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Mean-field limits for ultra-dense random-access networks

Citation for published version (APA):Cecchi, F. (2018). Mean-field limits for ultra-dense random-access networks. Technische Universiteit Eindhoven.

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Mean-Field Limitsfor Ultra-Dense

Random-Access Networks

is work was nancially supported by e Netherlands Organization for Sci-entic Research (NWO) through the TOP-GO grant 613.001.012.

© Fabio Cecchi, 2018

Mean-Field Limits for Ultra-Dense Random-Access Networks

A catalogue record is available from the Eindhoven University of TechnologyLibraryISBN: 978-90-386-4415-8

Cover design by Silvana Pianadei, “Soo Vitale”

Printed by Gildeprint Drukkerijen, Enschede

Mean-Field Limits for Ultra-DenseRandom-Access Networks

proefschrift

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnicus, prof.dr.ir. F.P.T. Baaijens, voor een

commissie aangewezen door het College voorPromoties in het openbaar te verdedigen

op donderdag 1 februari 2018 om 16.00 uur

door

Fabio Cecchi

geboren te Castelnuovo di Garfaganana, Ital̈ıe

iv

Dit proefschri is goedgekeurd door de promotoren en de samenstellingvan de promotiecommissie is als volgt:

voorzier: prof.dr. J.J. Lukkien1e promotor: prof.dr.ir. S.C. Borst2e promotor: prof.dr. J.S.H. van Leeuwaardenleden: prof.dr.ir. B.R.H.M. Haverkort (University of Twente)

dr. N. Gast (INRIA & University of Grenoble Alpes)prof.dr.ir. J.F. Grooteprof.dr. M.A. Peletierprof.dr. B. Van Houdt (University of Antwerp)

Het onderzoek dat in dit proefschri wordt beschreven is uitgevoerd inovereenstemming met de TU/e Gedragscode Wetenschapsbeoefening.

Acknowledgments

Four years have passed from my arrival in Eindhoven, a long journey sharedwith many persons that participated, consciously or not, in the development ofthis thesis. I would like to take advantage of these few lines to thank them.

First of all, I am immensely grateful to my supervisors Sem and Johan. Yourconstant guidance and insightful advices throughout these years have shaped meas a researcher and as a person, and I could not have asked for beer supervision.Sem, thank you for your passion, kindness, and thoroughness. I feel very luckyfor being granted the opportunity to work at your side, I will always look up toyou. Johan, thank you for your directness, constant motivation, and for alwaysnding time for me, no maer your uncountable number of tasks.

A special thanks goes to Phil, my unocial supervisor who led me throughthe maze of the technical details of this thesis. Many of the results achievedwould have not seen the light without your help. You taught me much morethan mathematics, I loved your anecdotes on Dutch windmills, British navy,Italian scientists, recipes, politics, and whatever else. I will always admire yournever-ending enthusiasm.

In these years I had the opportunity to work with a variety of people and Iam truly thankful for that. Florian, you have been a great host in Paris. Nidhiyou introduced me to statistical learning and I greatly enjoyed my internship atNokia Bell Labs. Peter, Seva, you welcome me in your project and Chapter 6 ofthis thesis is largely merit of yours.

Looking backw, I would have not been here if not for Peter Jacko. I have greatmemories of the internship at BCAM, and you instilled in me the condence Ineeded to embark on this journey, and I will always owe you for that.

I would like to express my sincere gratitude to Boudewijn Haverkort, NicolasGast, Jan Friso Grote, Mark Peletier, and Benny Van Houdt for agreeing to serve

v

vi Acknowledgements

on my doctoral commiee and for commenting my thesis.I have been lucky to get to know many colleagues here in the STO group.

Many already le and many just started, I learned a lot from all of you, and Iknow that the friendships established will survive the future unavoidable longdistance. A special mention goes to Alessandro, my big brother and constantsource of advice, to Carlo, for literally showing me the way home, to Gianmarco,Jori, Bri, and omas for having being there throughout this whole adventure,I could not have asked for beer fellows, and to Fiona for being the perfectfourth in our awesome oce. But these years would have not been the samewithout all of you, I will bring with me the lunches, the board game nights,the hipster events, the squashes and the basketball games, the italian dinners,Lunteren, the conference trips. ank you for the great moments, I will keepspamming the whatsapp group (not that you care…).

I have been fortunate to meet great friends outside the university as well.Salvatore, Egle, and Claudio, I loved our house, the bbq, and our lovely neighbour,Alberto, Antonio, Bea, Tommaso, GM, Nate, even if I might have stabbed youin the back a couple of times, I kind of like you all and I will truly miss ourevenings a lot.

It is not easy to see someone just a few times per year and still maintain asincere friendship. Cecca, Vale, you have been a constant presence by my sidefor my whole life, it will never change, Fana, Sara, Lisa, Aly, you kept me inthe loop of your crazy lives despite the distance and continuously supportedme, thank you! Nando’s folks, you have always been the best distraction forthe most stressful periods, Marco, Tommy, Leo, Pietro, Baei, Borgo, Sara, Ele,Giulia, Costi, Chiara, Vero, Mauro, Tome I have been so lucky to cross your pathin Pisa, looking forward to meeting you every single time, no maer where.

I am profoundly indebted to my family, for always trusting my decisions,never doubting, and unconditionally loving me. Grazie Signori, e grazie Paola peraverli sopportati e supportati senza di me. You have been more than wonderful.

Anna, my nal words are for you. You are my partner and my best friend, Iwould not be the same without you by my side. We shared every moment inthese years and I am eagerly looking forward to our future together.

Contents

Acknowledgments v

1 Introduction 11.1 Wireless networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Saturated CSMA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Unsaturated CSMA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Mean-eld asymptotic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Beyond the classical mean-eld scenario . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 roughput and Stability Analysis 232.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Structural characterization of the stability region . . . . . . . . . . . . . . 292.4 Boundaries of the stability region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 ree-node network illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.A Extended proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Mean-Field Analysis of Random-Access Networks 533.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3 Derivation of the mean-eld limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Analysis of the mean-eld limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

vii

viii Contents

3.6 Model extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.A Asymptotically Lipschitz in probability . . . . . . . . . . . . . . . . . . . . . . . . . . 853.B Extended proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Interchange of Limits and Performance Analysis 954.1 Model description and overview of the results . . . . . . . . . . . . . . . . . 964.2 Global stability of x∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.3 Positive recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4 Tightness and interchange of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.5 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.A Polling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.B Extended proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Optimal Activation Rates 1295.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.2 Stability and mean stationary performance . . . . . . . . . . . . . . . . . . . . . 1325.3 Multi-scale mean-eld limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.4 Aggregate back-o rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.A Connection with polling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.B Extended proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6 Mean-Field Limits for Multi-Hop Networks 1676.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.2 Mean-eld analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.3 Fixed-point approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.4 Performance analysis of linear networks . . . . . . . . . . . . . . . . . . . . . . . . 1786.5 Optimal back-o rates for linear networks . . . . . . . . . . . . . . . . . . . . . . . 1846.6 General networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.A Partial xed points as solution of a NLCP . . . . . . . . . . . . . . . . . . . . . . . 1926.B Extended proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7 Spatial Mean-Field Limits 2017.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057.3 Outline of the proof of the spatial mean-eld limit . . . . . . . . . . . . 209

Contents ix

7.4 Stochastic coupling: clustering approximation . . . . . . . . . . . . . . . . . . 2217.5 Analysis of the mean-eld limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.6 Unit circle with uniformly spaced nodes. . . . . . . . . . . . . . . . . . . . . . . . . 2307.7 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2457.A Extended proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Bibliography 261

Summary 273

About the author 275

Chapter 1

Introduction

Contents

1.1 Wireless networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Saturated CSMA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Unsaturated CSMA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Mean-eld asymptotic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Beyond the classical mean-eld scenario . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Wireless networks are already large and complex today and are expected togrow even denser in the future. e rapid increase in wireless applications hasled to a growing demand for scarce wireless spectrum, and hence it is necessaryto make the most ecient use of the limited available capacity. Obviously, whenthe number of nodes is large, in the hundreds or even thousands in connedareas, a dedicated channel cannot be assigned to each node, and nodes have toshare the medium. Simultaneous transmissions on the same channel will how-ever inevitably give rise to interference and loss of throughput. Medium AccessControl (MAC) mechanisms are crucial to resolve this contention. However, inlarge networks, a centralized control mechanism is hard to implement and tomaintain since it would require constant network status updates generatingprohibitive communication overhead. Hence, due to the massive number ofnodes, these large-scale networks will typically rely on the individual nodesto dynamically share the medium in a distributed fashion and for this reason,

1

2 Chapter 1. Introduction

the design and analysis of ecient distributed access schemes is of critical im-portance. Due to their low implementation complexity randomized algorithmshave emerged as a particularly popular class of distributed mechanisms.

Mathematical models for random-access networks face many obstacles andtractable analytical solutions are generally hard to obtain. Distributed MACalgorithms regulate the system behavior on a local level, but the evolution ofthe system on a global scale is intricate and dicult to characterize. e vastmajority of existing models assume constant presence of packets to transmit(saturated conditions) thus obtaining analytical tractability. Saturated modelsprovide a useful throughput characterization for persistent ows, but fail to de-scribe situations in which packets are generated sporadically. In these scenarios(unsaturated conditions), buer contents uctuate as packets are generated andtransmied over time. Specically, nodes may temporarily refrain from com-petition for the medium when the buers empty, yielding a intricate two-wayinteraction between the activity process and the buer content process. estudy of unsaturated models is of fundamental importance for delay-sensitiveapplications requiring agile medium access whenever packets are generated.

is thesis provides insights in key performance measures of random-accessnetworks such as throughput and packet delay, while accounting for the buerdynamics. Exact analytical solutions being out of reach, we focus on the asymp-totic regime where the number of nodes grows large, oen referred to as many-sources or mean-eld regime. In the context of random-access networks, themean-eld regime not only provides analytical tractability, but is also highlyrelevant for the large-scale networks which are envisioned to emerge in thefuture. In the mean-eld asymptotic regime the activity process evolves on afaster time scale than the population process and thus inuences the evolution ofthe laer only via its stationary distribution. e complicated relation betweenbuer dynamics and activity process drastically simplies and the queue lengthprocess may be described by the solution of an initial-value problem yieldingasymptotic approximations for the relevant performance metrics. In this chap-ter we briey introduce both wireless networks and the general concepts inmean-eld theory, discuss the relevant literature, and present an overview ofthe thesis content.

1.1 Wireless networks

A wireless network consists of a collection of devices which require access toa shared medium in order to exchange information. In each communication,a device might be either a transmier or an intended receiver, and we assumethat each data packet is to be sent by a transmiing device to a single receiver.

1.1. Wireless networks 3

Wireless networks are abstracted so as to be the collection of transmier/receiverpairs which will be commonly named nodes and denoted by N = {1, . . . , N}.When a node gains access to the medium, it is said to be active and packets aretransmied at the corresponding transmier-receiver pair. e joint activitystate of the network is described by ω = (ω1, . . . , ωN ) ∈ {0, 1}N where ωnindicates whether node n is active or not.

Each node is equipped with a buer where packets are queued before beingtransmied. When a node gains access to the medium, a packet leaves thebuer so as to be transmied. e queue length process describes the numberof packets in the buers of all nodes over time. is gives raise to an N -dimensional process. Moreover, since empty nodes temporarily refrain fromaccessing the medium, the activity process and the queue length process arestrongly intertwined. As mentioned earlier, when nodes sporadically generatepackets, thus having empty buers most of the time, the queueing state hasa fundamental impact on the activity dynamics of the network. In this thesiswe aim to gain a beer understanding of the intricate relationship linking theactivity process and queue length process, in order to evaluate the performanceof the network.

1.1.1 Interference model

When a transmier is active, the wireless signal propagates not only in the di-rection of the intended receiver but in an omni-directional manner. In particular,receivers may overhear the interference of other simultaneous transmissionsand perceive it as noise which could prevent a correct reception. Simultaneousactivity of nodes within close range can thus cause interference, preventingsuccessful reception of packets. e interference relations are described by theset Ω ∈ {0, 1}N , where ω ∈ Ω denotes a feasible activity state, in the sense thatevery node active in ω is able to successfully complete its transmission. In turn,Ω determines the capacity region of the network Γ = Conv(Ω), i.e.,

Γ ={γ =

∑ω∈Ω

αωω :∑ω∈Ω

αω ≤ 1, αω ≥ 0 ∀ω ∈ Ω}.

e set Γ is named capacity region since for any vector γ = (γ1, . . . , γN ) ∈ Γ,it is possible for each node n ∈ N to be active for a fraction γn of the time,while the activity state of the network always remains within Ω.

e set of feasible activity states Ω obviously depends on the interferencemodel chosen. Popular interference models include the physical model andthe protocol model [Gupta and Kumar, 2000]. e rst is based on the Signal-to-Interference-plus-Noise Ratio (SINR) conditions. In this thesis we use the


protocol model which is simpler and based on the idea that two nodes interfereif and only if they are located within a certain range of each other. In somesense, the protocol model approximates the physical model by neglecting thenoise generated by transmissions of nodes not within range. and it is generallyconsidered a good compromise between reality and accuracy. In fact, thismodel substantially simplies the mathematical analysis while it encapsulatesthe essential features of interference constraints in actual wireless networks.Moreover, seing a large interference range, yields a conservative estimate[Halldórsson and Tonoyan, 2015, Zhou et al., 2013].

e interference relations in the protocol model can be represented by aninterference graph G = (N , E) which characterizes the pairs of interferingnodes through the edge set E . Nodes n, n′ ∈ N , cannot transmit at the sametime if {n, n′} ∈ E . Hence, ω ∈ Ω if and only if ωnωn′ = 0 for every n′ ∈ Nnwhere

Nn = {n′ ∈ N : {n, n′} ∈ E} ⊆ N (1.1)

consists of all the nodes interfering with node n, the neighbors of n.

1.1.2 Medium Access Control

e MAC algorithm regulates the way nodes access the medium, and so thefraction of time they are active. An ecient access scheme should allow nodesto activate as much as possible while ensuring the activity state to alwaysremain in Ω. Within the various MAC algorithms one can distinguish betweencentralized algorithms which rely on global information of the network statusand distributed algorithms which only require local information.

Centralized algorithms have full knowledge of the network status and accord-ingly decide the most advantageous set of nodes to schedule for transmission atevery moment in time. e Max-weight scheduling algorithm [Tassiulas andEphremides, 1992] is an example of a centralized algorithm. At every decisioninstant, it selects the set of non-conicting nodes whose aggregate queue length(weight) is maximal and schedules their transmissions. e Max-weight algo-rithm has the remarkable ability of being maximally stable, meaning that itmanages to stabilize the system for any load within the interior of the capacityregion.

e full knowledge of the network status allows a centralized algorithm toalways make the best possible scheduling decision (according to some metrics).However, when the number of nodes is large, centralized algorithms can not beeasily implemented. In fact, to continually monitor the entire network, thesealgorithms require constant updates involving a prohibitive communicationoverhead. On the other hand, distributed algorithms operate locally and allow

1.2. Saturated CSMA model 5

each node to decide autonomously when to access the medium. e distributedoperation comes at the expense of optimality which might not be achievabledue to incomplete knowledge of the network status. e decisions of thesealgorithms oen involve some degree of randomness, hence the name random-access algorithms.

e rst random-access algorithm actually implemented was the ALOHAprotocol, developed in the seventies [Abramson, 1970, Roberts, 1975]. isscheme establishes that every node has to back-o from transmiing for randomperiods. e length of the back-o periods is node-dependent and is based onthe feedback received by the past transmission aempts.

e descendant of ALOHA is the Carrier-Sense Multiple-Access (CSMA)protocol [Kleinrock and Tobagi, 1975], which is currently at the core of the IEEE802.11 and 802.15.4 standards. e key feature of CSMA compared to ALOHAis the introduction of a sensing mechanism which facilitates the process ofcollision avoidance and renes the back-o mechanism inherited from theALOHA protocol. During its back-o period, a node also senses the mediumand checks whether the noise level would allow a transmission to start. When ithears too much noise, the node freezes its back-o period which is only resumedwhen the medium is sensed free again.

In this thesis we present mathematical models for various versions of theCSMA protocol. We develop methods to evaluate the performance measuresof CSMA networks, such as stability conditions, delay, and throughput of theindividual nodes.

1.2 Saturated CSMA model

As mentioned earlier, this thesis focuses on scenarios where packets exogenouslyarrive and need to be transmied before leaving the system. In particularnodes might have empty buers and refrain from medium competition fromtime to time. ere is a bulk of literature which instead addressed saturatedCSMA models, where nodes always have packets available to be transmied.Random-access networks in saturated conditions were already considered inthe eighties [Boorstyn and Kershenbaum, 1980, Boorstyn et al., 1987, Kelly,1985, Kershenbaum et al., 1987]. Similar models were further developed in[Wang and Kar, 2005, Durvy and iran, 2006], with several extensions andrenements in [Durvy et al., 2007, Gareo et al., 2008, Liew et al., 2010, Medepalliand Tobagi, 2006, Shi et al., 2008]. In these models queueing dynamics play norole and the analysis drastically simplies, allowing elegant results which wereview in this section.


Node n takes an exponentially distributed time with parameter µn to trans-mit a single packet. Aer the transmission is completed, node n has to obeya back-o period which is exponentially distributed with parameter νn. In asaturated model, the only process of interest is the activity process A(t) ∈ Ωwhich species which nodes have access to the medium (are active) at everymoment in time. e activity process is a reversible Markov process, wheregiven ω, ω′ ∈ Ω the transition rate between ω and ω′ is given by

r(ω, ω′) =

νn, if ω′ = ω − en,µn, if ω′ = ω + en,0, otherwise.

(1.2)

Denote σn = νn/µn, and σ = (σ1, . . . , σN ).A remarkable property of these models is that the activity process has the

elegant product-form stationary distribution [Boorstyn et al., 1987, Kelly, 1985]

π(ω;σ) := limt→∞

P {A(t) = ω} = 1Z(σ)

N∏n=1

σωnn (1.3)

with the normalizing constant

Z(σ) :=∑ω∈Ω

N∏n=1

σωnn . (1.4)

e product-form stationary distribution is insensitive to the distribution oftransmission times and back-o lengths [Liew et al., 2010, Van de Ven et al.,2010].

A fundamental performance metric in CSMA networks is the stationarythroughput, which is the rate at which packets are transmied in stationarity.e stationary throughput of node n is denoted by θn(σ) and given by

θn(σ) := µn∑

ω∈Ω+n

π(ω;σ), Ω+n = {ω ∈ Ω : ωn = 1}. (1.5)

Note that, due to (1.3), it holds as well that

µn∑

ω∈Ω+n

π(ω;σ) = νn∑

ω∈Ω−n

π(ω;σ), (1.6)

whereΩ−n =

{ω ∈ Ω : ωn = 0, ωn′ = 0 ∀n′ ∈ Nn

}. (1.7)

1.2. Saturated CSMA model 7

Figure 1.1: Interference graph for a three-node linear network.

Various papers have pursued the optimization of activation rates for sat-urated CSMA networks. e objective has mainly been concerned with opti-mizing some fairness criterion or global throughput utility metric [Jiang andWalrand, 2008, Marbach and Eryilmaz, 2008, Sanders et al., 2016].

In [Jiang and Walrand, 2010] it was shown that the range of the throughputmap θ(·) corresponds to the interior of the capacity region Γ = Conv(Ω). eyalso provided an adaptive distributed algorithm to determine the activationrates to achieve any target throughput vector γ ∈ int(Γ). In particular, forevery feasible target throughput vector γ ∈ int(Γ), there exists a unique vectorof activation rates σ(γ) ∈ RN+ such that

θ(σ(γ)) = γ. (1.8)

In [Van de Ven et al., 2011] numerical methods for the computation of σ(γ)were presented. e main drawback of such algorithms is the necessity to knowall the independent sets of the interference graph G, whose number may growexponentially in N . e numerical algorithms are thus implementable onlywhen the number of nodes is moderate. Due to the computational complexity ofexact numerical methods, there has recently been a lot of interest in obtainingaccurate approximations for σ(γ). e most popular methods are based onfree-energy approximations due to Bethe [Yun et al., 2015] and Kikuchi [Swamyet al., 2016, Van Houdt, 2017a] which are shown to provide exact formulae foracyclic and chordal networks, respectively. In [Van Houdt, 2017b] a theoreticalframework covering both methods is presented.

1.2.1 An illustrative example - ree-node linear network

To exemplify the model and the notation introduced so far, we now consideran illustrative example whose interference graph is displayed in Figure 1.1In this network there are three nodes and the interference graph is given byG = (N , E) with

N ={

1, 2, 3}, E =

{{1, 2}, {2, 3}

}.


e set of feasible activity states is given by

Ω ={

(0, 0, 0); (1, 0, 0); (0, 1, 0); (0, 0, 1); (1, 0, 1)}

={ω(k)

}k=1,...,5

,

and thus we have as capacity region of the system

Γ = Conv(Ω) ={γ ∈ R3 : γ1 + γ2 ≤ 1, γ2 + γ3 ≤ 1

}.

Moreover, we have that

Ω+1 ={ω(2);ω(5)

}, Ω+2 =

{ω(3)

}, Ω+3 =

{ω(4);ω(5)

},

Ω−1 ={ω(1);ω(4)

}, Ω−2 =

{ω(1)

}, Ω−3 =

{ω(1);ω(2)

}.

e stationary distribution of the activity process is given by π(ω(k);σ) with

π(ω;σ) =1

Z(σ)

3∏n=1

σω(k)nn , Z(σ) = 1 + σ1 + σ2 + σ3 + σ1σ3,

so that the stationary throughput of the various nodes θ(σ) is given by

θ1(σ) = µ1σ1 + σ1σ3Z(σ)

, θ2(σ) = µ2σ2Z(σ)

, θ3(σ) = µ3σ3 + σ1σ3Z(σ)

.

Finally, for every γ ∈ Int(Γ), it may be shown that the target-throughputback-o rates σ(γ) are given by

σ2(γ) =γ2(1− γ2)

(1− γ1 − γ2)(1− γ2 − γ3), σn(γ) =

γn1− γn − γ2

, n = 1, 3.

1.3 Unsaturated CSMA model

While saturated models provide a useful characterization for persistent owsthey fail to capture real-life scenarios where nodes generate packets only spo-radically. In particular, these models cannot be used to evaluate the performancein case nodes oen have empty buers and must comply with fairly tight delayconstraints. Such scenarios arise in an Internet-of-ings (IoT) context of emerg-ing applications in for example intelligent environments, smart energy grids,vehicular control, and industrial automation. One of the contributions of thisthesis is to show that in a certain sense the analysis of large-scale unsaturatednetworks can be reduced to the analysis of saturated counterparts, providing aperformance evaluation methodology for delay-sensitive applications.

1.3. Unsaturated CSMA model 9

Figure 1.2: Interference graph for an unsaturated CSMA network.

To go beyond the saturated CSMA networks described in Section 1.2, weallow exogenous arrivals of packets at the various nodes (unsaturated CSMAmodels). Packets arrive to the buer of node n as a Poisson process of rate λnand leave the system when the transmission is completed. us, each node isequipped with a buer as displayed in Figure 1.2. A node freezes its back-operiod whenever its buer is empty.

We denote the queue length process byQ(t) = (Q1(t), . . . , QN (t)), whereQn(t) ∈ N0 describes the number of packets waiting in the buer of node n attime t. e joint process

(Q(t), A(t)

)taking values in NN0 × Ω is Markovian

and characterizes the network evolution. Given that (q, ω), (q′, ω′) ∈ NN0 × Ω,the transition rates are

r((q, ω), (q′, ω′)

)=

λn, q

′ = q + en, ω′ = ω,

νn, q′ = q − en, ω′ = ω + en,

µn, q′ = q, ω′ = ω − en,

0, otherwise.

Observe that the state space of the process dened above is innite in as manydimensions as the number of nodes in the system and unfortunately a closed-form solution does not exist in general [Laufer and Kleinrock, 2016, Van deVen et al., 2010]. In particular, the product-form stationary distribution of theactivity process (1.3) does not hold anymore in unsaturated models. In Chapter2 we will thoroughly investigate this model and we will show that a closed-formstationary distribution of the activity process exists only when all the nodesinterfere with each other, i.e., in the case of a complete interference graph.

Even the basic conditions for the existence of a stationary distribution arenot known in general. Denote the load of node n ∈ N by ρn = λn/µn, and


observe that ρ = (ρ1, . . . , ρN ) ∈ Γ is a necessary (but not sucient) conditionfor positive recurrence of the queue length process. Only in case of interferencegraphs with specic structures the exact conditions can be established. Forinstance, when the interference is complete, it was proved in [Van de Ven et al.,2010] that the system is stable if and only if ρ ∈ Γ, i.e.,

∑n′∈N ρn′ < 1, and

maxn∈N

λnνn(1−

∑n′∈N ρn′)

< 1. (1.9)

However, for general interference graphs, closed-form stability conditions areas dicult to obtain as the entire stationary distribution. Surprisingly, even fora simple graph such as the one in Figure 1.1, explicit stability conditions are outof reach. As we will show in Chapter 2, nodes 1 and 3 are stable if

ρn <(1− ρ2

) νnµn + νn

, n = 1, 3,

but similar conditions for node 2 cannot be established. Intuitively, the fractionof time node 2 is not blocked (node 2 can thus back-o and transmit) stronglydepends on whether nodes 1 and 3 mostly transmit simultaneously or not, hencethe complexity of the model.

e above discussion assumed access schemes with xed activation rates,which are aractive due to their low implementation complexity. A parallel lineof work has focused on queue-based access mechanisms [Ghaderi and Srikant,2010, Jiang et al., 2010, Rajagopalan et al., 2009, Shah et al., 2011] where theback-o rates are queue-dependent. Remarkably, properly designed queue-based schemes achieve maximum stability [Jiang et al., 2010, Jiang and Walrand,2010, Shah and Shin, 2012], although being distributed. In this thesis we willmostly focus on schemes with xed activation rates, but some of our resultseasily generalize to scenarios with queue-dependent back-o rates.

1.3.1 Multi-hop CSMA model

So far, we have tacitly assumed that packets leave the system upon transmission.However, in device-to-device (D2D) communications [Asadi et al., 2014] or inmobile ad-hoc networks (MANETS), devices are used as relays and packetsare transmied along several links before reaching their nal destination andleaving the system. Regardless of the routing and the MAC algorithm used,multi-hop models extend single-hop models in a non-trivial way. Since packetsre-enter the system aer transmission, the interdependence between the queuelength process and the activity process intensies and the analysis of the modelgets even more complicated.

1.4. Mean-eld asymptotic regime 11

In multi-hop scenarios, we seek to understand in which way the networkdisposes of the incoming packets. Specically, it does not truly maer that anindividual node has excellent throughput performance if the packets it transmitsare then stuck at the following node. A relevant metric in multi-hop modelsis the end-to-end throughput, the rate at which packets reach their destination.Stability conditions for the network are important as well, but these beingalready dicult to establish in single-hop models, it is not surprising that thesehave remained elusive so far. Of particular interest is the analysis of multi-hopnetworks in over-saturation, meaning that packets are always present at thesource of the ow, i.e., the rst node in the multi-hop route, but not at the othernodes. is mixes saturated and unsaturated models and nodes may eitherbe able to sustain the incoming load (local stability) or act as bolenecks andbuild-up a queue in their buer [Adan et al., 2015]. Being able to locate theweak links along the path of the ows is of crucial importance in dealing withoverloaded situations.

Only partial results for multi-hop CSMA models have been obtained so far[Aziz et al., 2013, Denteneer et al., 2008, Laufer and Kleinrock, 2016, Shneerand Van de Ven, 2015]. In [Aziz et al., 2013] the authors studied a multi-hopCSMA network using simulations and experiments, and showed that the end-to-end throughput may decrease as the external arrival rate increases, due tocongestion. is phenomenon is unique to multi-hop networks, and cannotbe captured by single-hop saturated or unsaturated models. e behavior of athree-node linear multi-hop network in over-saturation is described in [Shneerand Van de Ven, 2015], but the approach used there cannot be extended to larger,more realistic networks.

1.4 Mean-eld asymptotic regime

In Section 1.3 we introduced the unsaturated CSMA model and observed thatan exact analysis is intractable due to complex interactions between the queuelength and activity process, except for full interference graphs. erefore,we resort to asymptotic analysis of unsaturated CSMA networks: we let thenumber of nodes grow large and examine the behavior in the so-called mean-eld asymptotic regime.

One motivation for a mean-eld analysis of CSMA networks is that theanalysis of the mean-eld limit is mathematically convenient as the complexinteraction between the queue length and activity processes drastically sim-plies. Another reason is that the mean-eld regime is highly relevant froma practical viewpoint as the number of nodes growing large reects the hugenumber of devices competing for medium access in real applications [Evans,


2011, Ericsson, 2011].In this section we introduce the terminology and the classical notation of

mean-eld theory in the context of random-access networks. We aim to conveythat the analysis of unsaturated CSMA networks simplies if certain symmetryconditions amongst the various nodes hold. An important instance is whenthere is a substantial number of nodes with similar trac and placement in thenetwork, so that the operation of one is equivalent to that of many others. Athorough description of the methodology will be provided in Chapter 3.

1.4.1 An illustrative example - Complete interference

To illuminate the methodology, we conne this preliminary analysis to a sym-metric scenario with N mutually interfering nodes and

λn =λ

N, νn =

ν

N, µn = µ,

for every node n ∈ N . ese assumptions will be relaxed in the detailed analysispresented in Chapter 3.

e various nodes are statistically indistinguishable and thus, to capturethe global behavior of the system, we only need to keep track of how manynodes are in each state and not of the specic states of individual nodes. isprocedure leads to a state aggregation which yields a simplied description ofthe model. Specically, the process

(Q(N)(t), A(N)(t)

)is replaced by(

X(N)(t), Y (N)(t)),

where X(N)(t) is the population process and Y (N)(t) the aggregate activityprocess. e population processX(N)(t) = {X(N)m (t)}m∈N0 is dened by

X(N)m (t) =1

N

N∑n=1

1

{Q(N)n (t) = m

}and takes values in

EN ={x ∈ R∞≥ : Nxm ∈ N0,

∞∑m=0

xm = 1}.

e aggregate activity process Y (N)(t) takes values in {0, 1}, and is dened as

Y (N)(t) =

N∑n=1

A(N)n (t).


Clearly,(Q(N)(t), A(N)(t)

)uniquely determines

(X(N)(t), Y (N)(t)

)but in

the laer process the specic state information of each node is lost. In fact, wekeep track only of the number of nodes in each state (population process) andof whether the medium is occupied or not (aggregate activity process).

Given the sequence of processes(X(N)(t), Y (N)(t)

), N > 0,

we aim to characterize the limit of the sequence asN →∞, i.e., the unsaturatedCSMA model in the mean-eld asymptotic regime. We will prove that

X(N)(Nt) ⇒ x(t), (1.10)

where ⇒ denotes “weak convergence” and applies to the entire path. elimiting process x(t) is continuous and takes values in

E ={x ∈ R∞≥ :

∞∑m=0

xm = 1},

and consists of the unique solution of a deterministic Initial-Value Problem (IVP)

dx(t)

dt= H(x(t)), x(0) = x∞, (1.11)

where the functionH(·) = (H0(·), H1(·), . . .) is dened by

H0(x) = −λx0 + νπ(0;σ(1− x0)

)x1,

Hm(x) = λ(xm−1 − xm

)− νπ

(0;σ(1− x0)

)(xm − xm+1

),

for m ≥ 1, andπ(0;σ(1− x0)

)=

1

1 + σ(1− x0). (1.12)

At rst sight, the aggregate activity process does not seem to play a rolein the evolution of x(t). However, the inuence of the activity process iscaptured by (1.12). Let us explain why by looking at the prelimit process withN nodes. e population process evolves on a time-scale of order 1/N (therescaled version evolves on a time-scale 1) while the aggregate activity processchanges on a time-scale of order 1. erefore, as N grows large, the processesexperience a time-scale separation. In (1.10) we consider a uid time-scale Nt,which captures the evolution of the population process but is too fast to describethe aggregate activity process. us, as N → ∞, the joint process exhibitsstochastic averaging [Feuillet and Robert, 2014, Hunt and Kurtz, 1994], in the


sense that the aggregate activity process reaches its stationary distributionbefore the population state changes. Note that from the point of view of theaggregate activity process, in between timeNt andNt+, the system looks static(the population state does not change).

In particular, the queueing dynamics at any given node are only aected bythe global network state through (1.12). e laer quantity corresponds to thefraction of time that no activity is present in case of a certain static activation rate.In fact, (1.12) coincides with the idle component of the stationary distributionof a saturated network with a single node with transmission rate µ and back-orate ν(1 − x0(t)). is time fraction thus encapsulates the global networkimpact and inherits the product-form stationary distribution (1.3). In this sense,mean-eld theory bridges the gap between saturated and unsaturated CSMAnetworks, reducing the analysis of the laer to the former.

1.4.2 Concepts in mean-eld theory

Mean-eld analysis has gained popularity in many dierent areas. e generalidea is that the local eects due to pairwise interactions on each tagged nodemay be approximated by the mean eld, a global averaged eect depending onthe current state of the entire system. Intuitively, the approximation is based ona law of large numbers kind of argument, where the eect of a single pairwiseinteraction is mitigated by that of many others, and is accurate when the systemis large [Als-Nielsen and Birgeneau, 1977, Ginzburg, 1961]. We now introducesome important concepts which are key to understanding the techniques andthe limitations of the mean-eld approach.

e nodes are exchangeable when the distribution of(Q(N)(t), A(N)(t)

)is

invariant under any permutation of the N nodes, see [Aldous, 1985, Grahamand Robert, 2009]. If the nodes are exchangeable at time t ≥ 0, the queue lengthprocess may be fully recovered via the population process. For any node n

P{Q(N)n (t) = m

}=

∑xN∈EN

P{Q(N)n (t) = m,X

(N)(t) = xN}

=∑

xN∈ENxNmP

{X(N)(t) = xN

}= E[X(N)m (t)], (1.13)

where the second equality is due to exchangeability.e system is chaotic if the state of the nodes is pairwise independent. Note

that even if the initial state is chaotic, the nodes interact with each other, and theself-organization of the network may not let chaos propagate. Specically, forevery n, n′ ∈ N the random variables Q(N)n (t) and Q(N)n′ (t) for t ≥ 0 are notnecessarily independent. e concept of propagation of chaos was introduced


in [Kac, 1959], where it was observed that the correlation between each pairof nodes remains low for longer periods of time as the number of nodes growslarge.

Mean-eld theory has been used to validate propagation of chaos and it hasbeen shown [Delcoigne and Fayolle, 1999, Graham and Méléard, 1994, Méléard,1996, Sznitman, 1991] that in the mean-eld asymptotic regime, for any nite setof nodes, the initial independence is maintained over any nite time horizon. In[Benaim and Le Boudec, 2008], the authors observed that when limt→∞ x(t) =x∗, chaoticity further propagates over an innite horizon.

A related concept is the decoupling assumption. e idea is that, in largesystems, the nodes decouple and become pairwise independent in stationarity.e decoupling assumption facilitates the stationary analysis of the system andhas been used in many notable papers. As an example, Bianchi’s celebratedformula [Bianchi, 2000], which has shaped the performance evaluation literaturefor the 802.11 MAC protocol, relies on the decoupling assumption. Mean-eldanalysis has been used to examine whether Bianchi’s formula is asymptoticallyexact as the number of nodes grows large and has been critical in variousscenarios [Bordenave et al., 2008, Duy, 2010, Cho et al., 2012, Michalopoulouand Mähönen, 2017, Sharma et al., 2009].

Mean-eld approximations have been shown to provide results that are notjust asymptotically exact but that are also extremely accurate for small values ofN . Recently there have been several studies quantifying the error made whenusing mean-eld approximations [Gast, 2017, Gast and Van Houdt, 2017, Ying,2015, Ying, 2017]. ese results are mostly based on Stein’s method [Bravermanand Dai, 2017, Braverman et al., 2017] and show how to compute the asymptoticerror of mean-eld approximation and how to correct them.

1.4.3 Fixed-point analysis

Mean-eld limits reveal important information on the transient behavior of thesystem when the number of nodes is large. e natural next step is to considerthe xed point of the mean-eld limit and leverage it so as to gain insights inthe system behavior in stationarity, and in particular derive approximationsfor the stationary performance metrics. is approach has been named xed-point method and has been extensively used in the literature [Bianchi, 2000,Bortolussi et al., 2013, Delcoigne and Fayolle, 1999, Fricker et al., 2012, Gast andGaujal, 2010, Van Houdt, 2014, Kolesnichenko et al., 2013, Mukhopadhyay et al.,2015, Van Spilbeeck and Van Houdt, 2015]. In the context of random-accessnetworks, this method has led to the identication of asymptotically exactstability conditions in [Bordenave et al., 2008] and to approximations for thethroughput performance in [Sharma et al., 2009].


e xed-point approximations based on the xed pointx∗ of the mean-eldlimit are asymptotically exact when

limN→∞

limt→∞

P{dE(X

(N)(t),x∗) > �}

= 0, ∀ � > 0, (1.14)

where dE is a distance metric for E. e derivation of sucient and necessaryconditions for (1.14) to hold have aracted major interest in recent years, see[Aghajani and Ramanan, 2016, Benaim and Le Boudec, 2011, Gamarnik andZeevi, 2006, Mukhopadhyay et al., 2015, Stolyar, 2015, Tsitsiklis and Xu, 2012].A common approach consists of showing that the stationary distribution ofthe prelimit process exists for every N and that the sequence of these distribu-tions converges to an invariant distribution for the system in the mean-eldregime. When the sequence of prelimit stationary distributions is tight and themean-eld invariant distribution is unique, it may be shown that the aboveargument applies. In [Kang and Ramanan, 2012] it is shown that tightnesswithout uniqueness is not sucient to establish (1.14).

It is oen technically challenging to prove (1.14), and when this cannot beestablished the xed-point approximations remain just heuristics. In Chapter 4we rigorously prove that the xed-point approximations for unsaturated CSMAmodels are asymptotically exact in case of complete interference graphs.

1.4.4 An illustrative example - Complete interference (cont’d)

To illustrate the concepts just introduced, let us now present the xed-pointmethod for the example introduced in Section 1.4.1. Given any node n ∈ N , weaim to obtain asymptotically exact approximations for

limN→∞

P{Q(N)n = m

}, lim

N→∞P{W (N)n > s

}, (1.15)

where Q(N)n and W (N)n are random variables distributed according to the sta-tionary queue length of node n and the stationary time spent in the buer by apacket before starting a transmission at node n, respectively.

For this example, we show that if

ξ :=λ

ν(1− ρ)< 1, (1.16)

then there exists a unique xed point x∗ = (x∗m)m∈N0 ∈ E which is given by

x∗m = (1− ξ)ξm, ∀m ∈ N0, (1.17)

and that x∗ is globally stable, thus having a unique mean-eld invariant distri-bution. It may be further shown that the sequence of stationary distributions of

1.5. Beyond the classical mean-eld scenario 17

the prelimit models is tight, and thus, thanks to the statistical exchangeabilityof the nodes in the system, we obtain the asymptotically exact approximations

Q(N)n ⇒ Geom(ξ),λ

NW (N)n ⇒ Exp

(1− ξξ

). (1.18)

1.5 Beyond the classical mean-eld scenario

e basic mean-eld theory developed for unsaturated CSMA networks can beused as a starting point to explore a broader set of problems which would beotherwise out of reach, as will be briey discussed below.

1.5.1 Optimization of the back-o rates

Consider the illustrative example with symmetric nodes and a full interferencegraph presented in Section 1.4.1. e mean-eld analysis revealed that, althoughthe load of the network remains constant, the stationary packet delay growslinearly in the number of nodes present in the network (1.18). In Chapter 5 wewill examine optimal activation-rate scalings in terms of the stationary packetdelay, and we will show that the linear growth may be signicantly reduced byproperly tuning the back-o rates.

More specically, we aim to address the issue of how to set the back-orate as a function of the network density and trac intensity. In order toavoid collisions, the value of the back-o rate should rst of all account forthe maximum signal propagation delay between interferers, which is mostlygoverned by the physical aributes of the network. However, as networks growincreasingly dense, the number of interferers can grow extremely large as well.us the aggregate back-o rate of the nodes within interference range can becorrespondingly large, which may also give rise to spurious collisions.

e above is countered by lowering the value of the back-o rate in densenetworks and for example seing it inversely proportional to the number ofnodes as in the illustrative example in Section 1.4.1 and in the mean-eldanalysis in Chapter 3. However, this implicitly relies on the assumption thatevery node always has packets to transmit which is pessimistic when nodesare only sporadically active. In this scenario, seing the back-o rate inverselyproportional to the number of nodes results in unnecessarily long delays whichare avoidable when the collective load is not particularly high.

In Chapter 5 we will look at the scenario where the mean nominal back-orate at each node is scaled by a factor f(N) as function of the total number ofnodes, instead of 1/N as in the illustrative example. In such a case, both theexpected stationary delay and the number of nodes with backlogged packets


scale as 1/f(N) as N → ∞. Hence, faster back-o rates may substantiallyimprove the delay performance.

1.5.2 Mean-eld analysis of multi-hop CSMA networks

In Section 1.3.1 we introduced the multi-hop CSMA model and observed thatvery few results are available in the literature. It is quite immediate to extend themean-eld theory developed for single-hop networks to the more complicatedmulti-hop scenario, and thus obtain insights in the network performance viathe xed-point method.

Figure 1.3: Device-to-device multi-hop wireless networks.

Consider a Device-to-device application as the one displayed in Figure 1.3.Packets need to be sent from the transmier to the target receiver and aresequentially transmied by intermediate nodes in a multi-hop fashion. enodes are naturally partitioned in various sets of similar nodes and a packettransmied by a node in the c-th subset joins the buer of a node in the (c+ 1)-th subset chosen uniformly at random. Aer C hops (three in the example inFigure 1.3) the packets eventually leave the system and are delivered to thetarget device.

In Chapter 6, we will consider the system in a mean-eld asymptoticregime where the number of nodes in each subset grows large, and derive aC-dimensional deterministic IVP with solution x(t). We will obtain conditionsfor the existence and uniqueness of a xed point x∗ and leverage it to obtainxed-point approximations for key performance metrics such as end-to-endthroughput and stability conditions.

As an example, we will approximate the end-to-end stationary throughputfor linear networks such as that in Figure 1.3 with uniform back-o rates. Weobserve that as the arrival rate grows, the end-to-end throughput increases upto when the second group of nodes is not able to sustain the load and acts asa boleneck. As the arrival rate further increases, the end-to-end throughput

1.5. Beyond the classical mean-eld scenario 19

decreases and eventually stabilizes when the rst group of nodes saturatesas well, entering the over-saturated regime. is prole of the end-to-endthroughput has been observed in experimental studies [Aziz et al., 2013], but tothe best of our knowledge was never captured by a theoretical model.

1.5.3 Mean-eld limits in continuous space

A crucial requirement for the classical mean-eld framework to apply is thatthe population of nodes can be partitioned into a nite number of classes ofstatistically indistinguishable nodes. e laer condition is a severe restrictionsince nodes typically have dierent locations, and hence are subject to dierentinterference constraints.

To tackle more realistic scenarios, we will consider in Chapter 7 nodeslocated within a given space and let their number grow large while keeping theinterference range xed. us the network becomes dense, in the sense that thenumber of interferers per node grows large, but each node has its own location,and we do not require any two nodes to be similar.

A natural approach to deal with a continuous space is to group nodeswith nearby locations and construct a set of C geographical classes (clusters)of quasi-identical nodes and apply the classical mean-eld theory so as toobtain a C-dimensional deterministic IVP [Bordenave et al., 2008, Chaintreauet al., 2009, Tschaikowski and Tribastone, 2017]. However, the aggregationprocedure involves a further level of approximation, and while the accuracymay be expected to improve when a ner spatial granularity is considered,the resulting increase in the number of clusters adds to the computationalcomplexity of obtaining the mean-eld solution. Intuitively, as C increases, theapproximation of the continuous space gets more accurate, but that comes atthe cost of an increasing dimensionality of the problem, and eventually makesthe problem intractable.

In our approach, we will use the mean-eld theory developed in Chapter 3as a stepping stone. Specically, we will show that as both N and C grow largethe unsaturated CSMA model in a continuous space is close to the aggregatecluster-based model. e mean-eld limit of the laer model can immediately bederived via the results in Chapter 3. Finally, we will show that as C grows large,the solution of the cluster-based mean-eld IVP converges to the solution of anIVP in continuous space which does not depend on the articial construction ofthe cluster model. In this way, we deduce a mean-eld limit for the unsaturatedCSMA model in a continuous space which does not involve any approximationdue to the space discretization.

We point out that the continuous-space mean-eld limit depends only onthe distribution of the nodes’ locations and their initial conguration. Similarly


to the classical mean-eld, the mean-eld version of the unsaturated model maybe interpreted as a saturated model where nodes may activate everywhere andwith reduced back-o rates due to the presence of nodes with empty buers.

1.6 Overview of the thesis

In Chapter 2 we present a preliminary analysis of random-access networks inunsaturated conditions. We establish that a complete solution is out of reach dueto the intrinsic complexity of the system in general interference congurations.e queueing dynamics and the activity process are strongly intertwined and donot admit closed-form stationary distributions. We obtain stability conditionsfor specic nodes in case the interference graph exhibits a locally completestructure, and explain why explicit conditions cannot be derived in the absenceof these structures. e chapter is based on [Cecchi et al., 2014].

To gain analytical tractability, we focus in Chapter 3 on the mean-eldasymptotic regime where the number of nodes grows large. e nodes arepartitioned in classes and within each class the nodes are statistically indistin-guishable. We prove that the population process weakly converges pathwise tothe unique solution of a deterministic initial-value problem (IVP). e mean-eld IVP is shown to have a unique xed point which brings out a connectionwith saturated models. e chapter expands the work in [Cecchi et al., 2015]and in the rst part of [Cecchi et al., 2016b].

We then describe in Chapter 4 how to derive approximations for the station-ary measures of the system with a nite number of nodes. e approximationsproposed are based on the xed point of the mean-eld IVP and were rstpresented in [Cecchi et al., 2016a]. We demonstrate that the approximations areasymptotically exact when the interference graph is complete. e argument isbased on the second part of [Cecchi et al., 2016b].

In Chapter 5, we investigate the impact of dierent back-o scalings on theperformance of large-scale networks. Via a novel multi-scale mean-eld limitapproach, we show that the packet delay scales inversely proportional to theback-o rate as the number of nodes grows large and that the vast majority ofnodes are empty for most of the time. More precisely, we prove a central limittheorem for the stationary number of backlogged packets, further rening themean-eld results. e chapter extends [Cecchi et al., 2018].

In Chapter 6 the mean-eld analysis is extended to accommodate morecomplicated multi-hop networks. We build on the results presented in [Cecchiet al., 2017c]. Approximations for key performance metrics of the system suchas stability conditions and end-to-end throughput are obtained via the xed-

1.6. Overview of the thesis 21

point method. e approximations derived are leveraged to describe a heuristicmethod for eciently choosing the back-o rates.

In Chapter 7, we consider ultra-dense networks in a continuous space andwe derive a novel mean-eld limit. In this scenario we allow each node to haveits own location and hence no pair of nodes necessarily have the same subsetof interferers. We show how the spatial distribution of the nodes impacts theperformance of the network and derive approximations for the performancemetrics, empirically showing their accuracy even when the number of nodesis moderate. is chapter is based on the results obtained in [Cecchi et al.,2017a, Cecchi et al., 2017b].

Chapter 2

Throughput and StabilityAnalysis

Contents

2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Structural characterization of the stability region . . . . . . . . . . . . . . 292.4 Boundaries of the stability region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 ree-node network illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.A Extended proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

In this chapter we consider the mathematical model for unsaturated CSMAnetworks introduced in Section 1.3. We provide a generic structural represen-tation of the throughput performance and corresponding stability region interms of the individual saturation throughputs of the various nodes. Whilein general the saturation throughputs are dicult to determine explicitly, weidentify certain cases where these values can be expressed in closed form. In thespecial case of a complete interference graph, this recovers the explicit stabilityconditions previously obtained in [Laufer and Kleinrock, 2013, Van de Ven et al.,2010].

For arbitrary interference graphs, we prove that various lower-dimensionalfacets of the stability region can be explicitly described, depending on the

23

24 Chapter 2. roughput and Stability Analysis

neighborhood structure of the graph. In particular, we show that all the ‘edges’(one-dimensional facets) of the stability region can either be expressed in closedform or numerically computed using matrix-analytic techniques. In passing,we reveal a aw in the throughput characterization presented in [Laufer andKleinrock, 2013] for noncomplete interference graphs.

e analysis in this chapter reveals the complex nature of this model. We dis-cuss what is analytically tractable and what is not, thus providing the rationalefor the asymptotic analysis we will perform in the next chapters to approximatekey performance metrics.

Chapter outline. e remainder of the chapter is organized as follows. InSections 2.1 and 2.2 we present a detailed model description and introducea few useful denitions and preliminary results. We provide the structuralrepresentation of throughput and stability region in Section 2.3. In Section 2.4we demonstrate that the boundaries as well as lower-dimensional facets ofthe stability region can be explicitly described in certain cases, dependingon the neighborhood structure of the interference graph. Section 2.5 focuseson a specic three-node network to illustrate the generic characterization ofthe stability region, accompanied by some numerical results in Section 2.6. InSection 2.7 we make concluding remarks and list some possible topics for furtherresearch.

2.1 Model description

We consider a network of N nodes as introduced in Section 1.3, where twonodes n, n′ ∈ N interfere with each other if and only if {n, n′} ∈ E , where(N , E) is the interference graph of the network.

If a node is in back-o, it may become blocked when any of its neighborsactivates. We distinguish two scenarios, depending on whether a back-o periodis started immediately aer a transmission that leaves the queue of a node empty,or only when the next packet arrives. In the former case, the queue may still beempty when the back-o period ends, and we assume that the node then startsthe next back-o period. For the purpose of the throughput analysis, it will beuseful to also allow for a node to behave in a greedy manner, and activate andtransmit a dummy packet when a back-o period ends and the queue is empty.We denote by G ⊆ N the subset of greedy nodes.

Since the back-o durations are exponentially distributed, it does not mat-ter whether a back-o period is started immediately aer a transmission thatleaves the queue of a node empty, or only when the next packet arrives. Equiva-lently, we could think of the potential activation epochs of a node as occurring

2.1. Model description 25

according to a Poisson process, and actual transmission periods starting when-ever a potential activation event occurs while the node is unblocked and has anonempty queue.

For every node n ∈ N , we dene Nn as in (1.1), the set of neighbors ofnode n, and denoteMn = N \ (Nn ∪{n}). A node is said to be blocked when-ever the node itself or any of its neighbors is active, and unblocked otherwise.In particular, when the activity state is ω ∈ Ω, the unblocked and the activenodes are given by

Uω :=⋂

n:ωn=1

Mn, Aω := {n : ωn = 1}.

e process is described by the joint Markov process {Q(t), A(t)}t≥0 as in-troduced in Section 1.3. Denote byπ(q, ω) = limt→∞ P {(Q(t), A(t)) = (q, ω)}the stationary probability that the joint activity and queue length state is(q, ω) ∈ NN0 × Ω, assuming it exists. If the stationary probabilities π(q, ω)exist, then they must satisfy the global balance equations

πG(q, ω)(∑n∈N

λn +∑

n∈Gc∩Uωqn>0

νn +∑

n∈G∩Uω

νn +∑n∈Aω

µn

)

=∑n∈Nqn>0

λnπG(q − en, ω) +

∑n∈Uω

µnπG(q, ω + en)

+∑n∈Aω

νnπG(q + en, ω − en) +

∑n∈G∩Aωqn=0

νnπG(q, ω − en), (2.1)

where Gc = N \ G denotes the subset of nodes that are not greedy. When noneof the nodes are greedy, the global balance equations slightly simplify to

π(q, ω)(∑n∈N

λn +∑n∈Uωqn>0

νn +∑n∈Aω

µn

)=∑n∈Nqn>0

λnπ(q − en, ω)

+∑n∈Uω

µnπ(q, ω + en) +∑n∈Aω

νnπ(q + en, ω − en),

with π(q, ω) = π∅(q, ω) for compactness.With minor abuse of notation, denote by πG(ω) = limt→∞ P {A(t) = ω}

the stationary probability that the activity state is ω ∈ Ω. Note that πG(ω) =∑q∈NN0

πG(q, ω) when the stationary probabilities πG(q, ω) exist, but thatπG(ω) may exist even when the laer is not the case, for instance when the


queues of some of the nodes are not ergodic. In the remainder of the chapterwe will always assume that the stationary probabilities πG(ω) exist, but donot suppose that the stationary probabilities πG(q, ω) exist, i.e., we will onlyassume ergodicity of the activity process.

When all the nodes behave in a greedy manner, i.e., G = N , the activityprocess is not aected by the queue length process at all. As we observed inSection 1.2, the activity process is reversible and that stationary distribution isof product form

πN (ω) =1

Z

N∏n=1

σωnn , ω ∈ Ω, (2.2)

where we have that πN (ω) = π(ω;σ) and Z = Z(σ) as dened in (1.3) and(1.4), and to ease the notation we omit the dependence on σ in the rest of thechapter.

2.2 Preliminary results

In the remainder of the chapter we will be mainly interested in analyzingthe throughput performance of the various nodes. We henceforth assume theactivity process to be ergodic so that the long-term throughput of node n maybe expressed in terms of the stationary distribution of the activity process as in(1.5), i.e.,

θn = µn∑

ω∈Ω+n

π(ω) (2.3)

with π(ω) = π∅(ω). Note that θn ≤ λn (unless node n is greedy), and we willbe particularly interested in obtaining conditions under which equality holds,implying that node n is rate stable. For any activity state ω ∈ Ω and n ∈ Gc,dene

π̃Gn(ω) = limt→∞

P {Qn(t) > 0, A(t) = ω} =∑q∈NN0

πG(q + en, ω),

assuming the relevant stationary probabilities to exist. For any activity stateω ∈ Ω, summing the global balance equations (2.1) over all the possible queuelengths q ∈ NN0 at the various nodes, we obtain the following set of aggregate

2.2. Preliminary results 27

balance equations:

πG(ω)( ∑n∈Aω

µn +∑

n∈G∩Uω

νn

)+

∑n∈Gc∩Uω

π̃Gn(ω)νn (2.4)

=∑n∈Uω

µnπG(ω + en) +

∑n∈G∩Aω

νnπG(ω − en) +

∑n∈Gc∩Aω

νnπ̃Gn(ω − en).

When all the nodes behave in a greedy manner, i.e., G = N , the above balanceequations simplify to

πN (ω)( ∑n∈Aω

µn +∑n∈Uω

νn

)=∑n∈Uω

µnπN (ω + en) +

∑n∈Aω

νnπN (ω − en). (2.5)

In this case, the stationary probabilities have the convenient product-form asstated in (2.2).

Motivated by this observation, we now introduce some useful coecientsin order to rewrite the aggregate balance equations (2.4) in a more compactway that resembles (2.5) more closely. For any non-greedy node n ∈ Gc andactivity state ω ∈ Ω, dene κn(ω) as the probability that the queue at node n isnonempty while the activity state is ω. us for any non-greedy node n ∈ Gcand activity state ω ∈ Ω,

κn(ω) :=limt→∞ P {Qn(t) > 0, A(t) = ω}

limt→∞ P {A(t) = ω}=π̃Gn(ω)

πG(ω), (2.6)

or equivalently, π̃Gn(ω) = κn(ω)πG(ω). In contrast, dene κn(ω) = 1 for everygreedy node n ∈ G and activity state ω ∈ Ω. While calculating the coecientsκn(ω) for a non-greedy node is not any simpler in general than determiningthe complete set of stationary probabilities πG(q, ω), they may be adopted torewrite the aggregate balance equations (2.4) as

πG(ω)( ∑n∈Aω

µn +∑n∈Uω

νnκn(ω))

=∑n∈Uω

µnπG(ω + en) +

∑n∈Aω

νnκn(ω − en)πG(ω − en). (2.7)

Comparing (2.5) and (2.7), we observe that the stationary distribution of theactivity process corresponds to that in a ctitious scenario where all the nodesbehave in a greedy manner and the back-o rates are state-dependent as captured


by the coecients κn(ω). While the state dependence destroys the product-form solution, and calculating the coecients κn(ω) remains dicult too ingeneral, the representation (2.7) will play a valuable role in the throughputanalysis.

Having introduced the coecients κn(ω), the long-term throughput ofnode n may equivalently be expressed as

θn = νn∑

ω∈Ω−n

κn(ω)π(ω), (2.8)

where Ω−n was dened in (1.7). It can be veried that the expressions (2.3)and (2.8) agree by summing the aggregate balance equations (2.7), as shown inAppendix 2.A.1.

As mentioned earlier, it must be the case that θn = λn in order for node nto be rate stable. In conjunction with the expressions for θn in (2.3) and (2.8),this yields the following two stability identities:∑

ω∈Ω+n

π(ω) =λnµn,

∑ω∈Ω−n

κn(ω)π(ω) =λnνn. (2.9)

If it were the case that the coecients κn(ω) do not depend on the activitystate ω, i.e., κn(ω) = κn for all ω with ω + en ∈ Ω, then the structure of (2.7)coincides with that of (2.5), and hence the corresponding stationary distributionresembles the product-form distribution in (2.2):

πκ(ω) =1

Zκ

N∏n=1

(κnσn)ωn , ω ∈ Ω, (2.10)

i.e., it holds that

πκ(ω) = π(ω;σ · κ), Zκ = Z(σ · κ),

where · denotes the componentwise product. If in addition all the nodes arerate stable, so that equations (2.9) hold, it can be shown that these determinea unique solution for the coecients κn. e fact that the laer coecientscannot exceed unity, is then used in [Laufer and Kleinrock, 2013] as a basis forderiving stability conditions.

However, the coecients κn(ω) in general do depend on the activity state ω,and hence the stability conditions in [Laufer and Kleinrock, 2013] are not validin general. An exception arises in the special case when the interference graphis complete, so that for each node n only the coecient κn = κn(0) appears in

2.3. Structural characterization of the stability region 29

(2.9). In that case, the product-form stationary distribution simplies to

π(0) =1

1 +∑Nn′=1 κn′σn′

, π(en) = κnσnπ(0), n ∈ N ,

and the rate stability identities reduce to

π(en) =λnµn, π(0) =

λnκnνn

, n ∈ N .

e constraint κn < 1 for all n ∈ N is then equivalent to

maxn∈N

λnνn

< 1−∑n′∈N

λn′

µn′, (2.11)

which agrees with the stability conditions provided in [Laufer and Kleinrock,2013] and previously obtained in [Van de Ven et al., 2010].

In certain cases it may be argued that κn(ω1) = κn′(ω2) for n 6= n′ orω1 6= ω2, so that the total number of distinct κn(ω) values is N or less. Inthat case, the stationary distribution will not necessarily have a product-formdistribution, but the equations (2.9) may still determine a unique solution forthe unknown coecients κn(ω), and provide a basis for deriving stabilityconditions.

2.3 Structural characterization of the stability region

In this section we provide a generic characterization of the long-term growthrates of the queues at the various nodes for arbitrary interference graphs. isyields an indication of the throughput performance of the various nodes and inparticular a representation of the stability region.

For any vector λ ∈ RN≥ and G ⊂ N , denote λ−G := (λn)n 6∈G ∈ RN−|G|≥ .

Dene θGn(λ−G) as the long-term throughput received by node n in a (ctitious)scenario where the nodes m ∈ G act in a greedy manner and the arrival ratesof the other nodes m′ 6∈ G are λm′ . In particular, with minor abuse of notation,dene the saturation throughput of node n as θ∗n(λ−n) := θ

{n}n (λ−{n}).

Like before, we assume that the activity process is ergodic, so that the long-term throughput values θGn(λ−G) may be expressed in terms of the stationarydistribution of the activity process as

θGn(λ−G) = µn∑

ω∈Ω+n

πG(ω;λ−G), (2.12)


where we denote πG(ω;λ−G) = πG(ω) with minor abuse of notation to explic-itly reect the dependence on λ−G . As before, the long-term throughput valuesθGn(λ−G) may equivalently be expressed as

θGn(λ−G) = νn∑

ω∈Ω−n

πG(ω;λ−G). (2.13)

It can be veried that the expressions (2.12) and (2.13) agree by summing theaggregate balance equations (2.7) as shown in Appendix 2.A.1.

In order to state the main result, we introduce the following sets for everyI ⊂ N in terms of the long-term throughput values θGn(λ−G) dened above,

ΛI = {λ ∈ RN≥ : λn ≤ θIn(λ−I) for all n ∈ I},Λ̄I = {λ ∈ RN≥ : λn = θIn(λ−I) for all n ∈ I},

andΛ =

⋂n∈N

Λn.

e next theorem establishes the relation between these sets and the functiongn(λn) = λn − θn, which denotes the asymptotic growth rate of the queue atnode n when its arrival rate is λn. In the notation of gn(λn) we suppress theimplicit dependence on the arrival rates λ−n at the other nodes.

eorem 2.1. For any arrival rate vector λ ∈ RN≥ , the long-term growth rategn(λn) of the queue at node n equals max{λn−θ∗n(λ−n), 0}, so that the through-put θn of node n equals min{λn, θ∗n(λ−n)}. us, if

λ ∈( ⋂n∈I

Λn

)∩( ⋂n′∈Ic

Λcn′), I ⊆ N ,

then the queues at the various nodes in I will be rate stable, while the queuesat the nodes in n′ ∈ Ic grow at a linear long-term rate λn′ − θ∗n′(λ−n′) > 0.Specically, for any arrival rate vector λ ∈ Λ, the queues of all the nodes are ratestable.

Proof. We focus on a specic node n, and x λ−n, the arrival rates at all theother nodes. Observe that strict positivity of the long-term growth rate of thequeue at node n, i.e., gn(λn) > 0, would mean that aer some nite time Tthe queue of node n will never empty again. is fact, in turn, implies that thelong-term throughput of node n must be the same as when node n acts in agreedy manner, so that

gn(λn) = λn − θn = λn − θ∗n(λ−n). (2.14)

2.3. Structural characterization of the stability region 31

At this point, we need to distinguish the cases λn ≤ θ∗n(λ−n) and λn >θ∗n(λ−n), and only need to show that the long-term growth rate is zero andstrictly positive in these two cases, respectively.

We rst consider the case λn ≤ θ∗n(λ−n). Assume that gn(λn) > 0 andobserve that this yields an immediate contradiction since, due to (2.14), thisimplies gn(λn) ≤ 0. erefore if λn ≤ θ∗n(λ−n), then gn(λn) = 0.

We now turn to the case λn > θ∗n(λ−n). e idea of the proof may beinformally described as follows. Observe that when the queue of node n isnonempty at some point, it will behave as if it were greedy for as long as thequeue remains nonempty. e probability that the transmissions occur at a rateless than θ∗n(λ−n) + �B during that period will be quite close to 1 when �B > 0and the initial queue length at node n is large enough. At the same time, theprobability that packets arrive at rate λn − �C or larger during that period willbe quite close to 1 as well when �C > 0. us, when �B + �C < θ∗n(λ−n)− λn,the probability that the queue of node n will never empty again (and in factgrow at a rate close to λn − θ∗n(λ−n)) will be close to 1 when the initial queuelength at node n is large enough.

We now proceed to provide a rigorous proof based on the above idea. Due toequation (2.14), it is sucient to prove that for every arrival rate λn > θ∗n(λ−n)the queue at node nwill never empty aer a certain nite time. In order to showthat, we dene the quantities Bn(t) and B∗n(t) as the cumulative number ofpacket transmissions at node n during the time interval [0, t] in the case whennode n behaves in a nongreedy and greedy manner, respectively. Dene alsothe quantities Cn(t) as the number of arrivals at node n during [0, t] and Qn(t)as the number of waiting packets at node n at time t. Note that, wheneverlimt→∞

B∗n(t)t exists, θ

∗n(λ−n) equals the value of this limit, so that for every

�B , δB > 0 there exists MB > 0 such that

P {B∗n(t) ≤ (θ∗n(λ−n) + �B)t+MB ,∀ t ≥ 0} > 1− δB . (2.15)

With exactly the same argument applied to the quantity Cn(t), we obtain thatfor every �C , δC > 0 there exists MC > 0 such that

P {Cn(t) ≥ (λn − �C)t−MC ,∀ t ≥ 0} > 1− δC . (2.16)

We now distinguish two cases. We rst consider the case Qn(0) ≥MB +MC . enQn(t) ≥ (λn−θ∗n(λ−n)−�B−�C)t for every t ≥ 0 with probabilityno less than (1− δB)(1− δC). In order to see that, rst observe that the event{Qn(t) ≥ (λn− θ∗n(λ−n)− �B− �C)t, ∀ t ≥ 0} is implied by the simultaneousrealization of the following events

{Qn(0) ≥MB +MC}, (2.17)


{Cn(t) ≥ (λn − �C)t−MC ,∀ t ≥ 0}, (2.18)

and{B∗n(t) ≤ (θ∗n(λ−n) + �B)t+MB ,∀ t ≥ 0}. (2.19)

Indeed, if we denote

t1 = inf{t ≥ 0 : Qn(t) ≤ (λn − θ∗n(λ−n)− �B − �C)t},

then, by denition, Qn(t) ≥ 1 for every t ≤ t1, which in turn means thatB∗n(t) = Bn(t) for every t ≤ t1, and thus

Qn(t) = Qn(0) + Cn(t)−Bn(t) = Qn(0) + Cn(t)−B∗n(t),

for all t ≤ t1. e independence of events (2.17)-(2.19) and the inequalities (2.15)and (2.16) then imply that t1 =∞with probability no less than (1−δB)(1−δC)as stated.

We now turn to the case Qn(0) < MB + MC . For every level M =MB +MC and parameter δ0 > 0, there exists a time Tδ0 1− δ0,

and we denote t2 = argmaxt∈[0,Tδ0 ]Qn(t). At this point we deduce that

Qn(t− t2) ≥ (λn − θ∗n(λ−n)− �B − �C)(t− t2), for every t ≥ t2

with probability larger than 1− δB − δC − δ0, independently of Qn(0). isimplies that with the same probability, we have that

lim inft→∞

Qn(t)

t≥ λn − θ∗n(λ−n)− �B − �C .

In particular, since suitableM,Tδ0 0and �B , �C > 0, it follows that

lim inft→∞

Qn(t)

t≥ λn − θ∗n(λ−n), a.s.

is shows that the long-term growth rate of the queue at node n is strictlypositive, and concludes the proof.

Inspection of the proof shows that eorem 2.1 does not rely on any specicproperty of the CSMA mechanism, and holds for any multi-queue system wherea node acts in a greedy manner as long as its queue is nonempty. e laer

2.4. Boundaries of the stability region 33

property is somewhat similar to condition P3 in Szpankowski [Szpankowski,1988, Szpankowski, 1994]. However, the proof of eorem 2.1 does not requirecondition P1 in [Szpankowski, 1988, Szpankowski, 1994], which entails a certainmonotonicity property that when a particular node acts in a greedy manner,all other nodes are worse o, which in fact may not hold for a CSMA networkwith an arbitrary interference graph. e statement of eorem 2.1 neverthelessapplies even when such a monotonicity property does hold, and then providesan alternative characterization of the stability region compared to [Szpankowski,1988, Szpankowski, 1994].

2.4 Boundaries of the stability region

In the previous section we have characterized the throughput performance andstability region of the system in terms of the saturation throughputs of thevarious nodes. Expressions for the saturation throughput θ∗n(λ−n) of node nare provided by either equation (2.12) or (2.13). However, the probabilitiesπn(ω;λ−n) in these expressions rely on the solution of the aggregate balanceequations (2.7), where G = {n}, i.e., node n is assumed to be greedy. ecoecients κn′(ω), n′ 6= n, that occur in these equations in turn involve thecomputation of the stationary distribution of the Markov process governed bythe global balance equations (2.1). Since the laer Markov process possessesan eective innite state space in possibly N − 1 dimensions, the stationarydistribution is in general not explicitly tractable for N ≥ 3. By eective statespace we mean the collection of states that inuence the stationary distributionof the joint activity process, which excludes the number of waiting packets atgreedy nodes.

In some cases, however, it is possible to derive closed-form expressions forthe relevant portions of the boundary Λ̄n, and circumvent the solution of theglobal balance equations (2.1). Consider for instance the scenario where all thenodes n′ ∈ Nn mutually interfere, i.e., form a clique in the interference graph.In that case, no two nodes in {n}∪Nn can be active simultaneously, i.e., ωn′ = 1for at most one n′ ∈ {n} ∪ Nn for any ω ∈ Ω. Exploiting this property, thesaturation throughput θ∗n(λ−n) can be determined explicitly by assuming all thenodes n′ ∈ Nn to be rate stable, which is the case for λ ∈ Λ̄n ∩ (∩n′∈NnΛn′),the relevant portion of the stability boundary Λ̄n. Indeed, when G = {n}, it


may be deduced from equations (2.12) and (2.13) that

1 =∑ω∈Ω

πn(ω;λ−n)

=∑

ω∈Ω+n

πn(ω;λ−n) +∑

ω∈Ω−n

πn(ω;λ−n) +∑ω∈Ω

∃ n′∈Nn:ωn′=1

πn(ω;λ−n)

=θ∗n(λ−n)

µn+θ∗n(λ−n)

νn+∑n′∈Nn

∑ω∈Ω+n′

πn(ω;λ−n)

=θ∗n(λ−n)

µn+θ∗n(λ−n)

νn+∑n′∈Nn

λn′

µn′,

where the last equality is due to the rate stability of the nodes n′ ∈ Nn andequation (2.9). us, we obtain that

θ∗n(λ−n) = (1−∑n′∈Nn

λn′

µn′)µnνnµn + νn

, (2.20)

so that Λ̄n ∩ (∩n′∈NnΛn′) can be explicitly identied. is equation maybe interpreted by recalling that node n can only be active when none of itsneighbors n′ ∈ Nn is active, which is the case a fraction of the time 1 −∑n′∈Nn

λn′µn′

when the laer nodes are rate stable. During these periods, node nwill be in back-o a fraction of time

1/νn1/µn + 1/νn

=µn

µn + νn,

and transmiing packets at rate µn the remaining fraction of time

1/µn1/µn + 1/νn

=νn

µn + νn.

In contrast, when the neighbors of node n do not form a clique in the inter-ference graph, the computation of the saturation throughput θ∗n(λ−n) does notseem tractable in general. In that case there are two nodes n1, n2 ∈ Nn thatdo not mutually interfere. e subgraph induced by the nodes n, n1, n2 thenresembles the three-node network depicted in Figure 2.1. Even in the absenceof any further nodes, the calculation of the saturation throughput of the centralnode in such a three-node network does

Mean-field limits for ultra-dense random-access networks · Mean-Field Limits for Ultra-Dense...

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