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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.
Document status and date:Published: 01/02/2018
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Mean-Field Limitsfor Ultra-Dense
Random-Access Networks
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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
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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
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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.
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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.
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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
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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
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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
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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
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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.
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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
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4 Chapter 1. Introduction
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.
-
6 Chapter 1. Introduction
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}
}.
-
8 Chapter 1. Introduction
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
-
10 Chapter 1. Introduction
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,
-
12 Chapter 1. Introduction
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).
-
1.4. Mean-eld asymptotic regime 13
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
-
14 Chapter 1. Introduction
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
-
1.4. Mean-eld asymptotic regime 15
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].
-
16 Chapter 1. Introduction
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
-
18 Chapter 1. Introduction
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
-
20 Chapter 1. Introduction
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
-
26 Chapter 2. roughput and Stability Analysis
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
-
28 Chapter 2. roughput and Stability Analysis
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)
-
30 Chapter 2. roughput and Stability Analysis
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)
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32 Chapter 2. roughput and Stability Analysis
{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
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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
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34 Chapter 2. roughput and Stability Analysis
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