Research ArticleUpdating Neighbour Cell List via Crowdsourced User ReportsA Framework for Measuring Time Performance

A Checco 1 C Lancia2 and D J Leith3

1University of Sheffield Sheffield UK2Leiden University Leiden Netherlands3Trinity College Dublin Dublin Ireland

Correspondence should be addressed to A Checco acheccosheffieldacuk

Received 19 July 2017 Accepted 27 November 2017 Published 21 January 2018

Academic Editor Stefano Savazzi

Copyright copy 2018 A Checco et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In modern wireless networks deployments each serving node needs to keep its Neighbour Cell List (NCL) constantly up to date tokeep track of network changesThe time needed by each serving node to update its NCL is an important parameter of the networkrsquosreliability and performance An adequate estimate of such parameter enables a significant improvement of self-configurationfunctionalities This paper focuses on the update time of NCLs when an approach of crowdsourced user reports is adopted Inthis setting each user periodically reports to the serving node information about the set of nodes sensed by the user itself We showthat bymapping the local topological structure of the network onto states of increasing knowledge a crispmathematical frameworkcan be obtained which allows in turn for the use of a variety of user mobility models Further using a simplifiedmobility model weshow how to obtain useful upper bounds on the expected time for a serving node to gain Full Knowledge of its local neighbourhood

1 Introduction

Neighbour Cell List Discovery (NCLD) is a core processof modern wireless networks especially when deployed inan unplanned and decentralised manner like WiFi hotspotsand LTE femtocells [1] In these scenarios each node needsto independently construct the NCL Further appropriateknowledge of network topology that is the neighbourhoodstructure of each node in the network allows the design ofmore efficient routing and interference-avoidance algorithmsand improved allocation of limited network resources In anumber of common situations relying on explicit commu-nication or on a central controller may be impractical oreven impossible for instance when neighbouring devicesbelong to a different operator Local knowledge of networktopology is enough to produce distributed algorithms forchannel allocation in WiFi networks code selection in smallcell networks and distributed graph colouring and routingand also for problems of joint power and channel allocationoptimisation (see Section 2)

Though related to location discovery the topic of thismanuscript is the discovery of existing neighbours without

targeting their actual geographical position We focus onthe process of NCLD via crowdsourcing meaning that thetask of detecting and reporting the existence of conflictingneighbours is delegated to users In this framework each userperiodically reports to the serving node information aboutthe set of neighbouring nodes observed see for exampleFigure 1 Exploiting User Equipment (UE) measurements isappealing because such technique is easy to implement andvirtually cost-free Nevertheless the information receivedfrom UE measurements is disregarded by the serving nodein most implementations [2 Section 741]

Keeping the NCL updated is fundamental for a numberof reasons

(i) Neighbouring cells can be added removed or tem-porarily offline

(ii) The handover to a new cell might be problematicwhenever it is not contained in theNCL of the servingcell

(iii) Some cells should not be added to the NCL listbecause they might reflect spurious measurementsyielding nonreliable handovers

HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 9028427 11 pageshttpsdoiorg10115520189028427

2 Wireless Communications and Mobile Computing

(iv) Neighbours with the same PCI should be handledwith specific solutions

With these reasons in mind this manuscript studies the time119879 necessary to achieve confident knowledge of the NCLthrough UE measurements Estimating 119879 enables optimaltuning of neighbour cell list management schemes 119879 is alsoimportant for the design and deployment of decentralisedoptimisation schemes A key example is self-organisation aproblem where the network nodes need to optimise theirconfiguration without a central controller that is relying onlocal information only There exist many fast and efficientdecentralised algorithms to self-organise a WiFifemtocellnetwork these algorithms are generally fed with a NCLwhich needs to be constantly kept up to date At implemen-tation level this means that each node needs to periodicallyestimate with a sufficient level of confidence which nodesof the network are potential conflicting neighbours Themajority of decentralised schemes require that each servingnode needs local knowledge of the local neighbourhood [3ndash6] and any attempt to relax this hypothesis comes at theexpense of performance as shown in Section 2

In the literature it is usually assumed that neighbourhoodinformation may be instantaneously acquired [7ndash13] that isthe time 119879 is considered negligible In fact this assumptionmay often not be valid either because it is necessary to listento the channel long enough to get a high-confidence estima-tion or because hidden nodessecond-hop NCLs need to beknown as well and thus it is necessary to use communicationwith users or other nodes to obtain such information Whenthe time needed by each serving node to update its NCL islarger than the time to execute the optimisation algorithm adecentralised approach might not be the best solution

In a framework where the NCL is built via crowdsourceduser reports our main goal is to rigorously characterise 119879and study its properties and bounds This is a problem thatto the best of our knowledge has not yet been addressed inthe literature Our main contributions are the following (i)the problem of user-reports-based NCLD is stated for thefirst time through a crisp mathematical formulation (ii) weintroduce a simple mobility model that is useful for gaininginsight into those situations where crowdsourcing via userreports is likely to yield the greatest benefit to a decentralisedapproach (iii) we show that this model can provide an upperbound on the time to topology discovery thus it can be usedas a design tool (see Section 55)

The rest of the paper is organised as follows in Section 2we present the related work then in Section 3 we show somepractical use cases where our approach can be applied InSection 4 we provide amathematical model for the discoveryprocess and in Section 5 we define the problem in the contextof such model and give some useful bounds We presentsimulation results to validate the model and show how it canbe used as a network design tool in Section 6 Finally inSection 7 we draw the conclusions

2 Related Work

In the field of decentralised algorithm design it has beenshown that local knowledge of network topology is enough to

Figure 1 Example of NCLD Each user report to the serving nodewhich neighbouring nodes they observe

produce a distributed algorithm for resource allocation suchlocal knowledge also allows the minimisation of scrambling-code collision and confusion in small cell networks see [14]and references therein This knowledge is sufficient and ina certain sense necessary to build efficient algorithms asthe attempts to relax the hypothesis that each serving nodeneeds knowledge of the local neighbourhood will result in anextreme loss of performances [15 16] which can be preventedonly in specific scenarios where the interferencemodel can bedescribed with a simple graph [17]

Perhaps the main motivation of this manuscript is thework of [18] where the authors propose a neighbour celllist management scheme based on the long-term statisticsof UE measurements A key parameter of this scheme isthe forgetting factor 120574 which weighs the longitudinal UEmeasurements Such parameter is clearly not trivial to tuneespecially in settings very common in wireless and cellularnetworking where instantaneous cell list acquisition cannotbe assumedThe optimal tuning 120574 is only possible by studyingthe statistical properties of the time 119879 necessary to achieveconfident knowledge of the NCL through UEmeasurements

The user report function is already available in commer-cial femtocells [19] and small cells networks and its imple-mentation for code confusion and interference reduction isrecommended in [14 20]

Crowdsourcing approaches have been investigated fordifferent applications for example for estimating both den-sity and number of attendees of large events [21] Many workspertain to the use of crowdsourcing for NCL discovery In[22] the use of mobile measurement to update the NCLof macrocells after deployment has been studied since theintrafrequency reporting function known as Detected SetReporting (DSR) is energetically costly for themobile devicethe use of it is suggested only in critical situations wherea problem with the current NCL is known A similar casewhere the NCL needs to be updated when a new macrocell

Wireless Communications and Mobile Computing 3

is deployed is studied in [23] Other ways to dynamicallybuild the NCL via crowdsourcing are presented in [18 24]A similar work applied to WiMax is presented in [25] and aclosely related approach for the femtocell case is presented in[26]

With this work we address the problem of estimatingthe NCL construction time which is necessary to assesswhether crowdsourcing is effective in a particular networkdeployment However to the best of our knowledge thisproblem has not been addressed in the literature yet

3 Use Cases

We show in this section some timely use cases where ourproposed framework can be applied as a network design tool

31 3G Network Optimisation In order to provide seamlessmobility and a satisfactory service the optimisation of thehandover function is fundamental in modern 3G cellularnetworks To achieve that the construction of a reliableNCL is one of the most critical tasks While in the past thiswas achieved by drive and walk testing the needs to adaptto changes in the network and to reduce the cost requiredifferent solutions [12 18 22]

The so-calledDetected Set Reporting (DSR) is an intrafre-quency 3GPP functionality that allows users to report cellsnot defined in the NCL In this way whenever a macrocelldetects a problem or when a new cell is deployed [23] such afunction can be activated The only disadvantage is that suchfunctionality is energetically costly for the mobile device soits use is recommended for short periods of time and only incritical situations where a problem with the current NCL isknown Therefore an estimation of the optimal time to keepthe DSR active is required Our work provides an effectiveframework to make such estimation possible

32 Small Cells Self-Configuration An important problemthat affects the small cells deployment for residential use iscode selection In 3G base stations have only few scramblingcodes available making the task of selecting the optimal allo-cation challenging Moreover communication with a centralcontroller is discouraged to avoid signalling overhead In 4Gand 5G Physical Cell Identity codes and 5G scrambling codeshave similar problems

A fully decentralised algorithm that can converge to theoptimal confusion- and collision-free code allocation hasbeen devised in [14] However it relies on the assumptionthat small cells are able to construct theirNCLUnfortunatelysmall cells are often not able to detect first- and second-hop neighbours reliably due to hidden-node effects andthe absence of an efficient sniffing Common Pilot Channel(CPICH) mechanism A technique to construct the NCLvia crowdsourcing has been proposed in [26] However theimplementation of such a technique would first require theevaluation of the time scale of the NCL construction and itscomparisonwith the time scale of the convergence of the codeallocation algorithm

A(a1)

A(a1)

A(a1)

a2

a1

A0

A1

A2

A3

A12

A23 A123

A13

a0

a3

Figure 2 Example of tessellation corresponding to scenario ofFigure 1 The scenario is modelled with a serving node 1198860 withthree interfering neighbours 1198861 1198862 and 1198863 the coverage area of theserving node 119860(1198860) can be tessellated with the sets 1198600 1198601 1198602 119860311986012 11986013 11986023 and 119860123 For simplicity of notation we will write1198600 fl 11986004 Neighbour Cell List Discovery Model

Given a set of wireless nodesA = 1198860 119886119873 let 119860(119886119894) sub R2denote the coverage area of access point 119886119894 Please note that119860(119886119894) generally depends on the transmission power of 119886119894 andon the radio propagation properties of themediumWe focuson serving access point 1198860 and letB denote the neighbouringnodes that have nonvoid intersection with 119860(1198860) that is

B = 119886119894 isin A 119894 gt 0 119860 (1198860) cap 119860 (119886119894) = 0 (1)

We will hereafter use the symbol119873 to denote the cardinalityofB that is119873 = |B|

LetP(B) denote the powerset ofB A tessellation of thearea 119860(1198860) is the collection of tiles 119860 iiisinP(B) such that119860 (1198860) = ⋃

iisinP(B)119860 i (2)

where 119860 i = ⋂119895isini119860(119886119895) cap 119860 (1198860) ⋃

119895notini119860(119886119895) i = 0

1198600 = 119860 (1198860) ⋃iisinP(B)0

119860 i (3)

In what follows each element 119860 j composing the tessellationis referred to as a tile and we will use the vector notation jto represent a set of neighbouring nodes Let us consider forexample i = 1198861 1198862 then the tile 119860 j is the portion of 119860(1198860)that is covered by 1198861 and 1198862 only see Figure 2

Whenever a user is in 119860 j it will report j to access point1198860 In other words 1198860 will be aware of the existence of those

4 Wireless Communications and Mobile Computing

neighbouring nodes 119886119894 isin j The rate of these reports dependson the mobility model assumed (see Section 5)

To keep the model as conservative as possible and toencompass the frequent case of half-duplex nodes we assume1198860 cannot detect the existence of any neighbour even though1198860 lies in one of the neighboursrsquo coverage area

Let K119905 denote the knowledge set of access point 1198860 thatis the set of neighbours that 1198860 is aware of at time 119905 Given asequence of reports j1 j119905 we have thatK119905 = ⋃119905119894=1 j119894K119905is a sequence of sets that satisfies

K119905 = 119905⋃119904=0

K119904 (4)

in particular |K119905| is nondecreasing in 119905 Clearly the knowl-edge state at time 119905K119905 takes values inP(B)Definition 1 (Full Knowledge) Given an integer119879 and a finitesequence of reports j1 j119879 the node 1198860 is said to have FullKnowledge (FK) of its neighbours at time 119879 if

K119879 = 119879⋃119904=1

j119904 =B (5)

Remark 2 If 1198860 has Full Knowledge (FK) of its neighbours attime 119879 so it has at all times 119879 + 119905 for 119905 ge 0 In other wordsonce 1198860 has reached FK it cannot lose it

Definition 3 (first time to FK) Given a sequence of reportsj1 j2 the first time to FK 120591 for the node 1198860 is the firsttime the latter reaches FK of its neighbours that is120591 fl min 119879 ge 0 such that K119879 =B (6)

Remark 4 The characterisation of the first time to FK gener-ally depends on the realisation of a sequence of user reportsthis means that 120591 is a random variable More precisely by (6)120591 is a stopping time see for example [27]

We end the section with a note on the tessellation

Remark 5 A generic tessellation ofB can be represented asa hypercube119867 = 0 1119873 by identifying the vertices of119867withthe tiles 119860 j that the tessellation is composed of The numberof tiles of a generic tessellation of B is 2119873 as well as thevertices of a hypercube represented as vectors of size119873 Thetiles of the tessellation can be mapped onto the vertices of thehypercube by identifying the 119894th component of the vertices119909 isin 119867 with 119886119894 isinB In other words119860 j larrrarr 119909 lArrrArr119909119894 = 1119886119894isinj 119894 = 1 119873 (7)

where 1 is the indicator functionWe define the order of a tileas the number of neighbours a report from that tilewould giveknowledge of the number of 119896th order tiles is (119873119896 ) A report

4th-order tile

3rd-order tiles

2nd-order tiles

1st-order tiles

0th-order tileLine of Full Knowledge

Figure 3 Hypercube representation of the tessellation for 119873 = 4There is one-zeroth order tile namely 119860119862 four first-order tiles 11986011198602 1198603 and 1198604 six second-order tiles 11986012 11986013 11986014 11986023 11986024 and11986034 four third-order tiles 119860123 119860124 119860134 and 119860234 and a fourth-order tile that is 1198601234from a 119896th order tile is equivalent to 119896 first-order reports Inparticular FK is attained with a report from the 119873th ordertile or at least two reports from two distinct (119873 minus 1)th tilesand so on This property can be graphically represented bywhat we call the Line of Full Knowledge see Figure 3The lineof FK is clearly not unique (eg there are119873 tiles of order119873minus1 but only 2 are part of a given line of FK) the aim of Figure 3is only to illustrate that a sequence of 119879 reports j1 j119879 isa path on the hypercube119867 and that FK is attained whenevera line of FK is reached at a time smaller than 119879

SinceK119905 the knowledge state at time 119905 takes on values inthe same setP(B) we can also map the knowledge states onthe hypercube119867 That is a sequence of reports j1 j2 j119905is equivalent to a single report from tile⋃119905119904=1 j119904 =K119905

We can now define the main problems of this work

Problem 6 (expected first time to Full Knowledge) Given anaccess point 1198860 a set B of neighbours with given positionand coverage area and a sequence of user reports we wantto characterise the expectation of the first time to FK that is

E (120591) = sum119905ge1

119905P (120591 = 119905) (8)

Obviously the way the user(s) moves inside the coveragearea 119860(1198860) heavily affects the difficulty of the problem andits answer However the formulation of Problem 6 has thegreat advantage of decoupling the notion of FK from the usermobility model addressing the mean value of the first timeto FK is also an enabler to the estimate of the tail of thedistribution of 120591mdashthrough Markovrsquos inequality for exampleFurther from a numerical point of view the expected time toFK may be achieved via a Monte Carlo simulation once thesetB and the mobility model in use are fixed

There may exist cases where it is only necessary tocharacterise the first time to attain partial knowledge of thelocal topology For example we may be interested in the firstmomentwhen the neighbouring nodes that have been alreadydiscovered that is the elements of the knowledge setK119905 are

Wireless Communications and Mobile Computing 5

enough to describe a given fraction of the local topologyThisidea motivates the following

Problem 7 (expected first time to 120575-knowledge) Let 984858 be ameasure over P(B) and fixed 120575 isin (0 1] Given an accesspoint 1198860 a set B of neighbours with given position andcoverage area and a sequence of user reports we want tocharacterise the expectation of the first time to 120575-knowledgeE(120591120575) where120591120575 = min119879 ge 0 such that

sumkisinP(K119879) 984858 (119860k)sumjisinP(B) 984858 (119860 j) ge 120575 (9)

When 120575 = 1 and 984858(119860 j) gt 0 for each j isinP(B) Problem 7 is equivalent to Problem 6 IndeedsumkisinP(K119879) 984858(119860k)sumjisinP(B) 984858(119860 j) ge 1 if and only ifK119879 equivB

We will hereafter consider the Lebesgue measure 984858(119860k) =119860kThis leads to the following interpretation 120575-knowledgeis attained when the knowledge set K119905 defines for the firsttime a tessellation that covers a fraction of119860(1198860) larger than orequal to 120575 Equivalently 120591120575 is the first time when the tiles thatwould give new information (in the sense that the cardinalityof the knowledge setK119879 would increase) cover a fraction of119860(1198860) that is smaller than 1 minus 120575Remark 8 The concept of 120575-knowledge is fundamental inthe simulation phase when we want to know whether userreports can effectively be used to give knowledge of thelocal topology Indeed it is likely that the neighbours 119886119894whose coverage area do not overlap with 119860(1198860) save for anearly negligible portion will be discovered after a very longtime in other words the leading contribution to E(120591) willbe represented by the mean first-visit time of the user(s)to 119860(119886119894) Discarding 119886119894 from the picture the concept of 120575-knowledge lets us focus on the quantitative analysis of NCLDsee Section 6

5 Teleport Mobility

The characterisation of 120591 the first time to FK depends on theassumed userrsquos mobility model it describes how users enterexit and move within 119860(1198860) The users evolution can then berepresented as a pair 119880119905 = (119899119905 119883119905) where 119899119905 is the numberof users that lie in 119860(1198860) at time 119905 and 119883119905 = (1199091119905 1199092119905 119909119899119905119905 )is a vector with the position of the 119899119905 users We assume theevolution of 119880119905 to be driven by a discrete-time Markov chain(MC) throughout the paper

The realisation of 1198801199050le119905le119879 completely determines thesequence of user reports j1 j119879 to the access point 1198860 (cfRemark 4) SinceK119905 only depends onK119905minus1 and 119880119905 then thebivariate process (119880119905K119905) is a MC

It will prove useful to consider a simplified mobilitymodel in which a single user continuously teleport betweentiles without leaving 119860(1198860)(this model will be extended tomany users and to more general models in Section 55)

Model 1 (teleport mobility) A single user moves within119860(1198860) according to a discrete-time MC taking on values inP(B) At any time the user cannot abandon the wholeregion that is it is constrained within 119860(1198860) At each stepthe user instantaneously teleports with a probability that isproportional to the measure of the destination tile (note thatthe actual position within a tile is undefined in this model)The destination tile can also be the same tile of previous stepmeaning that the user would remain on the same tile duringthat discrete-time step Assuming that all tiles are Lebesgue-measurable plane sets the transition probabilities are

P (i 997888rarr j) = 10038171003817100381710038171003817119860 j100381710038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 (10)

where sdot denotes the Lebesgue measure

Remark 9 Model 1 greatly simplifies the characterisation of120591 the first time to FK Indeed in this mobility model K119905 isindependent of119880119905 and the sole processK119905 is hence sufficientto describe the process of gathering knowledge from the userreports We will hereafter refer toK119905 as the knowledge chain

Assuming Model 1 we can easily describe the processof gathering knowledge from user reports as a discrete-time random walk on the hypercube 119867 = 0 1119873 (whichwe have introduced in Remark 5) having knowledge of 119899neighbouring nodes is in fact equivalent to receiving a reportfrom the 119899th order tile that gives information about all ofthem

Let 119875(sdot sdot) be the transition kernel of the knowledge chainIf k sube l then (4) guarantees that 119875(k l) = 0 because suchtransitionwouldmean a loss of knowledge Conversely whenk sube l a transition from k to l happens if the user moves to atile that contains the missing information (lk) and does notadd more information than that Therefore

119875 (k l) = summisinP(k)

1003817100381710038171003817119860 mcup(lk)10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 if k sube l0 otherwise (11)

The following result holds

Lemma 10 The matrix 119875 is upper triangularProof Let us consider the following partial ordering relationamong the states

k ⪯ llArrrArrk sube l (12)

6 Wireless Communications and Mobile Computing

By (11) 119875(k l) = 0 only if k ⪯ l Therefore any mapping

P (B) ni llarrrarr119897 isin 1 2 2119873 (13)

such that

k ⪯ llArrrArr119896 le 119897 (14)

will put the matrix 119875 into an upper triangular form Inparticular we can order the states by increasing cardi-nality and in lexicographic order (for 119873 neighbouringnodes that is with 2119873 different tiles this would mean thesequence 1 2 119873 1 2 119873minus1119873 1 2 119873)

The explicit computation of the whole matrix 119875 using (11)is expensive in general since 119875 is a 2119873 times 2119873 matrix Howeveras stated above 119875 is upper triangular In Section 53 weshow that it is possible to explicitly characterise its spectrumFor the readerrsquos reference the following equation shows theexample of transition matrix 119875 for119873 = 3

119875 = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817

sdot((((((((((((

100381710038171003817100381711986001003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986012310038171003817100381710038170 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 0 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 0 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 0 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 0 1

)))))))))))) (15)

51 Expected Time to Full Knowledge Let klowast = 1 2 119873be the state of FK By formula (11) 119875(klowast klowast) = 1 This meansthat the chain has an absorbing state and the hitting time ofthis state is just 120591 the first time to FK Hence we can computethe expected time to FK simply by

E [120591] = (119868 minus 119876)minus1 1 (16)

where 119876 is obtained from 119875 by removing the row and thecolumn relative to state klowast and 1 is the column vector ofones [28] In a similar way it is possible to compute the othermoments of 120591

Even if 119868 minus 119876 is upper triangular and can be blockdecomposed the computation of its inverse may not beaffordable when the cardinality of B grows In Section 54we will bound the probability of the event 120591 gt 11990552 Expected Time to 120575-Knowledge Regarding Problem 7 wecan easily modify matrix 119875 to obtain the expected time to 120575-knowledge Every state k isin P(B) such that

sumlisinP(k)

1003817100381710038171003817119860 l10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 ge 120575 (17)

can be aggregated in the absorbing state summing thecorresponding column of 119875 in the last column and theneliminating the column and row corresponding to state k Inthis way it is possible to compute E[120591120575] using (16)53 Eigenvalues The following result fully characterises thespectrum of the matrix 119875

Theorem 11 For k isin P(B) the eigenvalues of 119875 have theform

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 (100381710038171003817100381711986001003817100381710038171003817 + sumlsubk|l|=1

1003817100381710038171003817119860 l1003817100381710038171003817 + sdot sdot sdot + sum

lsubk|l|=119898

1003817100381710038171003817119860 l1003817100381710038171003817

+ sdot sdot sdot + sumlsubk|l|=|k|

1003817100381710038171003817119860 l1003817100381710038171003817)

(18)

Proof The matrix 119875 being upper triangular by Lemma 10the entries 119875(k k) are the eigenvalues of the matrix Let usthen imagine to have the knowledge chain in state k Theonly way for the chain to undergo a self-transition (k rarr k)is that the user reports any combination of neighbouringnodes that have already been discovered In other words theknowledge chain undergoes a self-transition if and only if theuser reports an element ofP(k) Therefore

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 sumlisinP(k) 1003816100381610038161003816119860 l1003816100381610038161003816 (19)

Last formula is equivalent to the thesis

Since each eigenvalue is a sum of positive elements thesecond-largest eigenvalue can be obtained by maximisingover the tiles of order119873 minus 1 = max

k|k|=119873minus1120582k (20)

Wireless Communications and Mobile Computing 7

54 Convergence Properties Bounds Using (20) it is possibleto obtain the following result

Lemma 12 Given 120576 gt 0 let119878 (1 minus 120576) = log 120576log (21)

Then 119878(1 minus 120576) reports are sufficient to achieve FK withprobability greater than or equal to (1 minus 120576)Proof Using Lemma 10 and (20) on 119875

P (120591 gt 119905) le 119905 (22)

For a small target tolerance 120576 of not achieving FKif 119905 ge log 120576

log 997904rArrP (120591 gt 119905) le 120576 (23)

120575-Knowledge Convergence Bounds Using the same manipu-lation of the matrix 119875 described in Section 52 Lemma 12 inSection 54 can be applied to the modified matrix to obtaina bound for the number of steps to have 120575-knowledge withhigh probability

55 120575-Knowledge and Other-Than-Teleport Mobility Model1 is equivalent to a single user teleporting instantaneouslyto a random point within the coverage area of the nodetime is discrete Thus at each time node 1198860 receives a reportfrom a location that is sampled from the uniform probabilitydistribution over the coverage area119860(1198860) Bearing inmind thenumerical characterisation of the first time to 120575-knowledgethe teleport model is particularly convenient This task couldbe in fact carried out within the Monte Carlo paradigm bysimply throwing sufficiently many points at random insidethe coverage area 119860(1198860) In other words it is possible tonumerically study the process through which 120575-knowledgeis achieved by sampling sufficiently many times a probabilitydensity function that is uniform over the coverage area119860(1198860)

Model 1 may prove itself unsatisfactory in a real lifescenario The main problem is that if we generate a sequenceof user reports according to it any two elements of thesequence are independent whereas in general they are not Ineach mobility model where the trajectory taken by the user isphysically feasible the user positions communicated by twosuccessive reports are in fact correlated due to the motionconstraints

Let us imagine that a single user travels inside the cov-erage area 119860(1198860) according to an unknown mobility modeland let 119879(119905) be the trajectory taken by the user Samplingthe trajectory at equally spaced discrete times we obtainan embedded sequence of user locations which correspondto an embedded sequence of user reports Next we cananalyse the sequence and understand after how many steps120575-knowledge has been reached By multiplying this numberof steps by the time lapse between two consecutive reports

(inter-report time) the time to 120575-knowledge can be obtainedfor that particular realisation of the user-reports sequenceFinally the procedure above can be repeated sufficientlymany times to estimate with a Monte Carlo method theexpected time necessary for 1198860 to reach 120575-knowledge

As mentioned above in a general mobility model it islikely that two successive user reports are correlated Thesecorrelations may decay as the inter-report time grows largerand larger As an example let us imagine that a single usertravels inside the coverage area 119860(1198860) according to a MCLet 120587 be the equilibrium probability measure of the chainand let 119905mix be the mixing time of the chain that is thetime needed for the chain to reach equilibrium If the inter-report time is chosen comparable to 119905mix then the timelapse between two successive reports will be sufficient forthe MC to forget the past trajectory in other words thecorrelations between consecutive reports will be negligibleAs a consequence the user locations will be independentlydrawn from the probability measure 120587 and the matrix 119875describing the knowledge evolution will become

119875 (k l) = summisinP(k)

120587 (119860 mcup(lk)) if k sube l0 otherwise (61015840)Therefore the formulation and the results developed in

Sections 51ndash54 are still valid if we consider a single-usermobility model based on a MC provided that the time lapsebetween two consecutive reports is of the order of the mixingtime of the chain Under the assumption that user reports aresent at a frequency comparable with the inverse mixing timeof the mobility MC we can compute an upper bound on thetime to 120575-knowledge Any reporting rate higher than 1119905mixwill in fact still guarantee that 1198860 achieves 120575-knowledge ofits neighbourhood in at most E[120591120575] sdot 119905mix seconds on average(recall that E[120591120575] is measured in number of reports)

551 Multiuser Scenario We end this section by brieflymentioning a straightforward application of Model 1 in amultiuser scenario Let us imagine that 119899 users may entermove within and exit 119860(1198860) according to a hidden mobilitymodel We assume that 119899 is a very large number and thatit is possible to statistically characterise the stationary user-density by means of a probability measure 120587 over 119860(1198860)At each time every user may independently send a reportwith a very small probability 119901 Then the number of reportsreceived by 1198860 in a given time interval is approximatelyPoissonian and the time lapse between two successive reportsis exponential with parameter 120582 = 119899119901 Next let 119898 = E[120591]be the expected time to FK expressed in number of reportsreturned by (61015840) and (16) the expected time to achieve FKis the expectation of the first time for a Poisson process ofparameter 120582 to hit the state 119898 A practical example for thiskind of scenario in presented in Section 562

56 Examples

561 Femtocells Deployment for Residential Use Regardingthe use case of femtocell self-organisation presented in

8 Wireless Communications and Mobile Computing

Section 32 each serves a very small number of devicesUsing data of typical residential densities and coverage areasa statistic of the tessellation can be devised If it is possibleto establish a time after which the user position can beconsidered as drawn from a uniformdistribution then 119878(120575) isan upper bound of the time to 120575-knowledge for all the inter-report times smaller than or equal to 562 Cells Deployed in Congested Areas Opposite to theprevious example cells deployed in congested places likea mall have an extremely large basin of potential usersHowever in situations where users main interest is otherthan connecting to the Internet it is reasonable to expect thesingle-user reporting-activity to be rather sporadic There-fore the Poissonian approximation that we have mentionedat the end of Section 55 may be applicable In this casecharacterising the time to achieve 120575-knowledge is possiblethrough a statistic of the typical (or worst case) tessellations

6 Simulations

61 Teleport Model on Random Positioned Nodes In thissection we offer a preliminary assessment of the possibility ofusing the machinery developed so far in real applications Tothis purpose we developed a simulation framework inMAT-LAB and studied a scenario where 8 nodes are positionedon a plane at random according to a uniform (bivariate)probability distribution that is uniformly at random Eachnode has a circular coverage area of the same size Weconsidered 350 different configurations with the constraintthat the coverage area of 1198860 has nonvoid intersection withthe coverage area of the remaining nodes meaning thatFK is achieved as soon as all 7 neighbours are reportedto 1198860 We compute the tessellation of each configurationusing a classical Monte Carlo sampler For each of these350 configurations we computed the expected time to 09-knowledge E[120591] together with the number of steps sufficientto guarantee 09-knowledge with 90 confidence that is119878(09) The inter-report time being fixed during this firstexperiment the amount of time in seconds to achieve 09-knowledge is directly proportional to the number of steps justevaluated

Figure 4 displays the empirical probability mass functionof these two quantities E[120591] is centred around 10 steps while119878(09) is shifted on higher values as expected being an upperbound Figure 5 shows the empirical cumulative distributionfunction of E[120591] and 119878(09) We see that 16 steps are sufficientto achieve 09-knowledge for nearly all scenarios (95) whilewe need 22 steps using 119878(09) We also notice that the boundobtained from (21) is a conservative estimation because ituses only the second-largest eigenvalue Indeed it takes intoaccount only the slowest way to reach the desired knowledgewhile the problem has a rich combinatorial structure thatcannot be completely captured by (22)

Roughly speaking a usermoving at 05ms according to arandom walk model and providing at least one report everyhour can guarantee the node will have 09-knowledge withhigh probability in less than 5 h and in less than two hours

5 10 15 200

005

01

015

Number of reports

5 10 15 200

005

01

015

Number of reports

E[]

S(09)

Figure 4 Empirical probability mass function of the expected timeto 09-knowledge and the number of steps to have 09-knowledgewith 90 confidence for the teleport model on random positionednodes Since the inter-report time is fixed the simulation time isdirectly proportional to the number of reports

0

02

04

06

08

1

0 5 10 15 20 25Number of reports

Empi

rical

cdf

E[]

S(09)

Figure 5 Empirical cumulative distribution function of theexpected time to 09-knowledge and the number of steps to have09-knowledge with 90 confidence for the teleport model onrandom positioned nodes Since the inter-report time is fixed thesimulation time is directly proportional to the number of reports

if reports are sent at least every 15 minutes (see next sectionfor a more detailed analysis on the interaction between

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

Wireless Communications and Mobile Computing 3

is deployed is studied in [23] Other ways to dynamicallybuild the NCL via crowdsourcing are presented in [18 24]A similar work applied to WiMax is presented in [25] and aclosely related approach for the femtocell case is presented in[26]

With this work we address the problem of estimatingthe NCL construction time which is necessary to assesswhether crowdsourcing is effective in a particular networkdeployment However to the best of our knowledge thisproblem has not been addressed in the literature yet

3 Use Cases

We show in this section some timely use cases where ourproposed framework can be applied as a network design tool

31 3G Network Optimisation In order to provide seamlessmobility and a satisfactory service the optimisation of thehandover function is fundamental in modern 3G cellularnetworks To achieve that the construction of a reliableNCL is one of the most critical tasks While in the past thiswas achieved by drive and walk testing the needs to adaptto changes in the network and to reduce the cost requiredifferent solutions [12 18 22]

The so-calledDetected Set Reporting (DSR) is an intrafre-quency 3GPP functionality that allows users to report cellsnot defined in the NCL In this way whenever a macrocelldetects a problem or when a new cell is deployed [23] such afunction can be activated The only disadvantage is that suchfunctionality is energetically costly for the mobile device soits use is recommended for short periods of time and only incritical situations where a problem with the current NCL isknown Therefore an estimation of the optimal time to keepthe DSR active is required Our work provides an effectiveframework to make such estimation possible

32 Small Cells Self-Configuration An important problemthat affects the small cells deployment for residential use iscode selection In 3G base stations have only few scramblingcodes available making the task of selecting the optimal allo-cation challenging Moreover communication with a centralcontroller is discouraged to avoid signalling overhead In 4Gand 5G Physical Cell Identity codes and 5G scrambling codeshave similar problems

A fully decentralised algorithm that can converge to theoptimal confusion- and collision-free code allocation hasbeen devised in [14] However it relies on the assumptionthat small cells are able to construct theirNCLUnfortunatelysmall cells are often not able to detect first- and second-hop neighbours reliably due to hidden-node effects andthe absence of an efficient sniffing Common Pilot Channel(CPICH) mechanism A technique to construct the NCLvia crowdsourcing has been proposed in [26] However theimplementation of such a technique would first require theevaluation of the time scale of the NCL construction and itscomparisonwith the time scale of the convergence of the codeallocation algorithm

A(a1)

A(a1)

A(a1)

a2

a1

A0

A1

A2

A3

A12

A23 A123

A13

a0

a3

Figure 2 Example of tessellation corresponding to scenario ofFigure 1 The scenario is modelled with a serving node 1198860 withthree interfering neighbours 1198861 1198862 and 1198863 the coverage area of theserving node 119860(1198860) can be tessellated with the sets 1198600 1198601 1198602 119860311986012 11986013 11986023 and 119860123 For simplicity of notation we will write1198600 fl 11986004 Neighbour Cell List Discovery Model

Given a set of wireless nodesA = 1198860 119886119873 let 119860(119886119894) sub R2denote the coverage area of access point 119886119894 Please note that119860(119886119894) generally depends on the transmission power of 119886119894 andon the radio propagation properties of themediumWe focuson serving access point 1198860 and letB denote the neighbouringnodes that have nonvoid intersection with 119860(1198860) that is

B = 119886119894 isin A 119894 gt 0 119860 (1198860) cap 119860 (119886119894) = 0 (1)

We will hereafter use the symbol119873 to denote the cardinalityofB that is119873 = |B|

LetP(B) denote the powerset ofB A tessellation of thearea 119860(1198860) is the collection of tiles 119860 iiisinP(B) such that119860 (1198860) = ⋃

iisinP(B)119860 i (2)

where 119860 i = ⋂119895isini119860(119886119895) cap 119860 (1198860) ⋃

119895notini119860(119886119895) i = 0

1198600 = 119860 (1198860) ⋃iisinP(B)0

119860 i (3)

In what follows each element 119860 j composing the tessellationis referred to as a tile and we will use the vector notation jto represent a set of neighbouring nodes Let us consider forexample i = 1198861 1198862 then the tile 119860 j is the portion of 119860(1198860)that is covered by 1198861 and 1198862 only see Figure 2

Whenever a user is in 119860 j it will report j to access point1198860 In other words 1198860 will be aware of the existence of those

4 Wireless Communications and Mobile Computing

neighbouring nodes 119886119894 isin j The rate of these reports dependson the mobility model assumed (see Section 5)

To keep the model as conservative as possible and toencompass the frequent case of half-duplex nodes we assume1198860 cannot detect the existence of any neighbour even though1198860 lies in one of the neighboursrsquo coverage area

Let K119905 denote the knowledge set of access point 1198860 thatis the set of neighbours that 1198860 is aware of at time 119905 Given asequence of reports j1 j119905 we have thatK119905 = ⋃119905119894=1 j119894K119905is a sequence of sets that satisfies

K119905 = 119905⋃119904=0

K119904 (4)

in particular |K119905| is nondecreasing in 119905 Clearly the knowl-edge state at time 119905K119905 takes values inP(B)Definition 1 (Full Knowledge) Given an integer119879 and a finitesequence of reports j1 j119879 the node 1198860 is said to have FullKnowledge (FK) of its neighbours at time 119879 if

K119879 = 119879⋃119904=1

j119904 =B (5)

Remark 2 If 1198860 has Full Knowledge (FK) of its neighbours attime 119879 so it has at all times 119879 + 119905 for 119905 ge 0 In other wordsonce 1198860 has reached FK it cannot lose it

Definition 3 (first time to FK) Given a sequence of reportsj1 j2 the first time to FK 120591 for the node 1198860 is the firsttime the latter reaches FK of its neighbours that is120591 fl min 119879 ge 0 such that K119879 =B (6)

Remark 4 The characterisation of the first time to FK gener-ally depends on the realisation of a sequence of user reportsthis means that 120591 is a random variable More precisely by (6)120591 is a stopping time see for example [27]

We end the section with a note on the tessellation

Remark 5 A generic tessellation ofB can be represented asa hypercube119867 = 0 1119873 by identifying the vertices of119867withthe tiles 119860 j that the tessellation is composed of The numberof tiles of a generic tessellation of B is 2119873 as well as thevertices of a hypercube represented as vectors of size119873 Thetiles of the tessellation can be mapped onto the vertices of thehypercube by identifying the 119894th component of the vertices119909 isin 119867 with 119886119894 isinB In other words119860 j larrrarr 119909 lArrrArr119909119894 = 1119886119894isinj 119894 = 1 119873 (7)

where 1 is the indicator functionWe define the order of a tileas the number of neighbours a report from that tilewould giveknowledge of the number of 119896th order tiles is (119873119896 ) A report

4th-order tile

3rd-order tiles

2nd-order tiles

1st-order tiles

0th-order tileLine of Full Knowledge

Figure 3 Hypercube representation of the tessellation for 119873 = 4There is one-zeroth order tile namely 119860119862 four first-order tiles 11986011198602 1198603 and 1198604 six second-order tiles 11986012 11986013 11986014 11986023 11986024 and11986034 four third-order tiles 119860123 119860124 119860134 and 119860234 and a fourth-order tile that is 1198601234from a 119896th order tile is equivalent to 119896 first-order reports Inparticular FK is attained with a report from the 119873th ordertile or at least two reports from two distinct (119873 minus 1)th tilesand so on This property can be graphically represented bywhat we call the Line of Full Knowledge see Figure 3The lineof FK is clearly not unique (eg there are119873 tiles of order119873minus1 but only 2 are part of a given line of FK) the aim of Figure 3is only to illustrate that a sequence of 119879 reports j1 j119879 isa path on the hypercube119867 and that FK is attained whenevera line of FK is reached at a time smaller than 119879

SinceK119905 the knowledge state at time 119905 takes on values inthe same setP(B) we can also map the knowledge states onthe hypercube119867 That is a sequence of reports j1 j2 j119905is equivalent to a single report from tile⋃119905119904=1 j119904 =K119905

We can now define the main problems of this work

Problem 6 (expected first time to Full Knowledge) Given anaccess point 1198860 a set B of neighbours with given positionand coverage area and a sequence of user reports we wantto characterise the expectation of the first time to FK that is

E (120591) = sum119905ge1

119905P (120591 = 119905) (8)

Obviously the way the user(s) moves inside the coveragearea 119860(1198860) heavily affects the difficulty of the problem andits answer However the formulation of Problem 6 has thegreat advantage of decoupling the notion of FK from the usermobility model addressing the mean value of the first timeto FK is also an enabler to the estimate of the tail of thedistribution of 120591mdashthrough Markovrsquos inequality for exampleFurther from a numerical point of view the expected time toFK may be achieved via a Monte Carlo simulation once thesetB and the mobility model in use are fixed

There may exist cases where it is only necessary tocharacterise the first time to attain partial knowledge of thelocal topology For example we may be interested in the firstmomentwhen the neighbouring nodes that have been alreadydiscovered that is the elements of the knowledge setK119905 are

Wireless Communications and Mobile Computing 5

enough to describe a given fraction of the local topologyThisidea motivates the following

Problem 7 (expected first time to 120575-knowledge) Let 984858 be ameasure over P(B) and fixed 120575 isin (0 1] Given an accesspoint 1198860 a set B of neighbours with given position andcoverage area and a sequence of user reports we want tocharacterise the expectation of the first time to 120575-knowledgeE(120591120575) where120591120575 = min119879 ge 0 such that

sumkisinP(K119879) 984858 (119860k)sumjisinP(B) 984858 (119860 j) ge 120575 (9)

When 120575 = 1 and 984858(119860 j) gt 0 for each j isinP(B) Problem 7 is equivalent to Problem 6 IndeedsumkisinP(K119879) 984858(119860k)sumjisinP(B) 984858(119860 j) ge 1 if and only ifK119879 equivB

We will hereafter consider the Lebesgue measure 984858(119860k) =119860kThis leads to the following interpretation 120575-knowledgeis attained when the knowledge set K119905 defines for the firsttime a tessellation that covers a fraction of119860(1198860) larger than orequal to 120575 Equivalently 120591120575 is the first time when the tiles thatwould give new information (in the sense that the cardinalityof the knowledge setK119879 would increase) cover a fraction of119860(1198860) that is smaller than 1 minus 120575Remark 8 The concept of 120575-knowledge is fundamental inthe simulation phase when we want to know whether userreports can effectively be used to give knowledge of thelocal topology Indeed it is likely that the neighbours 119886119894whose coverage area do not overlap with 119860(1198860) save for anearly negligible portion will be discovered after a very longtime in other words the leading contribution to E(120591) willbe represented by the mean first-visit time of the user(s)to 119860(119886119894) Discarding 119886119894 from the picture the concept of 120575-knowledge lets us focus on the quantitative analysis of NCLDsee Section 6

5 Teleport Mobility

The characterisation of 120591 the first time to FK depends on theassumed userrsquos mobility model it describes how users enterexit and move within 119860(1198860) The users evolution can then berepresented as a pair 119880119905 = (119899119905 119883119905) where 119899119905 is the numberof users that lie in 119860(1198860) at time 119905 and 119883119905 = (1199091119905 1199092119905 119909119899119905119905 )is a vector with the position of the 119899119905 users We assume theevolution of 119880119905 to be driven by a discrete-time Markov chain(MC) throughout the paper

The realisation of 1198801199050le119905le119879 completely determines thesequence of user reports j1 j119879 to the access point 1198860 (cfRemark 4) SinceK119905 only depends onK119905minus1 and 119880119905 then thebivariate process (119880119905K119905) is a MC

It will prove useful to consider a simplified mobilitymodel in which a single user continuously teleport betweentiles without leaving 119860(1198860)(this model will be extended tomany users and to more general models in Section 55)

Model 1 (teleport mobility) A single user moves within119860(1198860) according to a discrete-time MC taking on values inP(B) At any time the user cannot abandon the wholeregion that is it is constrained within 119860(1198860) At each stepthe user instantaneously teleports with a probability that isproportional to the measure of the destination tile (note thatthe actual position within a tile is undefined in this model)The destination tile can also be the same tile of previous stepmeaning that the user would remain on the same tile duringthat discrete-time step Assuming that all tiles are Lebesgue-measurable plane sets the transition probabilities are

P (i 997888rarr j) = 10038171003817100381710038171003817119860 j100381710038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 (10)

where sdot denotes the Lebesgue measure

Remark 9 Model 1 greatly simplifies the characterisation of120591 the first time to FK Indeed in this mobility model K119905 isindependent of119880119905 and the sole processK119905 is hence sufficientto describe the process of gathering knowledge from the userreports We will hereafter refer toK119905 as the knowledge chain

Assuming Model 1 we can easily describe the processof gathering knowledge from user reports as a discrete-time random walk on the hypercube 119867 = 0 1119873 (whichwe have introduced in Remark 5) having knowledge of 119899neighbouring nodes is in fact equivalent to receiving a reportfrom the 119899th order tile that gives information about all ofthem

Let 119875(sdot sdot) be the transition kernel of the knowledge chainIf k sube l then (4) guarantees that 119875(k l) = 0 because suchtransitionwouldmean a loss of knowledge Conversely whenk sube l a transition from k to l happens if the user moves to atile that contains the missing information (lk) and does notadd more information than that Therefore

119875 (k l) = summisinP(k)

1003817100381710038171003817119860 mcup(lk)10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 if k sube l0 otherwise (11)

The following result holds

Lemma 10 The matrix 119875 is upper triangularProof Let us consider the following partial ordering relationamong the states

k ⪯ llArrrArrk sube l (12)

6 Wireless Communications and Mobile Computing

By (11) 119875(k l) = 0 only if k ⪯ l Therefore any mapping

P (B) ni llarrrarr119897 isin 1 2 2119873 (13)

such that

k ⪯ llArrrArr119896 le 119897 (14)

will put the matrix 119875 into an upper triangular form Inparticular we can order the states by increasing cardi-nality and in lexicographic order (for 119873 neighbouringnodes that is with 2119873 different tiles this would mean thesequence 1 2 119873 1 2 119873minus1119873 1 2 119873)

The explicit computation of the whole matrix 119875 using (11)is expensive in general since 119875 is a 2119873 times 2119873 matrix Howeveras stated above 119875 is upper triangular In Section 53 weshow that it is possible to explicitly characterise its spectrumFor the readerrsquos reference the following equation shows theexample of transition matrix 119875 for119873 = 3

119875 = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817

sdot((((((((((((

100381710038171003817100381711986001003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986012310038171003817100381710038170 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 0 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 0 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 0 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 0 1

)))))))))))) (15)

51 Expected Time to Full Knowledge Let klowast = 1 2 119873be the state of FK By formula (11) 119875(klowast klowast) = 1 This meansthat the chain has an absorbing state and the hitting time ofthis state is just 120591 the first time to FK Hence we can computethe expected time to FK simply by

E [120591] = (119868 minus 119876)minus1 1 (16)

where 119876 is obtained from 119875 by removing the row and thecolumn relative to state klowast and 1 is the column vector ofones [28] In a similar way it is possible to compute the othermoments of 120591

Even if 119868 minus 119876 is upper triangular and can be blockdecomposed the computation of its inverse may not beaffordable when the cardinality of B grows In Section 54we will bound the probability of the event 120591 gt 11990552 Expected Time to 120575-Knowledge Regarding Problem 7 wecan easily modify matrix 119875 to obtain the expected time to 120575-knowledge Every state k isin P(B) such that

sumlisinP(k)

1003817100381710038171003817119860 l10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 ge 120575 (17)

can be aggregated in the absorbing state summing thecorresponding column of 119875 in the last column and theneliminating the column and row corresponding to state k Inthis way it is possible to compute E[120591120575] using (16)53 Eigenvalues The following result fully characterises thespectrum of the matrix 119875

Theorem 11 For k isin P(B) the eigenvalues of 119875 have theform

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 (100381710038171003817100381711986001003817100381710038171003817 + sumlsubk|l|=1

1003817100381710038171003817119860 l1003817100381710038171003817 + sdot sdot sdot + sum

lsubk|l|=119898

1003817100381710038171003817119860 l1003817100381710038171003817

+ sdot sdot sdot + sumlsubk|l|=|k|

1003817100381710038171003817119860 l1003817100381710038171003817)

(18)

Proof The matrix 119875 being upper triangular by Lemma 10the entries 119875(k k) are the eigenvalues of the matrix Let usthen imagine to have the knowledge chain in state k Theonly way for the chain to undergo a self-transition (k rarr k)is that the user reports any combination of neighbouringnodes that have already been discovered In other words theknowledge chain undergoes a self-transition if and only if theuser reports an element ofP(k) Therefore

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 sumlisinP(k) 1003816100381610038161003816119860 l1003816100381610038161003816 (19)

Last formula is equivalent to the thesis

Since each eigenvalue is a sum of positive elements thesecond-largest eigenvalue can be obtained by maximisingover the tiles of order119873 minus 1 = max

k|k|=119873minus1120582k (20)

Wireless Communications and Mobile Computing 7

54 Convergence Properties Bounds Using (20) it is possibleto obtain the following result

Lemma 12 Given 120576 gt 0 let119878 (1 minus 120576) = log 120576log (21)

Then 119878(1 minus 120576) reports are sufficient to achieve FK withprobability greater than or equal to (1 minus 120576)Proof Using Lemma 10 and (20) on 119875

P (120591 gt 119905) le 119905 (22)

For a small target tolerance 120576 of not achieving FKif 119905 ge log 120576

log 997904rArrP (120591 gt 119905) le 120576 (23)

120575-Knowledge Convergence Bounds Using the same manipu-lation of the matrix 119875 described in Section 52 Lemma 12 inSection 54 can be applied to the modified matrix to obtaina bound for the number of steps to have 120575-knowledge withhigh probability

55 120575-Knowledge and Other-Than-Teleport Mobility Model1 is equivalent to a single user teleporting instantaneouslyto a random point within the coverage area of the nodetime is discrete Thus at each time node 1198860 receives a reportfrom a location that is sampled from the uniform probabilitydistribution over the coverage area119860(1198860) Bearing inmind thenumerical characterisation of the first time to 120575-knowledgethe teleport model is particularly convenient This task couldbe in fact carried out within the Monte Carlo paradigm bysimply throwing sufficiently many points at random insidethe coverage area 119860(1198860) In other words it is possible tonumerically study the process through which 120575-knowledgeis achieved by sampling sufficiently many times a probabilitydensity function that is uniform over the coverage area119860(1198860)

Model 1 may prove itself unsatisfactory in a real lifescenario The main problem is that if we generate a sequenceof user reports according to it any two elements of thesequence are independent whereas in general they are not Ineach mobility model where the trajectory taken by the user isphysically feasible the user positions communicated by twosuccessive reports are in fact correlated due to the motionconstraints

Let us imagine that a single user travels inside the cov-erage area 119860(1198860) according to an unknown mobility modeland let 119879(119905) be the trajectory taken by the user Samplingthe trajectory at equally spaced discrete times we obtainan embedded sequence of user locations which correspondto an embedded sequence of user reports Next we cananalyse the sequence and understand after how many steps120575-knowledge has been reached By multiplying this numberof steps by the time lapse between two consecutive reports

(inter-report time) the time to 120575-knowledge can be obtainedfor that particular realisation of the user-reports sequenceFinally the procedure above can be repeated sufficientlymany times to estimate with a Monte Carlo method theexpected time necessary for 1198860 to reach 120575-knowledge

As mentioned above in a general mobility model it islikely that two successive user reports are correlated Thesecorrelations may decay as the inter-report time grows largerand larger As an example let us imagine that a single usertravels inside the coverage area 119860(1198860) according to a MCLet 120587 be the equilibrium probability measure of the chainand let 119905mix be the mixing time of the chain that is thetime needed for the chain to reach equilibrium If the inter-report time is chosen comparable to 119905mix then the timelapse between two successive reports will be sufficient forthe MC to forget the past trajectory in other words thecorrelations between consecutive reports will be negligibleAs a consequence the user locations will be independentlydrawn from the probability measure 120587 and the matrix 119875describing the knowledge evolution will become

119875 (k l) = summisinP(k)

120587 (119860 mcup(lk)) if k sube l0 otherwise (61015840)Therefore the formulation and the results developed in

Sections 51ndash54 are still valid if we consider a single-usermobility model based on a MC provided that the time lapsebetween two consecutive reports is of the order of the mixingtime of the chain Under the assumption that user reports aresent at a frequency comparable with the inverse mixing timeof the mobility MC we can compute an upper bound on thetime to 120575-knowledge Any reporting rate higher than 1119905mixwill in fact still guarantee that 1198860 achieves 120575-knowledge ofits neighbourhood in at most E[120591120575] sdot 119905mix seconds on average(recall that E[120591120575] is measured in number of reports)

551 Multiuser Scenario We end this section by brieflymentioning a straightforward application of Model 1 in amultiuser scenario Let us imagine that 119899 users may entermove within and exit 119860(1198860) according to a hidden mobilitymodel We assume that 119899 is a very large number and thatit is possible to statistically characterise the stationary user-density by means of a probability measure 120587 over 119860(1198860)At each time every user may independently send a reportwith a very small probability 119901 Then the number of reportsreceived by 1198860 in a given time interval is approximatelyPoissonian and the time lapse between two successive reportsis exponential with parameter 120582 = 119899119901 Next let 119898 = E[120591]be the expected time to FK expressed in number of reportsreturned by (61015840) and (16) the expected time to achieve FKis the expectation of the first time for a Poisson process ofparameter 120582 to hit the state 119898 A practical example for thiskind of scenario in presented in Section 562

56 Examples

561 Femtocells Deployment for Residential Use Regardingthe use case of femtocell self-organisation presented in

8 Wireless Communications and Mobile Computing

Section 32 each serves a very small number of devicesUsing data of typical residential densities and coverage areasa statistic of the tessellation can be devised If it is possibleto establish a time after which the user position can beconsidered as drawn from a uniformdistribution then 119878(120575) isan upper bound of the time to 120575-knowledge for all the inter-report times smaller than or equal to 562 Cells Deployed in Congested Areas Opposite to theprevious example cells deployed in congested places likea mall have an extremely large basin of potential usersHowever in situations where users main interest is otherthan connecting to the Internet it is reasonable to expect thesingle-user reporting-activity to be rather sporadic There-fore the Poissonian approximation that we have mentionedat the end of Section 55 may be applicable In this casecharacterising the time to achieve 120575-knowledge is possiblethrough a statistic of the typical (or worst case) tessellations

6 Simulations

61 Teleport Model on Random Positioned Nodes In thissection we offer a preliminary assessment of the possibility ofusing the machinery developed so far in real applications Tothis purpose we developed a simulation framework inMAT-LAB and studied a scenario where 8 nodes are positionedon a plane at random according to a uniform (bivariate)probability distribution that is uniformly at random Eachnode has a circular coverage area of the same size Weconsidered 350 different configurations with the constraintthat the coverage area of 1198860 has nonvoid intersection withthe coverage area of the remaining nodes meaning thatFK is achieved as soon as all 7 neighbours are reportedto 1198860 We compute the tessellation of each configurationusing a classical Monte Carlo sampler For each of these350 configurations we computed the expected time to 09-knowledge E[120591] together with the number of steps sufficientto guarantee 09-knowledge with 90 confidence that is119878(09) The inter-report time being fixed during this firstexperiment the amount of time in seconds to achieve 09-knowledge is directly proportional to the number of steps justevaluated

Figure 4 displays the empirical probability mass functionof these two quantities E[120591] is centred around 10 steps while119878(09) is shifted on higher values as expected being an upperbound Figure 5 shows the empirical cumulative distributionfunction of E[120591] and 119878(09) We see that 16 steps are sufficientto achieve 09-knowledge for nearly all scenarios (95) whilewe need 22 steps using 119878(09) We also notice that the boundobtained from (21) is a conservative estimation because ituses only the second-largest eigenvalue Indeed it takes intoaccount only the slowest way to reach the desired knowledgewhile the problem has a rich combinatorial structure thatcannot be completely captured by (22)

Roughly speaking a usermoving at 05ms according to arandom walk model and providing at least one report everyhour can guarantee the node will have 09-knowledge withhigh probability in less than 5 h and in less than two hours

5 10 15 200

005

01

015

Number of reports

5 10 15 200

005

01

015

Number of reports

E[]

S(09)

Figure 4 Empirical probability mass function of the expected timeto 09-knowledge and the number of steps to have 09-knowledgewith 90 confidence for the teleport model on random positionednodes Since the inter-report time is fixed the simulation time isdirectly proportional to the number of reports

0

02

04

06

08

1

0 5 10 15 20 25Number of reports

Empi

rical

cdf

E[]

S(09)

Figure 5 Empirical cumulative distribution function of theexpected time to 09-knowledge and the number of steps to have09-knowledge with 90 confidence for the teleport model onrandom positioned nodes Since the inter-report time is fixed thesimulation time is directly proportional to the number of reports

if reports are sent at least every 15 minutes (see next sectionfor a more detailed analysis on the interaction between

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

4 Wireless Communications and Mobile Computing

neighbouring nodes 119886119894 isin j The rate of these reports dependson the mobility model assumed (see Section 5)

To keep the model as conservative as possible and toencompass the frequent case of half-duplex nodes we assume1198860 cannot detect the existence of any neighbour even though1198860 lies in one of the neighboursrsquo coverage area

Let K119905 denote the knowledge set of access point 1198860 thatis the set of neighbours that 1198860 is aware of at time 119905 Given asequence of reports j1 j119905 we have thatK119905 = ⋃119905119894=1 j119894K119905is a sequence of sets that satisfies

K119905 = 119905⋃119904=0

K119904 (4)

in particular |K119905| is nondecreasing in 119905 Clearly the knowl-edge state at time 119905K119905 takes values inP(B)Definition 1 (Full Knowledge) Given an integer119879 and a finitesequence of reports j1 j119879 the node 1198860 is said to have FullKnowledge (FK) of its neighbours at time 119879 if

K119879 = 119879⋃119904=1

j119904 =B (5)

Remark 2 If 1198860 has Full Knowledge (FK) of its neighbours attime 119879 so it has at all times 119879 + 119905 for 119905 ge 0 In other wordsonce 1198860 has reached FK it cannot lose it

Definition 3 (first time to FK) Given a sequence of reportsj1 j2 the first time to FK 120591 for the node 1198860 is the firsttime the latter reaches FK of its neighbours that is120591 fl min 119879 ge 0 such that K119879 =B (6)

Remark 4 The characterisation of the first time to FK gener-ally depends on the realisation of a sequence of user reportsthis means that 120591 is a random variable More precisely by (6)120591 is a stopping time see for example [27]

We end the section with a note on the tessellation

Remark 5 A generic tessellation ofB can be represented asa hypercube119867 = 0 1119873 by identifying the vertices of119867withthe tiles 119860 j that the tessellation is composed of The numberof tiles of a generic tessellation of B is 2119873 as well as thevertices of a hypercube represented as vectors of size119873 Thetiles of the tessellation can be mapped onto the vertices of thehypercube by identifying the 119894th component of the vertices119909 isin 119867 with 119886119894 isinB In other words119860 j larrrarr 119909 lArrrArr119909119894 = 1119886119894isinj 119894 = 1 119873 (7)

where 1 is the indicator functionWe define the order of a tileas the number of neighbours a report from that tilewould giveknowledge of the number of 119896th order tiles is (119873119896 ) A report

4th-order tile

3rd-order tiles

2nd-order tiles

1st-order tiles

0th-order tileLine of Full Knowledge

Figure 3 Hypercube representation of the tessellation for 119873 = 4There is one-zeroth order tile namely 119860119862 four first-order tiles 11986011198602 1198603 and 1198604 six second-order tiles 11986012 11986013 11986014 11986023 11986024 and11986034 four third-order tiles 119860123 119860124 119860134 and 119860234 and a fourth-order tile that is 1198601234from a 119896th order tile is equivalent to 119896 first-order reports Inparticular FK is attained with a report from the 119873th ordertile or at least two reports from two distinct (119873 minus 1)th tilesand so on This property can be graphically represented bywhat we call the Line of Full Knowledge see Figure 3The lineof FK is clearly not unique (eg there are119873 tiles of order119873minus1 but only 2 are part of a given line of FK) the aim of Figure 3is only to illustrate that a sequence of 119879 reports j1 j119879 isa path on the hypercube119867 and that FK is attained whenevera line of FK is reached at a time smaller than 119879

SinceK119905 the knowledge state at time 119905 takes on values inthe same setP(B) we can also map the knowledge states onthe hypercube119867 That is a sequence of reports j1 j2 j119905is equivalent to a single report from tile⋃119905119904=1 j119904 =K119905

We can now define the main problems of this work

Problem 6 (expected first time to Full Knowledge) Given anaccess point 1198860 a set B of neighbours with given positionand coverage area and a sequence of user reports we wantto characterise the expectation of the first time to FK that is

E (120591) = sum119905ge1

119905P (120591 = 119905) (8)

Obviously the way the user(s) moves inside the coveragearea 119860(1198860) heavily affects the difficulty of the problem andits answer However the formulation of Problem 6 has thegreat advantage of decoupling the notion of FK from the usermobility model addressing the mean value of the first timeto FK is also an enabler to the estimate of the tail of thedistribution of 120591mdashthrough Markovrsquos inequality for exampleFurther from a numerical point of view the expected time toFK may be achieved via a Monte Carlo simulation once thesetB and the mobility model in use are fixed

There may exist cases where it is only necessary tocharacterise the first time to attain partial knowledge of thelocal topology For example we may be interested in the firstmomentwhen the neighbouring nodes that have been alreadydiscovered that is the elements of the knowledge setK119905 are

Wireless Communications and Mobile Computing 5

enough to describe a given fraction of the local topologyThisidea motivates the following

Problem 7 (expected first time to 120575-knowledge) Let 984858 be ameasure over P(B) and fixed 120575 isin (0 1] Given an accesspoint 1198860 a set B of neighbours with given position andcoverage area and a sequence of user reports we want tocharacterise the expectation of the first time to 120575-knowledgeE(120591120575) where120591120575 = min119879 ge 0 such that

sumkisinP(K119879) 984858 (119860k)sumjisinP(B) 984858 (119860 j) ge 120575 (9)

When 120575 = 1 and 984858(119860 j) gt 0 for each j isinP(B) Problem 7 is equivalent to Problem 6 IndeedsumkisinP(K119879) 984858(119860k)sumjisinP(B) 984858(119860 j) ge 1 if and only ifK119879 equivB

We will hereafter consider the Lebesgue measure 984858(119860k) =119860kThis leads to the following interpretation 120575-knowledgeis attained when the knowledge set K119905 defines for the firsttime a tessellation that covers a fraction of119860(1198860) larger than orequal to 120575 Equivalently 120591120575 is the first time when the tiles thatwould give new information (in the sense that the cardinalityof the knowledge setK119879 would increase) cover a fraction of119860(1198860) that is smaller than 1 minus 120575Remark 8 The concept of 120575-knowledge is fundamental inthe simulation phase when we want to know whether userreports can effectively be used to give knowledge of thelocal topology Indeed it is likely that the neighbours 119886119894whose coverage area do not overlap with 119860(1198860) save for anearly negligible portion will be discovered after a very longtime in other words the leading contribution to E(120591) willbe represented by the mean first-visit time of the user(s)to 119860(119886119894) Discarding 119886119894 from the picture the concept of 120575-knowledge lets us focus on the quantitative analysis of NCLDsee Section 6

5 Teleport Mobility

The characterisation of 120591 the first time to FK depends on theassumed userrsquos mobility model it describes how users enterexit and move within 119860(1198860) The users evolution can then berepresented as a pair 119880119905 = (119899119905 119883119905) where 119899119905 is the numberof users that lie in 119860(1198860) at time 119905 and 119883119905 = (1199091119905 1199092119905 119909119899119905119905 )is a vector with the position of the 119899119905 users We assume theevolution of 119880119905 to be driven by a discrete-time Markov chain(MC) throughout the paper

The realisation of 1198801199050le119905le119879 completely determines thesequence of user reports j1 j119879 to the access point 1198860 (cfRemark 4) SinceK119905 only depends onK119905minus1 and 119880119905 then thebivariate process (119880119905K119905) is a MC

It will prove useful to consider a simplified mobilitymodel in which a single user continuously teleport betweentiles without leaving 119860(1198860)(this model will be extended tomany users and to more general models in Section 55)

Model 1 (teleport mobility) A single user moves within119860(1198860) according to a discrete-time MC taking on values inP(B) At any time the user cannot abandon the wholeregion that is it is constrained within 119860(1198860) At each stepthe user instantaneously teleports with a probability that isproportional to the measure of the destination tile (note thatthe actual position within a tile is undefined in this model)The destination tile can also be the same tile of previous stepmeaning that the user would remain on the same tile duringthat discrete-time step Assuming that all tiles are Lebesgue-measurable plane sets the transition probabilities are

P (i 997888rarr j) = 10038171003817100381710038171003817119860 j100381710038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 (10)

where sdot denotes the Lebesgue measure

Remark 9 Model 1 greatly simplifies the characterisation of120591 the first time to FK Indeed in this mobility model K119905 isindependent of119880119905 and the sole processK119905 is hence sufficientto describe the process of gathering knowledge from the userreports We will hereafter refer toK119905 as the knowledge chain

Assuming Model 1 we can easily describe the processof gathering knowledge from user reports as a discrete-time random walk on the hypercube 119867 = 0 1119873 (whichwe have introduced in Remark 5) having knowledge of 119899neighbouring nodes is in fact equivalent to receiving a reportfrom the 119899th order tile that gives information about all ofthem

Let 119875(sdot sdot) be the transition kernel of the knowledge chainIf k sube l then (4) guarantees that 119875(k l) = 0 because suchtransitionwouldmean a loss of knowledge Conversely whenk sube l a transition from k to l happens if the user moves to atile that contains the missing information (lk) and does notadd more information than that Therefore

119875 (k l) = summisinP(k)

1003817100381710038171003817119860 mcup(lk)10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 if k sube l0 otherwise (11)

The following result holds

Lemma 10 The matrix 119875 is upper triangularProof Let us consider the following partial ordering relationamong the states

k ⪯ llArrrArrk sube l (12)

6 Wireless Communications and Mobile Computing

By (11) 119875(k l) = 0 only if k ⪯ l Therefore any mapping

P (B) ni llarrrarr119897 isin 1 2 2119873 (13)

such that

k ⪯ llArrrArr119896 le 119897 (14)

will put the matrix 119875 into an upper triangular form Inparticular we can order the states by increasing cardi-nality and in lexicographic order (for 119873 neighbouringnodes that is with 2119873 different tiles this would mean thesequence 1 2 119873 1 2 119873minus1119873 1 2 119873)

The explicit computation of the whole matrix 119875 using (11)is expensive in general since 119875 is a 2119873 times 2119873 matrix Howeveras stated above 119875 is upper triangular In Section 53 weshow that it is possible to explicitly characterise its spectrumFor the readerrsquos reference the following equation shows theexample of transition matrix 119875 for119873 = 3

119875 = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817

sdot((((((((((((

100381710038171003817100381711986001003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986012310038171003817100381710038170 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 0 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 0 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 0 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 0 1

)))))))))))) (15)

51 Expected Time to Full Knowledge Let klowast = 1 2 119873be the state of FK By formula (11) 119875(klowast klowast) = 1 This meansthat the chain has an absorbing state and the hitting time ofthis state is just 120591 the first time to FK Hence we can computethe expected time to FK simply by

E [120591] = (119868 minus 119876)minus1 1 (16)

where 119876 is obtained from 119875 by removing the row and thecolumn relative to state klowast and 1 is the column vector ofones [28] In a similar way it is possible to compute the othermoments of 120591

Even if 119868 minus 119876 is upper triangular and can be blockdecomposed the computation of its inverse may not beaffordable when the cardinality of B grows In Section 54we will bound the probability of the event 120591 gt 11990552 Expected Time to 120575-Knowledge Regarding Problem 7 wecan easily modify matrix 119875 to obtain the expected time to 120575-knowledge Every state k isin P(B) such that

sumlisinP(k)

1003817100381710038171003817119860 l10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 ge 120575 (17)

can be aggregated in the absorbing state summing thecorresponding column of 119875 in the last column and theneliminating the column and row corresponding to state k Inthis way it is possible to compute E[120591120575] using (16)53 Eigenvalues The following result fully characterises thespectrum of the matrix 119875

Theorem 11 For k isin P(B) the eigenvalues of 119875 have theform

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 (100381710038171003817100381711986001003817100381710038171003817 + sumlsubk|l|=1

1003817100381710038171003817119860 l1003817100381710038171003817 + sdot sdot sdot + sum

lsubk|l|=119898

1003817100381710038171003817119860 l1003817100381710038171003817

+ sdot sdot sdot + sumlsubk|l|=|k|

1003817100381710038171003817119860 l1003817100381710038171003817)

(18)

Proof The matrix 119875 being upper triangular by Lemma 10the entries 119875(k k) are the eigenvalues of the matrix Let usthen imagine to have the knowledge chain in state k Theonly way for the chain to undergo a self-transition (k rarr k)is that the user reports any combination of neighbouringnodes that have already been discovered In other words theknowledge chain undergoes a self-transition if and only if theuser reports an element ofP(k) Therefore

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 sumlisinP(k) 1003816100381610038161003816119860 l1003816100381610038161003816 (19)

Last formula is equivalent to the thesis

Since each eigenvalue is a sum of positive elements thesecond-largest eigenvalue can be obtained by maximisingover the tiles of order119873 minus 1 = max

k|k|=119873minus1120582k (20)

Wireless Communications and Mobile Computing 7

54 Convergence Properties Bounds Using (20) it is possibleto obtain the following result

Lemma 12 Given 120576 gt 0 let119878 (1 minus 120576) = log 120576log (21)

Then 119878(1 minus 120576) reports are sufficient to achieve FK withprobability greater than or equal to (1 minus 120576)Proof Using Lemma 10 and (20) on 119875

P (120591 gt 119905) le 119905 (22)

For a small target tolerance 120576 of not achieving FKif 119905 ge log 120576

log 997904rArrP (120591 gt 119905) le 120576 (23)

120575-Knowledge Convergence Bounds Using the same manipu-lation of the matrix 119875 described in Section 52 Lemma 12 inSection 54 can be applied to the modified matrix to obtaina bound for the number of steps to have 120575-knowledge withhigh probability

55 120575-Knowledge and Other-Than-Teleport Mobility Model1 is equivalent to a single user teleporting instantaneouslyto a random point within the coverage area of the nodetime is discrete Thus at each time node 1198860 receives a reportfrom a location that is sampled from the uniform probabilitydistribution over the coverage area119860(1198860) Bearing inmind thenumerical characterisation of the first time to 120575-knowledgethe teleport model is particularly convenient This task couldbe in fact carried out within the Monte Carlo paradigm bysimply throwing sufficiently many points at random insidethe coverage area 119860(1198860) In other words it is possible tonumerically study the process through which 120575-knowledgeis achieved by sampling sufficiently many times a probabilitydensity function that is uniform over the coverage area119860(1198860)

Model 1 may prove itself unsatisfactory in a real lifescenario The main problem is that if we generate a sequenceof user reports according to it any two elements of thesequence are independent whereas in general they are not Ineach mobility model where the trajectory taken by the user isphysically feasible the user positions communicated by twosuccessive reports are in fact correlated due to the motionconstraints

Let us imagine that a single user travels inside the cov-erage area 119860(1198860) according to an unknown mobility modeland let 119879(119905) be the trajectory taken by the user Samplingthe trajectory at equally spaced discrete times we obtainan embedded sequence of user locations which correspondto an embedded sequence of user reports Next we cananalyse the sequence and understand after how many steps120575-knowledge has been reached By multiplying this numberof steps by the time lapse between two consecutive reports

(inter-report time) the time to 120575-knowledge can be obtainedfor that particular realisation of the user-reports sequenceFinally the procedure above can be repeated sufficientlymany times to estimate with a Monte Carlo method theexpected time necessary for 1198860 to reach 120575-knowledge

As mentioned above in a general mobility model it islikely that two successive user reports are correlated Thesecorrelations may decay as the inter-report time grows largerand larger As an example let us imagine that a single usertravels inside the coverage area 119860(1198860) according to a MCLet 120587 be the equilibrium probability measure of the chainand let 119905mix be the mixing time of the chain that is thetime needed for the chain to reach equilibrium If the inter-report time is chosen comparable to 119905mix then the timelapse between two successive reports will be sufficient forthe MC to forget the past trajectory in other words thecorrelations between consecutive reports will be negligibleAs a consequence the user locations will be independentlydrawn from the probability measure 120587 and the matrix 119875describing the knowledge evolution will become

119875 (k l) = summisinP(k)

120587 (119860 mcup(lk)) if k sube l0 otherwise (61015840)Therefore the formulation and the results developed in

Sections 51ndash54 are still valid if we consider a single-usermobility model based on a MC provided that the time lapsebetween two consecutive reports is of the order of the mixingtime of the chain Under the assumption that user reports aresent at a frequency comparable with the inverse mixing timeof the mobility MC we can compute an upper bound on thetime to 120575-knowledge Any reporting rate higher than 1119905mixwill in fact still guarantee that 1198860 achieves 120575-knowledge ofits neighbourhood in at most E[120591120575] sdot 119905mix seconds on average(recall that E[120591120575] is measured in number of reports)

551 Multiuser Scenario We end this section by brieflymentioning a straightforward application of Model 1 in amultiuser scenario Let us imagine that 119899 users may entermove within and exit 119860(1198860) according to a hidden mobilitymodel We assume that 119899 is a very large number and thatit is possible to statistically characterise the stationary user-density by means of a probability measure 120587 over 119860(1198860)At each time every user may independently send a reportwith a very small probability 119901 Then the number of reportsreceived by 1198860 in a given time interval is approximatelyPoissonian and the time lapse between two successive reportsis exponential with parameter 120582 = 119899119901 Next let 119898 = E[120591]be the expected time to FK expressed in number of reportsreturned by (61015840) and (16) the expected time to achieve FKis the expectation of the first time for a Poisson process ofparameter 120582 to hit the state 119898 A practical example for thiskind of scenario in presented in Section 562

56 Examples

561 Femtocells Deployment for Residential Use Regardingthe use case of femtocell self-organisation presented in

8 Wireless Communications and Mobile Computing

Section 32 each serves a very small number of devicesUsing data of typical residential densities and coverage areasa statistic of the tessellation can be devised If it is possibleto establish a time after which the user position can beconsidered as drawn from a uniformdistribution then 119878(120575) isan upper bound of the time to 120575-knowledge for all the inter-report times smaller than or equal to 562 Cells Deployed in Congested Areas Opposite to theprevious example cells deployed in congested places likea mall have an extremely large basin of potential usersHowever in situations where users main interest is otherthan connecting to the Internet it is reasonable to expect thesingle-user reporting-activity to be rather sporadic There-fore the Poissonian approximation that we have mentionedat the end of Section 55 may be applicable In this casecharacterising the time to achieve 120575-knowledge is possiblethrough a statistic of the typical (or worst case) tessellations

6 Simulations

61 Teleport Model on Random Positioned Nodes In thissection we offer a preliminary assessment of the possibility ofusing the machinery developed so far in real applications Tothis purpose we developed a simulation framework inMAT-LAB and studied a scenario where 8 nodes are positionedon a plane at random according to a uniform (bivariate)probability distribution that is uniformly at random Eachnode has a circular coverage area of the same size Weconsidered 350 different configurations with the constraintthat the coverage area of 1198860 has nonvoid intersection withthe coverage area of the remaining nodes meaning thatFK is achieved as soon as all 7 neighbours are reportedto 1198860 We compute the tessellation of each configurationusing a classical Monte Carlo sampler For each of these350 configurations we computed the expected time to 09-knowledge E[120591] together with the number of steps sufficientto guarantee 09-knowledge with 90 confidence that is119878(09) The inter-report time being fixed during this firstexperiment the amount of time in seconds to achieve 09-knowledge is directly proportional to the number of steps justevaluated

Figure 4 displays the empirical probability mass functionof these two quantities E[120591] is centred around 10 steps while119878(09) is shifted on higher values as expected being an upperbound Figure 5 shows the empirical cumulative distributionfunction of E[120591] and 119878(09) We see that 16 steps are sufficientto achieve 09-knowledge for nearly all scenarios (95) whilewe need 22 steps using 119878(09) We also notice that the boundobtained from (21) is a conservative estimation because ituses only the second-largest eigenvalue Indeed it takes intoaccount only the slowest way to reach the desired knowledgewhile the problem has a rich combinatorial structure thatcannot be completely captured by (22)

Roughly speaking a usermoving at 05ms according to arandom walk model and providing at least one report everyhour can guarantee the node will have 09-knowledge withhigh probability in less than 5 h and in less than two hours

5 10 15 200

005

01

015

Number of reports

5 10 15 200

005

01

015

Number of reports

E[]

S(09)

Figure 4 Empirical probability mass function of the expected timeto 09-knowledge and the number of steps to have 09-knowledgewith 90 confidence for the teleport model on random positionednodes Since the inter-report time is fixed the simulation time isdirectly proportional to the number of reports

0

02

04

06

08

1

0 5 10 15 20 25Number of reports

Empi

rical

cdf

E[]

S(09)

Figure 5 Empirical cumulative distribution function of theexpected time to 09-knowledge and the number of steps to have09-knowledge with 90 confidence for the teleport model onrandom positioned nodes Since the inter-report time is fixed thesimulation time is directly proportional to the number of reports

if reports are sent at least every 15 minutes (see next sectionfor a more detailed analysis on the interaction between

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

Wireless Communications and Mobile Computing 5

enough to describe a given fraction of the local topologyThisidea motivates the following

Problem 7 (expected first time to 120575-knowledge) Let 984858 be ameasure over P(B) and fixed 120575 isin (0 1] Given an accesspoint 1198860 a set B of neighbours with given position andcoverage area and a sequence of user reports we want tocharacterise the expectation of the first time to 120575-knowledgeE(120591120575) where120591120575 = min119879 ge 0 such that

sumkisinP(K119879) 984858 (119860k)sumjisinP(B) 984858 (119860 j) ge 120575 (9)

When 120575 = 1 and 984858(119860 j) gt 0 for each j isinP(B) Problem 7 is equivalent to Problem 6 IndeedsumkisinP(K119879) 984858(119860k)sumjisinP(B) 984858(119860 j) ge 1 if and only ifK119879 equivB

We will hereafter consider the Lebesgue measure 984858(119860k) =119860kThis leads to the following interpretation 120575-knowledgeis attained when the knowledge set K119905 defines for the firsttime a tessellation that covers a fraction of119860(1198860) larger than orequal to 120575 Equivalently 120591120575 is the first time when the tiles thatwould give new information (in the sense that the cardinalityof the knowledge setK119879 would increase) cover a fraction of119860(1198860) that is smaller than 1 minus 120575Remark 8 The concept of 120575-knowledge is fundamental inthe simulation phase when we want to know whether userreports can effectively be used to give knowledge of thelocal topology Indeed it is likely that the neighbours 119886119894whose coverage area do not overlap with 119860(1198860) save for anearly negligible portion will be discovered after a very longtime in other words the leading contribution to E(120591) willbe represented by the mean first-visit time of the user(s)to 119860(119886119894) Discarding 119886119894 from the picture the concept of 120575-knowledge lets us focus on the quantitative analysis of NCLDsee Section 6

5 Teleport Mobility

The characterisation of 120591 the first time to FK depends on theassumed userrsquos mobility model it describes how users enterexit and move within 119860(1198860) The users evolution can then berepresented as a pair 119880119905 = (119899119905 119883119905) where 119899119905 is the numberof users that lie in 119860(1198860) at time 119905 and 119883119905 = (1199091119905 1199092119905 119909119899119905119905 )is a vector with the position of the 119899119905 users We assume theevolution of 119880119905 to be driven by a discrete-time Markov chain(MC) throughout the paper

The realisation of 1198801199050le119905le119879 completely determines thesequence of user reports j1 j119879 to the access point 1198860 (cfRemark 4) SinceK119905 only depends onK119905minus1 and 119880119905 then thebivariate process (119880119905K119905) is a MC

It will prove useful to consider a simplified mobilitymodel in which a single user continuously teleport betweentiles without leaving 119860(1198860)(this model will be extended tomany users and to more general models in Section 55)

Model 1 (teleport mobility) A single user moves within119860(1198860) according to a discrete-time MC taking on values inP(B) At any time the user cannot abandon the wholeregion that is it is constrained within 119860(1198860) At each stepthe user instantaneously teleports with a probability that isproportional to the measure of the destination tile (note thatthe actual position within a tile is undefined in this model)The destination tile can also be the same tile of previous stepmeaning that the user would remain on the same tile duringthat discrete-time step Assuming that all tiles are Lebesgue-measurable plane sets the transition probabilities are

P (i 997888rarr j) = 10038171003817100381710038171003817119860 j100381710038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 (10)

where sdot denotes the Lebesgue measure

Remark 9 Model 1 greatly simplifies the characterisation of120591 the first time to FK Indeed in this mobility model K119905 isindependent of119880119905 and the sole processK119905 is hence sufficientto describe the process of gathering knowledge from the userreports We will hereafter refer toK119905 as the knowledge chain

Assuming Model 1 we can easily describe the processof gathering knowledge from user reports as a discrete-time random walk on the hypercube 119867 = 0 1119873 (whichwe have introduced in Remark 5) having knowledge of 119899neighbouring nodes is in fact equivalent to receiving a reportfrom the 119899th order tile that gives information about all ofthem

Let 119875(sdot sdot) be the transition kernel of the knowledge chainIf k sube l then (4) guarantees that 119875(k l) = 0 because suchtransitionwouldmean a loss of knowledge Conversely whenk sube l a transition from k to l happens if the user moves to atile that contains the missing information (lk) and does notadd more information than that Therefore

119875 (k l) = summisinP(k)

1003817100381710038171003817119860 mcup(lk)10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 if k sube l0 otherwise (11)

The following result holds

Lemma 10 The matrix 119875 is upper triangularProof Let us consider the following partial ordering relationamong the states

k ⪯ llArrrArrk sube l (12)

6 Wireless Communications and Mobile Computing

By (11) 119875(k l) = 0 only if k ⪯ l Therefore any mapping

P (B) ni llarrrarr119897 isin 1 2 2119873 (13)

such that

k ⪯ llArrrArr119896 le 119897 (14)

will put the matrix 119875 into an upper triangular form Inparticular we can order the states by increasing cardi-nality and in lexicographic order (for 119873 neighbouringnodes that is with 2119873 different tiles this would mean thesequence 1 2 119873 1 2 119873minus1119873 1 2 119873)

The explicit computation of the whole matrix 119875 using (11)is expensive in general since 119875 is a 2119873 times 2119873 matrix Howeveras stated above 119875 is upper triangular In Section 53 weshow that it is possible to explicitly characterise its spectrumFor the readerrsquos reference the following equation shows theexample of transition matrix 119875 for119873 = 3

119875 = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817

sdot((((((((((((

100381710038171003817100381711986001003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986012310038171003817100381710038170 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 0 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 0 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 0 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 0 1

)))))))))))) (15)

51 Expected Time to Full Knowledge Let klowast = 1 2 119873be the state of FK By formula (11) 119875(klowast klowast) = 1 This meansthat the chain has an absorbing state and the hitting time ofthis state is just 120591 the first time to FK Hence we can computethe expected time to FK simply by

E [120591] = (119868 minus 119876)minus1 1 (16)

where 119876 is obtained from 119875 by removing the row and thecolumn relative to state klowast and 1 is the column vector ofones [28] In a similar way it is possible to compute the othermoments of 120591

Even if 119868 minus 119876 is upper triangular and can be blockdecomposed the computation of its inverse may not beaffordable when the cardinality of B grows In Section 54we will bound the probability of the event 120591 gt 11990552 Expected Time to 120575-Knowledge Regarding Problem 7 wecan easily modify matrix 119875 to obtain the expected time to 120575-knowledge Every state k isin P(B) such that

sumlisinP(k)

1003817100381710038171003817119860 l10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 ge 120575 (17)

can be aggregated in the absorbing state summing thecorresponding column of 119875 in the last column and theneliminating the column and row corresponding to state k Inthis way it is possible to compute E[120591120575] using (16)53 Eigenvalues The following result fully characterises thespectrum of the matrix 119875

Theorem 11 For k isin P(B) the eigenvalues of 119875 have theform

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 (100381710038171003817100381711986001003817100381710038171003817 + sumlsubk|l|=1

1003817100381710038171003817119860 l1003817100381710038171003817 + sdot sdot sdot + sum

lsubk|l|=119898

1003817100381710038171003817119860 l1003817100381710038171003817

+ sdot sdot sdot + sumlsubk|l|=|k|

1003817100381710038171003817119860 l1003817100381710038171003817)

(18)

Proof The matrix 119875 being upper triangular by Lemma 10the entries 119875(k k) are the eigenvalues of the matrix Let usthen imagine to have the knowledge chain in state k Theonly way for the chain to undergo a self-transition (k rarr k)is that the user reports any combination of neighbouringnodes that have already been discovered In other words theknowledge chain undergoes a self-transition if and only if theuser reports an element ofP(k) Therefore

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 sumlisinP(k) 1003816100381610038161003816119860 l1003816100381610038161003816 (19)

Last formula is equivalent to the thesis

Since each eigenvalue is a sum of positive elements thesecond-largest eigenvalue can be obtained by maximisingover the tiles of order119873 minus 1 = max

k|k|=119873minus1120582k (20)

Wireless Communications and Mobile Computing 7

54 Convergence Properties Bounds Using (20) it is possibleto obtain the following result

Lemma 12 Given 120576 gt 0 let119878 (1 minus 120576) = log 120576log (21)

Then 119878(1 minus 120576) reports are sufficient to achieve FK withprobability greater than or equal to (1 minus 120576)Proof Using Lemma 10 and (20) on 119875

P (120591 gt 119905) le 119905 (22)

For a small target tolerance 120576 of not achieving FKif 119905 ge log 120576

log 997904rArrP (120591 gt 119905) le 120576 (23)

120575-Knowledge Convergence Bounds Using the same manipu-lation of the matrix 119875 described in Section 52 Lemma 12 inSection 54 can be applied to the modified matrix to obtaina bound for the number of steps to have 120575-knowledge withhigh probability

55 120575-Knowledge and Other-Than-Teleport Mobility Model1 is equivalent to a single user teleporting instantaneouslyto a random point within the coverage area of the nodetime is discrete Thus at each time node 1198860 receives a reportfrom a location that is sampled from the uniform probabilitydistribution over the coverage area119860(1198860) Bearing inmind thenumerical characterisation of the first time to 120575-knowledgethe teleport model is particularly convenient This task couldbe in fact carried out within the Monte Carlo paradigm bysimply throwing sufficiently many points at random insidethe coverage area 119860(1198860) In other words it is possible tonumerically study the process through which 120575-knowledgeis achieved by sampling sufficiently many times a probabilitydensity function that is uniform over the coverage area119860(1198860)

Model 1 may prove itself unsatisfactory in a real lifescenario The main problem is that if we generate a sequenceof user reports according to it any two elements of thesequence are independent whereas in general they are not Ineach mobility model where the trajectory taken by the user isphysically feasible the user positions communicated by twosuccessive reports are in fact correlated due to the motionconstraints

Let us imagine that a single user travels inside the cov-erage area 119860(1198860) according to an unknown mobility modeland let 119879(119905) be the trajectory taken by the user Samplingthe trajectory at equally spaced discrete times we obtainan embedded sequence of user locations which correspondto an embedded sequence of user reports Next we cananalyse the sequence and understand after how many steps120575-knowledge has been reached By multiplying this numberof steps by the time lapse between two consecutive reports

(inter-report time) the time to 120575-knowledge can be obtainedfor that particular realisation of the user-reports sequenceFinally the procedure above can be repeated sufficientlymany times to estimate with a Monte Carlo method theexpected time necessary for 1198860 to reach 120575-knowledge

As mentioned above in a general mobility model it islikely that two successive user reports are correlated Thesecorrelations may decay as the inter-report time grows largerand larger As an example let us imagine that a single usertravels inside the coverage area 119860(1198860) according to a MCLet 120587 be the equilibrium probability measure of the chainand let 119905mix be the mixing time of the chain that is thetime needed for the chain to reach equilibrium If the inter-report time is chosen comparable to 119905mix then the timelapse between two successive reports will be sufficient forthe MC to forget the past trajectory in other words thecorrelations between consecutive reports will be negligibleAs a consequence the user locations will be independentlydrawn from the probability measure 120587 and the matrix 119875describing the knowledge evolution will become

119875 (k l) = summisinP(k)

120587 (119860 mcup(lk)) if k sube l0 otherwise (61015840)Therefore the formulation and the results developed in

Sections 51ndash54 are still valid if we consider a single-usermobility model based on a MC provided that the time lapsebetween two consecutive reports is of the order of the mixingtime of the chain Under the assumption that user reports aresent at a frequency comparable with the inverse mixing timeof the mobility MC we can compute an upper bound on thetime to 120575-knowledge Any reporting rate higher than 1119905mixwill in fact still guarantee that 1198860 achieves 120575-knowledge ofits neighbourhood in at most E[120591120575] sdot 119905mix seconds on average(recall that E[120591120575] is measured in number of reports)

551 Multiuser Scenario We end this section by brieflymentioning a straightforward application of Model 1 in amultiuser scenario Let us imagine that 119899 users may entermove within and exit 119860(1198860) according to a hidden mobilitymodel We assume that 119899 is a very large number and thatit is possible to statistically characterise the stationary user-density by means of a probability measure 120587 over 119860(1198860)At each time every user may independently send a reportwith a very small probability 119901 Then the number of reportsreceived by 1198860 in a given time interval is approximatelyPoissonian and the time lapse between two successive reportsis exponential with parameter 120582 = 119899119901 Next let 119898 = E[120591]be the expected time to FK expressed in number of reportsreturned by (61015840) and (16) the expected time to achieve FKis the expectation of the first time for a Poisson process ofparameter 120582 to hit the state 119898 A practical example for thiskind of scenario in presented in Section 562

56 Examples

561 Femtocells Deployment for Residential Use Regardingthe use case of femtocell self-organisation presented in

8 Wireless Communications and Mobile Computing

Section 32 each serves a very small number of devicesUsing data of typical residential densities and coverage areasa statistic of the tessellation can be devised If it is possibleto establish a time after which the user position can beconsidered as drawn from a uniformdistribution then 119878(120575) isan upper bound of the time to 120575-knowledge for all the inter-report times smaller than or equal to 562 Cells Deployed in Congested Areas Opposite to theprevious example cells deployed in congested places likea mall have an extremely large basin of potential usersHowever in situations where users main interest is otherthan connecting to the Internet it is reasonable to expect thesingle-user reporting-activity to be rather sporadic There-fore the Poissonian approximation that we have mentionedat the end of Section 55 may be applicable In this casecharacterising the time to achieve 120575-knowledge is possiblethrough a statistic of the typical (or worst case) tessellations

6 Simulations

61 Teleport Model on Random Positioned Nodes In thissection we offer a preliminary assessment of the possibility ofusing the machinery developed so far in real applications Tothis purpose we developed a simulation framework inMAT-LAB and studied a scenario where 8 nodes are positionedon a plane at random according to a uniform (bivariate)probability distribution that is uniformly at random Eachnode has a circular coverage area of the same size Weconsidered 350 different configurations with the constraintthat the coverage area of 1198860 has nonvoid intersection withthe coverage area of the remaining nodes meaning thatFK is achieved as soon as all 7 neighbours are reportedto 1198860 We compute the tessellation of each configurationusing a classical Monte Carlo sampler For each of these350 configurations we computed the expected time to 09-knowledge E[120591] together with the number of steps sufficientto guarantee 09-knowledge with 90 confidence that is119878(09) The inter-report time being fixed during this firstexperiment the amount of time in seconds to achieve 09-knowledge is directly proportional to the number of steps justevaluated

Figure 4 displays the empirical probability mass functionof these two quantities E[120591] is centred around 10 steps while119878(09) is shifted on higher values as expected being an upperbound Figure 5 shows the empirical cumulative distributionfunction of E[120591] and 119878(09) We see that 16 steps are sufficientto achieve 09-knowledge for nearly all scenarios (95) whilewe need 22 steps using 119878(09) We also notice that the boundobtained from (21) is a conservative estimation because ituses only the second-largest eigenvalue Indeed it takes intoaccount only the slowest way to reach the desired knowledgewhile the problem has a rich combinatorial structure thatcannot be completely captured by (22)

Roughly speaking a usermoving at 05ms according to arandom walk model and providing at least one report everyhour can guarantee the node will have 09-knowledge withhigh probability in less than 5 h and in less than two hours

5 10 15 200

005

01

015

Number of reports

5 10 15 200

005

01

015

Number of reports

E[]

S(09)

Figure 4 Empirical probability mass function of the expected timeto 09-knowledge and the number of steps to have 09-knowledgewith 90 confidence for the teleport model on random positionednodes Since the inter-report time is fixed the simulation time isdirectly proportional to the number of reports

0

02

04

06

08

1

0 5 10 15 20 25Number of reports

Empi

rical

cdf

E[]

S(09)

Figure 5 Empirical cumulative distribution function of theexpected time to 09-knowledge and the number of steps to have09-knowledge with 90 confidence for the teleport model onrandom positioned nodes Since the inter-report time is fixed thesimulation time is directly proportional to the number of reports

if reports are sent at least every 15 minutes (see next sectionfor a more detailed analysis on the interaction between

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

6 Wireless Communications and Mobile Computing

By (11) 119875(k l) = 0 only if k ⪯ l Therefore any mapping

P (B) ni llarrrarr119897 isin 1 2 2119873 (13)

such that

k ⪯ llArrrArr119896 le 119897 (14)

will put the matrix 119875 into an upper triangular form Inparticular we can order the states by increasing cardi-nality and in lexicographic order (for 119873 neighbouringnodes that is with 2119873 different tiles this would mean thesequence 1 2 119873 1 2 119873minus1119873 1 2 119873)

The explicit computation of the whole matrix 119875 using (11)is expensive in general since 119875 is a 2119873 times 2119873 matrix Howeveras stated above 119875 is upper triangular In Section 53 weshow that it is possible to explicitly characterise its spectrumFor the readerrsquos reference the following equation shows theexample of transition matrix 119875 for119873 = 3

119875 = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817

sdot((((((((((((

100381710038171003817100381711986001003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986012310038171003817100381710038170 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 0 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 0 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 0 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 1003817100381710038171003817119860121003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 0 0 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986011003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 0 100381710038171003817100381711986021003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 100381710038171003817100381711986001003817100381710038171003817 + 100381710038171003817100381711986021003817100381710038171003817 + 100381710038171003817100381711986031003817100381710038171003817 + 1003817100381710038171003817119860231003817100381710038171003817 100381710038171003817100381711986011003817100381710038171003817 + 1003817100381710038171003817119860121003817100381710038171003817 + 1003817100381710038171003817119860131003817100381710038171003817 + 100381710038171003817100381711986012310038171003817100381710038170 0 0 0 0 0 0 1

)))))))))))) (15)

51 Expected Time to Full Knowledge Let klowast = 1 2 119873be the state of FK By formula (11) 119875(klowast klowast) = 1 This meansthat the chain has an absorbing state and the hitting time ofthis state is just 120591 the first time to FK Hence we can computethe expected time to FK simply by

E [120591] = (119868 minus 119876)minus1 1 (16)

where 119876 is obtained from 119875 by removing the row and thecolumn relative to state klowast and 1 is the column vector ofones [28] In a similar way it is possible to compute the othermoments of 120591

Even if 119868 minus 119876 is upper triangular and can be blockdecomposed the computation of its inverse may not beaffordable when the cardinality of B grows In Section 54we will bound the probability of the event 120591 gt 11990552 Expected Time to 120575-Knowledge Regarding Problem 7 wecan easily modify matrix 119875 to obtain the expected time to 120575-knowledge Every state k isin P(B) such that

sumlisinP(k)

1003817100381710038171003817119860 l10038171003817100381710038171003817100381710038171003817119860 (1198860)1003817100381710038171003817 ge 120575 (17)

can be aggregated in the absorbing state summing thecorresponding column of 119875 in the last column and theneliminating the column and row corresponding to state k Inthis way it is possible to compute E[120591120575] using (16)53 Eigenvalues The following result fully characterises thespectrum of the matrix 119875

Theorem 11 For k isin P(B) the eigenvalues of 119875 have theform

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 (100381710038171003817100381711986001003817100381710038171003817 + sumlsubk|l|=1

1003817100381710038171003817119860 l1003817100381710038171003817 + sdot sdot sdot + sum

lsubk|l|=119898

1003817100381710038171003817119860 l1003817100381710038171003817

+ sdot sdot sdot + sumlsubk|l|=|k|

1003817100381710038171003817119860 l1003817100381710038171003817)

(18)

Proof The matrix 119875 being upper triangular by Lemma 10the entries 119875(k k) are the eigenvalues of the matrix Let usthen imagine to have the knowledge chain in state k Theonly way for the chain to undergo a self-transition (k rarr k)is that the user reports any combination of neighbouringnodes that have already been discovered In other words theknowledge chain undergoes a self-transition if and only if theuser reports an element ofP(k) Therefore

120582k = 11003817100381710038171003817119860 (1198860)1003817100381710038171003817 sumlisinP(k) 1003816100381610038161003816119860 l1003816100381610038161003816 (19)

Last formula is equivalent to the thesis

Since each eigenvalue is a sum of positive elements thesecond-largest eigenvalue can be obtained by maximisingover the tiles of order119873 minus 1 = max

k|k|=119873minus1120582k (20)

Wireless Communications and Mobile Computing 7

54 Convergence Properties Bounds Using (20) it is possibleto obtain the following result

Lemma 12 Given 120576 gt 0 let119878 (1 minus 120576) = log 120576log (21)

Then 119878(1 minus 120576) reports are sufficient to achieve FK withprobability greater than or equal to (1 minus 120576)Proof Using Lemma 10 and (20) on 119875

P (120591 gt 119905) le 119905 (22)

For a small target tolerance 120576 of not achieving FKif 119905 ge log 120576

log 997904rArrP (120591 gt 119905) le 120576 (23)

120575-Knowledge Convergence Bounds Using the same manipu-lation of the matrix 119875 described in Section 52 Lemma 12 inSection 54 can be applied to the modified matrix to obtaina bound for the number of steps to have 120575-knowledge withhigh probability

55 120575-Knowledge and Other-Than-Teleport Mobility Model1 is equivalent to a single user teleporting instantaneouslyto a random point within the coverage area of the nodetime is discrete Thus at each time node 1198860 receives a reportfrom a location that is sampled from the uniform probabilitydistribution over the coverage area119860(1198860) Bearing inmind thenumerical characterisation of the first time to 120575-knowledgethe teleport model is particularly convenient This task couldbe in fact carried out within the Monte Carlo paradigm bysimply throwing sufficiently many points at random insidethe coverage area 119860(1198860) In other words it is possible tonumerically study the process through which 120575-knowledgeis achieved by sampling sufficiently many times a probabilitydensity function that is uniform over the coverage area119860(1198860)

Model 1 may prove itself unsatisfactory in a real lifescenario The main problem is that if we generate a sequenceof user reports according to it any two elements of thesequence are independent whereas in general they are not Ineach mobility model where the trajectory taken by the user isphysically feasible the user positions communicated by twosuccessive reports are in fact correlated due to the motionconstraints

Let us imagine that a single user travels inside the cov-erage area 119860(1198860) according to an unknown mobility modeland let 119879(119905) be the trajectory taken by the user Samplingthe trajectory at equally spaced discrete times we obtainan embedded sequence of user locations which correspondto an embedded sequence of user reports Next we cananalyse the sequence and understand after how many steps120575-knowledge has been reached By multiplying this numberof steps by the time lapse between two consecutive reports

(inter-report time) the time to 120575-knowledge can be obtainedfor that particular realisation of the user-reports sequenceFinally the procedure above can be repeated sufficientlymany times to estimate with a Monte Carlo method theexpected time necessary for 1198860 to reach 120575-knowledge

As mentioned above in a general mobility model it islikely that two successive user reports are correlated Thesecorrelations may decay as the inter-report time grows largerand larger As an example let us imagine that a single usertravels inside the coverage area 119860(1198860) according to a MCLet 120587 be the equilibrium probability measure of the chainand let 119905mix be the mixing time of the chain that is thetime needed for the chain to reach equilibrium If the inter-report time is chosen comparable to 119905mix then the timelapse between two successive reports will be sufficient forthe MC to forget the past trajectory in other words thecorrelations between consecutive reports will be negligibleAs a consequence the user locations will be independentlydrawn from the probability measure 120587 and the matrix 119875describing the knowledge evolution will become

119875 (k l) = summisinP(k)

120587 (119860 mcup(lk)) if k sube l0 otherwise (61015840)Therefore the formulation and the results developed in

Sections 51ndash54 are still valid if we consider a single-usermobility model based on a MC provided that the time lapsebetween two consecutive reports is of the order of the mixingtime of the chain Under the assumption that user reports aresent at a frequency comparable with the inverse mixing timeof the mobility MC we can compute an upper bound on thetime to 120575-knowledge Any reporting rate higher than 1119905mixwill in fact still guarantee that 1198860 achieves 120575-knowledge ofits neighbourhood in at most E[120591120575] sdot 119905mix seconds on average(recall that E[120591120575] is measured in number of reports)

551 Multiuser Scenario We end this section by brieflymentioning a straightforward application of Model 1 in amultiuser scenario Let us imagine that 119899 users may entermove within and exit 119860(1198860) according to a hidden mobilitymodel We assume that 119899 is a very large number and thatit is possible to statistically characterise the stationary user-density by means of a probability measure 120587 over 119860(1198860)At each time every user may independently send a reportwith a very small probability 119901 Then the number of reportsreceived by 1198860 in a given time interval is approximatelyPoissonian and the time lapse between two successive reportsis exponential with parameter 120582 = 119899119901 Next let 119898 = E[120591]be the expected time to FK expressed in number of reportsreturned by (61015840) and (16) the expected time to achieve FKis the expectation of the first time for a Poisson process ofparameter 120582 to hit the state 119898 A practical example for thiskind of scenario in presented in Section 562

56 Examples

561 Femtocells Deployment for Residential Use Regardingthe use case of femtocell self-organisation presented in

8 Wireless Communications and Mobile Computing

Section 32 each serves a very small number of devicesUsing data of typical residential densities and coverage areasa statistic of the tessellation can be devised If it is possibleto establish a time after which the user position can beconsidered as drawn from a uniformdistribution then 119878(120575) isan upper bound of the time to 120575-knowledge for all the inter-report times smaller than or equal to 562 Cells Deployed in Congested Areas Opposite to theprevious example cells deployed in congested places likea mall have an extremely large basin of potential usersHowever in situations where users main interest is otherthan connecting to the Internet it is reasonable to expect thesingle-user reporting-activity to be rather sporadic There-fore the Poissonian approximation that we have mentionedat the end of Section 55 may be applicable In this casecharacterising the time to achieve 120575-knowledge is possiblethrough a statistic of the typical (or worst case) tessellations

6 Simulations

61 Teleport Model on Random Positioned Nodes In thissection we offer a preliminary assessment of the possibility ofusing the machinery developed so far in real applications Tothis purpose we developed a simulation framework inMAT-LAB and studied a scenario where 8 nodes are positionedon a plane at random according to a uniform (bivariate)probability distribution that is uniformly at random Eachnode has a circular coverage area of the same size Weconsidered 350 different configurations with the constraintthat the coverage area of 1198860 has nonvoid intersection withthe coverage area of the remaining nodes meaning thatFK is achieved as soon as all 7 neighbours are reportedto 1198860 We compute the tessellation of each configurationusing a classical Monte Carlo sampler For each of these350 configurations we computed the expected time to 09-knowledge E[120591] together with the number of steps sufficientto guarantee 09-knowledge with 90 confidence that is119878(09) The inter-report time being fixed during this firstexperiment the amount of time in seconds to achieve 09-knowledge is directly proportional to the number of steps justevaluated

Figure 4 displays the empirical probability mass functionof these two quantities E[120591] is centred around 10 steps while119878(09) is shifted on higher values as expected being an upperbound Figure 5 shows the empirical cumulative distributionfunction of E[120591] and 119878(09) We see that 16 steps are sufficientto achieve 09-knowledge for nearly all scenarios (95) whilewe need 22 steps using 119878(09) We also notice that the boundobtained from (21) is a conservative estimation because ituses only the second-largest eigenvalue Indeed it takes intoaccount only the slowest way to reach the desired knowledgewhile the problem has a rich combinatorial structure thatcannot be completely captured by (22)

Roughly speaking a usermoving at 05ms according to arandom walk model and providing at least one report everyhour can guarantee the node will have 09-knowledge withhigh probability in less than 5 h and in less than two hours

5 10 15 200

005

01

015

Number of reports

5 10 15 200

005

01

015

Number of reports

E[]

S(09)

Figure 4 Empirical probability mass function of the expected timeto 09-knowledge and the number of steps to have 09-knowledgewith 90 confidence for the teleport model on random positionednodes Since the inter-report time is fixed the simulation time isdirectly proportional to the number of reports

0

02

04

06

08

1

0 5 10 15 20 25Number of reports

Empi

rical

cdf

E[]

S(09)

Figure 5 Empirical cumulative distribution function of theexpected time to 09-knowledge and the number of steps to have09-knowledge with 90 confidence for the teleport model onrandom positioned nodes Since the inter-report time is fixed thesimulation time is directly proportional to the number of reports

if reports are sent at least every 15 minutes (see next sectionfor a more detailed analysis on the interaction between

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

Wireless Communications and Mobile Computing 7

54 Convergence Properties Bounds Using (20) it is possibleto obtain the following result

Lemma 12 Given 120576 gt 0 let119878 (1 minus 120576) = log 120576log (21)

Then 119878(1 minus 120576) reports are sufficient to achieve FK withprobability greater than or equal to (1 minus 120576)Proof Using Lemma 10 and (20) on 119875

P (120591 gt 119905) le 119905 (22)

For a small target tolerance 120576 of not achieving FKif 119905 ge log 120576

log 997904rArrP (120591 gt 119905) le 120576 (23)

120575-Knowledge Convergence Bounds Using the same manipu-lation of the matrix 119875 described in Section 52 Lemma 12 inSection 54 can be applied to the modified matrix to obtaina bound for the number of steps to have 120575-knowledge withhigh probability

55 120575-Knowledge and Other-Than-Teleport Mobility Model1 is equivalent to a single user teleporting instantaneouslyto a random point within the coverage area of the nodetime is discrete Thus at each time node 1198860 receives a reportfrom a location that is sampled from the uniform probabilitydistribution over the coverage area119860(1198860) Bearing inmind thenumerical characterisation of the first time to 120575-knowledgethe teleport model is particularly convenient This task couldbe in fact carried out within the Monte Carlo paradigm bysimply throwing sufficiently many points at random insidethe coverage area 119860(1198860) In other words it is possible tonumerically study the process through which 120575-knowledgeis achieved by sampling sufficiently many times a probabilitydensity function that is uniform over the coverage area119860(1198860)

Model 1 may prove itself unsatisfactory in a real lifescenario The main problem is that if we generate a sequenceof user reports according to it any two elements of thesequence are independent whereas in general they are not Ineach mobility model where the trajectory taken by the user isphysically feasible the user positions communicated by twosuccessive reports are in fact correlated due to the motionconstraints

Let us imagine that a single user travels inside the cov-erage area 119860(1198860) according to an unknown mobility modeland let 119879(119905) be the trajectory taken by the user Samplingthe trajectory at equally spaced discrete times we obtainan embedded sequence of user locations which correspondto an embedded sequence of user reports Next we cananalyse the sequence and understand after how many steps120575-knowledge has been reached By multiplying this numberof steps by the time lapse between two consecutive reports

(inter-report time) the time to 120575-knowledge can be obtainedfor that particular realisation of the user-reports sequenceFinally the procedure above can be repeated sufficientlymany times to estimate with a Monte Carlo method theexpected time necessary for 1198860 to reach 120575-knowledge

As mentioned above in a general mobility model it islikely that two successive user reports are correlated Thesecorrelations may decay as the inter-report time grows largerand larger As an example let us imagine that a single usertravels inside the coverage area 119860(1198860) according to a MCLet 120587 be the equilibrium probability measure of the chainand let 119905mix be the mixing time of the chain that is thetime needed for the chain to reach equilibrium If the inter-report time is chosen comparable to 119905mix then the timelapse between two successive reports will be sufficient forthe MC to forget the past trajectory in other words thecorrelations between consecutive reports will be negligibleAs a consequence the user locations will be independentlydrawn from the probability measure 120587 and the matrix 119875describing the knowledge evolution will become

119875 (k l) = summisinP(k)

120587 (119860 mcup(lk)) if k sube l0 otherwise (61015840)Therefore the formulation and the results developed in

Sections 51ndash54 are still valid if we consider a single-usermobility model based on a MC provided that the time lapsebetween two consecutive reports is of the order of the mixingtime of the chain Under the assumption that user reports aresent at a frequency comparable with the inverse mixing timeof the mobility MC we can compute an upper bound on thetime to 120575-knowledge Any reporting rate higher than 1119905mixwill in fact still guarantee that 1198860 achieves 120575-knowledge ofits neighbourhood in at most E[120591120575] sdot 119905mix seconds on average(recall that E[120591120575] is measured in number of reports)

551 Multiuser Scenario We end this section by brieflymentioning a straightforward application of Model 1 in amultiuser scenario Let us imagine that 119899 users may entermove within and exit 119860(1198860) according to a hidden mobilitymodel We assume that 119899 is a very large number and thatit is possible to statistically characterise the stationary user-density by means of a probability measure 120587 over 119860(1198860)At each time every user may independently send a reportwith a very small probability 119901 Then the number of reportsreceived by 1198860 in a given time interval is approximatelyPoissonian and the time lapse between two successive reportsis exponential with parameter 120582 = 119899119901 Next let 119898 = E[120591]be the expected time to FK expressed in number of reportsreturned by (61015840) and (16) the expected time to achieve FKis the expectation of the first time for a Poisson process ofparameter 120582 to hit the state 119898 A practical example for thiskind of scenario in presented in Section 562

56 Examples

561 Femtocells Deployment for Residential Use Regardingthe use case of femtocell self-organisation presented in

8 Wireless Communications and Mobile Computing

Section 32 each serves a very small number of devicesUsing data of typical residential densities and coverage areasa statistic of the tessellation can be devised If it is possibleto establish a time after which the user position can beconsidered as drawn from a uniformdistribution then 119878(120575) isan upper bound of the time to 120575-knowledge for all the inter-report times smaller than or equal to 562 Cells Deployed in Congested Areas Opposite to theprevious example cells deployed in congested places likea mall have an extremely large basin of potential usersHowever in situations where users main interest is otherthan connecting to the Internet it is reasonable to expect thesingle-user reporting-activity to be rather sporadic There-fore the Poissonian approximation that we have mentionedat the end of Section 55 may be applicable In this casecharacterising the time to achieve 120575-knowledge is possiblethrough a statistic of the typical (or worst case) tessellations

6 Simulations

61 Teleport Model on Random Positioned Nodes In thissection we offer a preliminary assessment of the possibility ofusing the machinery developed so far in real applications Tothis purpose we developed a simulation framework inMAT-LAB and studied a scenario where 8 nodes are positionedon a plane at random according to a uniform (bivariate)probability distribution that is uniformly at random Eachnode has a circular coverage area of the same size Weconsidered 350 different configurations with the constraintthat the coverage area of 1198860 has nonvoid intersection withthe coverage area of the remaining nodes meaning thatFK is achieved as soon as all 7 neighbours are reportedto 1198860 We compute the tessellation of each configurationusing a classical Monte Carlo sampler For each of these350 configurations we computed the expected time to 09-knowledge E[120591] together with the number of steps sufficientto guarantee 09-knowledge with 90 confidence that is119878(09) The inter-report time being fixed during this firstexperiment the amount of time in seconds to achieve 09-knowledge is directly proportional to the number of steps justevaluated

Figure 4 displays the empirical probability mass functionof these two quantities E[120591] is centred around 10 steps while119878(09) is shifted on higher values as expected being an upperbound Figure 5 shows the empirical cumulative distributionfunction of E[120591] and 119878(09) We see that 16 steps are sufficientto achieve 09-knowledge for nearly all scenarios (95) whilewe need 22 steps using 119878(09) We also notice that the boundobtained from (21) is a conservative estimation because ituses only the second-largest eigenvalue Indeed it takes intoaccount only the slowest way to reach the desired knowledgewhile the problem has a rich combinatorial structure thatcannot be completely captured by (22)

Roughly speaking a usermoving at 05ms according to arandom walk model and providing at least one report everyhour can guarantee the node will have 09-knowledge withhigh probability in less than 5 h and in less than two hours

5 10 15 200

005

01

015

Number of reports

5 10 15 200

005

01

015

Number of reports

E[]

S(09)

Figure 4 Empirical probability mass function of the expected timeto 09-knowledge and the number of steps to have 09-knowledgewith 90 confidence for the teleport model on random positionednodes Since the inter-report time is fixed the simulation time isdirectly proportional to the number of reports

0

02

04

06

08

1

0 5 10 15 20 25Number of reports

Empi

rical

cdf

E[]

S(09)

Figure 5 Empirical cumulative distribution function of theexpected time to 09-knowledge and the number of steps to have09-knowledge with 90 confidence for the teleport model onrandom positioned nodes Since the inter-report time is fixed thesimulation time is directly proportional to the number of reports

if reports are sent at least every 15 minutes (see next sectionfor a more detailed analysis on the interaction between

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

8 Wireless Communications and Mobile Computing

Section 32 each serves a very small number of devicesUsing data of typical residential densities and coverage areasa statistic of the tessellation can be devised If it is possibleto establish a time after which the user position can beconsidered as drawn from a uniformdistribution then 119878(120575) isan upper bound of the time to 120575-knowledge for all the inter-report times smaller than or equal to 562 Cells Deployed in Congested Areas Opposite to theprevious example cells deployed in congested places likea mall have an extremely large basin of potential usersHowever in situations where users main interest is otherthan connecting to the Internet it is reasonable to expect thesingle-user reporting-activity to be rather sporadic There-fore the Poissonian approximation that we have mentionedat the end of Section 55 may be applicable In this casecharacterising the time to achieve 120575-knowledge is possiblethrough a statistic of the typical (or worst case) tessellations

6 Simulations

61 Teleport Model on Random Positioned Nodes In thissection we offer a preliminary assessment of the possibility ofusing the machinery developed so far in real applications Tothis purpose we developed a simulation framework inMAT-LAB and studied a scenario where 8 nodes are positionedon a plane at random according to a uniform (bivariate)probability distribution that is uniformly at random Eachnode has a circular coverage area of the same size Weconsidered 350 different configurations with the constraintthat the coverage area of 1198860 has nonvoid intersection withthe coverage area of the remaining nodes meaning thatFK is achieved as soon as all 7 neighbours are reportedto 1198860 We compute the tessellation of each configurationusing a classical Monte Carlo sampler For each of these350 configurations we computed the expected time to 09-knowledge E[120591] together with the number of steps sufficientto guarantee 09-knowledge with 90 confidence that is119878(09) The inter-report time being fixed during this firstexperiment the amount of time in seconds to achieve 09-knowledge is directly proportional to the number of steps justevaluated

Figure 4 displays the empirical probability mass functionof these two quantities E[120591] is centred around 10 steps while119878(09) is shifted on higher values as expected being an upperbound Figure 5 shows the empirical cumulative distributionfunction of E[120591] and 119878(09) We see that 16 steps are sufficientto achieve 09-knowledge for nearly all scenarios (95) whilewe need 22 steps using 119878(09) We also notice that the boundobtained from (21) is a conservative estimation because ituses only the second-largest eigenvalue Indeed it takes intoaccount only the slowest way to reach the desired knowledgewhile the problem has a rich combinatorial structure thatcannot be completely captured by (22)

Roughly speaking a usermoving at 05ms according to arandom walk model and providing at least one report everyhour can guarantee the node will have 09-knowledge withhigh probability in less than 5 h and in less than two hours

5 10 15 200

005

01

015

Number of reports

5 10 15 200

005

01

015

Number of reports

E[]

S(09)

Figure 4 Empirical probability mass function of the expected timeto 09-knowledge and the number of steps to have 09-knowledgewith 90 confidence for the teleport model on random positionednodes Since the inter-report time is fixed the simulation time isdirectly proportional to the number of reports

0

02

04

06

08

1

0 5 10 15 20 25Number of reports

Empi

rical

cdf

E[]

S(09)

Figure 5 Empirical cumulative distribution function of theexpected time to 09-knowledge and the number of steps to have09-knowledge with 90 confidence for the teleport model onrandom positioned nodes Since the inter-report time is fixed thesimulation time is directly proportional to the number of reports

if reports are sent at least every 15 minutes (see next sectionfor a more detailed analysis on the interaction between

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

Wireless Communications and Mobile Computing 9

0

1

2

3

4

5

6

7

8

9

Model 1Random walkS(09)

Expe

cted

tim

e of 0

9-k

now

ledg

e (h)

0 01 02 03 04 05 06 07 08 09Inter-report period (h)

Figure 6 Empirical mean 09-knowledge time (in hours) of arandomwalk versus inter-report period comparedwithModel 1 andits bound 119878(09) in a femtocell grid of 8 nodes

report frequency and our bound) If the local topology is nottypically expected to change often these are acceptable times

To summarise simulation on random scenarios show thatour proposed bound can be used to estimate the time to 120575-knowledge Using realistic values the expected time to 120575-knowledge is reasonably small

62 Random Walk on a Grid In order to investigate andconfirm the ideas of Section 55 we simulated the reportssent with different inter-report times by a random walkerthat moves within 119860(1198860) under the condition of reflectiveboundary and compared this mobility model with Model 1(see Section 5) for a set of 8 nodes positioned as described atthe beginning of this section

In Figure 6 we let the inter-report time increase and com-pare the average time to achieve 09-knowledge according toboth the random walk (green line) and the teleport model(blue)We see that if the inter-report time is sufficiently largethe empirical mean time to achieve 09-knowledge for therandom walk model is well approximated by that of Model1

We assume typical femtocell parameters that is thatcoverage radius is 50m and that the user does a step in a gridof 25m every 5 s Figure 6 also shows that when reports aresent each 6min or less the time to 09-knowledge is smallerthan 1 h but at such high frequency the bound 119878(09) (redline) is not valid anymoreThe reason why more reports thanModel 1 are needed in the case of high-frequency reports isthe following since the inter-report time is short it is likelythat many reports will be sent from the same tile that is theknowledge chain will undergo many self-transitions

It is important to notice that the inter-report times usedin Figure 6 are far from the theoretical order of magnitudeof the random walk mixing time Yet Figure 6 suggests thatfor a family of scenarios it should be possible to determinethe value of the inter-report time such that the average time

AP 1AP 2

AP 3AP 4

Figure 7 Coverage areas at Hynes convention centre The colouredlines delimit the extension of the coverage areas Base stations aretransmitting at 21 GHz with a power of 34mW Point 119886 lies in thetile 119860123to achieve 09-knowledge may be well predicted by Model 1Once that value of the inter-report time is found the value ofE[12059109] returned by Model 1 may serve as an upper bound tothe actual time to achieve 09-knowledge when smaller inter-report times are implemented

To summarise simulation on random walks corroboratethe analysis of Section 55

63 A Realistic Scenario A received power map for 4 basestations in the Hynes convention centre have been generatedusing theWireless System Engineering (WiSE) [29] softwarea comprehensive 3D ray tracing based simulation packagedeveloped by Bell Laboratories Base stations are assumedtransmitting at a frequency of 21 GHz with a power of34mWWe assume there is a macrocell that covers the wholebuilding and we estimate its time to Full Knowledge Asbefore a Monte Carlo simulation has been made to estimatethe tessellation and then the expected time to 120575-knowledgehas been computed using a teleport mobility model (Model1) as explained in Section 52

Figure 7 shows the corresponding coverage areas whenthe power detection threshold is minus70 dBm Although theshape of the coverage areas and their intersection is muchmore complex than the simple scenario depicted in Sec-tion 61 it is still possible to construct the tessellation byconsidering which coverage areas each spatial point lies inFor example point 119886 lies in the coverage area of nodes 1 2and 3 so it belongs to the tile 119860123

Figure 8 displays the expected time to 120575-knowledgeE[120591120575] when 120575 is varied We notice a step-function-likebehaviour with a new step that is added every time a newstate becomes absorbing as explained in Section 52

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

10 Wireless Communications and Mobile Computing

05 06 07 08 09 12

4

6

8

10

12

14

E()

[num

ber o

f rep

orts]

Figure 8 Expected time to 120575-knowledge E[120591120575] using Model 1 inHynes convention centre for different values of the parameter 120575Theinter-report time is set to the very same value used for Figures 4 and5

0

2

4

6

8

10

12

14

Detection threshold (dBm)

E()

[num

ber o

f rep

orts]

minus60minus65minus70minus75minus80minus85minus90minus95minus100

Figure 9 Expected time to FK E[120591] using Model 1 in Hynes con-vention centre for different values of the user-detection thresholdThe inter-report time is set to the very same value used for Figures4 and 5

Figure 9 shows the behaviour of E[120591] the expected timeto FK when the user-detection threshold varies from a veryconservative value of minus60 dBm to a more realistic one ofminus100 dBm When the users are more sensitive the coverageareas and specifically the higher order tiles are biggerleading to better performance In particular we see that anaverage of 14 steps are enough to achieve FK

To summarise these results seem to confirm that thevalues obtained placing random nodes with circular coverageareas in Section 61 are compatible with real world scenariosso the use of statistics obtained frommacroscopic parametersas densities of deployment and distribution of coverage radiican be used as a tool to bound the time to 120575-knowledge7 Conclusions

In this paper we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provideda crisp mathematical formulation of it for a simple mobilitymodel We have also shown that such mobility model canbe effectively used as an upper bound for a wide range of

mobility models when the user-reports frequency is lowerthan the inverse mixing time of the Markov chain of theactual mobility model Additionally we have provided auseful method to estimate the time to 120575-knowledge when theproblem is too complex to be solved exactly

Simulations on random scenarios with typical small cellsparameters show that the expected number of reports inorder to have a high degree of knowledge of the local topologyis very small Roughly speaking a user moving at 05msaccording to a random walk model and providing at leastone report every hour can guarantee the serving node willhave 09-knowledge with high probability in less than 5 hand in less than two hours if reports are sent at least every 15minutes Since we do not expect the network topology to beaffected by high network dynamics these are acceptable timesfor the problems of interestWe encourage the adoption of thepresented framework to assess the possibility of employingcrowdsourced user reports in other self-configuration prob-lems comparing the time to 120575-knowledge with the expectedtime to convergence of a given decentralised algorithm

Simulations in more realistic scenarios show that thebounds obtained are compatible with the ones obtained fromstatistics on random scenarios with similar parameters Thisseems to confirm that the use of statistics obtained frommacroscopic parameters such as densities of deployment anddistribution of coverage radii can be used as a tool to boundthe time to 120575-knowledge

In conclusion we provide a useful tool to estimate thetime to NCL construction which is fundamental to assesswhether a decentralised algorithm can be employed in a givennetwork scenario

Conflicts of Interest

The authors declare that there are no conflicts of interest todisclose regarding the publication of this paper

Acknowledgments

The authors would like to thank Anna Zakrzewska fromBell Laboratories Alcatel Lucent Ireland for her insights anduseful suggestions

References

[1] Z Xiao T Li W Ding D Wang and J Zhang ldquoDynamicPCI allocation on avoiding handover confusion via cell statusprediction in LTE heterogeneous small cell networksrdquoWirelessCommunications and Mobile Computing vol 16 no 14 pp1972ndash1986 2016

[2] Technical Specification Group Radio Access Network ldquoHomeNode B (HNB) Radio Frequency (RF) requirements (FDD)rdquo3rd Generation Partnership Project TR 25967 2011

[3] O Johansson ldquoSimple distributed Δ+1-coloring of graphsrdquoInformation Processing Letters vol 70 no 5 pp 229ndash232 1999

[4] F Kuhn and R Wattenhofer ldquoOn the complexity of distributedgraph coloringrdquo in Proceeding of the twenty-fifth annual ACMsymposium on Principles of distributed computing (PODC rsquo06)pp 7ndash15 ACM Denver Colo USA July 2006

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

Wireless Communications and Mobile Computing 11

[5] M Luby ldquoRemoving randomness in parallel computation with-out a processor penaltyrdquo in Proceedings of the 29th Annual IEEESymposium on Foundations of Computer Science pp 162ndash173IEEE White Plains NY USA October 1988

[6] M Szegedy and S Vishwanathan ldquoLocality based graph color-ingrdquo in Proceedings of the twenty-fifth annual ACM symposiumon Theory of computing (STOC rsquo93) pp 201ndash207 ACM SanDiego Calif USA May 1993

[7] H Olofsson S Magnusson and M Almgren ldquoA concept fordynamic neighbor cell list planning in a cellular systemrdquo inProceeding of the Seventh IEEE International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo96) pp 138ndash142 IEEE Taipei Taiwan

[8] S Magnusson and H Olofsson ldquoDynamic neighbor cell listplanning in a microcellular networkrdquo in Proceedings of theICUPC 97 - 6th International Conference on Universal PersonalCommunications pp 223ndash227 San Diego Calif USA

[9] V M Nguyen and H Claussen ldquoEfficient self-optimization ofneighbour cell lists inmacrocellular networksrdquo in Proceedings ofthe 21st International Symposium on Personal Indoor andMobileRadio Communications (PIMRC rsquo10) pp 1923ndash1928 InstanbulTurkey September 2010

[10] M Amirijoo P Frenger F Gunnarsson H Kallin J Moe andK Zetterberg ldquoNeighbor cell relation list and measured cellidentity management in LTErdquo in Proceedings of the NOMS 2008- IEEEIFIP Network Operations and Management SymposiumPervasive Management for Ubiquitous Networks and Servicespp 152ndash159 Salvador Bahia Brazil April 2008

[11] R Atawia M El Azab T Elshabrawy and M Ashour ldquoRankedoverlapping coverage based construction of efficient neighbor-ing cell list for GSMUMTS cellular networksrdquo in Proceedings ofthe International Conference on Communications and Informa-tion Technology (ICCIT rsquo12) pp 254ndash259 Hammamet TunisiaJune 2012

[12] D Kim B Shin D Hong and J Lim ldquoSelf-configuration ofneighbor cell list utilizing E-UTRAN NodeB scanning in LTEsystemsrdquo in Proceedings of the 7th IEEE Consumer Communica-tions and Networking Conference (CCNC rsquo10) pp 1ndash5 IEEE LasVegas Nev USA January 2010

[13] D Aziz A Ambrosy L T W Ho L Ewe M Gruber andH Bakker ldquoAutonomous neighbor relation detection and han-dover optimization in LTErdquo Bell Labs Technical Journal vol 15no 3 pp 63ndash84 2010

[14] A Checco R Razavi D J Leith and H Claussen ldquoSelf-configuration of scrambling codes for WCDMA small cellnetworksrdquo in Proceedings of the IEEE 23rd International Sym-posium on Personal Indoor and Mobile Radio Communications(PIMRC rsquo12) pp 149ndash154 IEEE Sydney NSW AustraliaSeptember 2012

[15] K R Duffy C Bordenave and D J Leith ldquoDecentralizedconstraint satisfactionrdquo IEEEACMTransactions onNetworkingvol 21 no 4 pp 1298ndash1308 2013

[16] A Checco and D J Leith ldquoLearning-based constraint satisfac-tion with sensing restrictionsrdquo IEEE Journal of Selected Topics inSignal Processing vol 7 no 5 pp 811ndash820 2013

[17] A Checco and D J Leith ldquoFast Responsive DecentralizedGraph Coloringrdquo IEEEACM Transactions on Networking vol25 no 6 pp 3628ndash3640 2017

[18] Y Watanabe Y Matsunaga K Kobayashi H Sugahara andK Hamabe ldquoDynamic neighbor cell list management forhandover optimization in LTErdquo in Proceedings of the IEEE

73rd Vehicular Technology Conference (VTC rsquo11-Spring) pp 1ndash5IEEE Yokohama Japan May 2011

[19] P Sapiano ldquoldquoDiscovering neighbouring femto cellsrdquo USPatent EP2 214 434rdquo httpwwwfreepatentsonlinecomEP2214434A1html

[20] J Edwards ldquoImplementation of network listen modem forWCDMA femtocellrdquo in Proceedings of the IET Seminar onCognitive Radio and Software Defined Radio Technologies andTechniques IET London UK September 2008

[21] F M Naini O Dousse PThiran andM Vetterli ldquoOpportunis-tic sampling for joint population size and density estimationrdquoIEEE Transactions on Mobile Computing vol 14 no 12 pp2530ndash2543 2015

[22] D Soldani and I Ore ldquoSelf-optimizing neighbor cell listfor UTRA FDD networks using detected set reportingrdquo inProceedings of the IEEE 65th Vehicular Technology Conference- VTC2007-Spring pp 694ndash698 IEEE Dublin Ireland April2007

[23] F Parodi M Kylvaja G Alford J Li and J Pradas ldquoAnautomatic procedure for neighbor cell list definition in cellularnetworksrdquo in Proceedings of the IEEE International Symposiumon a World of Wireless Mobile and Multimedia Networks(WOWMOM rsquo07) pp 1ndash6 IEEE Espoo Finland June 2007

[24] MRHasanMTKawser andMR Islam ldquoAn automaticGSMneighbor cell list update procedure enhancing SON in LTErdquo inProceedings of the 6th International Conference on Electrical andComputer Engineering (ICECE rsquo10) pp 143ndash146 IEEE DhakaBangladesh December 2010

[25] J Li and R Jantti ldquoOn the study of self-configuration neighbourcell list for mobile WiMAXrdquo in Proceedings of the InternationalConference onNextGenerationMobile Applications Services andTechnologies (NGMAST rsquo07) pp 199ndash204 IEEE Cardiff UKSeptember 2007

[26] Z Becvar M Vondra and P Mach ldquoDynamic optimizationof neighbor cell list for femtocellsrdquo in Proceedings of the 77thVehicular Technology Conference (VTC Spring rsquo13) pp 1ndash6IEEE Dresden Germany June 2013

[27] D A Levin Y Peres and E L Wilmer Markov chains andmixing times AMS 2009

[28] J G Kemeny and J L Snell Finite Markov chains with a newappendix ldquoGeneralization of a fundamental matrixrdquo Springer-Verlag Berlin Germany 1976

[29] S J Fortune D M Gay B W Kemighan O Landron RA Valenzuela and M H Wright ldquoWISE design of indoorwireless systems practical computation and optimizationrdquoIEEE Computational Science amp Engineering vol 2 no 1 pp 58ndash68 1995

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