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Contents

1 Welcome Address 5

2 Program Committee and Organizing Committee 7

3 Sponsors 9

4 General Information 11

5 Program at a Glance 13

6 Plenaries and Tutorials 15

7 Social Events 19

8 Auditorium Layout 21

9 University Campus Layout 23

10 Sessions 25

11 Abstracts 49

12 Author Index 163

3

CONTENTS

4

1Welcome Address

Dear Participant,

It is our pleasure to welcome you to the 14th INFORMS Applied Probability Conference, to be held in theAuditorium of Eindhoven University of Technology.

Traditionally, emphasis in this conference is on applications of probability in operations research; typicalOR subjects are optimization of logistic networks under uncertainty, control of inventory systems, and per-formance analysis of communication networks. The analysis of the underlying probability models reliespredominantly on the theory of Markov chains, stochastic processes, queueing theory, and advanced simu-lation techniques. Due to the strong methodological connections, however, also various other applicationsare included in the conference, for instance those inspired by finance, physics and biology.

We are very pleased that the following researchers, leaders in our field, have agreed to give keynote lectures:Peter Glynn, Frank Kelly and Alain-Sol Sznitman. Another exciting prospect is formed by three tutorials ontimely subjects, to be presented by Maury Bramson, Paul Glasserman and Philippe Robert. Furthermore,about 350 twenty-minute talks (roughly as many as the number of registered participants) will be presentedin 90 sessions. Each time 9 sessions will run in parallel.

We are grateful to the many persons who have assisted us in organizing the conference. In particular, we’dlike to mention the Congress Office of Eindhoven University of Technology, the extremely able and helpfulstaff of EURANDOM, Josine Bruin, Hannie Horvath, Wim Senden, Erik Winands, and last but not leastMarko Boon, whose wizardry made the preparation of this Program book almost look easy.

We have chosen a schedule that leaves you with ample time to renew acquaintances, to make new friends,and to discuss problems and new ideas. We wish you a pleasant stay in Eindhoven!

July 2007The organizing committee

5

1 – Welcome Address

6

2Program Committee and Organizing

Committee

Co-Chairs

Onno Boxma (EURANDOM and TU Eindhoven, Eindhoven, The Netherlands, boxma@eurandom.tue.nl)Michel Mandjes (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, TheNetherlands, mmandjes@science.uva.nl)

Program Committee

Mor Armony (New York University, New York, USA)Jesus Artalejo (Complutense University of Madrid, Madrid, Spain)Rami Atar (Technion, Haifa, Israel)Francois Baccelli (INRIA & ENS, Paris, France)Sem Borst (Lucent, Murray Hill, USA & TU Eindhoven, Eindhoven, The Netherlands)Richard Boucherie (Twente University, Enschede, The Netherlands)B.D. Choi (Korea University, Seoul, Korea)Neil O’Connell (University of Cork, Cork, Ireland)Dee Denteneer (Philips Research, Eindhoven, The Netherlands)Doug Down (McMaster University, Hamilton, Canada)Paul Dupuis (Brown University, Providence, USA)Serguei Foss (Heriot-Watt University, Edinburgh, UK)Geoffrey Grimmett (University of Cambridge, Cambridge, UK)Peter Glynn (Stanford University, Stanford, USA)Shane Henderson (Cornell University, Cornell, USA)Remco van der Hofstad (TU Eindhoven & EURANDOM, Eindhoven, The Netherlands)Ingemar Kaj (Uppsala University, Uppsala, Sweden)Offer Kella (Hebrew University, Jerusalem, Israel)Ger Koole (Vrije Universiteit Amsterdam, Amsterdam, The Netherlands)Dirk Kroese (University of Queensland, Brisbane, Australia)Thomas Mikosch (University of Copenhagen, Copenhagen, Denmark)Ilkka Norros (VTT, Helsinki, Finland)Balaji Prabhakar (Stanford University, Stanford, USA)Kavita Ramanan (Carnegie Mellon University, Pittsburgh, USA)Ulrich Rieder (University Ulm, Ulm, Germany)Wim Schoutens (KU Leuven, Leuven, Belgium)Hanspeter Schmidli (University of Cologne, Cologne, Germany)Mark Squillante (IBM Thomas J. Watson Research Center, Yorktown Heights, USA)Tetsuya Takine (Osaka University, Osaka, Japan)

7

2 – Program Committee and Organizing Committee

Peter Taylor (University of Melbourne, Melbourne, Australia)Henk Tijms (Vrije Universiteit Amsterdam, Amsterdam, The Netherlands)Assaf Zeevi (Columbia University, New York, USA)

Organizing Committee

Ivo Adan (EURANDOM and TU Eindhoven, Eindhoven, The Netherlands)Onno Boxma (EURANDOM and TU Eindhoven, Eindhoven, The Netherlands)Lucienne Coolen (EURANDOM, Eindhoven, The Netherlands)Patty Koorn (EURANDOM, Eindhoven, The Netherlands)Michel Mandjes (Korteweg-de Vries Institute for Mathematics and University of Amsterdam, Amsterdam,The Netherlands)Jacques Resing (TU Eindhoven, Eindhoven, The Netherlands)

8

3Sponsors

The organizers gratefully acknowledge the support of the following organizations:

Stichting ABE

Co-sponsored by the INFORMS Applied Probability Society and The Institute of Mathematical Statistics.

informs®

9

3 – Sponsors

10

4General Information

Location and Conference Buildings

The conference will take place at the Auditorium of Eindhoven University of Technology (see maps inSection 8 and 9).

(Pre-)Registration and Conference Office

On Sunday July 8, the Conference Office is located in EURANDOM, which is housed in the LaplaceBuilding (LG on the map in Section 9). It will be open from 5:00pm till 7:00pm for pre-registration.Refreshments will be served during this meet and greet session. On Monday, Tuesday and Wednesday theConference Office is located on the first floor of the Auditorium (see map in Section 8). On Monday it isopen for registration from 7:30am until 6:00pm. On Tuesday and Wednesday it is open from 8:00am until6:00pm.

Badge

You are kindly requested to wear your key cord and badge during the conference as it grants you accessto the buildings and use of facilities such as catering, DAF-Museum, film and sports.

Computer Facilities

Wireless network access is available at several places in the Auditorium opposite the lecture rooms. Youcan plug in your notebook and get access with the personal login and password that you received, whenyou registered beforehand for this facility. If you had not registered beforehand, fill in the form “Wire-less Internet Access” in your conference documentation and deliver it at the reception on Monday after11:00am. If you don’t have a notebook with wireless connection possibility, you may use the EURAN-DOM notebooks, which are available in lecture room CZ 9 (see map in Section 8).

Refreshments and Meals

Coffee, tea and refreshments will be served during the breaks and before the first session of the day near thebook stands on Level 1 (see map in Section 8). Lunch is being offered in the Senaatszaal on Level 1, justopposite the book stands of Cambridge University Press and Springer. They are included in the registrationfee.

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4 – General Information

Parking

There are several car parks around the Auditorium. One needs a parking ticket to enter and another one toleave the University premises. Exit tickets can be obtained for free at the registration desk, when you handin your entrance ticket.

Sports Facilities

Eindhoven University of Technology has a very well equipped sports center .

INFORMS-participants who are interested in using some of these facilities are invited to visit the sportscenter (it is building SP on the campus map in Section 9). Your congress badge, to be shown at the receptiondesk, grants you access to the building and facilities.

The swimming pool is open daily for free swimming from 7:45am - 8:45am and from 12:00pm - 1:30pm.In the afternoon opening hours are:

July 9th Aqua jogging 5:00pm – 6:30pm, participation possible; if there are not enough participants,free swimming is possible.

July 10th Free swimming 4:30pm – 6:30pm.July 11th Aqua jogging 5:00pm – 6:30pm, participation possible; if there are not enough participants,

free swimming is possible.

Gyms for cardio fitness, fitness and other facilities are open from 8:30am until 7:00pm. Use of the facilitiesat the sports center is at your own risk.

Applied Probability Society Business Meeting

Time and date: Tuesday July 10, 6:15pm – 7:00pm.Location: CZ 4, Auditorium.

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5Program at a Glance

Monday July 9, 2007

9:00am – 09:15am Opening9:15am – 10:15am Plenary - Peter Glynn

10:45am – 12:15pm Parallel Sessions MA12:45pm – 1:45pm Tutorial - Philippe Robert

2:00pm – 3:30pm Parallel Sessions MB4:00pm – 5:30pm Parallel Sessions MC6:00pm – 10:00pm Reception / Dinner DAF Museum

Tuesday July 10, 2007

8:30am – 10:00am Parallel Sessions TA10:30am – 12:00pm Parallel Sessions TB12:20pm – 1:20pm Tutorial - Paul Glasserman

1:30pm – 2:30pm Plenary - Alain-Sol Sznitman2:45pm – 4:15pm Parallel Sessions TC4:45pm – 6:15pm Parallel Sessions TD6:15pm – 7:00pm APS Business Meeting6:30pm – 8:00pm Social Event

Wednesday July 11, 2007

8:30am – 10:00am Parallel Sessions WA10:30am – 12:00pm Parallel Sessions WB12:20pm – 1:20pm Tutorial - Maury Bramson

1:30pm – 3:00pm Parallel Sessions WC3:15pm – 4:15pm Plenary - Frank Kelly

Session Rooms

Opening and PlenaryBlauwe Zaal

Track 1, Tutorials andAPS Business MeetingCZ 4

Track 2CZ 5

Track 3CZ 10

Track 4CZ 11

Track 5CZ 12

Track 6CZ 13

Track 7CZ 14

Track 8CZ 15

Track 9CZ 16

13

5 – Program at a Glance

14

6Plenaries and Tutorials

Monday 9:15am – 10:15am Plenary in Blauwe Zaal

Recent Developments in Rare-Event Simulation

Peter W. Glynn, Stanford University, Stanford, USA

Computing rare-event probabilities is a problem that is of interest in many applied settings: loss in com-munications networks, bankruptcy in insurance risk, long customer fulfillment times in a supply chain,valuing an out-of-the money option, assessing system unavailability, etc. In this talk, we will describerecent progress in developing efficient algorithms for computing such rare-event probabilities. Importancesampling plays a key role, specifically the use of changes-of-measure that are "state-dependent" or "dy-namic". Appropriately designed Lyapunov functions are then needed to bound the variances that arise inthe analysis of such algorithms. In fact, the Lyapunov bound itself can play a role in the design of thealgorithm. This talk describes joint work with Jose Blanchet that is closely related to the Isaacs equationideas introduced by Dupuis and Wang.

Monday 12:45pm – 1:45pm Tutorial in CZ 4

A Survey on Tree Algorithms and their Analysis

Philippe Robert, INRIA-Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, France,Philippe.Robert@inria.fr

A tree algorithm is a procedure that recursively divides (splits) an initial set of n items into subsets untileach of the subsets obtained has a cardinality strictly less than some fixed number D. These algorithmshave a wide range of applications:

• access protocols in communication networks

• divide and conquer algorithms for data structures

• algorithms in distributed systems (leader election)

• statistical tests.

On the mathematical side, the analysis of tree algorithms is connected to classical problems of severalresearch domains such as:

• fragmentation processes

• random recursive decomposition

15

6 – Plenaries and Tutorials

• dynamical systems (Zeta functions)

• physics (complex dimensions of strings)

In this talk, we present a general survey of this area and present the mathematical methods used to analyzethe asymptotic behavior of additive functionals associated to these algorithms. Several approaches for thisanalysis will be discussed: Complex Analysis, Functional Operators and Probabilistic Methods (with anemphasis on the latter one).

The key result at the heart of the probabilistic analysis is the renewal theorem and its extensions when inde-pendence assumptions are relaxed. The purpose of this talk is also to stress the power and the complexityof this set of results in applied probability theory.

Tuesday 12:20pm – 1:20pm Tutorial in CZ 4

Monte Carlo Problems in Finance

Paul Glasserman, Columbia Business School Uris Hall 3022 Broadway New York, NY 10027, USA

This tutorial will discuss some applications and research problems arising in the use of Monte Carlo sim-ulation in finance. I will focus on two problems, in particular, that are of both practical and theoreticalinterest: the estimation of price sensitivities ("Greeks") for hedging, and the pricing of American options.

For the problem of estimating sensitivities, I will first review the pathwise method and the likelihood ratiomethod, highlighting their strengths and weaknesses for various types of financial problems. A more recentline of work (starting with Fournie et al. 1999) derives estimators using Malliavin calculus. I will discussconnections (based on joint work with Nan Chen) between the Malliavin estimators and the more traditionalmethods. In several cases, one may arrive at Malliavin estimators by combining the pathwise and likelihoodratio methods in a discrete-time approximation and then passing to the continuous-time limit.

An American option grants the holder the right to choose the time of exercise, so pricing an Americanoption entails solving an optimal stopping problem. I will review methods for approximating the prices ofAmerican options in complex models by simulation. I will give particular attention to the dual formulationsof Rogers (2001), Haugh and Kogan (2004) and Jamshidian (2003), which replace maximization overstopping times with minimization over martingales. I will compare additive and multiplicative forms ofduality, based on joint work with Nan Chen: we establish an equivalence in the quality of the boundsachieved by the two methods, but show that the variance of the multiplicative method is typically muchlarger.

Tuesday 1:30pm – 2:30pm Plenary in Blauwe Zaal

Random Walks and Interfaces

Alain-Sol Sznitman, ETH, Departement Mathematik, Rämistrasse 101 8092 Zürich, Switzerland

The general theme of this talk brings together random walks and percolation. We present results related tothe way paths of random walks can create interfaces. Such questions arise for instance when investigatingthe disconnection time by simple random walk of a discrete cylinder based on a large discrete torus (theproblem of the "termite in the wooden beam"), or when studying the percolative properties of the vacantset left by simple random walk on a large discrete torus by times comparable to the number of sites in thetorus. We also discuss a model describing the microscopic structure "in the bulk" in both examples. Part ofthe results presented here have been obtained in collaboration with A. Dembo or I. Benjamini.

16

Wednesday 12:20pm – 1:20pm Tutorial in CZ 4

Application of Fluid Limits to Queueing Networks and Associated Lyapunov Func-tions

Maury Bramson, University of Minnesota, Minneapolis, USA

Over the past decade, fluid limits and fluid models have become important tools for demonstrating the sta-bility of multiclass queueing networks. Fluid limits are, in essence, the deterministic limits of the queueingnetwork process obtained under law of large number scaling, and the corresponding fluid model is the setof equations satisfied by such limits. For a large family of disciplines, stability of the fluid model will implystability of the corresponding queueing network. This connection was first employed in works by Rybko,Stolyar, and, in particular, Dai in the early 1990’s. It is heavily used today.

The deterministic framework provided by fluid limits is easier to analyze than that of the correspondingqueueing networks. In some cases, Lyapunov functions have been derived; under certain settings, these aregiven by entropy functions. There is, however, no systematic procedure for analyzing fluid limits. More-over, although the framework considered in Dai applies to a large family of queueing networks, variousrecent networks of interest lie outside this family.

Here, we review the basic theory of fluid limits and give specific examples of their application. We alsopresent certain networks where the current theory of fluid limits does not apply, and discuss alternativeapproaches to these problems.

Wednesday 3:15pm – 4:15pm Plenary in Blauwe Zaal

Models of routing and congestion control

Frank Kelly, University of Cambridge, Cambridge, UK

The Internet has attracted the attention of many theoreticians, eager to understand the remarkable successof this diverse and complex artefact. One strand of this effort has been a framework that allows a congestioncontrol algorithm such as Jacobson’s TCP to be interpreted as a distributed mechanism solving a globaloptimization problem. The framework is based on fluid models of packet flows, and the form of theoptimization problem makes explicit the equilibrium resource allocation policy of the algorithm, whichcan often be restated in terms of a fairness criterion, such as proportional fairness or max-min fairness.The dynamics of the fluid models allow the machinery of control theory to be used to study stability, andto develop joint routing and rate control algorithms. A further strand of theoretical work has developeda connection level model, that represents the randomly varying number of flows present in a network.Stability on this time-scale is interpreted in terms of recurrence of a Markov chain. Under proportionalfairness and a local traffic condition, recent work with Kang, Lee and Williams has developed a heavytraffic diffusion approximation under which the workload process behaves like Brownian motion in theinterior of a polyhedral cone and is confined to the cone by reflection at the boundary. The diffusion has aproduct form invariant distribution with a strikingly simple interpretation in terms of dual random variables,one for each of the resources of the network. When routing is incorporated, the product form distributionhas a dual random variable for each of a set of generalised cut constraints.

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6 – Plenaries and Tutorials

18

7Social Events

Film, Monday July 9, 1:00pm – 1:50pm

THE PLEASURE OF FINDING THINGS OUT is fifty minutes of PUREFeynman! This is the original Horizon Nova interview filmed in 1981. Itwill delight and inspire anyone who would like to share something of thejoys of scientific discovery. Feynman is a master storyteller, and his tales– about childhood, Los Alamos, or how he won a Nobel Prize – are a vividand entertaining insight into the mind of a great scientist at work and play.

Location: the “Zwarte Doos” (ZD on the map in Section 9), filmzaal 1.02.

Reception at DAF-Museum, Monday July 9, 6:00pm – 10:00pm

After the last session, which ends at 5:30pm, we get together in front of theAuditorium (see campus map). Guides (with Grey EURANDOM T-shirt)will escort attendees and guests to the DAF-Museum (more information inthe flyer in the conference bag).

Hot and cold dishes are going to be served in an attractive atmosphere.

Your congress badge, to be shown at the entrance, will grant you access.

Route description for the 20-minute walk to the DAF-Museum:

• Walk in the direction of the railway station, follow for a short while the concrete Limbopath (com-bined cycle path and pedestrian path), take the first path on the left, just after the restaurant/movietheatre De Zwarte Doos (do not cross the Dommel river), go through the two tunnels until the Fuut-laan, cross the Fuutlaan, go more or less straight ahead (Parklaan), take the first street right (is alsocalled Parklaan) and cross the bridge.

• Just after the bridge on the left starts the path called Pad Marienhage. This path is part of theIndustrial Walk (Fabrieksroute, a walk along former industrial sites of Eindhoven). You will find acouple of plates (nrs. 16, 15, 14, 13 and 12) with a description (in Dutch) of these former sites.

16. Linen factory Van den Briel & Verster (now a youth centre called The Effenaar, including arestaurant)15. Linen weaving mill Ignatius de Haas (became part of the Dutch Textile Association in 1965)14. Soap-making factory/Soap works De Lelie (until 1968)13. Mock-velvet factory Schellens & Marto (moved to another factory site in Eindhoven)12. First gas plant of Eindhoven (until the introduction of natural gas in 1965)

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7 – Social Events

• Follow the path until the Kanaalstraat, turn left, cross the bridge and follow the Kanaaldijk Zuiduntil the next bridge, turn left, cross the bridge and follow the Tongelresestraat; on the left (almostimmediately) you will find the DAF-Museum.

Music and Get Together, Tuesday July 10, 6:30pm – 8:00pm

All participants are warmly invited to come to the open field in frontof the Auditorium to enjoy some drinks and snacks. In case of badweather (after all, this is the Netherlands!), we will meet in the Auditorium.Live music will be provided by the NEW ORLEANS DIXIE SKATERS(www.nods.nl).

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8Auditorium Layout

Opening hours: Monday 7:00am – 6:30pm, Tuesday 7:30am – 8pm (or 9pm in case the social event takesplace inside the Auditorium) and Wednesday 7:30am – 6:30pm.

Lecture rooms CZ 4 and CZ 5 are at level 0. The registration, plenary lectures, lunches and book standsare at level 1. Lecture rooms CZ 9 – CZ 16 and computer facilities are at level 2.

Level 0

CZ8CZ7CZ6

CZ3 CZ2 CZ1CZ 4

Track 1 and Tutorials

CZ 5

Track 2

21

8 – Auditorium Layout

Level 1

Senaatszaal

Voorhof

Blauwe zaal

Lunch

Book stands Registration desk

Opening and Plenary

Level 2

Blauwe zaal

CZ 12 CZ 11 CZ 10Track 5 Track 4 Track 3

CZ 13 CZ 14 CZ 15 CZ 16Track 6 Track 7 Track 8 Track 9

CZ 9

Computer Facilities

22

9University Campus Layout

Auditorium (Conference Site) Zwarte Doos (Movie Theatre)

Sports Centre

Laplace Gebouw (EURANDOM,Pre-registration)

RAILWAY & BUS-STATION

AUD: Auditorium (Conference site)ZD: Zwarte Doos (Movie theatre)SP: Sports centerLG: Laplace gebouw (EURANDOM, Pre-registration on Sunday)

23

9 – University Campus Layout

24

10Sessions

Monday 10:45am - 12:15pm

Session MA01 - Financial Engineering 1, Chair: Kou in CZ 4, see page 49

1. Optimal Consumption and Portfolio Management with Non-Observable Real PricesBensoussan, A.; Keppo, J.; Sethi, S.P.

2. Robustness and sensitivity analysis of risk measurement proceduresCont, R.; DeGuest, R.; Scandolo, G.

3. What Is a Good Risk Measure: Bridging the Gaps between Robustness, Subadditivity, and InsuranceRisk MeasuresHeyde, C.C.; Kou, S.G.; Peng, X.H.

4. Dynamic importance sampling for credit basketsWeber, S.; Liese, M.

Session MA02 - Random Fields, Chair: Kaj in CZ 5, see page 50

1. Operator Scaling Stable Random Fields: Theory and SimulationScheffler, H.P.; Biermé, H.; Meerschaert, M.M.

2. Random fields generated from multifractional pseudodifferential operators: Pseudostable randomfieldsRuiz-Medina, M.D.

3. Self-similar random fields as scaling limits of Poisson germ-grain modelsKaj, Ingemar; Estrade, Anne; Biermé, H.

4. Characterization of Banach Valued Stable Random MeasuresMahmoodi, S.; Soltani, A.R.

Session MA03 - Rare Event Simulation 1, Chair: Dupuis in CZ 10, see page 52

1. Fast Simulation of Brownian Motion Avoiding Poissonian ObstaclesBlanchet, Jose; Dupuis, Paul

2. Fast pricing of basket default swapsChen, Zhiyong; Glasserman, Paul

3. Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Random VariablesGlasserman, Paul; Juneja, Sandeep

25

10 – Sessions

4. Design and analysis of splitting algorithms for rare event estimationDupuis, Paul

Session MA04 - Asymptotic Analysis 1, Chair: Bassamboo / Randhawa in CZ 11,see page 53

1. Heavy Traffic Analysis of Maximum Pressure Policies for Stochastic Processing Networks withMultiple BottlenecksAta, Baris; Lin, Wuqin

2. Slow-server problem in many-server heavy traffic with non-exponential service timesTezcan, Tolga

3. Validity of heavy-traffic steady-state approximations in open queueing networks: sufficient condi-tions involving state-space collapseGurvich, I.; Zeevi, Assaf

4. Rental duration distributions and stock-levels: Why deadlines may be bad for rental firmsBassamboo, A.; Kumar, Sunil; Randhawa, R.S.

Session MA05 - Scheduling, Chair: Borst / Wierman in CZ 12, see page 54

1. New Approximations and Bounds for an M/G/s FIFO queueing systemGupta, Varun; Dai, Jim; Harchol-Balter, Mor; Zwart, Bert

2. Optimal scheduling discipline in an M/G/1 queue with Pareto-type service time distributionsAalto, S.; Ayesta, U.

3. Revisiting the performance of large jobs in the M/GI/1Wierman, Adam

4. Scheduling in polling systemsWinands, E.M.M.; Wierman, Adam; Boxma, O.J.

Session MA06 - Call Centers 1, Chair: Koole in CZ 13, see page 56

1. Density Forecasting of Intraday Call Center Arrivals using Models Based on Exponential SmoothingTaylor, J.W.; Snyder, R.D.

2. Short term dynamic prediction in Call CentersJongbloed, G.; Eveleens, J.L.

3. Forecast errors in service systemsSteckley, Samuel G.; Henderson, Shane G.; Mehrotra, Vijay

4. Shift Scheduling in Multi-skill Call CentersKoole, G.

Session MA07 - MDP 1, Chair: Feinberg in CZ 14, see page 57

1. The entangled scheduling and routing problemBispo, C.F.

2. The power of deterministic open-loop control policiesVan der Laan, D.A.

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3. A MDP approach to the control of traffic lightsHaijema, R.; Van der Wal, Jan

4. Splitting of randomized stationary policies in absorbing Markov Decision ProcessesDenardo, E.V.; Feinberg, E.A.; Rothblum, U.G.

Session MA08 - Retrial Queues, Chair: Artalejo in CZ 15, see page 59

1. Cellular mobile radio telephone networks with priority active customersMachihara, F.

2. A discrete-time queueing model with abandonmentsNobel, R.

3. M/M/s Retrial Queue with Balking and RenegingShin, Yang Woo; Choo, Taek Sik

4. Algorithmic analysis of the Geo/Geo/c retrial queueArtalejo, J.R.

Session MA09 - Inventory 1, Chair: Van Houtum in CZ 16, see page 60

1. Coordinated replenishment in a two echelon inventory system with transportation capacityBuyukkaramikli, N.C.; Gurler, U.; Alp, O.

2. A multi-item inventory system with multiple resource constraintsQiu, Minmin; Ding, Hongwei; Dong, Jin; Ren, Changrui; Wang, Wei

3. Coordination in divergent multi-echelon inventory systemsTimmer, J.

4. A product-form solution for two-echelon spare parts networks with emergency shipmentsVan Houtum, G.J.; Boucherie, R.J.; Heideveld, S.A.

Monday 2:00pm - 3:30pm

Session MB01 - Financial Engineering 2, Chair: Schoutens / Valdivieso in CZ 4, seepage 62

1. Optimal Portfolio Choice with AnnuitizationKoijen, Ralph S.J.; Nijman, Theo E.; Werker, Bas J.M.

2. Measuring comonotonicity in random vectorsKoch, I.

3. A tractable multi-firm default model with jumpsScherer, Matthias

4. Parameter estimation in fractionally integrated process of Ornstein-Uhlenbeck typeValdivieso, Luis

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Session MB02 - Percolation and Related Topics, Chair: Van den Berg in CZ 5, seepage 63

1. Exceptional times for the dynamical discrete web, part IFontes, L.R.G.; Newman, C.M.; Schertzer, E.; Ravishankar, K.

2. Exceptional times for the dynamical discrete web, part IIFontes, L.R.G.; Newman, C.M.; Schertzer, E.; Ravishankar, K.

3. Sharp phase transitions in divide-and-colour modelsMeester, R.J.; Balint, A.; Camia, Federico

4. Multiscale analysis of stochastic reaction networksKurtz, Thomas G.

Session MB03 - Simulation 1, Chair: Henderson in CZ 10, see page 64

1. Bandits and simulation selection optionsChick, S.E.; Gans, N.

2. On transience and recurrence in discrete-event simulationsHaas, P.J.; Glynn, P.W.

3. Analysis Of State-Dependent Importance Sampling AlgorithmsBlanchet, Jose

4. Adaptive control variates for steady-state simulationKim, Sujin; Henderson, Shane G.

Session MB04 - Stability of Stochastic Networks, Chair: Ramanan in CZ 11, seepage 65

1. Steady-state analysis of a multi-server queueing system in QED regimeGamarnik, D.; Momcilovic, P.

2. The heavy traffic limit of an unbalanced generalized processor sharing modelRamanan, K.; Reiman, Martin I.

3. Fluid model for a data network with alpha fair bandwidth sharing and general document size distri-butions: two examples of stabilityGromoll, H.C.; Lee, N.H.; Williams, R.J.

4. Stability of processor sharing networks with simultaneous resource requirementsHansen, J.; Reynolds, C.; Zachary, S.

Session MB05 - Communication Systems 1, Chair: Choi in CZ 12, see page 66

1. Performance Analysis of IEEE 802.11e EDCA in Non-saturation Condition by M/G/1 QueueBae, Y.H.; Kim, Kyung Jae; Choi, Bong Dae

2. Performance Analysis of IEEE 802.15.4 Unslotted CSMA/CAKim, Tae Ok; Park, Jin Soo; Kim, Kyung Jae; Choi, Bong Dae

3. Performance analysis of IEEE 802.15.4 Non-beacon Mode by Piggyback MethodPark, Jin Soo; Kim, Tae Ok; Kim, Kyung Jae; Choi, Bong Dae

4. Performance Analysis of Session-Level Load Balancing AlgorithmsRoubos, D.; Bhulai, S.; Van der Mei, R.D.

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Session MB06 - Many-server Models 1, Chair: Armony in CZ 13, see page 68

1. Optimal Control of Parallel Server Systems with Many ServersDai, Jim; Tezcan, Tolga

2. Fill rate constraint compliance and the many-machine repairman problemJennings, O.; Degrimenci, I.T.

3. Fair Dynamic Routing Policies in Large-Scale Service Systems with Heterogeneous ServersWard, A.R.; Armony, Mor

4. Large cross selling in call centersGurvich, I.; Armony, Mor; Maglaras, Constantinos

Session MB07 - Games 1, Chair: Economou in CZ 14, see page 69

1. Equilibrium customer strategies under various levels of information for the single server queue withunreliable server.Economou, A.; Kanta, S.

2. On the use of quasi birth-and-death processes in equilibrium selection in game theory and queueingnetworksAmir, Madjid

3. An analysis of oligopolistic competitive markets for queued services with QoS guaranteesDube, Parijat; Jain, Rahul; Touati, Corinne

4. Optimal balking strategies in the single server Markovian queue with compartmented waiting space.Economou, A.; Kanta, S.

Session MB08 - Fluid Queues, Chair: Adan in CZ 15, see page 70

1. On the stationary measure of a reflected Brownian motion in a wedge: some explicit resultsDieker, A.B.; Moriarty, J.

2. Transient Fluid Solutions and Queueing Networks with Infinite Virtual QueuesNazarathy, Y.; Weiss, G.

3. R, G and U and the time-inhomogeneous periodic Quasi-Birth Death ProcessMargolius, B.H.

4. Insurance Risk with Variable Number of PoliciesAdan, I.J.B.F.; Kulkarni, Vidyadhar

Session MB09 - Search Models, Chair: Klafter in CZ 16, see page 71

1. Intermittent search processesBenichou, Olivier

2. Search researchOshanin, Gleb

3. Parallel search of long circular strands: Modeling, analysis, and optimizationEliazar, Iddo

4. Stochasticity and search in gene regulation processesMetzler, Ralf

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Monday 4:00pm - 5:30pm

Session MC01 - Exit Times, Chair: Rolski in CZ 4, see page 73

1. On an exit problem of a two-dimensional risk process from the positive quadrantAvram, Florin; Palmowski, Z.; Pistorius, Martijn

2. Gaussian storage processes: long-time behaviorDebicki, K.

3. De Finetti’s dividend problem for a general Lévy insurance risk processPalmowski, Z.; Avram, Florin; Kyprianou, A.; Pistorius, Martijn

4. The exact asymptotics of the time to collision tail distributionPuchała, Z.; Rolski, T.

Session MC02 - Interacting Particle Systems, Chair: Van den Berg in CZ 5, see page74

1. The super-process limit of the critical spread-out contact process above 4 spatial dimensionsVan der Hofstad, R.

2. Duality and exact correlations in a model of heat conductionGiardina, C.; Kurchan, J.; Redig, F.

3. Approximate zero-one laws, spin-flip dynamics and Ising percolationVan den Berg, J.

Session MC03 - Stochastic Programming and Optimization 1, Chair: Gunawan inCZ 10, see page 75

1. Performance Evaluation of Interconnection Network System Based on Minimum Cut Sets MethodGunawan, Indra

2. Scheduling assembly flow-shops with resource leveling: a GA based heuristic approachWang, Qinhua; Ren, Changrui; Dong, Jin; Ding, Hongwei; Wang, Wei; Qiu, Minmin

3. A Fast Entropy Method for Solving Integer Programs: Part IRubinstein, R.Y.

4. A Fast Entropy Method for Solving NP-Hard Counting Problems: Part IIRubinstein, R.Y.

Session MC04 - Stochastic Networks and Diffusions, Chair: Budhiraja in CZ 11, seepage 76

1. Optimal buffer size for a stochastic processing network with a driftGhosh, A.P.; Weerasinghe, A.

2. Characterizations of the invariant measure for a class of reflected diffusions via the extended Sko-rokhod mapKang, Weining; Ramanan, K.

3. Diffusion limit for dynamic routing with random service ratesAtar, R.

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4. Convergence of Invariant Measures for Generalized Jackson NetworksBudhiraja, A.; Lee, Chihoon

Session MC05 - Flow-Level Models, Chair: Prabhakar in CZ 12, see page 78

1. Using the Cavity Method to Analyze Randomized Load Balancing Algorithms with General Flow-size DistributionsLu, Y.; Prabhakar, B.

2. Stability of Randomized Load Balancing Algorithms with General Flow-Size DistributionsBramson, M.

3. Proportional fairness: stability and insensitivity propertiesMassoulié, Laurent

4. Stochastic Stability Under Network Utility Maximization: General File Size DistributionChiang, M.; Tang, A.; Shah, D.

Session MC06 - Markov Processes, Chair: Van Doorn in CZ 13, see page 79

1. Level-crossing ordering of semi-Markov jump processes and applications to Whittle NetworksFerreira, F.; Pacheco, A.

2. Average optimality for risk-sensitive control with general state spaceJaskiewicz, A.

3. Stochastic binary relations of random variables and processesLeskelä, L.

4. Survival in a quasi-death processVan Doorn, E.A.

Session MC07 - Control, Chair: Lelarge in CZ 14, see page 80

1. A probabilistic analysis of a leader election algorithmMohamed, H.

2. Investigation of discrete stochastic systems with dynamic distributed routing and delayed informationYudaeva, N.V.

3. On asymptotic optimality of controls for time-varying queuesCudina, M.; Ramanan, K.; Shreve, S.

4. Heavy traffic limit for online bandwidth packingLelarge, Marc

Session MC08 - Queueing Models 1, Chair: Fralix in CZ 15, see page 81

1. The conditional distribution of the residual service time in the Mn/G/1 queueKerner, Yoav

2. Transient Solution to the Time Dependent Single Server Queue with BalkingAl-Seedy, R.O.; El-Sherbiny, A.A.; El-Shehawy, S.A.; Ammar, S.I.

3. Relating time- and customer- averages for queues using “forward” coupling from the pastPeköz, Erol A.; Ross, Sheldon

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4. A Time-Dependent View of ASTA and Little’s LawFralix, B.H.; Serfozo, R.F.; Riaño, G.

Session MC09 - Inventory 2, Chair: Yao in CZ 16, see page 82

1. An E O Q model with state dependence, random yield, and static pricingBaron, Opher; Berman, Oded; Perry, David

2. Periodic review inventory modelsBijvank, M.

3. Asymptotically Optimal Solutions to a Joint Replenishment-Transshipment ModelYao, D.D.; Xiao, Yongbo

4. Study on inventory management with existence of electronic marketsXie, Ming; Yin, Wenjun; Zhang, Bin; Bai, X.X.; Dong, Jin

Tuesday 8:30am - 10:00am

Session TA01 - Financial Engineering 3, Chair: Kou in CZ 4, see page 84

1. Pricing, hedging and calibrating credit from the top downGiesecke, Kay

2. Malliavin Greeks without Malliavin calculusChen, Nan; Glasserman, Paul

3. To be determinedCont, R.

4. Pricing Asian option via double Laplace transformCai, Ning; Kou, S.G.

Session TA02 - Stochastic Geometry, Chair: Van Lieshout in CZ 5, see page 85

1. Wiener sausage and sensor networksSpodarev, E.; Cerný, R.; Funken, S.; Meschenmoser, D.; Rataj, J.; Schmidt, V.

2. Maximum likelihood estimation for random sequential adsorptionVan Lieshout, M.N.M.

3. Random field model for fingerprint featuresThönnes, Elke

4. Opportunistic Routing on Signal-to-Interference-and-Noise Ratio GraphsBaccelli, F.; Blaszczyszyn, B.; Mühlethaler, P.

Session TA03 - Rare Event Simulation 2, Chair: Juneja in CZ 10, see page 86

1. Alternative efficiency proof and interpretations for a recent state-dependent importance samplingschemeDe Boer, P.T.; Scheinhardt, W.R.W.

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2. Importance sampling for heavy-tailed random walkBassamboo, A.; Juneja, Sandeep; Zeevi, Assaf

3. A Reduced-Form Model With Short-Time Contagion For Credit RiskBlanchet, Jose; Zhan, Li

4. Minimizing Tail Probabilities: Asymptotic Analysis and Efficient Simulation OptimizationJuneja, Sandeep; Kalra, Himanshu

Session TA04 - Scheduling and Stability, Chair: Hasenbein in CZ 11, see page 87

1. Fluid and Diffusion Limits of a Multiclass Overloaded Abandonment Queue under Priority Schedul-ingJennings, O.; Reed, J.E.

2. Stability of Non-random Critically Loaded Networks under MaxWeight SchedulingAndrews, D.M.; Jung, K.; Stolyar, A.L.

3. Undecidability results in the theory of queueing networks and Skorokhod problemBertsimas, D.; Gamarnik, D.; Katz, D.

4. Networks with Parameter Uncertainty: Stochastic Fluid Models with RecourseHasenbein, John J.; Buke, Burak; Morton, David P.

Session TA05 - Ad Hoc Networks, Chair: Van den Berg in CZ 12, see page 88

1. Efficient Resource Control for the EDCA and HCCA Mechanisms in Multi-Rate IEEE 802.11e Net-worksSiris, Vasilios A.; Courcoubetis, Costas

2. Performance Modeling of a Bottleneck Node in an IEEE 802.11 Ad-hoc NetworkRoijers, Frank; Van den Berg, J.L.; Mandjes, M.R.H.

3. Collaboration and incentives in wireless ad-hoc networksGibbens, R.J.; Key, P.B.

4. A fluid model for a relay node in an ad-hoc network: the case of heavy-tailed inputBekker, R.; Mandjes, M.R.H.

Session TA06 - Call Centers 2, Chair: Koole in CZ 13, see page 90

1. Shift scheduling for heterogeneous contact centers via mixed-integer programmingHelber, S.; Henken, K.

2. A real-time routing policy for call centersPot, S.A.

3. The Hayward Approximation with Queueing SystemsAksin, Z.; Van den Schrieck, Jean-Christophe

4. Staffing Call Centers with Differentiated Levels of Service: Approximate Solutions via ConstraintDualizationBassamboo, A.; Zeevi, Assaf

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Session TA07 - Statistics 1, Chair: Segers in CZ 14, see page 91

1. Martingale approach on the average run lengths of Exponentially EWMA control chartsAreepong, Y.

2. Comparative study of change detection properties of Poisson EWMA control procedureSukparungsee, S.; Novikov, A.

3. Benchmarking of Fitting Algorithms for Continuous Phase-Type DistributionsPérez, J.F.; Riaño, G.

4. Regularly varying multivariate time seriesBasrak, Bojan; Segers, Johan

Session TA08 - Polling Models, Chair: Winands in CZ 15, see page 92

1. Polling systems with parameter renewalsMacPhee, I.

2. M/G/∞ polling systems with random visit timesVlasiou, Maria; Yechiali, Uri

3. Polling Systems with Two-Phase Gated Service: Heavy Traffic Results for the Waiting-Time Distri-butionsVan der Mei, R.D.; Resing, J.A.C.

4. Branching-type polling systems with large setupsWinands, E.M.M.

Session TA09 - Switch Scheduling, Chair: Moallemi in CZ 16, see page 93

1. Switch scheduling via message passingBayati, M.

2. How to avoid doing research “10 years ahead of time”Firoozshahian, Amin; Hosseinkeh, Vahideh; Goel, Ashish ; Prabhakar, B.

3. Packet Election For Fair SchedulingJagabathula, S.; Doshi, V.; Shah, D.

4. Approximate Dynamic Programming for Switch SchedulingMoallemi, Ciamac C.; Kumar, Sunil; Van Roy, Benjamin

Tuesday 10:30am - 12:00pm

Session TB01 - Finance and Ruin, Chair: Collamore in CZ 4, see page 95

1. A Dirichlet bridge sampling of the variance gamma process: Pricing path-dependent optionsBallotta, Laura; Dimitrova, Dimitrina; Kaishev, Vladimir

2. Applications of series representations for Lévy processesMichna, Z.

3. Stochastic recurrence equations and ruin in a general Markovian environmentCollamore, J.F.; Nyrhinen, H.

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4. Managing Inventory of Items with Replacement WarrantyHuang, Wei; Kulkarni, Vidyadhar; Swaminathan, Jayashankar M.

Session TB02 - Random Partitions and Partition-valued Processes, Chair: Gnedinin CZ 5, see page 96

1. Small time asymptotics of Lambda-coalescentsBerestycki, J.; Berestycki, N.; Limic, V.

2. Gaussian fluctuations for Plancherel partitionsBogachev, L.

3. Ancestral processes in population genetics - exchangeable coalescentsMöhle, Martin

4. Random partitions and the site frequency spectrum in population geneticsBerestycki, N.; Berestycki, J.; Schweinsberg, Jason

Session TB03 - Gradient Estimation, Chair: Heidergott in CZ 10, see page 97

1. Policy Gradient Methods in Machine LearningPeters, Jan; Theodorou, Evangelos; Schaal, Stefan

2. Strong Properties of Weak DerivativesLeahu, Haralambie; Heidergott, B.; Hordijk, A.

3. Gradient Estimation for a Multi-Component Maintenance System Using Measure-Valued Differen-tiationYuan, Taoying; Heidergott, B.

4. Measure-Valued Differentiation for Random Horizon ExperimentsHeidergott, B.; Hordijk, A.

Session TB04 - Asymptotic Analysis 2, Chair: Xia in CZ 11, see page 99

1. Are end-to-end acknowledgements causing power law delays in large multi-hop networks?Jelenkovic, P.R.; Tan, J.

2. Stochastic ordering for regulated flows and the Better-than-Poisson PropertyGuillemin, F.; Mazumdar, R.R.; Rosenberg, C.P.; Ying, Yu

3. Sharpness: a sufficient and necessary condition for throughput scalabilityChaintreau, A.

4. Scalability of Fork/Join Queueing Networks with BlockingXia, Cathy H.; Liu, Zhen; Towsley, Don; Lelarge, Marc

Session TB05 - Randomized Algorithms and Resource Control Schemes, Chair:Borst in CZ 12, see page 100

1. Optimal peer-to-peer broadcasting schemesMassoulié, Laurent; Twigg, Andy

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2. Mean field limits of particle systems in varying environments, Application to modeling randommulti-access algorithmsBordenave, C.; McDonald, D.R.; Proutière, A.

3. Metastability of CDMA cellular systemsAntunes, N.; Fricker, C.; Robert, Ph.; Tibi, D.

4. Ranking and suggesting tags in collaborative tagging applicationsVojnovic, Milan

Session TB06 - Many-server Models 2, Chair: Van Leeuwaarden / Zwart in CZ 13,see page 102

1. Corrected diffusion approximations for many-server queues IJanssen, A.J.E.M.; Van Leeuwaarden, Johan S.H.; Zwart, Bert

2. Corrected diffusion approximations for many-server queues IIJanssen, A.J.E.M.; Van Leeuwaarden, Johan S.H.; Zwart, Bert

3. A new view on the G/GI/N queue in Halfin-Whitt regimeReed, J.E.

4. Moment Equivalences Between Multiserver and Single Server QueuesScheller-Wolf, A.; Vesilo, R.A.

Session TB07 - Stochastic Control and PDE, Chair: Atar in CZ 14, see page 103

1. Optimal Stopping and Free Boundary Characterizations for some Brownian Control ProblemsBudhiraja, A.; Ross, Kevin J.

2. A control-theoretic analysis of fast-sweeping methods for static Hamilton-Jacobi equationsDupuis, Paul

3. HJB equations, no arbitrage, and generalized Brownian networksAtar, R.; Budhiraja, A.; Williams, R.J.

4. A Representation for the Characteristic Functional of Banach Valued Stable Random MeasuresMahmoodi, S.; Soltani, A.R.

Session TB08 - Fluid Models, Chair: Scheinhardt in CZ 15, see page 104

1. Analysis of a resource sharing policy for computer systems using fluid queuesGautam, N.; Mahabhashyam, S.R.

2. A feedback fluid queue with two congestion control thresholdsMalhotra, R.; Mandjes, M.R.H.; Scheinhardt, W.R.W.; Van den Berg, J.L.

3. On shot noise fluid queues and infinite-server queues with batch arrivals: time-dependent behaviorScheinhardt, W.R.W.; Boucherie, R.J.; De Graaf, W.F.

4. Tandem Queue with Server Slow-downMiretskiy, D.I.; Scheinhardt, W.R.W.; Mandjes, M.R.H.

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Session TB09 - Supply Chains, Chair: Ettl / De Kok in CZ 16, see page 106

1. Centralized and decentralized vendor-managed inventory optimization under different levels of in-formation sharingEttl, M.; Lu, Y.; Squillante, M.S.

2. Optimal Policies for Joint Replenishment and SubstitutionXu, H.; Yao, D.D.; Zheng, S.

3. Benefits and risks of price protection in the computer industryEttl, M.; Huang, P.; Kapuscinski, R.; Sourirajan, K.

4. Period Order Quantities in Multi-Item Multi-Echelon Inventory SystemsDe Kok, Ton

Tuesday 2:45pm - 4:15pm

Session TC01 - Statistics 2, Chair: Kuhnt in CZ 4, see page 107

1. Statistical Analysis of Spatial Patterns of Customer BehaviorsBai, X.X.; Yin, Wenjun; Dong, Jin

2. On the performance of estimators of parameters in autoregressive model under asymmetric lossRoychowdhury, Soma; Bhattacharya, Debasis

3. On nonparametric inference for a class of queuing systemsPawlak, M.

4. An application of algebraic statistics to Bowker’s test for symmetryKuhnt, S.; Krampe, A.

Session TC02 - Anomalous Diffusion, Chair: Metzler in CZ 5, see page 108

1. Anomalous Subdiffusion RevisitedKlafter, Joseph

2. Anomalous Diffusion Leads to Weak Ergodicity BreakingBarkai, E.

3. Mesoscopic approach to reactions between subdiffusing particlesSokolov, Igor M.

4. Fractional Fokker-Planck dynamics: Stochastic representation and computer simulationWeron, Aleksander; Magdziarz, M.

Session TC03 - Cross-entropy, Chair: Kroese in CZ 10, see page 109

1. Parametric Minimum Cross-Entropy MethodRubinstein, R.Y.; Dolgin, A.; Kroese, D.P.; Glynn, P.W.

2. A Nonparametric Extension of the Cross-Entropy methodBotev, Z.I.; Taimre, T.; Kroese, D.P.

3. A Generalized Cross-Entropy Method: Discrete State SpacesTaimre, T.; Botev, Z.I.; Kroese, D.P.

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4. Adaptive importance sampling algorithms for rare-event simulations of the infinite-server queue.Ridder, A.

Session TC04 - Limit Theorems for Queueing Systems with Measure-valued StateDescriptors, Chair: Puha in CZ 11, see page 111

1. Limit Theorems for Many-Server QueuesRamanan, K.

2. Heavy Traffic Analysis for EDF Queues with RenegingKruk, Lukasz; Lehoczky, J.; Ramanan, K.; Shreve, S.

3. Central Limit Theorems for a large queuing network with resource poolingGraham, C.

4. The Fluid Limit of a Shortest Remaining Processing Time QueueDown, Douglas G.; Gromoll, H.C.; Puha, A.L.

Session TC05 - Communication Systems 2, Chair: Choi in CZ 12, see page 112

1. Performance analysis of small non-uniform packet switchesBeekhuizen, P.; Resing, J.A.C.

2. Performance analysis of sleep mode and idle mode for best effort traffic in IEEE 802.16eLee, Yong Hyun; Hwang, Eunju; Kim, Kyung Jae; Son, Jung Je; Choi, Bong Dae

3. Performance Analysis of Power Saving Class of Type 1 for both Downlink and Uplink Traffics inIEEE 802.16eBaek, Sangkyu; Son, Jung Je; Lee, Seon Mi; Choi, Bong Dae

4. A New Power Saving Mechanism with Periodic Traffic Indications in the IEEE 802.16eHwang, Eunju; Kim, Kyung Jae; Son, Jung Je; Choi, Bong Dae

Session TC06 - Matrix-Analytic Methods, Chair: Squillante / Taylor in CZ 13, seepage 114

1. Bounds and Decay Rate for a QBD Process with Countably Many Background StatesLi, H.; He, Q.-M.; Zhao, Y.Q.

2. Algorithmic approach to the extinction probability of branching processesHautphenne, S.; Latouche, Guy; Remiche, M.-A.

3. Physical interpretations of algorithms for stochastic fluid flowsBean, Nigel; O’Reilly, Małgorzata; Taylor, Peter

4. Invariant measures for transient fluid queuesVan Lierde, Sarah; Da Silva, Ana; Latouche, Guy

Session TC07 - MDP 2, Chair: Örmeci in CZ 14, see page 115

1. A Semi Markov Decision Approach for Train Conflict ResolutionAl-Ibrahim, Assil; Van der Wal, Jan

2. Exact finite approximations of average-cost countable Markov Decision ProcessesLeizarowitz, Arie; Shwartz, Adam

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3. A Controlled Markov Process Model for a Benchmark Reentrant Manufacturing Line with Infiniteand Finite Capacity BuffersRamírez-Hernández, José A.; Fernandez, Emmanuel

4. Effects of System Parameters on the Optimal Policy Structure in a Class of Queueing Control Prob-lemsOrmeci, E.L.; Cil, E.B.; Karaesmen, F.

Session TC08 - Queueing Models 2, Chair: Gómez-Corral in CZ 15, see page 116

1. Bad luck when joining the shortest queueBlanc, Hans

2. On the correlation structure of Gaussian queuesEs-Saghouani, A.; Mandjes, M.R.H.

3. Negative Queue Length:Queues for Quantum InformationToyoizumi, Hiroshi

4. On a batch Markovian arrival process subject to renewal catastrophesGómez-Corral, A.

Session TC09 - World Wide Web, Chair: Litvak in CZ 16, see page 117

1. PageRank of Scale-Free Growing NetworksAvrachenkov, K.; Lebedev, D.

2. Distances in Random GraphsVan der Hofstad, R.

3. Stochastic equations behind Web rankingVolkovich, Y.; Donato, D.; Litvak, Nelly; Scheinhardt, W.R.W.

4. A fair distribution of probability mass on the bow-tie graphsAvrachenkov, K.; Litvak, Nelly; Pham, Kim Son

Tuesday 4:45pm - 6:15pm

Session TD01 - Lévy Processes and Finance, Chair: Rieder / Bäuerle in CZ 4, seepage 119

1. Malliavin Calculus for Levy Processes and Applications in FinancePetrou, Evangelia

2. Extremal behavior of stochastic integrals driven by regularly varying Lévy processesHult, H.; Lindskog, F.

3. Dependence properties and comparison results for Lévy processesBäuerle, N.; Müller, A.; Blatter, A.

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Session TD02 - Spatial Models 1, Chair: Foss / Stolyar in CZ 5, see page 120

1. Large deviations of the interference in a wireless communication modelGanesh, A.J.; Torrisi, G.L.

2. M/D/1/1 loss system with interference and applications to transmit-only sensor networksBlaszczyszyn, B.; Radunovic, B.

3. Local and global properties of random directed treesRoy, Rahul

4. Near-Minimal Spanning Trees: a Scaling Exponent in Probability ModelsLelarge, Marc; Aldous, David; Bordenave, C.

Session TD03 - Asymptotics, Chair: Blanchet / Mikosch in CZ 10, see page 121

1. Towards Efficient Simulation of Long Delays for Heavy-tailed Multiserver QueuesBlanchet, Jose; Glynn, P.W.

2. Large deviations for point processes based on stationary sequences with heavy tailsHult, H.; Samorodnitsky, G.

3. Rare event simulation methodology for perpetuitiesBlanchet, Jose; Zwart, Bert

4. Boundary behavior and stability for multidimensional reflected diffusions with state-dependent jumpsin the positive orthantMazumdar, R.R.; Guillemin, F.; Piera, F.

Session TD04 - Large Deviations 1, Chair: Majewski in CZ 11, see page 122

1. Large deviations of multitype queuesDabrowski, A.; Lee, J.; McDonald, D.R.

2. Limiting distribution of the phase when level passes a thresholdKhanchi, Aziz; McDonald, D.R.

3. Large Deviations of Generalized Jackson NetworksMeiner, Silke

4. Functional asymptotics of a simple queueing network with short- and long-range dependent inputMajewski, Kurt

Session TD05 - Computer-communications, Chair: Hwang in CZ 12, see page 123

1. Asymptotic results in some random infinite urn modelsRobert, Ph.; Simatos, F.

2. Connectivity probability of one-dimensional ad hoc wireless networks with minimum hop pathAntunes, N.; Jacinto, Gonçalo; Pacheco, A.

3. Design and Analysis of a Packet Scheduler Exploiting the Multiuser Diversity for Performance En-hancementHwang, Gang Uk; Ishizaki, Fumio

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Session TD06 - Applied Probability, Chair: Hagwood in CZ 13, see page 124

1. Stochastic modeling of a dark matter experimentCoakley, Kevin

2. Stochastic Modeling in Gene Expression Measurements: Accounting for Both Background Errorand Instrument SaturationLu, Z.Q. John

3. Stochastic Modeling of a Carbon Nanotube in an Electrical FieldHagwood, Charles

4. Allocation Models and heuristics for the outsourcing of repairs for a dynamic warranty populationDing, L.; Glazebrook, K.D.; Kirkbride, C.

Session TD07 - Optimal Stopping 1, Chair: Gnedin in CZ 14, see page 125

1. Maximizing the expected duration time related to Poisson arrivalKurushima, A.; Ano, K.

2. A linear programming analysis of finite horizon optimal stopping problemsHelmes, K.

3. Optimal stopping with a rank-dependent lossGnedin, A.V.

4. Optimal Stopping of ”Seasonal” ObservationsPresman, E.L.; Sonin, Isaac

Session TD08 - Queueing in Telecom, Chair: Tijms in CZ 15, see page 127

1. A spectral method for a BMAP/D/c queue with synchronous serversNishimura, S.; Yatomi, N.

2. Differential Traffic TheoryGarcia, J.M.; Brun, O.

3. Moments of the duration of busy periods in oscillating queueing systemsPacheco, A.; Ribeiro, H.

4. A quick approximation for the state probabilities and waiting-time probabilities in the M/D/1 queueTijms, H.C.

Session TD09 - Engineering, Chair: Günalay in CZ 16, see page 128

1. Analysis of a Probabilistic Key Management Scheme for Vehicular CommunicationsVan den Berg, E.

2. Comparison of the effect of various forecasting methods on bullwhip effect in supply chainFarahani, R.Z.; Najafi, M.; Asgari, N.

3. Using Conditional Probabilities for Automatic New Topic Identification of Search Engine Transac-tion LogsOzmutlu, Seda; Ozmutlu, Huseyin C.; Buyuk, Buket

4. A queueing analysis of a hybrid manufacturing system with new and refurbished itemsGünalay, Y.; Gupta, D.

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Wednesday 8:30am - 10:00am

Session WA01 - Queues and Lévy Processes, Chair: Kella in CZ 4, see page 129

1. The M/G/1 queue with quasi-restricted accessibilityPerry, David

2. Lévy correlation cascadesEliazar, Iddo

3. Superposition of renewal processes and an application to multi-server queuesStadje, Wolfgang

4. Lévy driven fluid queues with negatively correlated service interruptions and busy periodsKella, Offer

Session WA02 - Random Graphs, Chair: Norros in CZ 5, see page 130

1. Two diameter results in the configuration modelHooghiemstra, Gerard

2. A local limit theorem for the critical random graphMüller, T.; Van der Hofstad, R.; Kager, W.

3. Percolation on the 2D Hamming graph: The supercritical phaseLuczak, M.J.; Van der Hofstad, R.

4. Robustness against core losses in a power-law random graphNorros, I.; Reittu, H.

Session WA03 - Simulation 2, Chair: Grassmann in CZ 10, see page 132

1. Zero-Variance Importance Sampling Estimators for Markov Process ExpectationsAwad, H.P.; Glynn, P.W.; Rubinstein, R.Y.

2. Fluid Limit for Hybrid MCMC samplersFort, G.; Moulines, E.; Priouret, P.

3. Applied Probability Contribution to the Burn-in Phase in Equilibrium SimulationGrassmann, W.K.; Sethi, S.

4. Ordering of time correlations in stationary Markov processes and queueing networksSzekli, R.; Daduna, H.

Session WA04 - Networks, Chair: Zazanis in CZ 11, see page 133

1. Functional diffusion approximation of the queue with an infinity of serversDecreusefond, L.; Moyal, P.

2. Diffusion approximations for reinsurance problemsZazanis, Michael; Yannacopoulos, A.N.; Papanicolaou, V.; Frangos, E.N.

3. Active-Idleness and the Multiclass Queueing Networks Optimization using Infinitesimal Perturba-tion AnalysisOrfão, N.M.; Bispo, C.F.

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4. On Stochastic Network CalculusJiang, X.

Session WA05 - Internet, Chair: Fricker in CZ 12, see page 135

1. Transient analysis and asymptotic of the mean of the first hitting time for a TCP window size processLöpker, Andreas H.; Van Leeuwaarden, Johan S.H.

2. Multiclass G/M/1 queueing system with self-similar input and non-preemptive priorityCaglar, M.; Iftikhar, M.; Landfeldt, B.; Singh, T.

3. Sampling Internet trafficAntunes, N.; Chabchoub, Y.; Fricker, C.; Guillemin, F.; Robert, Ph.

Session WA06 - Infinite Multidimensional Markov Chains, Chair: Squillante / Tay-lor in CZ 13, see page 136

1. Geometric decay in a QBD process with countable background states with applications to a join-the-shortest-queue modelLi, H.; Miyazawa, M.; Zhao, Y.Q.

2. Stability Analysis of a General Class of Markov ChainsGamarnik, D.; Squillante, M.S.

3. Some interesting decay rate behaviour in quasi-birth-and-death processes with infinite phase spacesMotyer, Allan; Taylor, Peter

4. Explicit solutions and decay rates for a general class of Markov chainsVan Leeuwaarden, Johan S.H.; Squillante, M.S.; Winands, E.M.M.

Session WA07 - Optimal Stopping 2, Chair: Sonin in CZ 14, see page 137

1. Fiducial approach in the invariant problem of optimal choiceBelenky, Vitali Z.

2. Bandwagon Random WalksQuinn, J.

3. A Generalized Gittins Index for Markov Chain and its Recursive CalculationSonin, Isaac

Session WA08 - Flexible Servers, Chair: Down in CZ 15, see page 138

1. Tandem Lines with Synergistic ServersAndradóttir, Sigrún; Ayhan, Hayriye; Down, Douglas G.

2. A call center model with upgradesLewis, Mark E.; Down, Douglas G.

3. Resource allocation in grid computingKoole, G.; Righter, R.

4. Linear Programming Based Affinity Scheduling for Heterogeneous Computing SystemsDown, Douglas G.; Al-Azzoni, Issam

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Session WA09 - Inventory 3, Chair: Van Houtum in CZ 16, see page 139

1. Approximate evaluation of order fill rates for an inventory system of service toolsVliegen, I.M.H.; Van Houtum, G.J.

2. Exact and Approximate Results for the Kit Management ProblemGullu, R.; Koksalan, M.

3. Valuation of downstream liquefied natural gas storageScheller-Wolf, A.; Wang, Xiaofeng (Mulan); Kekre, Sunder; Secomandi, Nicola

4. Analysis of an inventory model with lateral transshipmentKranenburg, A.A.; Van Houtum, G.J.

Wednesday 10:30am - 12:00pm

Session WB01 - Last Minute Talks, Chair: Boxma in CZ 4, see page 141

1. New Algorithms for Laplace Transform InversionDen Iseger, Peter

Session WB02 - Self-interacting Stochastic Processes, Chair: Van der Hofstad in CZ5, see page 141

1. An expansion for self-interacting random walksHolmes, M.

2. Parabolic Anderson model: Localisation of mass in random mediaKönig, W.; Mörters, P.; Sidorova, N.

3. Excited to the centerKozma, G.

4. Sampling for the maximum of Brownian motionVan Leeuwaarden, Johan S.H.

Session WB03 - Stochastic Programming and Optimization 2, Chair: Veatch in CZ10, see page 142

1. Efficient Allocation Under Incomplete InformationBurnetas, Apostolos N.

2. A problem of optimal sequential pricingHodge, D.J.

3. Cutting speed adjustment and tool magazine capacity allocation using stochastic dynamic program-mingLamond, B.F.

4. Approximation, structure and performance for average cost queueing network controlVeatch, Michael

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Session WB04 - Fluid Limits and Mean Fields, Chair: D’Auria in CZ 11, see page143

1. Mean field analysis of discrete-time closed queueing networks with geometric service times.Adan, I.J.B.F.; Beekhuizen, P.; D’Auria, B.

2. Spontaneous resonances and the onset of collective behavior in queuing networksVladimirov, A.A.; Rybko, A.N.; Shlosman, S.B.

3. Conic Instability of fluid limit modelsKim, Bara

Session WB05 - TCP, Chair: Baccelli in CZ 12, see page 144

1. Iterated function systems and ergodicity of AIMD networksKing, C.; Shorten, R.; Wirth, F.; Leith, D.

2. Queueing Theory for TCPWischik, D.

3. A stochastic model of Split TCPBaccelli, F.; Carofiglio, G.; Foss, S.

4. The Equations of TCPBaccelli, F.; Kim, Ki Baek; McDonald, D.R.

Session WB06 - Call Centers 3, Chair: Koole in CZ 13, see page 146

1. Modeling arrivalsChevalier, Philippe; Van den Schrieck, Jean-Christophe

2. An approximation method for evaluating the performances of a skill-based routing call center modelJouinni, Oualid; Salah Aguir, Mohamed; Dallery, Yves

3. Design and Capacity Planning in Multi-Skill Call CentersOrmeci, E.L.; Cakan, N.; Aksin, Z.; Karaesmen, F.

4. Dynamic call blending in the presence of time-varying arrivalsBalint, A.; Bhulai, S.

Session WB07 - Dynamic Control for Service Management, Chair: Hampshire inCZ 14, see page 147

1. A Modified Fluid Model for the Optimal Design of a Dynamic Call CenterMassey, Bill

2. Inventory rationing for a system with a lost-sales and a backordering customer classEnders, P.; Adan, I.J.B.F.; Van Houtum, G.J.

3. Dynamic Pricing and Capacity Planning of e-Services under Demand UncertaintyXiu, Cathy

4. Efficient Frontier Dynamic Pricing for On-Demand Music and Movie ServicesHampshire, Robert C.

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Session WB08 - Queueing Models 3, Chair: Brandt in CZ 15, see page 149

1. Necessary and Sufficient Conditions for Finite First Moment of the Workload Components in FIFOMultiserver QueuesScheller-Wolf, A.; Vesilo, R.A.

2. Comparison on Queueing Systems with Generalized Processor Sharing SchedulingKim, Sunggon; Lee, Ju Yong; Park, Wonyoung

3. Approximations for the M/G I/N + G I Type Call CenterBalcıoglu, Baris; Travani, Foad

4. Waiting times for M/M systems under generalized processor sharingBrandt, M.; Brandt, A.

Session WB09 - Combinatorial Structures, Chair: Gamarnik / Ramanan in CZ 16,see page 150

1. Probabilistic analysis of game tree evaluationNeininger, R.

2. On Temperate Regions in Random GrovesBaryshnikov, Y.

3. The mixing advantage is less than 2Hamza, Kais; Jagers, P.; Sudbury, A.; Tokarev, D.

4. Iterative Algorithm for Maximum Weight Independent SetSanghavi, S.; Shah, D.; Willsky, A.

Wednesday 1:30pm - 3:00pm

Session WC01 - Risk and Insurance, Chair: Kolbe in CZ 4, see page 151

1. Optimal Insurance Demand and Investment in a Dynamic Mean-variance FrameworkBiffis, E.

2. Long-term survivor mixture models for credit scoringMo, S.F.; Yau, K.K.W.

3. Stability and equilibria in incomplete marketsZitkovic, Gordan

4. Valuation and pricing of reverse mortgagesKolbe, Andreas; Zagst, Rudi

Session WC02 - Spatial Models 2, Chair: Foss / Stolyar in CZ 5, see page 152

1. Dynamics of pattern formation in multi-hop wireless networks using backoff-based MAC protocolsDurvy, Mathilde; Thiran, Patrick

2. Optimal throughput allocation and stability in a spatial random-access modelStolyar, A.L.

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3. A Random Multiple Access Protocol with Spatial InteractionsBordenave, C.; Foss, S.; Shneer, V.

4. Routing in Sensor Networks by Using Spatial DynamicsEphremides, Anthony

Session WC03 - Perturbation and Discrete Events, Chair: Heidergott in CZ 10, seepage 154

1. Law of Large Numbers for products of random matrices in the max-plus semiringMerlet, G.

2. Optimal short-term utilization bounds on multiclass networksOrfão, N.M.; Bispo, C.F.

3. Some new results on the performance optimization of queueing systemsXia, Li; Yin, Wenjun; Dong, Jin

4. Sensitivity analysis for Gaussian systems, a mathematical “urban legend” bustedHeidergott, B.; Vázquez-Abad, F.; Volk-Makarewicz, W.

Session WC04 - Large Deviations 2, Chair: Ignatiouk-Robert in CZ 11, see page 155

1. Large Deviation and Fluctuation of Diffusion Processes with discontinuous driftChiang, T.S.; Sheu, S.J.

2. Large deviations asymptotics of exit time and exit location for a class of Lagrangian systemsKovaleva, Agnessa

3. Martin boundary with large deviation technique for a random reflected random walks on a half-spaceIgnatiouk-Robert, Irina

4. Application of Tauberian theorem to the exponential decay of the tail probability of a random variableNakagawa, K.

Session WC05 - Bandwidth Sharing Networks, Chair: Borst in CZ 12, see page 157

1. Bandwidth-Sharing Networks in OverloadEgorova, R.; Borst, S.C.; Zwart, Bert

2. Stability of bandwidth sharing networks with monotone service ratesJonckheere, Matthieu

3. Importance sampling in rate-sharing networksLieshout, P.; Mandjes, M.R.H.

4. Flow-level Stability of Data Networks with Non-convex and Time-varying Rate RegionsLiu, J.; Proutière, A.; Yi, Y.; Chiang, M.; Poor, H.V.

Session WC06 - Biology, Chair: Spencer in CZ 13, see page 158

1. Determining Hidden Markov Models Efficacy in gene finding problemKazemnejad, A.; Hajizadeh, E.; Mirjafari, K.

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2. Count or control? Optimal monitoring for invasive species managementMcCarthy, M.A.; Moore, A.; Taylor, Peter

3. Using stochastic epidemic models to determine if emerging avian influenza can escape containmentSpencer, S.E.F.; O’Neill, P.D.

Session WC07 - Games 2, Chair: Weber in CZ 14, see page 159

1. Symmetric Rendezvous GamesFan, J.

2. On two-stopping and stopping game problemsSzajowski, K.

3. Symmetric rendezvous searchWeber, R.R.

Session WC08 - Queueing Models 4, Chair: Vlasiou in CZ 15, see page 160

1. Analysis of a queueing system with synchronized services and setup times.Economou, A.; Kapodistria, S.

2. A Geo/G/1/∞ queueing system with a generalized N -policy and setup-closedown timesMoreno, P.

3. Production inventory with service time and vacation to the serverKrishnamoorthy, A.; Narayanan, Viswanath C.

4. A multi-station queue with dependent interarrival and service timesVlasiou, Maria

Session WC09 - Health Care, Chair: Ziya in CZ 16, see page 161

1. Optimal outpatient appointment schedulingKaandorp, G.C.; Koole, G.

2. Nurse Staffing and Bed Capacity within Hospitals UnitsJennings, O.

3. Managing New Patient Admissions in Multi-site Healthcare FacilitiesTekin, E.

4. Identifying the Critical Patient: Finding the Balance between Undertriage and OvertriageZiya, S.; Argon, N.T.

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11Abstracts

Monday 10:45am - 12:15pm

Session MA01 - Financial Engineering 1, Chair: Kou in CZ 4

Optimal Consumption and Portfolio Management with Non-Observable Real Prices

A. Bensoussan, University of Texas at Dallas, USA, axb046100@utdallas.eduJ. Keppo, University of Michigan, USA, keppo@umich.eduS.P. Sethi, University of Texas at Dallas, USA, sethi@utdallas.edu

We consider the classical optimal consumption and portfolio investment problem subject to a random infla-tion in the consumption good prices over time. Both the cases when the investor observes the consumptiongood prices and when he receives noisy observations on those are treated. We derive the optimal policy andshow that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The formulas forthe compositions of the funds in the two cases are the same, but in general their actual compositions differbecause in the noisy observation case the consumption basket is a random variable. However, when theinvestor has a CRRA utility, then his optimal portfolio strategy is the same in the two cases. The additionalrisk due to the non observability decreases consumption and increases saving.

Robustness and sensitivity analysis of risk measurement procedures

R. Cont, IEOR Department – Columbia University, United States, Rama.Cont@columbia.eduR. Deguest, Centre de Mathématiques Appliquées – Ecole Polytechnique, France,deguest@cmapx.polytechnique.frG. Scandolo, Dipartimento di Matematica per le Decisioni – Università di Firenze, France,giacomo.scandolo@unifi.it

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the port-folio from available observations and computing a risk measure of this distribution which summarizes therisk of the portfolio. These two steps are usually considered and studied separately. We define the notionof risk measurement procedure, as a two–step procedure which yields a correspondence between a couple(DN , X), where X is a payoff and DN a data set on which the estimation is based, and a risk estimateR(DN , X) for X . This quantity is supposed to estimate the “abstract” risk measure ρ(P, X) where X ispayoff and P is the “true” probability measure from which the data set is generated. It allows us to studysome natural requirements of the risk measurement procedure: coherence, robustness and sensitivity. Westate some theoretical results which illustrate a conflict between the coherence of a risk measurement pro-cedure and its robustness. We complement this abstract result by computing measures of sensitivity, whichallow to quantify the robustness of various risk measures with respect to the data set used to compute them.

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In particular, we show that the same “risk measure” may exhibit quite different sensitivities depending onthe estimation procedure used.

What Is a Good Risk Measure: Bridging the Gaps between Robustness, Subaddi-tivity, and Insurance Risk Measures

C.C. Heyde, Columbia University, The United States, chris@stat.columbia.eduS.G. Kou, Columbia University, The United States, sk75@columbia.eduX.H. Peng, Columbia University, The United States, xp2102@columbia.edu

Two main axiomatically based risk measures are the coherent risk measure, which assumes subadditivityfor random variables, and the insurance risk measure, which assumes additivity for comonotonic randomvariables. We propose a new, data based, risk measure, called natural risk statistic, that is characterized bya new set of axioms. The new axioms only require subadditivity for comonotonic random variables, whichis consistent with the prospect theory in psychology. Comparing to two previous measures, the naturalrisk statistic includes the tail conditional median which is more robust than the tail conditional expectationsuggested by the coherent risk measure; and, unlike insurance risk measures, the natural risk statistics canalso incorporate scenario analysis. The natural risk statistic includes the VaR as a special case and thereforeshows that VaR, though simple, is not irrational.

Dynamic importance sampling for credit baskets

S. Weber, ORIE, Cornell University, 279 Rhodes Hall, Ithaca, NY 14853, sweber@orie.cornell.eduM. Liese, ETH Zurich, Switzerland, maren.liese@math.ethz.ch

We discuss dynamic importance sampling techniques which improve the efficiency of Monte Carlo estima-tors of the prices of credit derivatives like credit basket swaps and CDOs.

Session MA02 - Random Fields, Chair: Kaj in CZ 5

Operator Scaling Stable Random Fields: Theory and Simulation

H.P. Scheffler, University of Siegen, Germany, scheffler@mathematik.uni-siegen.deH. Biermé, MAP5 Université René Descartes, France, hermine.bierme@math-info.univ-paris5.frM.M. Meerschaert, Michigan State University, USA, mcubed@sstmsu.edu

A random field is a family of random variables indexed by points in an Euclidian parameter space. Classicalexamples such as the fractional Brownian field have stationary increments and are self-similar with Hurstindex H. Moreover they are isotropic, that is invariant under rotations of the underlying parameter space.In many applications, for example the modelling of fractured rock however, the field should be anisotropicand should have different self-similarity indices in different directions. Examples of such fields are theso-called operator scaling random fields. Two different and usually not equivalent representations of suchfields will be presented as well as an efficient simulation method for some important special cases.

Random fields generated from multifractional pseudodifferential operators: Pseu-dostable random fields

M.D. Ruiz-Medina, Granada University, Granada, Spain, mruiz@ugr.esJ.M. Angulo, Granada University, Granada, Spain, jmangulo@ugr.esV.V. Anh, Queensland University of Technology, Brisbane, Australia, v.anh@qut.edu.au

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In the Gaussian case, random fields defined by multifractional elliptic and parabolic equations are charac-terized in terms of Reproducing Kernel Hilbert Space (RKHS), and multifractional Sobolev space theory(Ruiz-Medina et al., 2004b, and Ruiz-Medina et al., 2007). Their local Hölder exponent is a functiondepending on space, and on space and time, respectively. On the other hand, under certain conditions, mul-tifractional pseudodifferential operators also generate Feller semigroups. Markov processes with transitionprobability densities having multifractional local Hölder exponent can then be introduced in this frame-work (see Jacob, 2005). An interesting special case is the so called pseudostable processes (Lévy-typeprocesses), which can be generated from the parametrix of the Riesz potential of variable order (see Zahle,2004, for the fixed-order case). We also consider the formulation of these models on non-regular compactdomains (Ruiz-Medina et al., 2004a). In particular, heterogeneous domains with variable local dimensionare studied in Ruiz-Medina et al. (2007). The multifractal domain case is introduced as a limiting casewhen regularity assumptions on the multifractional exponent defining the local dimension of the domain,the local Hölder exponent of the associated d(·)-measure, are relaxed .

Acknowledgements.

This work has been supported in part by projects MTM2005-08597 of the DGI and P05-FQM-00990 of the Andalou-sian CICE, Spain.

References

Jacob, N. (2005). Pseudodifferential Operators and Markov Processes, Vol. III. Imperial College Press.

Ruiz-Medina, M.D., Angulo, J.M. and Anh, V.V. (2007). Multifractality in Space-Time Statistical Models.Stochastic Environmental Research and Risk Assessment (to appear).

Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2004a). Fractal random fields on domains with fractalboundary. Infinite Dimensional Analysis, Quantum Probability and Related Topics 7, 395-417.

Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2004b). Fractional generalized random fields of variableorder. Stochastic Analysis and Applications 22, 775-800.

Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2007). Random fields with multifractional regularity or-der on multifractals. Infinite Dimensional Analysis, Quantum Probability and Related Topics (submitted).

Zahle, M. (2004). Riesz potentials and Liouville operators on fractals. Potential Analysis 21, 193-208.

Self-similar random fields as scaling limits of Poisson germ-grain models

Ingemar Kaj, Department of Mathematics, Uppsala University, Sweden, ikaj@math.uu.seAnne EstradeHermine Biermé

We study limiting random systems that arise by aggregation of spherical grains which are uniformly scat-tered according to a Poisson point process in d-dimensional space. The grains have random radius, in-dependent and identically distributed, with a distribution which is assumed to have a power law behavioreither in zero or at infinity. The resulting configurations of mass suitably centered and normalized exhibitlimit distributions under scaling, which are conveniently described in a random fields setting. In this man-ner we construct Gaussian self-similar random fields of index H , for arbitrary noninteger H > −d/2.We obtain also non-Gaussian random fields with interesting properties, in particular a model of the type"fractional Poisson motion".

References:

I.Kaj, L. Leskelä, I. Norros, V. Schmidt, Scaling limits for random fields with long-range dependence. Ann.Probab. 35:2 (2007), 528-550.

H. Biermé, A. Estrade, I. Kaj, About scaling behavior of random balls models. 6th Int. Conf. on Stereology,Spatial Statistics and Stochastic Geometry, Prague, June 2006.

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Characterization of Banach Valued Stable Random Measures

S. Mahmoodi, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran.mahmoodi@cc.iut.ac.irA.R. Soltani, Department of Statistics & Operations Research, Faculty of Science, Kuwait University,State of Kuwait. soltani@kuc01.kuniv.edu.kw

A Banach space (X: real separable) valued symmetric α stable random measure 8 with independentincrements on a measurable space (F,F ) is considered. A pair (k, µ), called here a control pair, forwhich k : X × F → R+, µ a positive measure on (F,F ), is introduced. It is proved that the law of 8is governed by a control pair; and every control pair will induce such 8. Moreover k is unique for a givenµ. Our derivations are based on the Generalized Bochner Theorem and the Radon- Nikodym Theorem forvector measures.

Session MA03 - Rare Event Simulation 1, Chair: Dupuis in CZ 10

Fast Simulation of Brownian Motion Avoiding Poissonian Obstacles

Jose Blanchet, blanchet@fas.harvard.eduPaul Dupuis, dupuis@dam.brown.edu

We are interested in developing efficient rare-event simulation methodology for the probability that, forlong time, Brownian motion avoids a cloud of obstacles that are distributed according to an homogeneousPoisson process. This problem, motivated from the physics literature, is related to the development of longtime asymptotics for the Laplace transform of the volume of the Wiener sausage. We investigate severalsimulation strategies that are suggested by celebrated large deviations results on the volume of the Wienersausage and discuss their theoretical performance.

Fast pricing of basket default swaps

Zhiyong Chen, Bear Stearns, New YorkPaul Glasserman, Columbia University, New York, pg20@columbia.edu

A basket default swap is a derivative security tied to an underlying basket of assets subject to credit risk.The value of the contract depends on the joint distribution of the default times of the underlying assets.Valuing a basket default swap often entails simulation of these default times. For baskets of high-qualitycredits and for swaps that require multiple defaults to trigger payment, pricing the swap is a rare-eventsimulation problem. Joshi and Kainth (2004) introduced an innovative importance sampling technique forthis problem that forces a predetermined number of defaults to occur on each path. This paper analyzes,extends and improves the Joshi-Kainth algorithm. We show that, in its original form, the algorithm canactually increase variance; we present an alternative that is guaranteed to reduce variance, even whendefaults are not rare. Along the way, we provide a rigorous underpinning in a setting sufficiently general toinclude both the original method and the version proposed here.

Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Ran-dom Variables

Paul Glasserman, Columbia University, pg20@columbia.eduSandeep Juneja, Tata Institute of Fundamental Research, India, juneja@tifr.res.in

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Successful efficient rare event simulation typically involves using importance sampling tailored to a specificrare event. However, in applications one may be interested in simultaneous estimation of many probabilitiesor even an entire distribution. In this talk, we address this issue in a simple but fundamental setting.Specifically, we consider the problem of efficient estimation of the probabilities P(Sn ≥ na) for largen, for all a lying in an interval A, where Sn denotes the sum of n independent, identically distributedlight-tailed random variables. Importance sampling based on exponential twisting is known to produceasymptotically efficient estimates when A reduces to a single point. We show, however, that this procedurefails to be asymptotically efficient throughout A when A contains more than one point. We analyze the bestperformance that can be achieved using a discrete mixture of exponentially twisted distributions and thenpresent a method using a continuous mixture. We show that a continuous mixture of exponentially twistedprobabilities and a discrete mixture with a sufficiently large number of components produce asymptoticallyefficient estimates for all a ∈ A simultaneously.

Design and analysis of splitting algorithms for rare event estimation

Paul Dupuis, Brown University, USA, dupuis@dam.brown.edu

Particle splitting methods are considered for the estimation of rare events with a large deviation scaling. Anotion of subsolution is defined for a related calculus of variations problem, and two main results are de-scribed that hold under mild conditions. The first is that the number of particles generated by the algorithmgrows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution.The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically)in terms of the subsolution.

Session MA04 - Asymptotic Analysis 1, Chair: Bassamboo / Rand-hawa in CZ 11

Heavy Traffic Analysis of Maximum Pressure Policies for Stochastic Processing Net-works with Multiple Bottlenecks

Baris Ata, Northwestern University, Evanston, IL, USA, b-ata@kellogg.northwestern.eduWuqin Lin, Northwestern University, Evanston, IL, USA, wuqin-lin@kellogg.northwestern.edu

A class of open processing networks operating under a maximum pressure policy is considered in the heavytraffic regime. We prove that the diffusion-scaled workload process for a network with several bottleneckresources converges to a semimartingale reflecting Brownian motion (SRBM) living in a polyhedral cone.We also establish a state space collapse result that the queue length process can be lifted from the lower-dimensional workload process.

Slow-server problem in many-server heavy traffic with non-exponential service times

Tolga Tezcan, Industrial and Enterprise Systems Engineering, University of Illinois atUrbana-Champaign, Urbana, Illinois, ttezcan@uiuc.edu

We consider the slow-server problem with non-exponential service times. We assume that service timeshave a certain phase-type distribution. Specifically, service time of a server is assumed to have an ex-ponential distribution with rate µi with probability pi . Specific values of the parameters for each serverdepend on the server pool it belongs to. We show that a static priority policy is asymptotically optimal inthe many-server heavy-traffic regime.

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11 – Abstracts

Validity of heavy-traffic steady-state approximations in open queueing networks:sufficient conditions involving state-space collapse

I. Gurvich, Columbia University, New York, ig2126@columbia.eduA. Zeevi, Columbia University, New York, assaf@gsb.columbia.edu

A class of diffusion processes known as semi-martingale reflecting Brownian motions (SRBMs) is oftenused to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson(1998) and Williams (1998) laid out a general and structured approach for proving the validity of suchheavy-traffic approximations, in which the SRBM model is obtained as a diffusion limit from a suitablynormalized queue-length process.

The Brownian system model provides, among other things, a simplified means for approximating thesteady-state behavior of the underlying queueing network. However, for the multiclass network modelsstudied by Bramson and Williams, it is not clear whether there is rigorous justification for such an approx-imation.

In this paper we give a broad set of sufficient conditions, building on the work of Bramson and Williams,under which the above question can be answered in the affirmative. The analysis illuminates some inter-esting connections between: stability of the pre- and post-limit processes; the fluid models correspondingto the two systems; state-space collapse; and the respective roles played by all of the above in establishingvalidity of heavy-traffic steady-state approximations.

Rental duration distributions and stock-levels: Why deadlines may be bad for rentalfirms

A. Bassamboo, Northwestern University, USA, a-bassamboo@northwestern.eduS. Kumar, Stanford University, USA, skumar@stanford.eduR.S. Randhawa, The University of Texas at Austin, USA, rsr@mail.utexas.edu

We study a rental system where a fixed number of heterogeneous users rent one product at a time from acollection of re-usable products. The online DVD rental firm Netflix provides the motivation. We assumethat rental durations of each user are i.i.d, non-lattice and have finite mean. We study transient behavior inthis system following the introduction of a new product that is desired by all the users. We represent theusage process for this new product in terms of an empirical distribution. This allows us to characterize theasymptotic behavior of the usage process as the number of users increases without bound, via appropriateversions of Glivenko-Cantelli and DonskerÂ’s theorems. Analyzing the usage process, we demonstrate thatan increase in the variability of the rental duration distribution can actually help the firm by allowing it toset lower capacity levels to provide a desired quality of service. Further, we show that the firm is better offnot imposing any deadlines for the return of the product.

Session MA05 - Scheduling, Chair: Borst / Wierman in CZ 12

New Approximations and Bounds for an M/G/s FIFO queueing system

Varun Gupta, Carnegie Mellon University, varun@cs.cmu.eduJim Dai, Georgia Institute of Technology, dai@gatech.eduMor Harchol-Balter, Carnegie Mellon University, harchol@cs.cmu.eduBert Zwart, Georgia Institute of Technology, bertzwart@gatech.edu

The M/G/s queueing system is the oldest multiserver system model, and has been the topic of performancepapers for almost half a century. However, even now, only coarse approximations exist for its mean delay.

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All the closed-form (non-numerical) approximations in the literature are based on the first two moments ofthe job size distribution. In this work we develop new techniques to obtain bounds and approximations forthe mean delay of an M/G/s queueing system.

Optimal scheduling discipline in an M/G/1 queue with Pareto-type service timedistributions

S. Aalto, Helsinki University of Technology, Finland, samuli.aalto@tkk.fiU. Ayesta, LAAS-CNRS, France, urtzi@laas.fr

We consider the mean delay optimization in an M/G/1 queue for some Pareto type service time distri-butions. If the Pareto distribution is defined in such a way that all positive values are allowed, then thedistribution has a decreasing hazard rate, which implies that the FB discipline is optimal. However, if it isdefined starting from a strictly positive value, the hazard rate is no longer monotone. We show that, not onlyfor this but for a whole class of service time distributions, the optimal discipline is a combination of FCFSand Foreground-Background (FB) disciplines, which gives full priority to the jobs with attained serviceless than some fixed threshold θ∗. These priority jobs are served in the FCFS manner. If there are no jobswith attained service less than θ∗, the full priority is given to the job with least amount of attained service.In our proof, we utilize the index approach developed by Gittins and Klimov. In addition to the ordinaryPareto type distributions with an infinite support, we also discuss the structure of the optimal schedulingdiscipline when dealing with bounded Pareto distribution with a finite support.

Revisiting the performance of large jobs in the M/GI/1

Adam Wierman, Carnegie Mellon University, USA, acw@cs.cmu.edu

In recent years, the response times (sojourn times) experienced by large job sizes have been the focus ofa large number of papers. Though results about many scheduling disciplines have appeared, to this point,results characterizing the response time of large job sizes have been limited to either mean value analysisor law of large numbers scalings. We will present a new framework that unifies these results and providesnew results characterizing the distributional behavior of large job sizes. Also, we will illustrate the impactthese new results have (i) for the analysis of the response time tail, (ii) for the analysis of busy periods, and(iii) for predicting response times of customers upon arrival.

Scheduling in polling systems

E.M.M. Winands, Eindhoven University of Technology, The Netherlands, e.m.m.winands@tue.nlA.C. Wierman, Carnegie Mellon University, USA, acw@cs.cmu.eduO.J. Boxma, Eindhoven University of Technology, The Netherlands, o.j.boxma@tue.nl

We present a simple mean value analysis (MVA) framework for analyzing the effect of scheduling withinqueues in classical asymmetric polling systems with gated or exhaustive service. Scheduling in pollingsystems finds many applications in computer and communication systems. Our framework leads not onlyto unification but also to extension of the literature studying scheduling in polling systems. It illustratesthat a large class of scheduling policies behaves similarly in the exhaustive polling model and the standardM/GI/1 model, whereas scheduling policies in the gated polling model behave very differently than in anM/GI/1.

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Session MA06 - Call Centers 1, Chair: Koole in CZ 13

Density Forecasting of Intraday Call Center Arrivals using Models Based on Expo-nential Smoothing

J.W. Taylor, Said Business School, University of Oxford, UK, james.taylor@sbs.ox.ac.ukR.D. Snyder, Monash University, Melbourne, Australia, ralph.snyder@buseco.monash.edu.au

Time series of intraday call center arrivals tend to consist of both intraweek and intraday seasonal cycles.In our previous work, we developed exponential smoothing formulations to accommodate the two seasonalcycles, with the emphasis being on point forecasting. In this paper, we adapt and extend these modelsfor the univariate density forecasting of count data. The result is a number of alternative single source oferror state space models with Poisson arrivals. We also consider single source of error state space quantilemodels, which have the appeal of avoiding distributional assumptions. Our empirical analysis evaluatespoint forecasting and density prediction up to two days ahead using call center arrivals for NHS Direct, the24 hour telephone helpline provided by the National Health Service in England and Wales.

Short term dynamic prediction in Call Centers

G. Jongbloed, Delft University of Technology, The Netherlands, g.jongbloed@tudelft.nlJ.L. Eveleens, VU University, The Netherlands, laurenz@few.vu.nl

In call centers it is important to have reliable predictions for the number of incoming calls in future periods.For human resource management, a long term prediction of the work load is of interest. Within a shorterperiod, say a day, predictions are important when other tasks can be assigned to agents within the center.

In this talk we consider prediction methods for the remaining number of incoming calls within the currentday, given the observed numbers up till some time point during the current day. Also historical data on theprocess of incoming calls are available.

First a stochastic model in the spirit of Whitt (1999) and Avramidis et al. (2003) is constructed and param-eters of interest are identified. An important parameter in the model is the distribution function G of thetotal number of calls for the present day. Avramidis et al. (2003) model this distribution using a Gammamixture of Poisson variables in the spirit of Jongbloed and Koole (2001).

We propose a more general and flexible model for the distribution of the number of daily incoming calls,a nonparametric mixture of Poisson distributions. This gain in flexibility is particularly useful in situationswhere there are unmeasured covariates that in fact define sub-populations of days.

ReferencesAvramidis, A.N., A. Deslauriers and L’Ecuyer. 2003. Modeling daily arrivals to a telephone call center.Management Science 50 896–908.Jongbloed, G. and G.M. Koole. 2001. Managing uncertainty in call centers using poisson mixtures. Ap-plied Stochastic Models in Business and Industry 17 307–318.Whitt, W. 1999. Dynamic staffing in a telephone call center aiming to immediately answer all calls. Oper-ation Research Letters 24 205–212.

Forecast errors in service systems

Samuel G. Steckley, The Mitre Corporation, U.S.A., sgsteckley@gmail.comShane G. Henderson, Cornell University, U.S.A., sgh9@cornell.eduVijay Mehrotra, San Francisco State University, U.S.A., vjm@sfsu.edu

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MA07

We investigate the presence and impact of forecast errors in the arrival rate of customers to a service system.Analysis of a large data set shows that forecast errors can be large relative to the fluctuations naturallyexpected in a Poisson process. We show that ignoring forecast errors typically leads to over-estimates ofperformance, and that forecast errors of the magnitude seen in our data set can have a practically significantimpact on predictions of long-run performance. We also define short-run performance as the randompercentage of calls received in a particular period that are answered in a timely fashion. We prove a centrallimit theorem that yields a normal-mixture approximation for its distribution for Markovian queues, andsketch an argument that shows that a normal-mixture approximation should be valid in great generality.Our results provide motivation for studying staffing strategies that are more flexible than the fixed-levelstaffing rules traditionally studied in the literature.

Shift Scheduling in Multi-skill Call Centers

Ger Koole, VU University Amsterdam, The Netherlands, koole@few.vu.nl

We describe simple methods for shift scheduling in call centers. Existing methods for single-skill callcenters are discussed and a new method for shift scheduling in multi-skill call centers is introduced. Themethod is easy to implement and has short computation times. A number of possible extensions to themodels is discussed. For example, we explain how to include other communication channels besidestelephone channels.

Joint work with Sandjai Bhulai and Auke Pot.

Session MA07 - MDP 1, Chair: Feinberg in CZ 14

The entangled scheduling and routing problem

C.F. Bispo, Instituto de Sistemas e Robótica, Instituto Superior Técnico, Lisbon, Portugal,cfb@isr.ist.utl.pt

The majority of the basic building blocks constructed in the context of queuing theory is such that thecentralized optimal decision policy, as computed by means of formulating a Markov Decision Problem(MDP), is non-idling when costs incurred are proportional to queue length. The consequences of this facthave been significant and profound in the research conducted for decades in this area.

A basic queuing network will be presented which poses some interesting challenges. It is a combinationof two classic decision problems in networks: the scheduling problem and the routing problem. For anetwork with three servers processing two customer classes one of the servers has to make schedulingdecisions which have a routing consequence outside of its control. Unlike many basic toy problems, thedistinguishing feature of the problem is the fact that, for some states, the optimal policy of this centralserver involves keeping it idle in the presence of customers. The problem will be formulated as an MDPand its optimal policy will be presented, as it is numerically produced for some instances of the problem.The consequences and challenges posed by this new problem, as well as its relevance in general queuingnetworks, will be discussed.

The power of deterministic open-loop control policies

D.A. van der Laan, Vrije University Amsterdam, The Netherlands, dalaan@feweb.vu.nl

We investigate the control of a call center handling two types of incoming calls. A fixed number of Cparallel homogeneous agents are available to respond incoming calls of both types. This can be modelledas controlling a discrete event dynamic system. We assume that type 1 (type 2) calls arrive at the system

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according to a Poisson process of rate λ1 (λ2) respectively. The time it takes for any agent to answer a type1 (type 2) call is generally distributed with means 1

µ1( 1µ2

) respectively. We assume there is an infinite FIFOqueue for waiting type 1 calls and there is a second infinite FIFO queue for waiting type 2 calls. Possibledecision events occur both at moments of arrival and service completions. After a service completion thecontrol policy should give a rule to decide whether the free agent should serve the (longest waiting) type 1or type 2 call. The same applies for moments of arrival if there is at least one idle agent at that time. If atany decision event the free agent is assigned to a type of calls for which no one is waiting then the agentwill be idling. Let E(W1) and E(W2) be the expected average waiting time for type 1 and type 2 callsrespectively. Then the performance measure we consider will have the form A1 E(W1) + A2 E(W2) forsome positive constants (weights) A1, A2. Typically reducing waiting times of type 1 jobs has much morepriority than reducing waiting times of type 2 jobs and thus we put A1 = 10, A2 = 1 or something like that.

In case of exponentially distributed service times controlling the system may be modelled as a MarkovDecision Problem (MDP) with a state space consisting of four integer components (x1, x2, x3, x4), wherex1 (x2) is the number of waiting type 1 (type 2) jobs respectively and x3 (x4) is the number of agentsserving type 1 (type 2) jobs. However, because of the curse of dimensionality solving this MDP by dynamicprogramming is awkward especially in case of a larger number of agents C . Moreover, this description isonly possible if exponentially distributed service times are assumed while in real-life systems service timesare more generally distributed. Finally the corresponding decision rules obtained by dynamic programmingare state dependent and therefore hard to implement in a real-life call center. These are all arguments tolook for a more robust (it should not be for only a particular case) approach and for that we consider controlpolicies which are not state dependent. Such policies are called open-loop control policies and the well-known Bernoulli policies fall in this class. For systems as described above a Bernoulli policy assigns atany decision event the free agent to a type 1 call with probability θ and to a type 2 call with the remainingprobability 1 − θ . Bernoulli policies are randomized policies and it is possible to optimize over θ by forexample simulation and a binary search. However, instead of the Bernoulli (randomized) policies it isalso possible to apply deterministic open-loop policies. The idea remains to optimize over θ , the long-runproportion of time an agent is assigned to a type 1 job, just as for Bernoulli policies. However, instead ofthrowing a "θ -coin" as Bernoulli policies do, a so-called balanced sequence of zeros and ones of rate θ isused which determines all assignments. Theoretical arguments for the superiority in performance of sucha deterministic open-loop policy of rate θ compared to the corresponding Bernoulli policy of rate θ will begiven. Moreover, we will confirm this by results from simulations in which we optimize over θ by lookingat the corresponding simulated performances for both deterministic and Bernoulli control policies.

A MDP approach to the control of traffic lights

René Haijema, University of Amsterdam, The Netherlands, r.haijema@uva.nlJan van der Wal, Eindhoven University of Technology, The Netherlands.

Optimal control of traffic lights, such that the overall average waiting time per car is minimized, appears tobe quite complicated. For large (networks of) intersections, the MDP becomes intractable due to the curseof dimensionality in the state space.

We suggest a one-step policy improvement approach based on the relative values of a (near-) optimal fixedcycle. Under the fixed cycle the value of a state at an intersection is decomposed into values for eachqueue. Key idea of our so-called relative value based rules is to interrupt the fixed cycle, by shortening orlengthening green periods and when all signals show red to prescribe which queue to serve next.

We consider several relative value based rules and compare them to the underlying fixed cycle, and actuatedcontrol policies and for tractable cases also to the optimal (MDP) policy. The rules can be extended to dealwith arrival information, and extensions to networks of intersections and appear to perform very well withwaiting time reductions in the order of 20% or even more.

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MA08

Splitting of randomized stationary policies in absorbing Markov Decision Processes

E.V. Denardo, Yale University, USA, eric.denardo@yale.eduE.A. Feinberg, Stony Brook University, USA, eugene.feinberg@sunysb.eduU.G. Rothblum, Technion, Israel, rothblum@ie.technion.ac.il

To split a randomized stationary policy is to construct a mixture of nonrandomized stationary policies withthe same occupation measure as the original policy. We discuss absorbing Markov Decision Processes(MDPs) which are generalizations of discounted MDPs. For absorbing MDPs with Borel state and actionsets, we present two results on splitting of a randomized stationary policy into a mixture of nonrandomizedstationary policies. The first result is an explicit formula for the splitting at one state. The second resultis a splitting of a K -randomized stationary policy into a mixture of (K + 1) stationary policies {φ1, i =0, 1, . . . , K } such that the policies φi and φi+1 differ at most at one state, i = 0, 1, . . . , K − 1. Theseresults are useful for studying of MDPs with multiple objectives and constraints. We discuss applicationsof these results to countable discounted MDPs and to constrained bandit problems.

Session MA08 - Retrial Queues, Chair: Artalejo in CZ 15

Cellular mobile radio telephone networks with priority active customers

F. Machihara, Tokyo Denki University, Japan, fumi@j.dendai.ac.jp

In cellular mobile radio telephone networks, hand-off failures produce retrials. In particular, networks be-ing congested, the influence upon the network performance becomes serious. We may give some prioritiesto mobile active customers over mobile inactive cutomers in order to protect against the deterioration. Thefirst method is a circuit reservation system. For mobile active inflow customers into the tagged cell, a fewcircuits are reserved. The second method is a soft-hand-off system. Even if a mobile active customer whois getting into the tagged cell meets all circuits being busy, his service is never interrupted. He can continueto hold his circuit in the pre-existing cell, until a circuit becomes free in the tagged cell. We quantify theeffect of these two methods.

A discrete-time queueing model with abandonments

Rein Nobel, Department of Econometrics, Vrije Universiteit, Amsterdam

This paper presents a one-server queueing model with abandonments in discrete time. In every time slot agenerally distributed number (batch) of customers arrives. The different numbers of arrivals in consecutiveslots are mutually independent. Each customer requires a generally distributed service time, counted inslots. Customers arriving in a slot can start their service only at the beginning of the next slot [i.e. delayedaccess]. When upon arrival customers find the server busy, they join a queue and wait for their service.When upon arrival customers find the server idle, then one of the incoming customers (randomly chosen)starts its service at the beginning of the next slot, whereas the other incoming customers, if any, join thequeue. The customers in the queue are served in the order of arrival. Every time slot each customer waitingin the queue decides to abandon the system for ever with a fixed (abandon) probability, i.e. independentlyfrom the other customers waiting in the queue, his position in the queue or the elapsed waiting time.Arrivals have precedence over departures [i.e. late arrivals]. The steady-state behaviour of this system isstudied.

M/M/s Retrial Queue with Balking and Reneging

Yang Woo Shin, Department of Statistics, Changwon National University, Changwon, Gyeongnam641-773, KOREA, ywshin@changwon.ac.kr

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11 – Abstracts

Taek Sik Choo, Department of Statistics, Changwon National University, Changwon, Gyeongnam641-773, KOREA

We consider an M/M/s retrial queue with balking and reneging as a call center model with impatientcustomers. The customer who has balked at entering the system or reneged on receiving service in serviceline may join the virtual pool of customers, called orbit and repeats its request after random amount oftime. The behaviors of retrial customers from orbit are the similar to those of primary customers that arrivefrom outside.

The system is described by a level-dependent-quasi-birth-and-death (LDQBD) process whose block matrixcomponents are of infinite size. We derive an algorithmic solution for the number of customers in thesystem. The algorithm is based on the generalized truncation method in principle. However, due to theinfinite size of block components, the truncation procedure is applied in two steps and the algorithm consistsof computing the stationary distributions of two-node network and retrial queue with finite capacity inservice facility.

Algorithmic analysis of the Geo/Geo/c retrial queue

J.R. Artalejo, Complutense University of Madrid, Spain, jesus_artalejo@mat.ucm.es

In this paper, we consider a discrete-time queue of Geo/Geo/c type with geometric repeated attempts. It isknown that its continuous counterpart, namely the M/M/c queue with exponential retrials, is analyticallyintractable due to the spatial heterogeneity of the underlying Markov chain, caused from the retrial feature.In discrete-time, the ocurrence of multiple events at each slot increases the complexity of the model andraises further computational difficulties. We propose several algorithmic procedures for the effcient com-putation of the main performance measures of the system. More specifically, we investigate the stationarydistribution of the system state, the busy period and the waiting time.

Session MA09 - Inventory 1, Chair: Van Houtum in CZ 16

Coordinated replenishment in a two echelon inventory system with transportationcapacity

N.C. Buyukkaramikli, Eindhoven University of Technology, The Netherlands,n.c.buyukkaramikli@tue.nlU. Gurler, Bilkent University, Turkey, ulku@bilkent.edu.trO. Alp, Bilkent University, Turkey, osmanalp@bilkent.edu.tr

In this study, we examine a coordinated replenishment problem in the presence of transportation capacity.The aim is to minimize the expected cost rate function with respect to several system parameters. We firststudy a single-echelon setting with multiple retailers under a quantity based joint replenishment policy. Alimited fleet of capacitated trucks is assumed for the transportation of the orders from the ample supplier,which is modeled as a queueing system. Operating characteristics are derived and the objective function isstudied using numerical methods. The model is then extended to a two-echelon inventory system, wherethe warehouse employs an (S − 1, S) inventory policy. The departure process of the warehouse inventorysystem is characterized, which constitutes the arrival process of the transportation unit that delivers thewarehouse orders to the retailers. This process is then approximated by an Erlang Process. Several nu-merical studies are conducted to assess the sensitivity of the expected total cost rate to the system and costparameters as well as the performance of the approximation.

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MA09

A multi-item inventory system with multiple resource constraints

Minmin Qiu,IBM China Research Laboratory, P.R.China, qiumm@cn.ibm.comHongwei Ding,IBM China Research Laboratory, P.R.China, dinghw@cn.ibm.comJin Dong,IBM China Research Laboratory, P.R.China, dongjin@cn.ibm.comChangrui Ren,IBM China Research Laboratory, P.R.China, rencr@cn.ibm.comWei Wang,IBM China Research Laboratory, P.R.China, wangwcrl@cn.ibm.com

Real life inventory systems are always capacitated by multiple resources such as storage areas and capitalsinvested. We model such an inventory system with multiple limited resources under (r ,Q) inventory policy(inventory is reviewed continuously) or (s,S) policy (inventory is reviewed periodically). The demandfor each item from customers is stochastic and stationary. All items in the inventory system are at unitform. Outside suppliers with infinite capacity replenish inventories to the system with constant lead timefor each item. The optimization problem is to find the best inventory policies for each item, satisfying allresource constraints. We first solve a relaxed optimization problem to obtain the initial inventory policiesfor all items, regardless of resource constraints. For each solution, eight neighbor solutions are defined.Then a heuristic algorithm is applied to the original optimization problem to improve the initial solution bysearching for the best neighbor solutions continuously. Numerical examples show that in most cases thealgorithm generates high quality solutions. A case study with actual data from a large distribution centershows the cost can be reduced by 8.1% compared with current inventory policies.

Coordination in divergent multi-echelon inventory systems

J. Timmer, University of Twente, The Netherlands, j.b.timmer@utwente.nl

This presentation is concerned with the coordination of inventory control in divergent multi-echelon inven-tory systems under decentralized control. All installations in these supply chains track echelon inventories.Under decentralized control the installations will decide upon base stock levels that minimize their owninventory costs. In general these levels do not coincide with the optimal base stock levels in the globaloptimum of the chain under centralized control. Hence, the total cost under decentralized control is largerthan under centralized control. To remove this cost inefficiency, a coordination mechanism is presented.This mechanism results in the global optimum of the chain being the unique Nash equilibrium of the corre-sponding strategic game. Furthermore, all installations agree upon the use of this mechanism because theyresult in lower costs per installation.

A product-form solution for two-echelon spare parts networks with emergency ship-ments

G.J. van Houtum, Technische Universiteit Eindhoven, The Netherlands, g.j.v.houtum@tue.nlR.J. Boucherie, University of Twente, The Netherlands, r.j.boucherie@utwente.nlS.A. Heideveld, University of Twente, The Netherlands, s.a.heideveld@utwente.nl

We consider an inventory model for repairable parts for capital goods with high down time costs. We haveone central warehouse, multiple local warehouses, and one central repair facility. When a part fails at acustomer, a ready-for-use part is immediately sent from the closest local warehouse if possible. Simulta-neously, the failed part is sent into repair. If the local warehouse is out of stock, then, via an emergencyshipment, a ready-for-use part is sent from the central warehouse if a part is available there, and otherwisefrom the central repair facility. We assume Poisson demand processes, generally distributed leadtimes forreplenishments, repairs, and emergency shipments, and a basestock policy for the inventory control. Viaa transformation into a network of Erlang loss queues with hierarchical jump-over blocking, we show thatthe steady-state behavior of our inventory system has a product-form solution. This main result has severalimportant implications.

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Monday 2:00pm - 3:30pm

Session MB01 - Financial Engineering 2, Chair: Schoutens / Valdiviesoin CZ 4

Optimal Portfolio Choice with Annuitization

Ralph S.J. Koijen,Finance Group, CentER, Tilburg University and Netspar, Tilburg, the Netherlands,R.S.J.Koijen@TilburgUniversity.nlTheo E. Nijman, Finance and Econometrics Group, CentER, Tilburg University and Netspar, Tilburg, theNetherlands, Nyman@TilburgUniversity.nlBas J.M. Werker, Finance and Econometrics Group, CentER, Tilburg University and Netspar, Tilburg,the Netherlands, Werker@TilburgUniversity.nl

We study the optimal life-cycle consumption and portfolio choice problem taking into account annuity riskat retirement. Optimally, the investor allocates retirement wealth to nominal, inflation-linked, and variableannuities and conditions this choice on the state of the economy. However, we also consider often-usedsub-optimal annuitization strategies. We determine subsequently how the investor optimally anticipatesannuitization before retirement. We find that i) information on term structure variables and risk premia sig-nificantly improves the optimal annuity choice, ii) annuity market incompleteness is economically costly,and iii) adjustments in the optimal investment strategy before retirement induced by the annuity demandare significant.

Measuring comonotonicity in random vectors

I. Koch, University of Antwerp, Belgium, inge.koch@ua.ac.be

In the last decades we observe an increasing interest from researchers and practitioners in the dependencestructure of a random vector. Many research is already performed in this field e.g. through the theory ofcomonotonicity. In fact the comonotonic dependence structure in particular has proved to be a very usefultool when approximating an unknown but strongly positively dependence structure. As a consequence ofthis evolution, there is a need for a measure reflecting how close a given dependence structure approachesthe comonotonic one.In this talk we design the comonotonicty coefficient. This coefficient can be calculated for multivariatevectors and examines to what extent the considered vector is comonotonic. This measure takes values inthe range [0, 1]. As we want to quantify the degree of comonotonicity, this measure is defined in such away that it equals 1 in case the random vector is comonotonic and equals 0 in case it is independent. Weshow how such a measure can be designed analytically, by making use of copulas for the modeling of thedependence structure and we discuss the main properties of this comonotonicity coefficient.

A tractable multi-firm default model with jumps

Matthias Scherer, HVB-Institute for Mathematical Finance, TU Munich, scherer@ma.tum.de

We present a multi-firm structural-default model which allows the pricing of bonds, CDS, and CDOs in aconsistent framework. Each firm-value process is modeled as the exponential of a jump-diffusion process,as proposed by Zhou (2001). Default is triggered by the first passage of the firm-value process belowsome default threshold. Dependence among the companies is introduced by means of a common marketfactor and by supporting common jumps. Jumps in the firm-value process of some company are triggeredby a thinned-out Poisson processes. This construction preserves the distribution of each individual firm-value process, which is a crucial property for a calibration of the model. On a firm-individual basis, one

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MB02

feature of our model are positive credit spreads for bonds and CDS with short maturities. This corrects amajor shortfall of pure-diffusion models. Moreover, it turns out that the term-structure of credit spreads, asimplied by the model, is extremely flexible and allows an excellent fit to observed prices. On a portfoliobasis, allowing simultaneous jumps provides an additional parameter to adjust the dependence among thecompanies. However, the distribution of first-passage times is not known for arbitrary jump-diffusionprocesses. To evaluate bond and CDS prices in our model, we present an efficient Monte Carlo simulationand a numerical algorithm based on the Laplace transform of first-passage times. Such transforms areavailable under the assumption of two-sided exponentially distributed jumps, due to Kou and Wang (2003).Model prices of CDOs and nth-to default contracts can then be found by Monte Carlo simulations. Finally,we present a calibration of the model in two steps. At first, each individual firm-value process is calibratedto CDS quotes. Then, the parameters of the dependence structure are calibrated using CDO quotes.

Parameter estimation in fractionally integrated process of Ornstein-Uhlenbeck type

Luis Valdivieso, Katholieke Universiteit Leuven, Belgium, lvaldiv@pucp.edu.pe

An estimation methodology to deal with fractionally integrated processes of Ornstein-Uhlenbeck type isproposed. The methodology is based on the continuous Whittle contrast. Simulations results are presentedfor a symmetric CGMY background driving Lévy process. An application to stochastic volatility modellingis also discussed.

Session MB02 - Percolation and Related Topics, Chair: Van den Bergin CZ 5

Exceptional times for the dynamical discrete web, part I + II

L.R.G. Fontes, Univ. of São Paulo, Brazil, lrfontes@gmail.comC.M. Newman, New York Univ., U.S.A., newman@courant.nyu.eduE. Schertzer, New York Univ., U.S.A., schertze@hotmail.comK. Ravishankar, SUNY New Paltz, U.S.A., ravishak@newpaltz.edu

Howitt and Warren introduced a system of coalescing one-dimensional simple symmetric random walksthat evolve in an extra continuous dynamical parameter s. The evolution is by independent updating of theBernoulli variables that describe the discrete web at any fixed s. Motivated by results of Häggstrom-Peres-Steif and Schramm-Steif on dynamical percolation, we have studied (ArXiv 0704.2706) exceptional timesfor this dynamical discrete web. One result is that there are exceptional values of s such that the walk fromthe origin Ss(n) is asymptotically bounded above by K

√n with a nontrivial dependence of the Hausdorff

dimension of the set of such s on the parameter K . If time permits, I will discuss the dynamical Brownianweb, a natural scaling limit of the dynamical discrete web.

Sharp phase transitions in divide-and-colour models

R.J. MeesterAndras BalintFederico Camia

We study a dependent percolation model, in which subcritical percolation clusters (say with parameter q)are independently coloured with two colours, black and white. We show that the phase transition is sharp.This implies that on the triangular lattice, the critical value for the colouring process is 1

2 , independently ofq , as long as q is subcritical.

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Multiscale analysis of stochastic reaction networks

Thomas G. Kurtz, University of Wisconsin-Madison, Madison, WI, USA, kurtz@math.wisc.edu

Attempts to model chemical reactions within biological cells have led to renewed interest in stochasticmodels for these systems. These models of cellular chemical reaction networks typically involve chemicalspecies numbers and reaction rates varying over several orders of magnitude. A number of researchershave proposed exploiting the“multiscale” nature of these models to reduce the complexity of the model tobe analyzed or simulated. Using elementary understanding of some basic stochastic processes, systematicapproaches to model reduction will be discussed.

Session MB03 - Simulation 1, Chair: Henderson in CZ 10

Bandits and simulation selection options

S.E. Chick, INSEAD, France, stephen.chick@insead.eduN. Gans, The Wharton School, The United States of America, gans@wharton.upenn.edu

This talk examines new work in the area of selecting the best of a finite set of alternatives, where bestis the maximum mean performance, and the mean is unknown but inferable from stochastic simulationexperiments. Unlike the usual approaches, which provide frequentist or Bayesian statistical properties,or which provide probabilistic convergence results, we link the economic costs (of sampling and timediscounting) of simulation experiments to the expected NPV of system implementation decisions that areimplied by the simulation analysis. For some special cases, we show how to express this problem as astoppable bandit process, prove that a Gittins index policy is optimal, and develop diffusion approximationsfor the optimal allocation indices.

On transience and recurrence in discrete-event simulations

P.J. Haas, IBM Almaden Research Center, USA, peterh@almaden.ibm.comP.W. Glynn, Stanford University, USA, glynn@stanford.edu

The generalized semi-Markov process (GSMP) is the usual model for the underlying stochastic process ofa complex discrete-event simulation. It is important to understand fundamental behavioral properties ofthe GSMP model, such as the conditions under which the states of a GSMP are recurrent. For example,the recurrence or non-recurrence of a specified “single state” determines whether or not the successive exittimes from the state can be used as a sequence of regeneration points. We review some sufficient conditionsfor recurrence in irreducible finite-state GSMPs; these conditions include requirements on the "clocks" thatgovern the occurrence of state transitions. For example, each clock-setting distribution must have finitemean. We then show that, in contrast to ordinary semi-Markov processes, an irreducible finite-state GSMPcan have transient states in the presence of clock-setting distributions with heavy tails.

Analysis Of State-Dependent Importance Sampling Algorithms

Jose Blanchet, blanchet@fas.harvard.edu

Importance Sampling (IS) is a popular variance reduction technique in simulation applications and is par-ticularly useful in rare-event simulation. Nevertheless, as several results and examples developed recentlyindicate, (IS) must be applied with care in the performance analysis via simulation of stochastic systems.In particular, it is often the case that one must resort to State-dependent Importance Sampling (SDIS)to obtain reliable estimators. In this talk, we discuss new techniques that allow both to design efficient

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SDIS estimators and theoretically verify their efficiency. We illustrate the techniques, which are based onLyapunov-type inequalities, with examples drawn from combinatorics (counting graphs), queueing theoryand risk analysis.

Adaptive control variates for steady-state simulation

S. Kim, Purdue University, U.S.A., sujin@purdue.eduShane G. Henderson, Cornell University, U.S.A., sgh9@cornell.edu

We introduce an adaptive control variate scheme for estimating steady-state performance measures forstochastic systems using simulation. We use a procedure based on sample average approximation fortuning the control variate estimators. The key assumption in our analysis is that the underlying process isregenerative.

Session MB04 - Stability of Stochastic Networks, Chair: Ramanan inCZ 11

Steady-state analysis of a multi-server queueing system in QED regime

D. Gamarnik, MIT, USA, gamarnik@mit.eduP. Momcilovic, University of Michigan Ann Arbor, USA, petar@eecs.umich.edu

We consider a multi-server GI/GI/N queueing system in Halfin-Whitt (QED) regime. The analysis of thissystem for the case of non-exponentially distributed service times presents a significant challenge. Weconsider the case of discrete service time distributions and show that in the large N limit the queue lengthprocess can be approximated by a simple discrete time continuous state Markov chain. We then establishan interchange of steady state and large N limits using tightness arguments. We further derive a simpleexpression for the decay rate of the limiting queue length process.

The heavy traffic limit of an unbalanced generalized processor sharing model

Kavita Ramanan, Carnegie Mellon University, Pittsburgh, PA, kramanan@math.cmu.eduMartin I. Reiman, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, marty@alcatel-lucent.com

We consider a server that processes J classes using the generalized processor sharing discipline. Extendinga previous paper where we assumed that the work arrival rate and nominal service rate for each class areequal, we consider the situation where the total arrival rate and service rate are equal, but the rates forindividual classes may not be. This gives rise to state space collapse. The dimension of the limit process isobtained by determining the invariant manifold of the fluid limit associated with this model. We show thatwhen the heavy traffic condition holds, the functional central limit of the scaled unfinished work process isa reflected diffusion process that lies in the invariant manifold. The reflected diffusion limit is characterizedby the extended Skorokhod map and may fail to be a semimartingale.

Fluid model for a data network with alpha fair bandwidth sharing and general doc-ument size distributions: two examples of stability

H.C. Gromoll, University of Virginia, USA, gromoll@virginia.eduN.H. Lee, University of California at San Diego, USA, nhlee@math.ucsd.eduR.J. Williams, University of California at San Diego, USA, williams@math.ucsd.edu

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We consider a fluid model for a data network with alpha fair bandwidth sharing and general documentsize distributions. This fluid model uses a measure valued state descriptor. The invariant states for thefluid model are characterized. Two examples are given of network topologies for which stability of thefluid model can be established under a nominal condition. The two types of networks are linear networksand tree networks. The latter is particularly interesting since for the original data network model, thedistribution of the number of documents process in steady state is expected to be sensitive to the (non-exponential) document size distribution. Related work on using the fluid model to study stability of theoriginal data network is also described.

Stability of processor sharing networks with simultaneous resource requirements

J. Hansen, Heriot-Watt University, Edinburgh, UK, j.hansen@hw.ac.ukC. Reynolds, Heriot-Watt University, Edinburgh, UK, c.j.reynolds@ma.hw.ac.ukS. Zachary, Heriot-Watt University, Edinburgh, UK, s.zachary@hw.ac.uk

We study the phenomenon of entrainment in processor sharing networks, whereby, while individual net-work resources have sufficient capacity to meet demand, the requirement for simultaneous availability ofresources means that a network may nevertheless be unstable. We show that instability occurs through poorcontrol, and that, for a variety of network topologies, only small modifications to controls are required inorder to ensure stability. For controls which possess a natural monotonicity property, we give some newresults for the classification of the corresponding Markov processes, which lead to conditions both forstability and for instability.

Session MB05 - Communication Systems 1, Chair: Choi in CZ 12

Performance Analysis of IEEE 802.11e EDCA in Non-saturation Condition by M/G/1Queue

Y.H. Bae, Department of Mathematics and Telecommunication Mathematics Research Center, KoreaUniversity, Seoul, Korea, unani96@korea.ac.krK.J. Kim, Department of Mathematics and Telecommunication Mathematics Research Center, KoreaUniversity, Seoul, Korea, kimkjae@korea.ac.krB.D. Choi, Department of Mathematics and Telecommunication Mathematics Research Center, KoreaUniversity, Seoul, Korea, queue@korea.ac.kr

We study performance analysis of IEEE 802.11e EDCA in non-saturation condition where 4 traffic classesare served. There have not been many analytic works in non-saturation mode due to analytic complexityof models. However, non-saturation mode is valid in most real WLAN networks. We deal with the non-saturation case that packets arrive according to Poisson process and queue up to a buffer at a station. Wederive probability generating function of HoL-delay Si of traffic class i . We model a station with trafficclass i as M/G/1 queueing system where the service time is the HoL-delay Si , and then obtain the severalperformance measures such as total access delay and throughput by applying the theory of M/G/1 queue.

Performance Analysis of IEEE 802.15.4 Unslotted CSMA/CA

Tae Ok Kim, Department of Mathematics and Telecommunication Mathematics Research Center, KoreaUniversity, Seoul, Korea, violetgl@korea.ac.krJin Soo Park, USN Service Division, KT, Seoul,Korea, vtjinsoo@paran.comKyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, kimkjae@korea.ac.kr

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Bong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, queue@korea.ac.kr

The standard of IEEE 802.15.4 MAC layer adopts beacon mode and non-beacon mode. Non-beacon modeis operated by unslotted CSMA/CA algorithm. This paper is the first work on performance of IEEE802.15.4 non-beacon mode. We develop a new method using the theory of regenerative process. Weobtain performance measures such as throughput, packet delay, packet loss probability and energy con-sumption. We show that numerical result is quite close to simulation result. Our performance measures areused for determining the optimal number of devices while supporting the required QoS constraints and forcalculating the life time of battery.

Performance analysis of IEEE 802.15.4 Non-beacon Mode by Piggyback Method

Jin Soo Park, USN Service Division, KT, Seoul,Korea, vtjinsoo@paran.comTae Ok Kim, Department of Mathematics and Telecommunication Mathematics Research Center, KoreaUniversity, Seoul, Korea, violetgl@korea.ac.krKyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, kimkjae@korea.ac.krBong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, queue@korea.ac.kr

We analyze the MAC performance of the IEEE 802.15.4 LR-WPAN non-beacon mode with piggybackdownload method in non-saturated condition. Our approach is to model stochastic behavior of one deviceas a discrete time Markov chain. First, we propose an analytical model describing the download behaviorof a device using piggyback method. Then, by combining the model of a device with upload traffic in[3] and one with download traffic in this paper, we obtain the performance measures such as throughput,packet delay, energy consumption and packet loss probability of a device with both upload and downloadtraffic. Numerical results and simulation results show that by removing a backoff procedure in the backoffmethod, the piggyback method can reduce the delay, loss probability and energy consumption comparedwith backoff method. Our results can be used to find optimal number of devices with some constraints onthese measures.

Performance Analysis of Session-Level Load Balancing Algorithms

Dennis Roubos, Vrije Universiteit Amsterdam, Faculty of Sciences, droubos@few.vu.nlSandjai Bhulai, Vrije Universiteit Amsterdam, Faculty of Sciences, sbhulai@few.vu.nlRob van der Mei, Vrije Universiteit Amsterdam, Faculty of Sciences, mei@few.vu.nl

Load balancing (LB) is crucial for the efficient operation of big server clusters. In the past, many differentLB strategies on the request level have been developed with great effectiveness in parallel applications.However, the LB problem is not yet solved completely; new applications and architectures require newfeatures. In particular, secure environments require that LB is done at session level instead of requestlevel; that is, once a session has been assigned to a server, all subsequent service requests are directed tothe assigned server. Despite the fact that many commercial products have been brought to the market toimplement LB at the session level, little insight has been obtained into the efficiency of such session-levelLB algorithms, leaving ample room for performance improvement and optimization.

Motivated by this, we study session-level LB with a focus on algorithms that are simple and easy to im-plement in real systems. The performance of the load balancer is highly dependent on the request profilesof the different sessions and the information that is available for decision making. We make this trade-offbetween the information that is available to the load balancer and the efficiency of the algorithm explicitby developing new algorithms, and compare their efficiency with existing algorithms. The algorithms aremainly based on the load of each server and the number of active sessions running on them. Extensive

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validation in an experimental setting shows that our algorithms outperform the existing ones, and as such,provide a simple, easy-to-implement yet effective means to improve the efficiency of large server clusters.

Session MB06 - Many-server Models 1, Chair: Armony in CZ 13

Optimal Control of Parallel Server Systems with Many Servers

Jim Dai, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute ofTechnology, Atlanta, Georgia 30332-0205, dai@gatech.eduTolga Tezcan, Industrial and Enterprise Systems Engineering, University of Illinois atUrbana-Champaign, Urbana, Illinois 61801, ttezcan@uiuc.edu

We consider parallel server systems that consist of several customer classes and server pools with expo-nential service times. We propose a control policy that dynamically routes customers. The policy usesminimal state information; in particular, it does not require arrival rate information. In the Hafin-Whittmany-server heavy traffic regime, we show that this policy is asymptotically optimal to minimize the totallinear holding and reneging costs. A key to the optimality proof is a state space collapse (SSC) result. Weshow how a framework of Bramson (1998) can be adapted, from conventional heavy traffic to many-serverheavy traffic, to prove the SSC result.

Fill rate constraint compliance and the many-machine repairman problem

O. Jennings, otisj@duke.eduIsilay Talay Degrimenci

The classic repairman problem with spares consists of a fixed number of failure-prone machines, clientswho use them, and repairmen responsible for fixing the broken machines. We assume that when theircurrent machine breaks, clients turn in the machine and desire an immediate replacement. Using a many-machine / many-repairman asymptotic approach, we consider how one would provision and staff such asystem so as to meet a target machine-replacement fill rate constraint.

Fair Dynamic Routing Policies in Large-Scale Service Systems with HeterogeneousServers

A.R. Ward, University of Southern California, amyward@marshall.usc.eduMor Armony, Stern School of Business, NYU, 44 West 4th Street, NY, NY 10012,marmony@stern.nyu.edu

Modern call centers typically have many agents answering calls with different skill levels. One naturalcontrol issue that arises is how to route customers to agents. It has been shown that routing customers tothe fastest agent first asymptotically minimizes steady-state customer waiting times. Unfortunately, thispolicy penalizes the faster agents by keeping them busy at all times, and only allowing the slower agentsto idle. We address the question of how to optimize system performance subject to fairness with respect toserver utilization.

Large cross selling in call centers

Itay Gurvich, Columbia Business School, 4I Uris Hall, 3022 Broadway, NY, NY 10027,ig2126@columbia.edu

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Mor Armony, Stern School of Business, NYU, 44 West 4th Street, NY, NY 10012,marmony@stern.nyu.eduConstantinos Maglaras, Columbia Business School, 409 Uris Hall, 3022 Broadway, NY, NY 10027,c.maglaras@gsb.columbia.edu

Cross selling is an increasingly prevalent practice in call centers. This is largely because it allows firmsto dynamically segment callers and customize product offerings accordingly. In this work we develop aframework under which both marketing decisions (such as: customer segmentation and pricing) and oper-ational decisions (such as: staffing, call sequencing and dynamic cross selling) may be considered jointly.We propose near-optimal pricing, staffing, sequencing and cross-selling policies under various forms ofcustomer segmentation. Finally, we characterize how the degree and timing of customer segmentationimpacts these policies and, correspondingly, the center’s profitability.

Session MB07 - Games 1, Chair: Economou in CZ 14

Equilibrium customer strategies under various levels of information for the singleserver queue with unreliable server.

A. Economou, University of Athens, Department of Mathematics, Section of Statistics and OperationsResearch, Greece, aeconom@math.uoa.grS. Kanta, University of Athens, Department of Mathematics, Section of Statistics and OperationsResearch, Greece, spkanta@math.uoa.gr

We consider the single server Markovian queue with an unreliable server, that is a server that alternatesbetween active and idle (repair) periods. The arriving customers decide whether to join or balk based on anatural reward-cost structure, which incorporates their desire for service as well as their unwillingness towait.

We examine customers’ behavior under various levels of information regarding the state of the system atarrival instances. More specifically, a customer may know or not know the state of the server and/or thenumber of present customers upon his arrival. We derive equilibrium strategies for the customers under thevarious information levels and we study the effect of the information to the system performance.

On the use of quasi birth-and-death processes in equilibrium selection in game the-ory and queueing networks

Madjid Amir, Sultan Qaboos University, Oman, maamir@squ.edu.om, P.O. Box 36, PC 123,DOMAS/SQU, Al Khod, OMAN

Quasi birth-and-death processes, a natural multi dimensional counterpart of the usual birth-death processes,are known to differ notably from the latter due to the fact that they fail to satisfy the so called detailedbalance equations. These processes are NOT reversible. I will show how these processes have successfullybeen used to investigate the crucial issue of equilibrium selection in Games. Furthermore, I intend to showthe use of quasi birth-and-death processes to investigate both the transient and long-run behavior of certainprocesses pertaining to Queueing systems.

An analysis of oligopolistic competitive markets for queued services with QoS guar-antees

Parijat Dube, IBM T. J. Watson Research Center, Hawthorne, NY, pdube@us.ibm.comRahul Jain, IBM T. J. Watson Research Center, Hawthorne, NY, rahul.jain@us.ibm.comCorinne Touati, INRIA - LIG, Grenoble, France, corinne.touati@inrialpes.fr

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We consider the problem where competing providers offer queued services with QoS guarantees. Usershave quasi-linear utility functions that depends on delay and the price paid. We first analyze the equilibriummarket shares of various providers. We assume that each provider has a service queue (such as an M/M/1FIFO or an M/G/1 processor sharing) with a given capacity. Each provider offers delay guarantees andannounces prices. Its objective is to maximize its revenue. We first observe that a competitive equilibriumdoes not exist in such a market in general. However, we show that the (price, delay) pair offered by aprovider is unique. We then model the interaction between the providers as a Stackelberg game. Ouranalysis seems to indicate non-existence of an equilibrium of this game. In fact, we observe a “price-war" phenomenon also seen in many other contexts. However, we conjecture that the allocations that theproviders offer in the Stackelberg game model, are in the core of the competitive exchange economy. Thisis part of on-going work.

Optimal balking strategies in the single server Markovian queue with compart-mented waiting space.

A. Economou, University of Athens, Department of Mathematics, Section of Statistics and OperationsResearch, Greece, aeconom@math.uoa.grS. Kanta, University of Athens, Department of Mathematics, Section of Statistics and OperationsResearch, Greece, spkanta@math.uoa.gr

We consider the single server Markovian queue and we assume that arriving customers decide whether tojoin or balk based on a natural reward-cost structure, which incorporates their desire for service as well astheir unwillingness to wait. We suppose that the waiting space of the system is partitioned in compartmentsof fixed capacity for a customers. We suppose that customers have partial information about the system.More specifically, the information is either the number of the compartment in which they enter or just theposition within it. We examine customers’ behavior under the various levels of information and we identifyequilibrium threshold strategies, thus extending the fundamental results of Naor and Edelson, Hildebrandwho studied the equilibrium strategies in the observable and unobservable M/M/1 queues. The social andprofit optimization problems are also solved.

Session MB08 - Fluid Queues, Chair: Adan in CZ 15

On the stationary measure of a reflected Brownian motion in a wedge: some explicitresults

A.B. Dieker, University College Cork, Ireland, t.dieker@proba.ucc.ieJ. Moriarty, University College Cork, Ireland, jm20@proba.ucc.ie

Although the stationary distribution of one-dimensional reflected Brownian motion with drift is exponen-tial, this is generally not true in a multidimensional setting. M. Harrison and R. Williams have characterizedthe class of reflected Brownian motions with exponential densities in terms of a so-called skew symmetrycondition on the reflection directions.

However, not much is known on the stationary measure (or its density) when the skew symmetry conditionfails to hold. For special multidimensional reflected Brownian motions, an intriguingly simple formulaarises from connections with reflection groups, which is a topic in algebra. In this formula, the stationarydensity is represented as a finite sum of exponential terms.

This talk reports an attempt to marry this formula with the literature on two-dimensional reflected Brownianmotion in a wedge.

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Transient Fluid Solutions and Queueing Networks with Infinite Virtual Queues

Y. Nazarathy, The University of Haifa, Israel, yonin@stat.haifa.ac.ilG. Weiss, The University of Haifa, Israel, gweiss@stat.haifa.ac.il

Fluid solutions of queueing networks over a finite time horizon often involve non-empty queues and fullutilization of some of the resources for substantial portions of the time horizon. This observation makes itappropriate to treat the queueing network for which the transient fluid solution is specified, as a networkwith infinite virtual queues (i.e. queues that never empty yet need to be served at specified nominal rates).Examples of such queueing networks have been previously analyzed under steady state by Weiss, Kopzonand Adan.

As an example of this interaction between the fluid solution and the queueing network, we present amethod for controlling a multi-class queueing network over a finite time horizon such that the holdingcosts (

∫ T0∑K

k=1 ck Qk(t)dt ) are approximately minimized by tracking an optimal fluid solution. Ourmethod is based on an adaptation of Dai and Lin’s results regarding rate stability of the maximum pressurepolicy for stochastic processing networks with ρ ≤ 1 to the case of a multi-class queueing network withinfinite virtual queues. We present the proof for the asymptotic optimality of our fluid tracking policy.

R, G and U and the time-inhomogeneous periodic Quasi-Birth Death Process

B.H. Margolius, Cleveland State University, Cleveland, OH, USA, b.margolius@csuohio.edu

We solve for the asymptotic periodic distribution and moments of the continuous time quasi-birth-and-death process with time-varying periodic rates in terms of R, G and U matrix functions which are analoguesof the R, G and U matrices of matrix analytic methods. We evaluate these QBDs numerically by matchingthe moments of the distribution of a generalized random walk with the same transition probabilities as theQBD, but with no boundary at level zero, and then use the approximate distribution for the generalizedrandom walk to evaluate the R, G and U matrix functions.

Insurance Risk with Variable Number of Policies

Ivo Adan, Eindhoven University of Technology, The Netherlands, iadan@win.tue.nlVidyadhar Kulkarni, University of North-Carolina, vkulkarn@email.unc.edu

In this talk we consider an insurance company selling life insurance policies. New policies are sold atrandom points in time, and each policy stays active for a random amount of time, during which the poli-cyholder pays premiums continuously at rate r . When the policy expires, the insurance company pays aclaim of random size. The aim is to compute the probability of eventual ruin starting with a given numberof policies and a given level of insurance reserves. We establish a remarkable result that, if the lifetimes ofpolicies are i.i.d. exponential random variables with rate µ, then the ruin probability is identical to the onein the standard compound Poisson model where the reserves increase at constant rate r and claims occuraccording to a Poisson process with rate µ. We conclude that the ruin probability does not depend on theinitial number of active policies, nor on the arrival process of new policies.

Session MB09 - Search Models, Chair: Klafter in CZ 16

Intermittent search processes

Olivier Benichou, CNRS, Paris VI, benichou@lptl.jussieu.fr

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Many physical, chemical or biological problems can be rephrased as search processes, involving a searcherand a target of unknown position. Intermittent search strategies, alternating active search phases and non-reactive displacement phases, are observed in various situations, ranging from the microscopic scale (as forthe search for a specific sequence on DNA) to the macroscopic scale (animals searching for food). Usingrepresentative analytical models, we show that these intermittent strategies are efficient and determinewhich durations of each phase optimize the search time.

Search research

Gleb Oshanin, CNRS, Paris VI, oshanin@lptl.jussieu.fr

We overview several recent results on search kinetics of an immobile target by a concentration of randomlymoving searchers. The searchers perform intermittent random walks in discrete time n = 0, 1, 2, ..., N ,where N is the maximal time the search process is allowed to run. With probability α the searchers stepon a nearest-neighbor, and with probability (1 − α) they leave the lattice and stay off until they land backon the lattice at a fixed distance L away from the departure point. We calculate the probability PN thatthe target remains undetected up to the maximal search time N , and seek to minimize this probability withrespect to α and L . We find that PN is a non-monotonic function of α, and show that there is an optimalchoice αopt(N ) of α well within the intermittent regime, 0 < αopt(N ) < 1, whereby PN can be orders ofmagnitude smaller compared to the "pure" random walk cases α = 0 and α = 1. Moreover, we show that ifwe are at liberty to choose L , we may further decrease PN by orders of magnitude by choosing an optimalvalue of this parameter.

Parallel search of long circular strands: Modeling, analysis, and optimization

Iddo Eliazar, Holon Institute of Technology and Tel Aviv University, Israel, eliazar@post.tau.ac.il

We introduce and explore a model of an ensemble of enzymes searching, in parallel, a circular DNAstrand for a target-site. The agents performing the search combine local scanning - conducted by a 1Dmotion along the strand, and random relocations on the strand - conducted via a confined motion in themedium containing the strand. Both the local scan mechanism and the relocation mechanism are consideredgeneral. The search-durations are analyzed, and their limiting probability distributions - for long strands -are obtained in closed-form. The results obtained: (i) encompass the cases of single, parallel and massively-parallel searches, taking place in the presence of either finite-mean or heavy-tailed relocation times; (ii)are applicable to a wide spectrum of local scan mechanisms including linear, Brownian, selfsimilar (e.g.,fractional Brownian and fractional Levy), and sub-diffusive motions; (iii) provide a quantitative theoreticaljustification for the necessity of the relocation mechanism; and, (iv) facilitate the derivation of optimalrelocation strategies.

Stochasticity and search in gene regulation processes

Ralf Metzler, Physics Dept, University of Ottawa, 150 Louis Pasteur, Ottawa, ON, K1N 6N5, Canada,metz@nordita.dk

The search of DNA-binding proteins for their specific binding site on the DNA molecule requires the scan-ning of megabases of the genetic code. Up until now, the detailed mechanisms are quite poorly understood.Recent developments in single molecule experimental techniques make it possible to study the search pro-cess of these proteins in real time. I discuss on the background of these experiments the stochastic processesinvolved in the search, namely, 3D bulk diffusion, 1D sliding motion along the DNA, and intersegmentaltransfers. The latter, under typical in vitro conditions, involve long-tailed jump lengths leading to a Levyflight. It is discussed how Levy statistics improve the search efficiency in that system. However, in vivo,

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the situation becomes more complicated, as the bulk motion due to molecular crowding by other macro-molecules in the cell has been shown to be subdiffusive. A simple model is discussed to investigate theeffects of subdiffusion to the target search, and it is argued that a weak ergodicity breaking arises that maybe of advantage for the search process.

References:

[1] M. Lomholt, T. Ambjoernsson, and R. Metzler, Phys. Rev. Lett. 95, 260603 (2005).

[2] I. M. Sokolov, R. Metzler, K. Pant, and M. C. Williams, Biophys. J. 89, 895 (2005).

[3] M. Lomholt, I. M. Zaid, and R. Metzler, Phys. Rev. Lett. 98, 200603 (2007).

Monday 4:00pm - 5:30pm

Session MC01 - Exit Times, Chair: Rolski in CZ 4

On an exit problem of a two-dimensional risk process from the positive quadrant

Florin Avram, Pau University, France, Florin.Avram@univ-Pau.frZbigniew Palmowski, University of Wrocław, Poland, zpalma@math.uni.wroc.plMartijn Pistorius, King’s College London, UK, Martijn.Pistorius@kcl.ac.uk

Consider two insurance companies that divide between them both claims and premia in some specifiedproportions. We model the occurrence of claims according to a renewal process. One ruin problem consid-ered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; anotheris that of entering the negative quadrant. When the claims arrive according to a Poisson process we obtaina closed form expression for the ultimate ruin probability. In the general case we analyze the asymptoticsof the ruin probability when the initial reserves of both companies tend to infinity.

Gaussian storage processes: long-time behavior

K. Debicki, University of Wrocław, Poland, debicki@math.uni.wroc.pl

Consider a fluid queue with service rate c > 0 and input process modelled by a centered Gaussian processwith stationary increments {η(t) : t ∈ R}. Let {W (t) : t ≥ 0} := {sups≤t (η(t)− η(s)− c(t − s)) : t ≥ 0}be the stationary buffer content process (storage process) for the system.

We focus on the asymptotic analysis of the tail distribution of

P( supt∈[0,T (u)]

W (t) > u)

as u → ∞. It appears that the presence of short range dependence or long range dependence for η(·)significantly influences the asymptotics.

Additionally, assuming that η(t) = BH (t) is a fractional Brownian motion with Hurst parameter H , weobtain an Erdös-Révész type of law of iterated logarithm for W (t).

De Finetti’s dividend problem for a general Lévy insurance risk process

Z. Palmowski, University of Wrocław, Poland, Zbigniew.Palmowski@math.uni.wroc.plFlorin Avram, Universite de Pau, France, Florin.Avram@univ-Pau.frA. Kyprianou, The University of Bath, UK, a.kyprianou@bath.ac.ukMartijn Pistorius, King’s College London, UK, Martijn.Pistorius@kcl.ac.uk

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In this talk we consider the solution to the classical control problem due to De Finetti (1957) and Gerber(1969) which concerns the optimal payment of dividends from an insurance risk process prior to ruin.The calculations we present go much further than existing literature in that our calculations are valid for ageneral spectrally negative Lévy risk process as opposed to the classical Cramér-Lundberg process. Relatedis the problem where we impose the restriction that ruin be prevented by bail-out loans. Drawing on thefluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of theoptimal strategy in the set of barrier strategies and the corresponding value function, for either of theproblems. Subsequently we investigate when the dividend policy that is optimal amongst all admissibleones takes the form of a barrier strategy. We also calculate distributional properties of the net present valueof dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplacetransform of the red period. The technique we use to derive these quantities appeals principally to excursiontheory rather than integro-differential equations and makes a new link with the distribution of integratedexponential subordinators. Finally, we also consider the problem with fixed cost for taking out dividends.

The exact asymptotics of the time to collision tail distribution

Z. Puchała, Wroclaw University, Poland, puchala@math.uni.wroc.plT. Rolski, Wroclaw University, Poland, rolski@math.uni.wroc.pl

Let W = {x ∈ IRn: x1 < . . . < xn} and X1

t , . . . , Xnt be independent, skip-free, strong Markov processes

such that x = (X10, . . . , Xn

0 ) ∈ W . By skip-free we mean that the exit of X t from W must be through W .We study the collision time τ = inf{t > 0 : X t /∈ W }. In the talk we discuss the following issues:

1. Karlin-McGregor theory for the distribution of τ for processes with the same transition law,

2. The exact asymptotics for IPx(τ > t) for X1t , . . . , Xn

t being independent Poisson processes, contin-uous time random walks or standard Brownian motions with identical transition law,

3. The exact asymptotics for IPx(τ > t) for X1t , . . . , Xn

t being independent Brownian motions withdifferent drifts.

Examples to Young tableaux and Markovian queues will be mentioned.

Session MC02 - Interacting Particle Systems, Chair: Van den Berg inCZ 5

The super-process limit of the critical spread-out contact process above 4 spatialdimensions

Remco van der Hofstad, Department of Mathematics and Computer Science, Eindhoven University ofTechnology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, rhofstad@win.tue.nl

We consider the contact process on Zd× R+, at the critical infection rate λc, for d > 4. The model is a

“spread-out” model having long range parameterized by L ≥ 1. We consider configurations in which thecluster of the origin survives to time t , and scale space by t1/2. We prove that for L sufficiently large allthe moment measures converge, as t →∞, to those of the canonical measure of super-Brownian motion.We use lace expansion methods on the discretized contact process, which is an oriented percolation model.The proofs are adaptations of those for oriented percolation.

This is joint work with Akira Sakai.

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Duality and exact correlations in a model of heat conduction

C. Giardina, Eindhoven University of Technology, The NetherlandsJ. Kurchan, CNRS-ESPCI, ParisF. Redig, Mathematical Institute, Leiden University, The Netherlands

We study a stochastic model of heat conduction, recently introduced in the study of Fourier law. We discussduality and show uniqueness and detailed properties of the non-equilibrium steady state, such as an exactexpression of the two-point correlation function. Some similarities with the simple symmetric exclusionprocess are discussed.

Approximate zero-one laws, spin-flip dynamics and Ising percolation

J. van den Berg, CWI and Vrije Universiteit, Amsterdam

In 1993 Higuchi showed that Ising percolation on the square lattice has a sharp transition. By using ap-proximate zero-one laws and the Ising spin flip dynamics, we show how this result fits as a special case ina more general framework. This more general framework involves models where the spin values can besuitably ‘finitarily’ represented by i.i.d. random variables.

Session MC03 - Stochastic Programming and Optimization 1, Chair:Gunawan in CZ 10

Performance Evaluation of Interconnection Network System Based on MinimumCut Sets Method

Indra Gunawan, Department of Mechanical and Production Engineering Auckland University ofTechnology, New Zealand, indra.gunawan@aut.ac.nz

Interconnection network system consists of layers of switching elements connected together in a prede-fined topology providing the connectivity between the set of processors and the set of memory modules.Performance evaluation of network system is important as it determines the usability and efficiency of thenetwork to provide services. A shuffle-exchange network (SEN), a specific type of Multistage Intercon-nection Network (MIN) that consists of 2x2 switching elements, has been widely considered as a practicalinterconnection system. This paper presents network reliability model based on minimum cut sets forSEN systems. A general equation for large-scale network reliability is derived subsequently to reduce thecomplexity of the reliability analysis for the interconnection network system.

Scheduling assembly flow-shops with resource leveling: a GA based heuristic ap-proach

Qinhua Wang, 1 - IBM China Research Laboratory, 2 - Department of Industrial Engineering, TsinghuaUniversity, P.R. China, wqh@mails.tsinghua.edu.cnChangrui Ren, Jin Dong, Hongwei Ding, Wei Wang, Minmin Qiu, IBM China Research Laboratory,P.R. China, {rencr, dongjin, dinghw, wangwcrl, qiumm}@cn.ibm.com

For large scale assembly manufacturing, global optimization approaches are time consuming and impracti-cable. A Genetic Algorithm (GA) based heuristic is proposed to make schedule for large scale of assemblytasks and level the resource requirement through scheduling operations in each working area separately.The variance of daily resource requirement is considered as the index for resource leveling. Firstly, a rolling

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back method is applied to ensure all orders are finished between their release dates and the due dates whileselecting corresponding operations as a group of scheduling targets in each step. Then the precedence andfloat interval are calculated for each operation through improved critical path method considering the re-markable constraints of parallel orders and sub-assembly relationships in assembly manufacturing. At lasteach group of operations is scheduled via GA with feasibility validation and the labor resource is leveled inthe corresponding makespan. Computational tests are carried out on three levels of actual data and demon-strate that the algorithm is much more efficient than global optimization, and for each production unit, thevariance of daily resource requirement decreases while there is also increase in global resource levelingcompared with experience-based and principle-based schedule result generated by simple heuristics.

A Fast Entropy Method for Solving Integer Programs: Part I

R.Y. Rubinstein, Technion, Israel, ierrr01@ie.technion.ac.il

We present a new method based on the classic minimum cross-entropy (MinxEnt) method, for approxi-mating a broad class of NP-hard integer programs including combinatorial optimization problems (COP’s). The main idea of our approach is to associate with each COP problem an auxiliary convex MinxEntprogram, which is easy to solve analytically and which has the so-called decomposability property in thesense that its optimal solution, the joint probability density function (pdf) presents a product of the marginalpdf’s. We show that the optimal marginal pdf’s converge to their associated degenerated counterparts. Thisallows to approximate the solution of COP’s deterministically, that is without resorting to simulation. Thecomplexity of our method is defined by the complexity of the MinxEnt program, that is by complexity ofa convex program with logarithmic objective function and linear constraints, the so-called log-linear con-vex program. Because of that our method allows to solve combinatorial optimization with thousands ofvariables and thousands of constraints. Supportive numerical results are presented.

A Fast Entropy Method for Solving NP-Hard Counting Problems: Part II

R.Y. Rubinstein, Technion, Israel, ierrr01@ie.technion.ac.il

We present a new method based on the classic minimum cross-entropy (MinxEnt) method for approximat-ing a broad class of NP-hard counting problems, like knapsack, satisfiability and cliques. The main idea ofour approach is to associate with each counting problem a convex MinxEnt program, which is easy to solveanalytically and which has the so-called decomposability property in the sense that its optimal solution, thejoint probability density function (pdf) presents a product of the marginal pdf’s. To approximate the desiredcounting quantity we use two approaches: Sanov’s theorem and importance sampling. In the former casethe desired counting quantity is approximated deterministically, that is without resorting to simulation. Thecomplexity of our method is defined by the complexity of the MinxEnt program, that is by complexity of aconvex program with logarithmic objective function and linear constraints, the so-called log-linear convexprogram. Because of that our method allows to solve counting problems with thousands of variables andthousands of constraints. Supportive numerical results are presented.

Session MC04 - Stochastic Networks and Diffusions, Chair: Budhi-raja in CZ 11

Optimal buffer size for a stochastic processing network with a drift

A. P. Ghosh, Statistics Department, Iowa State University, Ames, IA 50011-1210, USA,apghosh@iastate.eduA. Weerasinghe, Mathematics Department, Iowa State University

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MC04

We consider a one dimensional stochastic control problem that arises in queueing network applicationswhere the controller can choose the arrival and/or service rates and the buffer size b is finite (i.e. customersare rejected if the queue-length exceeds b). This model is similar to the model in Ata-Harrison-Shepp(Annals of Applied Probability, 2005) where the buffer size is kept fixed. We take a long run average cost(“ergodic cost") criterion, which has components corresponding to the cost for arrival/service rate control,holding cost for the customers in the queue as well as a penalty for rejecting customers when the buffer isfull. For this cost criterion, we obtain the optimal drift rate (i.e. optimal rates for arrivals and services) aswell as the optimal buffer size b*.

Our approach involves analyzing a family (indexed by the buffer size b) of ordinary differential equations,where the b-th equation is the Hamilton-Jacobi-Bellman (HJB) equation for the drift control problem instochastic processing networks with fixed buffer size b. We demonstrate a simple approach to constructsolution for this HJB equation (and thus the optimal drift rate) for each buffer size b. This analysis simplifiessome of the proofs in Ata-Harrison-Shepp. We make use of some specific characteristics of the family ofsolutions to choose the optimal buffer size b*.

Characterizations of the invariant measure for a class of reflected diffusions via theextended Skorokhod map

W. Kang, Carnegie Mellon University, U.S.A., weikang@andrew.cmu.eduK. Ramanan, Carnegie Mellon University, U.S.A., kramanan@math.cmu.edu

In this talk we focus on a class of reflected diffusions in polyhedrons defined via the extended Skorokhodmap. We show that the reflected diffusion has certain boundary property, i.e., the pushing process inthe reflected diffusion charges zero on the amount time the reflected diffusion spends at the boundary awayfrom a “bad” set V where two or more faces meet. Further, under suitable conditions, the reflected diffusionhas an unique invariant measure and such an invariant measure satisfies a basic adjoint relationship.

Diffusion limit for dynamic routing with random service rates

R. Atar, Technion, Israel, atar@ee.technion.ac.il

The diffusion approach to analyze queueing models has been very successful in providing approximatesolutions to complicated problems. In the case of many server limits, however, the ‘simplified’ diffusionmodels themselves are often hard.

I will derive a diffusion limit for a model with a single buffer, many servers, whose service rates are random,and natural assumptions on the routing discipline (i. maximum throughput, ii. fair distribution of idle timeamong servers), in the form of a particularly simple, one dimensional diffusion with random drift.

Convergence of Invariant Measures for Generalized Jackson Networks

Amarjit Budhiraja, Department of Statistics, University of North Carolina, budhiraj@email.unc.eduChihoon Lee, Department of Statistics, University of North Carolina,chlee@email.unc.edu

In a recent work Gamarnik and Zeevi(2006) established that, under certain exponential moment conditionson the network primitives, stationary distributions of suitably scaled critically loaded generalized Jacksonnetworks converge to the unique stationary distribution of a related reflected Brownian motion model. Inthis work we give a simple proof of the same result without making any exponential moment assumptions.Our moment requirements on the model are precisely those needed for the validity of a heavy traffic centrallimit theorem.

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Session MC05 - Flow-Level Models, Chair: Prabhakar in CZ 12

Using the Cavity Method to Analyze Randomized Load Balancing Algorithms withGeneral Flow-size Distributions

Y. LuB. Prabhakar, Stanford University, balaji@stanford.edu

Randomized load balancing schemes have been analyzed using the so-called supermarket model: Jobsarrive according to a rate nλ Poisson process at a bank of n rate 1, independent, exponential server queues.An arriving job joins the shortest of d randomly chosen queues. Mitzenmacher (1996) and Vvedenskaya,Dubroshin and Karpelevich (1996) have shown that there is an exponential improvement in the tail of thequeue-sizes when going from d = 1 to d > 1 in the above model. This demonstrated the idea that a smallamount of choice dramatically improves the performance of load balancing. However, their method ofanalysis relies on the Poisson arrival and exponential server assumption and doesn’t generalize.

In this talk we show how the Cavity Method of Statistical Physics could be used to analyze randomizedload balancing in more general settings as well as under changes to the basic model.

Stability of Randomized Load Balancing Algorithms with General Flow-Size Distri-butions

M. Bramson, University of Minnesota, bramson@math.umn.edu

In this talk, we consider randomized load balancing schemes where an arriving job joins the shortest ofd randomly chosen queues. As summarized in the related talk by B. Prabhakar, works by Mitzenmacher(1996) and Dubroshin et al. (1996) consider the case with Poisson input and exponentially distributedservice times.

Here, we consider the stability of generalizations of such systems with arbitrarily distributed interarrivaland service times. We discuss results that show that when such systems are subcritical, they are stable fora large family of disciplines and distributions, but that unstable examples exist.

Proportional fairness: stability and insensitivity properties

Laurent Massoulié, Thomson, Paris Research Lab, laurent.massoulie@thomson.net

In this talk a novel characterization of the proportional fairness bandwidth sharing criterion is proposed.Dynamics of concurrent data transfers in a network with such fair sharing are then considered. Using thenovel characterization, we show that the corresponding Markov process is in some sense approximatelyreversible, and use this to extend previously known ergodicity results to the case of phase-type service timedistributions.

Stochastic Stability Under Network Utility Maximization: General File Size Distri-bution

M. Chiang Princeton University, chiangm@princeton.eduA. Tang California Institute of Technology, aotang@its.caltech.eduD. Shah, Massachusetts Institute of Technology, devavrat@mit.edu

We prove the stochastic stability of resource allocation under Network Utility Maximization (NUM) undergeneral arrival process and file size distribution with bounded support, for α-fair utilities with α sufficiently

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MC06

small and possibly different for different sources’ utility functions. In addition, our results imply thatthe system operating under α-fair utility is 1/(1 + α)-approximate stable for any α ∈ (0,∞) for anyfile size distribution with bounded support. Our results are in contrast to the recent stability result ofBramson (2005) for max-min fair (i.e. α = ∞) under general arrival process and file size distribution,and that of Massoulie (2006) for proportional fair (i.e. α = 1) under Poisson arrival process and phase-type distributions. To obtain our results, we develop an appropriate Lyapunov function for the fluid modelestablished by Gromoll and Williams (2006).

Session MC06 - Markov Processes, Chair: Van Doorn in CZ 13

Level-crossing ordering of semi-Markov jump processes and applications to WhittleNetworks

F. Ferreira, CEMAT/Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal,mmferrei@utad.ptA. Pacheco, CEMAT/Instituto Superior Técnico, Lisboa, Portugal, apacheco@math.ist.utl.pt

In this talk we address the level-crossing order, proposed by A. Irle and J. Gani in 2001, which is suited todeal with scenarios in which we are interested on comparing random amounts of time stochastic processesneed to exceed certain levels. This stochastic order was investigated by A. Irle and J. Gani for skip-free tothe right discrete-time Markov chains (DTMCs), by A. Irle for skip-free to the right semi-Markov processes(SMPs) and continuous-time Markov chains (CTMCs), and by the authors for general DTMCs, SMPs andCTMCs, defined on state spaces order-isomorphic to some bounded or unbounded interval of integers. Weinvestigate the level-crossing ordering of Markov and semi-Markov jump processes defined on commonpartially ordered Polish spaces, and derive specific results for Whittle networks.

Average optimality for risk-sensitive control with general state space

A. Jaskiewicz, Wrocław University of Technology, Institute of Math. and Comp. Sci., 50-370 Wrocław,Poland, ajaskiew@im.pwr.wroc.pl

This talk concerns discrete-time Markov control processes on a general state space. A one-step cost func-tion is nonnegative and possibly unbounded. The decision maker is supposed to be risk-averse with theconstant risk coefficient γ > 0. The risk-sensitive average cost criterion is used as a performance mea-sure. We establish the optimality inequality for risk-sensitive dynamic programming and derive an optimalstationary policy. The result is proved under two different sets of compactness-continuity assumptions,namely, for Markov control processes with weakly continuous transition probabilities (Condition (W)) aswell as transition probabilities that are continuous with respect to setwise convergence (Condition (S)). Asimilar problem for risk-neutral stochastic control models has been examined by Schal (Math. Oper. Res.,1993) using the vanishing discount factor approach. This talk is based on the paper that appears in the Ann.Appl. Probab.

Stochastic binary relations of random variables and processes

L. Leskelä, Centrum voor Wiskunde en Informatica, The Netherlands, lasse.leskela@iki.fi

Stochastic ordering is a fundamental technique for approximating performance quantities of random sys-tems that are too complex to be analyzed exactly. In this talk I will sketch a definition of a stochastic binaryrelation that extends the notion of stochastic partial order into a relation between probability measures over

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two arbitrary state spaces. An application of Strassen’s coupling theorem then yields easily verifiable nec-essary and sufficient conditions for two Markov processes to preserve a given stochastic binary relation.These conditions turn out useful in approximating stationary distributions of vector-valued random walksand multiclass queueing systems.

Survival in a quasi-death process

E.A. van Doorn, University of Twente, The Netherlands, e.a.vandoorn@utwente.nl

We consider Markov chains in continuous time with an absorbing coffin state and a finite set S of transientstates. When S is irreducible the limiting distribution of the chain as t →∞, conditional on survival up totime t, is known to equal the (unique) quasi-stationary distribution of the chain. We address the problemof generalizing this result to a setting in which S may be reducible, and present a complete solution whenthe eigenvalue with maximal real part of the generator of the (sub)Markov chain on S has multiplicity one.The application of this result to pure death processes and, more generally, to quasi-death processes will bedescribed.

Session MC07 - Control, Chair: Lelarge in CZ 14

A probabilistic analysis of a leader election algorithm

H. Mohamed, INRIA Rocquencourt, France, Hanene.Mohamed@inria.fr

A leader election algorithm is an elimination process that divides recursively into tow subgroups an initialgroup of n items, eliminates one subgroup and continues the procedure until a subgroup is of size 1; theleader. In this paper the biased case is analyzed. We are interested in the cost of the algorithm, i.e. thenumber of operations needed until the algorithm stops. Using a probabilistic approach, the asymptoticbehavior of the algorithm is shown to be related to the behavior of a hitting time of two random sequenceson [0, 1].

Investigation of discrete stochastic systems with dynamic distributed routing anddelayed information

N.V. Yudaeva, Saratov State University, Russia, yudaevanv@sysan.ru

Discrete stochastic systems with network structure and dynamic distributed routing of some objects areinvestigated. Some elements of system make decisions about routing of processed objects. These decisionsare based on delayed information about current states of neighboring elements. We consider two models ofset of elements, one of which is decision making element. This set is called standard segment of discretestochastic system. In the first model we take account of processes of forming and processing of informationabout states of neighboring elements. In the second model we leave out of account of these processes andsuppose that this information is always available for decision making element. Queueing networks withdynamic distributed routing are used as models of segment. Expressions for steady-state characteristics ofconsidered systems with and without taking account of delay of information are derived.

On asymptotic optimality of controls for time-varying queues

M. Cudina, University of Texas at Austin, U.S.A., mcudina@math.utexas.eduK. Ramanan, Carnegie Mellon University, U.S.A., kramanan@math.cmu.eduS. Shreve, Carnegie Mellon University, U.S.A., shreve@andrew.cmu.edu

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MC08

We introduce a general framework for asymptotic optimality of controls for time-varying queues, motivatedby applications in manufacturing systems and telecommunications. In particular, we identify controls thatare fluid-optimal with respect to a class of performance measures and proceed to discuss the appropriateconditions on the performance measure which result in asymptotic optimality of fluid-optimal controls inthe uniform acceleration regime. This analysis is conducted in the settings of a single station and of atandem queue and uses the framework of strong approximations.

Heavy traffic limit for online bandwidth packing

M. Lelarge, INRIA-ENS, France, marc.lelarge@ens.fr

We consider the following stochastic bin packing process: the items arrive continuously over time to aserver and are packed into bins of unit size according to an online algorithm. The unpacked items form aqueue. The items have random sizes with symmetric distribution.

Our first contribution identifies some monotonicity properties of the queueing system that allow to derivebounds on the queue size for First Fit and Best Fit algorithms. As a direct application, we show how tocompute the stability region under very general conditions on the input process.

Our second contribution is a study of the queueing system under heavy load. We show how the monotonic-ity properties allow to derive bounds for the speed at which the stationary queue length tends to infinitywhen the load approaches one. In the case of Best Fit, these bounds are tight and show that the averagequeue size scales as2

(log(1/(1−ρ))

1−ρ

)when the load ρ approaches one. Our analysis shows connections be-

tween our dynamic model, average-case results on the classical bin packing problem and planar matchingproblems.

Session MC08 - Queueing Models 1, Chair: Fralix in CZ 15

The conditional distribution of the residual service time in the Mn/G/1 queue

Yoav Kerner, The Hebrew university of Jerusalem, mskerner@mscc.huji.ac.il

We observe an M/G/1 queue with the following variation: an arriving customer is rejected with probabilitywhich depends on the queue length. The waiting time of an arriving customer given the queue length is thesum of the service times required by those who stand in line, plus so residual service time of the one who isbeing served. As the distribution of the service time is given, we are after the conditional distribution of theresidual service time given the queue length. This distribution is a function of the rejection probabilities.We give a recursive formula for this distribution and for the corresponding Laplace-Stieltjes transform andmean value. We also give an application in decision making in queues.

Transient Solution to the Time Dependent Single Server Queue with Balking

R.O. Al-Seedy, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, EgyptA.A. El-Sherbiny, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia,EgyptS.A. El-Shehawy, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia,EgyptS.I. Ammar, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt,sherifamar2000@yahoo.com

The transient probabilities for single server queue with balking are derived in a form of an integral equation,when the arrival and departure rates are allowed to vary with time. The used method depends on the

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generating functions technique. In the current paper, the authors show that the transient probabilities forsingle server queue with balking are satisfying a Volterra integral equation of the second kind.

Relating time- and customer- averages for queues using “forward” coupling fromthe past

Erol A. Peköz, Department of Operations and Technology Management, Boston University, 595Commonwealth Avenue, Boston, MA 02215, pekoz@bu.eduSheldon Ross, smross@usc.edu

We give a new method for simulating the time-average steady-state distribution of a continuous-time queu-ing system, by extending a “read-once” or “forward” version of the coupling from the past (CFTP) al-gorithm developed for discrete-time Markov chains. We then use this to give a new proof of the “Pois-son Arrivals See Time Averages” (PASTA) property, and a new proof for why renewal arrivals see eitherstochastically smaller or larger congestion than the time average if inter-arrival times are respectively newbetter than used in expectation (NBUE) or new worse than used in expectation (NWUE).

A Time-Dependent View of ASTA and Little’s Law

B. H. Fralix, EURANDOM, Eindhoven, The Netherlands, fralix@eurandom.tue.nlR. F. Serfozo, Georgia Institute of Technology, Atlanta, GA, USA, rserfozo@isye.gatech.eduG. Riaño, Universidad de Los Andes, Bogotá, Colombia, griano@uniandes.edu.co

It is well known that many long-run characteristics of queueing systems (e.g. Little’s law and PASTA) canbe proved through the use of Palm probabilities induced by stationary systems. In this talk, we discuss howPalm theory can be used to show such phenomena can also more or less exist in queues before equilibriumis reached. We begin by showing that, in the absence of stationarity, there is still an interesting connectionbetween the stochastic intensity of a point process and the Palm measures induced by this process. Thisis used to provide necessary and sufficient conditions for a time-dependent version of “AST A” (our re-sults clearly do not involve time averages) that are very similar in structure to the well-known conditionsestablished by other authors in the stationary case. Consequences of this result include transient versionsof P AST A, conditional P AST A and Anti-P AST A. We’ll also discuss how Palm measures can be usedto quickly derive time-dependent versions of Little’s law. In particular, the theory can be used to relate anymoment of the queue-length at a time t to waiting times spent by customers in the system.

Session MC09 - Inventory 2, Chair: Yao in CZ 16

An E O Q model with state dependence, random yield, and static pricing

Opher Baron, Rotman School of Management, University of Toronto, 105 St. George Street, Toronto,Ontario, Canada M5S 3E6. opher.baron@rotman.utoronto.caOded Berman, Rotman School of Management, University of Toronto, 105 St. George Street, Toronto,Ontario, Canada M5S 3E6. berman@rotman.utoronto.caDavid Perry, Department of Statistics, University of Haifa. Haifa, Israel 31905. dperry@stat.haifa.ac.il

We consider EOQ models with state dependent demand, where the objective is to choose the optimalreorder point and order size to minimize the total inventory cost. We use level crossing methodologyto derive the holding cost rate. This approach simplifies significantly the analysis of these well studiedmodels and allows including more realistic demand models, random yield, and pricing. In the latter caseour objective is to maximize profits.

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Periodic review inventory models

M. Bijvank, VU University Amsterdam, The Netherlands, mbijvank@few.vu.nlI.F.A. Vis, VU University Amsterdam, The Netherlands, ivis@feweb.vu.nlG. Koole, VU University Amsterdam, The Netherlands, koole@few.vu.nl

In the current world of high service levels and tight delivery schedules companies need to adjust theiractions to deal with smaller orders that are made more frequently and shorter lead times. Existing inventorymodels are not designed to deal with these characteristics. Therefore, in this research we consider periodicreview models in which lead times are smaller than the length of a review period, i.e., at most one orderoutstanding. We model the inventory system as a Markov chain and present an approach to analyze theperformance of a certain replenishment policy. Based on this an optimal policy can be determined. Theapproach is general in the sense that it can be used for models with a cost objective or have a service levelconstraint, but also situations with backordering or lost sales can be considered.

Asymptotically Optimal Solutions to a Joint Replenishment-Transshipment Model

David D. Yao, IEOR Department, Columbia University, New York, NY 10027, USA, yao@columbia.eduYongbo Xiao, School of Economics and Management, Tsinghua University, Beijing, China

We study a model of two stores supplying seasonal demands allowing lateral transshipment. There is afinite planning horizon with a one-time replenishment opportunity before the season starts. While theoptimal transshipment policy can be derived by dynamic programming (DP), to optimize the replenishmentquantities on top of the DP value function results in a rather intractable problem. We construct an upperbound to the original problem, which takes the form of a concave optimization with a readily derivedsolution to replenishment quantities. We also propose a simple reservation policy for transshipment, leadingto a lower bound. Furthermore, we show that following the replenishment quantities from the upper-boundproblem and the reservation policy in the lower bound problem is asymptotically optimal, as the demandrates increase.

Study on inventory management with existence of electronic markets

Ming Xie, IBM China Research Laboratory,xieming@cn.ibm.comDiamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, ChinaWenjun Yin, IBM China Research Laboratory,yinwenj@cn.ibm.comDiamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, ChinaBin Zhang, IBM China Research Laboratory,zbin@cn.ibm.comDiamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, ChinaXinxin Bai, IBM China Research Laboratory, baixx@cn.ibm.comDiamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, ChinaJin Dong, IBM China Research Laboratory, dongjin@cn.ibm.comDiamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, China

In the past several years, Electronic Markets (EM) has been developing rapidly and became an importantchannel for firms to distribute their products. In this paper we consider a classical single period inventorycontrol problem of an in-store retailer who replenishes its stock from a supplier and sells it to the customer.The retailer jointly decides its ordering quantity and selling price to maximize its profit. Surplus inventoryis salvaged to the electronic markets with a dynamic e-price. Demand is determined by both the in-storeselling price and the e-price. Excess demand can be satisfied by replenishing inventory from electronicmarkets. To study this inventory control problem we build a simulation model. By which we investigateinventory policies and pricing strategies under different scenarios with and without electronic markets. Theresults provide valuable insights for inventory management with existence of electronic markets.

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Tuesday 8:30am - 10:00am

Session TA01 - Financial Engineering 3, Chair: Kou in CZ 4

Pricing, hedging and calibrating credit from the top down

Kay Giesecke, Management Science & Engineering, Stanford University, USA, giesecke@stanford.edu

A credit derivative is a contingent claim on the aggregate financial loss in a portfolio of credit-sensitiveinstruments such as loans, bonds or credit swaps. We summarize our recent results on the pricing, hedgingand calibration of credit derivatives using point process methods.

Malliavin Greeks without Malliavin calculus

Nan Chen, the Department of System Engineering and Engineering Management, the Chinese Universityof Hong Kong, Hong Kong, nchen@se.cuhk.edu.hkPaul Glasserman, the Columbia Business School, Columbia University, New York, USA,pg20@columbia.edu

We derive and analyze Monte Carlo estimators of price sensitivities (“Greeks”) for contingent claims pricedin a diffusion model. There have traditionally been two categories of methods for estimating sensitivities:methods that differentiate paths and methods that differentiate densities. A more recent line of work de-rives estimators through Malliavin calculus. The purpose of this article is to investigate connections be-tween Malliavin estimators and the more traditional and elementary pathwise method and likelihood ratiomethod. Malliavin estimators have been derived directly for diffusion processes, but implementation typi-cally requires simulation of a discrete-time approximation. This raises the question of whether one shoulddiscretize first and then differentiate, or differentiate first and then discretize. We show that in severalimportant cases the first route leads to the same estimators found through Malliavin calculus, but usingonly elementary techniques. Time-averaging of multiple estimators emerges as a key feature in achievingconvergence to the continuous-time limit.

To be determined

Rama Cont, CMAP Department, Ecole Polytechnique, IEOR Department, Columbia University

to be determined

Pricing Asian option via double Laplace transform

Ning Cai, Department of IEOR, Columbia University, USA, nc2118@columbia.eduS.G. Kou, Department of IEOR, Columbia University, USA, sk75@columbia.edu

In this paper, we present an analytic solution for double Laplace transform of the price of a continuousAsian option under the Black-Scholes model, which can be inverted numerically via two-sided Euler inver-sion algorithm. Error bound of the numerical inversion is derived theoretically. Numerical results indicatethat this method is efficient and accurate. In addition, a theoretical contribution of the paper is that we givea direct proof of our result by solving a PDE rather than using Bessel process and Lamperti’s relation.

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Session TA02 - Stochastic Geometry, Chair: Van Lieshout in CZ 5

Wiener sausage and sensor networks

E. Spodarev, Ulm University, Germany, evgueni.spodarev@uni-ulm.deR. Cerný, Charles University, Prague, Czech Republic, rostislav.cerny@mediaresearch.czS. Funken, Ulm University, Germany, stefan.funken@uni-ulm.deD. Meschenmoser, Ulm University, Germany, daniel.meschenmoser@uni-ulm.deJ. Rataj, Charles University, Prague, Czech Republic, rataj@karlin.mff.cuni.czV. Schmidt, Ulm University, Germany, volker.schmidt@uni-ulm.de

The parallel r -neighborhood of a path of the Brownian motion in d dimensions is called after NorbertWiener a Wiener sausage. This random compact set can serve as a model for the coverage area of aBrownian sensor able to detect a target within reach r . In this sense, Boolean models of Wiener sausagescan be used to describe the behavior of sensor networks trying to detect an immobile intruder.

In this talk, some properties of (Boolean models of) Wiener sausages are studied. In particular, the capacityfunctional, the volume fraction, the specific surface area, the convariance function and the spherical contactdistribution function are calculated. Approximations of the Wiener sausage by polyconvex sets and theconvergence of the corresponding curvature measures are addressed as well.

Maximum likelihood estimation for random sequential adsorption

M.N.M. van Lieshout,CWI,P.O. Box 94079, 1090 GB Amsterdam, The Netherlands,EURANDOM, P.O.Box 513, 5600 MB Eindhoven, The Netherlands

Packing problems occur in many physical and biological processes. Hard ‘particles’ such as cars, monomers,or proteins, arrive in some bounded Euclidean window according to a constant rate Poisson process, andselect a position for themselves according to some fixed distribution. If there would be no overlap with anestablished particle, the new one attaches itself to the selected location, otherwise it leaves the system. Theprocess continues for a given time, or until there is no room for another particle to establish itself withoutintersecting existing ones.

To date, most attention has focused on probabilistic work rather than inference. In this talk, we take thestatistical viewpoint. We place the RSA model in the context of Markov sequential spatial processes, deriveits Radon–Nikodym derivative, and consider maximum likelihood estimation of its parameters. Note thatmaximum likelihood is especially convenient in exponential family models, but that RSA exhibits a highlynon-linear dependence.

This research is supported by the Technology Foundation STW, applied science division of NWO, and thetechnology programme of the Ministry of Economic Affairs (project CWI.6156 ‘Markov sequential pointprocesses for image analysis and statistical physics’).

Reference: M.N.M. van Lieshout Maximum likelihood estimation for random sequential adsorption. Ad-vances in Applied Probability (SGSA) 38:889–898, December 2006.

Random field model for fingerprint features

Elke Thönnes, Department of Statistics and Centre for Scientific Computing, University of Warwick,Coventry, CV4 7AL, UK, E.Thonnes@warwick.ac.uk

Fingerprint experts classify information from human fingerprints into three categories: level one describesthe overall pattern of skin ridges; level two concerns singularities within the ridge flow such as ridgeendings or bifurcations; and finally, level three consists of high level features such as shape of ridge endingsor the location and shape of sweat pores.

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Sweat pores only occur on ridges and so there is a very intricate relationship between level two and levelthree features. In this talk I will discuss joint work with Mr J. Su and Prof. W.S. Kendall which representsfirst steps at modelling the ridge structure conditional on observed sweat pore locations. Using a non-lineartransformations on the pore pattern, a random tensor field is defined that indicates principal flow directions.The field can then be used to locate singularities within the ridge flow or to classify pores according to ridgemembership.

Opportunistic Routing on Signal-to-Interference-and-Noise Ratio Graphs

F. Baccelli, ENS/INRIA Paris, France, Francois.Baccelli@ens.frB. Błaszczyszyn, ENS/INRIA Paris, France and University of Wrocław, Poland,Bartek.Blaszczyszyn@ens.frP. Mühlethaler, INRIA Rocquencourt, France, Paul.Muhlethaler@inria.fr

In opportunistic routing in multi-hop mobile wireless networks, for each packet and each hop, one dy-namically selects as next relay the node that captures the packet transmission and which is the nearest tothe destination. Such a scheme allows each packet to take advantage of the local pattern of transmissionsand at any slot and at any hop. Simulations show that such routing schemes significantly improve the per-formance of multi-hop networks compared to conventional routing algorithms, where packets are routedon a pre-defined route obtained by a shortest path routing protocol. We propose a stochastic-geometryframework based on point maps to explain some observations made by simulation. In particular, we provethat our opportunistic routing algorithms can be optimized so as to minimize the average end-to-end delayincurred by a packet over long paths and that the optimum setting does not depend on the node density inthe homogeneous Poisson case. Various challenging problems remain open.

Session TA03 - Rare Event Simulation 2, Chair: Juneja in CZ 10

Alternative efficiency proof and interpretations for a recent state-dependent impor-tance sampling scheme

P.T. de Boer, University of Twente, The Netherlands, ptdeboer@cs.utwente.nlW.R.W. Scheinhardt, University of Twente, The Netherlands, w.r.w.scheinhardt@utwente.nl

Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queuehas been proposed by Dupuis, Sezer and Wang (2005). They derive it using game-theory, and prove itsasymptotic efficiency.

We present a simpler efficiency proof, by decomposing the likelihood ratio of a sample path into a factorthat only depends on the initial and final states, and a factor whose influence can be shown to be bounded.

Furthermore, we give intuitive interpretations for several of the abstract quantities involved in deriving thischange of measure. In particular, we show that the conditions derived using game theory are equivalent towell-known conditions on the likelihood-ratio of cyclic paths.

Importance sampling for heavy-tailed random walk

Achal Bassamboo, Northwestern University, Evanston, IL, USA, a-bassamboo@northwestern.eduSandeep Juneja, Tata Institute of Fundamental Research, Mumbai, India, juneja@tifr.res.inAssaf Zeevi, Columbia University, NY, NY, USA, assaf@gsb.columbia.edu

In this talk we consider a negative drift random walk with heavy-tailed increments. We are interestedin estimating the probability of this random walk exceeding a a large threshold starting from zero before

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taking a negative value, using simulation. We use Importance sampling technique for reducing the varianceof this estimator. Earlier results have shown that using a state-independent change-of-measure one cannotobtain an efficient measure. We present an asymptotically efficient state-dependent change-of-measure forestimating this probability.

A Reduced-Form Model With Short-Time Contagion For Credit Risk

Jose Blanchet, blanchet@fas.harvard.eduLi Zhan, zhan@stat.harvard.edu

Standard intensity default models do not capture contagion effects (basically the possibility of multiplesimultaneous defaults). We discuss a reduced form model that captures contagion effects in short periods oftime (a feature that can be useful to model “panic” situations). The proposed model has convenient analyticfeatures and induces loss distributions that possess heavier tails than standard intensity models. We alsodiscuss associated rare-event simulation methodology for estimating the tail of the loss distribution.

Minimizing Tail Probabilities: Asymptotic Analysis and Efficient Simulation Opti-mization

Sandeep Juneja, Tata Institute of Fundamental Research, India, juneja@tifr.res.inHimanshu Kalra, Tata Institute of Fundamental Research, India

We consider a resource allocation problem where the objective is a tail probability associated with max-imum of sums of a few random variables. Applications include stochastic activity networks where ourinterest is in large project delays, and portfolio optimization where we are concerned about the probabilityof large losses. We first consider a suitably scaled asymptotic of the tail probability as an approximateobjective function. The resulting approximate problem is a convex programming problem and is easilysolved providing insight into the nature of the solution to the original problem. We then solve the exactproblem using simulation optimization. Here we introduce a change of variable technique that allows asingle importance sampling probability measure to asymptotically efficiently estimate the objective overthe entire allocation space. We see that this considerably speeds up determining the optimal solution viasimulation particularly when insights from the asymptotic solution are utilized.

Session TA04 - Scheduling and Stability, Chair: Hasenbein in CZ 11

Fluid and Diffusion Limits of a Multiclass Overloaded Abandonment Queue underPriority Scheduling

O. Jennings, otisj@duke.eduJosh E. Reed

Consider a sequence of overloaded multiclass queues with abandonment, indexed by n. The customerclasses are split into priority subgroups and, within each subgroup, jobs are processed in a FIFO fashion.The sequence of arrival and service rates are linear in n, whereas the abandonment distribution is constant.All r.v.’s have general distributions. In the limit, only one of the subclasses has a nondegenerate waitingtime value. For this subclass, the limiting diffusion-scaled, abandonment-filtered virtual waiting time pro-cess is an OU process whose drift is proportional to the traffic intensity-weighted sum of the associatedabandonment densities, each evaluated at the limiting deterministic waiting time value.

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Stability of Non-random Critically Loaded Networks under MaxWeight Scheduling

D.M. Andrews, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA, andrews@research.bell-labs.comK. Jung, MIT, Cambrdge, MA, kmjung@mit.eduA.L. Stolyar, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA, stolyar@research.bell-labs.com

We study a queueing network, operating in discrete time, with deterministic (periodic) exogenous arrivalprocesses and fixed service rates at the nodes. The MaxWeight algorithm is used for the combined routingand scheduling. (Thus, the network evolution is deterministic.) We consider the case when network isexactly critically loaded, and prove that all queues remain bounded.

The result has implications for the running time of the MaxWeight algorithm, when it is used to solvemulticommodity flow problems. In particular, for a fixed problem instance, the algorithm can be appliedto find a flow that is within a (1 − ε)-factor of optimal in time O(1/ε) (improving the previous bound ofO(1/ε2)).

Undecidability results in the theory of queueing networks and Skorokhod problem

D. Bertsimas, MIT, USA, dbertsim@mit.eduD. Gamarnik, MIT, USA, gamarnik@mit.eduD. Katz, MIT, USA, dimdim@mit.edu

We consider a deterministic multiclass queueing network with a mixture of finite and infinite buffer capac-ities, operating under a static buffer priority scheduling rule and investigate the question of stability of suchclass of policies. To the day no tight conditions for stability of such policies are known. We establish thatstability is an algorithmically undecidable property, thus no constructive tight characterization is possible.We further extend this result to a deterministic version of a Skorokhod mapping problem, establishing asimilar undecidability result.

Networks with Parameter Uncertainty: Stochastic Fluid Models with Recourse

John J. Hasenbein, Graduate Program in Operations Research and Industrial Engineering,University of Texas at Austin, Austin, Texas, USA, jhas@mail.utexas.eduBurak Buke, David P. Morton, Graduate Program in Operations Research and Industrial Engineering,University of Texas at Austin, Austin, Texas, USA {bukeb,morton}@mail.utexas.edu

We consider fluid models in which there is uncertainty in the parameters defining the networks. We inves-tigate the problem of minimizing makespan when a stochastic programming decision structure is imposedon the control mechanisms. The controller must make a control decision before having full knowledgeof the parameters, and she may also update that decision after gaining some knowledge. The fluid modelproblem is in general not analytically tractable, and we present algorithms to numerically solve the problemefficiently.

Session TA05 - Ad Hoc Networks, Chair: Van den Berg in CZ 12

Efficient Resource Control for the EDCA and HCCA Mechanisms in Multi-RateIEEE 802.11e Networks

Vasilios A. Siris, Inst. of Comp. Science, FORTH and Dept. of Comp. Science, Univ. of Crete,vsiris@ics.forth.grCostas Courcoubetis, Department of Informatics, Athens Univ. of Economics and Business,courcou@aueb.gr

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We investigate the problem of efficient resource control for elastic traffic over the EDCA (Enhanced Dis-tributed Channel Access) and HCCA (Hybrid Coordination Function - HCF - Controlled Channel Access)mechanisms of IEEE 802.11e. Our approach considers an economic modelling framework based on con-gestion pricing that captures how various factors, such as the probability of attempting to transmit a frame,the use of the basic CSMA/CA or the RTS/CTS procedure, the transmission opportunity (TXOP), and thephysical layer transmission rate, contribute to congestion. Additionally, we consider the joint control of theEDCA and HCCA mechanisms, which allows us to determine the optimal sharing of the wireless channelbetween the two access mechanisms. We discuss the application of the framework for achieving class-based throughput differentiation, for performing explicit congestion notification (ECN) marking based onthe level of congestion in the wireless channel, and for modelling the performance of TCP congestioncontrol over EDCA. In these application scenarios we discuss how to estimate the optimum minimum con-tention window and the congestion prices based on the 802.11e parameters and actual measurements, inorder to achieve efficient channel utilization.

Performance Modeling of a Bottleneck Node in an IEEE 802.11 Ad-hoc Network

Frank Roijers, TNO Information and Communication Technology, the Netherlands, frank.roijers@tno.nlHans van den Berg, TNO Information and Communication Technology, the Netherlands,j.l.vandenberg@tno.nlMichel Mandjes, Korteweg-de Vries Institute, University of Amsterdam, the Netherlands,mmandjes@science.uva.nl

The IEEE 802.11 MAC-protocol, often used in ad-hoc networks, has the tendency to share the capacityequally amongst the active nodes, irrespective of their loads. An inherent drawback of this fair-sharing pol-icy is that a node that serves as a relay-node for multiple flows is likely to become a bottleneck. This paperproposes a flow-level performance model of such a bottleneck node using fluid-flow analysis. AssumingPoisson initiations of new flow transfers at the bottleneck node, we obtain insightful, robust, and explicitexpressions for characteristics related to the overall flow transfer time, the buffer occupancy, and the packetdelay at the bottleneck node. The analysis is enabled by a translation of the behavior of the bottleneck nodeand the source nodes in terms of an M/G/1 queueing model. We conclude the paper by an assessment ofthe impact of alternative capacity sharing amongst source nodes and the bottleneck in order to improve theperformance of the bottleneck.

Collaboration and incentives in wireless ad-hoc networks

R.J. Gibbens, University of Cambridge, United Kingdom, Richard.Gibbens@cl.cam.ac.ukP.B. Key, Microsoft Research Cambridge, United Kingdom, Peter.Key@microsoft.com

This talk considers how incentives for collaboration can be incorporated into a simplified model of com-munication in wireless ad-hoc networks. Such networks comprise heterogeneous, possibly mobile, sourcesand resources which are coupled through a common distributed wireless communication channel. We ex-amine some of the optimization and pricing methods proposed to coordinate jointly beneficial behaviourof the network and explore some of their game-theoretic consequences.

A fluid model for a relay node in an ad-hoc network: the case of heavy-tailed input

R. Bekker, Vrije Universiteit Amsterdam, The Netherlands, rbekker@few.vu.nlM. Mandjes, Korteweg-de Vries Institute for Mathematics, The Netherlands, mmandjes@science.uva.nl

Relay nodes in an ad-hoc network can be modelled as fluid queues, in which the available service capacityis shared by the input and output. In this paper such a relay node is considered; jobs arrive according to aPoisson process and bring along a random amount of work. The total transmission capacity is fairly shared,

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meaning that, when n jobs are present, each job transmits traffic into the queue at rate 1/(n + 1) while thequeue is drained at the same rate of 1/(n + 1). Where previous studies mainly concentrated on the caseof exponentially distributed job sizes, the present paper analyzes the case of regularly varying jobs. Thefocus lies on the tail asymptotics of the sojourn time S. Using sample-path arguments, it is proven thatP {S > x} behaves roughly as the residual job size, i.e., if the job sizes are regularly varying of index −ν,the tail of S is regularly varying of index 1− ν. The tail asymptotics of other performance metrics, such asthe workload in the queue, the flow transfer time and the queueing delay, are also addressed.

Session TA06 - Call Centers 2, Chair: Koole in CZ 13

Shift scheduling for heterogeneous contact centers via mixed-integer programming

S. Helber, Leibniz University Hannover, Germany,stefan.helber@prod.uni-hannover.deK. Henken, kirsten.henken@t-online.de

We present a discrete-time decision model to solve the shift-scheduling problem for a given day in a largeinbound contact center which uses different agent classes to serve different classes of impatient customerswith retrials. The dynamics of the contact center due to time-varying arrival rates and time dependentstaffing levels are modeled via a fluid model. Mixed-integer programming is used to approximate profitmaximizing staffing levels over the day. The results are compared to simulation.

A real-time routing policy for call centers

S.A. Pot, Vrije Universiteit of Amsterdam, The Netherlands, sapot@few.vu.nl

We consider a multi-skill call centers, having different job types and several agent groups. A routing policyis presented that deals with unexpected fluctuations in the arrival intensities. We show that the policy yieldsthe best achievable service levels under certain circumstances.

The Hayward Approximation with Queueing Systems

Zeynep Aksin, Koç University, Turkey, zaksin@ku.edu.trJean-Christophe Van den Schrieck, Université catholique de Louvain, Belgium,vandenschrieck@poms.ucl.ac.be

The Erlang-C formula is widely known in queueing theory. Its utilization is however limited to Poissonarrivals and exponential service time distribution. In this talk we propose to investigate if this formula canbe used in the case the arrival flow is an overflow. To be more specific, we want to check whether theHayward Approximation, which approximates the loss probability in case of overflow arrivals, could beextended to queueing systems.

Staffing Call Centers with Differentiated Levels of Service: Approximate Solutionsvia Constraint Dualization

Achal Bassamboo, Northwestern University, Evanston, IL, USA. a-bassamboo@northwestern.eduAssaf Zeevi, Columbia University, NY, NY, USA. assaf@gsb.columbia.edu

We consider a call center model with multiple customer classes and multiple server pools. Calls arriverandomly over time; customers are impatient and may abandon the system before their service commences.

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The objective is to minimize staffing costs subject to some quality of service constraints. We consider twospecific examples of the latter: i) an upper bound on the fraction of customers who abandon; and ii) anupper bound on the waiting time of the customer.

The approach proposed in this paper has two key ingredients. The first builds on suitably approximatingthe dynamics of the original system, which in turn simplifies the form of the constraint. The second stepuses duality theory to convert this constrained problem into an unconstrained (lagrangian) formulation;the Lagrange multipliers can be interpreted as penalties for constraint violation. This problem can besolved efficiently using saddlepoint algorithms. The resulting prescription for the staffing level (and thecorresponding routing control) can be shown to be almost optimal in a suitable mathematical sense, i.e.,its performance comes close to the best achievable performance in the original constrained optimizationproblem.

Session TA07 - Statistics 1, Chair: Segers in CZ 14

Martingale approach on the average run lengths of Exponentially EWMA controlcharts

Y. Areepong, University of Technology, Sydney, Australia, areepong@hotmail.comS. Sukparungsee, University of Technology, Sydney, Australia, saowanits@hotmail.comA. Novikov, University of Technology, Sydney, Australia, Alex.Novikov@uts.edu.au

Approximate computational methods, using the Martingale approach to determine the expected first pas-sage times in Exponentially EWMA charts, are demonstrated and extended to more general heavy-taileddistribution. Explicit formula are presented and compared with Markov Chain Approach and Monte CarloSimulation. The analytical and numerical results are given to show the efficacy of this approach.

Comparative study of change detection properties of Poisson EWMA control proce-dure

S. Sukparungsee, University of Technology, Sydney, Australia, saowanits@hotmail.comA. Novikov, University of Technology, Sydney, Australia, Alex.Novikov@uts.edu.au

This article aims to examine the analytical approximations called the ”Martingales Technique” to show howthe exact lower bounds of the Average Run Lengths (ARLs) of Poisson Exponentially Weighted MovingAverage (PEWMA). Based on this technique, it could be help to reduce the time of finding the ARLs. Theeffectiveness of this study will be done by comparison of the EWMA properties and CPU timing. Theoptimal design one-sided and two-sided case are provided as a combination of optimal parameters.

Benchmarking of Fitting Algorithms for Continuous Phase-Type Distributions

J.F. Pérez, Universidad de los Andes, Colombia, fern-per@uniandes.edu.coG. Riaño,Universidad de los Andes, Colombia, griano@uniandes.edu.co

Phase-Type distributions have shown to be useful in the stochastic modeling of real systems. Many effortshave been made to provide algorithms to fit the parameters of a Phase-Type distribution from raw data.In this paper, we present a comparative analysis of some recent fitting algorithms. These algorithms aretested against several data sets and their performance is measured in different ways. The results give insightabout their behavior and the situations in which one of them can overperform others. The analysis providesinformation that can be relevant when selecting one of those methods for a particular application.

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Regularly varying multivariate time series

Bojan Basrak, Zagreb University, Croatia, bbasrak@math.hrJohan Segers, Université catholique de Louvain, Belgium, segers@stat.ucl.ac.be

Extremes of univariate heavy-tailed first-order Markov chains are known to behave under general condi-tions as a multiplicative random walk, the tail chain. This property is found to be a particular case of a muchmore general one stating that a stationary multivariate time series is multivariate regularly varying if andonly if the following property holds: conditionally on the value of the process at a fixed time point beinglarge (in norm), the process converges in the sense of finite-dimensional distributions to a limit process, thetail process. This tail process is found to possess a number of remarkable properties. Moreover, all kindsof interesting quantities can be expressed in terms of this tail process: limit point processes, multivariateextremal indices, functionals of clusters of multivariate extremes. For general multivariate Markov chains,the tail process takes a special form, yielding neat formulas in certain examples, for instance for solutionsto stochastic difference equations or moving averages with random coefficients.

Session TA08 - Polling Models, Chair: Winands in CZ 15

Polling systems with parameter renewals

I. MacPhee, i.m.macphee@durham.ac.uk

Many widely used models of queueing systems assume time homogeneity for arrival streams and servicedistributions. This talk will discuss results about the long term behaviour of an exhaustive polling systemwith arrival rates and service time distribution that are randomly selected each time the server switches toa new queue. We suppose the server visits the stations in some cyclic ordering. Our main result establisheswhich moments of τ , the time taken to empty the system, are finite. The key parameter is the top Lyapunovexponent of a particular random matrix. It turns out that critical/null recurrent behaviour can occur in athick region of the parameter space.

This is joint work with M. Menshikov, D. Petritis and S. Popov. One paper, A Markov chain model of apolling system with parameter regeneration, is to appear in Annals of Applied Probability and we haverecently completed another.

M/G/∞ polling systems with random visit times

Maria Vlasiou, Georgia Institute of Technology, H. Milton Stewart School of Industrial & SystemsEngineering, 765 Ferst Drive, Atlanta GA 30332-0205, USA, vlasiou@gatech.eduUri Yechiali, Department of Statistics and Operations Research, School of Mathematical Sciences,Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel,uriy@post.tau.ac.il

We consider a polling system where a group of an infinite number of servers visits sequentially a setof queues. When visited, each queue is attended for a random time. Arrivals at each queue follow aPoisson process and service time requirement of each individual customer has a general pdf. Thus, eachof the queues comprising the system is, in isolation, an M/G/∞-type queue. A job that is not completedduring a visit will have a new service time requirement sampled from the service time distribution of thecorresponding queue. We derive the expected values and probability generating functions of the queuelengths, as well as the mean and Laplace-Stieltjes transform of the sojourn time of a customer. Moreover,we identify the policy that maximises the throughput of the system per cycle and conclude that under theHamiltonian-tour approach, the optimal visiting order is independent of the number of customers presentat the various queues at the start of the cycle. In other words (and somewhat surprisingly), additional

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information regarding the state of the system at the start of a cycle does not lead to an improvement of theoptimal policy.

Polling Systems with Two-Phase Gated Service: Heavy Traffic Results for the Waiting-Time Distributions

R.D. van der Mei, CWI and Vrije Universiteit, The Netherlands, R.D.van.der.Mei@cwi.nlJ.A.C. Resing, Eindhoven University of Technology, The Netherlands, j.a.c.resing@tue.nl

We consider an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and with so-called two-phase gated service at each queue, an interleaving schemethat aims to enforce fairness among the different customer classes. For this model, we use the classicaltheory of multi-type branching processes (MTBPs) to derive closed-form expressions for the Laplace-Stieltjes Transform (LST) of the waiting-time distributions when the load tends to 1, in a general parametersetting and under proper heavy-traffic (HT) scalings.

Branching-type polling systems with large setups

E.M.M. Winands, Eindhoven University of Technology, The Netherlands, e.m.m.winands@tue.nl

The present paper considers the class of polling systems that allow a multi-type branching process inter-pretation. This class contains the classical exhaustive and gated policies as special cases. We present anexact asymptotic analysis of the delay distribution in such systems, when the setup times tend to infinity.The motivation to study these setup time asymptotics in polling systems is based on the specific applicationarea of base-stock policies in inventory control. Our analysis provides new and more general insights intothe behavior of polling systems with large setup times.

Session TA09 - Switch Scheduling, Chair: Moallemi in CZ 16

Switch scheduling via message passing

M. Bayati, Microsoft Research, USA, bayati@microsoft.com

Max-product “belief propagation" is an iterative, local, message-passing algorithm for finding the maxi-mum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model.Despite the spectacular success of the algorithm in many application areas such as iterative decoding, com-puter vision and combinatorial optimization which involve graphs with many cycles, theoretical resultsabout both correctness and convergence of the algorithm are known in few cases.

On the other hand finding Maximum Weight Matching (MWM) in a weighted complete bipartite graphis a central problem in designing an input-queued switch for high speed internet routers. In this talkwe use max-product to design an iterative and distributed algorithm for finding MWM. Even though theunderlying bipartite graph has many short cycles, we find that surprisingly, the max-product algorithmalways converges to the correct answer. We also show that for a graph of size n, the computational cost ofthe algorithm scales as O(n3), which is the same as the computational cost of the best known algorithm.Finally, we establish the precise relation between the max-product algorithm and the celebrated auctionalgorithm proposed by Bertsekas. This suggests possible connections between dual algorithm and max-product algorithm for discrete optimization problems.

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How to avoid doing research “10 years ahead of time”

Amin FiroozshahianVahideh HosseinkehAshish GoelBalaji Prabhakar, Stanford University, balaji@stanford.edu

A thread of research dating to the mid-90s is concerned with the emulation of an output-queued (OQ) switchwith a combined input- and output-queued (CIOQ) switch. It showed that in order to emulate an N portOQ switch under adversarial inputs, it is necessary and sufficient for the CIOQ switch to run at a speedupof 2− (1/N ). This result validated a belief that was prevalent among implementors and researchers.

However, the Stable Marriage algorithms developed in this work were considered too complicated to imple-ment, and implementors contented themselves with the knowledge that 2 is “the right amount of speedup”and ignored the algorithms.

Recently, a combination of factors has made Stable Marriage algorithms attractive for implementation.This talk narrates the story and gives the technical details.

Packet Election For Fair Scheduling

S. Jagabathula, Massachusetts Institute of Technology, jskanth@mit.eduV. Doshi, Massachusetts Institute of Technology, vdoshi@mit.eduD. Shah, Massachusetts Institute of Technology, devavrat@mit.edu

In our previous work (2006), we conjectured that Max-Weight Max-Size Matching (MWMSM) schedulingalgorithm is both throughput and (average) delay optimal for an input-queued switch based on insightsobtained from the heavy traffic analysis of a class of Max-Weight Matching scheduling algorithms. In thistalk, we (almost) prove this conjecture: we show that MWMSM algorithm is a constant factor approxima-tion (with respect to average queue-size) of an optimal scheduling algorithm for any (time) i.i.d. arrivalprocess.

Approximate Dynamic Programming for Switch Scheduling

Ciamac C. Moallemi, Stanford University, USA, ciamac@stanford.eduSunil Kumar, Stanford University, USA, skumar@stanford.eduBenjamin Van Roy, Stanford University, USA, bvr@stanford.edu

We develop an approach based on temporal-difference learning to address scheduling problems in complexqueueing networks such as those arising in service, communication, and manufacturing systems. One novelfeature is the selection of basis functions, which is motivated by the gross behavior of the system in theasymptotic regime of heavy traffic. Another is the use of polytopic structure to efficiently identify desiredactions from an intractable set of alternatives. This metholodogy is applied to input-queued crossbar switchmodels with up to hundreds of queues and quadrillions of alternative actions. It is shown yield schedulingpolicies outperforming a heuristic recently shown to have certain optimality properties in the heavy trafficscale. We also extend the approach to a setting where aspects of the queueing network are not modeledand we must rely instead on empirical data. This data-driven approach is useful, for example, when thestatistical structure of arrivals is poorly understood but historical data traces are available.

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Tuesday 10:30am - 12:00pm

Session TB01 - Finance and Ruin, Chair: Collamore in CZ 4

A Dirichlet bridge sampling of the variance gamma process: Pricing path-dependentoptions

Laura Ballotta, Faculty of Actuarial Science and Insurance, Cass Business School London,L.Ballotta@city.ac.ukDimitrina Dimitrova, Faculty of Actuarial Science and Insurance, Cass Business School London,D.Dimitrova@city.ac.ukVladimir Kaishev, Faculty of Actuarial Science and Insurance, Cass Business School London,V.Kaishev@city.ac.uk

A new method for sampling a Variance Gamma (VG) process, called Dirichlet bridge sampling, is proposedand is shown to represent a generalization of the known gamma bridge sampling method. Dirichlet bridgesampling allows immediate generation of the entire trajectory of the process over a certain period of time,avoiding sequential sampling at arbitrary intermediate points in time as it is the case, instead, with thegamma bridge method. We explore the efficiency of the proposed simulation methodology and apply somevariance reduction techniques such as stratification. The proposed method is then tested against approachesalready existing in the literature, such as sequential sampling and bridge sampling of the VG process. Inparticular, we price path-dependent options such as Asian options, lookback options and barrier options.

Applications of series representations for Lévy processes

Z. Michna, Wroclaw University of Economics, Poland, zbigniew.michna@ae.wroc.pl

We consider α-stable Lévy processes and gamma Lévy process. We use a series representation of the α-stable Lévy process to condition on the largest jump. Then we get a certain decomposition into a simpleprocess and a Lévy process which has some finite exponential moments. We show that this Lévy processconverges to the α-stable Lévy process uniformly on compact sets with probability one. In other limitwe can approximate this process by Brownian motion. We apply this decomposition of an α-stable Lévyprocess to study asymptotic behavior of the distribution of supremum for certain α-stable processes.

Using the same technique as for α-stable Lévy processes we find an asymptotic behavior of finite time ruinprobability for a gamma Lévy process.

Stochastic recurrence equations and ruin in a general Markovian environment

J.F. Collamore, University of Copenhagen, Denmark, collamore@math.ku.dkH. Nyrhinen, University of Helsinki, harri.nyrhine@cc.helsinki.fi

We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochasticeconomic environment. Namely, we study the maximal behavior for the system of equations

Wn = B1 + A1 B2 + · · · + (A1 · · · An−1)Bn, n ≥ 1,

where (An, Bn) is a function of an underlying Markov chain in general state space; and show

P{

supn

Wn > u}∼ Cu−R as u →∞. (11.1)

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The left-hand side of (1) may be equated to the probability of ruin of an insurance company, given an initialcapital of u, with Markov-dependent stochastic investment returns.

In addition to (1), we also mention some related results for the study of perpetuities, and for the extremes ofthe ARCH(1) and GARCH(1,1) financial time series models. Our results build upon work of Goldie (Ann.Appl. Prob., 1991), who has developed tail asymptotics for independent sequences of random variablessubject to a random recurrence equation. We adopt a general approach based on the theory Harris recur-rent Markov chains and the associated theory of nonnegative operators, and meanwhile develop certainrecurrence properties for these operators.

Managing Inventory of Items with Replacement Warranty

Wei Huang, SAS Institute Inc, Cary, NC 27511Vidyadhar Kulkarni, Department of Statistics and Operations Research University of North Carolina,Chapel Hill, NC 27599Jayashankar M. Swaminathan, The Kenan-Flagler Business School University of North Carolina,Chapel Hill, NC 27599

In this paper we study a firm that faces demand from two sources: demand for new products and demand toreplace failed items under warranty. We model this setting as a multi-period single product inventory prob-lem where the new demand in different periods are independent and the demand for replacing failed itemsis dependent on the number and age of items under warranty. We consider backlogging and emergencysupply cases, and study both discounted cost and average cost cases. We prove the optimality of the w-dependent base stock ordering policy where the base stock level is a function of w, the vector representingthe number of items at different ages currently under warranty. For the special case, where the demand fornew products is stationary, we prove the optimality of a stationary w-dependent base stock policy for thefinite horizon discounted and the infinite horizon discounted and average cost cases. In our computationalstudy, we find that an optimal integrated policy can lead to 37% average improvement in expected costswhen compared to a policy that neglects warranty repairs.

Session TB02 - Random Partitions and Partition-valued Processes,Chair: Gnedin in CZ 5

Small time asymptotics of Lambda-coalescents

J. Berestycki, Université de Provence, jberest@cmi.univ-mrs.frN. Berestycki, University of British Columbia, nberestycki@math.ubc.caV. Limic, Université de Provence, vlada@cmi.univ-mrs.fr

Bertoin and Le Gall have recently remarked that there is a simple bijective transformation that maps themeasure 3 of a 3-coalescent onto the branching mechanism 9 of a continuum branching process suchthat Schweinsberg’s criterion on3 for coming down from infinity in the coalescence corresponds exactly toGrey’s criterion for extinction in finite time for the branching process. However no probabilistic explanationof this fact was known.

By shedding more light on the relations (which have been the focus of much attention lately) between 3-coalescent, continuous-state branching processes and so-called generalized Flemming-Viot super-processes,we are able to give such an explanation, and along the way we obtain an almost sure asymptotic formulafor the number of blocks at small-times, which until now was known only in very specific cases.

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TB03

Gaussian fluctuations for Plancherel partitions

L. Bogachev, University of Leeds, UK, bogachev@maths.leeds.ac.uk

The talk is based on joint work with Z. Su (Zhejiang University). We obtain a central limit theorem forYoung diagrams under the Plancherel measure in the bulk of the “spectrum” of partitions of large integers n.More specifically, we prove that, under a suitable normalization (growing like

√log n ), the corresponding

random process converges, in the FDD sense, to a Gaussian process with independent values. We alsodiscuss the link with an earlier result by S. Kerov (1993) on the convergence to a generalized Gaussianprocess. The proof is based on the Poissonization of the Plancherel measure and an application of a generalcentral limit theorem for determinantal point processes.

Ancestral processes in population genetics - exchangeable coalescents

Martin Möhle, University of Düsseldorf, Germany, moehle@math.uni-duesseldorf.de

Coalescents are partition-valued Markovian processes with a block-merging transition mechanism. Duringeach transition blocks merge into single blocks. These processes can arise in the limit as the total populationsize tends to infinity when studying the genealogy of a sample of individuals from an exchangeable finitepopulation.

Coalescents play an important role in the theory of coagulation and fragmentation. They have remarkablerelations to measure-valued processes, random recursive trees and other combinatorial objects such aspartition structures and sampling formulas.

This talk introduces into the theory of coalescents and reviews recent mathematical research and biologicalapplications around these stochastic processes.

Random partitions and the site frequency spectrum in population genetics

Nathanaël Berestycki, University of British Columbia, Canada, nberestycki@math.ubc.caJulien Berestycki, Université de Provence, France, jberest@cmi.univ-mrs.frJason Schweinsberg, University of California at San Diego, USA, jschwein@math.ucsd.edu

Suppose we take a sample of size n from a population and follow the ancestral lines backwards in timeuntil the most recent common ancestor. Under standard assumptions, this process can be approximated byKingman’s coalescent, in which two lineages merge at rate one. Mutations that have occurred since thetime of the most recent common ancestor will lead to segregating sites, which are positions in the DNAat which not all individuals in the sample are the same. If we denote by Mk the number of mutations thataffect k individuals in the sample, then the sequence M1, . . . ,Mn−1 is called the site frequency spectrum.We will explain how a result on random partitions of Karlin (1967) allows us to understand how the sitefrequency spectrum would be affected if some individuals in the population have unusually large numbersof offspring.

Session TB03 - Gradient Estimation, Chair: Heidergott in CZ 10

Policy Gradient Methods in Machine Learning

Jan Peters, University of Southern California, Los Angeles, USA, jrpeters@usc.eduEvangelos Theodorou, University of Southern California, Los Angeles, USA, etheodor@usc.eduStefan Schaal, University of Southern California, Los Angeles, USA, sschaal@usc.edu

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We present an in-depth survey of policy gradient methods as they are used in the machine learning com-munity for optimizing parameterized, stochastic control policies in Markovian systems with respect to theexpected reward. Despite having been developed separately in the reinforcement learning literature, policygradient methods employ likelihood ratio gradient estimators as also suggested in the stochastic simulationoptimization community. It is well-known that this approach to policy gradient estimation traditionallysuffers from three drawbacks, i.e., large variance, a strong dependence on baseline functions and a ineffi-cient gradient descent. In this talk, we will present a series of recent results which tackles each of theseproblems. The variance of the gradient estimation can be reduced significantly through recently introducedtechniques such as optimal baselines, compatible function approximations and all-action gradients. How-ever, as even the analytically obtainable policy gradients perform unnaturally slow, it required the step from‘vanilla’ policy gradient methods towards natural policy gradients in order to overcome the inefficiency ofthe gradient descent. This development resulted into the Natural Actor-Critic architecture which can beshown to be very efficient in application to motor primitive learning for robotics.

Strong Properties of Weak Derivatives

Haralambie Leahu, Vrije Universiteit, The Netherlands, hleahu@feweb.vu.nlBernd Heidergott, Vrije Universiteit, The Netherlands, bheidergott@feweb.vu.nlArie Hordijk, Leiden University, The Netherlands, hordijk@math.leidenuniv.nl

In this talk we provide bounds on cost functions over a finite collection of independent random variables.Our analysis is based on a study of the relation between weak differentiability and Fréchet differentiability.By applying a fundamental result from the theory of Banach spaces, we show that weak differentiabilityimplies norm Lipschitz continuity. This result leads to bounds on the sensitivity of finite products ofprobability measures in norm sense. We illustrate the application of these results by studying parameter-free perturbation analysis. Our main application is the expected transient waiting time in a G/G/1 queue.

Gradient Estimation for a Multi-Component Maintenance System Using Measure-Valued Differentiation

Taoying Yuan, Vrije Universiteit, The Netherlands, tyuan@feweb.vu.nlBernd Heidergott, Vrije Universiteit, The Netherlands, bheidergott@feweb.vu.nl

We consider a multi-component maintenance system with an age replacement policy: when a componentfails, it is replaced immediately, and all components older than a threshold age θ are preventively replaced.With each maintenance action, costs are associated. We derive three unbiased gradient estimators basedon the measure-valued differentiation approach for the derivative of the long-run average cost with respectto θ : the phantom estimator, the randomized phantom estimator and the combined phantom estimator.The combined estimator is the first unbiased gradient estimator that is feasible for systems with a largenumber of components. In addition, a new technique is introduced that allows to control the numberof phantoms to be simulated in parallel for obtaining the gradient estimator, and a novel approach forefficiently implementing the phantom generation is presented.

Measure-Valued Differentiation for Random Horizon Experiments

B. Heidergott, Vrije Universiteit, The Netherlands, bheidergott@feweb.vu.nlA. Hordijk, Leiden University, The Netherlands, hordijk@math.leidenuniv.nl

We consider the expected accumulated costs over a cycle of a phase-homogeneous random walk. For thismodel we establish sufficient conditions for the existence of the derivative of the cycle cost and we establishan unbiased gradient estimator. The main stability condition for our analysis is that the expected cycle costs

98

TB04

are finite. We thereby improve the results known in the literature so far, where usually finiteness of highermoments of the cycle length is assumed in order to establish unbiasedness of a particular gradient estimator.

Our analysis provides minimal conditions for unbiasedness of a gradient estimator for the derivative of theaccumulated costs. We will illustrate this with an example where we can establish the existence of thederivative of the expected accumulated costs even though the second moment of the cycle times fails toexist.

Session TB04 - Asymptotic Analysis 2, Chair: Xia in CZ 11

Are end-to-end acknowledgements causing power law delays in large multi-hop net-works?

P.R. Jelenkovic, Dept. of Electrical Engineering, Columbia University, New York City, USA,predrag@ee.columbia.eduJ. Tan, Dept. of Electrical Engineering, Columbia University, New York City, USA,jiantan@ee.columbia.edu

An end-user that is surfing the Web downloads documents from potentially diverse physical locations,implying that each of the downloads is likely conducted over a new routing path with a different numberof hops. Furthermore, the Web downloads are primarily carried over the TCP/IP networks that employthe end-to-end acknowledgements (ACKs), meaning that every time a packet is lost it needs to be resendfrom the source, which may be highly inefficient on long multi-hop routes that are typically encountered inwireless ad hoc networks. To capture the effects of the variability of routing paths and end-to-end ACKs,we study the delay in a sequence of random number, albeit fixed over each session, of tandem queues withfinite buffers and end-to-end ACKs. We show that this model may result in power law delays even if all theparameters of the model are exponentially bounded.

Stochastic ordering for regulated flows and the Better-than-Poisson Property

Fabrice M. Guillemin, France Telecon R & D, Lannion,France, fabrice.guillemin@orange-ftgroup.comRavi R. Mazumdar, University of Waterloo, Waterloo, Canada, mazum@ece.uwaterloo.caCatherine P. Rosenberg, University of Waterloo, Waterloo, Canada, cath@ece.uwaterloo.caYu Ying, Juniper Networks, Cupertino, Ca, USA, yu.ying06@gmail.com

Regulated flows are very useful and common models in the context of traffic control in networks. The mostcommon model is the leaky bucket or (σ, ρ) regulated flows due first analyzed by Cruz. In the deterministicor worst-case setting an extensive set of results based on the (min, plus) algebra, termed Network calculus,has been developed. From a practical standpoint, such an approach leads to very conservative designsand hence there is a need for developing a methodology to analyze the effects on queueing performance(particularly the end-to-end mean delay) that can exploit statistical independence between flows. One suchapproach is the Negligible Jitter (NJ) conjecture due to Roberts et al. that states that flows that are Better-than-Poisson (BTP) remain BTP within a network after multiplexing.

In this talk we present some partial answers to this question. Specifically we show that leaky-bucket reg-ulated flows satisfy the Better-than-Poisson property at the ingress. To do so we construct the appropriatestochastic ordering since the strong ordering does not hold. We then show that when a large number of inde-pendent flows are multiplexed, the regulated flows retain their initial regulation parameters with very highprobability within the network. In particular, for concentrator networks, one can show that the end-to-endmean delay can be bounded by using link independence coupled with the BTP property.

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Sharpness: a sufficient and necessary condition for throughput scalability

A. Chaintreau, Thomson, France, augustin.chaintreau@thomson.net

This paper describes a new class of distributed discrete-event systems described as infinite collections oftasks, each requiring a random time, that should follow some precedence rules. Our main assumptionsis that the tasks are organized along an euclidean lattice, and such that the precedence rules are invariantby translation. This model includes a broad class of regular communication networks such as tandem ofqueues, which may implement some local synchronization between servers.

Based on percolation results, we prove the following throughput scalability result: The subset of completedtasks grows with time according to a positive speed in any direction. The result is shown to derive fromonly two conditions: (1) a criterion called “sharpness” satisfied by the precedence relation and (2) a generalmoment condition on each task completion time, which only depends on the dimension of the lattice.As an application, the results we present offer general guidelines to design scalable reliable distributedapplications on overlay networks.

Scalability of Fork/Join Queueing Networks with Blocking

Cathy H. Xia, Zhen Liu, IBM T.J. Watson Research Center, Hawthorne, NY, USA,{cathyx,zhenl}@us.ibm.comDon Towsley, Dept. of Computer Science, University of Massachusetts, Amherst, MA, USA,towsley@cs.umass.eduMarc Lelarge, INRIA & Ecole Normale Supérieure, Paris, France, Marc.Lelarge@ens.fr

This paper investigates how the throughput of a general fork/join queueing network with blocking behavesas the number of nodes increases to infinity while the processing speed and buffer space of each nodestay unchanged. The problem is motivated by applications arising from distributed systems and computernetworks. One example is large-scale distributed stream processing systems where TCP is used as thetransport protocol for data transfer in between processing components. Other examples include reliablemulticast in overlay networks, and reliable data transfer in ad hoc networks. Using an analytical approach,the paper establishes bounds on the asymptotic throughput of such a network. For a subclass of networkswhich are balanced, we obtain sufficient conditions under which the network stays scalable in the sense thatthe throughput is lower bounded by a positive constant as the network size increases. Necessary conditionsof throughput scalability are derived for general networks. The special class of series-parallel networks isthen studied in greater detail, where the asymptotic behavior of the throughput is characterized.

Session TB05 - Randomized Algorithms and Resource Control Schemes,Chair: Borst in CZ 12

Optimal peer-to-peer broadcasting schemes

Laurent Massoulié, Thomson, Paris Research Lab, laurent.massoulie@thomson.netAndy Twigg

This talk focuses on the problem of relaying a live stream of information from a source to nodes of apeer-to-peer system. Scheduling rules based on local information are proposed, and shown to achieve thetheoretical broadcasting limit. A classical characterization, due to Edmonds, of the number of spanningtrees that can be packed in a graph is retrieved as a corollary. We next discuss a more general scenario whereseveral live streams of information are to be disseminated concurrently. Local rules are again proposed andshown to achieve the theoretical limit. As a byproduct we obtain an extension of Edmonds’ theorem onpacking of spanning trees to multiple commodities.

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TB05

Mean field limits of particle systems in varying environments, Application to mod-eling random multi-access algorithms

C. Bordenave, Department of Electrical Engineering, University of California, Berkeley, US.bordenav@eecs.berkeley.eduD. McDonald, Department of Mathematics and Statistics, University of Ottawa, Canada.dmdsg@uottawa.caA. Proutière, KTH, Radio Communication Systems, Sweden. alexandre.proutiere@radio.kth.se

We study an interacting particle system whose dynamics depends on an interacting random environment.As the number of particles grows large, the transition rate of the particles slows down (perhaps becausethey share a common resource of fixed capacity). The transition rate of a particle is determined by its state,by the empirical distribution of all the particles and by a rapidly varying environment. The transitionsof the environment are determined by the empirical distribution of the particles. We prove the propaga-tion of chaos on the path space of the particles and establish that the limiting trajectory of the empiricalmeasure of the states of the particles satisfies a deterministic differential equation. This deterministic dif-ferential equation involves the time averages of the environment process. We apply our results to analyzethe performance of communication networks where users access some resources using random distributedmulti-access algorithms. For these networks, we show that the environment process can be approximatedby a process describing the number of clients in a certain loss network, which allows us provide simple andexplicit expressions of the network performance.

Metastability of CDMA cellular systems

N. Antunes, Universidade do Algarve, Faculdade de Ciências e Tecnologia, Faro, Portugal,nantunes@ualg.ptC. Fricker, INRIA, domaine de Voluceau, 78153 Le Chesnay Cedex, France, Christine.Fricker@inria.frPh. Robert, INRIA, domaine de Voluceau, 78153 Le Chesnay Cedex, France, Philippe.Robert@inria.frD. Tibi, Université Paris 7, UMR 7599, 2 Place Jussieu, 75251 Paris Cedex 05, France,Danielle.Tibi@math.jussieu.fr

In this it is shown that the coexistence of a variety of different traffics in third generation cellular networksmay lead to a very undesirable behavior of the whole network: a metastability property. When this propertyholds, the state of the network fluctuates on a very long time scale between different set of states. Theselong oscillations of the network make impossible to predict the average performances of some of the keycharacteristics of the connections, such as the handoff blocking rate or the probability of call blocking. Asa consequence, the quality of service provided by such a network can be guaranteed only by, sometimespoor, lower bounds. Experiments of networks with this behavior are presented and the analysis of a corre-sponding simplified mathematical model is developped. The practical implications in the design of radioresource management for CDMA cellular networks are discussed.

Ranking and suggesting tags in collaborative tagging applications

Milan Vojnovic, Microsoft Research, United Kingdom, milanv@microsoft.com

We consider collaborative tagging systems where users can attach tags to information objects. Such sys-tems are widely used to add keywords meta-data to photos, videos, or web pages (social bookmarkingapplications). The meta-data is then used by information retrieval mechanism to provide accurate queryanswers. To that end, the goal of collaborative tagging systems is to quickly discover the true ranking of atag for an information object with respect to a given ranking criteria. In this paper, we consider the pop-ularity rank as a ranking criteria. Many collaborative tagging systems help users tagging of an object bymaking suggestions based on the tagging history of an information object. The problem with making tag

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suggestions is that they may reinforce some tags and the system fails to discover the true popularity rankof a tag for a given information object. To investigate this issue, we propose and study several algorithmsfor ranking and suggesting tags in collaborative tagging systems where we focus on the following designobjectives: (a) learn true popularity rank, (b) make relevant suggestions, and (c) learn fast. We find thatsimple incremental updates of the suggestion set, which besides suggestion set size require no configurationconstants, offer good performance. Performance is evaluated by analysis and numerical results for whichwe used a dataset of complete tagging histories of urls that we crawled from a popular social bookmarkingweb service over a month period.

Joint work with James Cruise, Dinan Gunawardena, and Peter Marbach.

Session TB06 - Many-server Models 2, Chair: Van Leeuwaarden /Zwart in CZ 13

Corrected diffusion approximations for many-server queues I

A.J.E.M. Janssen, Philips Research, The Netherlands, a.j.e.m.janssen@philips.comJohan S.H. van Leeuwaarden, Eindhoven University of Technology, The Netherlands,j.s.h.v.leeuwaarden@tue.nlBert Zwart, Georgia Institute of Technology, USA, bertzwart@gatech.nl

Queues with many servers are relevant in the performance analysis of large call centers. Since the com-plexity of such queueing models grows as the number of servers grows large, it is desirable to have scalableperformance analysis tools available. A popular tool that offers such scalability is the Halfin-Whitt approx-imation of the no-wait probability in the M/M/s queue.

The goal of this work is to improve the Halfin-Whitt approximation of the delay probability such that theresulting formula remains accurate when the number of servers is not necessarily large, and/or when thesystem is not necessarily in heavy-traffic. In addition, we give analytic estimates of the accuracy of thecelebrated square root staffing rule.

This talk will be the first in a series of two. In the second talk, we present similar results for the M/D/squeue.

Corrected diffusion approximations for many-server queues II

A.J.E.M. Janssen, Philips Research, The Netherlands, a.j.e.m.janssen@philips.comJohan S.H. van Leeuwaarden, Eindhoven University of Technology, The Netherlands,j.s.h.v.leeuwaarden@tue.nlBert Zwart, Georgia Institute of Technology, USA, bertzwart@gatech.nl

To obtain insight in the quality of heavy-traffic approximations for queues with many servers, we considerthe steady-state number of waiting customers in an M/D/s queue as s → ∞. In the Halfin-Whitt regime,it is well known that this random variable converges to the supremum of a Gaussian random walk. Wedevelop methods that yield accurate results in terms of series expansions and inequalities for the probabilityof an empty queue, and the mean and variance of the queue length distribution. The main idea is to viewthe M/D/s queue through the prism of the Gaussian random walk: as for the standard Gaussian randomwalk, we provide a scalable series expansion for the emptiness probability involving terms that include theRiemann zeta function.

This talk will be the second in a series of two. In the first talk, we present similar results for the M/M/squeue.

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A new view on the G/GI/N queue in Halfin-Whitt regime

J.E Reed, New York University, United States, je-reed@isye.gatech.edu

We study the G/G I/N queue from the perspective of the servers in the system. This point-of-view allowsus to obtain results for the G/G I/N queue in the Halfin-Whitt regime which were previously proventhrough different means. Our main results are to obtain weak limits for both the fluid and diffusion scaledqueue length process.

Moment Equivalences Between Multiserver and Single Server Queues

A. Scheller-Wolf, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA,awolf@andrew.cmu.eduR. Vesilo, Macquarie University, New South Wales, Australia, rein@elec.mq.edu.au

Generalizing techniques of Foss and Korshunov, we relate the delay-moment behavior of FIFO multiserverqueues to the delay-moment behavior of sets of independent FIFO single server queues. This provides (i)alternate derivations of known sufficient conditions for stationary multiserver delay moments; (ii) moregeneral necessary conditions, and; (iii) a structural interpretation underlying these moment conditions. Assuch we provide intuition about why multi-server queues behave as they do.

Session TB07 - Stochastic Control and PDE, Chair: Atar in CZ 14

Optimal Stopping and Free Boundary Characterizations for some Brownian Con-trol Problems

Amarjit Budhiraja, Department of Statistics, University of North Carolina, budhiraj@email.unc.eduKevin J. Ross, Department of Statistics, Stanford University, kjross@stanford.edu

We consider a two dimensional singular control problem with state constraints. By studying a relatedoptimal stopping problem we establish that the value function is C1. We then characterize an optimallycontrolled process as a diffusion reflected at an appropriate free boundary. This resolves a conjecture ofMartins-Shreve-Soner(1996) on the form of an optimal control for this class of control problems.

A control-theoretic analysis of fast-sweeping methods for static Hamilton-Jacobiequations

Paul Dupuis, Brown University, USA, dupuis@dam.brown.edu

The term “fast sweeping” refers to a set of numerical methods for static Hamilton-Jacobi equations, withthe property that the iterative scheme that is used to solve for the numerical approximation converges afteressentially a fixed number of sweeps through the numerical grid. In this talk we will discuss qualitativeproperties of these methods that can be proved using a control representation for the iterative scheme.

HJB equations, no arbitrage, and generalized Brownian networks

R. Atar, Technion, Israel, atar@ee.technion.ac.ilA. Budhiraja, University of North Carolina at Chapel Hill, NC, budhiraj@email.unc.eduR.J. Williams, University of California at San Diego, California, williams@math.ucsd.edu

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Let X (t) = X (0)+ B(t)+GU (t), t ≥ 0, where B is an n-dimensional Brownian motion, G a fixed matrix,and U a ‘singular control’, i.e., an adapted process that is locally of bounded variation and keeps X in agiven domain in Rn at all times t ≥ 0. Under some assumptions, a Hamilton-Jacobi-Bellman equationcharacterizes the value of a control problem in which a cost is to be minimized over all such controls U.The existence and uniqueness of solutions to this equation are shown to be connected to the intuitive ‘noarbitrage’ condition, that states that an instantaneous control action that does not affect the process X doesnot result in any immediate reduction in cost. Our results apply, in particular, to the diffusion controlproblems derived by Harrison and Williams (2005) for Generalized Brownian networks.

A Representation for the Characteristic Functional of Banach Valued Stable Ran-dom Measures

S. Mahmoodi, Isfahan University of Technology, Iran, mahmoodi@cc.iut.ac.irA.R. Soltani, Kuwait University, State of Kuwait, soltani@kuc01.kuniv.edu.kw

A Banach space (X: real separable) valued symmetric α stable random measure 8 with independentincrements on a measurable space (F,F ) is considered. A pair (k, µ), called here a control pair, forwhich k : X × F → R+, µ a positive measure on (F,F ), is introduced. It is proved that the law of 8is governed by a control pair; and every control pair will induce such 8. Moreover k is unique for a givenµ. Our derivations are based on the Generalized Bochner Theorem and the Radon- Nikodym Theorem forvector measures.

Session TB08 - Fluid Models, Chair: Scheinhardt in CZ 15

Analysis of a resource sharing policy for computer systems using fluid queues

N. Gautam, Texas A&M University, USA, gautam@tamu.eduS.R. Mahabhashyam, Oracle Corporation, USA, mahabhashyam@yahoo.com

A system consisting of two buffers, each with independent fluid sources is considered in this research. Theoutput capacity for the two buffers depend on the workload of only one of the buffers which is observable.A threshold-based policy is considered to dynamically assign output capacities for both buffers. Marginalworkload distributions for the two buffers need to be evaluated for this policy. The key contribution ofthis work is the performance analysis to derive the workload distribution in the two buffers. For the first(observable) buffer we use a spectral analysis by writing out the partial differential equations for the buffercontent. For the second (unobservable) buffer we model the random environment using a semi-Markovprocess and use a compensating source for performance analysis. In addition, the work also provides someguidelines to choose the output capacities for the two buffers as well as a mathematical program to deter-mine an optimal threshold to dynamically switch between output capacities. Further, various applicationsof such systems to computer-communication networks are discussed.

A feedback fluid queue with two congestion control thresholds

R. Malhotra, Bell Labs Europe, Alcatel-Lucent, Capitool 5, 7521 PL Enschede, The Netherlands,rimalhotra@alcatel-lucent.comM.R.H. Mandjes, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, TheNetherlands, mmandjes@science.uva.nlW.R.W. Scheinhardt, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands,w.r.w.scheinhardt@utwente.nlJ.L. van den Berg, TNO and University of Twente, P. O. Box 5050, 2600 GB Delft, The Netherlands, andj.l.vandenberg@tno.nl

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We present and analyze a fluid queue with a feedback-based traffic rate adaptation scheme which uses twothresholds. Previous work on feedback fluid queues has modeled queues with a single threshold to signalthe onset and end of congestion to the rate adapting traffic source. A major drawback of such systems isthat they impose significant bandwidth overhead due to signals sent to the source every time the thresholdis crossed. In addition, the single threshold makes it difficult to find a good trade-off between throughputand delay performance of the system. In our model with two thresholds, the higher threshold B1 is usedto signal the beginning of congestion and can be used to maximize throughput performance. The lowerthreshold B2 is used to signal the end of congestion and can be used to minimize delay. The differencebetween the two thresholds helps to control the amount of feedback signals sent to the traffic source. In ourmodel a fluid source is active which can behave like either of two Markov fluid processes. The first appliesas long as the upper threshold B1 has not been hit from below. As soon as that happens, the traffic sourceadapts and switches to the second process, until B2 is hit from above. We present the exact analysis of thestationary distribution of the buffer occupancy and performance measures. Important novelty of our modelcompared to earlier feedback models is that the traffic source may behave in two different ways betweenthe two thresholds, depending on the past of the buffer process.

On shot noise fluid queues and infinite-server queues with batch arrivals: time-dependent behavior

W.R.W. Scheinhardt, University of Twente, The Netherlands, w.r.w.scheinhardt@utwente.nlR.J. Boucherie, University of Twente, The Netherlands, r.j.boucherie@utwente.nlW.F. de Graaf, The Netherlands, W.F.deGraaf@phys.uu.nl

We consider a shot noise model, in which a fluid reservoir receives quantities of fluid (with some generalprobability distribution) that arrive according to a Poisson process. Moreover, the reservoir is continu-ously drained at a rate that is proportional to the current content of the reservoir. We are interested in thetime-dependent behavior of the content of the reservoir, where we also allow the arrival rate of the Pois-son process to depend on time. We derive the Kolmogorov forward (partial differential) equation for theLaplace-Stieltjes transform of the content at time t , and solve it. Interestingly, the model may be seen asthe limiting case of a certain sequence of infinite-server queues with batch arrivals. We show why this factholds, and how it enables us to find the solution also from the (known) generating function of the numberof jobs in the infinite-server queue. Some examples show how the moments of the content process canbehave as a function of time.

Tandem Queue with Server Slow-down

D.I. Miretskiy, University of Twente, The Netherlands, d.miretskiy@utwente.nlW.R.W. Scheinhardt, University of Twente, The Netherlands, w.r.w.scheinhardt@utwente.nlM.R.H. Mandjes, University of Amsterdam, The Netherlands, mmandjes@science.uva.nl

We study how rare events happen in the standard two-node tandem Jackson queue and in a generalization,the socalled slow-down network. In the latter model the service rate of the first server depends on thenumber of jobs in the second buffer: the first server slows down if the amount of jobs in the second bufferis above some threshold and returns to its normal speed when the number of jobs in the second bufferis below the threshold. This property protects the second buffer from frequent overflow. We provide the(non-trivial) stability condition for this model. Considering the joint queue length process for both modelswe construct the most probable path to some rarely visited subset of the state space, starting from anyinitial position, when the system is scaled with some large parameter. Based on this knowledge we con-struct state-dependent Importance Sampling schemes, which provide fast and precise results for estimatingprobabilities of rare events of interest (for example, overflows of a single node).

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Session TB09 - Supply Chains, Chair: Ettl / De Kok in CZ 16

Centralized and decentralized vendor-managed inventory optimization under dif-ferent levels of information sharing

M. Ettl, IBM Thomas J. Watson Research Center, USA, msettl@us.ibm.comY. Lu, IBM Thomas J. Watson Research Center, USA, yingdong@us.ibm.comM.S. Squillante, IBM Thomas J. Watson Research Center, USA, mss@watson.ibm.com

In this paper we study the trade-offs between serviceability, liability, and inventory in an outsourced supplychain with vendor-managed inventory (VMI). We formulate different stochastic optimization problemsto characterize the interactions between different supply chain echelons through their serviceability andliability arrangements under different types of control and information sharing assumptions. Structuraland algorithmic analysis of the optimization problems enables us to reveal the connections between theinterplay of serviceability and liability and the control of inventory at each echelon. We also presentextensive numerical experiments to illustrate and quantify our theoretical findings.

Optimal Policies for Joint Replenishment and Substitution

H. Xu, School of Management, Huazhong University of Science and Technology, Wuhan, ChinaD.D. Yao, IEOR Department, Columbia University, New York, NY 10027, USA, yao@columbia.eduS. Zheng, Department of Information and Systems Management, Hong Kong University of Science andTechnology, Clearwater Bay, Kowloon, Hong Kong,imzheng@ust.hk

We study an inventory model that allows substitution among products. Replenishment orders of productsmust be made at the beginning of the selling season. The products are then used to supply demand for theentire season of N periods. When a product is out of stock, the retailer may offer a substitution, which mayor may not be accepted by the customer. The decisions are the replenishment quantity for each productat the beginning of the season, and the dynamic substitution rule in each period of the season. We firstestablish the optimality of a threshold type substitution rule for a two-product model, and show that theobjective function is both concave and supermodular in the order quantities. We also consider constraintson the order quantities, and solve a related yield management problem. These results also form the basisfor a heuristic policy for the case of multiple products.

Benefits and risks of price protection in the computer industry

M. Ettl, IBM Thomas J. Watson Research Center, USA, msettl@us.ibm.comP. Huang, IBM Thomas J. Watson Research Center, USA, puhuang@us.ibm.comR. Kapuscinski, Stephen M. Ross School of Business, University of Michigan, Ann Arbor, MI 48109,USA, kapuscin@bus.umich.eduK. Sourirajan, IBM Thomas J. Watson Research Center, USA, ksourira@us.ibm.com

In high-tech industries where technological obsolescence is high, price protection has become a standardelement of contracts between manufacturers and distributors. The main idea of price protection is to creditdistributors for price erosion of goods that were purchased up to l weeks before a price drop but not yetsold to end customers. At the time price change is announced, the manufacturer reviews the distributor’spurchases over the last l weeks and at the distributor’s current inventory. The distributor is given a priceprotection credit equal to the product of the smaller of the two quantities and the price decrease. Whileprice protection was intended to give distributors sufficient incentives to stock enough inventory, empiricalevidence suggests that price protection often leads to overstocking. We analyze acceptable mechanisms toreduce excess inventory caused by price protection.

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Period Order Quantities in Multi-Item Multi-Echelon Inventory Systems

Ton de Kok, Technische Universiteit Eindhoven, The Netherlands, a.g.d.kok@tm.tue.nl

We consider multi-item multi-echelon inventory systems under periodic review echelon order-up-to poli-cies. The review periods of different items may differ. We assume that the review period of an item is amultiple of the review period of each of its parent items. When controlling multi-item multi-echelon sys-tems a key issue is the interaction between parent and child items in case of insufficient stock of child itemsto satisfy orders from their parents. In case of differences in review periods of parents of a particular childitem, this interaction is further complicated by the fact that some parent items may order at some point intime, while other parent items do not. This may lead to unwanted implicit prioritization of items that order"early" in the replenishment cycle of the child item. We propose a child item allocation mechanism thattakes differences in review periods of parent items into account, while allowing for rigorous mathematicalanalysis of the multi-item multi-echelon system.

Tuesday 2:45pm - 4:15pm

Session TC01 - Statistics 2, Chair: Kuhnt in CZ 4

Statistical Analysis of Spatial Patterns of Customer Behaviors

X.X. Bai, IBM China Research Laboratory, baixx@cn.ibm.com, ZhongGuanCun Software Park, 100094,ChinaW.J. Yin, IBM China Research Laboratory, yinwenj@cn.ibm.com, ZhongGuanCun Software Park,100094, ChinaJ. Dong, IBM China Research Laboratory, dongjin@cn.ibm.com, ZhongGuanCun Software Park,100094, China

Deep insights into customer behaviors is critical in marketing activities, e.g. targeting customers for pro-motion events, recognizing risk customers when competitor entries, etc. However, conventional customerbehavior analysis does not take into account of the geographic location and distribution of customers, e.g.typical customer behavior definitions by RFQ (Recency, Frequency and Quantity) can not indicate the factthat people in certain neighborhoods are more likely to shop in a similar way. This work studies the spatialpattern of customer behaviors and can help answer questions, i.e. what kind of customers prefer to buywhat merchandise at which format of stores how far to his residential areas. These spatial behaviors pat-terns are examined in a real china retailer case based on statistics on its member card info and transactiondata.

On the performance of estimators of parameters in autoregressive model underasymmetric loss

Soma Roychowdhury,Indian Institute of Social Welfare and Business Management and Visva-BharatiUniversityDebasis Bhattacharya, Indian Institute of Social Welfare and Business Management and Visva-BharatiUniversity, debasis_us@yahoo.com

In the AR(1) set-up it is well known that for different ranges of the parameters the distributions of differentquantities of interest (for example,the estimators,test statistics etc.)are different.There the equal amount ofover estimation and under estimation may not have the equal consequences.Thus the use of symmetric lossfunction in estimating the unknown parameters is not recommendable.Here we have shown that under

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asymmetric loss function Bayes estimator outperforms the classical point estimators when the operationalprior is sufficiently diffuse.Prediction problem will also be discussed using asymmetric loss funtion.

On nonparametric inference for a class of queuing systems

M.Pawlak, University of Manitoba, Canada, pawlak@ee.umanitoba.ca

Nonparametric methods for identification of the G/G/1 - type queuing system are studied. One of manypossible scenarios can be described as follows: given observations from interarrival and service time distri-butions of the G/G/1 queuing system we wish to estimate a stationary waiting time distribution function.Yet another case of interest is the form of service time density function given observations of the durationsof busy periods. In all these situations one has no any a priori knowledge about the underlying distributionsand nonparametric techniques for estimating the density and distribution functions are particularly suited.Unlike standard nonparametric estimation problems the queuing theory is characterized by the fact that wewish to infer about a distribution using indirect (incomplete) information. This can be viewed an inverseproblem and in this paper we examine a number of such problems resulting in the context of the G/G/1queuing system. Nonparametric estimation techniques for the studied inverse queuing problems are pro-posed and their asymptotic properties are established. In particular, expansions of the bias and variance ofthe estimators are given and asymptotic normality is established. The accuracy of the estimation techniquescritically depends on the degree of the invertibility of a given inverse problem.

An application of algebraic statistics to Bowker’s test for symmetry

S. Kuhnt, Eindhoven University of Technology, The Netherlands, s.kuhnt@tue.nlA. Krampe, University of Dortmund, Germany, krampe@statistik.uni-dortmund.de

Bowker (1948) proposed a commonly used test for axial symmetry in contingency tables. However, theused chi-squared approximation of the distribution of the test statistic might not always be adequate, forexample if the table is sparse. Therefore, modifications of the test statistic have been discussed. Wepropose a test of symmetry using results from algebraic statistics. Diaconis and Sturmfels (1998) combinecomputational commutative algebra and statistics to simulate samples of the conditional distribution ofa discrete exponential family with given sufficient statistic. In particular Groebner bases are utilized forthe construction of the Markov chain. We carry out a simulation study to determine and compare theperformances of the simulation test, the Bowker test and two modifications.

ReferencesBowker, A.H., 1948. A Test for Symmetry in Contingency Tables. Journal American Statistical Association 43, 572–574.Diaconis, P.; Sturmfels, B., 1998. Algebraic Algorithms for Sampling from Conditional Distributions. Annals of Statistics 26, 363–397.Krampe, A.; Kuhnt, S., 2007. Bowker’s Test for Symmetry and Modifications within the Algebraic Framework. To appear inComputational Statistics & Data Analysis.

Session TC02 - Anomalous Diffusion, Chair: Metzler in CZ 5

Anomalous Subdiffusion Revisited

Joseph Klafter, Tel Aviv University, klafter@post.tau.ac.il

An overview is given on anomalous diffusion processes with emphasis on subdiffusion, which is widespreadin many complex systems. We consider in particular the case of subdiffusion which corresponds to acontinuous-time random walk process in which the waiting time for a step is given by a probability distri-bution with a diverging mean value. Relationships to the fractional diffusion or Fokker-Planck equations

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and their generalizations will be discussed. Special attention will be given to the problem of the initialconditions and the corresponding aging processes. We will highlight recent results on linear response of acontinuous-time random walk system to a time-dependent field.

Anomalous Diffusion Leads to Weak Ergodicity Breaking

E. Barkai, Bar-Ilan University Ramat Gan 52900 Israel, barkaie@mail.biu.ac.il

Time averages of physical observables of several stochastic processes describing anomalous diffusion,remain random even in the limit of long measurement time. Examples include blinking quantum dots,anomalous diffusion of mRNA in the cell, and the sub-diffusive continuous time random walk model(CTRW). The basic question for such systems is what statistical theory replaces standard ergodic statisticalmechanics? A theory of weak ergodicity, based on the multi-dimensional arcsine distribution, gives thedistribution of the time averages of physical observables for such systems. When the diffusion turns normal,usual ergodic behavior is found.

Mesoscopic approach to reactions between subdiffusing particles

Igor M. Sokolov, Humboldt University, Berlin, igor.sokolov@physik.hu-berlin.de

We consider a system of subdiffusive particles which react locally according to the mass action law. Weshow that the correct system of reaction-subdiffusion equations for this case is not obtained by adding areaction term to the fractional diffusion equations describing subdiffusion and discuss the particular form ofreaction-subdiffusion equations for the case of irreversible reactions. We first consider the A→B reactionas the simplest example to check the formalism, since in this case the result can be obtained using analternative approach. We then turn to a more complex situation of a stationary A+B→0 reaction underconstant concentrations of reactants on both sides of the system in a slab geometry. We show that thestructure of reactants’ distribution in the case of subdiffusion differs vastly from the one in the normaldiffusion case. The most important differences correspond to the formation of complex reaction profileswith accumulation and depletion zones and to their extreme sensitivity to the boundary conditions.

Fractional Fokker-Planck dynamics: Stochastic representation and computer sim-ulation

Aleksander Weron, Wroclaw University of Technology, aleksander.weron@pwr.wroc.plM. Magdziarz

A computer algorithm for the visualization of sample paths of anomalous diffusion process is developed.It is based on the stochastic representation of the fractional Fokker-Planck equation describing anoma-lous diffusion in a nonconstant potential. We also answer positively a question raised by R. Metzler andJ.Klafter (2000): Can one see a competition between subdiffusion and Levy flights in the framework of thefractional Fokker-Planck dynamics ? Our method of Monte Carlo simulations demonstrates the competi-tion on the level of realizations as well as on the level of probability density functions of the anomalousdiffusion process. The introduced algorithm will surely provide tools for studying many relevant statisticalcharacteristics of the fractional Fokker-Planck dynamics.

Session TC03 - Cross-entropy, Chair: Kroese in CZ 10

Parametric Minimum Cross-Entropy Method

R. Y. Rubinstein, Technion, Israel Institute of Technology, Israel, ierrr01@ie.technion.ac.il

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A. Dolgin, Technion, Israel Institute of Technology, Israel, dandrey@techunix.technion.ac.ilD. P. Kroese, The University of Queensland, Australia, kroese@maths.uq.edu.auP. W. Glynn, Stanford University, USA, glynn@stanford.edu

We introduce a new method, called the parametric minimum cross-entropy (PME) method, for rare eventprobability estimation and counting with a particular emphasis on the satisfiability problem. The PMEmethod is derived from the well known Kullback’s minimum cross-entropy method, called MinxEnt. It isbased on the marginal distributions derived from the optimal joint MinxEnt distribution and is a parametricversion of it. Similar to cross-entropy (CE) method, the PME algorithm first casts the underlying countingproblem into an associated rare-event probability estimation problem, and then finds the optimal parametersof the importance sampling distribution to estimate efficiently the desired quantity. We present supportivenumerical results for counting the number of satisfiability assignments and compare PME with the CEmethod. Our numerical results suggest that the PME method is superior to CE. This is based on the factthat for the satisfiability problem and some other ones involving separable functions the optimal parametersof the importance sampling distribution can be estimated better with the MinxEnt type procedure, ratherthan with indicators.

A Nonparametric Extension of the Cross-Entropy method

Z.I. Botev∗, The University of Queensland, Australia, botev@maths.uq.edu.auT. Taimre, The University of Queensland, Australia, ttaimre@maths.uq.edu.auD.P. Kroese, The University of Queensland, Australia, kroese@maths.uq.edu.au

The Cross-Entropy method has been used extensively as an algorithm for multi-extremal optimization.One key characteristic of the method is the use of a parametric sampling density at each iteration, the pa-rameters of which are updated optimally in some well-defined sense. In this talk we propose the use ofa non-parametric kernel density estimator instead of a parametric sampling density. It is well known thatthe performance of a non-parametric estimator depends crucially on the appropriate choice of the so calledbandwidth parameters. One challenge then with the non-parametric model is to optimally select thesebandwidth parameters in the context of the Cross-Entropy method. We explain the theory behind the op-timal updating of the bandwidth parameters and provide numerical results demonstrating the effectivenessof the method. The numerical examples include rare-event estimation and continuous multi-dimensionaloptimization.

A Generalized Cross-Entropy Method: Discrete State Spaces

T. Taimre, The University of Queensland, Australia, ttaimre@maths.uq.edu.auZ. I. Botev, The University of Queensland, Australia, botev@maths.uq.edu.auD. P. Kroese, The University of Queensland, Australia, kroese@maths.uq.edu.au

Combinatorial optimization and difficult counting problems are encountered in many diverse fields. Recentwork on the cross-entropy method, the minimum cross-entropy method, and its parametric modification,has led to the development of easy-to-implement randomized algorithms for solving such problems. Onemajor feature of these existing techniques is the use of parametric sampling distributions. These lead toanalytic updating rules, but are often unable to adequately approximate the desired target densities. In thistalk, we extend the existing ideas of the cross-entropy method to formulate a generalized cross-entropymethodology, built around the idea of using non-parametric sampling distributions that are, in some sense,optimal. We illustrate this extension for optimization and estimation problems on discrete state spaces,with examples including algorithms for solving MAX-CUT and for estimating the number of solutions ofMAX-CUT.

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Adaptive importance sampling algorithms for rare-event simulations of the infinite-server queue.

A. Ridder, Vrije Universiteit Amsterdam, The Netherlands, aridder@feweb.vu.nl

In this paper we consider importance sampling simulations for the estimation of the transient probabilitythat the infinite-server queue with Poisson arrivals and general service-time distributions reaches high levelswithin a predefined time window. Under the light traffic assumption this is a rare event. Moreover, weassume that the service times are highly variable. In case of exponential servers we have showed that theoptimal path approach leads to a time-dependent, exponentially tilted, asymptotically optimal change ofmeasure. For general service times we have developed an adapted version of this change of measure whichgives good performances of the importance sampling estimators in case of low-variable service times. Forhigh-variable service times we consider a further adjustment of the algorithm by applying cross-entropytechniques.

Session TC04 - Limit Theorems for Queueing Systems with Measure-valued State Descriptors, Chair: Puha in CZ 11

Limit Theorems for Many-Server Queues

K. Ramanan, Carnegie Mellon University, Pittsburgh, kramanan@math.cmu.edu

Many-server queues arise in a variety of applications, including computer data networks and call centers.We uniquely characterise the strong law of large numbers or fluid limit (as the number of servers goesto infinity), of several functionals of this system, including in particular the total workload process. Wealso identify the large time behavior of the asymptotic limit process. The proof involves a measure-valueddescription of the Markovian dynamics and the analysis of a certain (distributional) partial differentialequation with boundary conditions. This is joint work with Haya Kaspi.

Heavy Traffic Analysis for EDF Queues with Reneging

Lukasz Kruk, Maria Curie-Sklodowska University, Lublin Poland, lkruk@hektor.umcs.lublin.plJohn Lehoczky, Carnegie Mellon University, Pittsburgh, PA USA, jpl@stat.cmu.eduKavita Ramanan, Carnegie Mellon University, Pittsburgh, PA USA, kramanan@math.cmu.eduSteven Shreve,Carnegie Mellon University, Pittsburgh, PA USA, shreve@cmu.edu

We present a heavy-traffic analysis of the behavior of a single server queueing system in which customershave deadlines, are served in EDF order, and renege at the moment their deadline elapses, even if serviceis in progress. The performance of the system is measured by the fraction of customer missing theirdeadline. This extends prior work in which customers were served to completion, even if they were late.With reneging the scaled workload process converges to a doubly-reflected Brownian motion process, andthe scaled measure-valued workload process converges to a nonrandom function, which is characterized.Simulation results are presented to compare the performance of the systems with and without reneging.

Central Limit Theorems for a large queuing network with resource pooling

C. Graham, CNRS and École Polytechnique, France, carl@cmapx.polytechnique.fr

Consider a network in which customers choose uniformly L ≥ 2 queues among N and join the shortest. Anatural state descriptor is the tail of the empirical measure of the queue lengths, for which Vvedenskaya,

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Dobrushin and Karpelevitch proved a law of large numbers (LLN) result in equilibrium as N goes toinfinity. The limit tail is hyper-exponentially decreasing, a fact of significant applied interest for such asimple resource pooling scheme.

We provide convergence rates and confidence intervals through a corresponding central limit theorem(CLT). The invariant laws are known only implicitly, and we first prove a functional CLT for transitoryregimes in which the initial conditions satisfy a CLT. We then study the long time behavior for the limitOrnstein-Uhlenbeck process. The functional CLT in equilibrium is eventually derived by proving rigor-ously that we may interchange the long time and the large network limits. An essential tool is a globalexponential stability result for the nonlinear dynamical system governing the LLN limit. It is based onoriginal comparison techniques with linear systems, combined with the study of the spectral decomposi-tion and spectral gap of the latter, performed in well-chosen Hilbert spaces.

The Fluid Limit of a Shortest Remaining Processing Time Queue

D.G. Down, McMaster University, Department of Computing and Software, 1280 Main Street West,Hamilton, ON Canada L8S 4K1, downd@univmail.cis.mcmaster.caH.C. Gromoll, Department of Mathematics, University of Virginia, Charlottesville, VA 22903,gromoll@virginia.eduA.L. Puha, Department of Mathematics, California State University San Marcos, San Marcos, CA92096-0001, apuha@csusm.edu

Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describethe evolution of this system, we use a measure valued process that keeps track of the residual service timesof all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, whichcan be tracked using the frontier process, the largest service time of any job that has ever been in service.We propose a fluid model and present a functional limit theorem justifying it as an approximation of thissystem. The fluid model state descriptor is a measure valued function for which the left edge of the supportis the fluid analog for the frontier process. Under mild assumptions, we prove existence and uniquenessof fluid model solutions. Furthermore, we are able to characterize the left edge of fluid model solutionsas the right continuous inverse of a simple functional of the initial condition, arrival rate, and service timedistribution. When applied to various examples, this characterization reveals the dependence on servicetime distribution of the rate at which the left edge of the fluid model increases.

Session TC05 - Communication Systems 2, Chair: Choi in CZ 12

Performance analysis of small non-uniform packet switches

P. Beekhuizen, Philips Research / Eurandom, The Netherlands, beekhuizen@eurandom.tue.nlJ.A.C. Resing, Eindhoven University of Technology, The Netherlands, resing@win.tue.nl

Due to the increasing number of modules on chips, e.g., memory blocks, processors, etc., traditional busesbetween such modules no longer suffice. As an alternative paradigm, so-called Networks on Chips (NoCs)are emerging, where modules are connected using switches or routers. These networks resemble telecom-munication networks such as LANs or ATM networks. One major difference, however, with LANs andATM networks is the fact that the number of input ports in a NoC switch is usually quite small (say 4 or 5).This especially invalidates the asymptotic analyses commonly performed.

In our talk we analyse the performance of a small non-uniform packet switch. First, we analyse the situationwhere all queues are saturated using a Markov chain approach. This analysis gives rise to a throughputapproximation for the non-saturated case, as well as an approximation of the stability conditions per queue.Second, we approximate the service rate in moderate traffic by a quadratic interpolation between its value

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in light traffic and in saturation. The service rate found is then used as a parameter of a mean waiting timeapproximation.

Performance analysis of sleep mode and idle mode for best effort traffic in IEEE802.16e

Yong Hyun Lee, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, violetgl@korea.ac.krEunju Hwang, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, violetgl@korea.ac.krKyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, kimkjae@korea.ac.krJung Je Son, Global Standards & Research Team, Telecom R&D Center, Samsung Electronics, Suwon,Korea, jungje.son@samsung.comBong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, queue@korea.ac.kr

The IEEE 802.16e is designed to use sleep mode and idle mode for power saving of a mobile station. Weapply the sleep mode operation of power saving class of type 1 when MS is in on-session and the idle modeoperation when MS is in off-session. We obtain the performance measures such as power consumptionratio and mean delay of packets. We show that analytic results match with simulation results quite well.Numerical examples show that the sleep mode and idle mode provide a considerable reduction on energyof mobile station.

Performance Analysis of Power Saving Class of Type 1 for both Downlink and Up-link Traffics in IEEE 802.16e

Sangkyu Baek, Korea University, Korea, myskcom@korea.ac.krJung Je Son, Samsung Electronics, Korea, jungje@samsung.comSeon Mi Lee, Korea University, Korea, wingco@korea.ac.krBong Dae Choi, Korea University, Korea, queue@korea.ac.kr

We consider the power saving class of type 1 in IEEE 802.16e where a mobile station communicates witha base station by both downlink and uplink traffics. This system is modelled as an embedded Markovchain and a semi-Markov process. We obtain power consumption ratio and queueing delay. We show thatour analytic results match with simulation results very well. Using our performance analysis, we find theoptimal system parameters such as the initial sleep window, the final sleep window and the close-downtime which minimize the power consumption of the mobile station while satisfying the required QoS onmean delay.

A New Power Saving Mechanism with Periodic Traffic Indications in the IEEE802.16e

Eunju Hwang, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, violetgl@korea.ac.krKyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, kimkjae@korea.ac.krJung Je Son, Global Standards & Research Team, Telecom R&D Center, Samsung Electronics, Suwon,Korea, jungje.son@samsung.comBong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center,Korea University, Seoul, Korea, queue@korea.ac.kr

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A new sleep mode scheme called the power saving mechanism with periodic traffic indications in theIEEE 802.16e. In the proposed scheme, traffic indication(TRF-IND) messages are regularly sent at everyconstant time to initiate transmission, and the bandwidth and energy can be saved by not sending sleeprequest(MOB-SLP-REQ) and sleep response(MOB-SLP-RSP) messages which are required in the originalpower saving classes in the IEEE 802.16e standard. We obtain sleep mode ratio and power consumptionratio by computing the Laplace Stieltjes transforms(LSTs) of the length of awake interval and sleep interval,and total delay by computing the LST of queueing delay. We show that our analytic results match withsimulation results very well. Using our performance analysis we find the optimal system parameters such asa TRF-IND interval and a close-down time which minimize the power consumption of MS while satisfyingthe required quality of service(QoS) on delay. Finally we show that the proposed scheme performs betterthen the power saving class of type 1 in the IEEE 802.16e standard.

Session TC06 - Matrix-Analytic Methods, Chair: Squillante / Taylorin CZ 13

Bounds and Decay Rate for a QBD Process with Countably Many BackgroundStates

H. Li, Mount Saint Vincent University, Canada, hli@msvu1.msvu.caQ.-M. He, Dalhousie University, Canada, Qi-Ming.He@Dal.CaY.Q. Zhao, Carleton University, Canada, zhao@math.carleton.ca

In this talk, we introduce upper and lower bounds for the tail in the stationary joint distribution and amarginal distribution for a quasi-birth-and-death process with countably many background (phase) states.We also discuss conditions for exactly geometric-tailed and light-tailed, but not exactly geometric, distri-butions for this type of system. Using examples, we demonstrate how to apply these conditions to modelsof interest.

Algorithmic approach to the extinction probability of branching processes

S. Hautphenne, Université Libre de Bruxelles, Belgium, shautphe@ulb.ac.beG. Latouche, Université Libre de Bruxelles, Belgium, latouche@ulb.ac.beM.-A. Remiche, Université Libre de Bruxelles, Belgium, mremiche@ulb.ac.be

A large number of questions in fields like biology and telecommunications may be modeled as continuous-time multi-type branching processes. In order to determine the extinction probability for this kind ofprocesses, it is necessary to find the minimal nonnegative solution of a non-linear matrix fixed point equa-tion. We examine the question of solving that equation, using a mixture of algorithmic and probabilisticarguments. This allows us to analyze the efficiency of various algorithms on the basis of the behavior of thebranching process itself. We focus our attention here on a particular class of continuous-time multi-typebranching processes, named Markovian binary trees (MBT), but our approach is amenable to more complexsystems. In the special case of the MBT, the fixed point equation turns out to be quadratic, which makes itsresolution particularly clear. Specifically, we analyze three linear algorithms, of which one is new, and, wepropose a quadratic algorithm arising from Newton’s iteration method for fixed-point equations. We alsogive them a probabilistic interpretation in terms of the MBT.

Physical interpretations of algorithms for stochastic fluid flows

Nigel Bean, University of Adelaide, Australia, nigel.bean@adelaide.edu.auMałgorzata O’Reilly, University of Tasmania, Australia, malgorzata.oreilly@utas.edu.auPeter Taylor, University of Melbourne, Australia, p.taylor@ms.unimelb.edu.au

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We discuss several algorithms, including quadratically convergent algorithms, which can be used to cal-culate the Laplace-Stieltjes transforms of the time taken to return to the initial level in the stochastic fluidflow model. We give physical interpretations of the algorithms within the fluid flow model. These physicalinterpretations are built upon the physical interpretations of the generator of the Laplace-Stieltjes trans-form of the total amount of fluid that flowed in and out the buffer. The major benefit of understanding thesephysical interpretations is the ablity to analyze the model using direct, elegant methods, within the fluidflows.

Invariant measures for transient fluid queues

Sarah Van Lierde, Université Libre de BruxellesAna da Silva Soares, Université Libre de BruxellesGuy Latouche, Université Libre de Bruxelles, Belgium

Positive recurrent fluid queues have a matrix-exponential stationary distribution. We show that a transientfluid queue also has a matrix-exponential invariant measure which is not summable. We give a construc-tion for this measure and we give the probabilistic interpretation for the key matrix which appears in theconstruction.

Session TC07 - MDP 2, Chair: Örmeci in CZ 14

A Semi Markov Decision Approach for Train Conflict Resolution

Assil Al-Ibrahim, University of Amsterdam, The Netherlands, a.alibrahim@uva.nlJan van der Wal, Eindhoven University of Technology, The Netherlands, jan.v.d.wal@tue.nl

Railways are a common way of transportation. The passengers that use railways demand an increasinglevel of service. Punctuality plays here a great role. This paper addresses the delays that are a consequenceof a conflict between two or more trains. Trains from more than one direction come together and from thatpoint onwards share what we will call the ’destination track’. When two or more trains try to enter the’destination track’, a decision is to be made about which train to move first. The optimal decision takesinto account several factors. We distinguish the train-speeds, the acceleration rate giving their mass, theurgency of trains, previous decisions and the trains that are still to arrive at the junction. To solve thisproblem we use a Semi-Markov Decision (SMD) approach. An advantage of the SMD-model is that it isable to deal with the dynamics of the train system. A disadvantage is that a description as a SMD, tendsto result in systems with tremendously large state spaces. As will be shown, it is possible to find a balancebetween these two aspects. By means of simulation, the performance of the SMD-model is compared tothat of FCFS and other rules.

Exact finite approximations of average-cost countable Markov Decision Processes

Arie Leizarowitz, Faculty of Mathematics, Technion, Haifa, Israel, la@techunix.technion.ac.ilAdam Shwartz, Faculty of Electrical Engineering, Technion, Haifa, Israel, adam@ee.technion.ac.il

For a countable-state, compact action space Markov decision process we introduce a method to derive em-beddings which produce finite-state Markov decision processes. Each finite-state embedded process hasthe same optimal cost as the original model—and similarly, costs of any stationary Markov policy agree.Moreover, the embedded Markov decision process agrees with the original process when restricting to theapproximating finite set, in the sense that the available actions are the same, and the use of correspondingactions yields identical transition probabilities and identical immediate rewards.

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The embedded process can be used as an approximation which, being finite, is more convenient for com-putation and implementation. Since the optimal costs are the same, it is an approximation only in the sensethat the process is different outside the finite set. The size of the approximation determines the range ofstates on which the approximation has the same dynamics as the original process.This method extends easily to constrained MDPs. The ideas also apply to optimal exit problems.

A Controlled Markov Process Model for a Benchmark Reentrant ManufacturingLine with Infinite and Finite Capacity Buffers

José A. Ramírez-Hernández, University of Cincinnati, USA, ramirejs@ececs.uc.eduEmmanuel Fernandez, University of Cincinnati, USA, emmanuel@ececs.uc.edu

We present results on the optimal job sequencing control policy for a benchmark reentrant manufactur-ing line, for Markov Decision Models with both infinite and finite capacity buffers, and under an infinitehorizon discounted cost criterion. For a set of one-stage cost functions the results obtained provide bothsufficient conditions and structural properties for the optimal policy.

Effects of System Parameters on the Optimal Policy Structure in a Class of Queue-ing Control Problems

E.L. Örmeci, Koç University, Turkey, lormeci@ku.edu.trE.B. Çil, Northwestern University, U.S.A., e-cil@kellogg.northwestern.eduF. Karaesmen, Koç University, Turkey, fkaraesmen@ku.edu.tr

We study a class of queueing control problems involving commonly used control mechanisms such asadmission control, pricing, and service rate control. Such controls are frequently employed in modelingservice systems, telecommunications, and also in inventory systems modeled as make-to-stock queues. It isby now well established that in a number of such important problems, the optimal policy is structured andis described by a few parameters. From a design point of view, it is useful to understand how the optimalpolicy changes when the system parameters such as arrival or service rates, or the number of servers vary.We present a general framework to investigate the policy implications of the system parameters by usingevent-based dynamic programming. In this framework, the value function of the control problem is firstrepresented by a number of common operators, and the effect of system parameters on the optimal policyis analyzed for each individual operator. We present this analysis for a set of frequently used operators.Whenever a queueing control problem can be modeled by these operators, the effects of system parameterson the optimal policy follows trivially from this analysis. This enables us to easily retrieve and generalizesome of the existing results in the literature. We also present examples in a number of challenging recentproblems.

Session TC08 - Queueing Models 2, Chair: Gómez-Corral in CZ 15

Bad luck when joining the shortest queue

Hans Blanc, Tilburg University, The Netherlands, blanc@uvt.nl

A frequent observation in service systems with queues in parallel is that customers in other queues tend tobe served faster than those in one’s own queue. We quantify the probability that the service of a taggedcustomer would have started earlier if that customer had joined another queue than the queue that wasactually chosen, for exponential multiserver systems with queues in parallel in which customers join oneof the shortest queues upon arrival and in which jockeying is not possible. We consider both symmetric

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and asymmetric systems, and also systems with additional dedicated traffic to one or more servers. Insymmetric systems, the probability of bad luck may well exceed 0.5 when there are three or more servers,but this only occurs if the load of the system is very close to 1. The approach of this probability to its heavytraffic limit is very steep, so that this limit, which is easily computable, will not be a good approximationfor most values of the load. Asymmetry in the service rates tends to increase the probability of bad luckin light traffic, but to decrease it in moderate to heavy traffic. Dedicated background traffic decreases thisprobability.

On the correlation structure of Gaussian queues

A. Es-Saghouani, University of Amsterdam, The Netherlands, aessagho@science.uva.nlM. Mandjes, University of Amsterdam, The Netherlands, mmandjes@science.uva.nl

We present results on the correlation structure of the workload process of a queue fed by Gaussian pro-cesses, such as fractional Brownian motion (fBm) and integrated Ornstein-Uhlenbeck (iOU) processes.The main question is: to what extent does the workload process inherit the correlation properties of the in-put process? We present an alternative definition of correlation that allows (in asymptotic regimes) explicitanalysis. For the special cases of fBm and iOU we analyze the behavior of this metric under a many-sources scaling. Relying on (the generalized version of) Schilder’s theorem, we are able to characterize itsdecay. We observe that, popularly speaking, the correlation structure of the input process carries over tothe workload process.

Negative Queue Length:Queues for Quantum Information

Hiroshi Toyoizumi, Waseda University, Japan, toyoizumi@waseda.jp

We study the detail of queues handling quantum communication, especially so-called quantum merging ofqubits. In quantum merging, by using EPR pairs (entangled pairs of quantum particles), jobs may bringextra service capacity to the server, which can be stored in the system for future use. We will model thisphenomena as queues that can have the negative queue length, and show the basic characteristics of thesequantum queues.

On a batch Markovian arrival process subject to renewal catastrophes

A. Gómez-Corral, Complutense University of Madrid, Spain, antonio_gomez@mat.ucm.es

The talk deals with a population of individuals that grows stochastically according to a batch Markovianarrival process and is subject to renewal generated geometric catastrophes. Our interest is in the semi-regenerative process describing the population size at arbitrary times. Towards this end, we study theunderlying Markov renewal process whose embedded Markov chain (at post-catastrophe epochs) may bethought of as a Markov chain of G I/G/1-type, which is indeed amenable to be studied through its R-and G-measures, and a suitably defined Markov chain of M/G/1-type. We present tractable formulas forthe equilibrium distribution of the population size and the distribution of the time to extinction for presentindividuals at post-catastrophe epochs, among other probabilistic descriptors.

Session TC09 - World Wide Web, Chair: Litvak in CZ 16

PageRank of Scale-Free Growing Networks

K. Avrachenkov, INRIA Sophia Antipolis, France, K.Avrachenkov@sophia.inria.frD. Lebedev, R&D France Télécom, France, dmitry.lebedev@orange-ftgroup.com

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PageRank is one of the principle criteria according to which Google ranks web pages. PageRank can beinterpreted as a frequency of webpage visits by a random surfer, and thus it reflects the popularity of awebpage. In the present work we find an analytical expression for the expected PageRank value in a scale-free growing network model as a function of the age of the growing network and the age of a particularnode. Then, we derive asymptotics that show that PageRank follows closely a power law in the middlerange of its values. The exponent of the theoretical power law matches very well the value found frommeasurements of the World Wide Web. Finally, we provide a mathematical insight for the choice of thedamping factor in PageRank definition.

Distances in Random Graphs

Remco van der Hofstad, Department of Mathematics and Computer Science, Eindhoven University ofTechnology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, rhofstad@win.tue.nl

The structure of real networks, such as the WWW, Internet and various biological networks, is receivingincreasing attention in both the mathematical and the networking community. Measurements have shownfascinating features, such as their ‘scale-free’ nature and the ‘small-world’ phenomenon. The small-worldphenomenon states that typical distances in large complex networks are small, while a complex network isscale free when it has a power law degree sequence, i.e., when the number of nodes with degree proportionalto k decays as an inverse power of k. It is difficult to model the topological structure of such networks, dueto their high (and ever increasing) complexity.

We shall discuss a few simple models for scale-free random graphs. We investigate the asymptotic law ofthe distances between typical pairs of vertices in the graph, and the structure of neighborhood exploration.For general complex networks, these distances are important features, since for example, the spread ofdiseases in a population is closely related to the topology of the social network describing the population.We hope that our findings prove relevant as well for the PageRank problem.

Stochastic equations behind Web ranking

Y. Volkovich, University of Twente, The Netherlands, y.volkovich@ewi.utwente.nlD. Donato, Yahoo! Research, Spain, debora@yahoo-inc.comN. Litvak, University of Twente, The Netherlands, n.litvak@ewi.utwente.nlW.R.W. Scheinhardt, University of Twente, The Netherlands, w.r.w.scheinhardt@ewi.utwente.nl

PageRank was introduced by Google as a popularity measure for Web pages. We study the probabilisticrelation between PageRank and other parameters of the Web such as in-degree, out-degree and the fractionof pages with out-degree zero. We model this relation through stochastic equations inspired by the originaldefinition of PageRank. We use the theory of regular variation to prove that the tail distributions of PageR-ank and in-degree follow power laws with the same exponent, which is in agreement with experiments. Wealso show that out-degree has a minor effect on the PageRank distribution.

A fair distribution of probability mass on the bow-tie graphs

Konstantin Avrachenkov, INRIA Sophia Antipolis, 2004 Route des Lucioles, 06902, FranceNelly Litvak,University of Twente, P.O.Box 513 7500 AE Enschede The NetherlandsKim Son Pham,St.Petersburg State University, 35 University Prospect, 198504, Peterhof, Russia

Link-based popularity measures are an important tool used by search engines for ranking Web pages. Weconsider the Google PageRank that is defined as a stationary distribution of an ‘easily bored surfer’ randomwalk on a Web Graph. At each step, with probability c, such random walk starts afresh from the uniformdistribution. The choice of the ‘damping factor’ c is not evident. The Google’s original choice c = 0.85

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was a compromise between the true reflection of the Web structure and numerical efficiency. However,due to the bow-tie structure of the Web, the Markov random walk on the original Web Graph does notreflect the importance of pages because it absorbs in dead ends. We propose new criteria for choosing thedamping factor based on the ergodic structure of the Web Graph. Specifically, we require that the corecomponent receives a fair share of the PageRank mass. Using singular perturbation approach we concludethat the value c = 0.85 is too high and suggest that the damping factor should be chosen around 1/2. Ouranalytical results are confirmed by experiments on two large samples of the Web Graph.

Tuesday 4:45pm - 6:15pm

Session TD01 - Lévy Processes and Finance, Chair: Rieder / Bäuerlein CZ 4

Malliavin Calculus for Levy Processes and Applications in Finance

Evangelia Petrou, HU Berlin, Germany, epetrou@mathematik.hu-berlin.de

Most of the literature on Malliavin Calculus and its applications to Finance focuses on markets that aregenerated by diffusion processes. Our main aim is to extend these results for markets that are generatedby Levy processes. Proving a chaos expansion for Levy spaces we are able to define directional Malliavinderivatives and to reach a Clark-Ocone- Haussman (COH) formula. Furthermore we can define the adjoint(Skorohod integral) of the directional derivatives and we prove an integration by parts formula. The abovetheory will be used to explicitly estimate hedging strategies and the Greeks in markets generated by Levyprocesses.

Extremal behavior of stochastic integrals driven by regularly varying Lévy pro-cesses

H. Hult, Brown University, USA, henrik_hult@brown.eduF. Lindskog, Royal Institute of Technology, Sweden, lindskog@kth.se

We study the extremal behavior of a stochastic integral driven by a multivariate Lévy process that is regu-larly varying with index α > 0. For predictable integrands with a finite (α + δ)-moment, for some δ > 0,we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Lévy pro-cess and we determine its limit measure associated with regular variation on the space of right-continuousfunctions with left limits. The connection to popular models in finance is discussed.

Dependence properties and comparison results for Lévy processes

N. Bäuerle, University of Karlsruhe, Germany, baeuerle@stoch.uni-karlsruhe.deA. Müller, Heriot-Watt University Edinburgh, GB, mueller@wior.uni-karlsruhe.deA. Blatter, University of Karlsruhe, Germany, blatter@stoch.uni-karlsruhe.de

We investigate dependence properties and comparison results for multidimensional Lévy processes. Asso-ciation, positive orthant dependence and positive supermodular dependence of Lévy processes are char-acterized in terms of the Lévy measure as well as in terms of the Lévy copula, a concept which has beenintroduced recently. As far as comparisons of Lévy processes are concerned we consider the supermodularand the concordance order and characterize them by orders of the Lévy measures and by orders of the Lévycopulas respectively. This generalizes recent results. Finally we show how these results can be applied in

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finance and insurance to obtain for example comparison statements for ruin times and probabilities as wellas for option prices.

Session TD02 - Spatial Models 1, Chair: Foss / Stolyar in CZ 5

Large deviations of the interference in a wireless communication model

A.J. Ganesh, Microsoft Research, Cambridge, UK, ajg@microsoft.comG.L. Torrisi, IAC, Rome, Italy, torrisi@iac.rm.cnr.it

Interference from other users limits the capacity, and possibly the connectivity, of wireless networks. Weconsider a simple model of a wireless ad-hoc network where node locations are described by a homo-geneous Poisson point process, and node transmission powers are random. For light-tailed transmissionpowers we obtain a large deviation principle for the total interference, and relate the speed of the large devi-ation principle to the tail decay rate of the distribution of transmission power. For heavy-tailed transmissionpowers we provide asymptotic upper and lower bound for the tail of the interference distribution.

M/D/1/1 loss system with interference and applications to transmit-only sensor net-works

B. Błaszczyszyn, ENS/INRIA Paris, France and University of Wrocław, Poland,Bartek.Blaszczyszyn@ens.frB. Radunovic, Microsoft Research, Cambridge, UK, bozidar@microsoft.com

We propose and analyze a probabilistic model of packet reception in the steady state regime of a non-slottedwireless communication channel. It is an extension of the classical M/D/1/1 Erlang’s loss model where theinterference created by different packet emissions is introduced by means of a shot-noise process. Moreprecisely, we assume that a given packet is admitted by the single receiver if this latter is idle at the packetarrival epoch and successfully received if, in addition, its signal-to-interference-and-noise ratio averagedover the reception period is large enough. As the main result we prove an analog of the Erlang’s formulafor the ergodic rate of the successfully received packets. Our work is motivated by some applications totransmit-only sensor networks.

Local and global properties of random directed trees

Rahul Roy, Indian Statistical Institute, New Delhi, rahul@isid.ac.in

Random directed trees are constructed in space as well as in the unit cube. For the spanning trees in spacewe study the geometric structure – in particular, when does the construction give us a connected graph, andwhen it gives a collection of disjoint trees. For the construction in the unit cube, we study the asymptoticproperties of the graph as the number of vertices goes to infinity.

Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models

M. Lelarge, INRIA-ENS, France, marc.lelarge@ens.frDavid AldousC. Bordenave, University of California at Berkeley, United States, bordenav@eecs.berkeley.edu

We study the relation between the minimal spanning tree (MST) on many random points and the “near-minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different

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from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio oflengths should scale as 1 + 2(δ2). We prove this scaling result in the model of the lattice with randomedge-lengths and in the 2-dimensional Euclidean model.

Session TD03 - Asymptotics, Chair: Blanchet / Mikosch in CZ 10

Towards Efficient Simulation of Long Delays for Heavy-tailed Multiserver Queues

Jose Blanchet, blanchet@fas.harvard.eduPeter Glynn, glynn@stanford.edu

We consider the question of computing via simulation the likelihood of observing at least one customerthat experiences a long delay within a busy period in a multiserver queue with heavy-tailed input. Weshall discuss a class of algorithms that can be argued heuristically to achieve asymptotic optimality formultiserver queues with regularly varying input throughout a wide range of stability conditions. In thecase of two server queues, our algorithms can be rigorously proved to achieve asymptotic optimality if thetraffic intensity is not equal to 1/2. In the single-server queue case, the algorithm is particularly easy toimplement and produces a new asymptotically optimal procedure for estimating the tail of the steady-statewaiting time.

Large deviations for point processes based on stationary sequences with heavy tails

H. Hult, Brown University, United States, henrik_hult@brown.eduG. Samorodnitsky, Cornell University, United States, gennady@orie.cornell.edu

A stationary sequence of random variables with regularly varying tails is considered. For this sequence itis possible that large values arrive in clusters. That is, there may be many large values in a relatively shortperiod of time. The aim is to give a detailed description of the occurrence of large values, not only the sizeof clusters but also within a single cluster. To do this a point process based on appropriately scaled pointsof the stationary sequence is constructed. A limiting measure, on the space of point measures, describes thejoint limiting behavior of all the large values of the sequence. From this limiting result one can proceed andobtain the functional large deviation for its partial sums, ruin probabilities, etc. Examples include, linearprocesses, random coefficient ARMA processes, and solutions to stochastic recurrence equations.

Rare event simulation methodology for perpetuities

Jose Blanchet, Harvard University, USA, blanchet@fas.harvard.eduBert Zwart, Georgia Institute of Technology, USA, bertzwart@gatech.nl

A perpetuity is the present value of a bond in the presence of possible random interest rates and premiums,in a discrete or continuous-time setting. Due to the nonlinear structure of a perpetuity, its tail behavior canbe heavy tailed, even if all input random variable are light-tailed. Large deviations estimates (see Goldie(1991) among many others) among many others) yield power tail behavior of the form Ct−κ , where κis easily computable, but C is not. Since the precise convergence rate towards this asymptote is also notclear, it is desirable to have efficient simulation techniques available. The goal of this talk is to presentseveral of such techniques. We consider several different cases and present simulation algorithms whichare asymptotically optimal. A typical feature of these algorithms is that they are state dependent, in contrastwith standard importance sampling techniques.

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Boundary behavior and stability for multidimensional reflected diffusions with state-dependent jumps in the positive orthant

Ravi R. Mazumdar, Department of Electrical and Computer Engineering, University of Waterloo,Waterloo, Canada

Reflected diffusions with jumps occur in many financial as well as queueing network models. There hasbeen much work by Williams and Harrison on Semi-martingale Reflected Brownian Motion (SRBM) (ICM1998) and the boundary characterization of such processes. However there are few results for reflected dif-fusions with jumps with non-constant drifts and diffusion coefficients and state-dependent reflections. Oneof the difficulties is that the associated Skorohod problem does not have a Lipschitz continuous reflectionmap. In the talk I will present 2 new sets of results for diffusions with jumps and state-dependent obliquereflections at the boundary. The first is a characterization of the boundary behavior in terms of the localtime and a result on the support of the distribution on the boundary surfaces. In particular the results gen-eralize some earlier results due to Taylor and Williams (PTRF 1993) and Guillemin and the speaker (Appl.Math. Opt. 1996). In the second part, we exploit the boundary characterization to obtain the necessaryand sufficient conditions for the existence of product-form stationary distributions and sufficient conditionsfor their existence. I will conclude with the general case with state-dependent reflections in a so-calledmonotone case.

Based on joint work with Dr. Fabrice Guillemin (CNET, France) and Francisco Piera (University of Chile).

Session TD04 - Large Deviations 1, Chair: Majewski in CZ 11

Large deviations of multitype queues

A. Dabrowski, University of Ottawa, CanadaJ. Lee, Yeungnam University, Republic of Korea, leejy@yu.ac.krD.R. McDonald, University of Ottawa, Canada, dmdsg@uottawa.ca

Consider a multitype queue where queued customers are served in their order of arrival at a rate whichdepends on the customer type. We derive sharp asymptotics of the steady state distribution of the queuesize by finding an appropriate harmonic function h for the Markov additive process. The associated h-transform produces a change of measure that increases the arrival rate of customers and decreases thedeparture rate thus making large deviations common. Using the change of measure we calculate the sharpasymptotics of the probability that the total number of customers in the queue reaches a high level beforeemptying and we also obtain a description of the queues when this large deviation occurs. We presentnumerical results obtained via our asymptotic expressions and compare those results with the simulationresults.

Limiting distribution of the phase when level passes a threshold

Aziz Khanchi, University of Ottawa, Canada, akhan037@uottawa.caDavid R. McDonald, University of Ottawa, Canada, dmdsg@uottawa.ca

Consider a network with n nodes. We provide the limiting distribution of the remaining nodes at themoment the first node crosses a threshold which tends to infinity. The performance of the system is studiedby twisting the associated Markov additive process using a suitable harmonic function. The current resultsare extended to the cases where the Markovian component of the twisted Markov additive process is notpositive recurrent.We apply our results to a Modified Jackson Network, where the second node helps the first one wheneverthe second node is idle. The existence of the limiting distribution for this network is rather surprising sincethe large deviation bridge path avoids emptying the second node.

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Large Deviations of Generalized Jackson Networks

Silke Meiner

Investigating generalized Jackson networks (not necessarily exponential iid inter event times and Bernoullirouting independent of everything else) I develop the rate function for the sample path Large Deviation ofqueue sizes. The rate function is identified on tubes around affine functions.My approach follows Ignatiouk-Robert (2000) and main stages in the argumentation are: an LD upperbound with the help of an exponential change of measure, the solution of an optimization problem in theupper bound and the interpretation of the optimizer. Associated with the optimizer is a change of measureunder which the network can be divided into two subnetworks: one in equilibrium the other behaving likea free process.Tightness of the upper bound follows from properties of the network under the change of measure corre-sponding to the optimizer found in the upper bound.

Functional asymptotics of a simple queueing network with short- and long-rangedependent input

Kurt Majewski, Siemens AG., Corp. Tech., München, Germany, Kurt.Majewski@siemens.com

We consider an exemplary queueing network with two nodes providing infinite buffer space and non-idlingfifo service at constant rate. Customers of a constant rate traffic have to visit both queues whereas thecustomers of a random cross traffic just visit the first queue before leaving. The queue length behavior ofthe second queue is a measure for the delay variation induced on the constant rate traffic by sharing the firstbuffer with the random cross traffic.

Functional weak heavy traffic (Peterson 1991), large deviation (Ramanan and Dupuis 1995), and moderatedeviation (Majewski 2006) asymptotics of this queue length behavior have been developed for suitablyscaled renewal cross traffic. We extend these three types of asymtptotics to certain appropriately scaledsequences of Markovian and long-range dependent cross traffic. The weak heavy traffic limits of thequeue length behavior for the three types of cross traffic processes can be characterized as twice reflectedBrownian motion, integrated Ornstein-Uhlenbeck process, and fractional Brownian motion, respectively.The minimizing paths leading to a large buffer content in the second queue in the large and moderatedeviation principles can be characterized explicitly and calculated numerically. They give insight into thetransition from short- to long-range dependent input in queueing networks.

Session TD05 - Computer-communications, Chair: Hwang in CZ 12

Asymptotic results in some random infinite urn models

P. Robert, INRIA-Rocquencourt, France, Philippe.Robert@inria.frF. Simatos, INRIA-Rocquencourt, France, Florian.Simatos@inria.fr

Motivated by the problem of diffusion of information in peer to peer networks, a stochastic model for whichN balls are thrown in an infinite number of urns is studied. The probability pi that a ball is in urn i is arandom number, which makes the balls dependent, and the asymptotic behaviors of the index of the firstempty urn as well as the index of the last occupied urn are investigated.

Connectivity probability of one-dimensional ad hoc wireless networks with mini-mum hop path

Nelson Antunes, CEMAT/University of Algarve, Faro, Portugal, nantunes@ualg.pt

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Gonçalo Jacinto, CIMA-UE/University of Évora, Évora, Portugal, gjcj@uevora.ptAntónio Pacheco, CEMAT/Instituto Superior Técnico, Lisboa, Portugal, apacheco@math.ist.utl.pt

In an ad hoc multi-hop network, the functionality of the network critically depends on the connectivityproperties of the network. These properties depend on the number of nodes, their transmission rangesand spatial distribution. This work investigates the connectivity probability in an one-dimensional ad hocnetwork from a different viewpoint: minimum hop path. We consider a source and a destination node anda finite set of relay nodes uniformly distributed between them. The exact probability distribution of theminimum number of hops (M) between the source and destination is derived from the density locationof relay nodes in the minimum hop path. The analysis is based on the Poissonification technique. Theprobability that the source and destination pair is connected (provided by Ghsemi and Nadser-Esfahani(2006)) can be obtained by summing the probability masses for each possible number of nodes in theminimum hop path. We provide an analytical expression for network design that enables the determinationof the required number (resp. transmission range) of relay nodes given a desired probability of connectivitywith the minimum value of M . Furthermore, we also study the distribution of the minimum hop pathduration and the mean length of connectivity periods for the different values of M . Finally, some numericalresults are presented.

Design and Analysis of a Packet Scheduler Exploiting the Multiuser Diversity forPerformance Enhancement

Gang Uk Hwang, Korea Advanced Institute of Science and Technology (KAIST), Republic of Korea,guhwang@amath.kaist.ac.krFumio Ishizaki, Nanzan University, Japan, fumio@ieee.org

In this paper, we consider a new packet scheduler exploiting the multiuser diversity in a wireless networkwith an AMC (Adaptive Modulation and Coding) scheme, and propose an analytic model based on theeffective bandwidth theory to investigate the joint effect of the AMC scheme and the packet scheduler onsystem performance. Based on our analytic results as well as numerical results, we show how to design thepacket scheduler to improve the spectral efficiency of the wireless channel.

Session TD06 - Applied Probability, Chair: Hagwood in CZ 13

Stochastic modeling of a dark matter experiment

Kevin Coakley, National Institute of Standards and Technology, kevin@boulder.nist.gov

In a proposed experiment mini-CLEAN (Cryogenic Low Energy Astrophysics with Noble gases), WeaklyInteracting Massive Particles (WIMPs) might be detected based on scintillation light produced when aWIMP deposits energy in a scintillator by nuclear recoil scattering. The WIMP is a candidate particle toexplain dark matter in the universe. In this experiment, any WIMP signal would be very weak comparedto a very intense background that produces scintillation light by electronic recoil scattering. I will discussstochastic models of observed pulses produced by both WIMP events and background events, and spatialand temporal methods to discriminate WIMP events from background events.

Stochastic Modeling in Gene Expression Measurements: Accounting for Both Back-ground Error and Instrument Saturation

Z.Q. John Lu, National Institute of Standards and Technology, john.lu@nist.gov

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There are multiple ways of measuring expressions of thousands of genes simultaneously through measuringthe amount of mRNAs by building probes using either the full complementary sequences or using sets ofprobes pairs based on short subsequences (e.g. Affymetrix arrays). Sources of error include both thenon-specific hybridization from using short oligonucleotide arrays to scanner performance limitations dueto using the two-color fluorescent dye effects, optical noises, and saturation at high intensity readings.NIST chemists have performed a careful study of two brand scanners over a long period, from which wewere able to characterize the scanner effects in the measurement process. In this paper, I describe somestatistical issues in fitting a general nonlinear measurement error model for microarray gene expressionmeasurements, which can account for both the background error and instrument saturation, while allowingthe signal to vary at a large exponential or Langmuir hyperbolic dynamic range.

Stochastic Modeling of a Carbon Nanotube in an Electrical Field

Charles Hagwood, National Institute of Standards and Technology, hagwood@nist.gov

We want to classify diameter selected carbon nanotubes by length. In this talk a stochastic differentialequation is used to describe the motion of nanotubes (or nanowires) undergoing Brownian rotation andtranslation in an electric field.

Allocation Models and heuristics for the outsourcing of repairs for a dynamic war-ranty population

L. Ding, The Management School, Edinburgh University, Edinburgh EH8 9JY, UKK.D. Glazebrook, Department of Management Science, Lancaster University, LA1 4YXC. Kirkbride, Department of Management Science, Lancaster University, LA1 4YX

We consider a scenario in which a large equipment manufacturer wishes to outsource the work involvedin repairing purchased goods while under warranty. Several external service vendors are available for thiswork. We develop models and analyses to support decisions concerning how responsibility for the warrantypopulation should be divided between them. These also allow the manufacturer to resolve related questionsconcerning, for example, whether the server capacities of the contracted vendors are sufficient to deliveran effective post sales service. Static allocation models yield information concerning the proportions ofthe warranty population for which the vendors should be responsible overall. Dynamic allocation modelsenable consideration of how such overall workloads might be delivered to the vendors over time in a waywhich avoids excessive variability in the repair burden. We apply DP policy improvement to develop aneffective dynamic allocation heuristic. This is evaluated numerically and is also used as a yardstick toassess simpler heuristics suggested by statis models. A dynamic greedy allocation heuristic is found toperform well. It is clear that dividing the workload equally among vendors with different service capacitiescan lead to serious losses.

Session TD07 - Optimal Stopping 1, Chair: Gnedin in CZ 14

Maximizing the expected duration time related to Poisson arrival

A. Kurushima, Tokyo University of Science, Japan, kul@ms.kagu.tus.ac.jpK. Ano, Institute of Applied Mathematics, Japan, kano@iapm.jp

Here we consider the following optimal stopping problem. Suppose the object arrives sequentially ac-cording to Poisson process with unknown intensity having Gamma prior density G(r, 1/a), where a isnonnegative. Let the time horizon be (0, T ] of finite length. Suppose the objects is rankable without equal

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rank and we observe only the relatively rank arrived so far. We want to maximize the expected durationtime owing the relatively best object, so that our problem is regarded as a variation of the secretary problem.For each r = 1,2 and 3, we derive the optimal stopping time. It is a threshold type which can be describedas τ ∗r = min{t ∈ [s(r)∗i , T ] : X i = 1}, where the event {X i = 1} means that i th object is the relativelybest object and s(r)∗i is a unique root of a certain equation for each r . We show the threshold sequence of{s(r)∗i }i≥1 is nonincreasing and the asymptotic value s(r)∗i goes to [(T − a)/e2

− a]+ as i → ∞ for allr = 1,2 and 3. Furthermore we shall investigate the nature of the optimal stopping time for the generalcase with a natural number r .

A linear programming analysis of finite horizon optimal stopping problems

K. Helmes, Humboldt University of Berlin, Germany, helmes@wiwi.hu-berlin.de

We numerically analyze finite horizon optimal stopping problems by exploiting an infinite dimensional lin-ear programming formulation of such problems. We approximate the infinite dimensional linear programs(LP) by particular finite dimensional LPs. The finite dimensional linear programs to be solved are basedon approximations of moment sequences of finite measures with compact support. The finite dimensionalLPs provide bounds on the optimal value as well as accurate information about the optimal stopping rules.

Optimal stopping with a rank-dependent loss

A.V. Gnedin, Utrecht University, The Netherlands, gnedin@math.uu.nl

For τ a stopping policy adapted to a sequence of n iid observations, we define the loss to be Eq(Rτ ), whereRτ is the rank of the τ th observation, and q is a nondecreasing function of the rank. This setting coversboth the best choice problem with q(r) = 1(r > 1), and Robbins’ problem with q(r) = r . We showthat for large n the optimal stopping problem has a scaling limit associated with the planar Poisson process.Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimalpolicy like complete history dependence.

Optimal Stopping of ”Seasonal” Observations

E.L. Presman, Central Economics & Mathematics Inst. of Russian Academy of Sci., Russia,presman@cemi.rssi.ruI.M. Sonin, Univ. of North Carolina at Charlotte, USA, imsonin@email.uncc.edu

Our talk has two main goals: first, to describe a new class of optimal stopping problems for which thesolutions can be found in a finite number of steps, and in many cases in an explicit form, and second, todemonstrate the potential of the Elimination algorithm (EA) developed by one of the authors earlier. Wepresent this algorithm in a modified form for an arbitrary state space.

Let B = {1, . . . ,m}, a statistician observes a random sequence Zn = (Xn, Y Xnn ), n ≥ 0, where (Y s

n )n≥0 foreach s ∈ B is an i.i.d. random sequence with distribution F s , and (Xn)n≥0 is a Markov chain independentof (Yn)n≥0, taking values in {e}

⋃B with transition probabilities P

[Xn+1 = j |Xn = i

]= pi j for i, j ∈

B, P[Xn+1 = e |Xn = e

]= 1. The matrix P =

(pi j)

i, j∈B is a strictly substochastic matrix, i.e. pi j ≥ 0

and 0 <∑j∈B

pi j < 1 for all i ∈ B, and e is an absorbing state. The goal is to maximize Eg(Zτ ) over

all possible stopping times τ , g( j, y) is a reward function. We prove that there exists a threshold vectorc = (c1, . . . , cm) such that S = {( j, y) : j ∈ B, g( j, y) ≥ c j } is an optimal stopping set. We give asufficient condition when the EA solves the problem of finding the value function and the vector c in afinite number of steps. The work of E.L.Presman was supported by RFBR grant 07-01-00541.

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TD08

Session TD08 - Queueing in Telecom, Chair: Tijms in CZ 15

A spectral method for a BMAP/D/c queue with synchronous servers

S. Nishimura, Tokyo University of Science, nishimur@rs.kagu.tus.ac.jpN. Yatomi, Tokyo University of Science, nao@astre.jp

A batch Markovian arrival process (BMAP) is great flexible and its queueing models are analyticallytractable for a modeling of IP traffic with a self-similar nature. Since IP traffic is continuous and data aretransmitted by synchronous multiple servers, we consider a BMAP/D/c queue with synchronous servers,where the arrival process to the system is a continuous-time BMAP defined by M × M rate matrices andthe beginnings of c servers with the constant service time h are synchronized as 0, h, 2h, . . .. We proposea spectral method for calculations of the stationary probability of the number of customers in the system atsynchronized epochs. A spectral method is fit to a BMAP with a large maximum batch size as IP traffic.First, by the modified Durand-Kerner method, the boundary vector with the size Mc is derived from theM zeros of the c nonlinear equations, independently. Next, by the inverse discrete Fast Fourier Transform(FFT) method for the vector probability generating function, the stationary probability is calculated. Ina numerical example, the stationary probability and its tail probability are efficiently calculated for theBMAP, which is estimated from real IP traffic and has a good fitness to traffic. Effects of the number ofservers on the stationary probabilities are discussed.

Differential Traffic Theory

J.M. Garcia, LAAS-CNRS, France, jmg@laas.frO. Brun, LAAS-CNRS, France, brun@laas.fr

Network operators have a growing need for efficient network simulation and planning tools (topologydesign, capacity planning, traffic engineering). One key component of such tools is the performance evalu-ation kernel. Given the limits of pure event-driven simulation methods, differential traffic modeling appearsas an efficient alternative for performance evaluation of large telecommunication systems. These modelsare based on approximations of the exact differential traffic equations obtained using the notion of implicittransient traffic. These models, have been introduced in commercial network planning tools (NetQUAD,NEST) and are extensively used for telephone and IP-MPLS networks. They proved their efficiency andaccuracy for large scale networks. We will present this modeling framework for circuit switched systems,multirate loss networks and packet-switched networks. Extension to an innovating hybrid simulation tech-nique will be developed.

Moments of the duration of busy periods in oscillating queueing systems

A. Pacheco, CEMAT/Instituto Superior Técnico, Lisboa, Portugal, apacheco@math.ist.utl.ptH. Ribeiro, CEMAT/Instituto Politécnico de Leiria, Leiria, Portugal, mhcr@estg.ipleiria.pt

We compute moments of the duration of busy periods in oscillating M/G/1 systems with finite capacityand batch arrivals, for which the service time distribution oscillates between two forms according to theevolution of the number of customers in the system. We work with an extended definition of busy period(that may be) initiated by multiple customers and derive an efficient recursive algorithm to compute mo-ments of the duration of a busy period starting with an arbitrary number of customers in the system. Wederive a recursive algorithm on the capacity of the system that is based on the fact that the duration of abusy period initiated by multiple customers in a system of some fixed capacity can be expressed in terms ofthe durations of busy periods of systems with smaller or equal capacities and initiated by a single customer.

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11 – Abstracts

A quick approximation for the state probabilities and waiting-time probabilities inthe M/D/1 queue

H.C. Tijms, Vrije University Amsterdam, The Netherlands, tijms@feweb.vu.nl

We present amazingly accurate approximations to the state probabilities and the waiting-time probabil-ities in the M/D/1 queue using a two-phase process with "branching probabilities" outside (0,1). Theapproximations are in the form of explicit expressions involving geometric and exponential terms. Theapproximations extend to the finite-capacity model

Session TD09 - Engineering, Chair: Günalay in CZ 16

Analysis of a Probabilistic Key Management Scheme for Vehicular Communications

E. van den Berg, Telcordia Technologies, USA, evdb@research.telcordia.com

Vehicular communication networks have drawn much attention recently, notably for their promise of en-hancing vehicle safety. By broadcasting warning messages, vehicles can notify each other of impendinghazards and recent accidents. Given the criticality of the safety applications, these messages must be en-crypted to enable secure communications. A simple probabilistic scheme exist for management of therequired cryptographic keys. The scheme has been studied extensively in the context of broadcast en-cryption. Of particular concern for practical deployment vehicular communications networks is vehicleprivacy: it is critical to avoid messages to and from vehicles being used to identify or track vehicles. Wedescribe a new method for a vehicle to select cryptographic certificates / keys which can be used with theprobabilistic key management scheme. The method allows a vehicle to use keys which it has in commonwith other vehicles in its direct neighborhood, thus increasing the privacy/anonymity of the vehicles. Weanalyze the performance of the method using random graphs and Markov chains.

Comparison of the effect of various forecasting methods on bullwhip effect in supplychain

Reza Zanjirani Farahani, Industrial Engineering Department, Amirkabir University of Technology,Tehran, IranMehdi Najafi, Industrial Engineering Department, Amirkabir University of Technology, Tehran, IranNasrin Asgari, Supply Chain Management Research Group, Tehran, Iran

Nowadays, enterprises in supply chain are attempting to increase their efficiency and effectiveness as wellas supply chain’s. Therefore, identification and consideration of factors which prevent enterprises to at-tain their expected effectiveness are very important. Inasmuch as one of these main factors is bullwhipeffect, being aware of its reasons help the enterprises to decrease the severity of bullwhip effect by optingproper decisions. Now that forecasting method is one of the most important factors in increasing or de-creasing of bullwhip effect, this paper consider and compromise the effect of various forecasting methodson bullwhip effect. In fact in this paper, the effect of various forecasting methods such as Moving Averageand Exponential Smoothing as well as Regression in bringing about bullwhip effect in a four layer supplychain -including: retailer, wholesaler, manufacturer and supplier- is considered. Then, the bullwhip effectmeasure is utilized to compromise the ineffectiveness of various forecasting methods. Owing to this, wegenerate random demands in two states including constant and increasing trend of demands. Eventually,we ranked the forecasting methods in two state of demand and utilize statistical method to certain aboutsignificance of difference among effect of various methods.

Keywords: Bullwhip Effect, Supply chain, Forecasting, Exponential Smoothing, Moving average, Simula-tion.

128

WA01

Using Conditional Probabilities for Automatic New Topic Identification of SearchEngine Transaction Logs

Seda Ozmutlu, Industrial Engineering Department Uludag University, Bursa, Turkey,seda@uludag.edu.trHuseyin C. Ozmutlu, Industrial Engineering Department Uludag University, Bursa, Turkey,hco@uludag.edu.trBuket Buyuk, Industrial Engineering Department Uludag University, Bursa, Turkey,buketbuyuk@uludag.edu.tr

One of the most important dimensions of search engine user information seeking behavior and searchengine research is content-based behavior. The purpose of this study is to perform automatic new topicidentification in search engine transaction logs using conditional probabilities of new topic arrivals. Sampledata logs from FAST and Excite are used in the study. Conditional probabilities of new topic arrivals andtopic continuations given query category are used to estimate new topic arrivals. The findings of thisstudy show that the conditional probability approach yielded favorable results in terms of the performancemeasures used for evaluating automatic new topic identification algorithms.

A queueing analysis of a hybrid manufacturing system with new and refurbisheditems

Y. Günalay, Bilkent University, Turkey, gunalay@bilkent.edu.trD. Gupta, University of Minnesota, USA, guptad@me.umn.edu

Consider a direct-to-market manufacturer which sells both new and refurbished versions of the same item.The firm uses make-to-stock policy for both product. Server performances are related through demand andperceived product substitutability by customers. The system is modeled as two interacting queues: fist as afinite population M/M/1 queue and the second as a Quasi-Birth-Death process. The system is analyzed fordifferent product characteristics and the best base-stock levels are calculated using exact or approximatenumerical methods.

Wednesday 8:30am - 10:00am

Session WA01 - Queues and Lévy Processes, Chair: Kella in CZ 4

The M/G/1 queue with quasi-restricted accessibility

David Perry, University of Haifa, Israel, dperry@stat.haifa.ac.il

We consider single-server queues of the M/G/1 type with a special kind of partial customer rejection calledquasi-restricted accessibility (QRA). Under QRA, the actual service time assigned to an arriving customerdepends on his service requirement, say x , the current workload, say w, and a prespecified thresholdb. If x + w ≤ b the customer is fully served. If w ≤< w + x , the customer receives service timeb − w + (w + x − b) f for some random number f ∈ [0, 1], while if w > b the actual service time isthe fraction f x of the requirement. The random fractions are assumed to be i.i.d. We derive the followingmeasures for the M/G/1 and the G/M/1 models: steady-state law, busy period analysis and cycle maximum.

Joint work with Onno Boxma, Wolfgang Stadje, and Shelley Zacks.

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11 – Abstracts

Lévy correlation cascades

Iddo Eliazar, Holon Institute of Technology and Tel Aviv University, Israel, eliazar@post.tau.ac.il

Random processes with heavy-tailed distributions have infinite variances. Thus, their correlation-structurescannot be investigated via their auto-covariance functions - which are undefined. We consider a large classof random processes driven by non-Gaussian Levy noises. Exploiting the underlying Poisson-superpositionstructure of the driving noises, we introduce a cascade of Poissonian Correlation Functions which character-ize the process-distribution of the random processes under consideration. Specific examples include: Levymotions; Levy-driven Ornstein-Uhlenbeck processes; Levy-driven Moving-Average processes; FractionalLevy motions; Fractional Levy noises; and, general Shot Noise processes. The "Poissonian methodology"developed enables a precise quantitative analysis of Levy-driven random processes displaying, simultane-ously, the Noah effect (heavy-tailed distributions) and the Joseph effect (long-range dependence).

Joint work with Joseph Klafter.

Superposition of renewal processes and an application to multi-server queues

Wolfgang Stadje, University of Osnabrück, Germany, wolfgang@mathematik.uni-osnabrueck.de

The aim of this talk is to compare the waiting times of customers in multiple-server queues, where theidle times are removed, with different numbers of servers. For this purpose we develop some resultsregarding the vector valued marked point process whose points are arrival epochs of the superpositionof renewal processes with different continuous inter-arrival distribution and the marks are the vectors offorward recurrence times of the various renewal processes at these arrival epochs.

Joint work with Offer Kella.

Lévy driven fluid queues with negatively correlated service interruptions and busyperiods

Offer Kella, The Hebrew University of Jerusalem, Israel, Offer.Kella@huji.ac.il

We consider a Lévy driven fluid queue which evolves as follows. Given some initial positive state, theprocess behaves like a Levy process with no negative jumps having a negative mean. At the instant whenthe process hits the origin the hitting time is compared with some independent exponential random variable.If the hitting time is smaller, then the station becomes idle for a time period which is a constant multipleof this hitting time. During this time the process behaves like a subordinator (input and no output). If thehitting time is larger then the system remains empty for an exponential time period at the end of whichthere is a jump of random length. We study the stability of this process and its steady state characteristics.The analysis involves the study of the iterations of some concave function and their limit.

Joint work with Onno Boxma and Michel Mandjes.

Session WA02 - Random Graphs, Chair: Norros in CZ 5

Two diameter results in the configuration model

Gerard Hooghiemstra, Delft University of Technology, The Netherlands, g.hooghiemstra@its.tuedelft.nl

This talk(based on joint work) gives two results on the diameter for the configuration model.

130

WA02

Denote by D a copy of the i.i.d. degree sequence D1, D2, . . . , DN , from which we built the random graph.We assume that D has a power law with exponent τ . Then the first result is:

For τ > 2, f1 + f2 = Pr(D = 1) + Pr(D = 2) > 0 and f1 < 1 we will show that α log N , for someconstant α, is a lower bound for the diameter. This result is most interesting for τ ∈ (2, 3), because we seethat in this case the diameter and the typical distance are of a different order of magnitude.

A second result gives an upper bound on the diameter for τ ∈ (2, 3). For f1 + f2 = 0 and τ ∈ (2, 3), aconstant times log log N is an upper bound for the diameter. If time allows I will sketch a proof of thesetwo results.

A local limit theorem for the critical random graph

T. Müller, EURANDOM / Technische Universiteit Eindhoven, the Netherlands, t.muller@tue.nlR. van der Hofstad, EURANDOM / Technische Universiteit Eindhoven, the Netherlands,r.w.v.d.hofstad@tue.nlW. Kager, EURANDOM / Technische Universiteit Eindhoven, the Netherlands, t.muller@tue.nlw.kager@tue.nl

The Erdos-Rényi random graph G(n, p) is constructed by taking n vertices and including each of the(n

2

)possible edges independently with probability p. In their seminal work around 1960 Erdos and Rényialready showed that if p = c

n (for c a fixed constant) then, as n tends to infinity, the largest component is oforder ln n with high probability if c < 1 and of order n with high probability if c > 1. Here we study theErdos-Rényi random graph inside the “critical window”, when p := 1

n +λ

n43

(where λ ∈ R is an arbitrary

constant). Previous work by various authors has shown that in this range the largest component is of ordern

23 with high probability, and that the number of vertices in the largest component (divided by n

23 ) has a

continuous limiting distribution. We will derive an expression for the probability density function of thelimiting distribution of the largest component, and a useful differential equation that the probability densityfunction satisfies.

Percolation on the 2D Hamming graph: The supercritical phase

M.J. Luczak, London School of Economics, United Kingdom, m.j.luczak@lse.ac.ukR. van der Hofstad, Eindhoven University of Technology, The Netherlands, r.w.v.d.hofstad@tue.nl

We study random subgraphs of the graph H(2, n), which is the Cartesian product of two complete graphson n vertices. Every edge is independently present with probability p = 1+ε

2(n−1) for some ε ∈ R. In asequence of three papers, Borgs, Chayes, van der Hofstad, Slade and Spencer studied phase transitions ina large class of random graphs including H(2, n), obtaining precise estimates for the maximum size of aconnected component in the subcritical and critical regimes, as well as upper bounds in the supercriticalphase. No general lower bounds on the largest supercritical component exist; indeed, it is thought thatthe geometry of a given graph will play a crucial role in lower bounding its giant component. Here, wecomplement the results of Borgs et al. for the supercritical phase in the Hamming graph H(2, n). We provethat, when ε � (log n)1/3n−2/3, then the largest connected component has size close to 2εn2 with highprobability. We thus obtain a law of large numbers for the largest connected component size, and showthat the corresponding values of p really are supercritical. Except for the factor (log n)1/3, this identifiesthe size of the largest connected component all the way down to the critical p window. This is another steptowards understanding phase transitions in graphs "with geometry" (not the complete graph).

Robustness against core losses in a power-law random graph

I. Norros, VTT, Finland, ilkka.norros@vtt.fiH. Reittu, VTT, Finland, hannu.reittu@vtt.fi

131

11 – Abstracts

We work on the conditionally Poissonian random graph model (Adv.Appl.Prob. 38:1) and focus on thecase of capacities (expected degrees) with finite mean and infinite variance. The typical distances withinthe largest component are then 2(log log N ), where N is the number of vertices. The shortest paths tendto have an up-down shape (‘up’ referring to higher capacity) and to travel through the top of the ‘core’consisting of vertices whose capacities exceed a certain level, growing with N . It is then interesting tostudy how the shortest paths change when a top part of the core is cut away. We consider the core asconsisting of 2(log log N ) ‘tiers’ and show that when a fixed number of top tiers is removed, the shortestpaths increase at most by a constant number of hops when N →∞. The main idea is that each tier has assubgraph a dense classical random graph, whose diameter is given by Bollobás’ well-known result. Whenthe whole core is removed, however, the nature of the graph changes more deeply, and the distances become2((log N )/`(N )), where `(N ) = o(log log log N ). It is also possible to follow the gradual change of pathlengths between these extremes.

Session WA03 - Simulation 2, Chair: Grassmann in CZ 10

Zero-Variance Importance Sampling Estimators for Markov Process Expectations

H.P. Awad, University of Miami, Florida, h.awad@miami.eduP.W. Glynn, Stanford University, California, glynn@stanford.eduR.Y. Rubinstein, Technion – Israel Institute of Technology, Israel, ierrr01@ie.technion.ac.il

We study the structure of zero-variance importance sampling estimators for expectations of functionals ofMarkov processes. For a class of expectations that can be characterized as solutions to linear systems, weshow that a zero-variance estimator can be constructed by using an importance distribution that preservesthe Markovian nature of the underlying process. This suggests that good practical importance samplingdistributions can be found by searching within the class of Markovian probability distributions. The classof expectations considered includes as particular cases, among others: the mean time until hitting a rareset, the expected cumulative discounted reward until hitting a set, the mean duration of an excursion, thetransient (deterministic horizon) expectation of a discounted final payoff, and steady-state expectations.

Fluid Limit for Hybrid MCMC samplers

G. Fort, LTCI, CNRS/GET, 46 rue Barrault, 75634 Paris Cedex 13, France, gfort@tsi.enst.frE. Moulines, LTCI, CNRS/GET, 46 rue Barrault, 75634 Paris Cedex 13, France, moulines@tsi.enst.frP. Priouret, LPMA, Université P.M.Curie, Boîte courrier 188, 75252 Paris Cedex 05, France,priouret@ccr.jussieu.fr

Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, withapplications to performance analysis, simulation, and optimization. In Fort et al. (2006), these techniqueshave been extended to a class of skip-free Markov chains : by scaling time, space, and the initial conditionby a large constant, a fluid approximation is obtained that describes the Markov dynamic in the macroscopicform. These extensions were then applied to characterize the stability of some Markov Chain Monte Carlo(MCMC) samplers, namely some Metropolis-Hastings algorithms.

MCMC algorithms depend on many design parameters that need to be finely tuned in order for the al-gorithm to converge efficiently. In this talk, we will caracterize the fluid limit of some hybrid MCMCsamplers. We will then discuss and show how the fluid limit yields information for a partial automation ofthe choice of the design parameters.

G. Fort, S. Meyn, E. Moulines and P. Priouret, ODE methods for skip-free Markov Chains stability withapplications to MCMC, arXiv math.PR/06072800, 2006.

132

WA04

Applied Probability Contribution to the Burn-in Phase in Equilibrium Simulation

W.K. Grassmann, University of Saskatchewan, Canada, grassman@cs.usask.caS. Sethi, University of Saskatchewan, Canada, sanjeevsethin@hotmail.com

In steady-state simulation, one typically uses time averages to estimate equilibrium expectations. Since forMarkovian systems with up to a million states, means and variances of time averages can be found by usingrandomization, applied probability can make and should make a contribution to this discussion. In our talk,we will, in particular, discuss the issue of the burn-in phase in simulation. Through experimentation, wefound that unless the initial state has a very low probability, using a burn-in phase reduces the mean squareddeviation between the time average and the equilibrium expectation, and one should therefore use a burn-inphase only to obtain a typical initial state. These experiments are then supported by logical arguments.

Ordering of time correlations in stationary Markov processes and queueing net-works

R. Szekli, University of Wroclaw, Poland, szekli@math.uni.wroc.plH. Daduna, Uni Hamburg, Germany, Hans.Daduna@math.uni-hamburg.de

We compare dependence in stochastically monotone Markov processes with partially ordered Polish statespaces using the concordance and supermodular orders. We show necessary and sufficient conditions forthe concordance order in terms of the one step transition probabilities for discrete time processes and interms of the corresponding infinitesimal generators for continuous time processes. We indicate relations ofdependence orderings to the asymptotic variance reduction effect in Monte Carlo Markov Chains (MCMC),to Cheeger’s constants, and to positive dependence for Markov processes. Further we show for X an ergodicJackson network processes with a routing matrix R and X another Jackson network processes having thesame service intensities but with the routing matrix R = [ri j ] such that the traffic solutions of the trafficequation for R and for R coincide, that for arbitrary real functions f, g

( f, QXg)π − ( f, QXg)π = λ/ξ0 Eπ (tr(W f,g(X t ) · diag(ξ) · (R − R))),

where diag(ξ) = (δi, j · ξi : 0 ≤ i, j ≤ J )) is the diagonal matrix with entries of the probability solution ofthe traffic equation, and W f,g(n) = [gi (n) f j (n)]i, j=0,1,...,J . Using this fact we compare time correlationsin networks with different routing but the same product form solution π .

Session WA04 - Networks, Chair: Zazanis in CZ 11

Functional diffusion approximation of the queue with an infinity of servers

L. Decreusefond, ENST Paris, France, decreuse@enst.fr,P. Moyal, Ceremade, Université Paris Dauphine, France, moyal@ceremade.dauphine.fr

We present the diffusion approximation of the measure-valued process ν representing the remaining pro-cessing times of the customers of a queue with an infinity of servers M/GI/∞. To that end we first deter-mine the mean field (ν∗) of a sequence {νn} after a Law of Large Numbers scaling, which is a deterministiccontinuous measure-valued function. Then we prove the weak convergence of the sequence of processes{√

n (νn− ν∗)

}to a local martingale, valued in the space S′ of tempered distribution. To prove this, we

show (i) the tightness of the sequence in the space of RCLL measure valued processes, (ii) the uniquenessof the subsequential limit by solving an integral equation in S′, the integrated transport equation. Wefinally apply these results to give fluid and diffusion approximations of the real-valued processes countingrespectively the congestion and the workload of the system.

133

11 – Abstracts

Diffusion approximations for reinsurance problems

Michael Zazanis, Athens University of Economics and Business, Athens, Greece, zazanis@aueb.grA.N. Yannacopoulos, University of the Aegean, Greece, ayannaco@aegean.grV. Papanicolaou, National Technical University of Athens, Athens, Greece, papanico@math.ntua.grE.N. Frangos, Athens University of Economics and Business, Athens, Greece, nef@aueb.gr

We study the problem of the joint risk process undertaken by an insurer and a reinsurer via a diffusionapproximation. For this model we obtain a partial differential equation, the solution of which provides theprobability that at least one of the two firms will be ruined. Using techniques from the theory of largedeviations for diffusions we obtain asymptotics for this ruin probability. These are compared to exactresults obtained for special cases of the parameters by relating them to the behavior of Brownian motionwith drift in Rd . Finally, the problem of optimal reinsurance in this context is also discussed.

Active-Idleness and the Multiclass Queueing Networks Optimization using Infinites-imal Perturbation Analysis

Nuno Órfão, Polytechnical Institute of Leiria, Portugal, nmorfao@estg.ipleiria.ptCarlos Bispo, Institute for Systems and Robotics, Instituto Superior Técnico, Portugal,cfb@omni.isr.ist.utl.pt

We address the issue of stochastic multiple class queueing networks optimization using Infinitesimal Per-turbation Analysis (IPA). Previous research introduced the concept of Active-Idleness (which consists onforcing a server to stay idle even in the presence of customers waiting to be served) and demonstrated thatis possible to stabilize unstable multiclass queueing networks by slightly changing the scheduling policy.A modified version of the Active-Idleness supervisory controller consisting on the control of relevant end-of-blocking time instants was implemented and lower levels of average work-in-process inventory wereachived. A simulation-based optimization algorithm was formulated for the class-fraction queueing net-work control variables, while the IPA is the methodology used to estimate the gradient of the objectivefunction with respect to it. Discrete versions of the derivative estimation equations and of the perturbationspropagation rules are provided. The optimization algorithm efficiency is evaluated with several numeri-cal experiments performed with classical multiclass queueing networks (Lu & Kumar, Dai and Seidmanmodels).

On Stochastic Network Calculus

Xiaoyue Jiang, Louisiana State University, USA, jiang@lsu.edu

This research is to develop a new form of stochastic network calculus for queuing system performanceguarantees by making use of the Legendre transform at the sample-path level. A symmetrical formulationis resulted in through separate modeling of traffic flows and service capacities. Performance measures atsingle nodes and the service capacities of tandem systems are explicitly bounded at the sample-path levelwhich consequently leads to bounds of the corresponding tail probabilities. Comparison and contrast to ex-isting (deterministic and stochastic) network calculus formulations and to the theory of large deviations areperformed. The applicability of the proposed approach is demonstrated by recovering the “bottleneck" phe-nomenon studied in large deviations under more general settings and by the improvement of scaling factorsknown to network calculus. Finally, perspectives on future research directions for tightening performancebounds are provided.

134

WA05

Session WA05 - Internet, Chair: Fricker in CZ 12

Transient analysis and asymptotic of the mean of the first hitting time for a TCPwindow size process

Andreas H. Löpker, EURANDOM, P.O. Box 513 - 5600 MB Eindhoven, The Netherlands.Email address: lopker@eurandom.tue.nlJohan S. H. van Leeuwaarden, Eindhoven University of Technology and EURANDOM, P.O. Box 513 -5600 MB Eindhoven, The Netherlands. Email address: j.s.h.v.leeuwaarden@tue.nl

The window size in TCP can be modeled as a piecewise deterministic Markov process that increases lin-early in time and experiences downward jumps at Poisson times. We present a transient analysis of thisprocess, the main result being a formula for the Laplace transform of the transient moments. Explicit re-sults for the integer and fractional moments are given, as well as an explicit characterization of the speedof convergence to steady-state. We further present results on the first hitting time of the process, includingpower series for the mean and the Laplace transform and an analysis of the asymptotic behavior. Analo-gous results are given for the recovery time, that is the time the process needs to recover from a downwardjump. Using these results the asymptotic behavior of the maximum process is investigated.

Multiclass G/M/1 queueing system with self-similar input and non-preemptive pri-ority

M. Çaglar, Koç University, Turkey, mcaglar@ku.edu.trM. Iftikhar, The University of Sydney, Australia, mohsinif@it.usyd.edu.auB. Landfeldt, The University of Sydney, Australia, bjornl@it.usyd.edu.auT. Singh, The University of Sydney, Australia

The Internet domains are tied together by service level agreements which are based on various QoS pa-rameters such as delay, jitter, packet-loss rate, throughput and availability. To offer tighter and more com-prehensive service level agreements, accurate modeling of IP traffic and its queuing behavior over theentire network domain is necessary. We present a novel analytical model of a single router which takesinto account multiple classes of self-similar traffic based on G/M/1 queueing system with non-preemptivepriority. Our long-range dependent traffic model is generated by infinitely many sources governed by aPoisson random measure. We derive exact expressions for the transition probabilities of the embeddedMarkov chain of G/M/1 by first deriving the interarrival distribution of the incoming traffic. Closed formexpressions for the expected waiting time of multiple classes have been derived. The queuing system isevaluated numerically for a typical router to show the effect of the Hurst parameter on several performancemeasures. Such work forms a basis for modeling the behavior of self-similar traffic accurately throughheterogenous network domains, eventually leading to the formation of realistic service level agreements.

Sampling Internet traffic

N. Antunes, Universidade do Algarve, Faculdade de Ciências e Tecnologia Campus de Gambelas,8005-139 Faro, Portugal, nantunes@ualg.ptY. Chabchoub, INRIA, Domaine de Voluceau, Rocquencourt BP.105, 78153 Le Chesnay Cedex, France,Yousra.Chabchoub@inria.frC. Fricker, INRIA, Domaine de Voluceau, Rocquencourt BP.105, 78153 Le Chesnay Cedex, France,Christine.Fricker@inria.frF. Guillemin, FT R&D, DAC Lannion, Technopole Anticipa, 2 Avenue Pierre Marzin, 22300 Lannion,France, Fabrice.Guillemin@orange-ftgroup.comPh. Robert, INRIA, Domaine de Voluceau, Rocquencourt BP.105, 78153 Le Chesnay Cedex, France,Philippe.Robert@inria.fr

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The aim of the study is to infer traffic characteristics from a partial knowledge of the traffic, a sampledtraffic. The target characteristic is the distribution of the volume of the flows. Such a problem is motivatedby pricing issues and detection of attacks. The main result is that statistics as the number of flows sampledk times on a given time interval, can be used to estimate the traffic parameters. Expressions involve meansof such variables, and burstiness of traffic has no impact on them. Moreover fluctuations of the statictics aregiven by a central limit theorem. It also explains why deterministic and random samplings are equivalent.The estimates have been tested on a different types of traffic including ADSL traffic and Abilene traces.

Session WA06 - Infinite Multidimensional Markov Chains, Chair: Squil-lante / Taylor in CZ 13

Geometric decay in a QBD process with countable background states with applica-tions to a join-the-shortest-queue model

H. Li, Mount Saint Vincent University, Canada, hli@msvu1.msvu.caM. Miyazawa, Tokyo University of Science, Japan, miyazawa@is.noda.tus.ac.jpY.Q. Zhao, Carleton University, Canada, zhao@math.carleton.ca

A geometric tail decay of the stationary distribution has been recently studied for the G I/G/1 type Markovchain with both countable level and background states. However, it can be only applied to the α-positivecase. In this paper, we specialize the G I/G/1 type to a quasi-birth-and-death (QBD) process. This not onlyrefines some expressions, but also allows us to extend the study, in terms of the matrix analytic method,to non-α-positive cases. We apply the result to a generalized join-the-shortest-queue model, which onlyrequires elementary computations. The obtained results enable us to discuss when the two queues arebalanced in the generalized join-the-shortest-queue model, and establish the geometric tail asymptoticsalong the direction of the difference between the two queues.

Stability Analysis of a General Class of Markov Chains

D. Gamarnik, MIT, USA, gamarnik@mit.eduM.S. Squillante, IBM Thomas J. Watson Research Center, USA, mss@watson.ibm.com

In this paper we consider a general class of infinite multidimensional Markov chains and derive a stabilityanalysis of these stochastic systems. More specifically, we establish exact conditions for the stability ofthis general class of Markov chains using both probabilistic arguments and Lyapunov function techniques,which also elucidates important connections between both sets of related stability arguments based onpurely probabilistic means and Lyapunov techniques. We provide large-deviations bounds and algorithmictechniques for the numerical computation of stability conditions and stationary distributions in order toobtain approximations within a given level of accuracy guarantee.

Some interesting decay rate behaviour in quasi-birth-and-death processes with infi-nite phase spaces

Allan Motyer, University of Melbourne, Australia, a.motyer@ms.unimelb.edu.auPeter Taylor, University of Melbourne, Australia, p.taylor@ms.unimelb.edu.au

Recently there has been considerable interest in the decay behaviour of quasi-birth-and-death processeswhen the phase space is countably-infinite. It has been observed that processes whose transition structurediffers only at level zero can have different decay rates as the level approaches infinity. Furthermore

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sequences of truncations in which the phase space is restricted to be finite, and then allowed to approachinfinity, can be shown to possess limiting decay behaviour that is not the same as that of the infinite system.

In this talk, we shall describe one approach to studying such phenomena and present some examples wherethis interesting behaviour occurs.

Explicit solutions and decay rates for a general class of Markov chains

J.S.H. van Leeuwaarden, EURANDOM, The Netherlands, leeuwaarden@eurandom.tue.nlM.S. Squillante, IBM Thomas J. Watson Research Center, USA, mss@watson.ibm.comE.M.M. Winands, Eindhoven University of Technology, The Netherlands, E.M.M.Winands@tue.nl

In this paper we consider a general class of infinite multidimensional Markov chains and derive explicit so-lutions for their stationary distribution and tail asymptotics. More specifically, we present an exact matrix-geometric solution for the stationary probability vector. We reformulate the probabilistic interpretations ofthe fundamental solution matrices in terms of Bernoulli excursions, leading to explicit expressions for thesematrix elements in terms of hypergeometric functions. We establish asymptotic results on the powers ofone of the fundamental solution matrices, which are used together with the corresponding matrix elementexpressions to obtain explicit large-deviation decay rates.

Session WA07 - Optimal Stopping 2, Chair: Sonin in CZ 14

Fiducial approach in the invariant problem of optimal choice

Vitali Z. Belenky, Central Economics & Mathematics Inst.,Russian Academy of Sci., Moscow,belenky@cemi.rssi.ru

Fiducial probabilities were introduced in 1920s by R. Fisher, and were defined rigorously in the works ofD. Frazer, R. Hora and R. Bueler and used extensively in the works by G. Klimov and A.Kuzmin. Our goalis to demonstrate that fiducial approach is the most fruitful when a decision-maker, who has to choose thebest object, faces a situation of complete uncertainty.

If the criterion of choice is invariant or equivariant then it is natural to require the strategy of choice to beinvariant as well. But if the criterion of choice has no invariant properties then the question "how to state aproblem of choice from distribution Fθ ∈ F with unknown parameter ?" is open. We propose to answerthis question we propose on the basis of fiducial probabilities. The main results are as follows:

1. The inertia principle for free random values is formulated and proved.

2. The fiducial sequence is introduced, and its properties are discussed.

3. We propose a logically noncontradictory statement for the problem of optimal choice from distributionFθ ∈ F with unknown parameter under arbitrary criterion function.

Bandwagon Random Walks

J. Quinn, UNC Charlotte, jequinn@uncc.edu

Some new, basic random walks with dependent increments are introduced. Consider two towers, of possi-bly fractional numbers of boxes, labelled with a ”1” and a ”-1.” Balls are placed in the towers sequentiallywith probabilities corresponding to the relative proportions of the boxes in the towers. If a ball goes in toone of the towers, a possibly fractional portion of a box is added to that tower for the next ball. The randomwalks are defined naturally with linear scaling. We determine the asymptotics for the walks, showing thatthey are governed by beta distributions. While the walks themselves are truly random, their asymptotic

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limits are deterministic after any positive initial time period. Possible applications include situations infinance where, after some initial, uncertain period, a drift direction is picked. Some issues connected withoptimal stoping are also discussed.

A Generalized Gittins Index for Markov Chain and its Recursive Calculation

Isaac Sonin, Univ. of North Carolina at Charlotte, USA, imsonin@email.uncc.edu

We discuss the generalization of the classical Gittins Index for a Markov chain and propose a transparentrecursive algorithm for its calculation. The foundation for this algorithm is a modified version of theElimination algorithm proposed earlier by the author to solve the problem of optimal stopping of a Markovchain in discrete time and a finite or countable state space.

[1] Sonin, I.M.,1999, The Elimination Algorithm for the Problem of Optimal Stopping, Math. Meth. ofOper. Res., 49, no.1, 111-123.[2] Presman E.L., Sonin I.M., Gittins Type Index Theorem for Randomly Evolving Graphs, pp. 567-588,in: From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, Kabanov, Y.; Lipster, R.;Stoyanov, J. (Eds.), Springer, 2006.

Session WA08 - Flexible Servers, Chair: Down in CZ 15

Tandem Lines with Synergistic Servers

Sigrún Andradóttir, School of Industrial and Systems Engineering, Gerogia Institute of Technology,Atlanta, GA 30332-0205, sa@isye.gatech.eduHayriye Ayhan, School of Industrial and Systems Engineering, Gerogia Institute of Technology, Atlanta,GA 30332-0205, hayhan@isye.gatech.eduDouglas G. Down, Department of Computing and Software, McMaster University, Hamilton, OntarioL8S 4L7, Canada, downd@univmail.cis.mcmaster.ca

Consider a tandem queueing network with N stations, infinite supply of jobs in front of station 1, infiniteroom for completed jobs after station N , finite buffers between stations, and M flexible servers. We assumethat several servers can work together on a single job and that the servers are synergistic. Our objective isto characterize the structure of the optimal dynamic server assignment policy and to investigate how theoptimal policy changes as a function of how well the servers work together.

A call center model with upgrades

Mark E. Lewis, Cornell University, Ithaca, NY, USA, mel47@cornell.eduDouglas S. Down, McMaster University, Hamilton, ON, Canada, downd@mcmaster.ca

We consider a call center with two parallel queues. The first queue is equipped with several flexibleservers that can serve either queue. The second is equipped with dedicated servers. Customers that arenot currently in service at the first queue receive upgrades to the second queue after an exponential time inqueue. We discuss structural results of optimal allocations of the flexible servers that are both congruentand incongruent with results in models without upgrades.

Resource allocation in grid computing

Ger Koole, Vrije Universiteit, The Netherlands, koole@few.vu.nlRhonda Righter, The University of California, Berkeley. USA, rrighter@ieor.berkeley.edu

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In grid computing, with many alternative computers available, each with varying extra capacity, and eachof which may connect or disconnect from the grid at any time, it may make sense to send the same task tomore than one computer. The application can then use the output of whichever computer finishes the taskfirst. Thus, the important issue of the dynamic assignment of tasks to individual computers is complicatedin grid computing by the option of assigning multiple copies of the same task to different computers.We show that when tasks have identical size and computer processing times are NWU, maximizing taskreplication stochastically maximizes the number of task completions by any time. That is, it is better todo the same task on as many computers as possible, rather than assigning different tasks to individualcomputers. Computers may be heterogeneous and their speeds may vary randomly, as is the case in gridcomputing environments. We also show that maximal task replication, along with a cµ rule, stochasticallymaximizes the successful task completion process when task processing times are exponential and dependon both the task and computer, and tasks have different probabilities of completing successfully.

Linear Programming Based Affinity Scheduling for Heterogeneous Computing Sys-tems

Douglas G. Down, McMaster University, Canada, downd@mcmaster.caIssam Al-Azzoni, McMaster University, Canada, alazzoi@mcmaster.ca

Resource management systems (RMS) are an important component in heterogeneous computing (HC) sys-tems. One of the jobs of an RMS is the mapping of arriving tasks onto the machines of the HC system. TheHC system is modelled as a system of parallel (heterogeneous) queues. Many different mapping heuris-tics have been proposed in recent years. However, most of these heuristics suffer from several limitations.One of these limitations is the performance degradation that results from using outdated global informationabout the status of all machines in the HC system. Motivated by recent work in the area of queues withflexible servers, we propose a heuristic which addresses this limitation by only requiring partial informationin making the mapping decisions and discuss its performance.

Session WA09 - Inventory 3, Chair: Van Houtum in CZ 16

Approximate evaluation of order fill rates for an inventory system of service tools

I.M.H.Vliegen, Technische Universiteit Eindhoven, The Netherlands, i.m.h.vliegen@tm.tue.nlG.J. van Houtum, Technische Universiteit Eindhoven, The Netherlands, g.j.v.houtum@tm.tue.nl

In this talk, we deal with the analysis of a single-location, multi-item inventory model for service tools,in which coupled demands and coupled returns occur. We distinguish multiple Poisson demand streams.Per stream there is a given set of tools that is requested per demand. We are interested in the order fillrates, i.e., the percentage of demands for which all requested tools are delivered from stock. Requestedtools that are not on stock are delivered via an emergency channel. For the warehouse under consideration,they may be considered as lost sales. Delivered tools are returned to the warehouse after a deterministicreturn time, that is equal for all tools. We develop three approximate models for the order fill rates, whichare all based on Markovian models. One approximate model has appeared to give an underestimation inall computational tests, while the second approximate model has led to an overestimation in all instancestested. The last approximate model combines the other two. This approximate model is very accurate andcan be computed efficiently. Hence, it is appropriate for usage in multi-item optimization algorithms.

Exact and Approximate Results for the Kit Management Problem

R. Güllü, Bogaziçi University, Industrial Engineering Department, refik.gullu@boun.edu.tr

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M. Köksalan, Middle East Technical University, Industrial Engineering Department,koksalan@ie.metu.edu.tr

In this talk we consider a kit planning problem where demand occurrences are not for individual items, butfor kits (a group of items). Each kit contains an arbitrary number of items. Kit demands occur accordingto a Poisson process. Whenever a kit demand occurs, only one item from the kit is used and the rest isreturned. The item that will be used from the kit is not known in advance and the whole kit has to stay atthe demand site for the whole duration. The used item is replenished through a stochastic supply system,with possible capacity limitation. We present exact results for the kit availability probability, and provideapproximate procedures for various performance measures.

Valuation of downstream liquefied natural gas storage

Alan Scheller-Wolf, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA,awolf@andrew.cmu.eduXiaofeng (Mulan) Wang, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA,xiaofenw@andrew.cmu.eduSunder Kekre, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA,sk0a@andrew.cmu.eduNicola Secomandi, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA,ns7@andrew.cmu.edu

The U.S. Energy Information Administration forecasts that local natural-gas production will be unable tomeet demand in the U.S. and most other developed countries, and expects liquefied natural gas (LNG)imports to play a major role in bridging this demand-supply gap. We present a practical real-option modelfor the valuation of LNG downstream storage. Our approach is based on a dynamic programming model todetermine the optimal release policy of the inventory held at the re-gasification terminal into the wholesalenatural-gas market. The model uses a closed queueing network model to represent upstream LNG produc-tion and shipping to the downstream re-gasification facility, and a reduced-form model of the evolution ofthe natural-gas spot price. We show that the optimal LNG inventory-management policy at the downstreamfacility has a basestock-target structure that depends on the natural-gas spot. The structure is nontrivial:When it is optimal to sell it is not necessarily optimal to sell down to zero. We also quantify the value ofthe real option to store LNG, and the relative benefits of this option for different parties involved in an LNGchain. We find that the option is equally valuable for LNG merchants and integrated producers, but that itsrelative benefit is significantly higher for the former than the latter parties.

Analysis of an inventory model with lateral transshipment

A.A. Kranenburg, Centre for Quantitative Methods CQM B.V., The Netherlands, kranenburg@cqm.nlG.J. van Houtum, Eindhoven University of Technology, The Netherlands, g.j.v.houtum@tm.tue.nl

In practice, spare parts supply chains often consist of multiple local warehouses and one or a few centralwarehouses. Local warehouses, located close to customers, are used to ensure prompt replenishment ofspare parts to customers. Stock replenishment for the local warehouses and emergency delivery to thecustomers in case local warehouses are out of stock, is done by the central warehouse(s). Since emergencydelivery from a central warehouse to a customer can cost a lot of time, often an additional option exists:lateral transshipment. In case a local warehouse is out of stock at the moment of a customer’s request, it isfirst tried to obtain the required spare part from a neighboring local warehouse, as this will cost less timethan an emergency delivery from the central warehouse.

If demand at each local warehouse follows a Poisson process and normal replenishment time from thecentral warehouse to the local warehouses is exponentially distributed, an exact analysis of the situationwith lateral transshipment can be done using a Markov Process description. However, this analysis can be

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done for situations with a limited number of local warehouses only, where in practice the number of localwarehouses typically is large. We present an approximate analysis based on the M/G/c/c-queueing model,and show that our results are close to the exact analysis.

Wednesday 10:30am - 12:00pm

Session WB01 - Last Minute Talks, Chair: Boxma in CZ 4

New Algorithms for Laplace Transform Inversion

Peter den Iseger, Cardano, The Netherlands, p.deniseger@cardano.com

The need for advanced numerical techniques in computational probability and finance is rapidly increasing.Many fast and efficient numerical transform inversion algorithms which can be employed to solve many ofthe problems arising in computational probability and finance have already been published in academic lit-erature. However flexible, fast and precise, these algorithms also have many limitations: they cannot tackleimportant problems such as iterative algorithms (for instance recursion, widely employed for dynamic pro-gramming style problems), Levy process based modeling, or problems where the inverse transform as wellas the transform itself needs to be computed (a good example is the waiting time distribution in a G/G/1queue). These inversion algorithms perform the inversion in a number of points at a time. On the otherhand, the numerical inversion method we are about to introduce is a piecewise inversion: we employ aGaussian quadrature rule in order to compute the coefficients of the Legendre expansions of the desiredfunction on all intervals [k, k + 1). The two key differences, which enable the new numerical technique totackle a much wider class of problems than its forebears, can be summarized as follows:

• The original function f (obtained after the numerical inversion) can be evaluated in arbitrary pointsxk (instead of only in points k, k = 0, 1, . . .).

• The new method works both ways: if some numerical values of the original function f are given,then the algorithm generates its Laplace transform L f (thus no closed-form analytical expressionis needed!), and vice versa: given some numerical values of the Laplace transform, the originalfunction f can be obtained in arbitrary points. These properties enable one to use the inversionalgorithm iteratively, hence solving problems such as pricing guaranteed return rate products, fastconvolution transforms, Ladder algorithms and the waiting time distribution in a G/G/1 queue. Highdimensional versions of the algorithms are very efficient since they exploit a embedded Kroneckerproduct structure and use efficient high dimensional versions of the fast Fourier transform. Theprecision and robustness of the algorithms is illustrated by a large number of numerical examples.

Session WB02 - Self-interacting Stochastic Processes, Chair: Van derHofstad in CZ 5

An expansion for self-interacting random walks

M. Holmes, EURANDOM, holmes@eurandom.tue.nl

I will briefly describe joint work with Remco van der Hofstad where we derive an expansion for self-interacting random walks. We will discuss how this expansion can be used to prove monotonicity ofthe drift for “excited random walk”, as a function of the excitement parameter, when the dimension issufficiently high.

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Parabolic Anderson model: Localisation of mass in random media

W. König, Universität Leipzig , Germany, koenig@math.uni-leipzig.deP. Mörters, University of Bath, United Kingdom, maspm@maths.bath.ac.ukN. Sidorova, University of Bath, United Kingdom, n.sidorova@maths.bath.ac.uk

We study the parabolic Anderson problem, i.e., the heat equation ∂t u = 1u + ξu on the d-dimentionalinteger lattice with independent identically distributed random potential {ξ(z) : z ∈ Zd

} and localised initialcondition u(0, x) = 10(x). Our interest is in the long-term behaviour of the random total mass of the uniquenon-negative solution, and we prove the complete localisation of mass for potentials with polynomial tails:We find a random process {Z t : t ≥ 0} such that at time t the mass of the system is completely localisedin Z t in the sense that limt↑∞ u(t, Z t )/

∑z∈Zd u(t, z) = 1, in probability. We also identify the asymptotic

distribution of Z t .

Excited to the center

Gady Kozma, gady.kozma@weizmann.ac.il

Place a cookie on every vertex on Zd and perform a random walk that, upon entering a vertex with a cookieon it, eats the cookie and then does a single step with a constant drift towards the center 0. When it returnsto the same vertex and there is no cookie there any more, it does simple random walk. We show that thismodel is recurrent in all dimensions.

Sampling for the maximum of Brownian motion

Johan S.H. van Leeuwaarden, Eindhoven University of Technology, The Netherlands,j.s.h.v.leeuwaarden@tue.nl

Say we observe a standard one-dimensional Brownian motion with negative drift at time points t0 =0, t1, t2, . . . only, where the times in between two consecutive points are i.i.d. distributed according tosome r.v. T with ET = ω−1. Here, ω represents the mean number of observations per unit of time, orthe sampling frequency. By increasing the sampling frequency the sampled version will converge to theBrownian motion. In this talk we investigate, depending on the choice of T , the speed of convergence as afunction of the sampling frequency.

We consider three types of sampling: (i) equidistant sampling, where T = ω−1, (ii) random sampling,where T follows an exponential distribution, and (iii) Erlangian sampling, where T follows an Erlangdistribution. For comparison, we consider the difference between the all-time maxima of the Brownianmotion and its sampled version. This difference tends to zero as ω→∞. All three sampling methods leadto a random walk of which the distribution of the maximum allows for an explicit solution.

Session WB03 - Stochastic Programming and Optimization 2, Chair:Veatch in CZ 10

Efficient Allocation Under Incomplete Information

Apostolos N. Burnetas, University of Athens, Greece, aburnetas@math.uoa.gr

Consider an allocation problem where one or more resources must be distributed to several activities sothat the total benefit is maximized. The benefit from each activity is characterized by a random coefficient

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with unknown distribution. We derive an adaptive policy that combines efficiently estimation of the alloca-tion benefits with expected profit maximization. The policy combines ideas from duality in optimization,stochastic approximation and multi-armed bandit theory.

A problem of optimal sequential pricing

D.J. Hodge, University of Cambridge, United Kingdom, D.Hodge@statslab.cam.ac.uk

We present an investigation into the optimal pricing regime for selling a set of n perishable goods via asequence of daily prices. The model is one which arises naturally online when purchasing items fromwebsites, where customers see today’s price and choose to either buy today, or shop around and checkback tomorrow hoping for a lower price. Under general assumptions we form the stochastic dynamicprogramming equations and describe how to solve them. We then take a highly plausible specific exampleand prove some results concerning the monotonicity, or not, of prices over time. Combining this with someresults concerning the character of the optimal value function, we can then quickly find the optimal pricingpolicy without lengthy search. The application of this problem to the Internet is particularly appropriate.A website administrator is now able, via site statistics, to determine how many customers have requestedto see the price and how many of those purchased or returned later.

Cutting speed adjustment and tool magazine capacity allocation using stochasticdynamic programming

B.F. Lamond, Laval University, Canada, bernard.lamond@fsa.ulaval.ca

We present a stochastic dynamic programming model for the optimal operation of a flexible machine tool,to find a sequence of cutting speeds for the successive tools used in a cutting operation, in order to min-imize expected makespan. We assume the tool life is random and the expected tool life decreases withcutting speed, and each tool requires a setup time when manually installed. Using a numerical method forcomputations, we compare the optimal dynamic policies with well-known static policies and we examinetheir impact on tool magazine capacity allocation among several part types and on expected makespan. Ourresults explain why dynamic adjustment of cutting speed is better with random tool life.

Approximation, structure and performance for average cost queueing network con-trol

Michael Veatch, Gordon College, Wenham, Massachusetts, mike.veatch@gordon.edu

Differential cost approximations yield tractable optimality equations, solved as an LP, for moderate sizequeueing networks. For small examples, analysis of the structure of this LP allows characterization of theaverage cost bound and the associated policy. This analysis and interpretation of the dual assist in choosingapproximations with good performance. Numerical results are given for larger networks.

Session WB04 - Fluid Limits and Mean Fields, Chair: D’Auria in CZ11

Mean field analysis of discrete-time closed queueing networks with geometric ser-vice times.

I.J.B.F. Adan, Eindhoven University of Technology, The Netherlands, iadan@win.tue.nl

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P. Beekhuizen, EURANDOM, The Netherlands, beekhuizen@eurandom.tue.nlB. D’Auria, Universidad Carlos III de Madrid, Spain, bernardo.dauria@uc3m.es

We consider a discrete-time closed queueing network with M servers and N circulating customers. At eachtime the servers have equal probability p > 0 to serve one customer from their queue if present; the servedcustomers are then routed with equal probability 1/M to any queue of the network. We study the meanfield dynamics of this network as the number of servers M → ∞ and N/M → ν = const. We provethat in the thermodynamical limit the random process corresponding to the network dynamic converges toa deterministic dynamical system that is the unique solution for the limiting infinite system of Kolmogorovdifferential equations. This model is relevant for the analysis of packet switches in ATM and Local AreaNetworks.

Spontaneous resonances and the onset of collective behavior in queuing networks

A.A. Vladimirov, Institute for Information Transmission, Russia, vladim@iitp.ruA.N. Rybko, Institute for Information Transmission, Russia, rybko@iitp.ruS.B. Shlosman, Centre de Physique Theorique, CNRS, Marseille, France, shlosman@cpt.univ-mrs.fr

Poisson Hypothesis assumes that a large network goes in a reasonable time to a state where the incomingflows to every node are Poisson ones with almost constant rate. We construct here a closed network, forwhich this is not the case: almost from every initial state we go to the synchronized state, in which the flowto all the nodes stays periodic in time. The fluid model of this network has a periodic attractor, and wedemonstrate that the non-linear Markov process in the mean field mode has a close attractor provided theload is high enough.

Conic Instability of fluid limit models

Bara Kim, Korea University, Korea, bara@korea.ac.kr

We introduce conic instability of fluid limit models and then we show that a stochastic network is unstablein the sense that a representing Markov process for the network is not Harris recurrent, if a correspondingfluid limit model is conically unstable.

This work is closely related to Meyn (Annals of Appl. Prob., 1995) and Dai (Annals of Appl. Prob., 1996).Meyn proved that if starting from any (nonzero) initial condition the fluid limit model explodes at a linearrate, then the associated queueing network explodes. Dai introduce weak instability of a fluid limit modelwhich means that every fluid limit starting from zero exits zero after a while, and then he showed thatif a fluid model is weakly unstable, then the associated queueing network explodes. The works of Meynand Dai have pros and cons: Meyn does not have to deal with fluid limits starting from zero, but needsto consider all other fluid limits. Dai has to consider only fluid limits that start from zero. However, fluidlimits starting from zero are most difficult to be characterized in some examples.

Conic instability introduced in this work is determined by only fluid limits starting from nonzero points ina conic region. We give some examples where the conic instability is suitably applied. The result of thiswork can be used as one of general tools that show instability of stochastic networks.

Session WB05 - TCP, Chair: Baccelli in CZ 12

Iterated function systems and ergodicity of AIMD networks

C. King, Northeastern University, USA, king@neu.eduR. Shorten, The Hamilton Institute, NUI Maynooth, Ireland, robert.shorten@nuim.ie

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F. Wirth, University of Bremen, Germany, fabian@math.uni-bremen.deD. Leith, The Hamilton Institute, NUI Maynooth, Ireland, doug.leith@nuim.ie

In this talk we consider communication networks that employ drop-tail queueing and Additive-IncreaseMultiplicative-Decrease (AIMD) congestion control algorithms. A basic assumption in the study of suchnetworks is that the underlying stochastic process is ergodic. Here we present a proof of ergodicity undervery general assumptions. Fairness formulae for the average congestion window size, and for throughputare also given.

Our starting point is a recently derived algebraic model that captures the essential features of networks inwhich TCP is deployed. This model may be analyzed using methods from matrix semigroup theory, inparticular results on paracontractive sets of matrices. This observation leads to the consideration of themodel as an iterated function system with a contractivity property for which several ergodicity results areavailable.

Extensions to nonlinear and adaptive versions of congestion control algorithms will be discussed.

Queueing Theory for TCP

D. Wischik, University College of London, D.Wischik@cs.ucl.ac.uk

Most traffic in the Internet is controlled by TCP, an algorithm which adjusts the transmission rate of a trafficflow in response to the congestion it perceives in the network. In the core of the Internet, where there aremany TCP flows, the aggregate traffic behaves predictably, both at a stochastic level and a fluid level. Thispredictability is the basis for queueing theory for TCP.

I will describe the natural limiting regime which permits us to obtain limit theorems for TCP, and showhow the stochastic and fluid parts of the limit mesh together. In fact, there are three different ways theycan mesh together, depending on how big the queue’s buffer is – by changing the buffer size we obtaindifferent features of the TCP models of Baccelli, Kelly, and Srikant. I will also describe the sorts of trafficdescriptors which are appropriate for capturing the burstiness of TCP traffic.

I will finish with practical application of this work, an answer to the question: how big should buffers be incore Internet routers?

A stochastic model of Split TCP

F. Baccelli, INRIA ENS, Paris, françois.baccelli@ens.frG. Carofiglio, Politecnico di Torino/INRIA ENS, Paris, giovanna.carofiglio@ens.frS. Foss, Heriot-Watt University, Edinburgh, s.foss@ma.hw.ac.uk

In wireless networks, TCP is proved to poorly perform by over-reacting to losses due to radio transmissionerrors that it wrongly attributes to congestion. The “split-connection” approach has been proposed in orderto achieve higher throughput in mixed wired-wireless scenarios, where it operates a separation at transportlayer between the wired and the wireless part. According to Split TCP, each TCP connection traversingtwo different media is split into two halves and a relay node is placed in the middle to temporarily storein-transit packets.In the model we developed we aimed at the characterization of TCP throughput on both links as wellas of buffer occupancy at the relay node. A representation of the system is given in terms of piecewise-deterministic Markov process and its stability is proved by exploiting some sample pathwise continuity andmonotonicity properties.We also analyzed the tail asymptotics for the buffer occupancy in the stationary regime.

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The Equations of TCP

Francois Baccelli, École normale supérieure, Paris France, Francois.Baccelli@ens.frKi Baek Kim, Samsung Electronics Co., Suwon, Korea, kkb@di.ens.frDavid McDonald, University of Ottawa, Ottawa Ontario, Canada, dmdsg@uottawa.ca

We present a class of transport equations arising from stochastic models of congestion control. This classcontains two cases of loss models as particular cases: the rate-independent case where the packet loss rateis independent of the throughput of the flow and the rate-dependent case where it depends on it. This classof equations covers both the case of persistent (TCP) flows and of non-persistent (HTTP) flows. There isa unique density solving the associated differential equation. This density and its mean value are providedas closed form expressions thereby generalizing the well know "square root formula" for persistent TCPflows to non-persistent HTTP flows.

Session WB06 - Call Centers 3, Chair: Koole in CZ 13

Modeling arrivals

Philippe Chevalier, Université catholique de Louvain, Belgium, philippe.chevalier@uclouvain.beJean-Christophe Van den Schrieck, Université catholique de Louvain, Belgium,vandenschrieck@poms.ucl.ac.be

The peakedness of an arrival flow is an indirect measure of the variability. It can be used to characterizearrivals in a much broader way than more conventional renewal models. In this talk we try to highlightsome properties of the peakedness as measured on some real data collected in different settings.

An approximation method for evaluating the performances of a skill-based routingcall center model

Oualid Jouini, Ecole Centrale Paris, France, walid.jouini@ecp.frMohamed Salah Aguir, Ecole Superieure de Technologie et d’Informatique, Tunisia, aguirms@yahoo.frYves Dallery, Ecole Centrale Paris, France,yves.dallery@ecp.fr

We consider a call center with multiple types of customers and multiple groups of specialized agents.Customers of a given type are only served by an agent of their corresponding group. In addition, thereis a flow of customers having “general" questions that can be served by agents of any group. Such asituation can be encountered in various applications such as bank call centers. First, we propose a skill-based routing architecture to operate our system. We develop a model in which we define the rule ofassigning “general" customers to specialists. Next, we develop an approximation method to numericallyevaluate the performance measures of each type of customers. The performance measures of interest arerelated to queueing delays. We conduct an extensive simulation study in order to examine the accuracyof the approximation method. The comparison between numerical and simulation results shows that theapproximation method is fairly accurate. Finally, we demonstrate how “general" customers are useful tocounterbalance the lack of pooling due to only having specialist groups. It appears, indeed, that a smallproportion of “general" customers allows to approximately attain the same performances as in a fullypooled structure.

Design and Capacity Planning in Multi-Skill Call Centers

E.L. Örmeci, Koç University, Turkey, lormeci@ku.edu.tr

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N. Çakan, Georgia Institute of Technology, U.S.A., Nesrin.Cakan@mgt.gatech.eduZ. Aksin, Koç University, Turkey, zaksin@ku.edu.trF. Karaesmen, Koç University, Turkey, fkaraesmen@ku.edu.tr

Call centers typically receive different types of calls, which require different skills from the agents. Atone extreme, agents can be trained to answer only one type of calls, which decreases the training costs atthe expense of operational inefficiency, while at the other extreme they can be trained to answer all typesto achieve high operational efficiency, only with high training costs. Obviously, a wide range of possibletraining options lies in between the two. Hence, one of the major issues in call centers is to decide on thecross-training level of agents, or equivalently to design the skill-sets available to the agents as well as thestaffing levels in each set. We first model the problem to determine the staffing levels for a given skill-setdesign as a two-stage stochastic program, which is solved by a simulation-based optimization procedureusing sample-path gradient estimation. Then we compare the performance of different skill-set designsunder different scenarios on the call rates, profits and training costs.

Dynamic call blending in the presence of time-varying arrivals

A. Balint, VU University Amsterdam, The Netherlands, abalint@few.vu.nlS. Bhulai, VU University Amsterdam, The Netherlands, sbhulai@few.vu.nl

Call blending (dynamically mixing inbound and outbound traffic) is an efficient method in call centers forobtaining high productivity and high service levels simultaneously. For time-homogeneous Poisson arrivalprocesses, a near optimal policy assigns agents to outbound calls if the number of idle agents exceeds acertain threshold. Based on this policy, we investigate how call blending can be implemented dynamicallyin the presence of time-varying arrival processes. We develop easily implementable methods resulting in aconsiderable improvement compared to the static near optimal policy with a fixed threshold and traditionalforecasting methods.

Session WB07 - Dynamic Control for Service Management, Chair:Hampshire in CZ 14

A Modified Fluid Model for the Optimal Design of a Dynamic Call Center

Bill Massey, Princeton University, wmassey@princeton.edu

A wide range of real time services can be modeled as loss systems. Numerous examples arise in the serviceindustry including: agent staffing for call centers, provisioning bandwidth for private line services, makingrooms available for hotel reservations and congestion pricing for parking spaces. In general these servicescharge arriving customers a price for admission and customers expect a specific quality of service level.Given that arriving customers make their decision to join the system based on the current service price, themanager may use price as a mechanism to control the utilization of the system. A major objective for themanager is then to find a pricing policy that maximizes total revenue while meeting the quality of servicetargets desired by the customers.

For systems with growing demand and service capacity, we provide a dynamic pricing algorithm. A keyfeature of our solution is congestion pricing. Our solution uses demand forecasts to anticipate future servicecongestion and set the present price accordingly. The optimal price is a function of the number of customersin the system and a “future congestion tax". We also quantify revenue tradeoffs between service capacityexpansion and quality of service changes.

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Inventory rationing for a system with a lost-sales and a backordering customer class

P. Enders, Carnegie Mellon University, USA, penders@cmu.eduI.J.B.F. Adan, Technische Universiteit Eindhoven, The Netherlands, i.j.b.f.adan@tue.nlG.J. van Houtum, Technische Universiteit Eindhoven, The Netherlands, g.j.v.houtum@tue.nl

We consider a single product that is kept on stock in a single stockpoint. A first class of customers has highpriority and demands from this class are lost if they are not immediately satisfied from stock. A secondclass of customers has a lower priority and those demands are backordered if they cannot be satisfied fromstock. For this class, one may even decide to backorder demands when the physical stock is low and onerather keeps those last items on stock for possible high priority demands. Demands arrive according toPoisson processes for both customer classes, we assume a deterministic replenishment leadtime, and wedistinguish linear inventory holding and penalty costs. We consider the class of critical level policies, underwhich the inventory position is kept at a fixed basestock level and low priority demands are backordered ifthe physical stock is at or below a fixed critical level at the moment of arrival. The objective is to minimizeaverage costs. We discuss the evaluation of a critical level policy via a matrix-geometric approach, wederive monotonicity results via Markov reward theory, we develop a method for the optimization withinthe class of critical level policies, and we study the distance to the real optimal solution.

Dynamic Pricing and Capacity Planning of e-Services under Demand Uncertainty

Cathy Xiu, IBM TJ Watson Research

In IT services markets, the demand for a service offering is generally governed by the price and the qualityof service. Customers have different profiles and they respond differently to price and service level offer-ings. The service level that a firm can offer depends upon its capacity and also the demand. Thereforeaccurate demand estimation requires the incorporation of the inherent interdependence between demand,price, quality of service and capacity. In this paper, we present a dynamic modeling framework to capturesuch interdependence. We further propose novel solutions for dynamic pricing and capacity planning underdemand uncertainty.

Efficient Frontier Dynamic Pricing for On-Demand Music and Movie Services

Robert C. Hampshire, Carnegie Mellon University, hamp@cmu.edu

The online distribution of music, television programming, and movies is growing. The recent emergenceand popularity of iTunes, Real, Napster, eMusic, YahooMusic, and a host of other digital media distributionservices is evidence of an expanding marketplace. However, these services face demand fluctuations,uncertain advertising revenue, and high costs for obtaining digital content.

Managers of digital media distribution services seek to increase the operational efficiency of their busi-nesses. A strategy for pricing the digital media content under dynamic demand and in the presence oflicensing costs are essential for survival. We develop dynamic pricing policies that directly consider de-mand fluctuations and the variability in the amount of content downloaded by customers. We view theservice price as a mechanism to hedge demand and purchasing uncertainty.

Pricing policies are presented for both the risk neutral and risk sensitive manager. In the risk neutral case,we analytically characterize the optimal service price for the bounded elastic demand case in terms ofa competitive service price . Additionally, for the risk neutral case, we provide a tool for managers todetermine whether a free advertisement supported service or a user supported service is most appropriate.

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Session WB08 - Queueing Models 3, Chair: Brandt in CZ 15

Necessary and Sufficient Conditions for Finite First Moment of the Workload Com-ponents in FIFO Multiserver Queues

A. Scheller-Wolf, Carnegie-Mellon University, USA, awolf@andrew.cmu.eduR.A. Vesilo, Macquarie University, Australia, rein@ics.mq.edu.au

In this talk, we derive first moment results for all the components of the stationary workload vector instable FIFO G I/G I/s queues (previous work has dealt only with the smallest component of the vector,the delay) . Our moment conditions incorporate the interaction between service time distribution, trafficintensity (ρ) and the number of servers in the queue. Sufficient conditions so that EWi <∞, where Wi isthe i’th smallest component of the ordered workload vector, depend crucially on the traffic intensity relativeto i , specifically on whether i ≤ k + 1 or i > k + 1, where ρ satisfies k < ρ < k + 1.

To be precise, if we denote a generic service time random variable by S, For i ≤ k+1, EWi <∞ providedthat E Sβ(i) <∞, where β(i) = (s − k + 1)/(s − k).

For i > k, EWi <∞ provided that E Sβ(i) <∞, where β(i) = (s − i + 1)/(s − i).

Finally for S in the class Lβ

1 (i), defined in [2], these conditions are also necessary.

References

[1] Scheller-Wolf, A., Necessary and Sufficient Conditions for Delay Moments in FIFO Multiserver Queues:Why s Slow Servers are Better than One Fast Server for Heavy-Tailed Systems. Operations Research, 51,pp. 748 - 758, 2003.

[2] Scheller-Wolf, A. and Vesilo, R., Structural Interpretation and Derivation of Necessary and SufficientConditions for Delay Moments in FIFO Multiserver Queues, Queueing Systems, 54, 221 - 232, 2006.

Comparison on Queueing Systems with Generalized Processor Sharing Scheduling

Sunggon Kim, Department of Information Statistics, Gyeongsang National UniversityJu Yong Lee, Department of Electrical & Computer Engineering, University of TorontoWonyoung Park, Department of Electrical Engineering and Computer Science, KAIST

Generalized processor sharing(GPS) service policy is a scheduling algorithm to allocate the bandwidth of aqueueing system with multi-class input traffic. The service policy is work-conserving and the backloggedqueues share the full bandwidth according to the preassigned ratios. Thus, the service policy guarantees aminimum bandwidth for each class of traffic while it yields the multiplexing gain. In a queueing systemwith single-class traffic, the performance such as waiting time or the overflow probability to a thresholdbecomes to be better if the bandwidth(i.e. the service rate) of the system increases. We consider thesame problem for GPS systems. We obtain a sufficient and necessary condition under which one queueingsystem shows performance at least as good as the other for any input process. We also consider the modegeneral problem. Suppose that some guaranteed bandwidth of a system is given and we want to determinethe other guaranteed bandwidths so the this GPS system shows performance at least as good as a givenGPS system. We obtain a condition so that we can calculate the other guaranteed bandwidths. In numericalresults, we show that all performance do not deteriorate if the guaranteed bandwidths satisfy the proposedcondition. We also investigate the sharpness of the sufficient conditions by numerical study. We expect thatthe proposed condition can be a guideline for changing GPS parameters when QoS requirements change.

Approximations for the M/G I/N + G I Type Call Center

Foad Iravani, University of Toronto, Department of Mechanical and Industrial Engineering, 5 King’sCollege Rd., Toronto, ON M5S 3G8, Canada, iravani@mie.utoronto.ca

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Barıs Balcıoglu, University of Toronto, Department of Mechanical and Industrial Engineering, 5 King’sCollege Rd., Toronto, ON M5S 3G8, Canada, baris@mie.utoronto.ca

In this study, we propose approximations to compute the steady-state performance measures of the M/G I/N+G I queue receiving Poisson arrivals with N identical servers, and general service and abandonment-timedistributions. The approximations are based on scaling a single server M/G I/1 + G I queue for whichwe suggest a practical way to compute the waiting time distributions and their moments. To do this, wemake use of the Laplace transform of the workload density function. Our first contribution is within thecontext of the M/G I/1+G I queue in obtaining the workload Laplace transform when abandonment-timedistributions are of Phase-type. Next, we scale-up the M/G I/1+ G I queue giving rise to our approxima-tions to capture the behavior of the multi-server system. As a practical application area, we further employthe approximations in determining the minimum staffing levels in a call center to meet the target valuesof certain performance measures, such as serving a proportion of customers within a specified time, andkeeping the portion of abandoning customers below set limits.

Waiting times for M/M systems under generalized processor sharing

M. Brandt, Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Germany, brandt@zib.deA. Brandt, Humboldt-Universität zu Berlin, Germany, brandt@wiwi.hu-berlin.de

We consider a system where the arrivals form a Poisson process and the required service times of therequests are exponentially distributed. According to the generalized processor sharing discipline, eachrequest in the system receives a fraction of the capacity of one processor which depends on the actualnumber of requests in the system. We derive systems of ordinary differential equations for the LST andfor the moments of the conditional waiting time of a request with given required service time as well asa stable and fast recursive algorithm for the LST of the second moment of the conditional waiting time,which in particular yields the second moment of the unconditional waiting time. Moreover, asymptoticallytight upper bounds for the moments of the conditional waiting time are given. The presented numericalresults for the first two moments of the sojourn times in the M/M/m − P S system show that the proposedalgorithms work well.

Session WB09 - Combinatorial Structures, Chair: Gamarnik / Ra-manan in CZ 16

Probabilistic analysis of game tree evaluation

R. Neininger, J.W. Goethe University Frankfurt, Germany, neiningr@math.uni-frankfurt.de

In the analysis of game-searching methods a basic problem is to determine the value of the root of aminimax tree with certain given numbers at its leaves (the input). Various models for the input and relatedevaluation algorithms have been proposed and analyzed.

We review some of these models, in particular models with input from the set {0, 1} and probabilistic mod-els such as Pearl’s model and the (random) incremental model. We discuss the complexity of evaluationalgorithms under these models: a new tail bound for the complexity of Snir’s randomized evaluation algo-rithm is given improving upon a Gaussian tail bound due to Karp and Zhang. Also a limit law for the root’svalue in Pearl’s model is given leading to a conjecture on the asymptotic distribution of the complexity ofα − β pruning.

The talk is based on joint works with Ali Khan and with Ali Khan and Devroye.

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WC01

On Temperate Regions in Random Groves

Y. Baryshnikov, Alcatel-Lucent, Bell Labs, New Jersey, ymb@research.bell-labs.com

Random groves process studied by Petersen and Speyer exhibits a frozen/temperate type of behavior similarto what is seen in random tilings of Aztec diamonds. In this work we justify these simulational observationsby finding the exact asymptotics for probabilities of edges using the complex analytic techniques. Joint withRobin Pemantle.

The mixing advantage is less than 2

Kais Hamza, kais.hamza@sci.monash.edu.au

Given n positive random variables X1, . . . , Xn , with specified expected values of n independent and iden-tically distributed copies of X1, . . . , Xn respectively, we obtain upper and lower bounds on the expectedvalue of the maximum of n independent copies of X1, . . . , Xn . We also present related comparison results.

Iterative Algorithm for Maximum Weight Independent Set

S. Sanghavi, Massachusetts Institute of Technology, sanghavi@mit.eduD. Shah, Massachusetts Institute of Technology, devavrat@mit.eduA. Willsky, Massachusetts Institute of Technology, willsky@mit.edu

Given a graph G = (V, E), an independent set I is a subset of V such that no two vertices in I areconnected in G. When vertices of G are assigned weights, the weight of I is the sum of weights of verticesin it. The question of finding an independent set with the maximum weight arises in communicationnetworks, distributed databases, parallel computation, etc. In these applications, algorithms are required tobe distributed for architectural reasons. The popular max-product algorithm for finding the max. weightindependent set (MWIS) is an iterative, simple heuristic. However, a little is known about convergence andcorrectness of the max-product algorithm in the context of MWIS.

In this talk, we will discuss the correctness and convergence of a variant of max-product to find max.wt. independent set for any bipartite graph with arbitrary non-negative weights. More generally, we willcharacterize the performance of this algorithm for arbitrary graph. We will also discuss some implicationsof convergence of max-product algorithm on integrality gap of a natural LP relaxation of MWIS.

Wednesday 1:30pm - 3:00pm

Session WC01 - Risk and Insurance, Chair: Kolbe in CZ 4

Optimal Insurance Demand and Investment in a Dynamic Mean-variance Frame-work

E. Biffis, Cass Business School, United Kingdom, enrico.biffis.1@city.ac.uk

We determine optimal insurance and investment strategies for mean-variance agents exposed to sourcesof risk only partially correlated with traded financial securities. We allow for constraints on the agents’strategies, uncertainty in model parameters and jumps in the dynamics of relevant risk factors. If the agentis an insurer, we obtain optimal retention levels for proportional reinsurance arrangements and examinewhen it is optimal to swap insurance risks for financial risks under regulatory constraints.

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Long-term survivor mixture models for credit scoring

S.F. Mo, City University of Hong Kong, Hong Kong SAR, msmosf@cityu.edu.hkK.K.W. Yau, City University of Hong Kong, Hong Kong SAR, mskyau@cityu.edu.hk

The long-term survivor mixture model, which is an extension of ordinary survival model, is applied forcredit risk management for individual borrowers by allowing risk-free (i.e. not at risk) borrowers. Notonly estimate the time-to-default of the borrowers, mixture model also predicts the probability of being atrisk. Comparing with traditional statistical models, mixture model shows better, or at least comparable,performance for credit scoring. It provides additional information about the borrowers’ default risk, whichassists the lending institutions to better manage credit risk.

Stability and equilibria in incomplete markets

Gordan Žitkovic, University of Texas at Austin, U.S.A., gordanz@math.utexas.edu

A study of continuity properties of demand functions of risk-averse agents investing in an incompletemarket has been initiated in the work of Larsen and Žitkovic (2006), and further developed in Kardaras andŽitkovic (2007). In this talk I will present a summary of those results and an extension of the fixed-pointtheorem of Knaster, Kuratowski and Mazurkiewicz beyond the realm of locally convex spaces. Finally, Iwill present an outline of a research program where the combination of the two is used to estabilsh existenceof financial equilibria in incomplete markets.

Valuation and pricing of reverse mortgages

Andreas Kolbe, HVB-Institute for Mathematical Finance,Munich University of Technology, Germany, kolbe@ma.tum.deRudi Zagst, HVB-Institute for Mathematical Finance,Munich University of Technology, Germany, zagst@ma.tum.de

Reverse mortgages are an innovative loan product for older homeowners. The homeowner receives either alump sum or a fixed annuity from the mortgage lender. The reverse mortgage contract is terminated whenthe mortgagor dies or sells the house. At this (random) point of time, all outstanding debt including allaccrued interest or, if the outstanding loan amount exceeds the house value, the house sale proceeds arepaid back to the mortgage lender.We develop a consistent valuation framework for reverse mortgage contracts based on reduced-form in-tensity models as used in credit risk modelling. Within our modelling framework we explicitly calculatethe probability that the total loan amount exceeds the house value at termination of the contract and derivethe maximum payment(s) which can be made to the homeowner under certain constraints. We apply ourresults to data from Germany, where reverse mortgages are not yet available, and discuss implications forthe design of reverse mortgages from a lender’s perspective.

Session WC02 - Spatial Models 2, Chair: Foss / Stolyar in CZ 5

Dynamics of pattern formation in multi-hop wireless networks using backoff-basedMAC protocols

Mathilde Durvy, LCA3-ISC-IC, Station 14, EPFL, CH-1015 Lausanne, Switzerland,mathilde.durvy@epfl.chPatrick Thiran, LCA3-ISC-IC, Station 14, EPFL, CH-1015 Lausanne, Switzerland,patrick.thiran@epfl.ch

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The design of Medium Access Control (MAC) protocols for multi-hop ad hoc networks is challenging, aseach network entity has only a local view of the network. Consequently, efficient, decentralized, MACprotocols are needed.

In this talk, we demonstrate that a protocol based on a local backoff mechanism (similar to the popularIEEE 802.11 protocol) can globally organize the transmissions in a network, in patterns of simultaneousnon interfering transmissions. We use a Markovian loss network model with local interactions to explainwhy such MAC protocols can achieve the maximal level of spatial reuse of the wireless medium (and athroughput close to optimum), but also why they can lead to severe unfairness among competing nodes –they can even favor spontaneously some connections to such an extent that the others suffer from virtuallycomplete starvation. Using this model, we analyze the impact of different parameters of the network andprotocol on the spatial reuse and fairness.

Optimal throughput allocation and stability in a spatial random-access model

A.L. Stolyar, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA, stolyar@research.bell-labs.com

We consider a model of random access (“Slotted-Aloha-type”) communication networks of general topol-ogy. Assuming network links receive exogenous arrivals of packets for transmission, we seek dynamicdistributed random access strategies whose goal is to keep all network queues stable. We consider twosuch strategies, both queue length based, and show their stability, as long as input rates are within thenetwork saturation rate region.

A Random Multiple Access Protocol with Spatial Interactions

C. Bordenave, University of California at Berkeley, USA, charles.bordenave@gmail.comS. Foss, Heriot-Watt University, Edinburgh, UK, foss@ma.hw.ac.ukV. Shneer EURANDOM, Eindhoven, The Netherlands, shneer@eurandom.tue.nl

We adopt the classical ALOHA scheme to the case when the transmissions in a multiple-access systemcreate noise in a certain neighbourhood of the receiver. We prove that the fluid limits of the system’sworkload satisfy a certain differential equation and, via an analysis of this differential equation, obtainconditions sufficient for the system to be stable. We also discuss the behaviour of the fluid limits.

Routing in Sensor Networks by Using Spatial Dynamics

Anthony Ephremides, University of Maryland, tony@eng.umd.edu

The problem of routing in ad hoc wireless networks has been heavily worked on but mainly from theperspective of traditional delay performance criteria. In wireless sensor networks the performance criteriaare different. There is a need to conserve energy and a need to perform the mission of the sensor network.We focus on networks that are deployed to detect the presence of targets. Thus the performance criterion isthe probability of correct detection and the concurrent achievement of energy efficiency.The entire networkmust be designed to support these objectives. The determination of optimal routes is prohibitively complexunless the performance criteria can be mapped into link metrics. In our approach we exploit the spatialdynamics of the signals emitted by the target (and measured by the sensors) to achieve such a mapping.We obtain actually a simple link metric as a result that is, however, predicated on a tandem arrangement ofsensors. As a result the use of this metric alone for performing routing is unsatisfactory. Heuristic "fixes"can be applied to translate the result to practical schemes. However, a truly two-dimensional approach isalso possible and we outline its preliminary formulation. This presentation is based on joint work withseveral people from Cornell University, notably Lang Tong and his students, or ex-students, YoungchulSung, Saswat Misra, and Animashree Anandkumar.

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Session WC03 - Perturbation and Discrete Events, Chair: Heidergottin CZ 10

Law of Large Numbers for products of random matrices in the max-plus semiring

G. Merlet, CEREMADE, Université Paris-Dauphine, FRANCE, glenn.merlet@gmail.com

We analyze the asymptotic behavior of the sequence of random variables (x(n, x0))n∈N defined by x(0, x0) =

x0 and x(n+ 1, x0) = A(n)x(n, x0), where (A(n))n∈N is a stationary and ergodic sequence of random ma-trices with entries in the semiring (R ∪ {−∞},max,+).Such sequences model a large class of discrete event systems, among which timed event graphs, 1-boundedPetri nets, some queuing networks, train, communications or computer networks. The sequence satisfy astrong law of large numbers (LLN) means that the asymptotic output or throughput exists. Thus, there hasbeen many paper about this subject, under various additional hypotheses.Using results by J. Mairesse and T. Boush on the backward scheme for such sequences, we give a set ofnecessary conditions for the LLN to be satisfied. Thanks to the interpretation in terms of paths on weightedgraph, techniques of subadditivity and results on finite state Markov chain, we prove a theorem to haveconverse results. This theorem implies all previous LLN for such sequences and allows us to prove thatone of the necessary condition is sufficient when the matrices are i.i.d. .

Optimal short-term utilization bounds on multiclass networks

N. M. Órfão, Polytechnical Institute of Leiria, Leiria, Portugal, nmorfao@estg.ipleiria.ptC. F. Bispo, Instituto de Sistemas e Robótica, Instituto Superior Técnico, Lisbon, Portugal,cfb@isr.ist.utl.pt

For multiclass queuing networks operated with distributed scheduling policies, previous work proposed theutilization of idling policies to address the stability problem. A controller, termed Time-Window Controller,was constructed where each class has a short-term bound on the amount of time it is allowed to occupyits server. Networks previously shown to be unstable were stabilized by an instance of this controller,not by changing policy, but by allowing the original policy to become non work-conserving. Also, somepreliminary simulation results showed the controller to have the potential to improve performance of anygiven distributed scheduling policy, even for stable networks.

To evaluate the controller’s ability to improve the performance of a given distributed scheduling policy,it is necessary to develop an efficient methodology to determine the optimal short-term bounds. We willaddress this problem using Infinitesimal Perturbation Analysis, IPA, to compute gradient estimates froma single simulation run. The expressions for the derivatives and their respective propagation rules willbe introduced. The validity of the IPA approach as a method to produce unbiased gradient estimateswill be addressed, and some preliminary computational studies will be presented, and their consequencesdiscussed.

Some new results on the performance optimization of queueing systems

Li Xia, IBM China Research Laboratory, xiali98@gmail.comWenjun Yin, IBM China Research Laboratory, yinwenj@cn.ibm.comJin Dong, IBM China Research Laboratory, dongjin@cn.ibm.com

Concerning a closed Jackson network with M servers and N customers, we propose a new method to op-timize the service rates. The objective is to minimize (or maximize) the customer-average performance,which is different from the traditional time-average performance. As we know, the steady-state distribu-tion of closed Jackson networks has the product-form solution. We use this special property to aggregate

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the perturbation realization factors, which are fundamental concepts of the Perturbation Analysis (PA) inqueueing systems. With this aggregated perturbation realization factors, we derive the performance differ-ence equation when the service rates are changed. Then, a type of policy iteration algorithms is developed.The complexity of this algorithm (the number of aggregated perturbation realization factors) is linear tothe system size, although the system state space will grow exponentially. Thus, our method survives the“dimensionality curse" of this type of problems. Simulation results demonstrate its efficiency. Our resultscan be viewed as a complementary method for the traditional Perturbation Analysis in queueing systems,which uses the gradient-based algorithms to do optimization.

Sensitivity analysis for Gaussian systems, a mathematical “urban legend” busted

B. Heidergott, Vrije Universieit, The Netherlands, bheidergott@feweb.vu.nlF. Vázquez-Abad, The University of Melbourne, Australia, vazquez@ms.unimelb.edu.auW. Volk-Makarewicz, The University of Melbourne, Australia, warrenvm@gmail.com

Infinitesimal Perturbation Analysis (IPA) is a methodology that uses a stochastic derivative to estimate thegradient of the expected value of a performance function, with respect to some parameter of the model.It is unbiased only when the expectation and derivative operations can be interchanged. For almost twodecades, many papers on gradient estimation for discrete event driven systems have claimed that IPA is thepreferred method, if applicable, and it is broadly accepted today as the method with minimal variance. TheScore Function (SF) and Weak Derivative (WD) methods can deal with discontinuities. They build directlythe derivative of a probability measure with respect to some parameter of the model.

In this talk we will focus on a ubiquitous class of problems where the dynamics of the system follows aGaussian model. They arise whenever the processes driving uncertainties are described by "cumulativesmall errors", and are used in Physics, Chemistry, Economics, Finance, Engineering, etc. We will presenttheoretical results that show that the WD estimator for the derivative with respect to the variance of anormal random variable actually has smaller variance than IPA, when the performance is any polynomialfunction of the variable, provided that the degree is higher than 1. For an exponential, the parameterspace is divided into two distinct regions where either IPA or WD exhibit smaller variance. These resultscontradict and provide the first rigourous proof that the mathematical "urban legend" is, in fact, unfounded.We will discuss the results for a more complex Gaussian system using a stochastic activity network, wherewe compare the three methods and we will discuss applications of our methodology to the problem ofsensitivity estimation when pricing financial derivatives.

The potential impact of our results is the encoding of a robust “one-size-fits-all" kind of algorithm toestimate the gradients of Gaussian systems performances with respect to either the mean or the varianceof the error terms. It is robust because WD (like SF) is a measure-valued derivative estimator that isconstructed from the derivative of the joint normal distribution, and it is unbiased for a large class ofperformance measures. To make our method efficient, we have optimised the CPU time for the generationof the random variables required, as well as introduced a particular correlation that may reduce the varianceof the final estimators.

Session WC04 - Large Deviations 2, Chair: Ignatiouk-Robert in CZ11

Large Deviation and Fluctuation of Diffusion Processes with discontinuous drift

T.S.Chiang, Academia Sinica, Taiwan, matsch@math.sinica.edu.twS.J.Sheu,Academia Sinica, Taiwan, Sheusj@math.sinica.edu.tw

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11 – Abstracts

For the system of d-dim stochastic differential equations

d X ε = b(X ε)dt + εdW (t), t∈[0, 1]X ε = x0

∈ Rd

where b(.) is smooth except possibly along the hyperplane x1 = 0, we shall present a large deviationtheorem for small ε. If in addition the drift satisfies the stability condition b1(x) < 0 if x1 > 0 andb1(x) > 0 if x1 < 0, we also present a fluctuation theorem for X ε(t), i.e., we shall find a deterministicfunction φ(t) such that

(X ε(t)− φ(t))/ε = ξ(t)+ Rε(t)

where ξ(t) is an Orstein-Uhlenbeck process and |Rε(t)|→0 in L1 as ε→0 for each t .

Large deviations asymptotics of exit time and exit location for a class of Lagrangiansystems

Agnessa Kovaleva, Space Research Institute, Russian Academy of Sciences, Russia, a.kovaleva@ru.net

We construct explicit asymptotic estimates of exit time and exit location for a class of the Lagrange systemswith weak additive noise. The appropriate mathematical basis to analyse the long-term escape processes inweakly perturbed systems is large deviations theory. Large deviations principle introduces a deterministiccost (action) functional that must be minimized by the “most likely” exit path. Derivation of the costcriterion and solution of the associated Hamilton-Jacobi equation are the central points of the theory.

We introduce the action functional for the Lagrange systems and derive the Hamilton-Jacobi equation of theassociated variational problem. The logarithmic asymptotics of the first exit time is found as the solutionof the latter equation. The closed-form solution is constructed as a sum of two terms, related to kinetic andpotential energy of the system, respectively. The exit location on the boundary of the reference domain andthe terminal state of the unconstrained coordinates at the moment of exit is defined. Applicability of thetechniques developed to the risk analysis and control problems is illustrated by examples.

Martin boundary with large deviation technique for a random reflected randomwalks on a half-space

Irina Ignatiouk-Robert, Irina.Ignatiouk@u-cergy.fr

To identify the Martin boundary for a transient Markov chain having Green’s function G(x, y) one haveto identify all possible limits limn→∞ G(x, yn)/G(x0, yn) with yn “tending to infinity”. For homogeneousrandom walks, these limits are usually obtained from the exact asymptotics of the Green’s function G(x, y).For non-homogeneous random walks, the exact asymptotics of the Green’s function is an extremely difficultthis problem. We identify Martin boundary for a reflected random walks on a half-space by using largedeviation technique combined with the ratio limit theorem. The full Martin compactification is in generalnot homeomorphic to the “radial” compactification obtained by Ney and Spitzer for the homogeneousrandom walks in Zd : convergence of a sequence of points yn to a point of on the Martin boundary does notimply convergence of the sequence yn/|yn| on the unit sphere. The existence of non-radial limits is relatedto non-linear optimal large deviation trajectories.

Application of Tauberian theorem to the exponential decay of the tail probability ofa random variable

K. Nakagawa, Nagaoka University of Technology, Nagaoka, Niigata 940-2188, Japan,nakagawa@nagaokaut.ac.jp

156

WC05

We give a sufficient condition for the exponential decay of the tail probability of a non-negative randomvariable. We consider the Laplace-Stieltjes transform of the probability distribution function of the randomvariable. In the previous work, we proved that if the singularities of the LS transform are only a finitenumber of poles, then the tail probability decays exponentially. However, this condition is not easy toverify because the axis of convergence is not a compact set. In this paper, we show a theorem that if theabscissa of convergence of the LS transform is negative finite and the real point on the axis of convergenceis a pole of the LS transform, then the tail probability decays exponentially. For the proof of the theorem,we extend and apply so-called a finite form of Ikehara’s complex Tauberian theorem by Graham-Vaaler.

Session WC05 - Bandwidth Sharing Networks, Chair: Borst in CZ 12

Bandwidth-Sharing Networks in Overload

R. Egorova, CWI/Eindhoven University of Technology, The Netherlands, r.egorova@cwi.nlS.C. Borst, Eindhoven University of Technology/CWI, The Netherlands, Bell Laboratories, USA,sem@win.tue.nlB. Zwart, Georgia Institute of Technology, USA, bertzwart@gatech.edu

Bandwidth-sharing networks as considered by Massoulié & Roberts provide a natural modeling frameworkfor describing the dynamic flow-level interaction among elastic data transfers. Under mild assumptions, ithas been established that a wide family of so-called α-fair bandwidth-sharing strategies achieve stability insuch networks provided that no individual link is overloaded.

In the present talk we focus on α-fair bandwidth-sharing networks where the load on one or several of thelinks exceeds the capacity. In order to characterize the overload behavior, we examine the fluid limit, whichemerges from a suitably scaled version of the number of flows of the various classes. We prove that thefluid limit is a linear function of time, and derive a fixed-point equation for the corresponding asymptoticgrowth rates. It is further shown that the fixed-point solution is also a solution to a related strictly concaveoptimization problem, and hence unique. We use the fixed-point equation to investigate the impact of thetraffic intensities and the variability of the flow sizes on the asymptotic growth rates.

Stability of bandwidth sharing networks with monotone service rates

Matthieu Jonckheere, CWI, The Netherlands, matthieu.jonckheere@cwi.nl

We examine the stability of multi-class queueing systems with the special feature that the service rates ofthe various classes depend on the number of users present of each of the classes. As a result, the vari-ous classes interact in a complex dynamic fashion. Such models arise in several contexts, especially inbandwidth sharing networks, and in wireless networks, as resource sharing algorithms become increas-ingly elaborate, giving rise to scaling efficiencies and complicated interdependencies among traffic classes.We summarize existing methods and results for establishing stability in this context and provide an exactcharacterization of the stability region under certain monotonicity assumptions. We also discuss weakernotions of stability like rate stability. The results are then illustrated for simple examples of networks withtwo or three classes of users.

Importance sampling in rate-sharing networks

P. Lieshout, CWI, The Netherlands, lieshout@cwi.nlM. Mandjes, University of Amsterdam, The Netherlands, mmandjes@science.uva.nl

We consider a single-node network, where classes share capacity in a Discriminatory Processor Sharingfashion. Assuming Poisson arrivals and exponentially distributed service requirements for each class, the

157

11 – Abstracts

dynamics of the user population may be described by a Markov process. We focus on the probability that,given the network is in some state at time zero, the network is in some other state at a different time. Inparticular, we assume that the underlying event is rare, such that this probability is extremely small. As noclosed-form expressions are available for this probability, an attractive approach may be to resort to Monte-Carlo simulation. Due to the rarity of the event under consideration, however, Monte-Carlo simulation isinfeasible. A natural approach to speed up the simulation is to use Importance Sampling (IS). We presentan IS algorithm to accelerate the simulation that is based on large-deviations results. Extensive simulationexperiments show that the algorithm performs well.

Flow-level Stability of Data Networks with Non-convex and Time-varying Rate Re-gions

J. Liu, Department of Electrical Engineering, Princeton University, NJ, US, jiapingl@princeton.eduA. Proutière, KTH, Radio Communication Systems, Sweden, alexandre.proutiere@radio.kth.seY. Yi, Department of Electrical Engineering, Princeton University, NJ, US, yyi@princeton.eduM. Chiang, Department of Electrical Engineering, Princeton University, NJ, US, chiangm@princeton.eduH. V. Poor, Department of Electrical Engineering, Princeton University, NJ, US, poor@princeton.edu

We investigate the flow-level stability and performance of utility-maximizing allocations in data networkswith non-convex and time-varying rate regions. We consider the Markovian flow-level model as adoptedin most of the contemporary work, with stationary and independent data arrivals and exponential file sizedistribution. In case of fixed and non-convex rate region or time-varying rate region, we provide the stabilityconditions for a large class of allocations, such as the MaxProjection policy and the α-fair policy, andfind out that the stability region depends on the particular allocation. Moreoever, for two-class networksunder α-fair allocations, we show that the stability region generally decreases when the allocation becomesfairer. This is in sharp contrast with the substantial existing literature on stability under fixed and convexrate regions, in which the stability region coincides with the rate region for many utility-based resourceallocation schemes, independent of the value of the fairness parameter. We further study the impact ofthis fairness-stability tradeoff on the system performance, e.g. average throughput and mean flow responsetime, and conduct numerical experiments to illustrate the new stability region and the performance vs.fairness trade-off.

Session WC06 - Biology, Chair: Spencer in CZ 13

Determining Hidden Markov Models Efficacy in gene finding problem

A. Kazemnejad , Tarbiat Modares University, Tehran, Iran, aklili@yahoo.comE. Hajizadeh , Tarbiat Modares University, Tehran, IranK. Mirjafari , Tarbiat Modares University, Tehran, Iran

One class statistical models is Hidden Markov Models (HMM) which are extensively used to analyzeseveral problems such as speech recognition and finding genes which are implicated in causing cancer.Now a day most of methods for prediction of genes are based on these models. In this paper we showhow weighting affects on Generalized HMM efficacy in gene finding problems in eukaryotes. For thisaim we choose a simple structure of HMM and entered Open Reading Frames (ORF) information witha weight matrix to model and the appropriate values of weights were estimated based on two criteria ofsensitivity and specificity in samples of DNA sequence. Results showed entering weights to model havecaused significant effect on efficacy and is comparable to existing gene finding methods.

158

WC07

Count or control? Optimal monitoring for invasive species management

M.A. McCarthy, University of Melbourne, Australia, mamcca@unimelb.edu.auA. Moore, University of Melbourne, Australia, a.moore@ms.unimelb.edu.auP.G. Taylor, University of Melbourne, Australia, p.taylor@ms.unimelb.edu.au

There exists an extensive body of research on mathematical models for the optimal control of biologicalpopulations. However, in order to apply these models to real world situations monitoring is required (to pa-rameterise models and track population size) and despite being far from trivial, monitoring costs are rarelyincluded when comparing management regimes. We consider the task of controlling an invasive pest popu-lation given a fixed budget which must be split between control and monitoring, where monitoring data arebinomial observations of an associated index of the population size. We present a relatively simple modelfor tracking this index and embed this in a Markov decision model to investigate the trade off betweenmonitoring and control. Preliminary results are somewhat counter-intuitive, and suggest that managersshould only monitor when they believe the population is small, that is, in a desirable state. Preliminaryresults have also demonstrated that a seemingly sensible objective function can lead to somewhat counter-intuitive results, suggesting that it is difficult to intuitively predict optimal monitoring-control regimes.This highlights the need for mathematical rigour in designing management regimes.

Using stochastic epidemic models to determine if emerging avian influenza can es-cape containment

S.E.F. Spencer, University of Nottingham, UK, simon.spencer@maths.nottingham.ac.ukP.D. O’Neill, University of Nottingham, UK, philip.o’neill@nottingham.ac.uk

Much research effort is currently focused on determining the effectiveness of control measures proposed tohalt the spread in humans of an emerging strain of avian influenza, such as the H5N1 subtype. This workuses a multi-type Branching Process of households to assess the probability that an outbreak will escapecontainment; even if the control measures are thought to be effective. The effect of different infectiousperiod distributional assumptions on the containment probability can also be explored using stochasticorderings.

Session WC07 - Games 2, Chair: Weber in CZ 14

Symmetric Rendezvous Games

J. Fan, Cambridge University, UK, jf328@cam.ac.uk

Two people are placed randomly on two distinct places(nodes) of a complete graph Kn or a circle Cn . Ateach step, they can move to any other place or stay put and are assumed to follow an identical randomstrategy. The aim is to rendezvous in a minimum time of steps. This is called a rendezvous problem andis like the case that we get separated in a park or a supermarket. It is easy to see that the best strategy whenthere are only two locations is to visit each one with equal probability at every step. However, the caseof 3 locations becomes difficult and is not solved until recently but by a strategy stated in 1990. It is alsoconjectured that the optimal strategy is not changed if the two players will overlook each other even whenthey are at the same place. This talk will discuss those various strategies on the rendezvous problem.

On two-stopping and stopping game problems

K. Szajowski, Institute of Mathematics and Computer ScienceWrocław University of Technology

159

11 – Abstracts

Wybrzeze Wyspianskiego 27, PL-50-370 Wrocław, Poland, Krzysztof.Szajowski@pwr.wroc.pl

The mathematical modeling of economic systems in stochastic environment leads to various mathematicaloptimization problems. If the decision problem consists in choice of intervention moment one can for-mulate the model of such case as the optimal stopping problem (see e.g. papers by Batabyal and Beladi,Clarke and Reed). If it is allowed to react more than once the approach depends on the number of decisionmakers and their aims. If there is one decision maker and two reactions we have the optimal two stoppingproblem. When there are two decision maker and their have prescribe aims we usually treat the problem asthe stopping game. In the game approach the communication problem between player should be taken intoaccount. It leads to various concepts of solutions. In this paper some double stopping problems and thetwo person stopping game models will be presented. The presentation is based on the results by Karpowiczand Szajowski.

Symmetric rendezvous search

R.R. Weber, University of Cambridge, UK, rrw1@cam.ac.uk

In the symmetric rendezvous search game played on K3 (the completely connected graph on 3 vertices)two players are initially placed at two distinct vertices (called locations). At each step a player can eitherstay where he is or move to a different location. The players wish to minimize the expected number ofsteps until they meet. Rendezvous search games of this type were first proposed by Steve Alpern in 1976.They are simple to describe, and model circumstances that most of us have met in real life. However, theirsolution has proved to be notoriously difficult. The symmetric rendezvous search game on K3 is the firstinteresting game of this type to have been solved. We discuss its solution, as well as several new toolsfor addressing such problems. One of these is semidefinite programming (which has been called “linearprogramming for the 21st century").

Session WC08 - Queueing Models 4, Chair: Vlasiou in CZ 15

Analysis of a queueing system with synchronized services and setup times.

A. Economou, University of Athens, Department of Mathematics Section of Statistics and OperationsResearch, Greece, aeconom@math.uoa.grS. Kapodistria, University of Athens, Department of Mathematics Section of Statistics and OperationsResearch, Greece, stellakap@math.uoa.gr

In recent years there is an emerging interest for queueing systems with synchronized services. Such sys-tems occur both in manufacturing (e.g. assembly lines) and in telecommunications (e.g. webseminars).Nevertheless, the literature on this kind of systems is scarce due to the mathematical intractability. In thistalk we will present a basic Markovian model which encompasses the characteristics of synchronized ser-vices and setup times. We obtain several performance descriptors concerning the number of customers, thebusy period and the waiting time distributions. We show that the theory of basic hypergeometric functions(also known as q-calculus) can be used to simplify some computations and to analyze the limiting behaviorof the performance measures.

A Geo/G/1/∞ queueing system with a generalized N -policy and setup-closedowntimes

P. Moreno, Pablo de Olavide University, Spain, mpmornav@upo.es

160

WC09

We analyze a discrete-time single-server queueing system whose arrival stream is a Bernoulli process andservice times are generally distributed. An extension of the N -policy is considered: the server remains idletill the queue length becomes i ≥ 1 with probability θi . After the idle (vacation) period, the server needs arandom amount of startup time before serving. Moreover, after the system is empty in a departure epoch,the service station employs a random time to close down the system, although the shutdown is interruptedand the service is resumed immediately without setup if an arrival takes place during such a period. Byusing the supplementary variable technique, we obtain the queue and system length distributions as well assome performance measures. We also find out the distributions and mean values of the different periods ina cycle and the waiting time. Finally, a cost model and several numerical results are presented.

Production inventory with service time and vacation to the server

A. Krishnamoorthy, Department of Mathematics, Cochin University of Science and Technology, Kochi682022, India, ak@cusat.ac.in,Viswanath C. Narayanan, Department of Mathematics, Government Engineering College,Sreekrishnapuram, India, viswam@cusat.c.in

A production inventory with positive service time is considered. Whenever the level of the inventoryreaches s due to demand (one at a time), a production mechanism is activated. Items are added one at a time.Production continues until inventory level is brought to S. When the inventory level is zero no customeris allowed to join (those who are already in the system are allowed to stay back. We obtain analyticalsolution to the system state distribution under the assumption of exponential distribution associated withinter-arrival, service and time between two successive additions to the inventory due to production. We thenextend the results to the case where arrival of external customers form an MAP; service times are iid PHdistributed,; lead times follow what we term as Markovian Lead Time process (MLP). Further no customeris permitted to join the system when the inventory level is zero and when there either no inventory or nocustomer or both are absent, then the server goes on a multiple vacation. Duration of each vacation is PHdistributed with appropriate representation. The system state probability distribution is derived. Severalperformance measures are computed. Numerical illustrations are provided. An optimization problem isalso attempted.

A multi-station queue with dependent interarrival and service times

Maria Vlasiou, Georgia Institute of Technology, USA, vlasiou@gatech.eduIvo Adan, Eindhoven University of Technology, The Netherlands, iadan@win.tue.nlOnno Boxma, Eindhoven University of Technology, The Netherlands, boxma@win.tue.nl

In this talk we discuss a single-server multi-station alternating queue where the interarrival times and theservice times are auto- and cross-correlated. We examine two cases. In the first case, interarrival andservice times depend on a common discrete time Markov chain. In the second case, we assume that theservice times depend on the previous interarrival time through their joint Laplace transform. The waitingtime process is directly analyzed by solving a Lindley-type equation by transform methods. Numericalexamples are included to demonstrate the effect of the autocorrelation of and the cross-correlation betweenthe inter-arrival and service times.

Session WC09 - Health Care, Chair: Ziya in CZ 16

Optimal outpatient appointment scheduling

G.C. Kaandorp, Erasmus Medical Center Rotterdam, The Netherlands, g.kaandorp@erasmusmc.nlG. Koole, VU University Amsterdam, The Netherlands, koole@few.vu.nl

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11 – Abstracts

We study optimal outpatient appointment scheduling. A local search procedure is derived that converges tothe optimal schedule with a weighted average of expected waiting times of patients, idle time of the doctorand tardiness (lateness) as objective. No-shows are allowed to happen. For certain combinations of param-eters the well-known Bailey-Welch rule is found to be the optimal appointment schedule. A tool in whichthis algorithm is implemented can be found on the internet: www.math.vu.nl/obp/healthcare/software/ges.

Nurse Staffing and Bed Capacity within Hospitals Units

O. Jennings, otisj@duke.edu

We model a hospital unit as a two-node queueing network. Patients arrive exogenously and, once space isavailable, are assigned beds. The two nodes represent the states between which patients alternate: needyand content. Needy patients require a nurse to restore them to the content state. Content patients eitherbecome needy or are discharged from the hospital. We consider the effect of staffing and bed capacity onsystem dynamics and quality of service.

Managing New Patient Admissions in Multi-site Healthcare Facilities

E. Tekin, Texas A&M University, United states, eylem@tamu.edu

Many healthcare providers own and operate multiple facilities that are geographically distributed, possiblywithin a metropolitan region. Due to random patient arrivals, the allocation of workloads to facilities maynot be always balanced, resulting in long waiting times. Hence, some new arriving patients would bewilling to switch from their primary choice of healthcare facility to avoid long waits. In this research, weinvestigate strategies to manage new patient admissions among multiple healthcare facilities to maximizelong-run average revenues while minimizing the expected waiting times.

Identifying the Critical Patient: Finding the Balance between Undertriage and Over-triage

S. Ziya, University of North Carolina, Chapel Hill, N.C, U.S.A., ziya@unc.eduN.T. Argon, University of North Carolina, Chapel Hill, N.C, U.S.A., nilay@unc.edu

Triage is the process of classifying patients into different priority groups according to their injuries. Mis-takes during triage are common and they fall under one of two categories: undertriage, which is assigning apatient a lower priority than what the patient’s condition actually requires, and overtriage, which is assign-ing a higher priority than what the patient’s actual condition calls for. In mass-casualty incidents, a certainlevel of undertriage and overtriage is unavoidable. However, the triage officers can choose to be eitherlenient towards undertriage or overtriage with their decisions. In this talk, we use simple queueing modelsto develop a better understanding of how the decisions need to be made depending on conditions like thepatient load and the characteristics of the patients’ injuries. We mainly consider two different scenarios:(i) a war-type environment where there is a constant flow of patients coming in and (ii) a single incidentisolated in time where there is some finite number of patients in need of treatment. We provide severalinsights on the “optimal” triage decisions. For example, we find that the effect of patient load on the triagedecisions depends on the injury characteristics of the patients. In particular, we find that when the patients’injuries are not deadly, with the increasing patient load, the triage officers should be more lenient towardsovertriage.

162

12Author Index

A

Aalto, S., MA05Adan, I.J.B.F., MB08, WB04,WB07Aksin, Z., TA06, WB06Al-Azzoni, Issam, WA08Aldous, David, TD02Al-Ibrahim, Assil, TC07Alp, O., MA09Al-Seedy, R.O., MC08Amir, Madjid, MB07Ammar, S.I., MC08Andradóttir, Sigrún, WA08Andrews, D.M., TA04Ano, K., TD07Antunes, N., TB05, TD05, WA05Areepong, Y., TA07Argon, N.T., WC09Armony, Mor, MB06, MB06Artalejo, J.R., MA08Asgari, N., TD09Ata, Baris, MA04Atar, R., MC04, TB07Avrachenkov, K., TC09, TC09Avram, Florin, MC01, MC01Awad, H.P., WA03Ayesta, U., MA05Ayhan, Hayriye, WA08

B

Baccelli, F., TA02, WB05, WB05Bae, Y.H., MB05Baek, Sangkyu, TC05Bai, X.X., MC09, TC01Balcıoglu, Baris, WB08Balint, A., MB02, WB06Ballotta, Laura, TB01Barkai, E., TC02

Baron, Opher, MC09Baryshnikov, Y., WB09Basrak, Bojan, TA07Bassamboo, A., MA04, TA03,TA06Bäuerle, N., TD01Bayati, M., TA09Bean, Nigel, TC06Beekhuizen, P., TC05, WB04Bekker, R., TA05Belenky, Vitali Z., WA07Benichou, Olivier, MB09Bensoussan, A., MA01Berestycki, J., TB02, TB02Berestycki, N., TB02, TB02Berman, Oded, MC09Bertsimas, D., TA04Bhattacharya, Debasis, TC01Bhulai, S., MB05, WB06Biermé, H., MA02, MA02Biffis, E., WC01Bijvank, M., MC09Bispo, C.F., MA07, WA04,WC03Blanc, Hans, TC08Blanchet, Jose, MA03, MB03,TA03, TD03, TD03Blaszczyszyn, B., TA02, TD02Blatter, A., TD01Bogachev, L., TB02Bordenave, C., TB05, TD02,WC02Borst, S.C., WC05Botev, Z.I., TC03, TC03Boucherie, R.J., MA09, TB08Boxma, O.J., MA05Bramson, M., MC05Brandt, A., WB08Brandt, M., WB08Brun, O., TD08

Budhiraja, A., MC04, TB07,TB07Buke, Burak, TA04Burnetas, Apostolos N., WB03Buyuk, Buket, TD09Buyukkaramikli, N.C., MA09

C

Caglar, M., WA05Cai, Ning, TA01Cakan, N., WB06Camia, Federico, MB02Carofiglio, G., WB05Cerný, R., TA02Chabchoub, Y., WA05Chaintreau, A., TB04Chen, Nan, TA01Chen, Zhiyong, MA03Chevalier, Philippe, WB06Chiang, M., MC05, WC05Chiang, T.S., WC04Chick, S.E., MB03Choi, Bong Dae, MB05, MB05,MB05, TC05, TC05, TC05Choo, Taek Sik, MA08Cil, E.B., TC07Coakley, Kevin, TD06Collamore, J.F., TB01Cont, R., MA01, TA01Courcoubetis, Costas, TA05Cudina, M., MC07

D

Dabrowski, A., TD04Daduna, H., WA03Dai, Jim, MA05, MB06Dallery, Yves, WB06

163

12 – Author Index

Da Silva, Ana, TC06D’Auria, B., WB04Debicki, K., MC01De Boer, P.T., TA03Decreusefond, L., WA04De Graaf, W.F., TB08Degrimenci, I.T., MB06DeGuest, R., MA01De Kok, Ton, TB09Denardo, E.V., MA07Den Iseger, Peter, WB01Dieker, A.B., MB08Dimitrova, Dimitrina, TB01Ding, Hongwei, MA09, MC03Ding, L., TD06Dolgin, A., TC03Donato, D., TC09Dong, Jin, MA09, MC03, MC09,TC01, WC03Doshi, V., TA09Down, Douglas G., TC04,WA08, WA08, WA08Dube, Parijat, MB07Dupuis, Paul, MA03, MA03,TB07Durvy, Mathilde, WC02

E

Economou, A., MB07, MB07,WC08Egorova, R., WC05Eliazar, Iddo, MB09, WA01El-Shehawy, S.A., MC08El-Sherbiny, A.A., MC08Enders, P., WB07Ephremides, Anthony, WC02Es-Saghouani, A., TC08Estrade, Anne, MA02Ettl, M., TB09, TB09Eveleens, J.L., MA06

F

Fan, J., WC07Farahani, R.Z., TD09Feinberg, E.A., MA07Fernandez, Emmanuel, TC07Ferreira, F., MC06Firoozshahian, Amin, TA09Fontes, L.R.G., MB02, MB02Fort, G., WA03Foss, S., WB05, WC02

Fralix, B.H., MC08Frangos, E.N., WA04Fricker, C., TB05, WA05Funken, S., TA02

G

Gamarnik, D., MB04, TA04,WA06Ganesh, A.J., TD02Gans, N., MB03Garcia, J.M., TD08Gautam, N., TB08Ghosh, A.P., MC04Giardina, C., MC02Gibbens, R.J., TA05Giesecke, Kay, TA01Glasserman, Paul, MA03, MA03,TA01Glazebrook, K.D., TD06Glynn, P.W., MB03, TC03,TD03, WA03Gnedin, A.V., TD07Goel, Ashish , TA09Gómez-Corral, A., TC08Graham, C., TC04Grassmann, W.K., WA03Gromoll, H.C., MB04, TC04Guillemin, F., TB04, TD03,WA05Gullu, R., WA09Günalay, Y., TD09Gunawan, Indra, MC03Gupta, D., TD09Gupta, Varun, MA05Gurler, U., MA09Gurvich, I., MA04, MB06

H

Haas, P.J., MB03Hagwood, Charles, TD06Haijema, R., MA07Hajizadeh, E., WC06Hampshire, Robert C., WB07Hamza, Kais, WB09Hansen, J., MB04Harchol-Balter, Mor, MA05Hasenbein, John J., TA04Hautphenne, S., TC06He, Q.-M., TC06Heidergott, B., TB03, TB03,TB03, WC03

Heideveld, S.A., MA09Helber, S., TA06Helmes, K., TD07Henderson, Shane G., MA06,MB03Henken, K., TA06Heyde, C.C., MA01Hodge, D.J., WB03Holmes, M., WB02Hooghiemstra, Gerard, WA02Hordijk, A., TB03, TB03Hosseinkeh, Vahideh, TA09Huang, P., TB09Huang, Wei, TB01Hult, H., TD01, TD03Hwang, Eunju, TC05, TC05Hwang, Gang Uk, TD05

I

Iftikhar, M., WA05Ignatiouk-Robert, Irina, WC04Ishizaki, Fumio, TD05

J

Jacinto, Gonçalo, TD05Jagabathula, S., TA09Jagers, P., WB09Jain, Rahul, MB07Janssen, A.J.E.M., TB06, TB06Jaskiewicz, A., MC06Jelenkovic, P.R., TB04Jennings, O., MB06, TA04,WC09Jiang, X., WA04Jonckheere, Matthieu, WC05Jongbloed, G., MA06Jouinni, Oualid, WB06Juneja, Sandeep, MA03, TA03,TA03Jung, K., TA04

K

Kaandorp, G.C., WC09Kager, W., WA02Kaishev, Vladimir, TB01Kaj, Ingemar, MA02Kalra, Himanshu, TA03Kang, Weining, MC04Kanta, S., MB07, MB07

164

Kapodistria, S., WC08Kapuscinski, R., TB09Karaesmen, F., TC07, WB06Katz, D., TA04Kazemnejad, A., WC06Kekre, Sunder, WA09Kella, Offer, WA01Keppo, J., MA01Kerner, Yoav, MC08Key, P.B., TA05Khanchi, Aziz, TD04Kim, Bara, WB04Kim, Ki Baek, WB05Kim, Kyung Jae, MB05, MB05,MB05, TC05, TC05Kim, Sujin, MB03Kim, Sunggon, WB08Kim, Tae Ok, MB05, MB05King, C., WB05Kirkbride, C., TD06Klafter, Joseph, TC02Koch, I., MB01Koijen, Ralph S.J., MB01Koksalan, M., WA09Kolbe, Andreas, WC01König, W., WB02Koole, G., MA06, WA08, WC09Kou, S.G., MA01, TA01Kovaleva, Agnessa, WC04Kozma, G., WB02Krampe, A., TC01Kranenburg, A.A., WA09Krishnamoorthy, A., WC08Kroese, D.P., TC03, TC03, TC03Kruk, Lukasz, TC04Kuhnt, S., TC01Kulkarni, Vidyadhar, MB08,TB01Kumar, Sunil, MA04, TA09Kurchan, J., MC02Kurtz, Thomas G., MB02Kurushima, A., TD07Kyprianou, A., MC01

L

Lamond, B.F., WB03Landfeldt, B., WA05Latouche, Guy, TC06, TC06Leahu, Haralambie, TB03Lebedev, D., TC09Lee, Chihoon, MC04Lee, J., TD04

Lee, Ju Yong, WB08Lee, N.H., MB04Lee, Seon Mi, TC05Lee, Yong Hyun, TC05Lehoczky, J., TC04Leith, D., WB05Leizarowitz, Arie, TC07Lelarge, Marc, MC07, TB04,TD02Leskelä, L., MC06Lewis, Mark E., WA08Li, H., TC06, WA06Liese, M., MA01Lieshout, P., WC05Limic, V., TB02Lin, Wuqin, MA04Lindskog, F., TD01Litvak, Nelly, TC09, TC09Liu, J., WC05Liu, Zhen, TB04Löpker, Andreas H., WA05Lu, Y., MC05, TB09Lu, Z.Q. John, TD06Luczak, M.J., WA02

M

Machihara, F., MA08MacPhee, I., TA08Magdziarz, M., TC02Maglaras, Constantinos, MB06Mahabhashyam, S.R., TB08Mahmoodi, S., MA02, TB07Majewski, Kurt, TD04Malhotra, R., TB08Mandjes, M.R.H., TA05, TA05,TB08, TB08, TC08, WC05Margolius, B.H., MB08Massey, Bill, WB07Massoulié, Laurent, MC05,TB05Mazumdar, R.R., TB04, TD03McCarthy, M.A., WC06McDonald, D.R., TB05, TD04,TD04, WB05Meerschaert, M.M., MA02Meester, R.J., MB02Mehrotra, Vijay, MA06Meiner, Silke, TD04Merlet, G., WC03Meschenmoser, D., TA02Metzler, Ralf, MB09Michna, Z., TB01

Miretskiy, D.I., TB08Mirjafari, K., WC06Miyazawa, M., WA06Mo, S.F., WC01Moallemi, Ciamac C., TA09Mohamed, H., MC07Möhle, Martin, TB02Momcilovic, P., MB04Moore, A., WC06Moreno, P., WC08Moriarty, J., MB08Mörters, P., WB02Morton, David P., TA04Motyer, Allan, WA06Moulines, E., WA03Moyal, P., WA04Mühlethaler, P., TA02Müller, A., TD01Müller, T., WA02

N

Najafi, M., TD09Nakagawa, K., WC04Narayanan, Viswanath C., WC08Nazarathy, Y., MB08Neininger, R., WB09Newman, C.M., MB02, MB02Nijman, Theo E., MB01Nishimura, S., TD08Nobel, R., MA08Norros, I., WA02Novikov, A., TA07Nyrhinen, H., TB01

O

O’Neill, P.D., WC06O’Reilly, Małgorzata, TC06Orfão, N.M., WA04, WC03Ormeci, E.L., TC07, WB06Oshanin, Gleb, MB09Ozmutlu, Huseyin C., TD09Ozmutlu, Seda, TD09

P

Pacheco, A., MC06, TD05, TD08Palmowski, Z., MC01, MC01Papanicolaou, V., WA04Park, Jin Soo, MB05, MB05Park, Wonyoung, WB08

165

12 – Author Index

Pawlak, M., TC01Peköz, Erol A., MC08Peng, X.H., MA01Pérez, J.F., TA07Perry, David, MC09, WA01Peters, Jan, TB03Petrou, Evangelia, TD01Pham, Kim Son, TC09Piera, F., TD03Pistorius, Martijn, MC01, MC01Poor, H.V., WC05Pot, S.A., TA06Prabhakar, B., MC05, TA09Presman, E.L., TD07Priouret, P., WA03Proutière, A., TB05, WC05Puchała, Z., MC01Puha, A.L., TC04

Q

Qiu, Minmin, MA09, MC03Quinn, J., WA07

R

Radunovic, B., TD02Ramanan, K., MB04, MC04,MC07, TC04, TC04Ramírez-Hernández, José A.,TC07Randhawa, R.S., MA04Rataj, J., TA02Ravishankar, K., MB02, MB02Redig, F., MC02Reed, J.E., TA04, TB06Reiman, Martin I., MB04Reittu, H., WA02Remiche, M.-A., TC06Ren, Changrui, MA09, MC03Resing, J.A.C., TA08, TC05Reynolds, C., MB04Riaño, G., MC08, TA07Ribeiro, H., TD08Ridder, A., TC03Righter, R., WA08Robert, Ph., TB05, TD05, WA05Roijers, Frank, TA05Rolski, T., MC01Rosenberg, C.P., TB04Ross, Kevin J., TB07Ross, Sheldon, MC08Rothblum, U.G., MA07

Roubos, D., MB05Roy, Rahul, TD02Roychowdhury, Soma, TC01Rubinstein, R.Y., MC03, MC03,TC03, WA03Ruiz-Medina, M.D., MA02Rybko, A.N., WB04

S

Salah Aguir, Mohamed, WB06Samorodnitsky, G., TD03Sanghavi, S., WB09Scandolo, G., MA01Schaal, Stefan, TB03Scheffler, H.P., MA02Scheinhardt, W.R.W., TA03,TB08, TB08, TB08, TC09Scheller-Wolf, A., TB06, WA09,WB08Scherer, Matthias, MB01Schertzer, E., MB02, MB02Schmidt, V., TA02Schweinsberg, Jason, TB02Secomandi, Nicola, WA09Segers, Johan, TA07Serfozo, R.F., MC08Sethi, S., WA03Sethi, S.P., MA01Shah, D., MC05, TA09, WB09Sheu, S.J., WC04Shin, Yang Woo, MA08Shlosman, S.B., WB04Shneer, V., WC02Shorten, R., WB05Shreve, S., MC07, TC04Shwartz, Adam, TC07Sidorova, N., WB02Simatos, F., TD05Singh, T., WA05Siris, Vasilios A., TA05Snyder, R.D., MA06Sokolov, Igor M., TC02Soltani, A.R., MA02, TB07Son, Jung Je, TC05, TC05, TC05Sonin, Isaac, TD07, WA07Sourirajan, K., TB09Spencer, S.E.F., WC06Spodarev, E., TA02Squillante, M.S., TB09, WA06,WA06Stadje, Wolfgang, WA01Steckley, Samuel G., MA06

Stolyar, A.L., TA04, WC02Sudbury, A., WB09Sukparungsee, S., TA07Swaminathan, Jayashankar M.,TB01Szajowski, K., WC07Szekli, R., WA03

T

Taimre, T., TC03, TC03Tan, J., TB04Tang, A., MC05Taylor, J.W., MA06Taylor, Peter, TC06, WA06,WC06Tekin, E., WC09Tezcan, Tolga, MA04, MB06Theodorou, Evangelos, TB03Thiran, Patrick, WC02Thönnes, Elke, TA02Tibi, D., TB05Tijms, H.C., TD08Timmer, J., MA09Tokarev, D., WB09Torrisi, G.L., TD02Touati, Corinne, MB07Towsley, Don, TB04Toyoizumi, Hiroshi, TC08Travani, Foad, WB08Twigg, Andy, TB05

V

Valdivieso, Luis, MB01Van den Berg, E., TD09Van den Berg, J., MC02Van den Berg, J.L., TA05, TB08Van den Schrieck,Jean-Christophe, TA06, WB06Van der Hofstad, R., MC02,TC09, WA02, WA02Van der Laan, D.A., MA07Van der Mei, R.D., MB05, TA08Van der Wal, Jan, MA07, TC07Van Doorn, E.A., MC06Van Houtum, G.J., MA09,WA09, WA09, WB07Van Leeuwaarden, Johan S.H.,TB06, TB06, WA05, WA06,WB02Van Lierde, Sarah, TC06Van Lieshout, M.N.M., TA02

166

Van Roy, Benjamin, TA09Vázquez-Abad, F., WC03Veatch, Michael, WB03Vesilo, R.A., TB06, WB08Vladimirov, A.A., WB04Vlasiou, Maria, TA08, WC08Vliegen, I.M.H., WA09Vojnovic, Milan, TB05Volk-Makarewicz, W., WC03Volkovich, Y., TC09

W

Wang, Qinhua, MC03Wang, Wei, MA09, MC03Wang, Xiaofeng (Mulan), WA09Ward, A.R., MB06Weber, R.R., WC07Weber, S., MA01Weerasinghe, A., MC04Weiss, G., MB08Werker, Bas J.M., MB01Weron, Aleksander, TC02

Wierman, Adam, MA05, MA05Williams, R.J., MB04, TB07Willsky, A., WB09Winands, E.M.M., MA05, TA08,WA06Wirth, F., WB05Wischik, D., WB05

X

Xia, Cathy H., TB04Xia, Li, WC03Xiao, Yongbo, MC09Xie, Ming, MC09Xiu, Cathy, WB07Xu, H., TB09

Y

Yannacopoulos, A.N., WA04Yao, D.D., MC09, TB09Yatomi, N., TD08Yau, K.K.W., WC01

Yechiali, Uri, TA08Yi, Y., WC05Yin, Wenjun, MC09, TC01,WC03Ying, Yu, TB04Yuan, Taoying, TB03Yudaeva, N.V., MC07

Z

Zachary, S., MB04Zagst, Rudi, WC01Zazanis, Michael, WA04Zeevi, Assaf, MA04, TA03,TA06Zhan, Li, TA03Zhang, Bin, MC09Zhao, Y.Q., TC06, WA06Zheng, S., TB09Zitkovic, Gordan, WC01Ziya, S., WC09Zwart, Bert, MA05, TB06, TB06,TD03, WC05

167

12 – Author Index

168

Tues

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