edges are removed uniformly at random,the contribution of edge failures that occur outside the giant is asymptotically

5


negligible. That is, we bound the probability that An,d ≥ k to occur due tothe cascading failure process by the same event in two related processes. For theupper bound, we consider the scenario where all edges in the small componentsthat disconnect from the giant immediately fail upon disconnection. For the lowerbound, we consider the scenario where none of the edges in the small componentsthat disconnect from the giant fail.

To make this rigorous, we first consider the probability that the number of edgefailures in the giant exceeds κ(k). That is, if we sequentially remove k edgesuniformly at random, a set of κ(k) edges were contained in the giant upon failure.We are interested in the probability that the coupled surplus capacities of all theseedges are insufficient to deal with the corresponding load surges. By translatingthis setting to a first-passage problem of a random walk bridge over a movingboundary, much in the spirit of the setting in Chapter 3, we show the followingresult in Section 5.4.3.

Proposition 5.2.1. If k = o(mα) for some α ∈ (0, 1), then as n→∞,

P(An,d ≥ κ(k)

)∼ 2θ√

2πk−1/2.

Second, we use this result to derive an upper bound for the failure size tail.Trivially, the failure size is bounded by the number of edges that are contained inthe giant component upon failure plus all the edges that exist outside the giantafter the cascade has stopped. We introduce the stopping time

υ(k) = minj ∈ N : j + |Em(j)| ≥ k, (5.2.5)

the minimum number of edges that need to be removed uniformly at random forthe number of edges outside the giant to exceed k− υ(k). In other words, after wehave removed υ(k) edges uniformly at random, the sum of the number of edgesoutside the giant and the number of removed edges exceeds k. The number ofedge removals in the giant is given by κ(υ(k)), and hence

An,d ≥ k ⊆An,d ≥ κ(υ(k))

.

We prove that υ(k) = k − o(k) with sufficiently high probability, and hence

P (An,d ≥ k) ≤ P(An,d ≥ κ(υ(k))

)∼ 2θ√

2π(k − o(k))

−1/2 ∼ 2θ√2πk−1/2.

Third, we derive a lower bound. We note that An,d ≤ An,d, and hence

An,d ≥ k ⊇An,d ≥ k

=An,d ≥ κ(%(k))

,

where

%(k) = minj ∈ N : κ(j) ≥ k. (5.2.6)

5


That is, %(k) is the number of edges that need to be removed uniformly at randomfor k failures to have occurred in the giant. We show that that %(k) = k+ o(k) withsufficiently high probability, and hence

P (An,d ≥ k) ≥ P(An,d ≥ κ(%(k))

)∼ 2θ√

2π(k + o(k))

−1/2 ∼ 2θ√2πk−1/2.

Since the upper and lower bounds coincide, this yields Theorem 5.1.1. We formal-ize this in Section 5.4.4.

5.3 Disintegration of the network

As explained in the previous section, the cascading failure process can be de-coupled in two elements of randomness. In this section, we study the first elementof randomness: the disintegration of the network as we sequentially remove edgesuniformly at random. In view of Theorem 5.1.1, our main focus is the case wherewe remove only O(m1−δ) edges with δ ∈ (0, 1). In particular, our goal is to proveTheorems 5.2.1-5.2.3.

This section is structured as follows. First, we show in Section 5.3.1 that the sequen-tial removal process is well-approximated by a process where we remove eachedge independently with a certain probability, also known as percolation. This is awell-studied process in the literature, and particularly in case of the configurationmodel [60]. In Section 5.3.2, we provide an alternative way to construct a percol-ated configuration model by means of an algorithm as developed in [60]. Thisalternative construction allows for simpler analysis, and is used in Section 5.3.3 toshow that for the percolated (connected) configuration model, typically, the com-ponents outside the giant component are either isolated nodes, line componentsor possibly cycle components. In Section 5.3.4, we derive large deviations boundson the number of edges outside the giant. We prove Theorem 5.2.1 in Section 5.3.5,and in addition, we provide a large deviations bound for the number of edges thatneed to be removed for the connected configuration model to become disconnec-ted. We prove Theorems 5.2.2 and 5.2.3 in Section 5.3.6. Although these sectionsfocus on the (connected) configuration model under a sublinear number of edgeremovals, we briefly recall results known in the literature involving the typicalbehavior of the random graph beyond the sublinear window in Section 5.3.7. Thiswill be important to obtain intuition of what happens to the failure size behaviorbeyond the sublinear case, the topic of interest in Section 5.5.

5.3.1 Percolation on the connected configuration model

To prove our results regarding the sequential removal process, we relate thisprocess to another one where each edge is removed independently with a certainremoval probability, also known as percolation. This is illustrated by the followinglemma.

Lemma 5.3.1. Let G = (V,E) be a graph, and write E′(G(q)) as the set of edges thathave been removed by percolation with parameter q ∈ [0, 1], and E′(i) as the set of edges

5

5.3. Disintegration of the network 117

when i edges are removed uniformly at random. It holds for every i ∈ [m], q ∈ [0, 1], andB ⊆ E such that |B| = i,

P(E′(i) = B) = P(E′(G(q)) = B

∣∣ |E′(G(q))| = i). (5.3.1)

Moreover, if q = qm = ω(m−1), then as m→∞,

|E′(G(q))|qm

P→ 1. (5.3.2)

Proof. Due to the i.i.d. property of the surplus capacities, it holds that E′(i) hasthe distribution of i edges chosen uniformly without repetitions from [m], soP(E′(i) = B) =

(mi

)−1.

Since all edges in CMn(d, q) are removed independently with probability q, itholds for all sets B ⊆ E with |B| = i that

P(E′(G(q)) = B) = qi(1− q)m−i, (5.3.3)

while

P(|E′(G(q))| = i) =

(m

i

)qi(1− q)m−i. (5.3.4)

Since E′(G(q)) = B ⊆ |E′(G(q))| = i, we obtain

P(E′(G(q)) = B | |E′(G(q))| = i) =

(m

i

)−1

, (5.3.5)

so (5.3.1) holds. From (5.3.4) we obtain (5.3.2) by concentration of Bin(m, q) ifqm→∞.

This lemma establishes that sequentially removing i := im = ω(1) edges uniformlyat random is well-approximated by a percolation process with removal probabilityq = i/m. In particular, this holds for the (connected) configuration model. Thisallows us to study many questions involving the connected configuration modelsubject to uniformly removing edges in the setting of percolation. The studyof percolation processes on finite (deterministic or random) graphs is a well-established research field, which dates back to the work of Gilbert [51] and is stillvery active these days. In particular, percolation on the configuration model is afairly well-understood process, as established by Janson in [60].

It is known that there exists a critical parameter qc := 1 − E[D]/E[D(D − 1)]such that if q < qc, then the largest component of the percolated (connected)configuration model contains a non-vanishing proportion of the vertices andedges [60, 81]. In particular, it is implied by the formula for qc that in order fora phase transition to appear, it is necessary that E[D(D − 1)]/E[D] ∈ (1,∞). IfE[D(D − 1)]/E[D] ≤ 1, then the largest component already has a sublinear sizein n even before percolation, while if E[D(D − 1)]/E[D] =∞, then there exists aconnected component of linear size in the percolated graph for every q ∈ (0, 1).

5


Typically, the (w.h.p. unique) component of linear size is referred to as the giantcomponent. All other components are likely to be much smaller, i.e. of sizeOP(log n)under some regularity conditions on the limiting degree sequence [81]. If q ≥ qc,there is no giant component in the percolated graph and all the components havesublinear size [60].

In order to derive Theorem 5.1.1, these results in the literature are not sufficient.In particular, in view of Theorems 5.2.1 and 5.2.2, a more detailed description ofthe network structure is needed in case that q = o(m). A critical tool to derive theresults in this setting is the explosion algorithm, as we describe next.

5.3.2 Explosion algorithm

Key to prove Theorems 5.2.1 and 5.2.2 is the explosion algorithm, designed by Jansonin [60]. This algorithm is a crucial element to understand how the network disin-tegrates. The explosion algorithm prescribes that instead of applying percolationon CMn(d) with removal probability q, we can run the procedure as described inAlgorithm 2.

Algorithm 2: Explosion algorithm [60].Input: A set of n vertices, such that for every j ∈ [n] the vertex vj has dj

half-edges attached to it according to the degree sequence d.Output: Graph CMn(d, q).

1. Remove each half-edge independently with probability (1− (1− q)1/2).Let Rn be the number of removed half-edges.

2. Add Rn degree-one vertices to the vertex set. Define the new degreesequence as d′ with N = n+Rn vertices.

3. Pair CMN (d′).

4. Remove Rn vertices of degree 1 from CMN (d′) uniformly at random.

Janson proved in [60] that the algorithm produces a random graph that is statist-ically indistinguishable from CMn(d, q), the graph obtained by removing everyedge in (not necessarily connected) CMn(d) with probability q ∈ [0, 1]. Yet, thegraph obtained by the explosion method is significantly easier to study as it issimply a configuration model with a new degree sequence and a couple of verticesof degree 1 have been removed, where the latter operation does not significantlychange the structure of the graph. Since the graphs obtained from the percolationprocess and the explosion method are identically distributed, we use the denom-ination CMn(d, q) both for the configuration model after percolation and for thegraph we obtain via the explosion algorithm.

Remark 5.3.1. The observant reader may notice that the explosion algorithm is designedfor the configuration model that is not necessarily initially connected. Fortunately, asestablished in [47], under Condition 5.1.1 connectivity has a non-vanishing probability

5


to occur as n → ∞. This is a crucial observation, since it implies that certain eventsthat happen w.h.p. on CMn(d) also happen w.h.p. on CMn(d). More specifically, for asequence of events (An)n∈N,

P(An | CMn(d) is connected) ≤ P(An)

P(CMn(d) is connected), (5.3.6)

where under Condition 5.1.1 [47],

lim infn→∞

P(CMn(d) is connected) > 0.

In particular, this implies that if for some sequence of random variables (Xn)n∈N it holdsthat Xn

P→ c for some constant c ∈ R in CMn(d), then the same statement holds for thegraph CMn(d). Similarly, if Xn = oP(an) for some sequence (an)n∈N in CMn(d), thenthis is also true for CMn(d).

Next, we point out some properties we use extensively in this section. First, weobserve that if q = o(1), then by Taylor expansion, the removal probability inthe explosion algorithm satisfies 1− (1− q)1/2 = (q/2)(1 + o(1)). Therefore, theprobability of a vertex of degree l to retain j half-edges satisfies

pl,j =

(l

j

)((1− q)−1/2

)j (1− (1− q)1/2

)l−j=

(l

j

)(q/2)l−j(1 + o(1)) (5.3.7)

if q = o(1) and j ≤ l. Moreover, let nl,j represent the number of vertices ofdegree that retain j half-edges. That is, nl,j are random variables with distribution

nl,jd= Bin(nl, pl,j). Due to Markov’s inequality, it holds that

P(nl,j > 0) ≤ nlpl,j . (5.3.8)

Moreover,

nl,jd→ Poi(a) if nlpl,j → a ∈ (0,∞),

nl,j = nlpl,j(1 + oP(1)) if nlpl,j →∞,(5.3.9)

due to the Poisson limit theorem and the law of large numbers, respectively. Notethat

E

( ∞∑

l=h

nl,0

)=

∞∑

l=h

nlpl,0 ≤ n(q/2)h

1− q/2(1 + o(1)) = O(nqh), h ≥ 2. (5.3.10)

By Markov’s inequality, it holds for every ε > 0

P

( ∞∑

l=h

nl,0 > ε

)= O(nqh), h ≥ 2. (5.3.11)

Similarly, it follows that for all q = o(1),

E

( ∞∑

l=h

nl,1

)≤ n

∞∑

l=h

l(q/2)l−1(1 + o(1)) = O(nqh−1), h ≥ 2, (5.3.12)

5


and hence for every ε > 0,

P

( ∞∑

l=h

nl,1 > ε

)= O(nqh−1), h ≥ 2, (5.3.13)

Finally, we observe that for every 1/m q 1, by the law of large numbers,

Rn = 2m(1− (1− q)−1/2)(1 + oP(1)) =nd

2q(1 + oP(1)) (5.3.14)

5.3.3 Typical structure of the percolated configuration model

We recall that our focus is on the case where the number of edges that are removedfrom the giant is of order o(m). In view of Lemma 5.3.1, this number is well-approximated by the number of edge removals in a percolation process withremoval probability q = o(1). In particular, in this regime there is a unique giantcomponent w.h.p. and other components are likely to be much smaller, i.e. w.h.p.the number of vertices and edges ourside the giant is of order o(m). Even morecan be said about the structure of these components. In this section, we show inthat these components are typically isolated nodes, line components or potentiallyisolated circles. More complex structures are relatively rare.

Remark 5.3.2. Remark 5.3.1 suggests that often it suffices to consider CMn(d) to provean analogous result for CMn(d). Moreover, the explosion algorithm is used to constructCMn(d, q) from CMN (d′) by the removal of Rn degree-one vertices, and hence we oftenalso focus onCMN (d′) in this section. We point out that the operation of removing degree-one vertices does not affect the connectivity of a component. Moreover, the probabilitythat the giant component in CMN (d′) is not unique is exponentially small [81]. Ifq = o(1), then the probability for Rn not to be of sublinear size is also exponentiallysmall. Therefore, the giant component in CMN (d′) remains the giant component inCMn(d, q) with extremely high probability, i.e. the probability that the complementis true has an exponentially decaying rate. Therefore, the number of edges outside thegiant in CMN (d′) provides an upper bound (w.h.p.) for the number of edges outsidethe giant in CMn(d, q). Moreover, since line components outside the giant in CMN (d′)either remain line components or become isolated nodes after the removal of degree-onevertices, the number of edges in CMn(d, q) (outside the giant) that are not contained inline components is bounded from above w.h.p. by the same quantity in CMN (d′).

First, we explore the degree sequence d′ of CMN (d′) in the explosion methodfrom Algorithm 2. Analogously to the notation for the original degree sequence d,we write n′i for the number of vertices of degree i in d′ and p′i := limn→∞ n′i/n asthe limiting fraction.

Lemma 5.3.2. Consider the explosion algorithm from Algorithm 2 with initial graphCMn(d) satisfying Condition 5.1.1 and q = i/m with 1 i m. The degree sequenced′ after explosion satisfies the following properties. For the number of vertices of degree

5


zero,

P(n′0 6= 0) = O(i2/m), if i m1/2,

n′0d→ Poi

(c2p22d

), if i = c

√m,

n′0 = p22d

i2

m (1 + oP(1)), if√m i m.

(5.3.15)

The number of degree-one vertices in CMN (d′) satisfies

n′1 =

(i+

2p2

di

)(1 + oP(1)). (5.3.16)

Finally, the fraction of vertices of degree k ≥ 2 in CMN (d′) converges to the limitingfraction of degree-k vertices in CMn(d), i.e.

p′k = pk, for all k ≥ 2. (5.3.17)

Proof. Recall that if q = o(1), then

1− (1− q)1/2 = (q/2)(1 + o(1)) = (i/(2m))(1 + o(1)),

pl,j =

(l

j

)(q/2)l−j(1 + o(1)),

and nl,jd= Bin(nl, pl,j). Using (5.3.9), we obtain

n2,0 =p2i

2

2dm(1 + oP(1)) if m1/2 i m, n2,0

d→ Poi(c2p2

2d

)if i = c

√m.

By (5.3.10) we know that these are in both cases the only leading-order contribu-tions. From (5.3.11), we obtain that

P

( ∞∑

l=h

nl,0 6= 0

)= O(i2/m),

if i √m. Similarly, it follows from (5.3.9), (5.3.13) and (5.3.14),

n′1 = (Rn + n2,1)(1 + oP(1)) =

(i+

2p2

di

)(1 + oP(1)).

Finally, we note that, since every removal of a half-edge in the first step of theexplosion algorithm changes the degree of one vertex,

nl −Rn ≤ n′l ≤ nl +Rn,

and hence p′lP→ pl for all l ≥ 2.

We use the degree sequence d′ to study the structure of the components outsidethe giant in CMn(d, q). We will show that these components are either isolatednodes, line components or possibly isolated cycles. More complex structures arerather unlikely to appear as the network disintegrates.

5


We begin by proving a bound on the number of edges belonging to componentsthat have a more complex structure, i.e. contain a vertex of degree at least threewhen q m−δ for some δ > 0. We show that this is not the leading-order termfor the number of edges outside the giant, since the number of lines and isolatedvertices is much larger. For a graph G, define dmax(G) as the largest degree ofany vertex in G and E(G) as the edge set of G. For every edge e and vertex v,write C(e) and C(v) for the connected component that contains e or v, respectively.Finally, let Cmax denote the largest component in a graph G.

Proposition 5.3.1. Consider the graphs CMn(d, q), CMn(d, q) and CMN (d′) withq = i/m, where mδ i m1−δ for some δ > 0. For all three graphs, if i = o(

√m),

then

P(#e /∈ Cmax : dmax(C(e)) ≥ 3 6= 0) = o(i2/m). (5.3.18)

For both graphs, if i √m, then

#e /∈ Cmax : dmax(C(e)) ≥ 3 = oP(i2/m

). (5.3.19)

Proof. We prove the result using the explosion algorithm. First, recall Remark 5.3.1.Applying (5.3.6) to the events

#e /∈ Cmax : dmax(C(e)) ≥ 3 6= 0 ,mi2

#e /∈ Cmax : dmax(C(e)) ≥ 3 > ε, ε > 0,

implies that in order to prove (5.3.18) and (5.3.19) for CMn(d, q), it suffices toshow (5.3.18) and (5.3.19) hold for CMn(d, q). In addition, recall Remark 5.3.2,implying that the number of edges inCMn(d, q)\Cmax in components containing anode with degree at least three is bounded by the same quantity in CMN (d′)\Cmaxwith sufficiently high probability. In other words, it suffices to prove that (5.3.18)and (5.3.19) hold for CMN (d′).

Recall the degree distribution of CMN (d′) from Lemma 5.3.2. It follows fromthe proof of [81, Lemma 11] that for every supercritical degree sequence d′

(i.e. E[D′n(D′n − 1)]/E(D′n) > 1) and any γ ∈ (0,∞), there exists c = c(d′) < 1such that in CMN (d′), P(∃C 6= Cmax : |E(C)| > γ log n) ≤ n2cγ logn. Therefore, forγ large enough,

P(∃ C 6= Cmax : |E(C)| > γ log n) = o(n−1). (5.3.20)

Consequently, since the number of edges in the giant component is much largerthan γ log n, it suffices to prove the claims (5.3.18) and (5.3.19) for

# e ∈ CMN (d′) : |E(C(e))| ≤ γ log n, dmax(C(e)) ≥ 3 . (5.3.21)

For this purpose, we use the standard exploration algorithm of CMN (d′) used inthe literature (see e.g. [36, 47] for some similar formulations). At each time t ∈ N,we define the sets of half-edges At,Dt,Nt as the active, dead and neutral sets,and explore them in the following way:

5


1. At step t = 0, pick a vertex v ∈ [n] with dv ≥ 3 uniformly at random andset all its half-edges as active. All other half-edges are set as neutral, andD0 = ∅.

2. At each step t, pick a half-edge e1(t) in At uniformly at random, and pairit with another half-edge e2(t) chosen uniformly at random in At ∪Nt. Sete1(t), e2(t) as dead. If e2(t) ∈ Nt, then find the vertex v(e2(t)) incident toe2(t) and activate all its other half-edges.

3. Terminate the process when At = ∅.

We observe that whenAt = ∅, we have exhausted the exploration of the connectedcomponent C(v), and the number of steps performed by the exploration algorithmis the number of edges in C(v) In order to prove the claim, we thus need to provethat there is no t ≤ γ log n such that At = ∅ with sufficiently high probability.Let (Z

(v)t )t≥0 count the number of active half-edges starting from a vertex v with

dv ≥ 3. Note that at step t the process can only go down if i) e2(t) ∈ Nt andits incident vertex has degree one, causing Z

(v)t = Z

(v)t−1 − 1, or ii) e2(t) ∈ At,

causing Z(v)t = Z

(v)t−1 − 2. We denote these events by A(t) and B(t), respectively.

Since Z(v)0 ≥ 3, this counting process needs to decrease by at least three in total

for the exploration process to die out. Moreover, the values of the countingprocess is small at the time steps where the process decreases. More specifically,Aγ logn = ∅ ⊆ F1 ∪ F2 ∪ F3, where

F1 =⋃

s1,s2,s3≤γ logn

A(s1) ∩A(s2) ∩A(s3) ∩ Z(v)

s1 , Z(v)

s2 , Z(v)

s3 ≤ 3,

F2 =⋃

s1,s2≤γ logn

A(s1) ∩B(s2) ∩ Z(v)

s1 , Z(v)

s2 ≤ 3,

F3 =⋃

s1,s2≤γ logn

B(s1) ∩B(s2) ∩ Z(v)

s1 , Z(v)

s2 ≤ 4.

We can bound the probabilities that these events occur by

P(F1) ≤ (γ log n)3

(1 +

2p2

d

)3i3

m3(1 + o(1)) = O

(i3 log3m

m3

),

P(F2) ≤ (γ log n)2

(1 +

2p2

d

)i

m(1 + o(1))

3

m− 2γ log n= O

(i log2m

m2

),

P(F3) ≤ (γ log n)2

(4

m− 2γ log n

)2

= O

(log2m

m2

).

Consequently, using the union bound, we obtain that for every i that satisfiesmε i m1−ε for some ε > 0,

E [# e : |E(C(e))| ≤ γ log n, dmax(C(e)) ≥ 3]

≤ nγ log n (P(F1) + P(F2) + P(F3)) = o

(i2

m

).

(5.3.22)

5


By Markov’s inequality, it follows that

P (# e : |E(C(e))| ≤ γ log n, dmax(C(e)) ≥ 3 6= 0) = o(i2/m),

and

# e : |E(C(e))| ≤ γ log n, dmax(C(e)) ≥ 3 = oP(i2/m

).

The following proposition specifies the number of vertices and edges in linesand the number of isolated nodes which are disconnected from the giant by apercolation process, which constitutes the leading-order term for CMn(d, q)\Cmax.

Proposition 5.3.2. Consider CMn(d, q) with q = i/m with√m i m. Define

Lk(n) as the number of isolated lines of length k andN0(n) the number of isolated vertices.Then,

m

i2

(N0(n) +

∞∑

k=2

kLk(n))

P→ 2dp2

(d− 2p2)2, (5.3.23)

m

i2

( ∞∑

k=2

(k − 1)Lk(n))

P→ 4p22

(d− 2p2)2. (5.3.24)

Instead, if i = o(√m), then

P(N0(n) +

∞∑

k=2

kLk(n) 6= 0)

= O(i2/m). (5.3.25)

Moreover, (5.3.23)-(5.3.25) hold also for CMn(d, q).

Before moving to the proof of Proposition 5.3.2, we first consider the highermoments of L′k(n), k ≥ 1, the number of isolated lines of length k in CMN (d′). Inparticular, we need the first and second moments to prove Proposition 5.3.2.

Lemma 5.3.3. For any sequence r = r2, ..., rk with k ≥ 2 of positive integer values, itholds as n→∞,

E[L′2(n)r2 · · ·L′k(n)rk ](mi2

)r2+...+rk→

k∏

j=2

(1

4

(1 +

2p2

d

)2(2p2

d

)j−2)rj

. (5.3.26)

Proof. We use the explosion algorithm from Algorithm 2. To illustrate the type ofarguments we use to prove this statement, we first consider the first moment only.

Define Vj as the set of all vertices of degree j in d′. Recall the degree sequence d′

from Lemma 5.3.2, and we define

Lk = v1, v2, ..., vk : v1, vk ∈ V1; v2, ..., vk−1 ∈ V2,

5


the set of all collections of k vertices that could form a line. Note that

L′k(n) =∑

l∈Lk1l forms a line.

Due to Lemma 5.3.2, we observe that

E[L′k(n)] = E

[∑

l∈Lk1l forms a line

]

= E[(n′12

)(n′2k − 2

)2k − 4

2m− 1

2k − 6

2m− 3· · · 2

2m− 2k + 5

1

2m− 2k + 3

]

= E

[n′1

2n′2k−2

2(k − 2)!

(2k − 4)!!

(2m)k−1

](1 + o(1)) =

i2

m

1

4

(1 +

2p2

d

)2(2p2

d

)k−2

(1 + o(1)).

Next, we generalize these arguments to higher and mixed moments of the vari-ables (L′k(n))n≥2. We follow the same approach used in [47] where the conver-gence of the number of lines to a sequence of Poisson variables was provedin the critical window for connectivity. Using the method of moments, in thiscase, we need to prove concentration of the number of vertices and edges in linecomponents.

We prove (5.3.26) by induction. With a slight abuse of notation, we start theinduction step at k = 1, or alternatively, at k = 2 with r2 = 0. Then, both sides in(5.3.26) are equal to one, and hence the induction hypothesis is satisfied.

Next, we show how to advance the induction hypothesis. We define

Wk(r) =

k⋃

j=2

lj(1), ..., lj(rj) : lj(h) ∈ Lj for all 1 ≤ h ≤ rj , 2 ≤ j ≤ k

,

the collection of sets of∑kj=2 rj possible lines. Moreover, for a set wk(r) ∈Wk(r),

we define E(wk(r)) as the event that all elements in the set wk(r) form a linecomponent in CMN (d′). Then, using the tower property, we can rewrite

E[L′2(n)r2 · · ·L′k(n)rh ] = E

∑

wk(r)∈Wk(r)

1E(wk(r))

= E

∑

wk−1(r)

1E(wk−1(r))E

∑

lk(1),...,lk(rk)∈Lk1lk(1)1lk(2) · · ·1lk(rk) | E(wk−1(r))

,

where 1lk(h) denotes the indicator of the event that the set lk(h) forms a line. Next,we show that for every wk(r) ∈Wk(r),

(mi2

)rk ∑

lk(1),...,lk(rk)∈LkE[1lk(1)1lk(2) · · ·1lk(rk)|E(wk−1(r))]

=

(1

4

(1 +

2p2

d

)2(2p2

d

)k−2)rk

(1 + o(1)).

5


Note that by induction, this suffices to conclude (5.3.26).

First, note that if any of the lk(j), 1 ≤ j ≤ rk, contains any of the vertices usedin wk−1(r), then it is not possible for it to form a line of length k as these ver-tices are already contained in line components of a smaller length. In that case,E[1lk(1)1lk(2) · · ·1lk(rk) | E(wk−1(r))] = 0. Therefore, we can consider the part ofthe graph excluding the line components in wk−1(r). This part of the graph isalso a configuration model, but with a slightly altered degree sequence. Notethat we exclude finitely many line components of finite length, and hence onlyexclude finitely many vertices and edges. With a slight abuse of notation, writer′ = r′(lk(1), ..., lk(rk)) as the number of mutually distinct lines. Then the probab-ility that r′ mutually distinct lines can be formed in the graph excluding the linecomponents in wk−1(r) is given by

E[1lk(1)1lk(2) · · ·1lk(rk)|E(wk−1(r))] =(2k − 4)!!r

′

(2m)r′(k−1)(1 + o(1)).

We sum these contributions over the number of possible sets that exist in thesubgraph. Write C(rk, r

′) as the number of distinct sets of rk lines that contain thesame set of r′ mutually distinct lines of length k. Importantly, C(rk, rk) = 1 andC(rk, r

′) is a finite integer if 1 ≤ r′ ≤ rk − 1. Recalling Lemma 5.3.2, we obtain(mi2

)rk ∑

lk(1),...,lk(rk)∈LkE[1lk(1)1lk(2) · · ·1lk(rk)|E(wk−1(r))]

=(mi2

)rkE

rk∑

r′=1

C(rk, r′)

(n′1

2n′2k−2

2(k − 2)!

(2k − 4)!!

(2m)k−1

)r′ (1 + o(1))

=

(1

4

(1 +

2p2

d

)2(2p2

d

)k−2)rk

(1 + o(1)),

concluding the proof.

From Lemma 5.3.3, we can bound the number of edges in isolated line componentsin CMN (d′).

Corollary 5.3.1. For every j ≥ 1, as n→∞,

E[(mi2

∞∑

k=2

kL′k(n))j]→( (d− p2)(d+ 2p2)2

2d(d− 2p2)2

)j. (5.3.27)

Proof. Note that it follows from Lemma 5.3.3

E[L′k(n)] =i2

m

1

4

(1 +

2p2

d

)2(2p2

d

)k−2

(1 + o(1)), k ≥ 2.

Consequently, for every ε > 0 there exists a N > 0 such that for all n ≥ N ,

m

i2E[L′k(n)] ≤ 1

4

(1 +

2p2 + ε

d− ε)2(2p2 + ε

d− ε)k−2

, k ≥ 2.

5


In particular, for ε small enough 2p2+εd−ε < 1 so that the sequence converges to zero

exponentially fast in k. We apply dominated convergence to obtain

E[mi2

∞∑

k=2

kL′k(n)]→ (d− p2)(d+ 2p2)2

2d(d− 2p2)2, (5.3.28)

E[mi2

∞∑

k=2

(k − 1)L′k(n)]→ (d+ 2p2)2

4(d− 2p2)2. (5.3.29)

In other words, we have derived the expected number of vertices and edges inline components in CMN (d′). Next, we prove (5.3.27) for every j ≥ 2. We definefor a sequence r of positive integer values, |r|1 =

∑kh=1 rh, i.e. its `1 norm. We

write

E[(mi2

∞∑

k=2

kL′k(n))j]

=mj

i2j

∑

r:|r|1=j

∏

h≥2

E[(hL′h(n))rh ].

For every ε ≥ 0, it holds for all n sufficiently large that(mi2

)|r|1 ∏

h≥2

E[(L′h(n))rh ] ≤ 1

4|r|1

(1 +

2p2 + ε

d− ε)2|r|1(2p2 + ε

d− ε)∑

h≥2(h−2)rh.

(5.3.30)

If ε is small enough, then 2p2+εd−ε < 1, and thus the sequence is decreasing exponen-

tially fast in∑h≥2 hrh. Applying dominated convergence thus yields (5.3.27).

Next, we prove Proposition 5.3.2.

Proof of Proposition 5.3.2. Again, note that it follows from Corollary 5.3.1 that

E[ ∞∑

k=2

kL′k(n)]

= O

(i2

m

).

By Markov’s inequality and Lemma 5.3.2, if i = o(√m),

P

(n′0 +

∞∑

k=2

kL′k(n) 6= 0

)= O

(i2

m

).

Recall that in the final step of the explosion algorithm, we only remove vertices ofdegree 1. Therefore, the only way for the number of vertices in line componentsand isolated vertices to increase is when a component in CMN (d′) with a vertexthat has a degree at least three turns into a line or an isolated node. By Proposi-tion 5.3.1, such components appear in CMN (d′) with probability oP(i2/m), andwe conclude that (5.3.25) holds.

In order to prove (5.3.23) and (5.3.24), we also need second moments. By Corol-lary 5.3.1,

E[(mi2

∞∑

k=2

kL′k(n))2]

= E[(mi2

∞∑

k=2

kL′k(n))]2

(1 + o(1)),

5


and thus, by the second moment method

m

i2

∞∑

k=2

kL′k(n)P→ (d− p2)(d+ 2p2)2

2d(d− 2p2)2.

Since n′0 = p2i2

2dm by Lemma 5.3.2, we obtain that

m

i2

(n′0 +

∞∑

k=2

kL′k(n))

P→ p2

2d+

(d− p2)(d+ 2p2)2

2d(d− 2p2)2.

The same arguments can also be used to prove concentration of∑∞k=2(k−1)L′k(n),

since it is dominated by∑∞k=2 kL

′k(n). That is,

m

i2

∞∑

k=2

(k − 1)L′k(n)P→ (d+ 2p2)2

4(d− 2p2)2.

We computed the number of vertices and edges that are contained in isolated nodecomponents or in line components inCMN (d′). To obtain the corresponding valuefor CMn(d, q) we need to subtract the number of degree one vertices that are partof line components and are removed in the last step of the explosion algorithm,and add the number of vertices whose components turn into a line or an isolatedvertex by the removal of some degree 1 vertices. By Proposition 5.3.1, the numberof components that can turn into a line or isolated vertex is bounded by oP(i2/m),and hence the contribution of these type of events is negligible. Therefore, itsuffices to consider the number of vertices and edges that are removed in thefinal step of the explostion algorithm from the line components in CMN (d′). Weobserve that there are in total

∞∑

k=2

2L′k(n) =(d+ 2p)2

2d(d− 2p)

i2

m(1 + oP(1)), (5.3.31)

vertices of degree one in the line components out of the i(1 + 2p2/d)(1 + oP(1))that exist in d′. We define LR(n) as the number of vertices removed from linecomponents in the last step of the explosion algorithm. We remove i(1 + oP(1))edges of degree one uniformly at random, so the number of the degree-one verticesremoved from lines is given by an hypergeometric variable with

E[LR(n)] =(d+ 2p2)2

2d(d− 2p2)

d

d+ 2p2(1 + o(1)) =

d+ 2p2

2(d− 2p2)(1 + o(1)). (5.3.32)

A hypergeometric random variable with diverging mean concentrates around themean, and hence by the law of large numbers,

m

i2

(N0(n) +

∞∑

k=2

kLk(n))

P→ p2

2d+

(d− p2)(d+ 2p2)2

2d(d− 2p2)2− d+ 2p2

2(d− 2p2)=

2dp2

(d− 2p2)2,

as claimed.

5


We do the same computation for the number of edges, this time accounting forthe fact that if both vertices of a line of length 2 are removed, only one edgeis removed, while all the other vertex and edge removals are in bijection. Thenumber of lines of length 2 which are removed is given by

L′2(n)

(1 +

2p2

d

)−2

=i2

m

1

4

(1 +

2p2

d

)2(1 +

2p2

d

)−2

(1 + oP(1)) =i2

4m(1 + oP(1)).

(5.3.33)

We conclude that

m

i2

∞∑

k=2

(k − 1)Lk(n)P→ (d+ 2p2)2

4(d− 2p2)2− d+ 2p2

2(d− 2p2)+

1

4=

4p22

(d− 2p2)2. (5.3.34)

Finally, we conclude that the claims also hold when forCMn(d, q) by Remark 5.3.1,i.e. by applying (5.3.6) to the events

∣∣∣mi2

(N0(n) +

∞∑

k=2

kLk(n))− 2dp2

(d− 2p2)2

∣∣∣ ≥ ε,

∣∣∣mi2

( ∞∑

k=2

(k − 1)Lk(n))− 4p2

2

(d− 2p2)2

∣∣∣ ≥ ε,

N0(n) +

∞∑

k=2

kLk(n) 6= 0.

Proposition 5.3.2 indicates that the typical number of isolated nodes and line com-ponents outside the giant component is of order Θ(i2/m). Naturally, the isolatednodes do not contribute to the number of edges outside the giant component, andhence we are mostly interested in the total number of edges in the line components,which is likely to be of order Θ(i2/m) due to (5.3.24).

Finally, we would like to comment on the number of isolated cycles in CMN (d′).Let C ′k(n), k ≥ 1, denote the number of isolated cycles with k edges. In view ofLemma 5.3.2, if q = o(1), then the degree distribution d′ satisfies all conditionsin [47] (with extremely high probability) except one, namely n′1 6= O(

√m). How-

ever, the proof of Theorem 3.3 in [47] does not use this condition to prove a boundon the number of isolated cycles: this condition was only needed to bound thenumber of line components. Therefore, it follows from [47, (5.18)],

limn→∞

E

∑

k≥1

kC ′k(n)

<∞, (5.3.35)

the expected number of edges outside the giant that are contained in cycle compon-ents is finite. Since CMn(d, q) is created from CMN (d′) by removing Rn verticesof degree one, we observe that all isolated cycles that exist in CMN (d′) remain

5


isolated cycles in CMn(d, q). Moreover, more isolated cycles can be formed frommore complex components in CMN (d′). Using Proposition 5.3.2 and (5.3.35) to-gether with Markov’s inequality, we observe that if q = i/m with

√m i mδ

for some δ ∈ (0, 1/2), then the number of edges in CMn(d, q), outside the giantthat are contained in cycle components is oP(i2/m). In view of Remark 5.3.1, thesame statement holds for CMn(d, q).

5.3.4 Probabilistic bounds on component sizes outside the giant

In this section, we provide large deviation bounds on the number of edges outsidethe giant for the percolated connected configuration model with removal prob-ability q = i/m with

√m i m1−δ for some δ ∈ (0, 1/2). Again, in view of

Remarks 5.3.1 and 5.3.2, it suffices to consider CMN (d′) only.

First, we provide a sharper bound on the probability of having a number of edgesin these complex components outside the giant that is even of a higher order ofmagnitude than O(i2/m).

Proposition 5.3.3. Consider CMN (d′) obtained by removal probability q = i/m with√m i m1−δ for some δ ∈ (0, 1/2). For every α > 0,

P(mi2

#e /∈ Cmax : dmax(C(e)) ≥ 3 ≥ mα)

= O(m−3). (5.3.36)

Proof. We note that, by the proof of [81, Lemma 11], for γ > 0 sufficiently large,

P(∃ C 6= Cmax : |E(C)| ≥ γ log n) = o(m−3). (5.3.37)

We are left to bound the contribution from components that contain at most γ log nedges. We use the method of moments. We observe that

(#Cl s.t. |E(Cl)| ≤ γ log n, dmax(Cl) ≥ 3)j

≤ (#v ∈ [n] : dv ≥ 3, |E(C(v))| ≤ γ log n)j

=∑

v1,...,vj∈[n]

1|E(C(v1))|≤γ logn : dv1≥3 · · ·1|E(C(vj))|≤γ logn : dvj≥3.

We stress that the vertices v1, ..., vj in the summation are not necessarily mutuallydistinct. For the purpose of exposition, write for a vertex v ∈ [n],

I(v) = dv ≥ 3, |E(C(v))| ≤ γ log n.Applying the tower property, we obtain

E[(#Cl s.t. |E(Cl)| ≤ γ log n, dmax(Cl) ≥ 3)j

]

≤ E

∑

v1,...,vj∈[n]

1I(v1)1I(v2) · · ·1I(vj)

= E

∑

v1,...,vj−1∈[n]

1I(v1)1I(v2) · · ·1I(vj−1)E

∑

vj∈[n]

1I(vj)

∣∣ I(v1), ..., I(vj−1)

.

5


For a graph (or component) G, write V (G) as the vertex set of that graph. Then,

E

∑

vj∈[n]

1I(vj)

∣∣ I(v1), I(v2), ..., I(vj−1)

= E

∑

vj∈[n]

1I(vj) : vj ∈j−1⋃

h=1

V (C(vh))∣∣ I(v1), I(v2), ..., I(vj−1)

+ E

∑

vj∈[n]

1I(vj) : vj 6∈j−1⋃

h=1

V (C(vh))∣∣ I(v1), I(v2), ..., I(vj−1)

.

The first term is trivially bounded by

E

∑

vj∈[n]

1I(vj) : vj ∈j−1⋃

h=1

V (C(vh))∣∣ I(v1), I(v2), ..., I(vj−1)

≤ (j − 1)γ log n.

For the second term, we note that we count the number of vertices outside thegiant component that have a degree at least three, while disregarding the set∪j−1h=1I(vh). We note that if we remove the components ∪j−1

h=1C(vh) from CMN (d′),the remaining graph is a configuration model but with a modified degree sequence.In other words, we remove components that have a total of at most O(log n) edges.This number is too small to change the degree sequence d′ much. That is, it followsfrom (the proof of) Proposition 5.3.1 that

E

∑

vj∈[n]

1I(vj) : vj 6∈j−1⋃

h=1

V (C(vh))∣∣ I(v1), I(v2), ..., I(vj−1)

= o

(i2

m

).

Iterating the argument, we obtain

E[(#Cl s.t. |E(Cl)| ≤ γ log n, dmax(Cl) ≥ 3)j

]= o

((j − 1) log n+ o

(i2

m

))j,

and hence

E[(#e : |E(C(e))| ≤ γ log n : dmax(C(e)) ≥ 3)j

]

≤(

(j − 1)(γ log n)2 + o( i2 log n

m

))j.

Finally, using Markov’s inequality, it holds for every j ∈ N,

P(m#e : |E(C(e))| ≤ γ log n : dmax(C(e)) ≥ 3

i2≥ mα

)

≤(

(j − 1)γ2(log n)2 + o( i2 log n

m

))jmj(1−α)

i2j

= O

(((log n)2 +

i2 log n

m

)m(1−α)

i2

)j= o

((log n)2m−α

)j.

Choosing j ≥ (3 + ε)/α for some ε > 0, the claim follows.

5


Next, we provide a result that shows that the probability for the number of edgesin line components in CMN (d′) to be of a higher order of magnitude than itsexpectation decays fast.

Lemma 5.3.4. Consider CMN (d′) obtained by removal probability q = i/m, where√m i m1−δ for some δ ∈ (0, 1/2). For every α > 0,

P(mi2

( ∞∑

k=2

(k − 1)L′k(n))≥ mα

)= O(m−3). (5.3.38)

Proof. For every j ∈ N, by Markov’s inequality,

P(mi2

( ∞∑

k=2

(k − 1)L′k(n) ≥ mα)≤ E

[(mi2

( ∞∑

k=2

kL′k(n))j]

m−αj .

By Corollary 5.3.1,

E[(mi2

( ∞∑

k=2

kL′k(n) ≥ mα)j]

m−αj = O(m−αj).

Choosing an integer j ≥ 3/α concludes the result.

5.3.5 First disconnections

In this section, we consider the question of how many edges need to be removed tocause the (initially connected) configuration model to become disconnected. Thatis, we would like to prove Theorem 5.2.1, and show that the most likely momentfor the first disconnection to occur is after ΘP(

√m) edges have been removed.

To prove Theorem 5.2.1, we use the equivalence between sequential edge removaland percolation as established in Lemma 5.3.1. More specifically, in order toprove Theorem 5.2.1, it follows from Lemma 5.3.1 that it suffices to show that in apercolation process with a removal probability of the order q = Θ(m−1/2) leads toa positive probability for the configuration model to remain connected.

Proposition 5.3.4. Consider the graphs CMn(d, q) and CMn(d, q). If q = c/√m with

c ∈ (0,∞), then

P(CMn(d, q) is connected)→(d− 2p2

d

)1/2

exp− 2c2p2

d− 2p2

, (5.3.39)

and

P(CMn(d, q) is connected)→ exp− 2c2p2

d− 2p2

. (5.3.40)

Proof. We build CMn(d, q) using the explosion algorithm from Algorithm 2. Toobtain the limiting probability that CMn(d, q) is connected, we first require moredetailed information on the degree sequence of d′. Observe that

q =c√m

=c√d/2

1√n.

5


Recall (5.3.11), (5.3.13), (5.3.14) and (5.3.9), and hence

n′0 =

∞∑

j=2

nj,0 = n2,0 + oP(1)d→ Poi

(c2p2

2d

), (5.3.41)

n′1 = Rn +

∞∑

j=2

nj,1 =√n

(c√d√2

+cp2√d/2

)(1 + oP(1)). (5.3.42)

In addition, we observe that nl − Rn ≤ n′l ≤ nl + Rn for every l ≥ 2, and hencewith high probability,

p′l = limn→∞

n′l/n = pl, l ≥ 2.

Moreover, we observe that the average degree of d′ converges in probability to d.

Secondly, we note that removing vertices of degree one cannot disconnect acomponent. Therefore, there are only two ways for CMn(d, q) to be connectedafter the explosion algorithm. That is, either CMN (d′) is already connected, orCMN (d′) consists of one (giant) component and other components that are linesof length two (the only component entirely made of vertices of degree one), whereall vertices of the line components are removed in the final step of the explosionalgorithm. In either case, we observe that one requires that n′0 = 0, which occurswith probability

P (n′0 = 0) = P(Poi

(c2p2

2d

)= 0)

(1 + oP(1)) = exp

−c

2p2

2d

(1 + oP(1)).

Theorem 2.2 of [47] implies that if n′0 = 0, then with high probability the graphCMN (d′) consists of a giant component, possible components that are lines, pos-sible components that are cycles, but no other type of components. Recall thatL′k(n) represents the number of components that are lines of length k ≥ 2 inCMN (d′), and C ′k(n) the number of components that are cycles of length k ≥ 1.We call component with a single vertex of degree two with a self-loop a cyclecomponent of length one, and a component with two vertices with multi-edgesbetween them a cycle of length two. Theorem 2.2 of [47] implies that

L′k(n)d→ Poi

(c2(d

2+ p2

)2(2p2)k−2

dk

), k ≥ 2,

C ′k(n)d→ Poi

((2p2)k

2kdk

), k ≥ 1,

and all limits are independent random variables. For CMn(d, q) to be connectedafter the explosion algorithm, no cycles should appear in CMN (d′), which occurswith probability

limn→∞

P (no cycle components in CMN (d′)) =

∞∏

k=1

limn→∞

P (C ′k(n) = 0)

=

∞∏

k=1

exp

− (2p2)k

2kdk

=

(d− 2p2

d

)1/2

.

5


Also, no lines of length more than three should appear, which occurs with probab-ility

limn→∞

P (L′k(n) = 0 ∀k ≥ 3) =

∞∏

k=3

limn→∞

P (L′k(n) = 0)

=

∞∏

k=3

exp

−c2(

d

2+ p2)2 (2p2)k−2

dk

= exp

− c

2

d2

(d

2+ p2

)22p2

d− 2p2

.

Finally, we require that all vertices in the line components are removed in thefinal step of the explosion algorithm. That is, a set of 2L′2(n) vertices need to beremoved out of the Rn randomly chosen vertices of degree one. Note that thenumber of vertices that are removed from line components in the last step of theexplosion algorithm is hypergeometrically distributed: we remove Rn verticesuniformly at random from n′1 vertices of which 2L′2(n) are in line components.Therefore, conditionally on n′1, Rn and L′2(n), the probability of all vertices to beremoved in the final step of the explosion algorithm is given by

(n′1 − 2L′2(n)

Rn − 2L′2(n)

)/(n′1Rn

)=Rn(Rn − 1) · · · (Rn − 2L′2(n) + 1)

n′1(n′1 − 1) · · · (n′1 − 2L′2(n) + 1)

In particular, if L′2(n) = 0, then with probability one all vertices in line com-ponents are removed in the last step of the algorithm. Using the tower prop-erty, (5.3.14), (5.3.42) and the observation that L′2(n) converges to a Poisson distri-bution, we observe that

P (all line components in CMN (d′) are removed in CMn(d, q))

= E[E[1all line components in CMN (d′) are removed in CMn(d,q) | n′1, Rn, L′2(n)

]]

=

∞∑

k=0

P(L′2(n) = k)E[(

n′1 − 2k

Rn − 2k

)/(n′1Rn

)]

=

∞∑

k=0

1

k!

(c2

d2

(d

2+ p2

)2)k

exp

− c

2

d2

(d

2+ p2

)2(

d

d+ 2p2

)2k

(1 + o(1))

= exp

− c

2

d2

(d

2+ p2

)2(

1−(

d

d+ 2p2

)2)

(1 + o(1))

= exp

−c

2p2(d+ p2)

d2

(1 + o(1)).

We conclude that

P(CMn(d, q) is connected) =

(d− 2p2

d

)1/2

(1 + o(1))

· exp

−c

2p2

2d− c2

d2

(d

2+ p2

)22p2

d− 2p2− c2p2(d+ p2)

d2

=

(d− 2p2

d

)1/2

exp

− 2c2p2

d− 2p2

(1 + o(1)).

5


For the second statement of the proposition, it is known from [76] that

P(CMn(d) is connected)→(d− 2p2

d

)1/2

,

and since CMn(d, q) is connected ⊆ CMn(d) is connected, the statement fol-lows.

Using the partial result of Proposition 5.3.4, we can prove Theorem 5.2.1.

Theorem 5.2.1. Suppose that we subsequently remove edges uniformly at random fromCMn(d). Define Tn,d as the smallest number of edges that need to be removed for thenetwork to be disconnected. Then,

m−1/2Tn,dd→ T (D),

where T (D) has a Rayleigh distribution with density function

fT (D)(x) =4p2x

d− 2p2e−

2x2p2d−2p2 . (5.2.1)

Proof. First, connectivity is a monotone property, and thus, once the graph is dis-connected it will stay disconnected for the rest of the process. From Lemma 5.3.1 itfollows that sequential removal of c

√m edges in CMn(d, q) is well-approximated

by a percolation process with removal probability q = cm−1/2(1 + oP(1)). Con-sequently, due to Proposition 5.3.4,

P(Tn,d ≥ x√m) = P

(CMn(d, x/

√m) is connected

)+ o(1)→ exp

− 2x2p2

d− 2p2

.

(5.3.43)

In other words,m−1/2Tn,d converges in distribution to a random variable T whosecomplementary cumulative distribution function is given by the expression onthe right-hand side of (5.3.43).

Proposition 5.3.4 implies that the percolated graph CMn(d, q) with removal prob-ability q = o(m−1/2) is disconnected with probability of order o(1). More detailedbounds can be provided for this range, as is illustrated by the next result.

Proposition 5.3.5. Consider CMn(d, q) with q = O(m−α) for some 1/2 < α < 1.Then,

P(CMn(d, q) is disconnected) = O(m1−2α). (5.3.44)

Consequently, if we remove i := im edges uniformly at random from CMn(d) withi = o(mβ) for some β ∈ (0, 1/2), then the probability of the resulting graph beingdisconnected is of order O(m2β−1).

5


Proof. If CMn(d, q) is disconnected, then there is at least one component outsidethe giant that is either an isolated node, a line, a cycle or a component that containsa vertex with degree at least three. To show the result, it therefore suffices to provethat each of these events occur with probability O(m1−2α).

First, it follows from Proposition 5.3.2 that in CMn(d, q),

P(N0(n) +

∞∑

k=2

kLk(n) 6= 0)

= O(m1−2α).

In other words, the event that there is an isolated node or a line component(outside the giant) occurs with probability O(m1−2α). Moreover, we can applyProposition 5.3.1 to obtain that in CMn(d, q),

P(∃ C 6= Cmax : dmax(C) ≥ 3 = o(m1−2α).

Finally, we show that with probability 1 − O(m1−2α) percolation on CMn(d, q)does not create cycles. We observe that

P

( ∞∑

k=1

kCk(n) 6= 0∣∣CMn(d) is connected

)

=P(∑∞

k=1 kCk(n) 6= 0: CMn(d) is connected)

P (CMn(d) is connected)≤

P(∑∞

k=1 kCk(n) 6= 0)

P (CMn(d) is connected),

where Ck(n) denotes the number of newly formed isolated cycles in CMn(d, q)that do not exist in the initial graph CMn(d). Since CMn(d) is connected withnon-vanishing probability, it is sufficient to show that

P

( ∞∑

k=1

kCk(n) 6= 0

)= O(m1−2α).

Again, we use the explosion method. We stress that running this algorithm ona sampled CMn(d) results in a graph that is not the same as the graph thatwould have been obtained if percolation had been applied on the same sampledCMn(d). Instead, sampling a CMn(d) and running the explosion algorithmresults in a graph that is statistically indistinguishable from one that is obtainedby sampling another CMn(d) and applying percolation on it. Therefore, we needto consider the question how to bound newly formed cycles in CMn(d, q). Thecrucial observation to answer this question is that such cycles contain at least onenode whose degree in d was at least three.

More specifically, the number of newly formed isolated cycles in CMn(d, q) isat most the number of cycles in CMn(d, q) that contain at least one node whosedegree in d is at least three. Write Zn for the number of vertices whose degree ind′ is two, but has degree at least three in d, i.e.

Zn =

∞∑

l=3

nl,2.

5


Every isolated cycle inCMn(d, q) that contains at least one node having an originaldegree at least three in d, either that cycle already exists in CMN (d′), or it wasformed from a component in CMN (d′) by the removal of degree one vertices. Inthe second case, this implies that there exists a component in CMN (d′) outsidethe giant with a maximum degree at least three, which occurs with probabilityo(m1−2α) by Proposition 5.3.1. What remains is to analyze the first case.

Write C ′k(n) for the number of cycles that exist inCMN (d′) and contain at least onenode whose original degree in d is at least three. For any k ≥ 1, a set of (distinct)degree-two vertices v1, ..., vk in d′ forms a cycle in CMN (d′) with probability

2(k − 1)

2m− 1

2(k − 2)

2m− 3· · · 2

2m− 2k + 3

1

2m− 2k + 1

if k ≥ 2, and 1/(2m−1) if k = 1. The number of sets of k vertices in d′ that all havedegree two of which at least one vertex has degree at least three in d is boundedby Zn

(n′2k−1

). Therefore,

E[C ′k(n)] ≤ E[Zn

(n′2k − 1

)(2(k − 1))!!(2m− 2k − 1)!!

(2m− 1)!!

].

We observe that n′2 ≤ n2 +Rn and Zn ≤ Rn, where Rn is a binomially distributedrandom variable with parameters 2m and 1−√1− q = i/(2m)(1+o(1)). Therefore,for every ε > 0, the probability that Rn ≥ (1 + ε)i occurs has an exponentiallydecaying tail. On the other hand, since n2/n→ p2, it holds for every ε > 0 and alln sufficiently large,

E[C ′k(n) | Rn ≤ (1 + ε)i] ≤ i(1 + ε)

(n2 + (1 + ε)i

k − 1

)(2(k − 1))!!(2m− 2k − 1)!!

(2m− 1)!!

= (1 + ε)i

2m

(2p2 + ε

d− ε

)k−1

and hence this sequence converges to zero exponentially fast in k. Applyingdominated convergence, we obtain

E

[ ∞∑

k=1

kC ′k(n) | Rn ≤ (1 + ε)i

]= O

(i

m

)= o(m1−2α).

By Markov’s inequality, we conclude that

P

( ∞∑

k=1

kCk(n) 6= 0

)= E

( ∞∑

k=1

kC ′k(n) 6= 0 | Rn ≤ (1 + ε)i

)+ o(m1−2α)

= o(m1−2α).

This proves the first statement of the proposition.

To prove the second statement of the proposition, we need to relate the percolationprocess to the process of removing edges uniformly at random from CMn(d).To each edge, attach a uniformly distributed random variable. An alternative

5


description of the percolation process with removal probability q is by removingall edges whose values of the random variable are below q. Let Um(1) ≤ ... ≤ Um(m)

denote m uniformly distributed order statistics. Then, the probability that i edgeremovals disconnects CMn(d) is given by P(CMn(d, Um(i)) is disconnected). Wenote that

P(CMn

(d, Um(i)

)is disconnected

)

≤ P(CMn

(d,mβ−1

)is disconnected

)+ P

(Um(i) > mβ−1

).

The above proof shows that

P(CMn

(d,mβ−1

)is disconnected

)= O(m2β−1).

The second term is negligible, since by the Chernoff bound,

P(Um(i) > mβ−1

)≤ (1 + o(1)) exp

−m

β

2

= O(m2β−1).

5.3.6 Number of edges outside the giant component

First, we prove Theorem 5.2.2 putting together the results proved in the Sec-tion 5.3.3.

Theorem 5.2.2. Suppose we remove i := im edges uniformly at random from the con-nected configuration model CMn(d) with

√m i m1−δ for some δ > 0. Write

Em(i) for the set of edges in the largest component of this graph, and let |Em(i)| denoteits cardinality. Then,

(m− i− |Em(i)|)mi2

P→ 4p22

(d− 2p2)2. (5.2.2)

Proof of Theorem 5.2.2. By Lemma 5.3.1, the number of edges outside the largestcomponent in CMn(d) after i failures can be derived by considering percolationon this graph with removal probability q = i/m. The edges outside the giant canbe divided into edges in line components, edges in cyclic components and edgesin more complex components (i.e. components which contain a vertex of degreeat least three). From Proposition 5.3.2 it follows that

m

i2

( ∞∑

k=2

(k − 1)Lk(n))

P→ 4p22

(d− 2p2)2.

Next we bound the number of edges outside the giant that are contained incomponents that are cycles or contained in more complex components. In view ofRemarks 5.3.1 and 5.3.2, we point out that this is bounded by the same quantityin CMN (d′). Therefore, to conclude the proof, it suffices to show that the total

5


number of edges contained in cycle components or more complex componentsoutside the giant is of order oP(i2/m) for CMN (d′).

By Proposition 5.3.1, the number of edges in CMN (d′) outside the giant thatis contained in a more complex component is of order oP(i2/m). To count thenumber of edges in cycle components, recall that C ′k(n) denotes the number ofcyclic components with k edges in CMN (d′), and that it satisfies (5.3.35), i.e.limn→∞ E[

∑k≥1 kC

′k(n)] <∞. Using Markov’s inequality, this implies that

m

i2

∑

k≥1

kC ′k(n) = oP(1).

Consequently, the number of edges in CMN (d′) outside the giant that are notcontained in line components is indeed of order oP(i2/m).

Theorem 5.2.2 prescribes the likely number of edges outside the giant componentas an effect of sequentially removing edges uniformly at random. The initiallyconnected configuration model is likely to disintegrate, as more edges are removed,in a unique giant and several small components, the majority of which are eitherlines or isolated nodes as long as the number of edge failures is sublinear.

Next, we show that the number of edges outside the giant component inCMn(d, q)is unlikely to be of a higher order of magnitude than its typical value during thecascade. We stress that this results concerns the percolation process, which canin turn be used to derive a large deviations bound for the sequential removalprocess.

Theorem 5.3.1. Consider CMn(d, q) and CMn(d, q) with q = i/m, with√m i

m1−δ for some δ > 0. Then, for every α > 0,

P(|E(CMn(d, q) \ Cmax)|m

i2≥ mα

)= O(m−3), (5.3.45)

and

P(|E(CMn(d, q) \ Cmax)|m

i2≥ mα

)= O(m−3). (5.3.46)

Proof. First, we show (5.3.45) by using the explosion algorithm. Again, in view ofRemarks 5.3.1 and 5.3.2, it suffices to show

P(|E(CMN (d′) \ Cmax)|m

i2≥ mα

)= O(m−3). (5.3.47)

We partition the different kinds of contributions to the total number of edgesoutside the giant of CMN (d′) as follows: edges can be contained in a line, a cycleor a more complex component. Due to Proposition 5.3.4, the number of edges inline components in CMN (d′) is bounded by

P( ∞∑

k=2

(k − 1)L′k(n)m

i2≥ mα

)= O(m−3). (5.3.48)

5


To bound the edges in cycles, we point out that if follows from [47, (3.7)] that allmoments of the number of cycles inCMN (d′) converge to a finite constant. That is,convergence of the first j factorial moments is shown, which implies convergenceof the first j moments. Again, this result is shown under the condition that thenumber of vertices of degree one is of order O(

√m) in [47], but this assumption

is not used for the results with respect to the isolated cycles. Consequently, forevery j, as long as p2 ∈ (0, 1),

limn→∞

E[( ∞∑

k=1

kC ′k(n))j]≤ E

[( ∞∑

k=1

Poi( (2p2)k

2kdk

)k)j]

<∞,

where the Poisson variables in the second term are all independent. Therefore, forevery α > 0, by applying Markov’s inequality, we obtain

P(mi2

∞∑

k=1

kC ′k(n) ≥ mα)≤ P

( ∞∑

k=1

kC ′k(n) ≥ mα)

≤ E[( ∞∑

k=1

kC ′k(n))3/α]

m−3 = O(m−3).

To bound the number of edges in other components we use Proposition 5.3.3 toobtain

P(m#e /∈ Cmax : dmax(C(e)) ≥ 3

i2≥ mα

)= O(m−3).

Thus we obtain (5.3.47) by summing the three different contributions.

Theorem 5.3.1 reveals that it is very unlikely for the number of edges outside thegiant to be larger than of order Θ(i2/m) when applying the percolation process.We can use this result to show that this also holds under a sequential removalprocess. That is, we can prove Theorem 5.2.3, which we recall next.

Theorem 5.2.3. Suppose i := im edges have been removed uniformly at random from theconnected configuration model CMn(d), where

√m i mα for some α ∈ (1/2, 1).

Then,

P(m− i− |Em(i)| > iα

)= O(m−3). (5.2.3)

Moreover, for every k := km for which k = o(mα) for some α ∈ (0, 1),

P(m− i− |Em(i)| > iα for some i ≤ k

)= o(m−1/2). (5.2.4)

Proof of Theorem 5.2.3. First, we prove (5.2.3) for i such that m1/2+ε i mα forsome ε ∈ (0, α− 1/2). Let Um(1) ≤ ... ≤ Um(m) denote m uniformly distributed orderstatistics, and note that

P(|Em(i)| ≤ m− i− iα

)= P

(∣∣E(CMn(d, Um(i))\Cmax)∣∣ > iα

).

5


Let ε > 0 (sufficiently) small, and note that by the Chernoff bound,

P(Um(i) 6∈

[i

mm−ε,

i

mmε

])= O(m−3).

Therefore,

P(E(∣∣CMn(d, Um(i))\Cmax)

∣∣ > iα)

≤ maxq∈[im−(1+ε),im−(1−ε)]

P(∣∣E(CMn(d, q)\Cmax)

∣∣ > iα)

+O(m−3).

We observe that for every q ∈[im−(1+ε), im−(1−ε)]with i = o(mα),

q2m

iα≤ i2−α

m1−2ε= o(m−(1−α)2+2ε) = o(1)

for all ε > 0 sufficiently small. By Theorem 5.3.1, we conclude that

P(|Em(i)| ≤ m− i− iα

)

≤ maxq∈[im−(1+ε),im−(1−ε)]


∣∣ > q2/m)

+O(m−3) = O(m−3).

To prove (5.2.4), note that by Proposition 5.3.5,

P(m− i− |Em(i)| > iα for some 1 ≤ i ≤ m1/8

)

≤ P(m− i− |Em(i)| 6= 0 for some 1 ≤ i ≤ m1/8

)= o(m−1/2).

Moreover, if k m1/2+ε for some ε ∈ (0, 1/2), using that k ≤ m, it follows directlyfrom (5.2.3),

P(m− i− |Em(i)| > iα for some m1/2+ε ≤ i ≤ k

)= O(km−3) = o(m−1/2).

Therefore, to conclude (5.2.4), it suffices to show that for some ε ∈ (0, 1/2) suffi-ciently small,

P(m− i− |Em(i)| > i1/8 for some m1/8 ≤ i ≤ m1/2+ε

)= o(m−1/2).

For convenience, write |Em(i)| = m− i− |Em(i)|, and consider values of i suchthat m1/8 ≤ i ≤ m1/2+ε for some ε ∈ (0, 1/2). Note that

P(|Em(i)| > i1/8

)= P

(|Em(i)| > i1/8, |Em(m1/2+ε)| < |Em(i)|/2

)

+ P(|Em(i)| > i1/8, |Em(m1/2+ε)| ≥ |Em(i)|/2

).

We observe that |Em(i)| is the number of edges outside the giant when removing iedges uniformly at random. Write ξ(i, j) as the number of edges that are removed

5


out of this set of |Em(i)| edges if we remove another j − i edges uniformly atrandom. Then, |Em(m1/2+ε)| ≥ |Em(i)| − ξ(i,m1/2+ε), and hence

P(|Em(i)| > i1/8, |Em(m1/2+ε)| < |Em(i)|/2

)

≤ P(|Em(i)| > i1/8, ξ(i,m1/2+ε) > |Em(i)|/2

).

Since ξ(i, j) ≤ j for every j ≥ i, it follows from the first claim of the corollary thatwith probability 1−O(m−3),

|Em(i)| ≤ |Em(m1/2+ε)|+ ξ(i,m1/2+ε) = o(m).

In other words, the probability that an edge out of the set of |Em(i)| is chosen tobe removed is strictly bounded by |Em(i)|/(m−m1/2+ε) = o(1) with probability1−O(m−3). In that case, the probability for more than half of the |Em(i)| > i1/8

edges to be removed has an exponentially decaying tail. In particular, this impliesthat

P(|Em(i)| > i1/8, |Em(m1/2+ε)| < |Em(i)|/2

)

≤ P(|Em(i)| > i1/8, ξ(i,m1/2+ε) > |Em(i)|/2

)= O(m−3).

For the other term, we observe that for every m1/8 ≤ i ≤ m1/2+ε,

P

(|Em(i)| > i1/8, |Em(m1/2+ε)| ≥ |Em(i)|

2

)≤ P

(|Em(m1/2+ε)| > i1/8

2

)

≤ P(|Em(m1/2+ε)| > m1/64

2

).

Similarly as in the proof of the first claim of the corollary, we observe that

P(Um(m1/2+ε) 6∈

[m−1/2−2ε,m−1/2+2ε

])= O(m−3),

and hence

P(|Em(m1/2+ε)| > 2m1/64

)≤ P

(Um(m1/2+ε) 6∈

[m−1/2−2ε,m−1/2+2ε

])

+ maxq∈[m−1/2−2ε,m−1/2+2ε]


∣∣ > 2m1/64).

It follows from Theorem 5.3.1 that the second term is also of order O(m−3) forevery ε < 1/256. We conclude that for every m1/8 ≤ i ≤ m1/2+ε with ε < 1/256,

P(|Em(i)| > i1/8

)= O(m−3),

from which (5.2.4) follows by the union bound. Moreover, it implies that (5.2.3)holds for all

√m i m1/2+ε for some ε ∈ (0, α− 1/2) as well.

5

5.4. Cascading failure process 143

5.3.7 Linear number of edge removals

For completeness, we provide a brief overview of known results about whatCMn(d, q) looks like when the removal probability is a fixed value q ∈ (0, qc), asstudied by Janson in [60]. It is shown that for any fixed q ∈ (0, qc), CMn(d, q) hasa unique giant component and many small components. However, in this phasethe giant does not contain n− o(n) vertices, as in the case when q → 0.

From [60], it is known that given a sequence CMn(d) there exists a function ξd(q)defined for q < qc such that in CMn(d, q)

|E(Cmax)|(1− q)m

P→ ξd(q), (5.3.49)

i.e., the proportion of edges in the giant component concentrates for every q < qc.The exact formula for ξd(q) comes from [60, Theorem 3.9],

ξd(q) =1− ρ√1− q +

(1− ρ)2

2, (5.3.50)

where ρ is defined as the solution of the equation

(1− q)1/2G′D(1− (1− q)1/2 − ρ(1− q)1/2

)+ (1− (1− q)1/2)d = ρd, (5.3.51)

where GD is the probability generating function of D. The same concentrationresult, with a different limit function, holds also for the number of vertices in thelargest component.

It is worth mentioning that in this case, lines and isolated vertices do not constitutethe vast majority of the small components anymore, but there is also a positivedensity of more complex small components to appear. These are mainly trees.

If q > qc, instead, there is no giant component that contains a non-vanishingproportion of the edges or vertices and usually the high-degree vertices determinethe size of the largest components, i.e. |Cmax| = ΘP(dmax) [61].

5.4 Cascading failure process

The results in the previous section explain the way the connected configurationmodel is likely to disintegrate as edge failures occur sequentially and uniformly atrandom. This yields the load surge values, and hence what remains to be done forproving Thoerem 5.1.1, is to compare the load surge values to the correspondingsurplus capacities at the edges.

To prove our main results as stated in Theorem 5.1.1, we follow the proof strategyas laid out in Section 5.2.3. Recall that the connected disintegrates in a giantcomponent and a sublinear number of edges outside the giant as long as no morethan o(m) edges are removed. We point out that, intuitively, this implies thatthe only dominant contribution to the total failure size comes from the number

5


of edges that are contained in the giant component upon failure. We make thisstatement rigorous in this section.

This section is structured as follows. In Section 5.4.1, we consider the settingwhere no disconnections takes place yet. Since it follows from Theorem 5.2.1that it is unlikely that the connected configuration model becomes disconnectedbefore Θ(

√m), we can show that Theorem 5.2.2 holds if k = o(

√m). For larger

thresholds, we consider the failure size tail of the giant component. To derivethis tail behavior, we translate this problem to a first-passage time problem over arandom moving boundary in Section 5.4.2. In Section 5.4.3, we derive the behaviorof this first-passage time. We conclude Theorem 5.2.2 in Section 5.4.4 by using thestrategy as laid out in Section 5.2.3.

5.4.1 No edge disconnections

Before we move to the tail of the failure size, we first consider the scenario whereno disconnections have occurred yet during the cascade. As long as edge failuresdo not cause disconnections in the graph, the load surge function remains thesame at every surviving edge. Recall that in this case it holds that |Emj (i)| = m− ifor all surviving edges ej ∈ [m], and hence recursion (5.1.1) is solved by

lmj (i) =θ

m+ (i− 1) · 1− θ/m

m. (5.4.1)

In other words, given that no disconnections have occurred after k edge failures,the load surge function behaves deterministically until step k (at every survivingedge). In this case, the problem reduces to a first-passage time problem, i.e. theevent An,d ≥ k corresponds to the event that the smallest k uniformly distrib-uted order statistics are below the linear load surge function. That is, this casereduces to the setting of Chapter 2. The following result follows by applyingTheorem 2.3.1 in this setting.

Proposition 5.4.1. Define A?n+1 as the number of edge failures in a star network withn + 1 nodes and m = n edges, and load surge function given by (5.4.1). For everyk := km satisfying k →∞ and m− k →∞ as m→∞,

P(A?n+1 ≥ k

)∼ 2θ√

2π

√m− km

k−1/2.

In particular, if 1 k m, it holds that

P(A?n+1 ≥ k

)∼ 2θ√

2πk−1/2.

A crucial argument used in the proof of Proposition 5.4.1 is that as n = m→∞,

P(A?n+1 ≥ k

)∼ P

(Um(i) ≤

θ + i− 1

m, i = 1, ..., k

).

5


The asymptotic behavior of the latter expression is obtained in Chapter 2 byobserving that the edge failure distribution is quasi-binomial for this particularload surge function, and exploiting the analytic expression for the probabilitydistribution function to derive the tail behavior. Alternatively, we related thisproblem to an equivalent setting in Chapter 3, where one is interested in thefirst-passage time of a random walk bridge. We use such a relation in the moreinvolved case where disconnections occur as well. Before moving to the generalcase, we briefly recall the translation to the equivalent random walk problemin the much simpler setting where no (edge) disconnections occur yet, also seeChapter 3.

Translation to random walk setting:Note that(Um(1), U

m(2), ..., U

m(m)

)

d=

Exp1(1)

m,

∑2j=1 Expj(1)

m, ...,

∑mj=1 Expj(1)

m

∣∣∣∣m+1∑

j=1

Expj(1) = m

,

(5.4.2)

where the exponentially distributed random variables are independent. Definethe random walk

Si =

i∑

j=1

(1− Expj(1)

), i ≥ 1, (5.4.3)

where S0 = 0. Then,

P(A?n+1 ≥ k

)∼ P

(Um(i) ≤

θ + i− 1

m, i = 1, ..., k

)

= P(Si ≥ 1− θ, i = 1, ..., k

∣∣Sm+1 = 1).

Define for every x ∈ R,

τ∗x := mini ≥ 1 : Si ≤ x. (5.4.4)

Then, the objective can be written as

P(A?n+1 ≥ k

)∼ P

(τ∗1−θ > k

∣∣Sm+1 = 1)∼ P

(τ∗1−θ > k

∣∣Sm = 0).

Due to Theorem 3.4.1, we observe that for k = o(m)

P(A?n+1 ≥ k

)∼ P

(τ∗1−θ > k

∣∣Sm = 0)∼√

2

πE(−Sτ1−θ

)k−1/2 =

2θ√2πk−1/2,

where the latter equality follows from the memoryless property of exponentials.

A similar random walk construction can be done for the case where disconnec-tions do take place. Before we consider this process, we first show the result of

5


Theorem 5.1.1 in the phase where it is unlikely to have any disconnections in theconnected configuration model. That is, removing k = o(

√m) edges uniformly at

random is unlikely to cause any disconnection by Theorem 5.2.1. Due to the coup-ling between the cascading failure process and sequential edge-removal process,Proposition 5.4.1 prescribes exactly the asymptotic behavior of the edge failuresize in that case.

Theorem 5.4.1. The probability that the cascading failure process disconnects the networkis given by

P(An,d ≥ Tn,d) ∼ 2θ√2π

(2p2

d− 2p2

)1/4

Γ

(3

4

)m−1/4, (5.4.5)

where Γ(·) denotes the gamma function. Consequently, for any threshold 1 k √m,

P (An,d ≥ k) ∼ 2θ√2πk−1/2. (5.4.6)

Proof. Note that

P(An,d ≥ Tn,d) =

∫ ∞

0

P(√mTn,d ∈ dt

)P(A?m+1 ≥ t

√m) dt

∼ 2θ√2πm−1/4E

[(√mTn,d)−1/2

]∼ 2θ√

2πm−1/4

∫ ∞

0

4p2

d− 2p2t1/2e−

2t2p2d−2p2 dt

=2θ√2π

(2p2

d− 2p2

)1/4

Γ

(3

4

)m−1/4,

where the second assertion follows due to the uniform convergence result ofA?m+1,see Theorem 4.1.1, and the third assertion follows from Theorem 5.2.1. For thesecond claim of the theorem, note that

P (An,d ≥ k) = P (An,d ≥ k,An,d < Tn,d) + P (An,d ≥ k,An,d ≥ Tn,d) ,

where due to Proposition 5.4.1,

P (An,d ≥ k,An,d < Tn,d) = P(An,d ≥ k

∣∣An,d < Tn,d)P (An,d < Tn,d)

= P(A?m+1 ≥ k

)P (An,d < Tn,d)︸︷︷︸

=1−o(1)

∼ 2θ√2πk−1/2,

and

P (An,d ≥ k,An,d ≥ Tn,d) ≤ P (An,d ≥ Tn,d) = O(m−1/4) = o(k−1/2).

5


5.4.2 Random walk formulation

This section is devoted to introducing a related random walk, and to showing thatthe number of edge failures in the giant asymptotically behaves the same as thefirst-passage time of a random walk bridge.

Recall that |Em(i)| denotes the number of edges in the giant when i edges havebeen removed uniformly at random, and let ei correspond to the edge corres-ponding to the i’th order statistic of the surplus capacities. Define the sequence ofprocesses Li,m : 1 ≤ i ≤ m+ 1,m ≥ 1, where L1,m = 1, and for 2 ≤ i ≤ m+ 1,

Li,m =

m+1−∑i−1

j=1 Lj,m

|Em(i−2)| if ei−1 ∈ Cmax,

0 if ei−1 6∈ Cmax.(5.4.7)

Note that this corresponds to the load surge increments in the giant if θ = 1, res-caled by a factorm+1. We consider a sequence of random walks (Si,m)m≥1,1≤i≤m+1

defined as

Si,m =

i∑

j=1

Xj,m (5.4.8)

with increments

Xi,m = Li,m − Expi,m(1), (5.4.9)

where Expi,m(1) are independent exponential random variables with unit rate.We note that em ∈ Cmax and |Em(m− 1)| = 1 by definition, since removing m− 1edges leaves only isolated nodes and one component with two nodes connected bya single edge which inherently is the component that contains the largest numberof edges. Therefore,

m+1∑

j=1

Lj,m =

m∑

j=1

Lj,m +m+ 1−∑m

j=1 Lj,m

1= m+ 1,

and hence the random walk satisfies the property

Sm+1,m =

m+1∑

j=1

Xj,m = m+ 1−m+1∑

j=1

Expj,m(1). (5.4.10)

Finally, for all m ≥ 1 define the stopping times

τm = min1 ≤ i ≤ m : Si,m < 1− θ, (5.4.11)

and τ = m+ 1 whenever Si,m ≥ 1− θ for all i = 1, ...,m.

In case of the star topology, i.e. no edge disconnections occur, it holds that|Em(i)| = m − i, causing Li,m = 1 for all 1 ≤ i ≤ m + 1. In that case we ob-served that the failure size tail could be written as the first-passage tail of therandom walk bridge. The above random walk formulation is a generalization thataccounts for edges that may possibly no longer be contained in the giant as edgefailures occur.

5


Proposition 5.4.2. Suppose mδ k m1−δ for some δ ∈ (0, 1/2). If

P(τm ≥ k

∣∣∣∣Sm+1,m = 0

)∼ 2θ√

2πk−1/2, (5.4.12)

then

P(An,d ≥ κ(k)

)∼ P

(τm ≥ k

∣∣∣∣Sm+1,m = 0

).

Proof of Proposition 5.4.2. Write l(·) for the load surge function corresponding tothe edges in the giant. Note that by construction,

P(An,d ≥ κ(k)

)= P

(Um(i)1ei∈Cmax ≤ l(κ(i)), i = 1, ..., k

).

In other words, whenever an edge is contained in the giant, one checks whetherthis edge has sufficient capacity to deal with the load surge function. Instead oflooking only at those instances, we would like to compare all order statistics to anappropriately chosen function. For this purpose, we introduce the function l∗(·)defined as

l∗(i) = l(κ(i− 1) + 1), i = 1, ...,m.

We note this function satisfies two important properties:

(p1) l∗(i) = l(κ(i)) if ei ∈ Cmax;

(p2) l∗(i) = l∗(i− 1) if ei−1 6∈ Cmax.

Moreover, as l(·) is non-decreasing for all i ≥ 2, this holds as well for l∗(i). Wedefine the two stopping times,

T = min1 ≤ i ≤ m : Um(i)1ei∈Cmax > l(κ(i))

and

T ∗ = min1 ≤ i ≤ m : Um(i) > l∗(i).

We observe that the first property (p1) implies that T ∗ ≤ T , and T ∗ = T ifeT∗ ∈ Cmax. Together with the second property (p2) and the observation thatUm

(T )≥ Um(T∗), this implies

T = minj ≥ T ∗ : ej ∈ Cmax. (5.4.13)

Therefore,

P(An,d ≥ κ(k)

)= P

(T > k

)= P (T ∗ > k) + P

(T ∗ ≤ k; T > k

). (5.4.14)

5


To conclude the proof, we relate the random walk to the stopping time T ?, andshow that the second contribution in (5.4.14) is negligible. For the first claim,we consider the perturbed increments Li,m(θ); 1 ≤ i ≤ m + 1,m ≥ 1 withL1,m(θ) = θ and

Li,m(θ) =

m+1−∑i−1

j=1 Lj,m(θ)

|Em(i−2)| if ei−1 ∈ Cmax,

0 if ei−1 6∈ Cmax.(5.4.15)

Note that this corresponds to the load surge increments rescaled by a factor m+ 1.In particular, we observe L·,· = L·,·(1), and

(m+ 1)l∗(i) = θ

(m+ 1

m− 1

)+

i∑

j=1

Lj,m(θ) =θ

m+

i∑

j=1

Lj,m(θ).

Note that θ/m = O(1/m), and

maxi=1,...,k

∣∣∣∣∣∣

i∑

j=1

(Lj,m(θ)− Lj,m)− (θ − 1)

∣∣∣∣∣∣

≤ max

i=1,...,k

|1− θ||Em(i)|

,

which is of order O(1/m) with probability 1− o(m−2) by Theorem 5.2.3. Since

P (T ∗ > k) = P(

(m+ 1)Um(i) ≤ (m+ 1)l∗(i), i = 1, ..., k)

= P

i∑

j=1

Expj(1) ≤ θ

m+

i∑

j=1

Lj,m(θ), i = 1, ..., k∣∣m+1∑

j=1

Expj(1) = m+ 1

= P

i∑

j=1

Xj,m ≥ −θ

m+

i∑

j=1

(Lj,m − Lj,m(θ)), i ≤, k∣∣m+1∑

j=1

Expj(1) = m+ 1

,

it follows that

P (T ∗ > k)

= P

i∑

j=1

Xj,m ≥ 1− θ + o(1), i = 1, ..., k∣∣m+1∑

j=1

Expj(1) = m+ 1

+ o(m−2).

Due to our hypothesis (5.4.12), it follows that as m→∞,

P (T ∗ > k) ∼ P

i∑

j=1

Xj,m ≥ 1− θ, i = 1, ..., k∣∣m+1∑

j=1

Expj(1) = m+ 1

= P(τm ≥ k

∣∣∣∣Sm+1,m = 0

).

To conclude the result, it remains to be shown that the second term in (5.4.14) isof order o(k−1/2). Since we assumed that mδ k m1−δ for some δ ∈ (0, 1/2),

5


we observe that there exists an α ∈ (0, 1) such that both k2/m mα k. For allsuch α ∈ (0, 1), it holds that

P(T ∗ ≤ k; T > k

)= P

(T ∗ ∈ [k −mα, k]; T > k

)+ P

(T ∗ < k −mα; T > k

).

We note that by our hypothesis and our previous result,

P(T ∗ ∈ [k −mα, k]; T > k

)≤ P (T ∗ > k −mα)− P

(T ∗ > k; T > k

)

∼ 2θ√2π

(k −mα)−1/2 − 2θ√2πk−1/2 = o(k−1/2).

Finally, we observe that by (5.4.13),

P(T ∗ < k −mα; T > k

)≤k−mα∑

j=1

P(T ∗ = j; T − T ∗ > mα

)

≤k−mα∑

j=1

m∑

r=0

(m− j + 1− r

m− k

)mαP(|Em(j − 1)| = r

)

≤ o(m−1/2) +

k−mα∑

j=1

(jα

m− k

)mα

= o(m−1/2) +O

(k

(kα

m− k

)mα)= o(m−1/2),

where the third assertion follows from Theorem 5.3.1.

To derive the asymptotic probability of the event An,d ≥ κ(k) to occur fork m1−δ for some δ ∈ (0, 1), Proposition 5.4.2 implies that it suffices to showthat the asymptotic behavior of the first-passage time of the defined random walkis given by (5.4.12).

5.4.3 Behavior of the number of edge failures in the giant

We start the analysis by showing that if k = o(mα) for some α ∈ (0, 1), thenProposition 5.2.1 holds for the number of failures in the giant. We recap thisproposition next.

Proposition 5.2.1. If k = o(mα) for some α ∈ (0, 1), then as n→∞,

P(An,d ≥ κ(k)

)∼ 2θ√

2πk−1/2.

For k = o(√m), this result is already proven in Theorem 5.4.1. For the remainder

of the proofs in this section, we therefore assume k = Ω(√m).

To prove Proposition 5.2.1, we will extensively use the random walk

Si =

i∑

j=1

(1− Expj(1)

), i ≥ 1,

5


where S0 = 0. This is related to τm as defined in (5.4.11) through the relation

τm = min1 ≤ i ≤ m : Si < 1− θ +

i∑

j=1

(1− Lj,m), (5.4.16)

and τm = m+ 1 if Si ≥ 1− θ+∑ij=1 (1− Lj,m) for all 1 ≤ i ≤ m. Moreover, for a

sequence g = gii∈N, let Tg correspond to the first-passage time of the randomwalk Si over this sequence, i.e.,

Tg = mini ∈ N : Si < gi.

We use the following strategy to prove Proposition 5.2.1. First, we show that for aparticular class of (deterministic) boundary sequences, it holds that

P(Tg > k) ∼ 2θ√2πk−1/2

as k →∞. Next, we show that the boundary as given in (5.4.16) falls in this classof boundary sequences with sufficiently high probability, and hence

P(τm > k) ∼ 2θ√2πk−1/2

as k →∞. Finally, we show that conditioning on the event that the random walkreturns to zero at time m+ 1 does not affect the tail behavior, i.e. for all k = o(mα)for some α ∈ (0, 1), it holds that as m→∞,

P(An,d ≥ κ(k)

)∼ P

(τm > k

∣∣Sm+1 = 0)∼ P(τm > k) ∼ 2θ√

2πk−1/2.

First-passage time for moving boundaries initially constant for sufficient time

Before moving to the proof of Proposition 5.2.1, we consider the first-passagetime behavior of (Si)i∈N for a particular class of moving boundaries. The nextlemma shows that the first-passage time over a boundary that is monotone non-decreasing, grows slower than

√i, and is initially constant for a sufficiently large

time, behaves the same as the first-passage time over the constant boundary.

Lemma 5.4.1. Suppose l := lk is such that kα l k as k →∞ for some α ∈ (0, 1).Define the boundary sequence

g+i,l =

1− θ if i ≤ l,iγ if i > l,

with γ ∈ (0, 1/2). Then, as k →∞,

P(Tg+ > k

)∼ P (T1−θ > k) .

5


Remark 5.4.1. We point out that the class of boundary sequences as described inLemma 5.4.1 is not covered by the literature. That is, almost all related literature considermoving boundaries that can be described by sequences of the form (gi)i∈N, i.e. that do notdepend on k. The exception is [42], but this paper restricts to constant boundaries only.

Still, the literature offers some partial results. First, since Tg+ > k ⊆ T1−θ > k,

P(Tg+ > k

)≤ P (T1−θ > k) .

For a lower bound, we would like to remark that g+i,l is a non-decreasing sequence in i ≥ 1,

where g+i,l ≤ iγ for all i ∈ N. Therefore, due to Proposition 1 in [125] (or Theorem 2

in [53]),

P(Tg+ > k

)≥ P (Tiγ > k) ∼ cγP (T0 > k) ∼ c′γP (T1−θ > k) ,

where cγ , c′γ ∈ [0,∞). Moreover, since

∞∑

i=1

iγ

i3/2<∞,

it holds that cγ , c′γ > 0 [53]. In order to prove Lemma 5.4.1, we therefore need to showthat c′γ = 1.

Proof. First, recall that Tg+ > k ⊆ T1−θ > k, and hence

P(Tg+ > k

)≤ P (T1−θ > k) .

Therefore, it suffices to show that the reversed inequality holds asymptotically.

We bound the moving boundary g+ by an appropriate piecewise constant bound-ary with finitely many jumps. For each of these constant intervals, we use theresults in [42] to show that the trajectory of the random walk is asymptoticallyindistinguishable with respect to the boundary g+ and the piecewise constant one.Finally, we glue the intervals together to conclude the result.

First, note that since γ ∈ (0, 1/2), we can assume without loss of generality thatα > 0 is such that kα(1+η) l kα((2γ)−1−η) with η = ((2γ)−1 − 1)/4 > 0. Todefine the piecewise constant boundary, let

r := minj ∈ N : α(2γ)−j > 1,

and note that 1 ≤ r < ∞ since 2γ ∈ (0, 1) and α ∈ (0, 1). Choose a fixed ε > 0sufficiently small such that

• ε < αη, which implies that l = o(kα(2γ)−1−ε

);

• α/(2γ)− ε < α/(2γ)2 − 2ε < ... < α/(2γ)r − rε;

• ε < (1− α(2γ)−r)/r.

5


Define tεj,k, j ≥ 0 with tε0,k = l and

tεj,k = kα(2γ)−j−jε, 1 ≤ j ≤ r.

We point out that r corresponds to the number of times the piecewise constantboundary makes a jump, and the values tεj,k, 0 ≤ j ≤ r−1, correspond to the timeswhere the piecewise constant boundaries jump. Since ε > 0 is chosen sufficientlysmall, l = tε0,k tε1,k ... tεr−1,k k tεr,k as k →∞. Write

h(j) =

1− θ if j = 0,

kα(2γ)−(j−1)/2−jε/2 if 1 ≤ j ≤ r,

and define the boundary sequence as

hεi,k =

h(0) = 1− θ if i ≤ tε0,k = l,

h(j) if tεj−1,k < i ≤ tεj,k, 1 ≤ j ≤ r − 1,

h(r) if i > tεr−1,k.

We point out that by construction,

h(j)/√tεj−1,k = k−ε/2 =⇒ h(j) = o

(√tεj−1,k

), 1 ≤ j ≤ r,

and hence hεi,k = o(√i) for all i ≤ k as k →∞. Moreover,

h(j)/(tεj,k)γ = kjε(γ−1/2) =⇒ h(j) ≥ (tεj,k)γ 1 ≤ j ≤ r.

Consequently,

hεi,k ≥ g+i,l, 1 ≤ i ≤ k,

and therefore we obtain the lower bound

P(Tg+ > k

)≥ P (Thε > k) .

Next, we provide a lower bound for the tail behavior of Thε . Fix δ > 0, and notethat

P (Thε > k) ≥ P(Thε > k;Stεj,k ∈

(δ√tεj,k, 1/δ

√tεj,k

)∀ 0 ≤ j ≤ r − 1

).

Conditioning on the position of the random walk at the times tεj,k, 0 ≤ j ≤ r − 1yields

P (Thε > k) ≥∫ 1/δ

√tε0,k

u0=δ√tε0,k

· · ·∫ 1/δ

√tεr−1,k

ur−1=δ√tεr−1,k

P(Th(r) > k − tεr−1,k

∣∣S0 = ur−1

)

·r−1∏

j=0

P(Stεj,k−tεj−1,k

∈ duj ;Th(j) > tεj,k − tεj−1,k

∣∣S0 = uj−1

),

5


where we write t−1 = 0 and u−1 = 0 for convenience. In other words, we partitionthe trajectory of the random walk in intervals where the boundary is constant.Recall that for every 0 ≤ j ≤ r − 1, it holds that tεj,k − tεj−1,k = tεj,k(1 + o(1)), andh(j) = o(

√tεj−1,k) for every 1 ≤ j ≤ r. Applying Proposition 18 in [42], we obtain

uniformly in uj = Θ(√tεj,k), 1 ≤ j ≤ r − 1,

P(Stεj,k−tεj−1,k

∈ duj ;Th(j) > tεj,k − tεj−1,k

∣∣S0 = uj−1

)

duj

∼√

2

π

V (uj−1 − h(j))√tεj,k − tεj−1,k

uj − h(j)

tεj,k − tεj−1,k

exp

−(uj − h(j)

)2

2(tεj,k − tεj−1,k)

,

where V (·) denotes the renewal function corresponding to the decreasing lad-der height process of the random walk. The behavior of this function is well-understood: it is non-decreasing and V (t) ∼ t/E(−ST0

) = t as t → ∞. Since byconstruction, h(j) = o(uj−1) = o(uj) for every 1 ≤ j ≤ r − 1, we obtain

P(Stj−tj−1

∈ duj ;Th(j) > tj − tj−1

∣∣S0 = uj−1

)

duj

= (1 + o(1))

√2

π

V (uj−1 − (1− θ))√tj − tj−1

uj − (1− θ)tj − tj−1

exp

− (uj − (1− θ))2

2(tj − tj−1)

∼ P(Stj−tj−1 ∈ duj ;T1−θ > tj − tj−1

∣∣S0 = uj−1

)

duj.

Similarly, it holds uniformly in ur−1 = Θ(√tεr−1,k) [42, Proposition 18],

P(Th(r) > k − tεr−1,k

∣∣S0 = ur−1

)∼ P

(T1−θ > k − tεr−1,k

∣∣S0 = ur−1

).

Then,

P (Thε > k) ≥ (1 + o(1))P

(T1−θ > k;Stεj,k ∈

(δ√tεj,k,

√tεj,kδ

)∀ 0 ≤ j ≤ r − 1

).

Conditioning on staying above the constant boundary and applying the unionbound yields

P (Thε > k) ≥ (1 + o(1))P (T1−θ > k)

·

1−

r−1∑

j=0

P(Stεj,k 6∈

(δ√tεj,k, 1/δ

√tεj,k

) ∣∣T1−θ > k)

= (1 + o(1))(

1− r(

1− e− δ2

2

)− re− 1

2δ2

)P (T1−θ > k) .

Letting δ ↓ 0, we find that for every ε > 0 sufficiently small,

lim infk→∞

P(Tg+ > k

)

P (T1−θ > k)≥ lim inf

k→∞P (Thε > k)

P (T1−θ > k)= 1,

from which we conclude that the result holds.

5


The next lemma shows a similar result as Lemma 5.4.1, yet with a boundary thatis monotone non-increasing and initially constant for a sufficiently large time.

Lemma 5.4.2. Suppose l := lk is such that kα l k for some α ∈ (0, 1) as k →∞.Define the boundary sequence

g−i,l =

1− θ if i ≤ l,−iγ if i > l,

with γ ∈ (0, 1/2). Then, as k →∞,

P(Tg− > k

)∼ P (T1−θ > k) .

Proof. First, we observe that

P(Tg− > k

)≥ P (T1−θ > k) ,

and hence it suffices to show that the reversed inequality holds asymptotically.Our proof will be similar to the proof of Lemma 5.4.1, but adapted to provide anupper bound.

Fix ε > 0 small as in Lemma 5.4.1, and define the piecewise constant boundary

hεi,k =

h(0) = 1− θ if i ≤ tε0,k = l,

−h(j) if tεj−1,k < i ≤ tεj,k, 1 ≤ j ≤ r − 1,

−h(r) if i > tεr−1,k.

for r, times tεj,k and levels h(j) defined as in the proof of Lemma 5.4.1. We notethat for every fixed δ > 0,

P(Tg− > k

)≤ P

(Thε > k;Stεj,k ∈

(δ√tεj,k, 1/δ

√tεj,k

), ∀ 0 ≤ j ≤ r − 1

)

+

r∑

j=0

P(Tg− > k;Stεj,k 6∈

(δ√tεj,k, 1/δ

√tεj,k

)).

Using analogous arguments as in Lemma 5.4.1, we obtain

P(Thε > k;Stεj,k ∈

(δ√tεj,k, 1/δ

√tεj,k

), ∀ 0 ≤ j ≤ r − 1

)

≤ (1 + o(1))P (T1−θ > k) .

For the other terms, define the sequence (hi)i∈N with hi = min1− θ,−iγ. Then,due to Theorem 1 of [34],

r∑

j=0


(δ√tεj,k, 1/δ

√tεj,k

))

≤r−1∑

j=0

P(Stεj,k 6∈

(δ√tεj,k, 1/δ

√tεj,k

) ∣∣Th > k)P(Th > k

)

≤ (1 + o(1))r(

1− e− δ2

2 + e−1

2δ2

)P(Th > k

).

5


Letting δ ↓ 0 yields [34]

r∑

j=0


(δ√tεj,k, 1/δ

√tεj,k

))= o

(P(Th > k

))= o

(k−1/2

).

Since

P (T1−θ > k) ∼ 2θ√2πk−1/2,

we conclude that

lim supk→∞

P(Tg− > k

)

P (T1−θ > k)≤ 1.

From Lemmas 5.4.1 and 5.4.2, the following corollary follows directly:

Corollary 5.4.1. Suppose l := lk is such that both kα l k for some α ∈ (0, 1), andthe boundary sequence satisfies

gi,l =

1− θ if i ≤ l,o(iγ) if i > l,

for some θ > 0 and γ ∈ (0, 1/2). Then, as k →∞,

P (Tg > k) ∼ P (T1−θ > k) ∼ 2θ√2πk−1/2.

Proof of Proposition 5.2.1

To prove Proposition 5.2.1, we first show that the tail of τm behaves the same asthat of T1−θ, after which we use the relation in Proposition 5.4.2 to derive the tailof An,d. In view of (5.4.16), we need to understand the behavior of the randomwalk

Yi,m =

i∑

j=1

(1− Lj,m), 1 ≤ i ≤ m+ 1,

where Y0,m = 0. In order to apply Corollary 5.4.1, we therefore need to show thatthe random walk is likely to be close to zero for a sufficiently long time l, andwithin [−iγ , iγ ] for all l ≤ i ≤ k for some 0 < γ < 1/2.

Proposition 5.4.3. Suppose k = o(mα) for some α ∈ (0, 1) and γ ∈ (α/2, 1/2). Then,as m→∞,

P

∣∣∣∣∣∣

i∑

j=1

(1− Lj,m)

∣∣∣∣∣∣> iγ for some 1 ≤ i ≤ k

= o(m−1/2).

5


Proof. From Proposition 5.3.5, it follows that the probability that there are discon-nections when removing less than o(m1/4−ε) edges with ε ∈ (0, 1/4) is of ordero(m−1/2). Therefore, for every l = o(m1/4−ε) with ε ∈ (0, 1/4), it is likely thatLi,m = 0 for every i ≤ l , and hence

P

∣∣∣∣∣∣

i∑

j=1

(1− Lj,m)

∣∣∣∣∣∣> iγ for some 1 ≤ i ≤ l

≤ P

∣∣∣∣∣∣

i∑

j=1

(1− Lj,m)

∣∣∣∣∣∣6= 0 for some 1 ≤ i ≤ l

= o(m−1/2).

Therefore, to prove the proposition, it suffices to show that for every k for whichm1/4−ε k mα for some α ∈ (0, 1) and ε ∈ (0, 1/4),

P

∣∣∣∣∣∣

i∑

j=l

(1− Lj,m)

∣∣∣∣∣∣> iγ for some l ≤ i ≤ k

= o(m−1/2),

where e.g. l = m(1/4−ε)/2. Write π1 = 1, and

πi =|Em(i− 2)|m− i+ 2

, 2 ≤ i ≤ m+ 1,

a random variable representing the probability that edge ei−1 is in the giant. LetBer(π) denote a Bernoulli distributed random variable with success probability π,and note that

Li,m =

(1

πi+

Yi−1,m

|Em(i− 2)|

)Ber(πi) ≥ 0. (5.4.17)

In view of Theorem 5.2.3, we observe that πi is likely to be

πi ≥m− i+ 2− (i− 2)α

m− i+ 2≥ 1− iα−1.

More precisely, let E = πi = 1 ∀i < l, πi ≥ 1− iα−1, ∀l ≤ i ≤ k. Then,

P

∣∣∣∣∣∣

i∑

j=1

(1− Lj,m)


≤ P

∣∣∣∣∣∣

i∑

j=1

(1− Lj,m)


∣∣∣∣ E

+ P(Ec),

where due to Theorem 5.2.3, it holds that P(Ec) = o(m−1/2). Next, we show thatthe summed probabilities have an exponentially decaying tail. Define the stoppingtime

σi = sup j ∈ N : j ≤ i, Yi,m ≥ 0 .

5


We remark that σi ≥ l. Due to (5.4.17), it holds for every l ≤ i ≤ k,

P(Yi,m < −iγ

∣∣∣∣E)≤

i−1∑

r=l

P

i∑

j=1

Lj,m > i+ iγ ;σi = r

∣∣∣∣E

≤i−1∑

r=l

P

−Yr,m +

i∑

j=r+1

1

πiBer(πi) > (i− r) + iγ ;σi = t

∣∣∣∣E

(1 + o(1))

≤i−1∑

r=l

P

i∑

j=r+1

1

πiBer(πi) > (i− r) + iγ

∣∣∣∣E

(1 + o(1)).

Applying Chernoff’s bound, we obtain for every t > 0

P

i∑

j=r+1

1


∣∣∣∣E

≤ e−t(i−r+iγ)E

exp

t

i∑

j=r+1

1

πjBer(πj)

∣∣∣∣E

.

Although the random variables π1, ..., πi are not independent, they are conditionedto be close to one and satisfy a Markovian property. Let p = 1 − iα−1, and notethat the conditional event E implies that πj ≥ p for all 1 ≤ j ≤ i. Define Fi asthe filtration generated by removing i edges uniformly at random. Applying thelaw of total expectation and noting that 1 + x(et/x − 1) is a (strictly) decreasingfunction for all t > 0, we observe that

E

exp

t

i∑

j=1

1

πjBer(πj)

∣∣∣∣E

= E

E

exp

t

i∑

j=1

1

πjBer(πj)

∣∣∣∣Fi−1; E

= E

exp

t

i−1∑

j=1

1

πjBer(πj)

E

[1 + πi(e

t/πi − 1)

∣∣∣∣Fi−1; E]

≤(

1 + p(et/p − 1))E

exp

t

i−1∑

j=1

1

πjBer(πj)

∣∣∣∣E

.

Applying the same argument recursively yields the bound

P

i∑

j=r+1

1


∣∣∣∣E

≤ e−t(i−r+iγ)

(1 + p(et/p − 1)

)i−r

for every t ≥ 0. We point out that the right-hand side corresponds exactly to theChernoff bound that would have been obtained if we consider the sum of i− rBernoulli distributed independent random variables with parameter p. It follows

5


that

P

i∑

j=r+1

1


∣∣∣∣E

≤ exp

− i2γ

2(i− r)p(1− p)

= exp

−1

2i2γ−α

(1 + o(1)).

We conclude that

P(Yi,m < −iγ

∣∣∣∣E)≤ i exp

−1

2i2γ−α

(1 + o(1)).

On the other hand, we can use analogous arguments to bound P(Yi,m > iγ

∣∣E).

This would yield

P(Yi,m > iγ

∣∣∣∣E)≤ i exp

−1

4i2γ−α

(1 + o(1)).

We conclude that

P

∣∣∣∣∣∣

i∑

j=l

(1− Lj,m)


≤k∑

i=l

P

∣∣∣∣∣∣

i∑

j=1

(1− Lj,m)

∣∣∣∣∣∣> iγ

∣∣∣∣E

+ o(m−1/2)

≤k∑

i=l

2i exp

−1

8i2γ−α

+ o(m−1/2) = o(m−1/2).

The tail behavior of τ follows directly by combining Proposition 5.4.3 and Corol-lary 5.4.1.

Corollary 5.4.2. If k = o(mα) for some α ∈ (0, 1), then as m→∞,

P (τm > k) ∼ P (T1−θ > k) ∼ 2θ√2πk−1/2.

Proposition 5.2.1 follows from Corollary 5.4.2 if we show that conditioning on theevent that Sm+1,m = Sm+1 = 0 does not change the tail of the stopping time τm.Indeed, this turns out to be the case.

Proof of Proposition 5.2.1. We show that asymptotically the behavior of the condi-tioned stopping time τm|Sm+1 = 0 is determined solely by what happens for the

5


increments until time k. Note that by Proposition 5.4.2

P(An,d ≥ κ(k)

)∼ P

Si ≥ 1− θ +

i∑

j=1

(1− Lj,m) , i = 1, ..., k

∣∣∣∣Sm+1 = 0

= P(τm > k

∣∣Sm+1 = 0).

Fix ε ∈ (0, 1). We bound the probability terms both from above and below, andshow that these bounds asymptotically behave the same as ε ↓ 0. Denote by fi(·)the density of the random walk Si at time i ≥ 1. Since the density of the randomwalk is bounded, we point out that it holds that [93]

limm→∞

supx∈R

∣∣√mfm(√mx)− φ(x)

∣∣ = 0, (5.4.18)

where φ(·) denotes the standard normal density function. Note that

P(τm > k

∣∣Sm+1 = 0)

= P(τm > k;Sk ≤ ε

√m∣∣Sm+1 = 0

)

+ P(τm > k;Sk > ε

√m∣∣Sm+1 = 0

).

For the first term, we observe

P(τm > k;Sk ≤ ε

√m∣∣Sm+1 = 0

)

=1

fm+1(0)

∫ ε√k

−∞P (τm > k;Sk ∈ du) fm+1−k(−u)

≤ 1

fm+1(0)P (τm > k) sup

u∈[1−θ+∑kj=1(1−Lj,m),ε

√k]

fm+1−k(−u).

Due to (5.4.18),

fm+1(0) =(1 + o(1))√

2πm

and

supx∈R

fi(√ix) ≤ 1 + o(1)√

2πi

as i→∞. This yields the upper bound

lim supm→∞

P(τm > k;Sk ≤ ε

√m∣∣Sm+1 = 0

)

P (τm > k)≤ lim sup

m→∞

√2πm√

2π(m+ 1− k)= 1.

For the second term, we show it is negligible. Note that

P(τm > k;Sk > ε

√m∣∣Sm+1 = 0

)≤ P (Sk > ε

√m)

fm+1(0)

= (1 + o(1))√

2πmP(Sk > ε

√m).

5


Applying Chernoff’s bound, it holds for every t ≥ 0,

P(Sk > ε

√m)≤ exp

−tε√m+ kt+ k log

(1

1 + t

).

In particular, this holds for t = ε√m/(k − ε√m) > 0 (for m large enough). Using

this choice of t and applying series expansions, we derive

P(Sk > ε

√m)≤ exp

−(ε2m

k+ ε√m

)1

1− ε√m/k + k log

(1− ε

√m

k

)

= exp−ε

2m+ ε√mk

k

(1 +

ε√m

k+O

(mk2

))− ε√m− ε2m

2k−O

(m3/2

k2

)

= exp−ε

2m

k+ o(1)

.

Due to Corollary 5.4.2,

P (τm > k) = Θ(k−1/2

),

and henceP(τm > k;Sk > ε

√m∣∣Sm+1 = 0

)

P (τm > k)= O

(√km exp

−ε

2m

k+ o(1)

)= o(1).

We conclude the upper bound

lim supm→∞

P(τm > k

∣∣Sm+1 = 0)

P (τm > k)≤ 1.

For a lower bound, we observe

P(τm > k

∣∣Sm+1 = 0)≥ P

(τm > k;Sk ≤ ε

√m∣∣Sm+1 = 0

)

≥ (1 + o(1))√

2πmP (τm > k) infu∈[1−θ+∑k

j=1(1−Lj,m),ε√k]fm+1−k(−u).

Due to Proposition 5.4.3, it holds with probability 1− o(m−1/2) that∣∣∣∣k∑

j=1

(1− Lj,m)

∣∣∣∣ = o(√k).

Combining this observation with (5.4.18) yields

infu∈[1−θ+∑k

j=1(1−Lj,m),ε√k]fm+1−k(−u) = (1 + o(1))

1√2πm

e−ε2

2 .

We conclude that

lim infm→∞

P(τm > k

∣∣Sm+1 = 0)

P (τm > k)≥ e− ε

2

2 .

As ε ↓ 0, the lower bound tends to one as well. We conclude that as m→∞,

P(τm > k

∣∣Sm+1 = 0)∼ P (τm > k) ∼ 2θ√

2πk−1/2

due to Corollary 5.4.2.

5


5.4.4 Proof of main result

As is laid out in the proof strategy described in Section 5.2.3, it only remains tobe shown that the stopping times υ(k) and %(k), as defined in (5.2.5) and (5.2.6)respectively, are close to k. This follows from the extremely likely events thatonly a few components disconnect from the giant, and that such components arerelatively small. In particular, it is likely that υ(k) = k − o(k) and %(k) = k + o(k).

Lemma 5.4.3. Suppose k = o(mα) for some α ∈ (0, 1). Then,

P (υ(k) ≤ k − kα) = o(m−1/2).

Proof. This statement is a consequence of Theorem 5.2.3. Recall that by definition,

υ(k) + |Em(υ(k))| ≥ k,

and υ(k) ≤ k. Moreover,

|Em(υ(k))| = m− υ(k)− |Em(υ(k))| ≤ max1≤j≤k

m− j − |Em(j)|

.

It follows that

P(|Em(υ(k))| ≥ kα

)≤ P

(max

1≤j≤k

m− j − |Em(j)|

≥ kα

)

≤ P(

max1≤j≤k

m− j − |Em(j)|

≥ jα

)= o(m−1/2)

by Theorem 5.2.3. We conclude that

P (υ(k) ≤ k − kα) ≤ P(|Em(υ(k))| ≥ kα

)= o(m−1/2).

Lemma 5.4.4. Suppose k = o(mα) for some α ∈ (0, 1). Then,

P(%(k) > k + k

α+12

)= o(m−1/2).

Proof. Again, this is a consequence of Theorem 5.2.3. We note that for every1 ≤ l ≤ m,

κ(l) =

s∑

i=1

Ber(πi+1),

where

πi =|Em(i− 2)|m− i+ 2

5


is a random variable. Due to Theorem 5.2.3,

P(πi ≤ 1− iα

m− i+ 2for some 2 ≤ i ≤ k + 1

)= o(m−1/2).

Since for every α ∈ (0, 1), the function iα−1/(m− i+ 1) is an increasing functionin i, it follows that

P(πi ≤ 1− kα−1 for some 2 ≤ i ≤ k + 1

)= o(m−1/2).

We derive the bound

P (%(k) > k + kα) = P(κ(k + k(α+1)/2

)≤ k

)≤ P

k+k(α+1)/2∑

i=1

Ber(πi+1) ≤ k

≤ P

k+k(α+1)/2∑

i=1

Ber(kα−1) > k(α+1)/2

≤ exp

−1

2k(1−α)/2(1 + o(1))

= o(m−1/2),

where the last inequality is due to the Chernoff bound.

Using these lemmas, we can prove our main result.

Proof of Theorem 5.1.1. Using the proof strategy as laid out in Section 5.2.3, wehave the upper bound

P (An,d ≥ k) ≤ P(An,d ≥ κ(υ(k))

)

≤ P(An,d ≥ κ(υ(k))

∣∣ υ(k) ≥ k − kα)

+ P (υ(k) ≤ k − kα) ,

where, due to Proposition 5.2.1,

P(An,d ≥ κ(υ(k))

∣∣ υ(k) ≥ k − kα)≤ P

(An,d ≥ κ(k − kα)

)

∼ 2θ√2π

(k − kα)−1/2 ∼ 2θ√2πk−1/2.

Due to Lemma 5.4.3,

P (υ(k) ≤ k − kα) = o(m−1/2) = o(k−1/2).

For the lower bound, we observe

P (An,d ≥ k) ≥ P(An,d ≥ κ(%(k))

)

≥ P(An,d ≥ κ(%(k))

∣∣ %(k) ≤ k + k(α+1)/2)P(%(k) ≤ k + k(α+1)/2

).

5


By Proposition 5.2.1,

P(An,d ≥ κ(%(k))

∣∣ %(k) ≤ k + k(α+1)/2)≥ P

(An,d ≥ κ

(k + k(α+1)/2

))

∼ 2θ√2π

(k + k(α+1)/2

)−1/2

∼ 2θ√2πk−1/2,

and due to Lemma 5.4.4,

P(%(k) ≤ k + k(α+1)/2

)= 1− o(m−1/2).

5.5 Universality principle

Theorem 5.2.2 described the tail behavior of the failure size in case of sublinearthresholds. In this section, we consider whether the scale-free behavior prevailsif the threshold is of linear size, i.e. the threshold is of the same order as thenumber of vertices/edges. We conjecture that the scale-free behavior prevails upto a critical point. Also, we provide an explanation why the scale-free behaviorcan extend to a wide class of other graphs as well. We stress that the argumentsthat are provided in this section are intuitive in nature, and rigorous proofs of theclaims remain to be established.

The proof of Theorem 5.2.2 relies on the translation to a first-passage time of therandom walk bridge Si,m over a (moving) boundary that is close to constant 1− θ.We prove that the difference Si,m − Si is small (Proposition 5.4.3) if k is sublinear,causing the random walk Si,m to be asymptotically indistinguishable from Si upto time k. That is, the random walk bridge is asymptotically indistinguishablefrom the one corresponding to the star topology. Since Si is a random walkwith independent identically distributed increments with zero mean and finitevariance, it is well-known by Donsker’s theorem that appropriately scaling therandom walk bridge Si yields convergence to a Brownian bridge. Therefore, theprobability that An,d exceeds k asymptotically behaves the same as the probabilitythat a Brownian bridge stays above zero until time k, multiplied by a constant thatrelates to the translation of the boundary to any other (constant) boundary. Werecall that in case of Si =

∑ij=1(1− Expj(1)) and a boundary (close to) 1− θ, this

constant is given by θ.

When the threshold k := km is of the same order as m, this analysis does notfollow through. The (maximum) difference between the two random walks upto time k is likely to become of order Θ(

√k) = Θ(

√m), an order of magnitude

that affects the asymptotic behavior. Moreover, the number of edges outside thegiant is also no longer likely to be of size o(m), and hence we need to understandwhat the failure behavior typically is in these components as well. The naturalquestion that comes to mind is whether the scale-free behavior in the failure sizetail prevails or not for linear-sized thresholds. Next, we argue heuristically whythis type of behavior prevails up to a certain critical point.

If we remove k = βm edges uniformly at random, where β ∈ (0, qc), then there isa non-vanishing proportion of edges outside the largest (giant) component [81].

5

5.5. Universality principle 165

Nevertheless, the sizes of the components outside the giant are relatively small, i.e.the number of edges in such a component is at most OP(logm). It turns out thatthis causes it to be likely that whenever a component detaches from the giant, thecascade stops in the small component. More specifically, suppose that the edgewith the i’th smallest surplus capacity is contained in the giant upon failure, andcauses a component to detach a component from the giant. Due to the way that thetotal load surge is defined, we conjecture that it is likely that the total load surge isclose to i/m+ Θ(1/

√m) upon failure. Since the number of edges in the smaller

component is likely to be O(logm), all the surplus capacities of these edges arelikely to be at least i/m + Ω(1/ logm), which is much larger than the total loadsurge. That is, all edges are likely to have sufficient capacity to deal with the load,and no more failures occur in the smaller detached component. Moreover, even ifthe cascade continues, it would contribute at most a logarithmic number of edgesto the total failure size. These observations lead to the claim that the dominantcontribution in the failure size comes from the number of edges that are containedin the giant upon failure.

To track the failure behavior in the giant component, one can use the sequence ofrandom walks Si,m. That is, the translation of the failure size to the first-passagetime of the random walk bridge over the constant boundary 1− θ remains (likelyto be) true if k ≤ βm with β ∈ (0, qc). Although the random walk is no longerclose to Si, the increments of Si,m do have zero mean and a variance that is likelyto be finite (and non-constant), and hence satisfy a martingale property. Therefore,we conjecture that the probability that the failure size in the giant componentexceeds κ(k) behaves the same as the probability that a Brownian bridge (withnon-constant variance) stays above zero until time k, multiplied by a constant.

Since we argued that it should hold that An,d ≈ An,d, this leads to the followingconclusion. Write k = αm with α ∈ (0, 1). In view of (5.3.49), we observethat for all i := im sufficiently smaller than the critical point qc, it holds thatκ(i) ≈ m

∫ i/m0

ξd(q)dq. Write

βα := minx∈(0,1)

∫ x

0

ξd(q)dq = α

. (5.5.1)

Then %(k) ≈ βαm, since

κ(%(k)) = k = αm ≈ κ(βαm).

Therefore,

P(An,d ≥ k) ∼ P(An,d ≥ k) ∼ P(An,d ≥ κ(βαm)).

Summarizing, we have the following conjecture for CMn(d, q). Suppose k = αmwith α ∈ (0, 1) such that βα < qc is satisfied. Then,

P(An,d ≥ k) ∼ f(α)k−1/2.

To support this conjecture, we performed a Monte-Carlo simulation experiment.In particular, we tested the conjecture against the erased configuration model.

5


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

k/m

k1/2P(A

G≥k)

Simulation n = 500Simulation n = 1000Simulation n = 1500Simulation n = 2000

2θ/√2π√

(m− k)/m

Figure 5.2: Erased configuration model.

That is, we create this graph according to the configuration model mechanismwith a prescribed degree sequence. We merge multiple edges and erase self-loops.Moreover, after sampling such a graph, we remove any smaller componentssuch that the final graph is a simple connected graph. Due to the properties ofthe configuration model, the number of self-loops and multiple edges is verysmall (if any), and only a finite number of vertices and edges are not containedin the giant. As a result, this graph and the configuration model conditionedon connectivity are indistinguishable asymptotically, and will lead to the sameasymptotic result for the number of edge failures. In our simulations we choosen ∈ 500, 1000, 1500, 2000, and a degree sequence with n1 = dn1/3e vertices ofdegree one, n2 = n3 = n/2 − dn1/3e and n4 = dn1/3e. Therefore, the number ofedges is (close to) m = 5/4n. The results are displayed in Figure 5.2. Indeed, itappears that our conjecture holds in this case.

Not only do we believe that this conjecture holds for the connected configurationmodel, we argue that the scale-free behavior may hold for a wider range of graphs.In particular, the relevant properties of the configuration model that we used inthe analysis are the following. First, it is likely that no (significant) disconnectionsoccur at the beginning of the cascading failure process. For example, in the case ofa configuration model, we showed that the first disconnection is likely to occurafter Θ(

√m) edge failures. Secondly, whenever the cascading failure process

causes disconnections to occur, a giant component appears and disconnectionsonly create relatively small components. It is well-known that this property issatisfied up to a certain critical threshold qc for the configuration model, but thisholds in fact for many more types of random graphs. In other words, for our resultto prevail in other graph topologies, the graph should satisfy the following twoproperties:

• The first disconnection (if any) is only likely to occur after Ω(mδ) failures,where δ > 0;

5


• There exists a critical parameter qc such that if q < qc, the largest componentof the percolated graph is unique w.h.p. and contains a non-vanishingproportion of the vertices and edges. Moreover, all other components arelikely to be relatively small for q < qc, e.g. the second largest componentcontains at most OP((logm)c) number of edges for some c <∞.

Whenever a graph G = (V,E) satisfies these two properties, we conjecture thatthe number of edge failures AG exhibits scale-free behavior. That is, for a range ofthresholds k := km, it holds that

P (AG ≥ k) ∼ fG(α)k−1/2, (5.5.2)

where fG(·) > 0 and α = limm→∞ k/m. In particular, fG(0) = 2θ/√

2π. Thefunction fG depends on the specifics of the graph, such as average degree andmore detailed connectivity properties. The range of values for k for which (5.5.2)holds depends on the critical threshold qc and the typical number of edges thatare outside the giant component.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

k/m

k1/2P(A

G≥k)

Simulation n = 500

Simulation n = 1000

Simulation n = 1500

Simulation n = 2000

2θ/√2π√

(m− k)/m

Figure 5.3: d√ne × d√ne square lattice graph.

To test conjecture (5.5.2), we first consider a d√ne × d√ne square lattice graphwhere the opposite boundaries are not connected with n ∈ 500, 1000, 1500, 2000.On the square lattice it is known that there is a phase transition for the existenceof a giant component when qc = 1/2 [67]. Moreover, significant disconnectionsoccur only after quite a significant number of failures have occurred. Indeed, todisconnect one edge e (not on the boundary) from the giant we need to removeat least six edges (the ones that share an end-vertex with e). This suggests thatsince there are roughly 2n edges in the graph, the first time the process producesan edge disconnected from the giant should be of order Θ(n5/6). Moreover, itis known that in this regime the second largest component in a box of volumen is polylogarithmic in n [68]. Thus, it satisfies the conditions we conjecturefor a graph to be in the same mean-field universality class as the configurationmodel. In Figure 5.3, we observe that indeed the first significant disconnections

5


happen after a much longer time than in CMn(d), and the limiting function fork1/2P (ALattice ≥ k) remains very close to the setting where no disconnection takesplace for a relatively long time.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

k/mER

k1/2P(A

G≥k)

Simulation n = 500Simulation n = 1000Simulation n = 1500Simulation n = 2000

2θ/√2π√(mER − k)/mER

Figure 5.4: Giant component of the Erdös-Rényi random graph withedge retention probabability 2/n, where mER = λ(1− η2

λ)n/2.

Secondly, we consider the giant component of the Erdös-Rényi random graph.That is, for every pair of the n vertices, there exists an edge with probability λ/n,and we consider the cascading failure process on the giant component. The giantis uniquely defined asymptotically: it is well-known that for every λ > 1 thereexists a unique giant component Cmax for which holds that [58],

|v : v ∈ Cmax|n

P−→ ζλ,

where ζλ = 1− ηλ > 0 and ηλ satisfies the fixed-point equation

ηλ = E(η

Pois(λ)λ

)= 1− e−ληλ .

In our simulation experiment, we choose n ∈ 500, 1000, 1500, 2000 possiblevertices, and λ = 2. Therefore, the graph on which we perform the cascadingfailure process is likely to have around ζλn vertices and λ(1− η2

λ)n/2 edges (withΘ(√n) fluctuations). From the definition of the Erdös-Rényi random graph it is

clear that if we run a percolation process on it, the resulting graph is again anErdös-Rényi random graph, but with a smaller λ. It is known that for every λ > 1all the components outside the giant are at most of size Θ(log n). Therefore, thedisintegration of the network is similar to the one of the configuration model, yetwith the critical edge-removal probability corresponding to qc = (λ− 1)/λ in thiscase.

However, the first disconnection is likely to occur after finitely many edge failures,since the number of vertices with degree one in the giant of an Erdös Rényi graphis likely to be of order Θ(n). In other words, this graph violates the condition that

5


the first disconnections should occur after Ω(mδ) edge failures for some δ > 0.Nevertheless, it appears from our simulation result that (5.5.2) still prevails, seeFigure 5.4. In other words, the condition on no early disconnections can possiblybe relaxed. The analysis would require significant changes, and particularly, theresults on the distribution of first-passage times over moving boundaries such asLemmas 5.4.1 and 5.4.2, as the load surge is no longer almost deterministic at thebeginning of the process.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

k/m

k1/2P(A

G≥k)

Simulation n = 250

Simulation n = 500

Simulation n = 750

Simulation n = 1000

2θ/√2π√

(m− k)/m

Figure 5.5: Graph G = CS(n, 4).

In contrast, we consider a graph G = CS(n, 4) consisting of n star-componentswith 4 edges each, connected by a single path of edges connecting all the com-ponents. This graph thus consists of 5n vertices and also m = 5n edges. Weobserve that as soon as one of the n edges on the single path fails, the remaininggraph is a (connected) tree. Therefore, with high probability, the graph woulddisconnect in two components both of order Θ(m) after removing only a fixednumber of edges. This effect is likely to occur again in both components whenedges are removed uniformly at random in these components. This violates bothproperties we needed to prove our result for the configuration model. We observein Figure 5.5 that indeed k1/2P (ACS ≥ k) does not seem to converge to a singlefunction as n→∞.

6

Chapter 6

Relation between city populationsizes and blackout sizes

Based on:Emergence of scale-free blackout sizes in power grids

T. Nesti, F. Sloothaak, and B. Zwart

As mentioned in the introduction of this thesis, the models in Chapters 2-5 relyon the premise that power outages are governed by critical phenomena. In thischapter, we take a fundamentally different perspective by hypothesizing that thescale-free nature of city sizes drives the scale-free behavior of the blackout sizes.

6.1 Introduction

The emergence of scale-free phenomena in complex networks has received alot of attention in the last two decades [12, 52, 91, 99]. Typically, the scale-freenature relates to the topology of the network, and particularly, the nodal degreedistribution. In Chapter 5, we introduced a mathematical framework that resul-ted in scale-free behavior for the failure size, while the network itself does not(necessarily) possess heavy-tailed properties. In this chapter, we introduce a com-pletely different framework that provides a relatively simple explanation for theemergence of scale-free blackout sizes in power grids.

Not only does the number of affected customers during a blackout exhibit scale-free behavior [24, 26, 38, 57], this type of behavior also arises for city populationsizes [13, 103]. Remarkably, there appears to be a strong similarity betweenthe exponents of power-law distributed city populations and the number ofaffected customers during a blackout. To the best of our knowledge, this strikingconnection has not been explored yet in the literature. To corroborate this claimwith historical data, see Figure 6.1. More specifically, we hypothesize that both thenumber of affected customers during a blackout, as well as the population size,

171

6

172 Chapter 6. Relation between city population sizes and blackout sizes

100 102 104 106

x

10−4

10−3

10−2

10−1

100

P(X

>x) City sizes:

α = 1.37 ± 0.08

xmin = (52.46 ± 11.88) × 103

Blackout sizes:α = 1.31 ± 0.08

xmin = (140 ± 34.03) × 103

Figure 6.1: Tail indices US population sizes and blackout sizes.

has a Pareto (power-law) tail, i.e. as x→∞,

P (X > x) ∼(

x

xmin

)−α= Kx−α, (6.1.1)

for some exponent α > 0 and constant K = xminα > 0. We employ the PLFIT

method [31] to estimate α and xmin from blackout data [121] and city populationsizes [31]. Indeed, the exponents α of the Pareto tails are remarkably close.

To provide an explanation for this relation, we develop a novel framework thataccounts for the physical laws in power systems. We model the power grid as aconnected graph G = G(N ,L), where the set of nodes N represents the n buses inthe system, and the set of edges L corresponds to the m transmission lines. Weconsider a static setting where each city inhabitant requires one unit of energy,and we assume that the size of a randomly chosen city X has a Pareto tail withexponent α > 0. We adopt the DC approximation to model the power flows inthe grid, which is briefly described in the introduction of this thesis. We provide amore detailed description of our framework in Section 6.2.

In this chapter, we are interested in the amount of power that cannot be deliveredto the customers after the cascade, also referred to the total amount of load that isshed. This event occurs whenever the power grid becomes disconnected through-out the cascade. This is why grid operators always aim for the power network tobe m− 1 reliable, i.e. the failure of any single line does not lead to instability/dis-connections in the power grid. Our goal is to show that whenever the power gridis not m− 1 reliable, then there exists a constant C > 0 such that

P (total load shed > x) ∼ Cx−α (6.1.2)

as x→∞.

This chapter is structured as follows. We describe the DC approximation andthe model framework in detail in Section 6.2, and we describe our main result inSection 6.3. In Section 6.4, we show that whenever there is major power outage,

6


the only likely setting it occurred is when there is a single city with a large powerdemand. In Sections 6.5 and 6.6, we derive the consequences in this special case,e.g. the power generations and the cascade sequence. We conclude that thisleads to scale-free behavior in the blackout size, formally stated in our main result(Theorem 6.7.1) in Section 6.7. To illustrate how to apply the main result to a fixedtopology, we consider a specific network topology in Section 6.8. We point outthat we exclude certain network topologies from our framework for technicalconvenience. In Section 6.9, we discuss how our results can be generalized to holdfor all network topologies.


In our framework, we combine large deviations theory with a well-establishedpower flow model, commonly used in the power engineering community. Wetherefore adopt some conventions in our notation from the physics/power engin-eering literature. For example, we write G = G(N ,L) as the graph representingthe power grid, and we make symbols bold whenever they represent a vector or amatrix. Moreover, we write ei for the unit vector where the ith entry is one, and edenotes the all-one vector. Finally, J denotes the all-one matrix.

The framework we consider in this chapter makes use of the DC approximation,which is briefly described in the introduction of this thesis. Before moving toa detailed model description, we first discuss the DC approximation and itsproperties in more detail.

6.2.1 Preliminaries: DC approximation

We describe the DC power flow model. Let g,d ∈ Rn represent general nodalgeneration and load vectors, respectively, and let p = g − d be the net powerinjections vector. The DC approximation models the relationship between activepower injections p ∈ Rn and active line power flows f ∈ Rm, and is commonlyused in high-voltage transmission system analysis [111]. It can be representedthrough the linear mapping

f = V(g − d), (6.2.1)

where the matrix V ∈ Rm×n is known as the Power Transfer Distribution Factors(PTDF) matrix and is constructed as follows.

PTDF matrix

Choosing an arbitrary but fixed orientation of the transmission lines, the networkstructure is described by the edge-vertex incidence matrix C ∈ Rm×n defined as

C`,i =

1 if ` = (i, j),

−1 if ` = (j, i),

0 otherwise.

6


Denote by β` > 0 the weight of edge ` ∈ L, corresponding to the susceptance of thattransmission line. Note that β` = x−1

` , where x` is the reactance of line `. Denoteby B the m×m diagonal matrix defined as B = diag(β1, . . . , βm). The networktopology and weights are simultaneously encoded in the weighted Laplacian matrixof the graph G, defined as L = C>BC or entry-wise as

Li,j =

−βi,j if i 6= j,∑k 6=j βi,k if i = j.

All the rows of L sum up to zero and thus the matrix L is singular.

Let L+ ∈ Rn×n denote the Moore-Penrose pseudo-inverse of matrix L. That is,for real-valued matrices, the Moore-Penrose pseudo-inverse L+ of a matrix L isdefined as a matrix that satisfies the following four conditions:

1. LL+L = L ,

2. L+LL+ = L+ ,

3. (LL+)> = LL+ ,

4. (L+L)> = L+L,

where L> denotes the transpose of matrix L. It is well-known that the pseudo-inverse exists and is unique [92].

According to the DC approximation, the relation between any zero-sum vectorof power injections p ∈ Rn and the phase angles θ ∈ Rn can be written in matrixform as p = Lθ. Using the Moore-Penrose pseudo-inverse, we can rewrite this as

θ = L+p. (6.2.2)

The line power flows f are related to the phase angles θ via the linear relationf = BCθ. In view of (6.2.2), the line power flows f can be written as a lineartransformation of the power injections p, i.e.

f = Vp, (6.2.3)

where V := BCL+ is the PTDF matrix.

The following lemma is based on a well-known result in graph theory (e.g. see [80]).

Lemma 6.2.1. If G is a connected graph, rk (V) = rk (C) = rk (L) = rk (L+) = n− 1,and the null space of V is the one-dimensional subspace generated by e = (1, . . . , 1) ∈ Rn,i.e.

Ker (V) = Ker (C) = Ker (L) = Ker (L+) =< e > .

The following lemmas are technical results which will be needed later in thederivation of our results.

6


Lemma 6.2.2. Changing the orientation of a subset of lines L′ ⊂ L has the effect ofswapping the sign of the corresponding rows of the PTDF matrix V. In particular, it isalways possible to choose the orientation such that Ve1 ≥ 0.

Proof. Changing the orientation of a line from lk = (i, j) to lk = (j, i), by definition,amounts to swapping the sign of the k-th row of matrix C, yielding a modifiedmatrix C = I(k)C, where I(k) is a diagonal matrix with I

(k)ii = 1 if i 6= k and

I(k)kk = −1. Since L = C>C = C>I(k)I(k)C = C>C = L, the matrices L and L+

are not affected by the change. As a consequence, the modified PTDF matrixV = BCL+ = I(k)V differs from V only by the swapped signs on the k-throw.

Lemma 6.2.3. Let G be assigned the orientation such that the set of edges incident tonode 1 is L1 = (1, j) | j is adjacent to 1, i.e. C`,1 = 1 = −C`,j for all ` = (1, j) ∈ L1.Then, V`,1 ≥ 0 for every ` ∈ L1. The converse is also true.

Proof. First, note that the largest element in each row of L+ is its diagonal entry(Corollary 1 in [122]), i.e. L+

1,1 − L+1,j ≥ 0 for every ` = (1, j) ∈ L1. For any

line ` = (1, j) ∈ L1, we have V`,1 = (CL+)`,1 = C`,1L+1,1 + C`,jL

+1,j , where

C`,1 = −C`,j = ±1 depending on the orientation of line `. Thus, V`,1 ≥ 0 if andonly if C`,1 = 1 = −C`,j .

Optimal Power Flow

The Optimal Power Flow (OPF) program is an optimization problem that determ-ines the generation schedule minimizing the total system generation cost whilesatisfying demand/supply balance and network physical constraints [14]. In itsfull generality, the OPF is a nonlinear, nonconvex optimization problem. For thepurpose of this chapter, we focus on a tractable approximation based on the DCpower flow equations referred to as DC-OPF, which can be formulated as thefollowing optimization problem:

ming∈Rn

n∑

i=1

Ci(gi) (6.2.4)

s.t.n∑

i=1

gi =

n∑

i=1

di, (6.2.5)

gi≤ gi ≤ gi, i ∈ N , (6.2.6)

−f` ≤ V`(g − d) ≤ f`, ` ∈ L, (6.2.7)

The function Ci(·) : R → R denotes the costs of generation at node i, g, g ∈ Rndenote the vector of nodal minimum and maximum generation capacities, re-spectively, and f ∈ Rm denotes the vector of line limits. We assume that Ci(·)is an increasing quadratic function, i.e. we assume Ci(gi) = g2

i /2, i = 1, ..., n.Finally, we assume that generator limits do not impose an effective constraint, i.e.gi

= −∞, gi =∞, i = 1, ..., n.

6


Power flow redistribution

Suppose that a subset of transmission lines L′ ⊂ L fail, and causes no discon-nections. Then, the power flows will redistribute among the remaining linesaccording to power flow physics. The way the power flows redistribute is gov-erned by the new PTDF matrix V, which can be constructed analogously to V,mapping the (unchanged) power injections to the new power flows. We assumethat the redistribution occurs instantaneously, without any transient effects.

As an illustration, we show how the redistributed power flows can be calculatedin the special case of an isolated failure L′ = `. In this case, it is enough tocalculate the vector φ(`) ∈ Rm−1 of redistribution coefficients, known as line outagedistribution factors. The quantity φ(`)

j takes values in [−1, 1], and |φ(`)j | represents

the percentage of power flowing in line ` that is redirected to line j after the failureof the former. In particular, the new power flow configuration after the failure ofline ` = (i, j), denoted by f (`) ∈ Rm−1, is given by

f(`)k = fk + f

(`)` φ

(`)k , ∀` 6= k, (6.2.8)

where, for k = (a, b) and ` = (i, j), the coefficient φk,` ∈ R can be computed as

φk,` = φ(i,j),(a,b) = β−1` ·

Ra,j −Ra,i +Rb,i −Rb,j2(1− x−1

i,jRi,j), (6.2.9)

where Ri,j is the effective resistance between nodes i and j, given by

Ri,j = (ei − ej)TL+(ei − ej) = (L+)i,i + (L+)j,j − 2(L+)i,j . (6.2.10)

6.2.2 Our framework

In view of the DC-OPF, in order to obtain a fundamental understanding of thecausal relation between blackout sizes and city sizes using the DC approximationmodel, we require a framework that adequately sets the power demands, thetransmission line limits, generation limits, and the cost function for any fixedtopology G = (N ,L). In addition, we need to specify a mechanism that causesthe initial line failure, as well as which lines possibly fail next after the powerflow redistribution. For this purpose, we consider a framework that consists ofthree problems: the planning problem, the operational problem, and the emergencyproblem.

The planning problem

The planning problem refers to how the generation limits, the line limits and powerdemand are determined with respect to the city sizes X1, ..., Xn. We assumethat each node represents a city with a size of Xi inhabitants, and for ease ofpresentation, we consider a framework with a static setting where each inhabitantdemands one unit of energy, i.e. di = Xi for every i = 1, ..., n. The city sizesare independently distributed, and have a Pareto-tail with exponent α > 0, i.e.given by (6.1.1). We stress that we write X when we specifically assume that

6


the demands at the cities have a Pareto tail, and d if we discuss results for ageneral/normalized/other demand vector in the DC approximation framework.

Recall that we assume that the cost function is an increasing quadratic functionand that generator limits do not pose an effective constraint in the DC-OPF. Theline limits are set as a fraction of the absolute power flow in a setting where alsothe line limits pose no effective constraint. More specifically, in the absence of anygenerator and transmission line limits, the solution of the DC-OPF is

g∗ =1

n

n∑

i=1

Xie.

The associated flow vector is given by

f∗ = V(g∗ −X) = −VX,

where we used that Vg∗ = 0, see Lemma 6.2.1. For a safety tuning parameterλ ∈ [0, 1], the operational line limits are set as

fj = λ |(VX)j |, j = 1, ...,m. (6.2.11)

The operational problem

In the operational problem, we solve the DC-OPF for an increasing quadratic costfunction and line limits as in (6.2.11) to obtain the generation vector g. That is, wesolve

ming∈Rn

n∑

i=1

g2i /2 (6.2.12)

s.t.n∑

i=1

gi =

n∑

i=1

Xi, (6.2.13)

VX− λ |VX| ≤ Vg ≤VX + λ |VX| , (6.2.14)

where |VX| denotes the vector with elements (|VX|)j = |(VX)j |, j = 1, ...,m.

The emergency problem

In the emergency problem, we focus on the failure process after an initial disturb-ance. We assume that the initial failure is caused by a single line failure, chosenuniformly at random over all lines. We point out that our framework can beextended to multiple initial line failures, or adapted to deal with generator failures.The initial failure may cause a cascading effect that leads to multiple line failuresthat disintegrate the network. A consecutive line failure occurs whenever there isat least one line such that its emergency line limit is exceeded. That is, instead ofconsidering the conservatively chosen operational line limits f`, we take the linelimits to be

F` = λ∗f`, ` = 1, ...,m,

6


for some constant λ∗ > 1. A canonical choice is λ∗ = 1/λ. We assume that linefailures occur sequentially, and occurs at the line where its relative exceedance islargest.

Whenever line failures cause the network to disconnect in multiple islands, weassume that the energy balance is restored by proportionally lowering eithergeneration or demand at all nodes. Naturally, this alters the line power flows.More specifically, before the initial disturbance occurs, the network flows are givenby V(g −X), where g is the solution of the DC-OPF in the operational problem.After any line failure, we check whether this causes the network to disconnect, andif so, we proportionally lower the generation in one component and the demandin the other component such that demand and generation are balanced in the twodisconnected components. The network flows are updated according to the lawsof physics in every component. That is, the removal of one or more lines yields amodified matrix V, and possibly modified generation g and demand d. The lineflows are given by V(g − d). This failure process continues until the line limits F`of all surviving lines are sufficient to carry the power flows.

6.3 Main result and road map of the proof

Whenever the network disintegrates in multiple components, we alter the genera-tion and demand to restore the power balance in every component. The decreaseof the power demand reflects the event that customers are affected: the total powerdemand cannot be met and load is shed. Our object of interest in this chapter isthe total amount of load that is shed during the cascade. More specifically, wewant to show that (under some technical conditions) this quantity has a Paretotail with exponent α > 0, see (6.1.2).

In order to show (6.1.2), we need to understand which scenarios are the mostlikely ones to be in whenever there is a large-scale blackout. It turns out that thedominant scenarios are those where there is a single city that has a large powerdemand, while the demands of other cities are relatively negligible. This is acrucial observation: in order to understand how large blackouts occur, it sufficesto grasp the settings where Xi > 0 for a city i, and Xj < εXi for j 6= i for someε > 0 small. Consequently, (almost) all power generation needs to be transmittedto city i, and the total amount of load that is shed is the generation that cannotreach city i after the cascade. That is, writeA1 for the set of nodes in the componentthat contains the city with the largest demand after the cascade has taken place.We point out that the set A1 is a random variable, and in particular, A1 = 1, ..., nif the cascade stops without causing network disconnections. Then,

total shed load ≈ S :=

∣∣∣∣∣∣∑

i 6∈A1

(gi −Xi)

∣∣∣∣∣∣=

∣∣∣∣∣∑

i∈A1

(Xi − gi)∣∣∣∣∣ .

In other words, the total amount of shed load is approximately equal to themismatch between generation and demand in the component containing the citywith highest power demand.

6

6.3. Main result and road map of the proof 179

In view of the DC approximation framework, we observe that we can normalizeall quantities, i.e. demands, line limits, power generation and power flows, bythe sum of the demands. In particular, if the city i has the largest power demand,then this gives rise to a setting where the normalized demand vector is given byd ≈ ei. In this special case, the DC-OPF can be solved explicitly: we will showthat gj = λ/n all j 6= i and gi = 1− λ(n− 1)/n. In turn, the emergency problemalso has an explicit solution. That is, conditioning on the position of the first linefailure, the sequence of failures that takes place is deterministic. Under sometechnical conditions, we show that the cascade sequence under demand vectord = X, satisfying Xj < εXi for j 6= i for some ε > 0, converges to the cascadesequence under demand vector d = ei.

These observations lead to the following results. For Pareto tails [101], it holdsthat

P

(n∑

i=1

Xi > x

)∼ nP(Xi > x) ∼ nKx−α.

Therefore,

P(S > x) =

n∑

j=1

P (S > x | |A1| = j)P (|A1| = j)

∼n−1∑

j=1

P

((n− j)λ

n

n∑

i=1

Xi > x | |A1| = j

)P (|A1| = j) ∼ Cx−α,

where

C = nK

n−1∑

j=1

P (|A1| = j)λα(1− j/n)α ∈ [0,∞).

In particular, if the network topology is not m − 1 reliable under unit demandvectors, then C > 0, and we obtain a Pareto tail for the amount of load that is shed.In summary, we would obtain the following result.

Theorem 6.3.1. Suppose there is a fixed topology G = (N ,L) and a fixed λ ∈ (0, 1).Moreover, suppose that some technical conditions are satisfied, which will be specified inTheorem 6.7.1. If G is not m− 1 reliable under unit demand vectors, then there exists aconstant C ∈ (0,∞) such that

P(S > x) ∼ Cx−α (6.3.1)

as x→∞. Otherwise, as x→∞,

P(S > x) = O(x−2α). (6.3.2)

6


6.4 Principle of a single city with large demand

A vital property in our framework is that whenever there is a major blackout,then the dominant scenario that constitutes such an event is when there is asingle city that has a large power demand. To formalize this notion, write X1 =maxX1, ..., Xn with Xi, i = 1, ..., n, independent and identically Pareto-taileddistributed power demands. That is, label the vertices such that the city withlargest demand has city label 1. Note that for every ε > 0,

P (S > x) = P

(S > x;

n∑

i=2

Xi < εX1

)+ P

(S > x;

n∑

i=2

Xi ≥ εX1

). (6.4.1)

It turns out that we can show that (in certain settings) the first term on the right-hand side has a Pareto tail, and the second term is negligible. More specifically,the following result can be shown.

Lemma 6.4.1. Suppose Xi, i = 1, ..., n, are independent and identically Pareto distrib-uted with tail exponent α > 0, and write X1 = maxX1, ..., Xn. For every ε > 0, asx→∞,

P

(S > x;

n∑

i=2

Xi ≥ εX1

)= O

(x−2α

). (6.4.2)

Proof. We observe that the total mismatch can never exceed the sum of all de-mands, and hence

S ≤n∑

i=1

Xi ≤ nX1.

Therefore,

P

(S > x;Xi >

n∑

i=2

Xi ≥ εX1

)≤ P

(X1 >

x

n;Xi > ε

X1

nfor some i = 2, ..., n

)

≤ P(Xi > ε

x

n2for some i = 2, ..., n.

)

Write I(y) = |i : Xi > y|. Since for every η > 0,

P (I(ηx) ≥ 2) = O(x−2α

)

as x→∞, the result follows.

In other words, Lemma 6.4.1 implies that if for all ε > 0 sufficiently small,

P

(S > x;

n∑

i=2

Xi < εX1

)∼ Cx−α

holds for some constant C ∈ (0,∞), then in view of (6.4.1), it constitutes thedominant term.

6

6.5. Closed-form solution for the operational OPF in a special case 181

6.5 Closed-form solution for the operational OPF in a specialcase

Note that without loss of generality, we can always do some rescaling in ourframework by dividing all parameters (e.g. generation, line limits, etc.) by thesum of all power demands. This yields an equivalent setting where the total powerdemand equals one. In view of Lemma 6.4.1, it is natural to consider the specialcase where the demand vector is given by d = e1. That is, node 1, henceforthreferred to as the sink node, has unit demand, while all other nodes have zerodemand. For this special case, a closed-form solution exists for the generationvector in the operational OPF.

First, we consider the planning problem. As stated in the model description, in theabsence of any generator and transmission line limits, the solution of the planningOPF is g∗ = 1

ne. The associated flow vector is given by f∗ = V(g∗ − e1) = −Ve1,where we used that Vg∗ = 0 (Lemma 6.2.1). Therefore, the operational prob-lem (6.2.4)-(6.2.7) reduces to

ming∈Rn

∑ni=1 g

2i /2

s.t. e>g = e>e1 = 1,−λ|Ve1| ≤ V(g − d) ≤ λ|Ve1|,

which we will denote by P (λ). Lemma 6.5.1 shows that the solution of P (λ) is ofclosed form.

Lemma 6.5.1. Let λ ∈ (0, 1). Let G be assigned the orientation such that Ve1 ≥ 0.Then, the solution of P (λ) is given by

g(λ) = λ1

ne + (1− λ)e1,

i.e. g1(λ) = 1− λn−1n and gi(λ) = λ 1

n for all i = 2, . . . , n. The corresponding line flowsare at capacity and are given by f(λ) = −λVe1.

Proof. First, we note that the selected orientation on G implies that the set of edgesincident to node 1 is L1 = (1, j) | j is adjacent to 1 (i.e. the edges in L1 exit node1), or, in terms of the edge-node incidence matrix C, that C`,1 = 1 = −C`,j for all` = (1, j) ∈ L1. This is proved in Lemma 6.2.3 in Section 6.2.1.

Due to the chosen orientation, f(λ) = λ |Ve1| = λVe1 and the line limit constraintsin P (λ) can be rewritten as

(1− λ)Ve1 ≤ Vg ≤ (1 + λ)Ve1.

The problem P (λ) is a strictly convex optimization problem with linear equalityand inequality constraints. Therefore, in order to show that g(λ) is the uniqueoptimal solution, it is sufficient to show that it satisfies the KKT conditions for

6


P (λ), which read

g + V>(µ+ − µ−) + γe = 0, (6.5.1)

µ+ ≥ 0,µ− ≥ 0, , (6.5.2)

µ+l (Vg − (1 + λ)Ve1)` = 0∀` ∈ L, (6.5.3)

µ−l (−Vg + (1− λ)Ve1)` = 0∀` ∈ L, (6.5.4)

e>g = 1, (6.5.5)(1− λ)Ve1 ≤ Vg ≤ (1 + λ)Ve1, (6.5.6)

where γ is the Lagrange multipliers for the equality constraint and µ+,µ− ∈ Rmare the Lagrange multipliers for the inequality constraints.

Since Vg(λ) = (1 − λ)Ve1 and e>g(λ) = e>e1 = 1, the candidate solutiong(λ) clearly satisfies the feasibility conditions (6.5.5) and (6.5.6), and also thecomplementary slackness condition (6.5.4). Moreover, condition (6.5.3) is satisfiedif we choose µ+ = 0.

Using the facts that Vg(λ) = (1 − λ)Ve1 and Ker(V) =< e >, pre-multiplyingequation (6.5.1) by V yields (1− λ)e1 + V>µ ∈ Ker(V). This is equivalent to

(1− λ)e1 + V>µ =(1− λ)

ne,

where in the last equality we used again the property that Ve = 0. To concludethe proof, it remains to be shown that there exist a nonnegative solution µ− ≥ 0of the matrix equation

V>(−µ−) = (1− λ)(e/n− e1). (6.5.7)

We construct a non-negative solution µ− as follows:

µ−` := (1− λ)eL1 =

(1− λ) l ∈ L1

0 l /∈ L1,,

where eL1is a m- dimensional vector containing ones in positions given by L1,

and 0 elsewhere. Invoking Lemma 6.2.3 we see that Ce1 = eL1, which yields

µ− = (1− λ)Ce1. Using the definition of V = CL+,L = C>C, and the propertyL+L = (I− J/n) (see [122]), we observe that Eq. (6.5.7) is indeed satisfied:

V>µ− = (1− λ)V>Ce1 = (1− λ)(L+L)e1 = (1− λ)(e/n− e1).

Setting γ = −1/n completes the proof.

6.6 Convergence of the cascade sequence

In this section, we consider the emergency problem. In the special case that d = e1,we observe that whenever there is a network disconnection, the component thatdoes not contain the sink node has no power demand, and hence the generation at

6

6.6. Convergence of the cascade sequence 183

every node in that component is reduced by λ/n to zero. Evidently, no consecutivefailures occur in this component. On the other hand, the demand at the sink nodeis reduced by the number of nodes that disconnect from this component timesλ/n. Therefore, the total amount of load that is shed is exactly equal to the totalamount of reduced power generation at the sink node (power imbalance), whichis given by

S =∑

i 6∈A1

λ

n=λ(n− |A1|)

n. (6.6.1)

Naturally, the way the failure process cascades through the network after the initialdisturbance is highly dependent on the network topology. The redistribution ofpower flow takes place as described in Sections 6.2.1 and 6.2, and we stress that thisis a deterministic process. In this special case, the only sources of randomness comefrom the choice of the initial line failure, and possibly the choice of subsequent linefailure whenever the redistribution of power flow causes the relative exceedanceto be the same at multiple lines. Therefore, given a network topology and the linethat initially fails, we can determine exactly how the failure process propagatesthrough the network.

It may be apparent from Lemma 6.4.1 that the special case where d = e1 describessome form of limiting behavior. That is, as Lemma 6.4.1 holds for every ε > 0,we observe that the normalized demand vector d converges to the unit vector e1

as ε ↓ 0. Next, we show that for almost all values of λ, for all demand vectors dfor which d→ e1 as ε ↓ 0, the order at which line failures occur converges to thesequence of line failures as if the demand vector would have been d = e1.

Note that the operational OPF (6.2.4)-(6.2.7) with no generation limits is given by

ming∈Rn

1

2g>g (6.6.2)

s.t. e>g = e>d, (6.6.3)|V(g − d)| ≤ λ |Vd| . (6.6.4)

It is a strictly convex optimization problem, and since g = d is a feasible point, thefeasible set of this optimization problem is nonempty. Therefore, for each demandvector d, there exists a unique optimal solution g∗(d).

If we view d as a parameter of the problem, then (6.6.2)-(6.6.4) is an instance ofa multi-parametric quadratic programming (mp-QP) problem with a strictly convexobjective function, for which it is known that the optimal solution g∗(d) is acontinuous function of the parameter vector d [120, Theorem 1]. This continuityproperty will be used extensively in this section.

We assume in our framework that line failures occur sequentially, i.e. a next linefailure occurs at the line where the line limit is relatively most exceeded. Recallthat Fj denotes the emergency line limit of line j ∈ L, and is given by

Fj = λ∗λ|(Vd)j | = |(Vd)j |,

6


where we take the canonical choice λ∗ = 1λ . We write f (m)

j for the flow on line jafter the failure of the first m − 1 lines and after the load/generation sheddingtook place, where we use the convention that f (1)

j denotes the flow on line j when

no initial disturbance has occurred yet, and f(m)j = 0 if line j has already failed

before the m-th step of the cascading failure process. The cascade is initiated bythe random failure of line ` = `(1). The m-th line to fail, for m ≥ 2, is given by

`(m) = arg maxj∈A(m)

|f (m)j | − FjFj

= arg max

j∈A(m)

|f (m)j |Fj

, (6.6.5)

where A(m) = j : |f (m)j | ≥ Fj is the set of lines that exceed the limit.

Remark 6.6.1. Note that the line limits and line flows depend on d and λ through theoperational OPF, so that the sequence of subsequent failures depends on d, λ, and on theinitial failure ` = `(1). That is,

Fj = Fj(d), f(m)j = f

(m)j (d, λ),A(m) = A(m)(`, λ,d), `(m) = `(m)(`, λ,d).

For the sake of exposition, we do not write the dependency on d, λ and `.

Let C = `(1), . . . , `(T ) be a cascade sequence, where `(T ) is the last failure beforethe cascade stops. Such a sequence is uniquely determined by the first failure `(1)

and by the demand vector d and by λ, i.e. C = C(d, λ, `). In view of Lemma 6.4.1and the normalization property, the goal of this section is to show that if d→ e1,then the cascade sequence does not depend on d anymore, i.e.

C(d, λ, `) = C(e1, λ, `) if d→ e1.

We observe that if |A(m)| = 0, no more line failures occur. Technically, it is alsopossible that |A(m)| > 1 and hence the subsequent line failure next needs to bechosen out of a set of multiple lines. We exclude the settings that do not yieldunique maximizers from our framework.

Assumption 6.6.1. For all lines j, the ratios between redistributed flows and line limits

|f (m)j (e1)|Fj(e1)

=λ|(V(m)e1)j ||(Ve1)j |

are all different for all m ≥ 2, where V(m) denotes the PTDF matrix for the remainingnetwork after m− 1 failures have taken place.

Assumption 6.6.1 is needed to ensure the uniqueness of the maximizer in (6.6.5).It ensures that whenever the first line failure ` and the parameter λ is known, thecascade sequence is unique and deterministic for demand vector d = e1. We pointout that this assumption implies that certain network topologies are excludedfrom our framework. This assumption is for technical convenience, and we stressthat our results hold more generally. In particular, our analysis follows through

6


for network topologies with some form of symmetry, and we can slightly adaptthe framework to deal with these cases as well.

More specifically, suppose that |A(m)| > 1 for some m ∈ N and the set A(m)

consists only of lines that are indistinguishable from one another (lines that are‘symmetric’). Since nodal demands are independent and identically distributed,this implies that each of these lines has an equal probability of being the line thatfails next. By the symmetry of the network topology, regardless of which line ischosen to fail next, the resulting networks after the cascade are indistinguishable.We illustrate this notion for a symmetric 6-node topology in Section 6.8.

To analyze the power imbalance in this framework, we need to introduce somenotation as well as formally define the shedding rule and the redistribution ofpower flows.

Definition 6.6.1 (Uniform shedding rule). Let g(1) = g∗,d(1) = d be the initialgeneration and demand vectors. Assume that the removal of lines `(1), . . . , `(m), m ≥ 1,disconnects the network in components G(m)

i = (N (m)i ,L(m)

i ), i = 1, . . . , hm. Define thepower imbalance in component G(m)

i as

YG(m)i

=∑

k∈N (m)i

(g(m)k − d(m)

k ).

In order to re-achieve power balance, generation and demand in each component aremodified iteratively according to the following uniform shedding rule

d(m+1)k =

(1−

YG(m)i∑

l∈N(m)i

d(m)l

)d

(m)k if YG(m)

i< 0 (shed load)

d(m)k if YG(m)

i≥ 0

, k ∈ N (m)i ,

g(m+1)k =

g(m)k if YG(m)

i< 0(

1−YG(m)i∑

l∈N(m)i

g(m)l

)g

(m)k if YG(m)

i≥ 0 (shed gener.), k ∈ N (m)

i .

Definition 6.6.2 (Power flow redistribution). Suppose that the lines `(1), . . . , `(m),m ≥ 1, are removed, and disconnects the network in components G(m)

i = (N (m)i ,L(m)

i ),i = 1, . . . , hm. Then, the line flows in component G(m)

i are given by

f(m+1)Li = V(m+1,Gi)(g(m+1)

Ni − d(m+1)Ni ),

where V(m+1,Gi) is the PTDF matrix for the subgraph G(m)i , and g(m+1)

Ni , d(m+1)Ni are

defined as in Definition 6.6.1.

A second assumption we require to show the convergence of the cascade sequenceinvolves the following.

6


Assumption 6.6.2. For all m ≥ 2,

maxj

|f (m)j (e1)|Fj(e1)

6= 1.

That is, for d = e1 it is not possible for the candidate line to fail next has a flow |f (m)j |

that is exactly equal to its limit. In terms of PTDF matrices and λ, this assumption reads

maxj

λ|(V(m,Gi)e1)j ||(Ve1)j |

6= 1, m ≥ 2.

Assumption 6.6.2 means that we exclude finitely many λ-s from our analysis,which correspond to phase transitions. In order to prove the convergence of thecascade sequence, we also need a continuity property of the line flows at everystage with respect to the demand vector.

Lemma 6.6.1 (Continuity of f (m)j with respect to d). At each stage m of the cascade,

the redistributed power flows f (m)j are continuous in the initial demand vector d for all

j = 1, ...,m.

Proof. Assume that the removal of lines `(1), . . . , `(m), m ≥ 1, disconnects thenetwork in components G(m)

i = (N (m)i ,L(m)

i ), i = 1, . . . , hm. According to Defini-tion 6.6.2,

f(m+1)Li = V(m+1,Gi)(g(m+1)

Ni − d(m+1)Ni ),

for each connected component G(m)i , so f (m+1) is continuous in g(m+1),d(m+1).

Moreover, according to Definition 6.6.1, g(m+1),d(m+1) are continuous functionsof g(m),d(m). By expanding the recursion, and using that g∗(d) is continuous in d,we see that f (m+1) is continuous in d.

Finally, we can show the main result of this section.

Proposition 6.6.1. Assume that Assumptions 6.6.1 and 6.6.2 hold, and let C(d, λ, `) =`(1), . . . , `(T ) be a cascade sequence initiated by ` = `(1). Then, there exists an ε > 0such that

d1 = 1, dj < ε, ∀j ≥ 2 =⇒ C(d, λ, `) = C(e1, λ, `).

Proof. Let `(1) be the first failure, and consider

`(2) = arg maxj∈A(2)

|f (2)j |Fj

,

where A(2) = A(2)(d, λ) = j : |f (2)j | ≥ Fj. It follows from Lemma 6.6.1 that

f(2)j (d)→ f

(2)j (e1) as d→ e1, so by continuity and Assumption 6.6.2

|f (2)j (d)| − Fj(d)→ |f (2)

j (e1)| − Fj(e1) 6= 0.

6


Consequently, there exists an ε > 0 such that, if dj < ε for all j ≥ 2, then

|f (2)j (d)| > Fj(d) ⇐⇒ |f (2)

j (e1)| > Fj(e1).

In other words, a line limit is exceeded for d = e1 (which, due to our assumption,implies that it is strictly exceeded) if and only if it is also (strictly) exceeded whend is close enough to e1, implying that A(2)(d) = A(2)(e1) .

Moreover, there exists an ε(1) ≤ ε such that, if dk < ε for k ≥ 2, then

maxj∈A(2)(d1,λ)

|f (2)j (d, λ)|Fj(d)

= maxj∈A(2)(e1,λ)

|f (2)j (d, λ)|Fj(d)

= maxj∈A(2)(e1,λ)

|f (2)j (e1, λ)|Fj(e1)

,

where in the second equality we used that A(2)(d) = A(2)(e1), and in the thirdequality we used again continuity. Finally, Assumption 6.6.1 allows us to concludethat the max is unique and that the (unique) second failure `(2)(d, λ) = `(2)(e1, λ)does not depend on d if dk < ε(1), k ≥ 2.

As Lemma 6.6.1 holds for every stage of the cascade, we can repeat the steps aboveto construct a sequence ε(T ) ≤ . . . , ε(2) ≤ ε(1) such that the cascade sequence C iswell-defined and does not depend on d if dj < ε(T ) for all j ≥ 2.

Example 6.6.1. To illustrate how one can easily derive the phase transition values, weconsider the 4-node cycle topology. With the standard clock-wise orientation, we have

V(clock) =1

8

3 −3 −1 11 3 −3 −1−1 1 3 −3−3 −1 1 3

.

For d1 d2, d3, d4, we can change the orientation such that Vd ≥ 0, which is given bythe edge-list (1, 2), (2, 3), (4, 3), (1, 4). Then the matrix V reads

V =1

8

3 −3 −1 11 3 −3 −11 −1 −3 33 1 −1 −3

,

In this case,

φk,` = −sk =

−1 if k ∈ (1, 2), (2, 3)1 if k ∈ (4, 3), (1, 4) .

Assume that the first failure is ` = (1, 2), so that the power flow redistribution is

λ ((Ve1)k + φk,`(Ve1)`) = λ ·

1/4 if k = (2, 3)

−1/2 if k = (4, 3)

−3/4 if k = (1, 4).

Then, the critical values of λ are given by

λ = ± (Ve1)k(Ve1)k + φk,`(Ve1)`

, k 6= `, (6.6.6)

6


and we find that they are λ = 14 ,

12 . Moreover, if λ < 1

4 , then the cascade stops immediatelyafter the failure of the first line. If 1

4 < λ < 12 , then line (4, 3) fails afterwards and if

λ > 12 lines (2, 3), (4, 3), (1, 4) fail afterwards. Therefore, λ = 1

4 ,12 can be seen as phase

transition points.

6.7 Asymptotic behavior of power imbalance

In the previous sections, we showed that if there is a large blackout, then the onlylikely scenario it occurred is when there is a single city that has a significantlylarger demand than all other cities. Under technical conditions, given the positionof the city with largest demand (i.e. labeling this as city 1) and the first line failure,the cascade sequence is deterministic and identical to the one as if the demandvector would have been d = e1. We exploit these properties to derive the tailbehavior of S, or equivalently, the amount of load that is shed/the number ofaffected customers.

We point out that the demands are independent and identically distributed, sothe probability that a city has the largest demand equals 1/n. To obtain the tailbehavior of S, we need that Assumptions 6.6.1 and 6.6.2 to hold regardless ofwhich city has the largest power demand.

Assumption 6.7.1. Assumptions 6.6.1 and 6.6.2 hold for any relabeling of the vertices.

Note that since the number of cities n is finite, and inherently also the number ofthe possible lines where the first failure occurs, Assumption 6.7.1 excludes only afinite number of possible values of λ from our framework. Next, we specify themain result of this chapter, i.e. we recall Theorem 6.3.1 and specify the technicalconditions.

Theorem 6.7.1. Suppose there is a fixed topology G = (N ,L) and a fixed λ ∈ (0, 1), forwhich Assumption 6.7.1 holds. Write Z(i, `), i = 1, ..., n, ` = 1, ...,m, as the numberof cities that are not in the same component as city i after the cascade under demandvector d = ei and first line failure `. If there exists a (i, `) such that Z(i, `) = 0 for somei ∈ 1, ..., n and ` ∈ 1, ...,m, then as x→∞,

P(S > x) ∼ Cx−α, (6.7.1)

where

C =

n∑

i=1

m∑

`=1

K

m

(Z(i, `)

λ

n

)α∈ (0,∞).

Otherwise, as x→∞,

P(S > x) = O(x−2α). (6.7.2)

Proof. First, since the demands are independent and identically distributed, weobserve that each city has an equal probability of being the city with the largestdemand. That is, if B denotes the city that has the largest demand, then

P(B = i) = 1/n, i = 1, ..., n.

6

6.7. Asymptotic behavior of power imbalance 189

By the law of total probability,

P (S > x) =

n∑

i=1

1

nP(S > x

∣∣B = i).

Fix some ε > 0 (sufficiently small), and note that for all i = 1, ...n,

P(S > x

∣∣B = i)

= P

S > x;

n∑

j 6=iXj < εXi

∣∣B = i

+ P

S > x;

n∑

j 6=iXj ≥ εXi

∣∣B = i

.

Due to Lemma 6.4.1, we observe that the second term is of order O(x−2α) for alli = 1, ..., n, and hence

n∑

i=1

1

nP

S > x;

n∑

j 6=iXj ≥ εXi

∣∣B = i

= O(x−2α).

For the first term, note that Assumption 6.7.1 ensures that Z(i, `) is well-definedfor all i = 1, ..., n and ` = 1, ...,m. Since we choose our first failure uniformly atrandom among all lines, we observe that by the law of total probability, it holdsthat

P

S > x;

n∑

j 6=iXj < εXi

∣∣B = i

=

m∑

l=1

1

mP

S > x;

n∑

j 6=iXj < εXi

∣∣ `(1) = l, B = i

for all i = 1, ..., n.

In case that Z(i, `) = 0 for all i = 1, ..., n and ` = 1, ...,m, it follows from Propos-ition 6.6.1 that for all ε > 0 sufficiently small, the cascade sequence causes nodisconnections for every city with largest demand and first line failure `(1). Thatis, for all i = 1, ..., n, l = 1, ...,m, x > 0 and ε > 0 sufficiently small,

P

S > x;

n∑

j 6=iXj < εXi

∣∣ `(1) = l, B = i

= 0.

Therefore, if Z(i, `) = 0 for all i = 1, ..., n and ` = 1, ...,m, then for all ε > 0sufficiently small,

n∑

i=1

1

nP

S > x;

n∑

j 6=iXj < εXi

∣∣B = i

= 0,

6


and we conclude that (6.7.2) holds.

Next, suppose that Z(i, `) 6= 0 for at least some i ∈ 1, ..., n and ` ∈ 1, ...,m.It follows from Proposition 6.6.1 and the normalization observation that for alli ∈ 1, ..., n and ` ∈ 1, ...,m for which Z(i, `) 6= 0, it holds for all ε > 0sufficiently small that the cascade sequence is the same as the one when thedemand vector would have been d = ei. In particular, whenever

∑nj 6=iXj < εXi,

it holds for all ε > 0 sufficiently small that the set A1 is deterministic and is thesame set of nodes as if demand would have been d = ei, and the number ofcities disconnected from city i equals Z(i, `). Recall Lemma 6.5.1 and the propertythat the generator vector g is a continuous function of d. Consequently, for alli ∈ 1, ..., n and ` ∈ 1, ...,m for which holds that Z(i, `) 6= 0, and for all ε > 0sufficiently small,

P

S > x,

n∑

j 6=iXj < εXi

∣∣ `(1) = `, B = i

= P

∑

i6∈A1

(gi −Xi) > x,

n∑

j 6=iXj < εXi

∣∣ `(1) = `, B = i

≤ P

Z(i, `)

(λ

n+ c1(ε)

)Xi > x,

n∑

j 6=iXj < εXi

∣∣ `(1) = `, B = i

,

where c1(ε) is a strictly positive function with c1(ε)→ 0 as ε ↓ 0. For independentand identically Pareto-distributed random variables X1, ..., Xn, it holds that asx→∞, [101]

P (maxX1, ..., Xn ≥ x) ∼ nP(Xi > x) = nKx−α.

Therefore, for all i ∈ 1, ..., n and ` ∈ 1, ...,m for which Z(i, `) 6= 0,

limε↓0

limx→∞

xαP

S > x,

n∑

j 6=iXj < εXi

∣∣ `(1) = `, B = i

≤ limε↓0

n

(Z(i, `)

(λ

n+ c1(ε)

))α= nK

(Z(i, `)

λ

n

)α.

Similarly, we can obtain the same lower bound, i.e.

limε↓0

limx→∞

xαP

S > x,

n∑

j 6=iXj < εXi

∣∣ `(1) = `, B = i

≥ nK

(Z(i, `)

λ

n

)α.

We conclude that as x→∞,

P (S > x) =

n∑

i=1

m∑

`=1

K

m

(Z(i, `)

λ

n

)αx−α.

Note that term in front of x−α is a double sum of finitely many terms, and hencewe can also conclude that (6.7.1) holds.

6

6.8. Cascade analysis for 6-node topology 191

6.8 Cascade analysis for 6-node topology

524

524

524

524

124

124

124

124

(a) Case A: the city with the largestdemand is one of two cities thatconnects to four other cities.

512

112

112

112

512

112

112

112

(b) Case B: the city with the largest de-mand is one of four cities that connectsto two other cities.

Figure 6.2: Line limits in the 6-node example.

To illustrate how to derive the asymptotic behavior of the amount of load that isshed using our framework, we consider a network topology that consists of sixnodes and eight lines, see Figure 6.2. It follows from our results that in order tounderstand the behavior for large blackouts, it suffices to consider the behaviorunder unit demand vectors. Due to the highly symmetric structure of the networktopology in this example, there are only two relevant options for the position ofthe city with the highest demand, as is illustrated in Figure 6.2. The red noderepresents the city that has unit demand (largest), while the other nodes have zerodemand. Note that cases A and B occur with probability 1/3 and 2/3, respectively.In each case, one can solve the operational problem to determine the emergencyline limits, which are also depicted in Figure 6.2. We illustrate how the cascadingfailure processes evolve in these cases next.

6.8.1 Case A

The first line failure is chosen uniformly at random among all eight lines. Again,due to the symmetries of the network topology, we only need to consider twopossibilities: when the first failure occurs at a top line, or when it occurs at one ofthe bottom lines. Figure 6.3 illustrates the possible cascade when the initial failureis at a top line. The subsequent line failure occurs at the line for which the ratioof flow and line limit is largest, and therefore these values are depicted next toall (remaining) lines in Figure 6.3. Only if the maximum ratio is strictly largerthan one, another line fails. If the ratio is less than one, no consecutive line failureoccurs, and if it equals one, then it corresponds to a phase transition case.

We observe that after the initial failure at a top line, the ratio of flow and line limitis highest at the corresponding bottom line. The ratio is 4λ, and hence λ = 1/4is the first phase transition value. If λ < 1/4 the cascade ends immediately,otherwise this line fails. In step 2 (which is relevant for all value of λ > 1/4), weobserve that the bottom lines all have the same maximum ratio of 4/3λ, violating

6


43λ 4

3λ 4

3λ

4λ 83λ 8

3λ 8

3λ

(a) Initial failure.

1615

λ 1615

λ 1615

λ

43λ 4

3λ 4

3λ

(b) Step 2.

45λ 6

5λ 6

5λ

2λ 2λ

(c) Step 3.

45λ 4

5λ 8

5λ

4λ

(d) Step 4.

45λ 4

5λ 4

5λ

(e) Step 5.

Figure 6.3: First line failure at a top line.

Assumption 6.6.1. Yet, demands are independent and identically distributed,and the symmetric structure of this network topology ensures that each of theremaining bottom lines have equal probability to fail next. This also illustrateswhy Assumption 6.6.1 is rather restrictive, and why our framework can still dealwith network topologies that have these types of symmetries as well. Regardlessof the actual line that is chosen to fail next, the networks that appear in the nextstages of the cascade are indistinguishable from one another. The second stepalso yields the second phase transition value: if 1/4 < λ < 3/4, the cascade stops,and if λ > 3/4 another bottom line fails. In the latter case, also steps 3 and 4 are

6

6.8. Cascade analysis for 6-node topology 193

observed, where the network turns stable at step 5.

45λ 16

15λ 16

15λ 16

15λ

43λ 4

3λ 4

3λ

(a) First steps.

45λ 4

5λ 4

5λ 4

5λ

(b) Network after cascade.

Figure 6.4: First line failure at a bottom line.

If the first line failure occurs at a bottom line, then the corresponding top lineis stable, and a possible subsequent failure occurs at one of the three remainingbottom lines. Again, consecutive line failures occur when λ > 3/4, and the cascadestops if λ < 3/4. Using an analogous analysis as in the previous case, we wouldend up with a network where all bottom lines have failed, and all top lines areintact when λ > 3/4. An illustration of this case is given in Figure 6.4.

6.8.2 Case B

We can perform a similar analysis in this case. Again, due to the symmetries, thereare only two truly different possibilities for the first line failure to occur: one ofthe two top lines, or one of the six other lines in Figure 6.2(b).

In Figure 6.5 we illustrate the possible cascades if the initial failure occurs at oneof the two top lines. In this case, the cascade only continues if λ > 3/8, and stopsif λ < 3/8. That is, we observe another phase transition value, i.e. λ = 3/8. Ifλ > 3/8, then after the initial line failure three more line failure occurs, which afterthe cascade stops. In Figure 6.6 we illustrate the possible cascades if the initialfailure occurs at one of the six bottom lines. Then, there is only a possible secondline failure when λ > 1/2, after which the cascade stops. If λ < 1/2, the cascadestops immediately after the initial line failure. Consequently, we obtain λ = 1/2as a fourth phase transition value.

6.8.3 Tail behavior of blackout size

To derive the tail behavior of the total amount of load shed, we need to determinethe number of combinations that leads to j cities disconnected from the city withlargest demand j = 0, 1, ..., n − 1. That is, we count the number of tuples (i, `)such that Z(i, `) = j, j = 0, 1, ..., n − 1, where i denotes the city label and ` thefirst line failure. Since the network consists of six nodes and eight lines, thereare a total of 48 possible tuples. It follows from the previous sections that inthis example, there are four phase transition values of λ, namely 1/4, 3/8, 1/2

6


23λ

23λ

23λ

2λ

83λ

83λ

83λ


λ

λ

2λ

2λ

4λ

4λ

(b) Step 2.

6λ

2λ

2λ

2λ

8λ

(c) Step 3.

0

0

0

25λ

(d) Step 4.

Figure 6.5: First line failure at a top line.

1415

λ

43λ

43λ

1615

λ

23λ

23λ

2λ


45λ

λ

λ

45λ

λ

λ

(b) Step 2.

Figure 6.6: First line failure at a bottom line.

and 3/4. Therefore we need to distinguish between five possible intervals of λ. InTable 6.1 we provide an overview, which follows directly from the derivations inthe previous sections. A direct consequence is the following corollary.

Corollary 6.8.1. Consider the 6-node network topology. If λ ∈ (0, 1/4), then as x→∞,

P(S > x) ∼ O(x−2α).

6

6.9. Discussion 195

#(i, `) : Z(i, `) = j j = 0 j = 1 j = 2 j = 3 j = 4 j = 50 < λ < 1/4 48 0 0 0 0 01/4 < λ < 3/8 40 8 0 0 0 03/8 < λ < 1/2 32 8 0 0 8 01/2 < λ < 3/4 8 32 0 0 8 03/4 < λ < 1 0 32 8 0 8 0

Table 6.1: Overview of number of tuples that lead to the disconnectionof j cities from the city with largest demand.

Otherwise, as x→∞,

P(S > x) ∼ C(λ)Kx−α,

where

C(λ) =

(λ/6)α if 1/4 < λ < 3/8,

(λ/6)α

+ (2λ/3)α if 3/8 < λ < 1/2,

4 (λ/6)α

+ (2λ/3)α if 1/2 < λ < 3/4,

4 (λ/6)α

+ (λ/3)α

+ (2λ/3)α if 3/4 < λ < 1.

6.9 Discussion

In Theorem 6.7.1, we assume that Assumptions 6.6.1 and 6.6.2 hold for any re-labeling of vertices. These assumptions exclude certain network topologies andfinitely many values of λ (the phase transition values) from our framework. Al-though we illustrate in Section 6.8 that particularly Assumption 6.6.1 can oftenstill be dealt with, we point out that we believe that a completely different type ofanalysis can overcome both technical assumptions. In this section, we provide adirection that one may take to develop an alternative proof.

First, we need to determine the optimal solution of the DC-OPF in the pre-limit.We point out that the DC-OPF is a strictly convex optimization problem, and itsfeasible set is nonempty (since d is a feasible point). Therefore, for each demandvector d, there exists a unique optimal solution g := g∗(d). We write the feasibleset of possible generation vectors as

Fd =g ∈ Rn : e>g = e>d, |V(g − d)| ≤ λ |Vd|

. (6.9.1)

First, we point out that minimizing the objection function over the feasible are Fd

is equivalent to minimizing the Euclidean distance between de 6∈ Fd and a pointin the feasible area Fd.

Lemma 6.9.1. Vector g∗ is the optimal solution of the problem

ming∈Rn

∥∥g − de∥∥

2(6.9.2)

s.t. e>g = e>d, (6.9.3)|V(g − d)| ≤ λ |Vd| , (6.9.4)

if and only if g∗ is the optimal solution of the OPF.

6


Proof. Observe that for all g ∈ Fd,

‖g − de‖22 =

n∑

i=1

(gi − d)2 =

n∑

i=1

g2i − 2d

n∑

i=1

gi +

n∑

i=1

d2 =

n∑

i=1

g2i − nd2,

where the final equality follows since e>g = e>d for all g ∈ Fd. Since d is aconstant, and taking the root is a concave function, we conclude that the minimumof ‖g − de‖2 is attained at the same point as the minimum of g>g/2 over thefeasible area Fd.

Next, we take a closer look at the feasible area Fd. We observe that for everyg ∈ Fd,

(1− λ)(Vd)j ≤ (Vg)j ≤ (1 + λ)(Vd)j , if (Vd)j ≥ 0, (6.9.5)(1 + λ)(Vd)j ≤ (Vg)j ≤ (1− λ)(Vd)j , if (Vd)j < 0 (6.9.6)

for all j = 1, ...,m. In other words, the feasible are is described by a set of2m parallel hyperplanes. Moreover, the optimal solution needs to lie on thehyperplane Hd := g ∈ Rn : e>g = e>d due to (6.6.3). Since rk(V) = n − 1and e ∈ Ker(V), the feasible set Fd therefore describes a convex polytope in thespaceHd. Consequently, the optimal solution to the DC-OPF is the point on the(boundary of) the convex polytope Fd ∈ Hd that is closest to the point de ∈ Hd.

We stress that the arguments that follow in this section are heuristic. In view ofLemma 6.4.1 and the normalization property, we only need to consider demandvectors d = (1, d2, ...., dn) such that

∑ni=2 di < ε for ε ≥ 0 small. Conditioning on

the ratio between the demands d2, ..., dn, one would be able to write the DC-OPFsolution as an affine function of the demands.

Conjecture 6.9.1. Suppose it holds that

(X2, ..., Xn)∑ni=2Xn

= (γ2, ..., γn),

n∑

i=2

γi = 1,

and consider the scaled demand vector d = (1, γ2ε, ..., γnε). Then, there exist constantsaij := a(γ2, ..., γn)ij , i, j = 1, ..., n, such that for all ε > 0 sufficiently small, the optimalsolution of the DC-OPF with demand vector d is given by

g∗i =

n∑

j=1

aijdj , i = 1, ..., n.

Conjecture 6.9.1 may be proved using the geometric interpretation, exploiting thenotion that the optimal solution to the DC-OPF is the point on the (boundary of)the convex polytope Fd that is closest to the point de, where the limiting case (asε ↓ 0) with point 1/ne and polytope Fe1

is known.

The next step is determining the cascade sequence under the ratio for demandvector (1, γ2ε, ..., γnε)X1 for all sufficiently small ε > 0. Since the flow and line

6

6.9. Discussion 197

limits are affine functions of the demand vector, the ratio

ψ(m)j (γ, ε) :=

|f (m)j ((1, γ2ε, ..., γnε)X1) |Fj ((1, γ2ε, ..., γnε)X1)

=|f (m)j (1, γ2ε, ..., γnε)|Fj(1, γ2ε, ..., γnε)

is a function in ε > 0. We point out that it may be possible that for some vectors(γ2, ..., γn), it holds for two lines j,m ∈ L that

ψ(m)j (γ, ε) = ψ

(m)l (γ, ε)

for all ε > 0. We claim that this only occurs for a negligible number of vectors(γ2, ..., γn) with respect to all possible proportions. That is, with (sufficiently) highprobability, the ratios yield a unique ordering from largest to smallest ratio for allε > 0 sufficiently small, i.e. there is an ordering (j1, ..., jm) such that

ψ(m)j1

(γ, ε) > ... > ψ(m)jm

(γ, ε)

for all ε > 0 sufficiently small. This is the way Assumption 6.6.1 is overcome: with(sufficiently) high probability the ratio between flow and line limit yield uniquemaximizers. Consequently, given the city with largest demand, the line where thefirst failure takes place, (almost all) vector(s) (γ2, ..., γn), the cascade sequence isdeterministic and uniquely defined.

The final step to prove the result is quite similar to the proof of Theorem 6.7.1. Wecondition on the position of the city i with largest demand and the line ` wherethe first failure takes place, where each possible tuple (i, `) occurs with probability1/(nm). Given these events, we observe that there are finitely many possiblecascade sequences. One can determine all possible proportions (γ2, ..., γn) thatgive rise to a particular cascade sequence. Using a Lebesgue-type of measure, itassigns a probability for every possible cascade sequence to take place. Summingover all possible tuples (i, `) yields a result similar to Theorem 6.7.1, but whereAssumption 6.7.1 is removed.

It may be clear that the type of analysis as outlined above can become computa-tionally intensive. The analysis for the main setting of this chapter only uses thefirst column of the PTDF matrix during the cascade, making the computations easy,quick and more tractable. Therefore, one should only resort to the more involvedanalysis for a tuple (i, `) at step m ≥ 2 of the cascade whenever Assumptions 6.6.1and/or 6.6.2 are violated.

7

Chapter 7

Battery swapping dynamicswithin a single facility

Based on:Complete resource pooling of a load balancing policy for a network of battery swapping stations

F. Sloothaak, J.R. Cruise, S. Shneer, M. Vlasiou, and B. ZwartPreprint avalaible on: arXiv:1902.04392

In the first part of this thesis, we focused on the question why scale-free behaviorappears in blackout sizes. This topic is relevant on the level of the high-voltagetransmission system, and we were interested in what measures can be taken toprevent a small disruption from cascading to a large-scale power outage. In theremainder of this thesis, we shift our attention to the power demand on the levelof distribution systems. In particular, we consider the effect of Electric Vehicles(EVs), a source of high variability in power demand on a local level.

7.1 Introduction

A steady energy transition is taking place due to the decarbonization of theeconomy, leading to many significant challenges and research opportunities [85].The integration of more sustainable power sources, e.g. renewables, brings aboutan increasing amount of variability in power generation, making an adequateoperation of the grid that guarantees a secure electricity delivery more difficult.On the other hand, it also becomes more important to manage increasing demandfluctuations caused by e.g. EVs. A key challenge in the deployment and take upof EVs by society is the provision of a scalable charging infrastructure.

As explained in the introduction of this thesis, the adoption of EVs has been slowinitially due to various practical challenges. Examples include high purchasecosts of an EV, battery life problems and long battery charging times [114]. Aviable solution to many of these issues is the development of a battery swappingnetwork. That is, every EV contains a battery that serves as the fuel for users to

199

7

200 Chapter 7. Battery swapping dynamics within a single facility

drive around. Whenever the battery is close to depletion, the EV user can move toa facility to swap its battery for one that is fully charged.

In recent years, there has been a growing amount of research on both the plan-ning/design as well as the operation/scheduling in battery swapping systems,see [118] for an overview. Most papers employ robust optimization techniques tofind optimal solutions for certain objectives, while only few studies focus on thequality-of-service for EV users. The exception are a collection of papers writtenby a set of authors [113–115, 117, 118], that use asymptotic analysis and MarkovDecision Process techniques to propose suitable solutions.

In this chapter, we are interested in providing adequate provisioning rules for thenumber of assets in a single battery swapping facility. This serves as a steppingstone for a network setting that accounts for occupation level correlations betweenstations. To the best of our knowledge, this effect has not been studied before inother literature in the context of battery swapping systems.

This chapter is structured as follows. We recall the model description as outlinedin the introduction of this thesis in Section 7.2. This model involves a novel closedqueueing model, and we propose a square-root provisioning rule for the numberof assets in the battery swapping facility. This rule stems from the Quality-and-Efficiency-Driven (QED) regime known from asymptotic many-server queueingtheory. In Section 7.3, we review the typical properties that the QED regime bringsabout for system performance, as well as related literature. In Section 7.4, wederive the steady-state distribution of the queue length, i.e. the total number ofbatteries that are being charged or waiting to be charged. We derive the fluidand diffusion limit of the queue length process in Section 7.5, and we show thatan interchange of limits is justified in Section 7.6. In Section 7.7, we use thelimiting queue length distribution to approximate performance measures such asthe waiting probability. We consider two related settings in Sections 7.8 and 7.9.Finally, we comment on how to establish an adequate provisioning scheme toattain a targeted waiting probability in Section 7.10.


We recall the model for EVs utilizing battery swapping technology as outlined inthe introduction of this thesis. There are a number of vehicles that always moveto the same station, leading to the conservation of batteries. This gives rise to aclosed Markovian queueing network model.

More specifically, recall that we consider the setting where there are r EV usersthat return a to battery swapping facility after an exponentially distributed timewith mean λ. The battery swapping facility has three types of resources: Bspare batteries, F charging points and G swapping systems. We assume thatthe charging of a battery takes an exponential amount of time with rate µ. Theswapping time is relatively short compared to charging, we assume it to benegligible. Moreover, we point out that due to the conservation of batteries, thetotal number of batteries in the system is equal to r +B, where the batteries areeither contained in an EV, being charged, or already fully-charged at the swappingfacility. An illustration of this model is given in Figure 7.1.

7


r +B batteries

∞

Exp(λ)

F

B

Exp(µ)

Figure 7.1: Visual representation of the battery swapping facility.

The observant reader may notice that the number of swapping servers G is notaccounted for in Figure 7.1. We point that the role of the swapping servers onlyhas an effect in the sense that it implies a bound for the number of charging points.More specifically, whenever there is an EV arrival at the facility, a swapping servertakes out the almost depleted battery and exchanges it for a fully-charged one ifavailable. As long as the exchange has not finished, the EV cannot move from theswapping server until a fully-charged battery becomes available. The EVs thatmay arrive in the meantime have to wait until the exchange has taken place, andthe batteries in those waiting EVs cannot be moved to a charging point yet. Asa consequence, having more than B + G charging points creates no additionalcharging capacity, i.e. we obtain the bound F ≤ B +G. Typically, due to the highpurchase costs, the number of swapping servers G equals one.

The main quantity of interest in this chapter is the number of batteries that are inneed of charging, which we also refer to as the queue length. Let Q(t) denote thenumber of batteries in need of charging at time t ≥ 0. We point out that from thequeue length behavior, the following three performance measures can be derived:the waiting probability of an arbitrary EV user, the expected waiting time and theresource utilization of the system. We consider these performance measures underthe square-root slack provisioning policy

Br =λr

µ+ β

√λr

µ, β ∈ R,

F r =λr

µ+ γ

√λr

µ, γ ≤ β.

(7.2.1)

We point out that γ ≤ β is a consequence of the bound F ≤ B + G, where Gis a small fixed integer. We denote by Qr(t) the queue length at time t ≥ 0 ifscaling rules (7.2.1) are applied. The idea behind the square-root provisioningpolicy stems from the Quality-and-Efficiency-Driven (QED) regime, known fromasymptotic many-server queueing theory [55].

7


7.3 QED regime

The square-root staffing provisioning policy originates back to the work of Erlanghimself in 1923 [46], and was first rigorously analyzed in the pioneering work ofHalfin and Whitt [55]. In this work, they consider the Erlang-C model as illustratedin Figure 7.2. That is, the arrival rate is denoted by λ, the service rate by µ and thenumber of servers by s.

s

Exp(µ)

Poisson(λ)

Figure 7.2: The Erlang-C model.

It is shown in [55] that the probability of waiting has a non-degenerate limit if andonly if (1− λ/(µs))√s→ β as s→∞ for some fixed parameter β > 0. This givesrise to the square-root provisioning rule

s =λ

µ+ β

√λ

µ, β > 0.

That is, to ensure a nondegenerate limit for the waiting probability, the number ofresources needs to be set as the minimally required amount λ/µ (needed to ensurestability) with an additional hedge of β

√λ/µ. Let φ and Φ denote the density and

the cumulative distribution function of a normal distribution, and let W denotethe waiting time of an arbitrary EV user. Then, the waiting probability convergesto

P(W > 0) ∼(

1 +βΦ(β)

φ(β)

)−1

,

a strictly decreasing function in β > 0 with range (0, 1). Consequently, anytargeted delay probability can be achieved by adjusting β > 0. Moreover themean delay is of order 1/

√s and hence asymptotically negligible, and the service

utilization is given by 1 − O(1/√s). In other words, the QED policy leads to a

waiting probability that tends to a value strictly between zero and one, the waitingtime vanishes, and a near-optimal resource utilization is achieved. While the QEDregime gives precise limits when the system size s goes to infinity, it is well-knownthat the asymptotic behavior kicks in quickly, so that QED limits serve as sharpapproximations already for relatively small systems [49, 64, 104, 133].

System performance under a square-root provisioning policy has been studiedfor many other systems in the last few decades. Most examples were inspiredby call centers [19, 69, 132, 133], where abandonment plays a central role. Theeffect of finite-sized features was first studied in [123], and appears to be of criticalimportance in different healthcare systems [74, 124, 131]. In this chapter, we studythe performance of a system in a novel application area, where finite-sized effectsplay a role in a completely different way.

7

7.4. Steady-state distribution 203

7.4 Steady-state distribution

In this setting, the queue length process is a simple birth-death process for whichthe steady-state distribution is easily derived by standard theory for Markovchains, irrespective of whether the QED scaled provisioning rules (7.2.1) hold.More specifically, the queue length Q(t), t ≥ 0 is a birth-death process with statespace Q(t) ∈ 0, 1, ..., B + r for all t ≥ 0, with birth rate λ(r − (Q(t)−B)

+) and

death rate µminQ(t), F. Let

π(B,F,r)k = P (Q(∞) = k)

denote the steady-state distribution of the number of batteries in need of charging.

Lemma 7.4.1. Consider a single swapping station with F charging points and B sparebatteries, where F ≤ B +G. The steady-state distribution is given by

π(B,F,r)k =

(λr/µ)k

k! π(B,F,r)0 if 0 ≤ k ≤ minB,F

(λr/µ)k

F !Fk−Fπ

(B,F,r)0 if F ≤ k ≤ B

rBr!(r+B−k)!

(λ/µ)k

k! π(B,F,r)0 if B ≤ k ≤ F

rBr!(r+B−k)!

(λ/µ)k

F !Fk−Fπ

(B,F,r)0 if maxB,F ≤ k ≤ B + r,

(7.4.1)

where

π(B,F,r)0 =

(B∑

k=0

(λr/µ)k

k!+

B−1∑

k=F+1

(λr/µ)k

F !F k−F+

B+r∑

k=B

rBr!

(r +B − k)!

(λ/µ)k

F !F k−F

)−1

(7.4.2)

if F ≤ B, and

π(B,F,r)0 =

(F∑

k=0

(λr/µ)k

k!+

F−1∑

k=B+1

rBr!

(r +B − k)!

(λ/µ)k

k!

+

B+r∑

k=B

rBr!

(r +B − k)!

(λ/µ)k

F !F k−F

)−1 (7.4.3)

if B ≤ F .

Although the steady-state distribution is an exact, closed-form expression, itprovides little qualitative insight on the queue length behavior, and in particular,the behavior of the process when it has not reached steady state yet. Therefore, weresort to fluid and diffusion limits which in practice serve as good approximationsfor moderate to large-scale systems. This allows us to provide approximationsfor the performance measures of interest, e.g. the waiting probability and theexpected waiting time.

7


7.5 Limiting queue length behavior

Due to the curse of dimensionality, it is very challenging to gain a qualitativeinsight in the (transient) behavior of processes in large-scale systems. Therefore,we consider the fluid and diffusion limits. Recall that Qr(t) corresponds to thequeue length process (the number of batteries in need of charging) under thescaling rules (7.2.1) with r cars at time t ≥ 0. We consider the fluid scaling

Qr(t) =Qr(t)

r, r ≥ 1, t ≥ 0. (7.5.1)

The fluid-scaled process converges to a deterministic, continuous monotone pro-cess with a single fixed steady-state value.

Proposition 7.5.1. Consider a single swapping station operating where the number ofresources is given by scaling rules (7.2.1). If Qr(0)→ Q(0) as r →∞ with Q(0) a finiteconstant, then Qr → Q in distribution as r →∞, where Q satisfies the ODE

dQ(t)

dt=

λ− µQ(t) if Q(t) < λ/µ,λ2/µ− λQ(t) if Q(t) ≥ λ/µ

and has the steady-state value

limt→∞

Q(t) =λ

µ.

Proposition 7.5.1 implies that the number of batteries in need of charging can beapproximated by

Qr(t) ≈ rQ(t),

where Q(t) = limr→∞ Qr(t) is a solution of an ODE. It describes the approximatetransient behavior before reaching steady state.

Proof of Proposition 7.5.1. We observe that Qr(·) is a birth-death process with statespace Qr = 0, 1, ..., Br + r, arrival (birth) rate λr(j) = λ (r − (j −Br)+) andservice (death) rate µr(j) = µminj, F r for all j ∈ Qr. The fluid-scaled processQr therefore has state space 0, 1/r, ..., (Br + r)/r, and drift and infinitesimalvariance functions

mr(x) =λr(brxc)

r− µr(brxc)

r= λ− λ(brxc −Br)+

r− µminbrxc, F r

r

and

σ2r(x) =

λr(brxc)r2

+µr(brxc)

r2= λ− λ(brxc −Br)+

r+µminbrxc, F r

r.

Taking the limit yields under scaling rule (7.2.1)

m(x) := limr→∞

mr(x) =

λ− µx if x < λ/µ,−λx+ λ2/µ if x ≥ λ/µ,

7

7.5. Limiting queue length behavior 205

and

σ2(x) := limr→∞

σ2r(x) = 0.

That is, we obtain a degenerate limiting diffusion, and hence the limiting initialpoint will yield a deterministic path for the limiting process Q. The mean directlyprovides the ODE that the limiting process Q satisfies. The limiting value (ast→∞) of the limiting process is also correct due to the following reasoning. Forall t ≥ 0 for which Q(t) < λ

µ , we observe dQ(t)/dt > 0. Hence, if Q(0) < λ/µ, thenthe limiting process follows by the deterministic path

Q(t) =λ

µ+

(Q(0)− λ

µ

)e−µt.

Similarly, for all t ≥ 0, if Q(t) > λ/µ, then dQ(t)/dt < 0. Hence if Q(0) > λ/µ ,then the limiting process follows the deterministic path

Q(t) =λ

µ+

(Q(0)− λ

µ

)e−λt.

Finally, if Q(0) = λ/µ, then dQ(t)/dt = 0 for all t ≥ 0, which yields Q(t) = λ/µ forall t ≥ 0.

We point out that whenever the queue length is near its steady-state value, itremains close to that value from that time onward. That is, if Qr(t0) ≈ λr/µfor some t0 ≥ 0, then Qr(t) ≈ λr/µ for all t ≥ t0. From that point on, the fluidlimit becomes a rather rough estimate for the number of batteries in need ofcharging that allows for further investigation of the fluctuations around this value.Therefore, we turn our focus to the diffusion scaling

Qr(t) =Qr(t)− λr/µ√

λr/µ, r ≥ 1, t ≥ 0. (7.5.2)

This scaling provides more sensitive approximations, as it captures fluctuations oforder

√r. The diffusion-scaled process will tend to a piecewise linear Ornstein-

Uhlenbeck processes, with a steady-state distribution that can be expressed ana-lytically.

Theorem 7.5.1. Consider a single swapping station that operates under rules (7.2.1).If Qr(0) → Q(0) in distribution as r → ∞, then Qr → Q in distribution as r → ∞.The process Q is a diffusion process with drift

m(x) = −λ(x− β)+ − µminx, γ,

and constant infinitesimal variance 2µ. The steady-state density of Q(∞) = limt→∞ Q(t)is given by

f(x) =

α1φ(x)Φ(γ) if x < γ,

α2

(γe−γ(x−γ)

) (1− e−γ(β−γ)

)−1if γ ≤ x < β,

α3

√λµφ

(x−(β−µλγ)√

µ/λ

)Φ(−√

µλγ)−1

if x ≥ β,(7.5.3)

7


where αi = ri/(r1 + r2 + r3), i = 1, 2, 3, with

r1 = 1,

r2 =

φ(γ)Φ(γ)

−1 1γ

(1− e−γ(β−γ)

)if γ 6= 0,√

2πβ if γ = 0,

r3 =φ(γ)

Φ(γ)e−γ(β−γ)

√µ

λφ

(√µ

λγ

)−1

Φ

(−√µ

λγ

).

Proof. The infinitesimal mean of the centered scaled process is given by

mr(x) =λr

(bλr/µ+ x

√λr/µc

)− µr

(bλr/µ+ x

√λr/µc

)

√λr/µ

→ −λ(x− β)+ − µminx, γ,

and the infinitesimal variance is given by

σ2r(x) =

λr

(bλr/µ+ x

√λr/µc

)+ µr

(bλr/µ+ x

√λr/µc

)

λr/µ→ 2λr

λr/µ= 2µ.

Using Equations (28) and (33) in [21], this implies that the limiting process is apiecewise-linear diffusion process with steady-state density given by (7.5.3), whereαi, i = 1, 2, 3, is the probability that the steady-state process is in that interval.Equations (5) and (6) from [21] yield the steady-state probabilities.

7.6 Interchange of limits

Equation (7.5.3) in Theorem 7.5.1 is obtained by taking the limit of the scaleddiffusion process (as r →∞), and finding its steady-state distribution (as t→∞).However, in order to obtain a good approximation of the steady-state distributionwith a fixed number of cars r, it is arguably more reasonable to consider thesteady-state distribution of the scaled diffusion process (as t→∞) and next takethe limit as r → ∞. Fortunately, the following theorem shows that the order inwhich one takes the limit does not affect the result.

Theorem 7.6.1. Consider the single swapping station operating under the policy (7.2.1).The steady-state distribution of the diffusion scaled process Qr(∞) converges in distribu-tion to Q(∞) as in Theorem 7.5.1.

Since the steady-state distribution is given by Lemma 7.4.1, in order to proveTheorem 7.6.1 it suffices to derive the limiting behavior of this distribution un-der scaling rules (7.2.1). First we derive the limiting steady-state distributionconditioned to be in one of the intervals [0, F r], [F r, Br], and [Br, Br + r].

7

7.6. Interchange of limits 207

Lemma 7.6.1. Under scaling rules (7.2.1), the following limiting results hold as r →∞.If x ≤ γ, then

P

(Qr <

λr

µ+ x

√λr

µ

∣∣Qr < F r

)→ Φ(x)

Φ(γ).

If γ ≤ x < β, then

P

(Qr <

λr

µ+ x

√λr

µ

∣∣F r ≤ Qr < Br

)→(

1− e−γ(x−γ))(

1− e−γ(β−γ))−1

.

Finally, if x ≥ β, then

P

(Qr ≥ λr

µ+ x

√λr

µ

∣∣Qr ≥ Br)→ Φ

((β − x)

√λ

µ− γ√µ

λ

)Φ

(−γ√µ

λ

)−1

.

Proof. The first expression follows from the central limit theorem and the proper-ties of the Poisson distribution,

P

(Qr <

λr

µ+ x

√λr

µ

∣∣∣∣Qr < F r

)=

P(Qr < λr

µ + x√

λrµ

)

P (Qr < F r)

=

∑λr/µ+x√λr/µ−1

k=0 πk∑F r−1k=0 πk

=P(

Pois(λr/µ) ≤ λr/µ+ x√λr/µ− 1

)

P(

Pois(λr/µ) ≤ λr/µ+ γ√λr/µ− 1

) → Φ(x)

Φ(γ).

The second expression can be obtained using geometric series,

P

(Qr <

λr

µ+ x

√λr

µ

∣∣∣∣F r ≤ Qr < Br

)=


k=F r πk∑Br−1k=F r πk

=


k=F r

(λrµF r

)k

∑Br−1k=F r

(λrµF r

)k =1−

(λrµF r

)λr/µ+x√λr/µ

− 1 +(λrµF r

)F r

1−(λrµF r

)Br− 1 +

(λrµF r

)F r

=

(1 + γ√

λr/µ

)−x√λr/µ−(

1 + γ√λr/µ

)−γ√λr/µ

(1 + γ√

λr/µ

)−β√λr/µ−(

1 + γ√λr/µ

)−γ√λr/µ

→ e−γx − e−γ2

e−γβ − e−γ2 =1− e−γ(x−γ)

1− e−γ(β−γ).

7


The final expression again uses the central limit theorem and the properties of thePoisson distribution. That is,

P

(Qr ≥ λr

µ+ x

√λr

µ

∣∣∣∣Qr ≥ Br)

=P(Qr ≥ λr

µ + x√

λrµ

)

P (Qr ≥ Br)

=

∑Br+r

k=λr/µ+x√λr/µ

1(r+Br−k)!

(λµF r

)k

∑Br+rk=Br

1(r+Br−k)!

(λµF r

)k

=

∑Br+r−λr/µ−x√λr/µ

k=01k!

(λµF r

)Br+r−k

∑rk=0

1k!

(λµF r

)Br+r−k =

∑r+(β−x)√λr/µ

k=01k!

(µF r

λ

)ke−µF

r/λ

∑rk=0

1k!

(µF r

λ

)ke−µF r/λ

.

Since for every z ∈ R,

P(

Pois (µF r/λ) ≤ r + z√λr/µ

)

= P

(Pois (µF r/λ)− r − γ

√µrλ√

r≤ z√λ

µ− γ√µ

λ

)→ Φ

(z

√λ

µ− γ√µ

λ

)

as r →∞, we obtain

P

(Qr ≥ λr

µ+ x

√λr

µ

∣∣∣∣Qr ≥ Br)→ Φ

((β − x)

√λ

µ− γ√µ

λ

)Φ

(−γ√µ

λ

)−1

.

To obtain the limiting distribution of the steady-state distribution as r → ∞, itremains to derive the asymptotic behavior of the probabilities that the queuelength is in the intervals [0, F r], [F r, Br], and [Br, Br + r]. To do so, we use thefollowing three lemmas.

Lemma 7.6.2. Under scaling rules (7.2.1), as r →∞,

F r−1∑

k=0

(λrµ

)k

k!e−λr/µ → Φ(γ).

Proof. This follows from the central limit and the properties of the Poisson distri-bution,

F r−1∑

k=0

(λrµ

)k

k!e−λr/µ = P

(Pois(λr/µ) < λr/µ+ γ

√λr/µ

)→ Φ(γ).

7


Lemma 7.6.3. Under scaling rules (7.2.1) with γ 6= 0, as r →∞,

Br−1∑

k=F r

(λr

µF r

)kF rF

r

F r!e−

λrµ → 1√

2π

1

γ

(e−γ

2/2 − e−βγ+γ2/2)

=φ(γ)

γ

(1− e−γ(β−γ)

).

If γ = 0, then

Br−1∑

k=F r

(λr

µF r

)kF rF

r

F r!e−

λrµ → β√

2π.

Proof. If γ 6= 0, we observe that

Br−1∑

k=F r

(λr

µF r

)k=

(λrµF r

)F r+1

−(λrµF r

)Br

1−(λrµF r

)

=

(λr

µF r

)λrµ

(λrµF r

)γ√λr/µ+1

−(λrµF r

)β√λr/µ

1−(λrµF r

)

due to geometric series. Moreover,

(λr

µF r

)γ√λr/µ+1

−(λr

µF r

)β√λr/µ

=

(1 +

γ√λr/µ

)−γ√λr/µ+1

−(

1 +γ√λr/µ

)−β√λr/µ

→ e−γ2 − e−βγ ,

and due to geometric series,

1−(λr

µF r

)= 1−

(1 +

γ√λr/µ

)−1

= − γ√λr/µ

+O(r−1).

In addition,

(λr

µF r

)λrµ

=

(1 +

γ√λr/µ

)λrµ

= exp

−λrµ

(γ√λr/µ

− 1

2

γ2

λr/µ+O(r−3/2)

)

= exp

−√λr

µγ +

γ2

2+O(r−1/2)

.

7


Finally, using Stirling’s approximation

F rFr

F r!e−

λrµ ∼ 1√

2πF re−

λrµ +F r ∼ 1√

2π√λr/µ

eγ√λrµ .

Combining the expressions yields

Br−1∑

k=F r

(λr

µF r

)kF rF

r

F r!e−

λrµ → 1√

2π

1

γeγ

2/2(e−γ

2 − e−βγ)

=1√2π

1

γ

(e−γ

2/2 − e−βγ+γ2/2).

If γ = 0, using Stirling’s approximation, we directly obtain

Br−1∑

k=F r

(λr

µF r

)kF rF

r

F r!e−

λrµ = (Br − F r − 1)

F rFr

F r!e−

λrµ → β√

2π.


rBr

r!F rFr

F r!e−

λrµ

Br+r∑

k=Br

1

(r +Br − k)!

(λ

µF r

)k→√µ

λeγ2

2 −βγ+µλγ2

2 Φ

(−√µ

λγ

)

= φ(γ)e−γ(β−γ)

√µ

λφ

(√µ

λγ

)−1

Φ

(−√µ

λγ

).

Proof. We observe that

(λ

µF r

)−(Br+r)

e−µFr

λ

Br+r∑

k=Br

1

(r +Br − k)!

(λ

µF r

)k=

r∑

k=0

1

k!

(µF r

λ

)ke−

µFr

λ

= P(

Pois(µF r

λ

)≤ r)

= P

(Pois (µF r/λ)− µF r/λ√

µF r/λ≤ r − µF r/λ√

µF r/λ

)→ Φ

(−γ√µ

λ

).

7


Moreover, applying Stirling’s approximation twice,

rBr

r!F rFr

F r!e−

λrµ

(λ

µF r

)Br+r

eµFr

λ ∼√

r

F r

(λr

µF r

)Br+r

e−r+Fr−λµ r+

µλF

r

∼√µ

λ

(1 +

γ√λr/µ

)Br+r

eγ√λrµ +√

µrλ

∼√µ

λeγ√λrµ +√

µrλ

· exp

−(r +

λr

µ+ β

√λr

µ

)(γ√λr/µ

− 1

2

γ2

λr/µ+O(r−3/2)

)

→√µ

λexp

γ2

2− βγ +

µ

λ

γ2

2

.

Multiplying the two expressions yields the result.

Lemma 7.6.5. Under scaling rules (7.2.1), as r →∞:

P (Qr < F r)→ r1

r1 + r2 + r3

P (F r ≤ Qr < Br)→ r2

r1 + r2 + r3

P (Qr ≥ Br)→ r3

r1 + r2 + r3,

where

r1 = 1,

r2 =

φ(γ)Φ(γ) 1

γ

(1− e−γ(β−γ)

)if γ 6= 0,√

2πβ if γ = 0,

r3 =φ(γ)


√µ

λφ

(√µ

λγ

)−1

Φ

(−√µ

λγ

).

Proof. It follows from Lemmas 7.6.2-7.6.4 that

π(Br,F r,r)0 eλr/µ → Φ(γ)(r1 + r2 + r3),

and applying Lemmas 7.6.2-7.6.4 again yields

P (Qr < F r)→ Φ(γ)

Φ(γ)(r1 + r2 + r3),

P (F r ≤ Qr < Br)→ Φ(γ)r2

Φ(γ)(r1 + r2 + r3),

P (Qr ≥ Br)→ Φ(γ)r3

Φ(γ)(r1 + r2 + r3).

7


Consequently, it follows that Theorem 7.6.1 holds.

Proof of Theorem 7.6.1. This result follows directly from Lemmas 7.6.1 and 7.6.5.

7.7 Performance measures

Since it follows from Theorem 7.6.1 that the order in which the limits are taken doesnot impact the result, we use the limiting process Q(∞) to obtain approximationsfor the performance measures.

In open queueing systems where the arrival process is a time-homogeneousPoisson process, the steady-state value of any quantity is the same as at arrivalinstant. This property is known as the PASTA (Poisson Arrivals See Time Averages)property. In particular, the waiting probability equals the steady-state probabilitythat the number of fully-charged batteries is zero, or equivalently, the number ofbatteries in need of charging is at least B. Unfortunately, the arrival process inour closed setting is state-dependent. Yet, Theorem 7.5.1 shows that the changesin arrival rate are only of order O(

√r), i.e. the arrival rate is λr − O(

√r) (with

high probability). These small changes will therefore become negligible as r →∞in the sense that the PASTA property remains valid asymptotically. This notioncan be formalized similarly as is done in [123]. Summarizing, if W denotes thewaiting time of an arriving EV user, then

P(W > 0) = limr→∞

P (Qr(∞) ≥ Br) = P(Q(∞) ≥ β

),

where Q(∞) is as in Theorem 7.5.1.

The key concept to derive the expected waiting time is Little’s law, stating that thelong-term average number of waiting cars, denoted by QrW , equals the long-termthroughput multiplied by the average waiting time. In other words,

E(QrW ) = θE(W ),

where the throughput θ can be viewed as the long-term average rate at whichEVs arrive, and hence also leave the battery swapping station. We can express thethroughput as

θ = λr − λE(QrW ),

since the long-term average number of batteries not in need of charging is infact the expected number of cars not waiting at the station in this closed system.Therefore, it follows that

E(W ) =E(QrW )

λ(r − E(QrW )). (7.7.1)

7

7.7. Performance measures 213

In turn, the expected number of waiting cars can be derived by invoking The-orem 7.5.1 and the observation QrW = (Qr(∞)−Br)+,

E(QrW ) =

r∑

k=Br+1

(k −Br)P (Q(∞) = k)

=

√λr

µ

r∑

k=Br+1

k −Br√λr/µ

P

(Q(∞) =

k − λr/µ√λr/µ

)∼√λr

µ

∫ ∞

β

(x− β)f(x) dx

as r →∞. We point out that E(QrW ) is consequently of order Θ(√r), and together

with (7.7.1) this implies that E(W ) is of order Θ(1/√r) and hence vanishes in the

limit.

The resources will be fully utilized under the proposed QED policy as r → ∞.Theorem 7.5.1 implies that that at most O(

√r) charging points are not utilized,

and the number of fully-charged batteries is also of order O(√r). Therefore, as

r →∞,

ρF r = 1−O(1/√r), ρBr = 1−O(1/

√r). (7.7.2)

Theorem 7.7.1. Consider a swapping station operating under (7.2.1). Then the followingproperties hold as r →∞. The waiting probability has a non-degenerate limit given by

P(W > 0) ∼ P(Q(∞) ≥ β

)=

(1 +

√λ

µ

φ(√µ/λγ)

φ(γ)eγ(β−γ) Φ(γ)

Φ(−√µ/λγ)

+

√λ

µ

φ(√µ/λγ)

γ

(eγ(β−γ) − 1

)Φ

(−√µ

λγ

)−1)−1

.

The expected waiting time behaves as

E(W )√r∼ α3√

λµ

(√µ

λφ(µλγ)

Φ

(−√µ

λγ

)−1

− µ

λγ

).

Finally, the resource utilizations behave as

ρF r → 1, ρBr → 1.

Proof. Since the PASTA property holds asymptotically, we find that the probabilityan arriving car has to wait for a fully-charged battery is asymptotically equivalentto the probability that the system is in a state where more than Br batteries arecharging at the same time, which gives the first asymptotic relation. Then,

P(Q(∞) ≥ β

)= r3/(r1 + r2 + r3)

where ri, i = 1, 2, 3, are as in Theorem 7.5.1, from which the result follows directly.

7


For the expected waiting time, we consider the expected number of waiting carsat the charging station. We observe that

E(QW )√λr/µ

→∫ ∞

β

(x− β)f(x) dx = α3Φ

(−√µ

λγ

)−1 ∫ ∞

0

√λ

µyφ

(y + µ/λγ√

µ/λ

)dy

= α3

(√µ

λφ

(√µ

λγ

)Φ

(−√µ

λγ

)−1

− µ

λγ

),

where α3 is as in Theorem 7.5.1. We remark that this expression is always positive:trivially for γ ≤ 0, and also for γ > 0 since for every x > 0

Φ(−x) ≤ φ(x)

x,

which can be shown by partial integration. Due to Little’s law, we note that

E(W ) =E(QW )

λ (r − E(QW )),

which yields the result for the expected waiting time.

Finally, for the utilization levels, the scaling of Q(∞) implies that

E(# Idle Charging Points) = Θ(√r),

E(# Fully-Charged Batteries) = Θ(√r), E(Qr) = Θ(r).

Therefore, the utilization levels satisfy (7.7.2) and the result follows.

7.8 Unlimited number of charging points for single stationsystem

In this section we consider the effect of having a limited number of charging pointsf and swapping servers G. That is, we consider a related setting where there arean unlimited number of charging points (F =∞) and also an unlimited numberof swapping servers G =∞.

We point out that Lemma 7.4.1 implies the steady-state distribution, which is givenby

π(B,r)k =

(λr/µ)k

k! π(B,r)0 if 0 ≤ k ≤ B

r!rB

(r+B)!

(r+Bk

) (λµ

)kπ

(B,r)0 if B ≤ k ≤ B + r,

(7.8.1)

where

π(B,r)0 =

(B−1∑

k=0

(λr/µ)k

k!+

r!rB

(r +B)!

B+r∑

k=B

(r +B

k

)(λ

µ

)k)−1

(7.8.2)

7

7.8. Unlimited number of charging points for single station system 215

Also in this particular case one can pose a QED provisioning rule for the numberof spare batteries alone, and derive the asymptotic properties. That is, we considerthe performance of the system under the following scaling rule:

Br =λr

µ+ β

√λr

µ. (7.8.3)

7.8.1 Fluid and diffusion limits

Although our main focus is performance measures in steady state, it is instructiveto see the transient behavior for this large-scale system. To that end, we explorethe fluid and diffusion limits.

Consider the fluid-scaled process Qr(t), t ≥ 0, where Qr(t) = Qr(t)/r for allr ≥ 1. The following theorem holds under the QED scaling.

Theorem 7.8.1. Suppose the system operates under the scaling rule (7.8.3), and thatQr(0)→ Q(0) in distribution with Q(0) a finite constant, then Qr → Q in distributionas r →∞, where Q satisfies the ODE

dQ(t)

dt= λ− µQ(t)− λ

(Q(t)− λ

µ

)+

,

and has the unique steady-state value

Q(∞) =λ

µ.

Proof. We follow the framework of [21] for birth-death processes. We observe thatQr is a birth-death process with state space Qr = 0, 1, ..., Br + r, and arrivalrates λr(j) = λ (r − (j −Br)+) and service rate µr(j) = jµ for all j ∈ Qr. Thefluid-scaled process Qr therefore has state space 0, 1/r, ..., (Br + r)/r, and driftand diffusion functions

mr(x) =λr(brxc)

r− µr(brxc)

r= λ− λ(brxc −Br)+

r− µbrxc

r

and

σ2r(x) =

λr(brxc)r2

+µr(brxc)

r2= λ− λ(brxc −Br)+

r+µbrxcr

.

Taking the limit yields

m(x) := limr→∞

mr(x) = λ− µx− λ(x− λ

µ

)+

and

σ2(x) := limr→∞

σ2r(x) = 0.

7


That is, we obtain a degenerate limiting diffusion (since the limiting infinitesimalvariance is zero), and hence the limiting initial point will yield a deterministic pathfor the limiting process Q. The mean directly provides the ODE that the limitingprocess Q satisfies. What remains to be shown is the limiting value (as t→∞) .

For all t ≥ 0 for which Q(t) < λµ , we have

dQ(t)

dt= λ− µQ(t) > 0,

and for all t ≥ 0 for which Q(t) > λµ , we obtain

dQ(t)

dt=λ(λ+ µ)

µ− (λ+ µ)Q(t) < 0.

Moreover, if Q(t) = λ/µ for some t ≥ 0, then dQ(t)/dt = 0. This implies thatthe initial limiting value Q(0) determines the unique deterministic path for thelimiting process. That is, if Q(0) < λ/µ, then the limiting process satisfies

Q(t) =λ

µ+

(Q(0)− λ

µ

)e−µt,

a strictly increasing function that tends to λ/µ as t→∞. Similarly, if Q(0) > λ/µ,then the limiting process satisfies

Q(t) =λ

µ+

(Q(0)− λ

µ

)e−(λ+µ)t,

a strictly decreasing function that tends to λ/µ as t→∞. Finally, if Q(0) = λ/µ,then dQ(t)/dt = 0 and hence Q(t) = λ/µ for all t ≥ 0.

We now zoom in on the fluctuations around the fluid limit by taking the diffusionlimit. We consider the centered scaled process

Qr(t) :=Qr(t)− λr/µ√

λr/µ, t ≥ 0. (7.8.4)

Under the QED scaling rule (7.8.3), we find the following diffusion limiting pro-cess.

Theorem 7.8.2. Suppose the system operates under the scaling rule (7.8.3). If it holdsthat Qr(0)→ Q(0) in distribution as r →∞, then Qr → Q in distribution as r →∞,where Q can be described as a two pieced-together Ornstein-Uhlenbeck (OU) process.More precisely, Q is a diffusion process with mean

m(x) = −λ(x− β)+ − µx,

and constant infinitesimal variance 2µ. The steady-state density Q(∞) is given by

f(x) =

(1− α) φ(x)Φ(β) if x < β,

α√

λ+µµ φ

(x−βλ/(λ+µ)√

µ/(λ+µ)

)Φ(−β√

µλ+µ

)−1

if x ≥ β, (7.8.5)

7


where

α =

1 +

√λ+ µ

µeβ2

2λ

λ+µΦ(β)

Φ(−β√

µλ+µ

)

−1

.

Proof. The proof is similar to the techniques used in the proof of the fluid limits.The infinitesimal mean of the centered scaled process is given by

mr(x) =λr

(bλr/µ+ x

√λr/µc

)− µr

(bλr/µ+ x

√λr/µc

)

√λr/µ

=λr − λ

(bλr/µ+ x

√λr/µc −Br

)+

− µ(bλr/µ+ x

√λr/µc

)

√λr/µ

,

which converges to

m(x) = limr→∞

mr(x) = −λ(x− β)+ − µx

as r →∞. The infinitesimal variance tends to

σ2r(x) =

λr

(bλr/µ+ x

√λr/µc

)+ µr

(bλr/µ+ x

√λr/µc

)

λr/µ→ 2λr

λr/µ= 2µ,

as r →∞. Using Equation (28) from [21], this implies that the limiting process isa piecewise-linear diffusion process with steady-state density given as in (7.8.5),where α is the probability that the steady-state process is above β. Equations (5)and (6) from [21] yield

α =

(1 +

f(β+)/α

f(β−)/(1− α)

)−1

=

1 +

√λ+ µ

µeβ2

2λ

λ+µΦ(β)

Φ(−β√

µλ+µ

)

−1

.

7.8.2 Steady-state limits

We point out that the fluid and diffusion limits are useful for studying the systemin transience. The steady-state limiting distribution is then obtained by lettingt→∞with respect to the limiting diffusion process. In other words, the steady-state density function (7.8.5) is obtained by first taking the limit of the centeredscaled process as n→∞, after which its steady-state distribution is obtained byletting t→∞. The following result shows that interchanging the limits does notchange the result.

7


Theorem 7.8.3. As r →∞,

Qr(∞)→ Q(∞)

in distribution, where Q(∞) is the steady-state distribution as in Theorem 7.8.2.

Before moving to the proof of this result, we need to understand the limitingbehavior of the steady-state probability that more than Br batteries are in need ofrecharging. To do so, we first determine the asymptotic behavior of the normaliza-tion constant in (7.8.2).

Lemma 7.8.1. Under scaling rule (7.8.3), as r →∞,

π(Br,r)0 e

λµ r →

(Φ(β) +

√µ

λ+ µe−

β2

2λ

λ+µΦ

(−β√

µ

λ+ µ

))−1

.

Proof. Note that

(π

(Br,r)0

)−1

e−λµ r =

Br∑

k=0

(λr/µ)k

k!e−

λrµ +

Br+r∑

k=Br+1

r!rBr

(r +Br)!

(r +Br

k

)(λ

µ

)ke−

λrµ .

The convergence of the first term follows directly from the CLT for a Poissondistributed random variable,

Br∑

k=0

(λr/µ)k

k!e−

λrµ = P (Pois (λr/µ) ≤ Br)

= P

(Pois (λr/µ)− λr/µ√

λr/µ≤ β

)→ Φ(β).

We extract from the second term

Br+r∑

k=Br+1

(r +Br

k

)(λ

µ

)k (µ

λ+ µ

)Br+r

= P (Bin(Br + r, λ/(λ+ µ)) ≥ Br + 1)

= P

Bin(Br + r, λ/(λ+ µ))− (Br + r) λ

λ+µ√(Br + r) λµ

(λ+µ)2

≥ µBr − λr + λ+ µ√λµ√Br + r

→ Φ

(−β√

µ

λ+ µ

),

7


where we used the CLT for a sum of independent Bernoulli distributed randomvariables. Finally, what is left of the second term is given by

(µ

λ+ µ

)−(Br+r) r!rB

r

(r +Br)!e−

λrµ ∼

√µ

λ+ µ

(1 +

β√λµ

λ+µ√r

)λ+µµ r+β

√λrµ

eβ√λrµ

=

√µ

λ+ µExp

(λ+ µ

µr + β

√λr

µ

)(β

√λµ

λ+ µ

1√r− β2

2

λµ

(λ+ µ)2

)

+β

√λr

µ

1

r+O(r−3/2)

=

√µ

λ+ µExp

−β

2

2

λ

λ+ µ+O(r−1/2)

,

where we used Stirling’s approximation twice and a Taylor expansion for thelogarithm term. Combining the three expressions yields the results.

Using Lemma 7.8.1, we can derive the asymptotic behavior of having more thanBr batteries in need of recharging in steady state.

Proposition 7.8.1. Under scaling rule (7.8.3), the probability that an arriving car has towait for a battery behaves as

P(Qr(∞) ≥ Br)→

1 +

√λ+ µ

µeβ2

2λ

λ+µΦ(β)

Φ(−β√

µλ+µ

)

−1

as r →∞.

Proof. Note that

P(Qr(∞) ≥ Br) =

Br+r∑

k=Br

π(Br,r)k

= π(Br,r)0 e

λµ r

Br+r∑

k=Br+1

r!rBr

(r +Br)!

(r +Br

k

)(λ

µ

)ke−

λrµ .

It follows from the proof of Lemma 7.8.1 that, as r →∞,

Br+r∑

k=Br+1

r!rBr

(r +Br)!

(r +Br

k

)(λ

µ

)ke−

λrµ

→ Φ

(−β√

µ

λ+ µ

)√µ

λ+ µExp

−β

2

2

λ

λ+ µ

.

Rewriting the terms yields the result.

7


Next, we prove Theorem 7.8.3.

Proof of Theorem 7.8.3. By definition,

Q(∞) = limt→∞

limr→∞

Qr(t),

and we want to show that changing the order of the limits yields the same distri-bution. The steady-state distribution for a system with r cars is given by (7.8.1).Due to Proposition 7.8.1, it suffices to show that for every x < β

limr→∞

P(Qr(∞) ≤ x

∣∣Qr(∞) < β)→ Φ(x)

Φ(β),

and for every x ≥ β,

limr→∞

P(Qr(∞) > x

∣∣Qr(∞) ≥ β)→ Φ

(−x− βλ/(λ+ µ)√

µ/(λ+ µ)

)Φ

(−β√

µ

λ+ µ

)−1

.

Using the steady-state distribution given in (7.8.1), we observe that for every β,

P(Qr(∞) ≤ x

∣∣Qr(∞) < β)

=P(Qr(∞) ≤ x

)

P(Qr(∞) < β

)

= P

(Pois(λr/µ)− λr/µ√

λr/µ≤ x

)P

(Pois(λr/µ)− λr/µ√

λr/µ< x

)−1

→ Φ(x)

Φ(β)

as r →∞. For every x ≥ β,

P(Qr(∞) > x

∣∣Qr(∞) ≥ β)

=P(Qr(∞) > x

)

P(Qr(∞) ≥ β

)

=P(

Bin(Br + r, λ

λ+µ

)> λr

µ + x√

λrµ

)

P(

Bin(Br + r, λ

λ+µ

)> λr

µ + β√

λrµ

) .

Similarly to the proof of Proposition 7.8.1, The CLT implies

P

(Bin

(Br + r,

λ

λ+ µ

)>λr

µ+ x

√λr

µ

)→ Φ

(−x− βλ/(λ+ µ)√

µ/(λ+ µ)

)

and

P

(Bin

(Br + r,

λ

λ+ µ

)>λr

µ+ β

√λr

µ

)→ Φ

(−β√

µ

λ+ µ

)

as r →∞.

7


7.8.3 Performance measures

We consider the waiting probability, the waiting time and the utilization level ofthe spare batteries. We note that α in (7.8.5) corresponds to the probability thatQ(∞) ≥ β. Although the PASTA property does not hold in this closed system, itdoes hold asymptotically, providing an appropriate approximation for the waitingprobability.

Corollary 7.8.1. Under scaling rule (7.8.3), as r →∞,

P(W > 0)→ P(Q(∞) ≥ β) =

1 +

√λ+ µ

µeβ2

2λ

λ+µΦ(β)

Φ(−β√

µλ+µ

)

−1

.

For the waiting time, we point out that in this case the identity

E(W ) =E(QW )

λ (r − E(QW ))

still holds due to Little’s law. Using this relation, we can obtain the expectedwaiting time.

Proposition 7.8.2. Under scaling rule (7.8.3), as r →∞,

E(QW )√r→

√λµ

λ+µ e− β22 λ

λ+µ

Φ(β) +√

µλ+µe

− β22 λλ+µΦ

(−β√

µλ+µ

)

·(

1√2πe−

β2

2µ

λ+µ − β√

µ

λ+ µΦ

(−β√

µ

λ+ µ

)).

Consequently, the waiting time of a car behaves as

E(W )√r =

E(QW )√r

λr − λE(QW )→

√µλ

1λ+µe

− β22 λλ+µ

Φ(β) +√

µλ+µe

− β22 λλ+µΦ

(−β√

µλ+µ

)

·(

1√2πe−

β2

2µ

λ+µ − β√

µ

λ+ µΦ

(−β√

µ

λ+ µ

))

as r →∞.

Proof. We note that as r →∞,

E(QW )√λr/µ

→∫ ∞

β

(x− β)f(x) dx =

∫ ∞

0

yf(y + β) dy.

Elementary calculus yields∫ ∞

β

yφ

(y − βµ/(λ+ µ)√

µ/(λ+ µ)

)dx

=µ

λ+ µφ

(β

√µ

λ+ µ

)− β

(µ

λ+ µ

)3/2

Φ

(−β√

µ

λ+ µ

).

7


Then as r →∞, we obtain

E(QW )√λr/µ

→ α

√λ+ µ

µ

(µ

λ+ µφ

(β

√µ

λ+ µ

)

−β(

µ

λ+ µ

)3/2

Φ

(−β√

µ

λ+ µ

))Φ

(−β√

µ

λ+ µ

)−1

,

where

α =Φ(−β√

µλ+µ

)

Φ(−β√

µλ+µ

)+√

λ+µµ e

β2

2λ

λ+µΦ(β).

Simply rewriting the terms yields the result.

Finally, the utilization level follows directly from the scaling of the diffusion scaledprocess. More specifically, the scaling implies that


Corollary 7.8.2. Under scaling rule (7.8.3),

ρBr → 1 as r →∞.

7.9 Unlimited number of swapping servers for single stationsystem

In this section, we consider the effect of the number of swapping servers G bylooking at a related setting. More specifically, we consider a single station wherethe swapping servers pose no effective condition, i.e. G = ∞ or equivalentlyG ≥ max1, F r −Br. That is, every battery of an EV that is at the station can betaken to a charging point (if available). We point out that for all F r ≤ Br, thisreduces to the main setting in this chapter. Therefore, in this section, we consideronly

Br =λr

µ+ β

√λr

µ, β ∈ R, (7.9.1)

F r =λr

µ+ γ

√λr

µ, γ > β. (7.9.2)

7.9.1 Diffusion limits

The steady-state distribution is given in Lemma 7.4.1. Moreover, the fluid-limitresult of Proposition 7.5.1 and its proof remain valid in this case, since the fluctu-ations of order Θ(

√r) are too small to be of any impact on the fluid scale. There

will however be a notable effect on the diffusion scale.

7

7.9. Unlimited number of swapping servers for single station system 223

Theorem 7.9.1. Suppose the system operates under (7.9.2). If Qr(0)→ Q(0) in distri-bution as r →∞, then Qr → Q in distribution as r →∞. The process Q(t), t ≥ 0 isa diffusion process with mean

m(x) = −λ(x− β)+ − µminx, γ,

and constant infinitesimal variance 2µ. The steady-state density Q(∞) is given by

f(x) =

α1φ(x)Φ(β) if x < β,

α2

√λ+µµ φ

(x− λ

λ+µβ√µ/(λ+µ)

)η if β ≤ x < γ,

α3

√λµφ


µ/λ

)Φ

(−γ−(β−µλγ)√

µ/λ

)−1

if x ≥ γ,

(7.9.3)

where

η =

(Φ

(γ − λ

λ+µβ√µ/(λ+ µ)

)− Φ

(√µ

λ+ µβ

))−1

and αi = ri/(r1 + r2 + r3), i = 1, 2, 3 with

r1 = 1,

r2 =φ(β)

Φ(β)

√µ

λ+ µφ

(√µ

λ+ µβ

)−1(

Φ

(γ − λ


)− Φ

(√µ

λ+ µβ

)),

r3 =φ(β)

Φ(β)φ

(γ − λ


)φ

(√µ

λ+ µβ

)−1√µ

λφ

(γ − (β − µ

λγ)√µ/λ

)−1

· Φ(−γ − (β − µ

λγ)√µ/λ

).

Proof. The infinitesimal mean of the centered scaled process is given by

mr(x) =λr

(bλr/µ+ x

√λr/µc

)− µr

(bλr/µ+ x

√λr/µc

)

√λr/µ

→ −λ(x− β)+ − µminx, γ,

and the infinitesimal variance is given by

σ2r(x) =

λr

(bλr/µ+ x

√λr/µc

)+ µr

(bλr/µ+ x

√λr/µc

)

λr/µ∼ 2λr

λr/µ= 2µ.

Using Equation (28) and (33) from [21], this implies that the limiting process isa piecewise-linear diffusion process with steady-state density given as in (7.9.3),where αi, i = 1, 2, 3 is the probability that the steady-state process is in thatinterval. These steady-state probabilities follow directly from Equations (5) and(6) in [21].

7


7.9.2 Steady-state limits

The main goal of this section is to show that the limiting steady-state distributionis the same process as the diffusion limiting process in steady state, i.e. theinterchange of limits is allowed (t→∞ and r →∞). Again, we observe that theform of the formula is different in the intervals [0, F r], [Br, F r], and [F r, Br + r].First we derive the limiting steady-state distribution conditioned to be in one ofthese intervals.

Lemma 7.9.1. Suppose scaling rules (7.9.2) hold. As r →∞,

P

(Qr <

λr

µ+ x

√λr

µ

∣∣Qr < Br

)→ Φ(x)

Φ(β)

if x < β,

P

(Qr <

λr

µ+ x

√λr

µ

∣∣Br ≤ Qr < F r

)

→ Φ

(x− λ


)(Φ

(γ − λ


)− Φ

(√µ

λ+ µβ

))−1

if β ≤ x < γ, and

P

(Qr ≥ λr

µ+ x

√λr

µ

∣∣Qr ≥ F r)→ Φ

(−x− (β − µ

λγ)√µ/λ

)Φ

(−γ − β + µ

λγ√µ/λ

)−1

if x ≥ β.

Proof. The first expression follows, similarly to the case of F r < Br, from the CLTand the properties of the Poisson distribution:

P

(Qr <

λr

µ+ x

√λr

µ

∣∣∣∣Qr < Br

)=

P(Qr < λr

µ + x√

λrµ

)

P (Qr < Br)

=P(

Pois(λr/µ) ≤ λr/µ+ x√λr/µ− 1

)

P(

Pois(λr/µ) ≤ λr/µ+ β√λr/µ− 1

) → Φ(x)

Φ(β).

The second expression follows similarly as the proof of Lemma 7.8.1, using thebinomial distribution and the CLT:

P

(Qr <

λr

µ+ x

√λr

µ

∣∣Br ≤ Qr < F r

)

→ Φ

(x− λ


)(Φ

(γ − λ


)− Φ

(β − λ


))−1

.

7


The final expression again uses the CLT and the properties of the Poisson distribu-tion:

P

(Qr ≥ λr

µ+ x

√λr

µ

∣∣∣∣Qr ≥ F r)

=P(Qr ≥ λr

µ + x√

λrµ

)

P (Qr ≥ F r)

=

∑Br+r

k=λr/µ+x√λr/µ

1(r+Br−k)!

(λµF r

)k

∑Br+rk=F r

1(r+Br−k)!

(λµF r

)k =

∑r+(β−x)√λr/µ

k=01k!

(µF r

λ

)ke−µF

r/λ

∑r+(β−γ)√λr/µ

k=01k!

(µF r

λ

)ke−µF r/λ

.

Since for every z ∈ R,

P(

Pois (µF r/λ) ≤ r + z√λr/µ

)→ Φ

(z

√λ

µ− γ√µ

λ

)= Φ

(z − γ µλ√µ/λ

)

as r →∞, the result follows immediately.


P (Qr < F r)→ r1

r1 + r2 + r3,

P (F r ≤ Qr < Br)→ r2

r1 + r2 + r3,

P (Qr ≥ Br)→ r3

r1 + r2 + r3,

where

r1 = 1,

r2 =φ(β)

Φ(β)

√µ

λ+ µφ

(√µ

λ+ µβ

)−1(

Φ

(γ − λ


)− Φ

(√µ

λ+ µβ

)),

r3 =φ(β)

Φ(β)φ

(γ − λ


)φ

(√µ

λ+ µβ

)−1√µ

λφ

(γ − (β − µ

λγ)√µ/λ

)−1

· Φ(−γ − (β − µ

λγ)√µ/λ

).

Proof. Applying Lemmas 7.9.3-7.9.5 below yields the result.


Br−1∑

k=0

(λrµ

)k

k!e−λr/µ → Φ(β).

7


Proof. This follows from the CLT and the properties of the Poisson distribution.


F r−1∑

k=Br

rBr

r!

(r +Br − k)!

(λ/µ)k

k!

→√

µ

λ+ µe−

β2

2λ

λ+µ

(Φ

(γ − λ


)− Φ

(√µ

λ+ µβ

))

= φ(β)

√µ

λ+ µφ

(√µ

λ+ µβ

)−1(

Φ

(γ − λ


)− Φ

(√µ

λ+ µβ

)).

Proof. The proof is similar to that of Lemma 7.8.1, using the properties of thebinomial distribution and the CLT, and Stirling’s approximation.


rBr

r!F rFr

F r!e−

λrµ

Br+r∑

k=F r

1

(r +Br − k)!

(λ

µF r

)k→√µ

λeγ2

2 −βγ+µλγ2

2 Φ

(−√µ

λγ

)

= φ(γ)e−γ(β−γ)

√µ

λφ

(√µ

λγ

)−1

Φ

(−√µ

λγ

).

Proof. We observe that

(λ

µF r

)−(Br+r)

e−µFr

λ

Br+r∑

k=F r

1

(r +Br − k)!

(λ

µF r

)k

= P

(Pois (µF r/λ)− µF r/λ√

µF r/λ≤ r +Br − F r − µF r/λ√

µF r/λ

)

→ Φ

((β − γ)

√λ

µ− γ√µ

λ

).

Moreover, applying Stirling’s approximation twice,

rBr

r!F rFr

F r!e−

λrµ

(λ

µF r

)Br+r

eµFr

λ →√µ

λexp

γ2

2− βγ +

µ

λ

γ2

2

= φ(β)φ

(γ − λ


)φ

(√µ

λ+ µβ

)−1

·√µ

λφ

(γ − (β − µ

λγ)√µ/λ

)−1

Φ

(−γ − (β − µ

λγ)√µ/λ

).

Multiplying the two expressions yields the result.

7


7.9.3 Performance measures

Theorem 7.9.2. Under scaling rules (7.9.2),

P(W > 0)→ P(Q(∞) ≥ β

)= 1− r1/(r1 + r2 + r3),

where ri, i = 1, 2, 3 are as in Theorem 7.9.1. The expected waiting time behaves as

E(W )√r→ α2√

λµ

√

µ

λ+ µ

φ

(γ− λ


)− φ

(√µ

λ+µβ)

Φ

(γ− λ


)− Φ

(√µ

λ+µβ) −

µ

λ+ µβ

+α3√λµ

√µ

λ

φ

(γ−β+µ

λγ√µ/λ

)

Φ

(−γ−β+µ

λγ√µ/λ

) − µ

λγ

,

with αi as in Theorem 7.9.1. Finally, the utilization levels behave as

ρF r → 1, ρBr → 1 as r →∞.

Proof. Since the PASTA property holds asymptotically, we find that the probabilityan arriving car has to wait for a fully-charged battery is asymptotically equal to theprobability that the system is in a state where more than Br batteries are chargingat the same time, which gives the first asymptotic relation. Thus,

P(Q(∞) ≥ β

)→r3/(r1 + r2 + r3) if γ < β,1− r1/(r1 + r2 + r3) if β ≤ γ,

where ri, i = 1, 2, 3, are as in Theorem 7.9.1, from which the result follows directly.

For the expected waiting time, note that

E(QW )√λr/µ

→∫ ∞

0

yf(y + β) dy

= α3Φ

(−γ − (β − µ

λγ)√µ/λ

)−1 ∫ ∞

γ−β

√λ

µφ

(y + µ

λγ√µ/λ

)dy + α2

·(

Φ

(γ − λ


)− Φ

(√µ

λ+ µβ

))−1 ∫ β−γ

0

√λ+ µ

µφ

(y + µ


)dy

= α2

√

µ

λ+ µ

φ

(γ− λ


)− φ

(√µ

λ+µβ)

Φ

(γ− λ


)− Φ

(√µ

λ+µβ) −

µ

λ+ µβ

+ α3

√µ

λ

φ

(γ−β+µ

λγ√µ/λ

)

Φ

(−γ−β+µ

λγ√µ/λ

) − µ

λγ

.

7


Due to Little’s law, we have the identity

E(W ) =E(QW )

λ (r − E(QW )),

which yields the results for the expected waiting time.

Finally, for the utilization levels, the scaling of Q(∞) implies that

E(# Idle Charging Points) = Θ(√r),


Therefore, the utilization levels satisfy (7.7.2) and the result follows.

7.10 Provisioning scheme

In the Erlang-C model, the QED scaling only has a single square-root provisioningrule for the number of servers. Under this scaling rule, the waiting probabilitytends to a strictly decreasing function in β > 0 with range (0, 1). Consequently, anytargeted delay probability can be achieved by adjusting β > 0. In our main setting,there are two types of resources that are scaled as in (7.2.1). Therefore, determininga pair (β, γ) such that a predetermined target delay probability is achieved is notstraightforward, since the pair (β, γ) is not uniquely defined. Moreover, we havethe additional constraint that γ ≤ β, since F ≤ B +G with G small and fixed.

WriteW (β, γ) as the (limiting) waiting probability of an arbitrary EV user when thesystem operates under (7.2.1) with pair (β, γ), and let W (β) denote the (limiting)waiting probability in the setting where there are an unlimited number of chargingpoints and swapping servers (i.e. the setting of the previous section). Since γ ≤ βand W (β, γ) are strictly decreasing in both β and γ, it holds that

W (β, γ) ≥W (β, β) ≥ W (β), (7.10.1)

and similarly,

W (β, γ) ≤W (γ, γ). (7.10.2)

Consider the problem of finding a pair (β, γ) such that W (β, γ) = α, where α isthe targeted delay probability. We point out that the function W (x, x) is a strictlydecreasing function in x with range (0, 1), and hence the equation W (x, x) = αhas a unique solution x∗. Moreover, the following property holds for any pair(β, γ) satisfying W (β, γ) = α.

Lemma 7.10.1. Let α ∈ (0, 1) and suppose x∗ is the unique solution of the equationW (x, x) = α. Then, for any (β, γ) satisfying W (β, γ) = α it holds that β ≥ x andγ ≤ x.

Proof. This is a consequence of the lower bound (7.10.1), the upper bound (7.10.2),and the property γ ≤ β.

7

7.10. Provisioning scheme 229

Note that for any fixed β ≥ x∗, it holds thatW (β, γ) is a strictly decreasing functionin γ ≤ β with range [W (β, β), 1). Since α ∈ [W (β, β), 1], we can determine aunique γ ≤ x∗ such that W (β, γ) = α. Similarly, for any fixed γ ≤ x∗, it holdsthat W (β, γ) is a strictly decreasing function in β ≥ γ with range (0,W (γ, γ)].Since α ∈ (0,W (γ, γ)], we can determine a unique β ≥ x∗ such that W (β, γ) = α.Depending on the costs of charging point and battery collections, one can derivean optimal trade-off between the two resources such that a target delay probabilityis achieved.

Finally, we would like to remark that the related settings discussed in the previoustwo sections (i.e. if G =∞ and γ > β, and if G =∞ and F =∞) provide lowerbounds for the waiting probability. It captures the effect of having finitely manyswapping serves. Using the results on e.g. the delay probability, swapping facilityoperators can also study the trade-off between performance and additional costsof having more swapping servers.

8

Chapter 8

Battery swapping dynamicsin a network setting

Based on:Complete resource pooling of a load balancing policy for a network of battery swapping stations

F. Sloothaak, J.R. Cruise, S. Shneer, M. Vlasiou, and B. ZwartPreprint avalaible on: arXiv:1902.04392

In this chapter, we consider a network of battery swapping facilities, modeledas a closed Markovian queueing network that operates under a dynamic arrivalpolicy. We propose a provisioning rule for the capacity levels by adopting the(Quality-and-Efficiency-Driven) QED regime, as introduced in Section 7.3. Key inthe derivations is to prove a state-space collapse result, which in turn implies thatcomplete resource pooling is achieved. That is, performance levels are as good asif there would have been a single station with an aggregated number of resources.

8.1 Introduction

There has been work done on the operation and control of a stand-alone batteryswapping station, see e.g. [118] for an overview, but there is a clear gap in theliterature when extending this to the operation a wider network of stations. Inthis chapter, we introduce a model for EVs utilizing battery swapping techno-logy within the context of a fixed region. Within the region there are a numberof charging/swapping stations and vehicles that, in general, do not leave theregion, leading to the conservation of batteries. This gives rise to a class of closedMarkovian queueing network models, which we use in a novel way to model theevolution of the battery population within a city.

With the advancement of smartphones and online technologies, a range of serviceproviders provide occupancy level information to customers to improve delayperformance. In a battery swapping system, such information can motivate EVusers to visit the least busy location in the direct vicinity. In this way, the total

231

8

232 Chapter 8. Battery swapping dynamics in a network setting

(charging) work is shared among multiple stations in a more balanced way. Weintegrate a load-balancing policy in our framework to incorporate this phenomenon.An intrinsic problem is to establish a proper load-balancing strategy, as well assuitable capacity levels, that ensures that the total work is divided among allstations as fair as possible, while also accounting for the inherent tradeoff betweenEV users’ quality-of-service and operational costs.

To adequately balance service performance and resource utilization, we adopt theQuality-and-Efficiency-Driven (QED) regime as introduced in Chapter 7. Recallthat this regime typically gives rise to a square-root slack provisioning policy forthe capacity levels. This policy leads to favorable performance for large systems:as the number of customers r grows large, the waiting probability tends to a valuestrictly between zero and one, the waiting time vanishes with a rate 1/

√r, and

near-optimal resource utilization of 1 − O(1/√r) is achieved. To inherit such

properties for the battery swapping framework, we adopt a similar capacity leveldesign policy for both the number of charging servers and the number of sparebatteries relative to the expected offered load under the load-balancing strategy.

To add to the favorable properties of delay performance in the QED regime, theload-balancing strategy ensures that the relative charging loads at the differentstations do not grow apart too much since arriving EV users always move to theleast-loaded station. This phenomenon has been observed in a number of settingsand is referred to as state space collapse, see [20, 129] for an overview and [33] forwork most closely related to this work. In fact, when capacity levels are chosenappropriately, this effect is so strong that complete resource pooling takes place:the system behaves as if there is only a single station with an aggregated numberof resources. It ensures that it is unlikely that EV users are waiting for a battery atone station, while another is readily available at any other station, even amongthose stations that are far from their direct vicinity.

Load-balancing policies have attracted a lot of attention in recent years due toextremely relevant applications in large data centers, see [18] for an overview.Typically, these systems comprise many single-server stations where a centraldispatcher decides where to allocate incoming tasks. In contrast, our frameworkinvolves a network of (a fixed number of) multiserver stations for which weintroduce a unique feature: an arriving EV user restricts itself to move only toone of the stations in his direct vicinity. By appropriately setting the capacitylevels according to the QED provisioning rule, we show that despite this ratherrestrictive constraint, the resource pooling effect can still be achieved.

The mathematical tools used in this chapter are completely different from themore elementary analysis employed for the stand-alone station case in Chapter 7.This is since a direct analysis of the steady-state distribution of the queue length isintractable under the load-balancing strategy in case of multiple stations. Instead,we resort to a fluid and diffusion limit approach. We derive the existence ofthe fluid limit and identify its unique invariant state. Using a diffusion-scaledqueue length process, we zoom in on the fluctuations around the invariant state.We prove a state space collapse (SSC) result by showing that in the limit (as thenumber of EVs grows large) the diffusion-scaled queue lengths tend to becomearbitrarily close almost instantaneously and stay that way for any fixed interval.

8

8.2. Model Description 233

This property can be exploited to derive the limiting queue length behavior atevery station, and show that it implies the complete resource pooling effect.

The derivations of our results rely heavily on the framework developed by Dai andTezcan [33], that in turn can be seen as an extension of [20]. More specifically, Daiand Tezcan [33] show that if an open Markovian system satisfies four assumptions,then their framework can be applied to conclude that a particular (multiplicative)state-space collapse result holds. We stress that to show that all four assumptionsare satisfied is far from trivial, as is illustrated by [119] for an open Markoviannetwork operating under a specific load-balancing policy. In particular, oneshould introduce an appropriately chosen state-space-collapse (SSC) function,and show that the four assumption are satisfied for this SSC function. Moreover,to show that this implies strong state-space collapse for the queue lengths, acompact containment condition needs to be proved. In this chapter, we extend theframework in [33] to incorporate a closed network setting. Moreover, we show thenecessary steps to derive a strong state-space collapse result for the queue lengthsunder our novel load-balancing policy.

8.2 Model Description

We consider a queueing network with S battery swapping stations and r EVs.Each EV has one battery providing the energy for the car to drive. Every stationi ∈ 1, ..., S has three types of assets: Fi charging points, Bi spare batteries andGi swapping servers. Whenever there is an EV arrival at a station, a swappingserver takes out the almost depleted battery and exchanges it for a fully-chargedone if available. The swapping time is relatively very short (compared to chargingtimes), and therefore we assume it to occur instantaneously. Batteries in needof charging are being recharged whenever a charging point is available, andwe assume every recharge to take an exponential amount of time with rate µ,independent of everything else. Whenever a battery is fully charged, it is placedin an EV immediately if one is waiting, and otherwise stocked for a future EVarrival. After receiving a fully-charged battery, the EV requires recharging afteran exponential amount of time with rate λ. With probability pij stations i and jare in the EV user’s direct vicinity. We assume that EV users consult some onlinedevice, and are motivated to move to the station that is relatively least-loaded. Wespecify the meaning of this statement below in (8.2.3).

Similarly for the case with only a single station, we can make the following twoobservations. We point out that batteries are always exchanged, and thereforethe number of batteries physically present at a station can never be below thisstation’s number of spare batteries. In fact, this observation implies that thequeueing model is closed, where the total number of batteries is given by

Total # batteries in system = r +

S∑

j=1

Bi.

The second observation concerns the role of the swapping servers. Whenever abattery is taken out of the EV, the car cannot move from the swapping server untilan exchange of batteries has taken place. Therefore, no more than Bi+Gi batteries

8


can be charged simultaneously at a station i ∈ 1, ..., S. Having more chargingpoints creates no additional charging capacity, and hence we can assume that

Fi ≤ Bi +Gi, i = 1, ...S. (8.2.1)

In addition, we assume that the number of such expensive swapping technologiesis small at every station, i.e. Gi < G for all i = 1..., S, with 1 ≤ G < ∞ being asmall fixed integer.

To achieve favorable performance levels, we propose an associated QED-scaledcapacity level for the resources at the stations. More specifically, we consider a se-quence of systems indexed by the number of cars r, where we write a superscript rfor processes and quantities to stress the dependency on r. Let pi =

∑Sj=1 pij/2,

i = 1, ..., S, be the effective arrival probability at station i. For a system with r cars,we set the capacity levels of the number of charging points and the number ofspare batteries as

Bri = pi

(λrµ + β λrµ

), β ∈ R,

F ri = pi

(λrµ + γ

√λrµ

), γ ≤ β,

(8.2.2)

for all i = 1, ..., S. We remark that the bound for the number of charging points ori-ginates from (8.2.1). Since the number of swapping servers is fixed and small andthe number of cars r grows large, this condition reduces to the γ ≤ β requirementin (8.2.2).

The main quantity of interest in this chapter is the number of batteries that are inneed of charging, which we will also refer to as the queue length. Let Qi(t) denotethe number of batteries in need of charging at station i at time t ≥ 0, and denoteQ(t) = (Q1(t), ..., QS(t)). The queue length also defines the occupancy level (load)of a station. An EV in need of charging closest to station i and j at time t ≥ 0moves to stations i iff

Qi(t)

pi<Qj(t)

pj, (8.2.3)

where ties are broken evenly. That is, we measure the occupation levels of stationsrelative to its effective arrival rate in (8.2.3). The reason for this is that we considerthe occupation level of a station as a fraction of the queue length and the numberof assets, and since we also provision the number of assets of a station relativeto its effective arrival rate in (8.2.2), this property is also inherited by the load-balancing strategy. Figure 8.1 illustrates the closed queueing model under thisload-balancing arrival mechanism.

Besides the queue length process, we focus on three performance measures in thischapter: the waiting probability of an arbitrary EV, its expected waiting time andthe resource utilization levels of the stations. We consider the resources of theswapping stations to be the charging points and the spare batteries. We define theutilization level of the charging points to be the fraction of charging points thatare busy with charging, and the utilization level of the spare batteries to be thefraction of batteries that are not fully-charged with respect to the total number ofbatteries at the station. In steady state, the latter corresponds to the fraction oftime at a station that a battery is expected to wait for an arriving EV.

8

8.3. System dynamics 235

r +∑S

j=1Bj batteries

∞

Exp(λ) pij

FS

BS

Exp(µ)

Fj

Bj

Exp(µ)

Fi

Bi

Exp(µ)

F1

B1

Exp(µ)

Figure 8.1: Illustration of a closed queueing network with multiplestations.

8.3 System dynamics

When the number of stations S ≥ 2, the analysis of the system behavior needs toaccount for the underlying routing mechanism of arriving EVs. Whenever an EVis in need of recharging, stations i and j are in its direct vicinity with probabilitypij , and it chooses to move the station i if (8.2.3) holds. For a resource poolingeffect to occur, we require that there are a sufficient number of pairs (i, j) for whichpij > 0. For example, if the network consists of four stations with p12 = p34 = 1/2,there are no arrivals that can choose between one station in the set 1, 2 andanother in the set 3, 4. Therefore, possible discrepancies in queue lengths are notleveled by the arrival mechanism between these two sets. Therefore, we assumethat for every non-empty set S of stations, there is at least one pair (i, j) withi ∈ S and j 6∈ S for which pij > 0. This statement is equivalent with the followingassumption.

Assumption 8.3.1. Let G = (V,E) be a graph with node set V = 1, ..., S and edgeset E = (i, j) : pij > 0. We assume that the graph G is connected.

8


There are many processes that are of interest in this system, and in particular, thequeue length process at each station. In our analysis, we consider Xr(t), t ≥ 0with

Xr = (Ar, Ard, Qr, Zr, Y r, T r, Dr, Lr) ,

where

• Ar =(Arij ; i, j ∈ E

), where Arij(t) is the number of arrivals that are closest

to stations i and j until time t ≥ 0 in the r’th system;

• Ard =(Arij,i; i, j ∈ E

), where Arij,i(t) is the number of arrivals that are

closest to stations i and j and are routed to station i until time t ≥ 0 in ther’th system;

• Qr =(Qrj ; 1 ≤ j ≤ S

), where Qrj(t) is the number of batteries in need of

charging at time t ≥ 0 in the r’th system;

• Zr =(Zrj ; 1 ≤ j ≤ S

), where Zrj (t) is the number of busy servers (charging

points) at time t ≥ 0 in the r’th system;

• Y r, where Y r(t) is the aggregated time of all cars that are not waiting atsome station until time t ≥ 0 in the r’th system;

• T r =(T rj ; 1 ≤ j ≤ S

), where T rj (t) is the aggregated time of all servers at

station j that were charging until time t ≥ 0 in the r’th system;

• Dr =(Drj ; 1 ≤ j ≤ S

), where Dr

j (t) is the number of service completions atstation j until time t ≥ 0 in the r’th system;

• Lr, where Lr(t) is the number of batteries that are positioned in an EV notwaiting at a station in the r’th system at time t ≥ 0.

Clearly, there are strong relations between the individual processes in Xr. For ex-ample, there is a routing policy that dictates where a car in need of a fully-chargedbattery drives to the least-occupied station in its direct vicinity in order to swap itsbattery. This notion is captured by the arrival processesAr (the classification of thedifferent arrival types) and Ard (the routing decision). To generate the arrival andservice completion processes, we introduce a set of independent Poisson processes.Let Λij(t), t ≥ 0 for all i, j ∈ E be independent Poisson processes with ratepijλ and Sj(t), t ≥ 1 for all 1 ≤ j ≤ S be independent Poisson processes withrate µ. The system dynamics satisfy the following identities.

8

8.4. Fluid limit 237

Arij(t) = Arij,i(t) +Arij,j(t) ∀i, j ∈ E, (8.3.1)

Arij(t) = Λij (Y r(t)) , ∀i, j ∈ E, (8.3.2)

Qrj(t) = Qrj(0) +∑

i:i,j∈EArij,j(t)−Dr

j (t), ∀j = 1, ..., S, (8.3.3)

Drj (t) = Sj

(T rj (t)

), ∀j = 1, ..., S, (8.3.4)

Y r(t) =

∫ t

0

Lr(s) ds (8.3.5)

T rj (t) =

∫ t

0

Zrj (s) ds, ∀j = 1, ..., S, (8.3.6)

Zrj (t) = minQrj(t), F rj , ∀j = 1, ..., S, (8.3.7)

Lr(t) = r −S∑

j=1

(Qrj(t)−Brj

)+, (8.3.8)

Aij,i(t) can only increase when Qri (t)/pi ≤ Qrj(t)/pj ∀i, j ∈ E. (8.3.9)

We refer to these equations as the system identities, and they prove to be centralfor deriving our results. The derivations use the framework set out in [33], whichin turn is based on [20]. We adopt much of the notation and definitions, and beforestating our main results, we repeat them in order for the description to be self-contained. For each positive integer d, we denote by Dd[0,∞] the d-dimensionalSkorohod path space. For x, y ∈ Dd[0,∞] and T > 0, let

‖x(·)− y(·)‖T = sup0≤t≤T

|x(t)− y(t)|,

where |z| = maxi=1,...,d |zi| for any z = (z1, ..., zd) ∈ Rd. The space Dd[0,∞]is endowed with the J1 topology, and the weak convergence in this space isconsidered with respect to this topology. We say that a sequence of functionsxn ∈ Dd[0,∞] converges uniformly on compact sets (u.o.c) sets to x ∈ Dd[0,∞]as n→∞ if for each T ≥ 0,

‖xn(·)− x(·)‖T → 0

as n → ∞. Moreover, we say that t ≥ 0 is a regular point of a function x if xis differentiable at t ≥ 0, and denote its derivative by x′(·). We assume that therandom variables in Xr live on the same probability space (Ω,F ,P). Often, weconsider sample paths of stochastic processes, and whenever we want to makethe dependence on the sample path explicit, we write Xr(·, ω) for the sample pathassociated with ω ∈ Ω for a stochastic process Xr.

8.4 Fluid limit

To capture the rough system dynamics, we consider the fluid-scaled process

X = limr→∞

Xr, Xr =Xr

r.

8


For each process Xr in Xr, we define similarly its fluid equivalent as Xr = Xr/rand its limiting process X = limr→∞ Xr. We adopt the definition of a fluid limitand its invariant state(s) from [33]. That is, we consider A ⊂ Ω such that theFSLLN holds, i.e.

Λij(rx)

r→ pijλx, i, j ∈ E

Sj(rx)

r→ µx, j = 1, ..., S,

u.o.c. as r → ∞. Due to the FSLLN, we observe that one can choose A largeenough such that P(A) = 1.

Definition 8.4.1. We call X a fluid limit of Xr if there exists an ω ∈ A and a(sub)sequence rl with rl → ∞ as l → ∞, such that Xrl(·, ω) converges u.o.c. toX(·, ω). Moreover, let q = (q1, ..., qS) be an invariant state of the fluid limits if for anyfluid limit X, Q(0) = (Q1(0), ...., QS(0)) = (q1, ..., qS) = q implies that Q(t) = q forall t ≥ 0.

We point out that in the previous chapter, we considered the fluid-scaled queuelength process in Proposition 7.5.1 only for S = 1 and the sequence rl = l.Definition 8.4.1 allows for a more general setting. In particular, instead of requiringQr(0) → Q(0) with Q(0) a finite constant, Definition 8.4.1 allows for Q(0) to berandom. Proposition 7.5.1 implies that in case that S = 1, the fluid limits exist andare deterministic, (Lipschitz) continuous paths that depend only on the realizationof Q(0). Moreover, there is a single unique invariant state given by λ/µ. A similarresult holds when S ≥ 2.

Theorem 8.4.1. Let Xr be a sequence of systems. Then the fluid limits exist, where eachcomponent is Lipschitz continuous. Each fluid limit X satisfies the following equations forall t ≥ 0:

Aij(t) = Aij,i(t) + Aij,j(t) ∀i, j ∈ E, (8.4.1)Aij(t) = pijλY (t) ∀i, j ∈ E, (8.4.2)

Qj(t) = Qj(0) +∑

i:i,j∈EAij,j(t)− Dj(t), ∀j = 1, ..., S, (8.4.3)

Dj(t) = µTj(t), ∀j = 1, ..., S, (8.4.4)

Y (t) =

∫ t

0

L(s) ds, ∀j = 1, ..., S, (8.4.5)

Tj(t) =

∫ t

0

Zj(s) ds, ∀j = 1, ..., S, (8.4.6)

Zj(t) = minQj(t), pjλ/µ, ∀j = 1, ..., S, (8.4.7)

L(t) = 1−S∑

j=1

(Qj(t)− pjλ/µ

)+. (8.4.8)

Also, for every i, j ∈ E, if t is a regular point of X, then

A′ij,i(t) = λpijL(t), and A′ij,j(t) = 0 ifQj(t)

pj>Qi(t)

pi. (8.4.9)

8


Finally, there is a unique invariant state given by q = (q1, ..., qS) with qi = piλ/µ fori = 1, ..., S.

Proof. First, we show that the fluid limits exist, where all components are Lipschitzcontinuous. For this purpose, we show that for all ω ∈ A the sequence X(·, ω)has a convergent subsequence, where every component in the limiting process isLipschitz continuous.

Fix ω ∈ A. We observe that for every 0 ≤ t1 < t2,

∣∣∣∣T rj (t2, ω)

r−T rj (t1, ω)

r

∣∣∣∣ ≤F rj (t2 − t1)

r<

(λ

µ+ |γ|

√λ

µ

)(t2 − t1).

Therefore, there exists a subsequence rl such that T rlj (·, ω) converges u.o.c. tosome Tj(·, ω) as l→∞ for every j = 1, ..., S, which is Lipschitz continuous.

Using Lemma 11 in [3], Equation (8.3.4) and the fact that ω ∈ A, it follows thatDrj (·, ω) also converges u.o.c. to Dj(·, ω) for every j = 1, ..., S. In fact, it follows

that Dj(·, ω) = µTj(·, ω), and is therefore also Lipschitz continuous.

Next, we consider the arrival processes. First, we observe that Lr(t) ≤ r for everyt ≥ 0. Therefore,

∣∣∣∣Y r(t2, ω)

r− Y r(t1, ω)

r

∣∣∣∣ ≤ t2 − t1,

for all 0 ≤ t1 < t2, and hence there exists a subsequence rl such that Y rl(·, ω)converges u.o.c. to some Y (·, ω) as l → ∞ for every j = 1, ..., S, which is againLipschitz continuous.

Moreover, it follows from (8.3.2) for all 0 ≤ t1 < t2,

Arij(t2, ω)− Arij(t1, ω) ≤ Λij (rt2)− Λij (rt1)

r.

As ω ∈ A and the FSLLN applies, it follows from Theorem 12.3 in [15] that thereis some subsequence rl such that Arij(·, ω) converges u.o.c. as l → ∞ to someprocess Aij(·, ω). In particular, it holds for all 0 ≤ t1 < t2 that

Aij(t2, ω)− Aij(t1, ω) ≤ pijλ(t2 − t1),

and Aij is hence Lipschitz continuous. Similarly, we can show the same conver-gence result for the processes Arij,i(·, ω) to Aij,j(·, ω).

By (8.3.3), it follows also that Qrlj (·, ω) is precompact, which in turn impliesthat Zrlj (·, ω) is precompact due to (8.3.7). Moreover, Lrl(·, ω) is precompactby (8.3.8). In conclusion, the fluid limit exists with each component being Lipschitzcontinuous.

Fluid equations (8.4.1)-(8.4.8) follow from the FSLLN and applying Lemma 11 of[3]. Equation (8.4.9) requires additional arguments. Suppose X to be a fluid limitwith corresponding ω ∈ A and subsequence rll∈N. If for some t > 0 we have

8


that Qj(t)/pj > Qi(t)/pi, then it follows by the continuity of the fluid limit thatthere exists a δ > 0 such that

Qj(s)

pj>Qi(s)

pi

for all s ∈ [t− δ, t+ δ]. By definition of the fluid limit, it holds for large enough rlthat

Qrlj (s, ω)

pj>Qrli (s, ω)

pi

for all s ∈ [t− δ, t+ δ]. In this case, the routing policy states that all arrivals of typei, jmove to station station i. Therefore, Arlij,j remains constant on [t− δ, t+ δ],and hence A′ij,j(t) = 0. Moreover, station i receives all arrivals and by the FSLLNand (8.4.2),

Aij,i(t2, ω)− Aij,i(t1, ω) = pijλL(t, ω)(t2 − t1).

for all t1 < t2 with t1, t2 ∈ [t − δ, t + δ]. It follows that A′ij,i(t, ω) = pijλL(t, ω)by (8.3.1).

Finally, we show that q = (p1λ/µ, ..., pSλ/µ) is the unique invariant state in thissystem. Introduce the function

h(t) = max1≤j≤S

Qj(t)

pj

− min

1≤j≤S

Qj(t)

pj

,

and write

Smax(t) = arg max1≤j≤S

Qj(t)

pj

, Smin(t) = arg min

1≤j≤S

Qj(t)

pj

.

Trivially, h(t) ≥ 0. Since Q(·) is Lipschitz continuous, so is h(·), and hence it isdifferentiable almost everywhere. To show that h(0) = 0 implies h(t) = 0 for allt ≥ 0, it therefore suffices to show that if h(t) > 0 then h′(t) < 0 for every regularpoint t of X. By (8.4.3), we observe that for every j = 1, ..., S and regular pointt ≥ 0,

Q′j(t) =∑

i:i,j∈EA′ij,j(t)− D′j(t).

Due to (8.4.4)-(8.4.7), D′j(t) = minµQj(t), pjλ. In particular, we observe thatD′i(t)/pi = D′j(t)/pj for all i, j ∈ Smax(t), as well as for all i, j ∈ Smin(t). Due toLemma 2.8.6 from [32], as t is a regular point, it follows that

∑i:i,j∈E A

′ij,j(t)

pj=

∑k:k,l∈E A

′kl,l(t)

pl

8


for every j, l ∈ Smax(t), as well as for every j, l ∈ Smin(t). Due to (8.4.1), (8.4.2)and (8.4.9), we conclude that for every j ∈ Smax(t),

∑i:i,j∈E A

′ij,j(t)

pj=

1

pj

pj∑i∈Smax(t) pi

∑

i,j∈E,i,j∈Smax(t)

pijλL(t)

< λL(t).

On the other hand, for every j ∈ Smin(t),

∑i:i,j∈E A

′ij,j(t)

pj=

1

pj

pj∑i∈Smin(t) pi

∑

i,j∈E,i∈Smin(t)∪j∈Smin(t)

pijλL(t)

> λL(t).

Observing that D′j(t)/pj > D′i(t)/pi for every j ∈ Smax(t) and i ∈ Smin(t), wetherefore conclude that if h(t) > 0 with t a regular point, then

h′(t) < λL(t)− λL(t) = 0.

In other words, for every invariant state of the fluid limit, it must hold thatQi/pi(t) = Qj/pj(t) for every i 6= j. We observe, in view of (8.4.3), that

Q′j(t)

pj=

∑i:i,j∈E A

′ij,j(t)− D′j(t)pj

,

where for every 1 ≤ j ≤ S,

D′j(t)

pj= µminQj(t)/pj , λ/µ,

and hence in view of (8.4.1) and (8.4.2),

∑

i:i,j∈EA′ij,j(t) = pjλL(t).

That is, Q′j(t) = 0 if and only if

µminQj(t)/pj , λ/µ = pjλL(t).

In view of (8.4.8), this occurs if and only if Qj(t) = pjλ/µ for every j = 1, ..., S.

The invariant-state result for the fluid limit is central for the existence of a properly-defined diffusion process, as it states that if Q(0) = (p1λ/µ, ..., pSλ/µ), then thefluid limits are time-invariant.

8


8.5 Diffusion limit and system performance

Due to the policy governing which station a car drives to in order to replace abattery, one observes the so-called load-balancing effect. By setting the number ofresources as in (8.2.2), this load-balancing effect is so strong that in fact completeresource pooling occurs. In other words, the system behaves as if there is a singlelarge swapping station where the number of resources equals the aggregated totalof the individual stations. This appealing consequence ensures that there are noidle resources at one station, while at another there are possibly long lines of carsthat are waiting for a battery exchange.

The key concept to derive this effect is to show a state space collapse (SSC) result.That is, we consider the diffusion-scaled queue length process defined as

Qri (t) =Qri (t)− piλr/µpi√λr/µ

, i = 1, ..., S.

In our model, the SSC result states that (almost instantaneously) the diffusion-scaled queue length processes are arbitrarily close at all stations, and stay closeduring any fixed interval.

Theorem 8.5.1. Suppose

Qr(0)d→ Q(0),

as r →∞, where Q(0) is a random vector. Then, for every Kr = o(√r) with Kr →∞

as r →∞, and for every T > 0 and ε > 0,

P

(sup

Kr/√r≤t≤T

|Qi(t)− Qj(t)| > ε

)→ 0 (8.5.1)

for every i, j ∈ 1, ..., S as r →∞. If, in addition for every i, j ∈ 1, ..., S,

|Qri (0)− Qrj(0)| P→ 0,

then

P(‖Qi(·)− Qj(·)‖T > ε

)→ 0 (8.5.2)

for every i, j ∈ 1, ..., S as r →∞.

The proof of Theorem 8.5.1 is given in the next section. This SSC result reveals thatinstead of considering the individual queue length processes, it suffices to trackthe total queue length process. More specifically, define the sequence of randomprocesses QrΣ(t), t ≥ 0with r ∈ N, where QrΣ(t) =

∑Sj=1Q

rj(t), and

QrΣ(t) =

∑Sj=1(Qrj(t)− pjλr/µ)

√λr/µ

=

S∑

j=1

pjQrj(t).

8

8.5. Diffusion limit and system performance 243

As the state space collapse implies that Qri (t) ≈ Qrj(t) for all i, j ∈ 1, ..., S (fort ≥ Kr/

√r), we can approximate the queue length at an individual station by

Qrj(t) = pj

(λr

µ+ Qrj(t)

√λr

µ

)≈ pj

(λr

µ+ QrΣ(t)

√λr

µ

)

for all j = 1, ..., S. The limiting process of the total queue length can be derivedusing the SSC result.

Theorem 8.5.2. Suppose Qr(0)→ Q(0) in distribution as r →∞, and∣∣∣Qri (0)− Qrj(0)

∣∣∣ P→ 0

for all i, j ∈ 1, ..., S. Then, QrΣ → QΣ in distribution as r → ∞, where QΣ is adiffusion process with drift

m(x) = −λ(x− β)+ − µminx, γ,

and constant infinitesimal variance 2µ. The steady-state density QΣ(∞) is given by

fΣ(x) =

α1φ(x)Φ(γ) if x < γ,

α2

(γe−γ(x−γ)

) (1− e−γ(β−γ)

)−1if γ ≤ x < β,

α3

√λµφ


µ/λ

)Φ(−√

µλγ)−1

if x ≥ β,(8.5.3)

where αi = ri/(r1 + r2 + r3), i = 1, 2, 3, with

r1 = 1,

r2 =

φ(γ)Φ(γ) 1

γ

(1− e−γ(β−γ)

)if γ 6= 0,√

2πβ if γ = 0,

r3 =φ(γ)


√µ

λφ

(√µ

λγ

)−1

Φ

(−√µ

λγ

).

Proof. We observe that the steady-state density is a direct consequence of thediffusion process [21]. What remains to be shown is that QrΣ converges to thedescribed diffusion process as r →∞. Equivalently, we need to show that

dQΣ(t) = −λ(QΣ(t)− β

)+

− µminQΣ(t), γ

+√

2µdW (t), (8.5.4)

where W (t), t ≥ 0 is a standard Brownian motion. We note that due to thesystem identities,

QΣ(t) = QΣ(0) +

∑i,j∈E A

rij(t)−

∑Sj=1D

rj (t)√

λr/µ,

8


where∑

i,j∈EArij(t) = Λ

(∫ t

0

Lr(s) ds

),

andS∑

j=1

Drj (t) =

S∑

j=1

Sj

(∫ t

0

Zrj (s) ds

).

We observe that due to the FCLT and Theorem 8.4.1,

Λ(∫ t

0Lr(s) ds

)− λ

∫ t0Lr(s) ds

√λr/µ

=Λ(r∫ t

0Lr(s) ds

)− λr

∫ t0Lr(s) ds

√1/µ√λr

d→ BMA(t),

where BMA(t), t ≥ 0 is Brownian motion with mean zero and variance µ. Simil-arly, due to the FCLT and Theorem 8.4.1,

S∑

j=1

Sj

(∫ t0Zrj (s) ds

)− µ

∫ t0Zrj (s) ds

√λr/µ

=

S∑

j=1

Sj

(r∫ t

0Zr(s) ds

)− µr

∫ t0Zr(s) ds

√1/(pjµ)

√pjλr

d→ BMD(t),

where BMD(t), t ≥ 0 is an (independent) Brownian motion with mean zeroand variance µ. The sum of these two processes is equal to (in distribution) aBrownian with mean zero and variance 2µ, which contributes the

√2µdW (t) term

in (8.5.4). Next, we observe that due to the system identities and the definition ofthe diffusion scaling,

λ∫ t

0Lr(s) ds−∑S

j=1 µ∫ t

0Zrj (s) ds

√λr/µ

= −λS∑

j=1

pj

∫ t

0

(Qrj(s)− β

)+

ds− µS∑

j=1

pj

∫ t

0

minQrj(s), γ

ds.

Since

min1≤j≤S

Qr(t) ≤ QrΣ(t) ≤ max1≤j≤S

Qr(t)

for all t ∈ [0, T ], and due to Theorem 8.5.1,∥∥∥QrΣ(·)− Qrj(·)

∥∥∥T≤ ε(r), j = 1, ..., S,

where the sequence ε(r), r ∈ N can be chosen such that ε(r) → 0 as r → ∞.Then, as r →∞,

λ∫ t

0Lr(s) ds−∑S

j=1 µ∫ t

0Zrj (s) ds

√λr/µ

P→ −λ∫ t

0

(QΣ(s)− β

)+

ds− µ∫ t

0

minQΣ(s), γ

ds.

8


This contributes the first two terms in (8.5.4). Applying the continuous-mappingtheorem [15] concludes the result.

Another consequence of the state space collapse result is that the waiting probabil-ities and expected waiting times are equal at all stations, as well as the resourceutilization levels. In fact, it exhibits the same behavior as if there would be a singlestation due to the complete resource pooling effect.

Corollary 8.5.1. Suppose the system is operating under (8.2.2). Then the followingproperties hold as r →∞ for all i = 1, ..., S. The waiting probability has a non-degeneratelimit given by

P(W ri > 0)→ P

(QΣ(∞) ≥ β

)

=

(1 +

√λ

µ

φ(√µ/λγ)

φ(γ)eγ(β−γ) Φ(γ)

Φ(−√µ/λγ)

+

√λ

µ

φ(√µ/λγ)

γ

(eγ(β−γ) − 1

)Φ

(−√µ

λγ

)−1)−1

.

The expected waiting time behaves as

E(W ri )√r→ α3√

λµ

(√µ

λφ(µλγ)

Φ

(−√µ

λγ

)−1

− µ

λγ

).

Finally, the resource utilizations behave as

ρF ri → 1, ρBri → 1.


To prove Theorem 8.5.1, we use a framework similar to that of [20], and [33]. Theconstruction consists of several steps, which we lay out next.

1. Divide interval [0, T ] into T√r intervals of length 1/

√r, indexed by m. In

each interval, consider the hydrodynamically-scaled process of X. For each ofthese intervals, we

a) show the scaled process is "almost" Lipschitz continuous;b) show convergence to some hydrodynamic limiting process for a suffi-

ciently large part of the state space;c) derive the hydrodynamic limit equations.

2. Relate the hydrodynamic scaling to the diffusion scaling, using a SSC func-tion to deal with complications regarding the range of the time variable.Transferring the results appropriately, we show multiplicative SSC with re-spect to the SSC function.

3. Using a compact containment condition, we show that this implies strongSSC.

8


8.6.1 Hydrodynamic scaling and its limiting process

In order to introduce the hydrodynamic scaling, we use a diffusion scaling for thevalues of the process but we slow the process down in time in order to analyzewhat occurs initially (what would happen instantaneously on diffusive scale).That is, we divide the interval [0, T ] in T

√r intervals of length 1/

√r, indexed by

m. We write p = (p1, ..., pS), and

xr,m = max

∣∣∣∣Qr(m√r

)− pλr

µ

∣∣∣∣2

,

∣∣∣∣Lr(m√r

)− r∣∣∣∣2

, r

. (8.6.1)

For the processes in X, we introduce the following hydronimically-scaled variants.For Qr, Zr and Lr, let

Qr,m(t) =1

√xr,m

(Qr(m√r

+

√xr,mt

r

)− pλr

µ

), (8.6.2)

Zr,m(t) =1

√xr,m

(Zr(m√r

+

√xr,mt

r

)− pλr

µ

), (8.6.3)

Lr,m(t) =1

√xr,m

(Lr(m√r

+

√xr,mt

r

)− r), (8.6.4)

the deviations of these processes with respect to their fluid limits. For the processesAr, Ad, Y r, T r and Dr, we introduce

Ar,m(t) =1

√xr,m

(Ar(m√r

+

√xr,mt

r

)−Ar

(m√r

)), (8.6.5)

Ar,md (t) =1

√xr,m

(Ard

(m√r

+

√xr,mt

r

)−Ard

(m√r

)), (8.6.6)

Y r,m(t) =1

√xr,m

(Y r(m√r

+

√xr,mt

r

)− Y r

(m√r

)), (8.6.7)

T r,m(t) =1

√xr,m

(T r(m√r

+

√xr,mt

r

)− T r

(m√r

)), (8.6.8)

Dr,m(t) =1

√xr,m

(Dr

(m√r

+

√xr,mt

r

)−Dr

(m√r

)). (8.6.9)

In other words, we track the increase of these processes during the interval[m/√r,m/

√r +√xr,mt/r]. By definition of xr,m, we note that

|Xr,m(0)| ≤ 1,

which will be a required compactness property when we prove convergence to ahydrodynamic limit. Moreover, due to our fluid-limit results, we can show that√xr,m/r is very small for all ω ∈ A.

Lemma 8.6.1. Suppose Qr(0) → Q(0) for some random vector Q(0), and let M > 0fixed. For every ε > 0 and ω ∈ A,

maxm<√rT

√xr,m

r‖Qr,m(t)‖M ,

√xr,m

r‖Lr,m(t)‖M

≤ ε

for r large enough.

8


Proof. Due to our fluid limit result in Theorem 8.4.1 and the definition of xr,min (8.6.1), we observe that

√xr,m

r≤ max‖Qr(t)− pλr/µ‖T /r, ‖Lr(t)− r‖T /r, 1/

√r ≤ ε

for r large enough. Moreover, for r large enough,

max‖Qr(t)− pλr/µ‖T+Mε /r, ‖Lr(t)− r‖T+Mε/r

≤ ε.

We conclude that for every m ≤ √rT ,√xr,m

r‖Qr,m(t)‖M =

‖Qr(m/√r +√xr,m/rt)− pλr/µ‖Mr

≤ ε,

and√xr,m

r‖Lr,m(t)− r‖M ≤ ε.

For the hydrodynamically scaled process, the system identities translate to

Ar,mij (t) = Ar,mij,i (t) +Ar,mij,j (t) ∀i, j ∈ E, (8.6.10)

Ar,mij (t) =Λij(Y r(m/

√r) +

√xr,mY

r,m(t))− Λij (Y r(m/

√r))

√xr,m

, ∀i, j ∈ E,

(8.6.11)

Qr,mj (t) = Qr,mj (0) +∑

i:i,j∈EAr,mij,j (t)−Dr,m

j (t), ∀j = 1, ..., S, (8.6.12)

Dr,mj (t) =

Sj(T rj (m/

√r) +

√xr,mT

r,mj (t)

)− Sj

(T rj (m/

√r))

√xr,m

, ∀j = 1, ..., S,

(8.6.13)

Y r,m(t) = t+

√xr,m

r

∫ t

0

Lr,m(s) ds, ∀j = 1, ..., S, (8.6.14)

T r,mj (t) = pjλ

µt+

√xr,m

r

∫ t

0

Zr,mj (s) ds, ∀j = 1, ..., S, (8.6.15)

Zr,mj (t) = min

Qr,mj (t),

1√xr,m

γp

√λr

µ

, ∀j = 1, ..., S, (8.6.16)

Lr,m(t) = −S∑

j=1

(Qr,mj (t)− 1

√xr,m

βpj

√λr

µ

)+

(8.6.17)

Ar,mij,i (t) can only increase whenQr,mi (t)

pi≤Qr,mj (t)

pj∀i, j ∈ E. (8.6.18)

In order to show that Xr,m is almost (with the exception of certain events) Lipschitzcontinuous, we would like to exclude these events, i.e. show that such events areunlikely to occur.

8


Lemma 8.6.2. Fix ε > 0, M > 0 and T > 0. For r large enough, there exists a constantN > 0 (only depending on λ, µ, and pij ; i, j ∈ E) such that

P(

maxm<√rT

sup0≤t1≤t2≤M

|Ar,m(t2)−Ar,m(t1)| −N(t2 − t1) ≥ ε)≤ ε, (8.6.19)

P(

maxm<√rT

sup0≤t1≤t2≤M

|Dr,m(t2)−Dr,m(t1)| −N(t2 − t1) ≥ ε)≤ ε. (8.6.20)

Moreover,

P(

maxm<√rT

∥∥∥∥Y r,m(t)− 1

pijλAr,mij (t)

∥∥∥∥M

≥ ε)≤ ε, i, j ∈ E (8.6.21)

P(

maxm<√rT

∥∥∥∥Tr,mj (t)− 1

µDr,mj (t)

∥∥∥∥M

≥ ε)≤ ε, j = 1, ..., S, (8.6.22)

Proof. First, we note that due to the memoryless property which both the arrivaland service completion processes satisfy, the choice of m is irrelevant and thus canbe made arbitrarily. To prove (8.6.19), we first show that for every i, j ∈ E,

P(

sup0≤t1≤t2≤M

Ar,mij (t2)−Ar,mij (t1)− 1

pijλ(t2 − t1)

≥ ε)≤ ε.

From (8.6.14) and (8.6.17), we observe that Yr,m(t) ≤ t and non-decreasing. Dueto the properties of Poisson processes,

Ar,mij (t2)−Ar,mij (t1)d=

Λij(√xr,mY

r,m(t2))− Λij

(√xr,mY

r,m(t1))

√xr,m

d≤ Λij

(√xr,mt2

)− Λij

(√xr,mt1

)√xr,m

.

Therefore,

P(

sup0≤t1≤t2≤M

Ar,mij (t2)−Ar,mij (t1)− t2 − t1

pijλ

≥ ε)

≤ P

(sup

0≤t1≤t2≤M

Λij(√xr,mt2

)− Λij

(√xr,mt1

)√xr,m

− t2 − t1pijλ

≥ ε)

≤ P

(∥∥∥∥∥Λij(√xr,mt

)√xr,m

− t

pijλ

∥∥∥∥∥M

≥ ε/2)≤ ε

2M2√xr,m≤ ε

2M2√r,

where the final to last inequality follows from Proposition 4.3 in [20]. ChoosingN = 1/(λmini,j∈E pij) and applying the union bound twice, we obtain

P(

maxm<√rT

sup0≤t1≤t2≤M

|Ar,m(t2)−Ar,m(t1)| −N(t2 − t1) ≥ ε)≤ εT |E|

2M2,

which yields (8.6.19).

8


The proof for (8.6.20) is completely analogous, but with minor adaptations asone uses T r,mj (t) ≤ pjλ/µt instead of Y r,m(t) ≤ t. We conclude that (8.6.19)and (8.6.20) show that the hydrodynamically scaled arrival process and servicecompletion process are almost Lipschitz continuous.

In order to prove (8.6.21) and (8.6.22), we introduce the following processes. Letuij(l), l ≥ 1 be independent exponentially distributed random variables withrate pijλ, representing the time that a car has before it needs recharging at eitherstation i or j. Let vj(l), l ≥ 1 be independent exponentially distributed randomvariables with rate µ, representing the service requirement (recharging time) of abattery at station j. Define

Uij(n) =

n∑

l=1

uij(l), i, j ∈ E

Vj(n) =

n∑

l=1

vj(l), j = 1, ..., S,

the aggregated interarrival time of n cars that will choose between stations i and j,and the total service requirement of n batteries at station j, respectively. Weobserve the identities

Λij(t) = maxn : Uij(n) ≤ t, Sj(t) = maxn : Vj(n) ≤ t, t ≥ 0.

Moreover, due to (8.3.2) and (8.3.4), we observe

Uij(Arij(t)) ≤Y r(t) ≤ Uij(Arij(t) + 1), i, j ∈ E (8.6.23)

Vj(Drj (t)) ≤T rj (t) ≤ Vj(Dj(t) + 1), j = 1, ..., S. (8.6.24)

As in [33], we define for notational convention, for b = (b1, b2) ∈ N,

Ur,mij(Ar,mij (t), b

)

=1

√xr,m

(Urij

(Arij

(m√r

+

√xr,mt

r

)+ b1

)− Urij

(Arij

(m√r

)+ b2

)),

V r,mj

(Dr,mj (t), b

)

=1

√xr,m

(V rj

(Drj

(m√r

+

√xr,mt

r

)+ b1

)− V rj

(Drj

(m√r

)+ b2

)).

In view of (8.6.23) and (8.6.24), this yields the inequalities

Ur,mij (Arij(t), (0, 1)) ≤Y r,m(t) ≤ Ur,mij (Arij(t), (1, 0)), i, j ∈ E (8.6.25)

V r,mj (Dj(t), (0, 1)) ≤T r,mj (t) ≤ V r,mj (Dj(t), (1, 0)), j = 1, ..., S. (8.6.26)

Using these processes, we first prove the following bounds:

P(

maxm<√rT

∥∥∥∥Uij(Ar,mij (t), b)− 1

pijλAr,mij (t)

∥∥∥∥M

≥ ε)≤ ε, ∀i, j ∈ E,

(8.6.27)

P(

maxm<√rT

∥∥∥∥Vj(Dr,m(t), b)− 1

µDr,mj (t)

∥∥∥∥M

≥ ε)≤ ε, j = 1, ..., S,

(8.6.28)

8


for b = (1, 0) and b = (0, 1). The proof is similar to that of (78) in [119]. We observethat the proof of (8.6.19) implies that in particular

P(Arij

(√xr,mM

r

)≥ 2M

pijλ

√xr,m

)≤ ε

M2√r,

and hence also

P(Arij

(√xr,mM

r

)+ 1 ≥ 3M

pijλ

√xr,m

)≤ ε

M2√r

for r large enough. Proposition 4.2 of [20] states

P(∥∥∥∥Uij(l)−

l

pijλ

∥∥∥∥n

≥ εn)≤ ε

n.

Therefore, it follows that

P

(∥∥∥∥Uij(Arij(t))−Arij(t)

pijλ

∥∥∥∥√xr,mM/r

≥ ε2M√xr,m

pijλ

)≤ ε√

r

(1

M2+pijλ

2M

),

and

P

(∥∥∥∥Uij(Arij(t) + 1)−Arij(t)

pijλ

∥∥∥∥√xr,mM/r

≥ ε3M√xr,m

pijλ

)≤ ε√

r

(1

M2+pijλ

3M

).

Increasing ε appropriately, we obtain

P

(∥∥∥∥∥Ur,mij (Ar,mij (t), b)−

Ar,mij (t)

pijλ

∥∥∥∥∥M

≥ ε)≤ ε

T√r

for both b = (0, 0) and b = (1, 0). Using the union bound yields

P

(max

m<√rT

∥∥∥∥∥Ur,mij (Ar,mij (t), b)−

Ar,mij (t)

pijλ

∥∥∥∥∥M

≥ ε)≤ ε.

for both b = (0, 0) and b = (1, 0). To conclude the proof for b = (0, 1) as well, weobserve

P(

maxm<√rT

∥∥Ur,mij (Ar,mij (t), b)− Ur,mij (Ar,mij (t), (0, 0))∥∥M≥ ε)

≤ P(

maxm<√rT

∣∣∣∣Uij(Arij

(m√r

)+ 1

)− Uij

(Arij

(m√r

))∣∣∣∣ ≥ ε√xr,m

)

≤ P(ur,T,maxij ≥ √xr,mε

)≤ ε,

where the final inequality follows from Lemma 5.1 in [20] with

ur,T,maxij = maxuij(l) : Ui(l − 1) ≤ rT.

The proof of (8.6.28) is analogous to (8.6.27), replacing the arrival processes by theservice processes. Equations (8.6.21) and (8.6.22) are then a direct consequenceof (8.6.25) and (8.6.26).

8


Using the previous result, we can show that X is almost Lipschitz continuous.

Proposition 8.6.1. Fix ε > 0, M > 0 and T > 0. For r large enough,

P(

maxm<√rT

sup0≤t1≤t2≤M

|Xr,m(t2)− Xr,m(t1)| −N(t2 − t1) ≥ ε)≤ ε,

where N <∞ is constant (depending only on λ, µ and pij ; i, j ∈ E).

Proof. This follows in a straightforward way from Lemma 8.6.2 and the hydro-dynamically scaled system equations. That is, let Vr denote the intersection of thecomplements of the events given in equations (8.6.19)-(8.6.22), so P(Vr) ≤ 1−N0εwithN0 the number of equations in Lemma 8.6.2. We note that in order to prove theproposition, it suffices to show that for every ω ∈ Vr, and for every t1, t2 ∈ [0, T ]and m <

√rT ,

|Xr,m(t2)− Xr,m(t1)| ≤ N1(t2 − t1) +N2ε, (8.6.29)

where N1 and N2 are only dependent on the system parameters (i.e. λ, µ, p). Lett1, t2 ∈ [0, T ] with t1 ≤ t2. By definition of Vr,

|Ar,m(t2)−Ar,m(t1)| ≤ N(t2 − t1) + ε,

and

|Dr,m(t2)−Dr,m(t1)| ≤ N(t2 − t1) + ε,

for N as in Lemma 8.6.2. Due to (8.6.10),

|Ar,md (t2)−Ar,md (t1)| ≤ |Ar,m(t2)−Ar,m(t1)| ≤ N(t2 − t1) + ε.

In view of (8.6.12) and (8.6.10), we observe

|Qr,m(t2)−Qr,m(t1)| ≤ |E| |Ar,m(t2)−Ar,m(t1)|+ S |Dr,m(t2)−Dr,m(t1)|≤ (|E|+ S)N(t2 − t1) + 2ε.

Due to (8.6.21),

|Y r,m(t2)− Y r,m(t1)| ≤∑

i,j∈E

|Ar,m(t2)−Ar,m(t1)|pijλ

+ 2ε

≤∑

i,j∈E

N

pijλ(t2 − t1) +

∑

i,j∈E

1

pijλ+ 2

ε.

and similarly, due to (8.6.22),

|T r,m(t2)− T r,m(t1)| ≤ 1

µ|Dr,m(t2)−Dr,m(t1)|+ 2ε ≤ N

µ(t2 − t1) +

(1

µ+ 2

)ε.

In view of (8.6.16),

|Zr,m(t2)− Zr,m(t1)| ≤ |Qr,m(t2)−Qr,m(t1)| ≤ (|E|+ S)N(t2 − t1) + 2ε.

8


Finally, due to (8.6.17),

|Lr,m(t2)− Lr,m(t1)| ≤ S |Qr,m(t2)−Qr,m(t1)| ≤ S(|E|+ S)N(t2 − t1) + 2Sε.

We conclude that (8.6.29) is satisfied, as each process in Xr,m satisfies this property.

As is done in [20, 33], one can take ε appropriately small for every system. That is,for fixed M > 0, N > 0 and T > 0, let

Kr0 =

max

m<√rT

sup0≤t1≤t2≤M

|Xr,m(t2)− Xr,m(t1)| ≥ N(t2 − t1) + ε(r)

,

where ε(r) → 0 as r → ∞ is a sequence of positive real numbers. Moreover, inview of Lemma 8.6.1, let for that same sequence ε(r)r∈R,

Hr =

max

m<√rT

√xr,m

r‖Qr,m(t)‖M ,

√xr,m

r‖Lr,m(t)‖M

≤ ε(r)

.

Let Kr denote the intersection of Kr0, Hr, and the complements of the eventsin Lemma 8.6.2. We note that Lemma 8.6.1, Lemma 8.6.2 and Proposition 8.6.1continue to hold for the sequence ε(r) if ε(r)→ 0 sufficiently slow. We concludethat P(Kr)→ 1 as r →∞.

Corollary 8.6.1. Fix M > 0 and choose N > 0 and ε(r) as above. Then,

limr→∞

P(Kr) = 1.

Following the framework of [20], we can use these results to state that the hydro-dynamically scaled system converges to a hydrodynamic limit. Fix M > 0 and letE be the set of right-continuous functions x : [0,M ]→ Rd with left limits. Let

E′ = x ∈ E : |x(0)| ≤ 1, |x(t2)− x(t1)| ≤ N |t2 − t1| ∀t1, t2 ∈ [0,M ].

Moreover, we set

Er = Xr,m,m <√rT, ω ∈ Kr,

and

E = Er, r ∈ N.

We remark that these definitions are not related to E, the set of all possible pairsof stations where cars can move to.

Definition 8.6.1. A hydrodynamic limit of E is a point x ∈ E such that for all ε > 0 thereexists a r0 ∈ N so that for every r ≥ r0 there is some y ∈ Er such that |x(·)−y(·)|M < ε.

Since |Xr,m(0)| ≤ 1, the following result is a consequence of Proposition 4.1 in [20].

8


Corollary 8.6.2. Let E, Er, E be as above. Fix ε > 0, M > 0, T ≥ 0, and choose r largeenough. Then, for ω ∈ Kr and any m <

√rT ,

‖Xr,m(·)− X(·)‖M ≤ ε

for some hydrodynamic limit X(·) ∈ E′ of E .

Finally, to conclude this section, we derive the equations that are satisfied by anyhydrodynamic limit.

Proposition 8.6.2. Let M > 0 be fixed, and let X be a hydrodynamic limit of E over[0,M ]. Then X satisfies the following equations:

Aij(t) = Aij,i(t) + Aij,j(t) ∀i, j ∈ E, (8.6.30)

Aij(t) = pijλY (t) = pijλt ∀i, j ∈ E, (8.6.31)

Qj(t) = Qj(0) +∑

i:i,j∈EAij,j(t)− Dj(t), ∀j = 1, ..., S, (8.6.32)

Dj(t) = µTj(t) = pjλt, ∀j = 1, ..., S, (8.6.33)

Y (t) = t, ∀j = 1, ..., S, (8.6.34)

Tj(t) = pjλ/µt, ∀j = 1, ..., S, (8.6.35)

A′ij,i(t) =

pijλ if Qi(t)pi<

Qj(t)pj

0 if Qj(t)pj> Qi(t)

pi

∀i, j ∈ E. (8.6.36)

Remark 8.6.1. We cannot provide such general equations for Z(·) or L(·), sincethese limits depend on xr,m. That is, the processes Zr,m(·) and Lr,m(·) converge toa limit, but the limiting process may differ for different m. In the proof, we specifythe limiting equations of these processes as well.

Proof of Proposition 8.6.2. Let X be a hydrodynamic limit of E . For a given δ > 0,choose (r,m) such that ε(r) ≤ δ, and

‖X(t)− Xr,m(t, ω)‖M ≤ δ.

Due to (8.6.14), (8.6.15) and ω ∈ Hr, we derive

‖Y (t)− t‖M ≤ (1 +M)δ, ‖Tj(t)− pjλ/µt‖M ≤ (1 +M)δ, j = 1, ..., S.

From (8.6.21) and (8.6.22), we obtain

‖Aij(t)− pijλt‖M ≤ (2 +M)δ, i, j ∈ E,

and

‖Dj(t)− pjλt‖M ≤ (2 +M)δ, j = 1, ..., S.

8


Equation (8.6.30) is a clear consequence of (8.6.10). Combining the above equa-tions, we observe

∥∥∥∥∥∥Qj(t)− Qj(0)−

∑

i:i,j∈EAij,j(t) + Dj(t)

∥∥∥∥∥∥M

≤ 2(|E|+ S)(2 +M)δ.

These bounds imply that any hydrodynamic limit satisfies equations (8.6.30)-(8.6.35). Finally, we still have to show (8.6.36). If for some t ∈ [0,M ],

Qj(t)

pj>Qi(t)

pi,

then by continuity of X, there exists a η > 0 such that this also holds for alls ∈ [t− η, t+ η), as well as

Qr,mj (t)

pj>Qr,mi (t)

pi.

Due to (8.6.18), this implies that Ar,mij,i (s) is constant on s ∈ [t−η, t+η]). Therefore,its limit is also constant on [t− η, t+ η], and hence the derivative is zero. On theother hand, if

Qj(t)

pj<Qi(t)

pi,

then by continuity of X, there exists a η > 0 such that this also holds for alls ∈ [t− η, t+ η]), and

Qr,mj (t)

pj<Qr,mi (t)

pi.

Since A′ij,j = 0, and due to (8.6.10) with limiting process (8.6.31),

A′ij,i = limη↓0

A′ij(t+ η)− A′ij(t)η

= pijλ.

8.6.2 The SSC function

In this section, we introduce the state space collapse (SSC) function under whichwe show multiplicative state space collapse. The SSC function we use in thischapter is g : RS → R, defined as

g(q) = max1≤j≤S

qjpj− min

1≤j≤Sqjpj

(8.6.37)

where q = (q1, ..., qS). We note that g(·) is a non-negative continuous function andsatisfies

g(αq) = αg(q)

for every α > 0.

8


Lemma 8.6.3. Suppose g : RS → R is defined as in (8.6.37). Then,

g(Q(t)) ≤ H(t) ∀t ≥ 0,

for every hydrodynamic model solution X satisfying |X(0)| ≤ 1, where

H(t) =

(2

min1≤j≤S pj− ht

)+

, (8.6.38)

with h > 0 some constant that depends only on λ and pij , i, j ∈ E. Moreover, ifg(Q(0)) = 0 and |X(0)| ≤ 1, then g(Q(t)) = 0 for all t ≥ 0.

Proof. The proof relies heavily on the ideas used in the proof for the fluid limit.

h = min

λ∑i∈I pi

∑

i,j∈E,i∈I∪j∈I

pij −λ∑

j∈J pj

∑

i,j∈E,i∈J∩j∈J

pij : I,J 6= ∅, I ∩ J = ∅

,

where I,J ⊆ 1, ..., S. Since

1∑i∈I pi

∑

i,j∈E,i∈I∪j∈I

pij > 1,1∑

j∈J pj

∑

i,j∈E,i∈J∩j∈J

pij < 1,

for any non-empty I,J ⊂ 1, ..., S with I ∩ J = ∅, we observe that h < 0. For ahydrodynamic limiting process X, let HX(·) be given by

HX(t) =(g(Q(0))− ht

)+

.

We note that this function is non-negative, and HX(t) = 0 for all t ≥ g(Q(0))/h.

To show that g(Q(t)) is bounded by this function, we note that it suffices to showthat whenever g(Q(t)) > 0 with t ≥ 0 being a regular point of X,

g′(Q(t)) ≤ −h.

For this purpose, let

Smax(t) =

i ∈ 1, ..., S : Qi(t)/pi = max

1≤j≤SQj(t)/pj

,

and

Smin(t) =

i ∈ 1, ..., S : Qi(t)/pi = min

1≤j≤SQj(t)/pj

.

Due to Lemma 2.8.6 in [32], it holds for all i, j ∈ Smax(t) that Q′i(t)/pi = Q′j(t)/pj ,and similarly, for all i, j ∈ Smin(t) it holds that Q′i(t)/pi = Q′j(t)/pj . Therefore,

8


due to hydrodynamic limit equations (8.6.30)-(8.6.36) and the observation thatthere is at least one station j 6∈ Smax(t) such that i, j ∈ E, it follows that

Q′i(t)pi

<λ∑

i∈Smax(t) pi

∑

i,j∈E,i,j∈Smax(t)

pij − λ

for all i ∈ Smax(t). Similarly, for all i ∈ Smin,

Q′i(t)pi

>λ∑

i∈Smin(t) pi

∑

i,j∈E,i∈Smin(t)∪j∈Smin(t)

pij − λ.

We conclude that g′(Q(t)) ≤ −h, and hence g(Q(t)) ≤ HX(t) for all t ≥ 0. Inparticular, if g(Q(0)) = 0 and |X(0)| ≤ 1, it follows from the definition of HX(·)that g(Q(t)) = 0 for all t ≥ 0.

The first statement of the lemma follows since for every hydrodynamic modelsolution X satisfying |X(0)| ≤ 1, it holds that g(Q(0)) ≤ 2/min1≤j≤S pj . Hence,HX(·) ≤ H(·) for every hydrodynamic model solution X satisfying |X(0)| ≤ 1.

This result implies that the hydrodynamically scaled queue length almost satisfiesthis property as well. The next result is an immediate consequence of Corol-lary 8.6.2.

Corollary 8.6.3. Fix ε > 0, M > 0 and T > 0. Then, for every ω ∈ Kr,g(Qr,m(t)) ≤ H(t) + ε

for all t ∈ [0,M ] and m <√rT , where H(·) is as in Lemma 8.6.3. Moreover, if

g(Q(0))→ 0 in probability as r →∞, then for all ω ∈ Lr with

Lr = Kr ∩∣∣g(Qr,0(0))

∣∣ ≤ ε

it holds that

‖g(Qr,0(t))‖M ≤ ε,and

limr→∞

P(Lr) = 1.

8.6.3 Multiplicative state space collapse

The goal of this section is to show multiplicative state space collapse for the SSCfunction defined in (8.6.37). To do so, we first need to relate the hydrodynamicand diffusion scaling. That is, we observe that

Qr,mj (t) = pj

√λr/µ

xr,mQrj

(m√r

+

√xr,mt

r

)=pj√λ/µ

yr,mQrj

(1√r

(m+ yr,mt)

),

8


where

yr,m =

√xr,mr

= max

∣∣∣∣∣p√λ

µQr(m√r

)∣∣∣∣∣ ,∣∣∣∣Lr

(m√r

)∣∣∣∣ , 1,

with

Lr(t) =Lr(t)− r√

r.

Corollary 8.6.3 can be translated to the diffusion scaled process. Consider the SSCfunction g : RS → R defined as

g(q) = max1≤j≤S

qj − min1≤j≤S

qj

with q = (q1, ..., qS).

Corollary 8.6.4. Fix ε > 0, M > 0 and T > 0. Then for r large enough, and ω ∈ Kr,

g(Qr(t)) ≤ yr,m√λ/µ

H

(1

yr,m(√rt−m)

)+ ε

yr,m√λ/µ

for all t ∈ [0, T ] with m ∈ N such that

m√r≤ t ≤ m+ yr,mM√

r.

Also, for all ω ∈ Lr,

‖g(Qr(t)‖Myr,0/√r ≤ ε

yr,0√λ/µ

.

Since H(·) is given by (8.6.38), it holds that H(t) = 0 for all t ≥ 2/(hmin1≤j≤S pj).We would like to show that (

√rt−m)/yr,m can be chosen large enough to obtain a

very small upper bound, and use that property to show multiplicative state spacecollapse.

Lemma 8.6.4. Suppose M ≥ 2(N + 2) is fixed, and let

mr(t) = min

m ∈ N :

m√r≤ t ≤ m+ yr,mM√

r

.

Then for r large enough,√rt−mr(t)

yr,mr(t)≥ M

2(N + 2)

for every ω ∈ Kr and t ∈ (Myr,0/√r, T ].

8


Proof. For every ω ∈ Kr, by definition of the set,

|Xr,m(t2)− Xr,m(t2)| ≤ N |t2 − t1|+ ε,

for t1, t2 ∈ [0,M ] and m <√rT . In particular, for t2 = 1/yr,m, t1 = 0 and ε ≤ 1,

max

∣∣∣∣Qr(m+ 1√

r

)−Qr

(m√r

)∣∣∣∣ ,∣∣∣∣Lr

(m+ 1√

r

)− Lr

(m√r

)∣∣∣∣

≤ √xr,mN

yr,m+√xr,m.

Applying the reverse triangle inequality, we observe∣∣∣∣∣p√λ

µQr(m+ 1√

r

)∣∣∣∣∣−∣∣∣∣∣p√λ

µQr(m√r

)∣∣∣∣∣ ≤∣∣∣∣∣p√λ

µQr(m+ 1√

r

)− p√λ

µQr(m√r

)∣∣∣∣∣≤ N + yr,m ≤ (N + 1)yr,m,

and similarly,∣∣∣∣Lr

(m+ 1√

r

)∣∣∣∣−∣∣∣∣Lr

(m√r

)∣∣∣∣ ≤ (N + 1)yr,m.

Therefore, it always holds that

yr,m+1 ≤ yr,m + (N + 1)yr,m = (N + 2)yr,m.

For every t ∈ (Myr,0/√r, T ], it follows by definition of mr(t) that√rt ≥ mr(t)− 1 + yr,mr(t)−1M.

In particular,√rt−mr(t)

yr,mr(t)≥ yr,mr(t)−1M − 1

yr,mr(t)≥ M

N + 2− 1

yr,mr(t)≥ M

2(N + 2),

where the last inequality follows since M ≥ 2(N + 2).

Next, we show the main result of this section.

Theorem 8.6.1. Suppose Qr(0)→ Q(0) for some random vector Q(0). For every T > 0,ε > 0 and M <∞ with

M ≥ max

4(N + 2)

hmin1≤j≤S pj, 2(N + 2), 1

,

it holds that

P

(supMyr,0/

√r≤t≤T g(Qr(t))

max‖Qr(t)‖T , 1> ε

)→ 0, (8.6.39)

as r →∞. If, in addition, g(Qr(0))→ 0 in probability as r →∞, then for every T > 0,

‖g(Qr(t))‖Tmax‖Qr(t)‖T , 1

P→ 0, (8.6.40)

as r →∞.

8


Proof. Fix η > 0 and note that by construction there exists a r0 such that for allr > r0,

P(Kr) > 1− η.

For every ω ∈ Kr, we have derived bounds that only require that M is bounded.We note that for M as in the statement of the theorem allows for every t ∈ [0, T ],m/√r ≤ t ≤ (m + yr,mM)/

√r for some m <

√rM . Moreover, it follows from

Lemma 8.6.4 and (8.6.38), that

H

(1

yr,mr(t)(√rt−mr(t))

)= 0

for all t ∈ (Myr,0/√r, T ]. In view of Corollary 8.6.4, we obtain for every ε > 0,

g(Qr(t)) ≤ εyr,mr(t)√λ/µ

for all t ∈ (Myr,0/√r, T ]. Since for all t ∈ [0, T ],

yr,mr(t) = max

∣∣∣∣∣p√λ

µQr(mr(t)√

r

)∣∣∣∣∣ ,∣∣∣∣Lr

(mr(t)√

r

)∣∣∣∣ , 1

≤ max

∥∥∥∥∥p√λ

µQr (t)

∥∥∥∥∥T

,∥∥∥Lr (t)

∥∥∥T, 1

,

we obtain for every ω ∈ Kr,

supMyr,0√r≤t≤T g(Qr(t))

max∥∥∥p

√λµ Q

r (t)∥∥∥T,∥∥∥Lr (t)

∥∥∥T, 1 ≤ ε√

λ/µ.

Note that for every t ≥ 0,

|Lr(t)| =

∣∣∣∣∣∣

S∑

j=1

(pj

√λ

µQrj(t)− β

√λ

µ

)+∣∣∣∣∣∣≤√λ

µ

(|Qr(t)|+ S|β|

). (8.6.41)

Moreover, since ε > 0 is arbitrary, we can conclude that (8.6.39) holds.

If |Qr(0)| P→ 0, it follows from Corollary 8.6.4, that

g(Qr(t)) ≤ ε√λ/µ

yr,0 ≤ε√λ/µ

max

∥∥∥∥∥p√λ

µQr (t)

∥∥∥∥∥T

,∥∥∥Lr (t)

∥∥∥T, 1

for all ω ∈ Lr and t ∈ [0,Myr,0/√r]. Since ε > 0 is arbitrary, together with (8.6.39)

and (8.6.41), we obtain (8.6.40).

Remark 8.6.2. Note that the bounds in Theorem 8.6.1 are obtained for every fixedT > 0. Yet, from the proof it is clear that one has the following slightly more

8


general result. Suppose Qr(0) → Q(0) for some random vector Q(0). For everyε > 0, M <∞ as in Theorem 8.6.1, and tr ∈ (Myr,0/

√r,∞),

P

(supMyr,0/

√r≤s≤tr g(Qr(s))

max‖Qr(s)‖tr , 1> ε

)→ 0, (8.6.42)

as r → ∞. If, in addition, g(Qr(0)) → 0 in probability as r → ∞, then for everytr ∈ (Myr,0/

√r,∞),

‖g(Qr(s))‖trmax‖Qr(t)‖tr , 1

P→ 0, (8.6.43)

as r → ∞. In other words, the interval over which the state space collapse isconsidered can also be chosen as a sequence of intervals indexed by r.

8.6.4 Strong state space collapse

Although Theorem 8.6.1 shows multiplicative state space collapse, our interestlies in the strong state space collapse as is stated in Theorem 8.5.1. To show strongstate space collapse, it suffices to show that the denominators in Theorem 8.6.1 arebounded in a probabilistic sense. More specifically, ‖Qr(t)‖T should satisfy thecompact containment property. Before doing so, we prove a result that shows thateven if the diffusion-scaled queue lengths are initially not necessarily close to oneanother, these queue lengths do not explode for a sufficiently short period of time.

Lemma 8.6.5. Suppose Qr(0)→ Q(0) for some random vector Q(0), and M ∈ [0,∞).Then,

limK→∞

limr→∞

P(‖Qr(t)‖Myr,0/

√r > K

)= 0.

Proof. Fix ε ∈ (0, 1) small. First, note that

P(‖Qr(t)‖Myr,0/

√r > K

)≤ P

(max

|Qr(0)|, yr,0

> εK

)

+ P(‖Qr(t)‖Myr,0/

√r > K; max

|Qr(0)|, yr,0

≤ εK

).

(8.6.44)

By definition,

yr,0 = max

∣∣∣∣∣p√λ

µQr (0)

∣∣∣∣∣ ,∣∣∣Lr (0)

∣∣∣ , 1.

Since Qr(0)→ Q(0) for some random vector Q(0),

limK→∞

limr→∞

P(|Qr(0)| > εK

)= 0,

8


and due to the definition of yr,0 and (8.6.41) for t = 0, this implies

limK→∞

limr→∞

P(

max|Qr(0)|, yr,0

> εK

)= 0.

To bound the first term in (8.6.44) as well, we observe that the queue length atsome time is trivially bounded by

|Qr(t)| ≤ |Qr(0)|+ max

∑

i,j∈EΛij(rt), max

1≤j≤SSj(F rj t)

.

We observe that F rj ≤ (1 + ε)λr/µ for r large enough. Moreover, if Λ(t), t ≥ 0denotes a Poisson process with rate λ, then due to the properties of the Poisson

process it holds that∑i,j∈E Λij(·) d

= Λ(·). In terms of the diffusion scaling, theabove bound yields for all t ≥ 0,

|Qr(t)| ≤ |Qr(0)|+ max Λ(rt),max1≤j≤SSj((1 + ε)λr/µt)min1≤j≤S pj

√λr/µ

.

Therefore, using this bound for t = Myr,0/√r ≤ εMK/

√r and noting that Poisson

processes are (non-decreasing) counting processes,

P(‖Qr(t)‖Myr,0/

√r > K; max

|Qr(0)|, yr,0

≤ εK

)

≤ P

(max Λ(εMK

√r),max1≤j≤S Sj (ε(1 + ε)λ/µMK

√r)

min1≤j≤S pj√λr/µ

> (1− ε)K).

Due to the LLN, we observe that

limK→∞

limr→∞

P

(Λ(εMK

√r)

min1≤j≤S pj√λ/µK

√r> 1− ε

)= 0

for ε > 0 small enough (e.g. for ε < 1−M/(M +√λ/µmin1≤j≤S pj)). Similarly,

due to the LLN,

limK→∞

limr→∞

S∑

j=1

P

(Sj (ε(1 + ε)λ/µMK

√r)

min1≤j≤S pj√λ/µK

√r> 1− ε

)= 0

for ε > 0 small enough (e.g. for ε < min1≤j≤S pj/(M√λ/µ + min1≤j≤S pj)). We

conclude that the first term in (8.6.44) converges to zero, i.e.

limK→∞

limr→∞

P(‖Qr(t)‖Myr,0/

√r > K; max

|Qr(0)|, yr,0

≤ εK

)= 0,

and hence the result follows.

Next, we show that the process Qr(·) satisfies the compact containment property.

8


Proposition 8.6.3. Suppose |Qr(0)| → Q(0) for some random vector Q(0). Then, forevery T > 0 and ε > 0,

limK→∞

limr→∞

P(‖Qr(t)‖T > K

)= 0. (8.6.45)

Proof. Fix ε ∈ (0, 1/3) small, and let K > max2|β| + 2, 2|γ| + 2. Introduce thesequence of stopping times

τ rK = inf

t ≥ 0 : max

1≤j≤SQrj(t) > K

,

T rK = sup

0 ≤ t ≤ τk : min

1≤j≤SQrj(t) ≤ K/2

,

and similarly,

τ rK = inf

t ≥ 0 : min

1≤j≤SQrj(t) < −K

,

T rK = sup

0 ≤ t ≤ τk : max

1≤j≤SQrj(t) ≥ −K/2

.

Clearly,

P(‖Qr(t)‖T > K

)≤ P (τ rK ≤ τ rK ≤ T ) + P (τ rK ≤ τ rK ≤ T ) . (8.6.46)

In order to improve the readability of the proof, we first present a proof in the casewhen g(Qr(0)) → 0 in probability as r → ∞. We then comment on the changesneeded to adapt the proof to the case when this condition does not necessarilyhold.

Case I: g(Qr(0))→ 0 in probability as r →∞.

Since |Qr(0)| → Q(0) for some random vector Q(0), it holds that

limK→∞

limr→∞

P(|Qr(0)| > K/2

)= 0,

and hence we can assume that both τ rK > T rK > 0 and τ rK > T rK > 0 (for K largeenough). Moreover, for every t ≤ minτ rK , τ rK ,

‖g(Qr(t))‖minτrK ,τrK

max‖Qr(t)‖minτrK ,τrK, 1≤ ε ⇒ max

1≤i≤SQri (t)− min

1≤i≤SQri (t) ≤ εK. (8.6.47)

Next, we provide bounds for the two ways of crossing the boundary K separately.First, we consider the first term in (8.6.46). We observe

P (τ rK ≤ τ rK ≤ T )

≤ P

(τ rK ≤ τ rK ≤ T ;

‖g(Qr(t))‖τrKmax‖Qr(t)‖τrK , 1

≤ ε)

+ P

(‖g(Qr(t))‖τrK

max‖Qr(t)‖τrK , 1> ε

).

8


Due to Theorem 8.6.1 and (8.6.43),

limr→∞

P

(‖g(Qr(t))‖τrK


)= 0.

To bound the first term, define the process QrΣ(t), t ≥ 0with

QrΣ(t) =

∑Sj=1

(Qrj(t)− pjλr/µ

)√λr/µ

=

S∑

j=1

pjQrj(t).

We observe that

min1≤j≤S

Qrj(t) ≤ QrΣ(t) ≤ max1≤j≤S

Qrj(t). (8.6.48)

Due to the system identities, we observe that for every t ∈ [T rK , τrK ],

QrΣ(t) = QrΣ(T rK) +

∑i,j∈E A

rij(t)−Arij(T rK)√λr/µ

−∑Sj=1D

rj (t)−Dr

j (TrK)

√λr/µ

.

We note that due to the properties of the Poisson process,∑

i,j∈EArij(t)−Arij(T rK) ≤ST

∑

i,j∈EΛrij(rt)− Λrij(rT

rK)

d= Λ(rt)− Λ(rT rK),

with Λ(t), t ≥ 0 an (independent) Poisson process with rate λ. Moreover, sincefor all t ∈ [T rK , τ

rK ] it holds that Qrj(t) ≥ γ for every i ∈ 1, ..., S,

S∑

j=1

Drj (t)−Dr

j (TrK) =

S∑

j=1

Sj(Frj t)− Sj(F rj T rK).

Using the FCLT, we observe that

Λ(rt)− Λ(rT rK)−∑Sj=1 Sj(F

rj t)− Sj(F rj T rK)

√λr/µ

d→ BM(t)− BM(T rK)− γµ(t− T rK)

as r →∞, where BM(t), t ≥ 0 is a Brownian motion with zero mean and finitevariance (independent of K). Finally, by the definitions of the stopping times, andin view of (8.6.47) and (8.6.48), for all t ∈ [T rK , τ

rK ],

QrΣ(τ rK)− QrΣ(T rK) ≥ (1− ε)K − (1 + ε)K/2 = (1− 3ε)K/2.

We conclude that

limr→∞

P

(τ rK ≤ τ rK ≤ T ;


≤ ε)

≤ P(

sup0≤s≤t≤T

BM(t)− BM(s)− γµ(t− s) ≥ 1− 3ε

2K

),

8


which converges to zero as K →∞ since ε ∈ (0, 1/3).

The analysis of the second term in (8.6.46) uses similar arguments as the first term.We observe

P (τ rK ≤ τ rK ≤ T ) ≤ P

(τ rK ≤ τ rK ≤ T ;


≤ ε)

+ P

(‖g(Qr(t))‖τrK


).

Again, due to Theorem 8.6.1 and (8.6.43),

limr→∞

P

(‖g(Qr(t))‖τrK


)= 0.

Due to the system identities, we observe that for every t ∈ [T rK , τrK ],

QrΣ(t) = QrΣ(T rK) +

∑i,j∈E A

rij(t)−Arij(T rK)√λr/µ

−∑Sj=1D

rj (t)−Dr

j (TrK)

√λr/µ

.

Due to the definitions of the stopping times, we observe that for all t ∈ [T rK , τrK ], it

holds that Qrj(t) ≤ β for every j ∈ 1, ..., S, and hence Lr(t) = r. Therefore, dueto the properties of the Poisson process,

∑

i,j∈EArij(t)−Arij(T rK) =

∑

i,j∈EΛrij(rt)− Λrij(rT

rK)

d= Λ(rt)− Λ(rT rK),

with Λ(t), t ≥ 0 a Poisson process with rate λ. Moreover,

S∑

j=1

Drj (t)−Dr

j (TrK) ≤ST

S∑

j=1

Sj(Frj t)− Sj(F rj T rK).

Using the FCLT, we observe again that

Λ(rt)− Λ(rT rK)−∑Sj=1 Sj(F

rj t)− Sj(F rj T rK)

√λr/µ

d→ BM(t)− BM(T rK)− γµ(t− T rK)

as r → ∞, where we recall that BM(t), t ≥ 0 is a Brownian motion with zeromean and finite variance (independent of K). Finally, by definition of the stoppingtimes, and in view of (8.6.47) and (8.6.48), it holds for all t ∈ [T rK , τ

rK ],

QrΣ(τ rK)− QrΣ(T rK) ≤ −(1− ε)K − (−(1 + ε)K/2) = −1− 3ε

2K.

We conclude that

limr→∞

P

(τ rK ≤ τ rK ≤ T ;


≤ ε)

≤ P(

sup0≤s≤t≤T

BM(t)− BM(s)− γµ(t− s) ≤ −1− 3ε

2K

),

8

8.7. Simulation experiments 265

which also converges to zero as K → ∞ since ε ∈ (0, 1/3). Since this holds forboth of the two summed probabilities in (8.6.46), we conclude that (8.6.45) holds.

Case II: general case, i.e. when we do not assume that g(Qr(0)) → 0 in probability asr →∞.

Let M ∈ [1,∞) be fixed and satisfy the property as in 8.6.1. Since |Qr(0)| → Q(0)

for some random vector Q(0) and due to Lemma 8.6.5, it holds that

limK→∞

limr→∞

P(‖Qr(t)‖Myr,0/

√r > K/2

)= 0.

Therefore, we can assume that both τ rK > T rK > Myr,0/√r and τ rK > T rK >

Myr,0/√r (for K large enough). The proof in this general case is then completely

analogous to that in the previous one: ‖g(Qr(t))‖τrK needs to be replaced withsupt∈(Myr,0/

√r,τrK ]|g(Qr(t))|, and ‖g(Qr(t))‖τrK – with supt∈(Myr,0/

√r,τrK ]|g(Qr(t))|.

Next, we prove our main result stated in Theorem 8.5.1.

Proof of Theorem 8.5.1. Equation (8.5.2) follows from Theorem 8.6.1 and Proposi-tion 8.6.3. To prove (8.5.1), note that for every ε > 0 and any sequence Kr, r ∈ N,

P

(sup

Kr/√r≤t≤T

g(Qr(t)) > ε

)

= P

(sup

Kr/√r≤t≤T

g(Qr(t)) > ε ;Kr > Myr,0

)+ P (Kr ≤Myr,0)

≤ P

(sup

Myr,0/√r≤t≤T

g(Qr(t)) > ε

)+ P (Kr ≤Myr,0) .

Theorem 8.6.1 and Proposition 8.6.3 imply that for every ε > 0,

limr→∞

P

(sup

Myr,0/√r≤t≤T

g(Qr(t)) > ε

)= 0.

Moreover, for any sequence Kr, r ∈ N for which Kr = o(√r) with Kr →∞ as

r →∞, it holds that

limr→∞

P (Kr ≤Myr,0) = 0,

by definition of yr,0, (8.6.41) and since Qr(0) → Q(0) for some random vectorQ(0). We conclude that (8.5.1) holds as well.

8.7 Simulation experiments

The results presented are given in an asymptotic regime where the charging timesare exponentially distributed. In this section, we conduct simulation experiments

8


to evaluate the quality of our approximations and the robustness of the state-spacecollapse result against more general charging times. Throughout the experiments,we consider a network with 5 stations where the arrival probabilities are given byp12 = p24 = p34 = p35 = 0.1, p23 = 0.4 and p45 = 0.2, see Figure 8.2. This resultsin an effective arrival probability p = (p1, ..., p5) = (0.05, 0.3, 0.3, 0.2, 0.15) at thestations.

1

2 3

4 50.2

0.1 0.1

0.4

0.1

0.1

Figure 8.2: Illustration of the battery swapping network with arrivalstreams used in the simulation experiments.

As a battery swapping infrastructure currently does not exist yet in real-life, thereis no (significant) data that can be exploited to obtain useful parameter choices.Instead, we discuss an adequate provisioning strategy under the assumption thatrecharging takes one hour on average (µ = 1), and that every EV user returns forrecharging services after every 40 hours on average (λ = 0.025). In addition, westress that our results are based on an asymptotic regime, and therefore requirethe system to be sufficiently large for the approximation to become meaningful.We allow for (at least) r = 50000 EV users in this infrastructure in order for thenumber of resources to be sufficiently large for the asymptotic results to becomemeaningful. The effective loads at the stations in this case are

(pjλr

µ

)

j=1,...5

= (62.5, 375, 375, 250, 187.5),

and we note that due to the QED provisioning policy in (8.2.2), the numbersof charging points and spare batteries are close to these values. Obviously, thenumber of resources are integer values, and in our simulation experiments wechoose

F r =

(⌈λr

µ+ γ

√λr

µ

⌉)

j=1,...,5

, Br =

(⌈λr

µ+ β

√λr

µ

⌉)

j=1,...,5

8.7.1 State space collapse for exponential charging times

A first-order approximation for the queue length process is implied by the fluidresult in Theorem 8.4.1. We validate this approximation for the above-described

8


setting, with initial queue length Qri (0) = 150 for all stations i = 1, ..., 5. Thatis, only station 1 is initially overloaded, while all other stations are underloaded.The equations in Theorem 8.4.1 together with the Lipschitz continuity describe aunique fluid limit with the given initial queue length. This yields the approxima-tions

Qri (t) ≈ rQ(t), j = 1, ..., S.

In particular, in the case when the initial queue length is Qri (0) = 150 for allstations i = 1, ..., 5, this results in approximations

Qr1(t) ≈

150− 62.5t if t ≤ t3,62.5e1.4−t if t3 ≤ t ≤ t4,62.5 + 195

64 e1.4−t − 517

16 e−t otherwise,

Qr2(t) ≈ Qr3(t) ≈

498.5 + 0.625t− 348.5e−t if t ≤ t2,75t152 + 14955

38 − 775538 e−t if t2 ≤ t ≤ t3,

750019 − 75

152e1.4−t − 7755

38 e−t if t3 ≤ t ≤ t4,375 + 585

32 e1.4−t − 1551

8 e−t otherwise,

Qr4(t) ≈

249.25 + 0.3125t− 99.25e−t if t ≤ t1,9977 + 5

28 t+ 29e−t if t1 ≤ t ≤ t2,498519 + 25

76 t− 258519 e−t if t2 ≤ t ≤ t3,

500019 − 25

76e1.4−t − 2585

19 e−t if t3 ≤ t ≤ t4,250 + 195

16 e1.4−t − 129.25e−t otherwise,

Qr5(t) ≈

150e−t if t ≤ t1,15t112 + 2991

28 + 874 e−t if t1 ≤ t ≤ t2,

1495576 + 75

304 t− 775576 e−t if t2 ≤ t ≤ t3,

375019 − 75

304e1.4−t − 7755

76 e−t if t3 ≤ t ≤ t4,187.5 + 585

64 e1.4−t − 1551

16 e−t otherwise,

where t1 ≈ 0.1826, t2 ≈ 0.3189, t3 = 1.4 and t4 ≈ 1.4758. Times ti, i = 1, 2, 4,correspond to the moments where two stations (approximately) have the samerelative queue lengths, and t3 is the moment where the number of EVs in need ofrecharging is (approximately) equal to the number of stations/spare batteries. Asample path comparison with its fluid approximation (dotted lines) is graphicallyillustrated in Figure 8.3. We observe that the fluid limit approximations capturethe typical values of the actual queue length process quite accurately. We observeapparent fluctuations around its approximation, and we note that these becomerelatively small as r grows large.

To observe the state-space collapse, we plot the same sample path by its diffusionscaling, see Figure 8.4. Indeed, around t4 ≈ 1.4758 the diffusion-scaled queuelengths appear to become close and remain nearly equal to one another after this

8


0 1 2 3 40

100

200

300

400

Time

Que

ueLe

ngth

Simulation station 1Simulation station 2Simulation station 3Simulation station 4Simulation station 5

Figure 8.3: Sample path of the queue lengths when Qr(0) =(150, 150, 150, 150, 150).

0 1 2 3 4

−4

−2

0

2

4

6

8

Time

Qi(t)

=Q

i(t)−

piλr/µ

pi

√λr/µ


Figure 8.4: Sample path of the diffusion-scaled queue lengths whenQr(0) = (150, 150, 150, 150, 150).

time. In addition, as time moves, on the diffusion-scaled queue lengths fluctuatearound zero.

The differences between the diffusion-scaled queue lengths are small and fluctuateerratically among each other when one would zoom in on this domain. Obviously,the differences between the diffusion-scaled queue lengths are not arbitrarilysmall since r is finite. Even if the diffusion-scaled queue lengths at all stationsare the same, and an arriving EV moves to station 1, this causes a discrepancy of1/(p1

√λr/µ) ≈ 0.5657 in the described setting. Theorem 8.5.1 implies that the

distance between the queue lengths become smaller as the number of EV usersr grows large. To illustrate this notion, we consider the maximum differencebetween the queue lengths over a finite interval T = 1 in Figure 8.5, which ismonotonically decreasing in r. In addition, we observe that the average max-imum distance, i.e. 1/T

∫ T0

max1≤i<j≤S|Qi(t)−Qj(t)| dt, is also monotonicallydecreasing in r, and is not excessively smaller than the maximum distance of the

8


interval.

0 20 40 60 80 100 ×1030

1

2

3

4

r

max0≤t≤T

max1≤i<j≤5

|Qi(t)−Qj(t)|1T

∫ T

0max

1≤i<j≤5|Qi(t)−Qj(t)| dt

Figure 8.5: Maximum queue length measures for T = 1 averaged over10 000 samples.

It turns out that the maximum distance is usually caused by the diffusion-scaledqueue length at station 1 being smaller than the queue length at another station(often station 2 or station 3). This discrepancy is also reflected in the waiting timeprobabilities of the stations. In Figure 8.6, we plotted the waiting probabilities ofall stations in the case of 2500000 EV arrivals averaged over 20 samples. We pointout that the stair-type effect appearing in the waiting probabilities is due to theceiling of the number of resources at the stations. Moreover, as r is finite and weuse the ceiling function, the waiting times are not all exactly equal, which is mostapparent for station 1. As r grows large, the waiting probabilities do grow closerand move near to their asymptotic expressions. Still, the waiting probabilities aretypically below their asymptotic expressions. This implies that the provisioningrules (8.2.2) guarantee that a desired waiting probability is achieved.

−0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

γ

P(W

>0)

Simulation station 1Simulation station 2Simulation station 3Simulation station 4Simulation station 5Asymptotics

Figure 8.6: Waiting probabilities with respect to its asymptotic expres-sion when β = 1.

8


8.7.2 Non-Markovian charging time distribution

In order to be able to rigorously prove the state-space results, we assumed expo-nential charging times in our framework. Yet, extensive simulation experimentssuggest that these results hold for any charging time distribution with finite meanand variance. In Figures 8.7 and 8.8, it appears that similar behavior occurs onfluid scale in case of deterministic and uniformly distributed charging times as forthe exponential case.

0 1 2 3 40

100

200

300

400

Time

Que

ueLe

ngth


Figure 8.7: Sample paths of the queue lengths for charging times equalto one when Qr(0) = (0, 0, 0, 0, 0).

0 1 2 3 40

100

200

300

400

Time

Que

ueLe

ngth


Figure 8.8: Sample paths of the queue lengths for charging distributionuniform U(0.75, 1.25) when Qr(0) = (0, 0, 0, 0, 0).

When the queue lengths are initially zero, the system behaves close to its invariantstate for t ≥ 1. Similarly to the setting with exponential charging times, themaximum difference between the diffusion-scaled queue length behaves quiteerratically, see Figure 8.9. Still, the differences are very small, and grow smalleras r grows larger suggesting that state-space collapse also holds in this setting.That is, the system behaves similarly to the situation when there would be asingle station with an aggregated number of charging points and spare batteries

8.8. Discussion 271

and a charging time distribution as at the individual stations. Consequently,performance measures such as waiting probability and expected waiting time areapproximately equal to their counterparts in a single-station system.

3.5 3.6 3.7 3.8 3.9 40

0.1

0.2

0.3

0.4

0.5

Time

max

1≤i<

j≤5|Q

i(t)−Q

j(t)|

DeterministicUniform

Figure 8.9: Maximum distance between queue lengths for non-exponential charging times.

8.8 Discussion

The introduction of this novel framework opens up a wide range of possibleavenues one can pursue in future research. Next, we highlight a few possibledirections we consider important and challenging future steps.

Firstly, the framework can be significantly enriched by including multiple cus-tomer types to model a range of car brands using different battery systems.Secondly, the model can incorporate a swapping time to model performancemore accurately. Thirdly, there is a delay between the moment an EV user consultsqueue length information and the actual arrival due to transportation time. Asis perceived in health care settings and bike-sharing systems, this can have aconsiderable effect on the queue length behavior. A fourth enhancement would beto incorporate a time-inhomogeneous demand rate to better simulate the expecteddiurnal variation. This will lead to a varying amount of slackness in the capacitywithin the QED regime.

A final aspect we would like to address is the following. There is a substantial un-derlying variability in the fluctuating energy prices, which sharply rise wheneverthe energy grid is more strained and vice versa. A battery swapping infrastructurewill be sensitive to these prices changes and can provide an indispensable assetfor supporting a stable grid in the future. Adequate management of chargingschedules can contribute immensely to relieving strain during peak moments,e.g. by deferring the moment of charging or even let batteries return electricityto the grid. It is beyond the scope of this work to provide efficient and adequateprovisioning rules in these challenging settings, yet offers intriguing avenues topursue in future research. The main insight provided in the present study is theeffectiveness of simple load balancing policies, and can prove useful in at least theplanning stage of a swapping network.

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Summary

Power grids are indispensable infrastructures that are of critical importance tomodern-day society, and have fortunately proven to be capable of providing anincredibly steady and secure supply of electricity. Nevertheless, in the last twodecades, several highly visible blackouts have taken place, and the frequency atwhich these catastrophic events occur increases rapidly. Technological advances,such as the rise of renewable energy sources and electric vehicles, have increaseduncertainties and volatility in power generation and demand. Power grids needto evolve and become more resilient to prevent large-scale blackouts, creating acritical need for stochastic approaches for capacity dimensioning and planning asopposed to the traditional deterministic optimization-based framework.

Typically, a blackout occurs as a result of a small disturbance that cascades through-out the network due to (sequential) overloading of components. Analysis of his-toric blackout data suggests that the amount of lost power, as well as the number ofaffected customers, exhibits power-law behavior. A fundamental question is whatcauses this heavy-tailed feature to appear in the context of power outages. In thisthesis, we study the blackout propagation in the energy grid through macroscopiccascading failure models. In these models, the power grid is typically representedby a graph, where vertices correspond to cities and/or power generation sourcesand edges represent transmission lines over which electricity is transmitted.

In Chapters 2 and 3, we study a stylized cascading failure model that capturessome salient elements of power outages. We focus on the failure size, i.e. thenumber of edge failures, and identify which range of parameters leads to a power-law tail for the failure size distribution. It turns out that this framework can berelated to a random walk bridge setting, where the failure size corresponds tothe first-passage time over a moving boundary. The conditions for the movingboundary directly imply a rather general description under which the failure sizetail exhibits power-law behavior. We identify an intriguing phase transition whenconsidering the probability that the failure size exceeds a threshold relatively closeto the initial total number of edges.

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We study the effect of network disintegration in Chapters 4 and 5. Wheneverthe cascade causes a disconnection in the power grid, the failure propagation inone component no longer affects the dynamics in the rest of the network. Werigorously derive the impact of a single immediate disconnection, and show whythe power-law tail property typically prevails for the failure size distribution.Yet, it affects the prefactor of the power-law tail, and may even change the expo-nent. In addition, we extend our framework by generalizing the cascading failuremechanism to complex networks, and in particular, a connected configurationmodel. An intrinsic feature of the failure mechanism is that the propagation offailures occurs non-locally and depends on the global network structure, whichcontinuously changes during the cascade. Well-established results from percol-ation theory suggest that the network disintegrates in a giant component andseveral significantly smaller components up to a certain critical threshold. Weexploit this observation by noting that the dominant behavior comes from whathappens in the giant component, for which we derive a power-law tail for thefailure size.

In Chapter 6, we introduce a framework that yields a completely different ex-planation for the power-law tail in the blackout size. We hypothesize that thepower-law tail for the number of affected customers during a blackout is drivenby the power-law tailed nature of city population sizes. Our framework accountsfor the laws of electricity, and more specifically, we adopt the Direct-Current (DC)power flow model. Combining the DC power flow model with large deviationstheory, we show that the dominant scenario in which a large-scale power outageoccurs is when there is a single city whose power demand is so excessive that thepower demands of other cities are relatively negligible. Our insights illustrate thatinstead of investing in network upgrades, it would be more effective to enhancethe resilience of cities, e.g. expanding energy storage possibilities and findingmeasures to quickly reduce power demand on a local level.

A significant role for reducing power demand volatility can be played by electricvehicles (EVs), which are expected to critically contribute to reducing carbonemission in the transportation sector. The batteries can relieve the strain on the gridin times of high demand, especially when the control can be centralized. That canbe achieved by the construction and operation of a battery swapping infrastructurerather than the in-vehicle charging of batteries. Swapping station managementis faced with a tradeoff between EV users’ service quality and operational costs.In Chapters 7 and 8, we adhere to a square-root slack provisioning policy for thenumber of resources at the various stations. This policy ensures a steady-statewaiting probability strictly between zero and one, a vanishing waiting time, anda near-optimal resource utilization as the number of EV users grows large. Tofurther improve system efficiency, electronic devices can offer information on theoccupancy level at stations in the neighborhood. When an EV user has an incentiveto move to the least-loaded station in the direct vicinity, a load-balancing effecttakes place and a significant improvement of the delay performance is achieved.We show that a complete resource pooling effect takes place, ensuring that it isunlikely that EV users are waiting for a battery at one station while another isreadily available at a different station.

About the author

Fiona Sloothaak was born in Lemmer, The Netherlands, on April 6, 1991. Aftercompleting her secondary education at Zuyderzee College in Emmeloord in 2008,she studied Mathematics at Utrecht University. In 2012, she received her Bachelor’sdegree with highest honors (cum laude). She then pursued a Master’s degree inapplied mathematics at Eindhoven University of Technology (TU/e), where shewrote a Master’s thesis in collaboration with Philips Research, and obtained herdegree with highest honors (cum laude) in 2015.

Fiona continued as a PhD student with the Stochastic Operations Research groupat TU/e under the supervision of Bert Zwart and Sem Borst. In her research, shedeveloped macroscopic stochastic models to obtain a fundamental understandingof why large-scale blackouts occur in the power transmission system. The resultsof this research are presented in this dissertation.

During her PhD, Fiona has been involved in several teaching activities regardingstochastic processes, queueing theory and stochastic decision theory. In 2015-2018,she also acted as a mentor for students at the Industrial and Applied Mathematicsmaster’s program. She was a member of the Department Council in 2016 and2017. She also co-organized a winter school on energy systems in 2017, as a specialedition of the yearly organized Young European Queueing Theorists (YEQT)workshop.

Fiona will defend her PhD thesis on January 16, 2020. In November 2019, she startsworking as a postdoctoral researcher at the Industrial Engineering & InnovationSciences department at TU/e.

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