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Markov Random Field Models of Multicasting in Tree Networks Markov Random Field Models of Multicasting in Tree Networks Author(s): Kavita Ramanan, Anirvan Sengupta, Ilze Ziedins and Partha Mitra Source: Advances in Applied Probability, Vol. 34, No. 1 (Mar., 2002), pp. 58-84 Published by: Applied Probability Trust Stable URL: http://www.jstor.org/stable/1428372 . Accessed: 18/09/2013 14:55 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Advances in Applied Probability. http://www.jstor.org This content downloaded from 128.148.231.12 on Wed, 18 Sep 2013 14:55:03 PM All use subject to JSTOR Terms and Conditions Adv.Appl. Prob. 34, 58-84 (2002) Printed in NorthernIreland @ AppliedProbability Trust 2002 MARKOV RANDOM FIELD MODELS OF MULTICASTING IN TREE NETWORKS KAVITARAMANAN * ** AND ANIRVANSENGUPTA,* Bell Laboratories,Lucent Technologies ILZE ZIEDINS,***University ofAuckland PARTHAMITRA,* Bell Laboratories,Lucent Technologies Abstract In this paper,we analyse a model of a regulartree loss networkthat supportstwo types of calls: unicast calls that requireunit capacity on a single link, and multicastcalls that require unit capacity on every link emanating from a node. We study the behaviour of the distributionof calls in the core of a large networkthat has uniform unicast and multicastarrival rates. At sufficientlyhigh multicastcall arrivalrates the networkexhibits a 'phasetransition', leading to unfairnessdue to spatialvariation in the multicastblocking probabilities. We study the dependenceof the phase transitionon unicast arrivalrates, the coordinationnumber of the network, and the parity of the capacity of edges in the network. Numericalresults suggest that the natureof phase transitionsis qualitatively differentwhen thereare odd and even capacitieson the links. These phenomenaare seen to persist even with the introductionof nonuniformarrival rates and multihopmulticast calls into the network. Finally, we also show the inadequacyof approximationssuch as the Erlangfixed-point approximations when multicastingis present. Keywords:Multicasting; unicasting; broadcasting; Cayley trees; Bethe lattice; loss networks;blocking probabilities;phase transitions;Erlang fixed-point approximations; reduced load approximations;statistical mechanics; spin models; hard core models; symmetrybreaking AMS 2000 SubjectClassification: Primary 60G60; 90B 15 Secondary68M20; 60K35 1. Introduction In this paper, we present and analyse a model of multicast and unicast calls arriving at an idealized loss network which has the form of a symmetric tree. Multicasting arises in both queueing and loss networks. Instead of having a simple end-to- end connection, a transmission is made to a group of individuals from a single site. An example of this in the loss network setting is a conference call. In a queueing setting this arises, for instance, when multiple copies of a message on the Internet are broadcast from one person to a number of sites via a mailing list distribution. In recent years there has been a growing interest in multicasting applications that require reliable data delivery such as software distribution and news broadcasts. This has given rise to several interesting problems. One major problem is the issue of how to construct connections between individuals in a way that makes communication Received 4 December2000; revision received 2 January2002. * Postal address:Bell Laboratories,Lucent Technologies, 600 MountainAvenue, Murray Hill, NJ 07974, USA. ** Email address:[email protected] ***Postal address:Department of Statistics,University of Auckland,Private Bag 92019, Auckland,New Zealand. 58 This content downloaded from 128.148.231.12 on Wed, 18 Sep 2013 14:55:03 PM All use subject to JSTOR Terms and Conditions Markovrandom field modelsof multicastingin treenetworks 59 efficient and reliable and minimizes demands on network resources. Much effort has been expended on this [1], [6], [16]. Before this can be answered, however, we need to be able to analyse networkperformance in the presence of multicastcalls, a task that is considerably more complex than the performanceanalysis of a purely unicast network. In this paper we study performancein the context of a symmetric loss networkin the form of a tree. This is a very simple networkwhich neverthelessexhibits interestingbehaviour, and raises questions aboutthe performanceanalysis of more complicatednetworks using traditionaltechniques and approximations.We expect thatthe insightgleaned here will be useful in moregeneral contexts, especially in view of the fact that many of the qualitativeproperties observed in the case of a symmetricnetwork persist even when asymmetryis introduced. A general loss network without controls can be described as follows. Let J be the finite collection of resources(or links) in the network,and C = {Cj, j E J} where Cj is the capacity of resource j. Let R denote the finite set of possible call (or customer)types in the network. Calls of each type r E R arriveas a Poisson process of rate Vrand have identicallydistributed holding or service times that we assume, without loss of generality,to have mean 1 (see [5]). Each call of type r requirescapacity Ajr from resource j E J for the durationof its holding time. If one or more of the resources(links) j for which Ajr > 0 does not have sufficientfree capacityto carrythe call of type r, then the call is blocked and consideredlost. Otherwisethe call is accepted. All arrivalprocesses and holding times are assumedto be independentof one another.One of the standardmeasures of performancefor a loss system is the probabilitythat a call is blocked or lost (the loss, or blocking probability). Loss networkshave been widely studied, as models of circuit-switchedand ATM networks, and in other contexts. Excellent introductionsto this general model are found in [13] and [17]. We confine our attentionin this paper to a special kind of symmetricloss networkwith a regulartree structure.Let Toobe an infinite tree with q + 1 edges emanatingfrom each node. Let SL be the finite sphericalsubtree of radiusL which consists of a single centralnode and all nodes no more thana distanceL from it, where the distancebetween two nodes is the minimum numberof edges in any pathconnecting the two nodes. There are q + 1 edges emanatingfrom each node in SL except the terminalnodes, which are attachedto just one edge. We assume that each edge has the same capacity,C. We let E8(T)denote the set of edges in a given tree T and use 8 (t) to denote the set of edges that are incidentwith the node t. For each node t E T, we define XA(t) := {t' E T : tt' e 8(t)} to be the set of immediateneighbours of t. We allow two types of calls in the network. Single link calls, requiringcapacity on just a single edge, arriveat each edge as a Poisson process of rate k. Single link calls connect two neighbouring nodes. Multicast calls, on the other hand, are centred at a node, and connect that node to all of its immediateneighbours. Thus a call centredat an internalnode t e SL requirescapacity on each of the q + 1 edges in JN(t), while a call centredat a terminalnode requirescapacity on just one edge. We assume that multicastcalls arriveat each node t as a Poisson process of rate vt = v. We will assume that all calls requirejust a single unit of capacity on each edge of their connection. Throughoutthis paperwe will be concentratingon assessing the blocking probabilitiesfor large sphericaltree networks. Much of the treatmentbelow can be applied in the more complex multiratesetting, but in orderto clarify the presentationwe do not do so here. Althoughthe tree networkthat we have describedhere is clearlyunrealistic, it is nevertheless a useful tool for studying some of the behavioursthat multicastingmight introduceto a loss network.We show below thatmulticasting can introduceunfairness in the symmetricspherical tree that we study, as the blocking probabilitiesseen by the same call types at differentnodes This content downloaded from 128.148.231.12 on Wed, 18 Sep 2013 14:55:03 PM All use subject to JSTOR Terms and Conditions 60 K. RAMANANETAL. may vary considerably. These differences are a consequence of a phase transitioneffect that arises when the arrivalrate for multicast calls is sufficientlyhigh and that of unicast calls is sufficiently low. Thus we observe that the presence of unicast calls assists in maintaining fairness and homogeneity of the blocking probabilitiesin the system. In addition, we study the trade-offbetween unicast and multicastcall arrivalrates for a given fixed unicastblocking probability.Finally, we also shed some light on approximationsof networkswith multicasting. A commonly made assumptionwhen calculating approximateblocking probabilitiesis that resources (edges or links) behave as though they are blocking independentlyof one another. This assumptionunderpins approximations such as the Erlangfixed-point approximation, also knownas the reducedload approximation,and refinements of it-approximations thathave been widely andvery successfullyused, even whereit is clearthat the
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