In this paper we consider a stochastic model for a frequency-agile CSMA
protocol for wireless networks where multiple orthogonal frequency channels are
available. Even when the possible interference on the different channels is
described by different conflict graphs, we show that the network dynamics can
be equivalently described as that of a single-channel CSMA algorithm on an
appropriate virtual network. Our focus is on the asymptotic regime in which the
network nodes try to activate aggressively in order to achieve maximum
throughput. Of particular interest is the scenario where the number of
available channels is not sufficient for all nodes of the network to be
simultaneously active and the well-studied temporal starvation issues of the
single-channel CSMA dynamics persist. For most networks we expect that a larger
number of available channels should alleviate these temporal starvation issues.
However, we prove that the aggregate throughput is a non-increasing function of
the number of available channels. To investigate this trade-off that emerges
between aggregate throughput and temporal starvation phenomena, we propose an
analytical framework to study the transient dynamics of multi-channel CSMA
networks by means of first hitting times. Our analysis further reveals that the
mixing time of the activity process does not always correctly characterize the
temporal starvation in the multi-channel scenario and often leads to
pessimistic performance estimates.Comment: 15 pages, 4 figures. Accepted for publication at IFIP Performance
  Conference 201

Zocca, Alessandro

English

arXiv

In this paper, we consider a stochastic model for a frequency-agile CSMA protocol for wireless networks where multiple orthogonal frequency channels are available. Even when the possible interference on the different channels is described by different conflict graphs, we show that the network dynamics can be equivalently described as that of a single-channel CSMA algorithm on an appropriate virtual network. Our focus is on the asymptotic regime in which the network nodes try to activate aggressively in order to achieve maximum throughput. Of particular interest is the scenario where the number of available channels is not sufficient for all nodes of the network to be simultaneously active and the well-studied temporal starvation issues of the single-channel CSMA dynamics persist. For most networks, we expect that a larger number of available channels should alleviate these temporal starvation issues. However, we prove that the aggregate throughput is a non-increasing function of the number of available channels. To investigate this trade-off that emerges between aggregate throughput and temporal starvation phenomena, we propose an analytic framework to study the transient dynamics of multi-channel CSMA networks by means of first hitting times. Our analysis further reveals that the mixing time of the activity process does not always correctly characterize the temporal starvation in the multi-channel scenario and often leads to pessimistic performance estimates

Caltech Authors - Main

Temporal starvation in multi-channel CSMA networks: an analytical framework

In this paper we consider a stochastic model for a frequency-agile CSMA protocol for wireless networks where multiple orthogonal frequency channels are available. Even when the possible interference on the different channels is described by different conflict graphs, we show that the network dynamics can be equivalently described as that of a single-channel CSMA algorithm on an appropriate virtual network. Our focus is on the asymptotic regime in which the network nodes try to activate aggressively in order to achieve maximum throughput. Of particular interest is the scenario where the number of available channels is not sufficient for all nodes of the network to be simultaneously active and the well-studied temporal starvation issues of the single-channel CSMA dynamics persist. For most networks we expect that a larger number of available channels should alleviate these temporal starvation issues. However, we prove that the aggregate throughput is a non-increasing function of the number of available channels. To investigate this trade-off that emerges between aggregate throughput and temporal starvation phenomena, we propose an analytical framework to study the transient dynamics of multi-channel CSMA networks by means of first hitting times. Our analysis further reveals that the mixing time of the activity process does not always correctly characterize the temporal starvation in the multi-channel scenario and often leads to pessimistic performance estimates.</p

Zocca, A.

VU Research Portal

VU Research PortalTemporal starvation in multi-channel CSMA networksZocca, A.published inPerformance Evaluation Review2019DOI (link to publisher)10.1145/3308897.3308923document versionPublisher's PDF, also known as Version of recorddocument licenseArticle 25fa Dutch Copyright ActLink to publication in VU Research Portalcitation for published version (APA)Zocca, A. (2019). Temporal starvation in multi-channel CSMA networks: An analytical framework. PerformanceEvaluation Review, 46(3), 52-53. https://doi.org/10.1145/3308897.3308923General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 27. May. 2021Temporal starvation in multi-channel CSMA networks:an analytical frameworkAlessandro ZoccaDepartment of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadena, Californiaazocca@caltech.eduABSTRACTIn this paper we consider a stochastic model for a frequency-agile CSMA protocol for wireless networks where multipleorthogonal frequency channels are available. Even when thepossible interference on the different channels is described bydifferent conflict graphs, we show that the network dynam-ics can be equivalently described as that of a single-channelCSMA algorithm on an appropriate virtual network. Our fo-cus is on the asymptotic regime in which the network nodestry to activate aggressively in order to achieve maximumthroughput. Of particular interest is the scenario where thenumber of available channels is not sufficient for all nodes ofthe network to be simultaneously active and the well-studiedtemporal starvation issues of the single-channel CSMA dy-namics persist. For most networks we expect that a largernumber of available channels should alleviate these tempo-ral starvation issues. However, we prove that the aggre-gate throughput is a non-increasing function of the num-ber of available channels. To investigate this trade-off thatemerges between aggregate throughput and temporal star-vation phenomena, we propose an analytical framework tostudy the transient dynamics of multi-channel CSMA net-works by means of first hitting times. Our analysis furtherreveals that the mixing time of the activity process does notalways correctly characterize the temporal starvation in themulti-channel scenario and often leads to pessimistic perfor-mance estimates.Categories and Subject DescriptorsG.3 [Mathematics of Computing]: Probability and Statis-tics—Stochastic Networks; G.2.2 [Mathematics of Com-puting]: Discrete Mathematics—Graph TheoryKeywordsCSMA algorithm, throughput analysis, Markov chain, hit-ting times, mixing times1. INTRODUCTIONThe Carrier-Sense Multiple-Access (CSMA) algorithm is apopular distributed medium-access control mechanism andindeed various incarnations of which are currently imple-mented in IEEE 802.11 WiFi networks. CSMA is a random-access algorithm, in the sense that it relies on random-IFIP WG 7.3 Performance 2018 Toulouse, FranceCopyright is held by author/owner(s).ness to both avoid simultaneous transmissions and share themedium in the most efficient way. Being intrisincally a ran-domized scheme, many stochastic models have been devel-oped in the literature to study its performance in terms ofthroughput, stability, delay, and spatial fairness.It is well-known that the delay performance of CSMA al-gorithms can be rather poor and much worse than for othermechanisms, such as MaxWeight. One of the root causesfor these poor delay performances has been identified in thetemporal starvation phenomenon: even in scenarios whereall nodes have a good long-term throughput, they may indi-vidually experience long sequences of transmissions in rapidsuccession, interspersed with extended periods of starvation.These starvation effects are particularly pronounced whenthe nodes become more aggressive in trying to activate,which is the regime in which the network should operateto achieve maximum throughput.Most of the literature focuses on single-channel CSMA al-gorithms, in which all the transmission occur on the samefrequency. In this paper we consider the natural general-ization in which multiple orthogonal frequency channels areavailable; various variations of this model have been stud-ied in the literature. Our ultimate goal is understandingwhether temporal starvation effects can be effectively miti-gated by the usage of multiple channels. In order to make afair comparison, we do not assume that additional channelscan be added to the existing one, but rather that the totalavailable wireless spectrum could be divided into C non-overlapping channels with smaller capacity on which nearbynodes can transmit without interfering with each other. Inthis way, at the cost of a potentially lower throughput, thestarvation effects in the network can be alleviated or eveneliminated and in this way obtain substantial improvementsin delay performance. The aim of the present paper is toinvestigate this non-trivial trade-off and for this reason weanalyze the aggregate throughput of multi-channel CSMAnetworks and develop an analytical framework for quantify-ing temporal starvation effects for these networks.2. MULTI-CHANNEL CSMA NETWORKWe consider a network of transmitter-receiver pairs shar-ing a wireless medium according to a CSMA-type algorithm.A node indicates potential data transmission between a trans-mitter and a receiver. We assume each node can sense theinterference and transmit on any of the C available channels,but at most on one at a time. For every c = 1, . . . , C all pos-sible conflicts between nodes on channel c are described bya conflict graph Gc = (V,Ec). Node i has a different back-Performance Evaluation Review, Vol. 46, No. 3, December 2018 65    . Dec. 5–7, 2018, Toulouse, France    .52 Performance Evaluation Review, Vol. 46, No. 3, December 2018off timer for each of the C available channels and we modelthese timers as C independent Poissonian clocks, ticking atrates νi,1, . . . , νi,C . When the first of these C clocks rings foran inactive node, it activates on the corresponding channel,say c, if and only if the neighboring nodes of i in Gc are notactive on the same channel. The transmission times of nodei on channel c are independent and exponentially distributedwith mean 1/µi,c. A network activity state is described by avector x ∈ {0, 1, . . . , C}N where xi = 0 if node i is inactiveand xi = c if node i is active on channel c with 1 ≤ c ≤ C.Let XC be the collection of feasible network activity states,which consists of all the vectors x ∈ {0, 1, . . . , C}N suchthat for every c = 1, . . . , C two neighboring nodes in Gc arenot simultaneously active on channel c. If the vector XC(t)describes the activity state of the network at time t, then{XC(t)}t≥0 is a Markov process on the state space XC .We show that the multi-channel CSMA dynamics can berepresented as single-channel CSMA dynamics on an appro-priate virtual network, even when the possible interferenceon the different channels is described by different conflictgraphs. This equivalent representation allows us to immedi-ately derive that the multi-channel CSMA network activityprocess {XC(t)}t≥0 has the following product-form station-ary distributionπC(x) = Z−1N∏i=1C∏c=1(νi,cµi,c)1{xi=c}, x ∈ XC ,and to prove its insensitivity to back-off and transmissiontime distributions.3. MAIN CONTRIBUTIONSEven if most of the results generalized to heterogeneousrate, for simplicity we assume homogeneous activation ratesνi,c ≡ ν and unit transmission rates µi,c ≡ 1. We focuson the regime in which the activation rate ν grows large, inwhich the CSMA dynamics favors the activity states witha maximum number of active nodes, to which we will referas dominant states. These activity states play a crucial rolein our analysis, since the timescales at which transitionsbetween them occur are intimately related to the magnitudeof the temporal starvation effects.If the number C of available channels is larger than orequal to the chromatic number of the corresponding conflictgraph χ(G), then every proper C-coloring of the graph Gcorresponds to a dominant state for the network dynamicsand the throughput analysis becomes trivial. Thus, we fo-cus on the most relevant scenario where C < χ(G), in whichthere are no activity states where all nodes are simultane-ously active. In this scenario all admissible activity statescorrespond to partial C-colorings of the graph G.Our contributions in [1] can be summarized as follows.• Assuming nodes are saturated, we prove various prop-erties of the asymptotic aggregate throughputΘ(C) := limν→∞∑i∈V1C∑x∈XCπC(x)1{xi 6=0},and, in particular, we show that that is a non-increasingfunction of the number C of available channels.• We show how the temporal starvation can be evalu-ated in the high-activation limit ν → ∞ by studyingthe expected transition times between the dominantactivity states. In particular, we introduce the starva-tion index Υ(C) of the network, that characterizes thetimescale of temporal starvation phenomena asmaxi∈V(limν→∞logν(Eτi(ν)))= Υ(C)− 1,where τi is the first hitting time of a dominant statewhere node i is active starting from the worst pos-sible starting state. The starvation index Υ(C) canbe calculated in terms of the structure of the statespace XC of the Markov process describing the CSMAdynamics. More precisely, it can be obtained solvinga max-min problem that involves the communicationheight ∆(x, y) between any pair of states x, y ∈ XC .This quantity is defined the minimum number of nodes(with respect to that of any dominant state) that needto become simultaneously inactive at some point alongany path from x to y in XC , namely∆(x, y) := minω:x→ymaxz∈ω(A(C)− a(z)), (1)where a(x) :=∑Ni=1 1{xi 6=0} is the number a(x) ofactive nodes in x, regardless of the channels they areactive on, and A(C) := maxx∈XC a(x) is the maximumnumber of active nodes in conflict graph G when Cchannels are available.• We analyze the mixing time tmix(ν) of multi-channelCSMA dynamics using the same framework built tostudy the transition times between dominant statesand derived the following asymptotic inequalitylimν→∞logν tmix(ν) ≥ Γ(C)− 1,where Γ(C) ≥ 1 is the worst communication heightbetween any pair of dominant states. We comparethis new index Γ(C) with the starvation index of thenetwork and show thatΥ(C) ≤ Γ(C).These two indices can be equal, but for most con-flict graphs and choices of the number C of availablechannels the inequality is strict, as we show with acounterexample. This fact suggests that in the high-activation regime the mixing time does not always cor-rectly characterize the temporal starvation timescalein multi-channel CSMA networks, often leading to pes-simistic performance estimates.• By means of various counterexamples, we show thatmany desirable properties and performance indices arenot monotone in the number C of available channels,revealing the difficulty of finding analytically the besttrade-off between throughput and temporal starvationeffects.AcknowledgmentsThe author is supported by NWO Rubicon grant 680.50.1529and is grateful to S.C. Borst and J.S.H. van Leeuwaardenfor the precious feedback on the early stages of this work.4. REFERENCES[1] A. Zocca, Temporal starvation in multi-channel CSMA net-works: an analytical framework, arXiv:1808.03602, 2018.66 Performance Evaluation Review, Vol. 46, No. 3, December 2018Performance Evaluation Review, Vol. 46, No. 3, December 2018 53

Temporal starvation in multi-channel CSMA networks:An analytical framework

In this paper we consider a stochastic model for a frequencyagile CSMA protocol for wireless networks where multiple orthogonal frequency channels are available. Even when the possible interference on the different channels is described by different conflict graphs, we show that the network dynamics can be equivalently described as that of a single-channel CSMA algorithm on an appropriate virtual network. Our focus is on the asymptotic regime in which the network nodes try to activate aggressively in order to achieve maximum throughput. Of particular interest is the scenario where the number of available channels is not sufficient for all nodes of the network to be simultaneously active and the well-studied temporal starvation issues of the single-channel CSMA dynamics persist. For most networks we expect that a larger number of available channels should alleviate these temporal starvation issues. However, we prove that the aggregate throughput is a non-increasing function of the number of available channels. To investigate this trade-off that emerges between aggregate throughput and temporal starvation phenomena, we propose an analytical framework to study the transient dynamics of multi-channel CSMA networks by means of first hitting times. Our analysis further reveals that the mixing time of the activity process does not always correctly characterize the temporal starvation in the multi-channel scenario and often leads to pessimistic performance estimates

In this paper we consider a stochastic model for a frequency-agile CSMA protocol for wireless networks where multiple orthogonal frequency channels are available. Even when the possible interference on the different channels is described by different conflict graphs, we show that the network dynamics can be equivalently described as that of a single-channel CSMA algorithm on an appropriate virtual network. Our focus is on the asymptotic regime in which the network nodes try to activate aggressively in order to achieve maximum throughput. Of particular interest is the scenario where the number of available channels is not sufficient for all nodes of the network to be simultaneously active and the well-studied temporal starvation issues of the single-channel CSMA dynamics persist. For most networks we expect that a larger number of available channels should alleviate these temporal starvation issues. However, we prove that the aggregate throughput is a non-increasing function of the number of available channels. To investigate this trade-off that emerges between aggregate throughput and temporal starvation phenomena, we propose an analytical framework to study the transient dynamics of multi-channel CSMA networks by means of first hitting times. Our analysis further reveals that the mixing time of the activity process does not always correctly characterize the temporal starvation in the multi-channel scenario and often leads to pessimistic performance estimates

Temporal starvation in multi-channel CSMA networks: an analytical framework

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

VU Research Portal

Caltech Authors - Main

NARCIS