Markov Random Field Models of Multicasting in Tree Networks

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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

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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.

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 independenceassumption does not hold. Thereis now considerabletheoretical justification for the use of these approximations (see e.g. [13] for details). However,it seems thatwhen appliedto multicastnetworks, although the approximationmay still be very good for low values of the multicast arrivalrate, it may considerablyunderestimate the point at which phase transitionoccurs, and cannot be applied directlybeyond thatpoint. Karvoet al. [10] have also appliedthe reducedload approximation to a multicastnetwork, and aftercomparing it with simulationscomment that it performsworse at higher values of the arrivalrate. The special structureof the tree networkgreatly simplifies the studyof these questions,since the removalof any single edge from the tree splits it into two disjoint subgraphs.This in turn means thatwe can very easily develop recursionsthat give us the exact blocking probabilities. The recursionsthat we obtainin this paperfor theblocking probabilities are related to those given in [22] and are given in the form developedin the statisticalmechanics literature (see e.g. [2]). Kelly [12], [13] previouslyconsidered the purelymulticast tree network model for C = 1 and obtaineda recursionwhich gives the probabilityof a call being accepted. An earlierpaper by Spitzer[18] also gives resultsfor tree networksthat apply to the C = 1 case. Zachary[20], [21] analyses the more generaltree network,which includes as a special case the model considered here with C U but k = 0 the multicastnetwork with no unicast calls). In • E {oo}, (i.e. pure particular,given a homogeneousMarkov specification on the Cayley tree, in [20, Theorem4.1] Zacharyestablishes a one-one correspondencebetween Markov chains associated with that specificationand solutions to an associatedrecursion problem, which is related to recursions consideredin this paper. Georgii [8] serves as a good introductionand overview of some of these earlier results. More recently, van den Berg and Steif [19] obtain results for general bipartitegraphs, but again they apply only to the equivalentof the C = 1 and 'k = 0 case in our model. Louth [14] has studied a relatedmodel on a squarelattice, and van den Berg and Steif also applied their results to the squarelattice (again under a condition equivalentto the requirementthat C = 1 and k = 0). Brightwell andWinkler [3], [4] have appliedthe methods of graphtheory and combinatoricsto the tree networkwith the equivalentof C > 1 and X = 0. Observe that in all cases, the conditions placed on these models previously consideredin the literatureare equivalentto assumingthat there is no unicasttraffic in the network. In Section 2, we introduce the basic model and provide an explicit expression for the stationarydistribution on a finite tree. In Section 2.1, we derive a recursion to calculate the normalizationconstant, and in Section 2.2, we provide expressions for the unicast and multicastblocking probabilities. In Section 3, we study these quantitiesfor the collection of nodes located deep within a large network. In Sections 3.1 and 3.2, we calculatethe stationary distributionand blocking probabilitiesfor these core nodes. In Section 3.3, we specialise to the case C = 1 and study the trade-offsbetween unicastand multicastarrival rates. In Section 3.4,

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 models of multicastingin tree networks 61 we consider the case C = 2, and comment on the general C case in Section 3.5. We analyse the Erlangfixed-point approximation for this networkin Section 4 and show that it providesan inadequatedescription of the networkwhen multicastingis present. In Section 5, we show that phase transitionspersist even for generalizationsof the model that allow multihop multicast calls and heterogeneousarrival rates. In contrast,as discussed in the conclusions in Section 6, it is conjecturedthat heterogeneity destroys phase transitionsfor integerlattice networks[19].

2. Model description As explained in the introduction,we consider a loss network that has the structureof a sphericalCayley tree (see, for example, Figure 2 below). The radiusof such a tree is defined to be the distance (or numberof edges) between the centralnode and any terminalnode. The nodes in a tree can be divided into even and odd lattices in the following fashion. Designate an arbitrarynode as being 'even'. Then all nodes at an even distance from it belong to the even lattice, and those at an odd distancefrom it belong to the odd lattice. For a finite tree, the referencenode is often takento be one of the terminalnodes. The resourcesin the networkT are its edges 8 (T), each of which has capacity C. Note that the capacity C may also be even or odd. The total numberof types of calls in any Cayley tree T is given by MT := IT I + 18(T)I, comprisingmulticast calls that arriveat each node t with Poisson rate v (and requireunit capacity from each edge incident to that node), and unicast calls that arriveat an edge with Poisson rateX (andrequire unit capacityfrom that edge for the period of its holding time). Let Z := {0, 1 ...., C}. The set of feasible configurationson T (subjectto the blocking constraint)is given by

:= E : + nt + ntt, C for every t', t E T, t' E (2.1) O2 {n •T nt, < N(t)}. The dynamicsof the model can then be describedby a Markovprocess n(s), where for each s e [0,9o), n(s) := {nt(s), ne(s), t E T, e e 8(T)} E 2T and nt(s) and ne,(s) are, respectively the randomvariables whose distributionsdescribe the numberof multicastand unicastcalls in progress at node t and edge e at time s. Under the assumptionsstated above, it is well known [13] thatthe process n (s) has a stationarydistribution 7r, which has the truncatedproduct form

1 Vnt ne 7r(n)= H-y h (2.2) Z fnET, nt-tenT et nnH(T) where Z, is the normalizingconstant z=> H (2.3) ZHnt+ nee nE2 tETtET eEE(T) 2.1. Recursion relation As discussed above, we have an explicit expression (2.2) for the stationarydistribution of interest. However,this expression is not always very useful because the calculationof the normalizationconstant in (2.3) often becomes infeasiblefor networksof realisticsize. Note that the stationarydistribution w defines a Markovrandom field on T (see [22] for a discussion on the connectionbetween loss networksand Markovrandom fields). Therefore,not surprisingly, analogous problems of calculating normalizationconstants of the form (2.3) often arise in statistical mechanics in the calculation of Gibbs measures for Markov randomfields. From

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root node

FIGURE1: A rooted(2,3) Cayleytree.

the statisticalmechanics perspective, the networkmodel describedabove (with X = 0) can be interpretedas a 'spin model' on the Cayley tree with hardconstraints [2], [15], and is analogous to the hard-coremodels studied in [19]. Given a tree T, and n E T, the weight of the configurationn for this model takes the form

W(n) := Hg(nt) h(nt, nt, nt), tET tt'E&9(T) where Vi ;k g(i) := h(i, j, k) := x(C - (i + j + k)), i!' 'k! and

x(X : 0 if x < 0, 1 otherwise.

The weight is proportionalto the probabilityof a configuration. The normalizationconstant definedin (2.3) can be expressedin termsof the weight as Z, = EnEn, W(n), and is referred to in the physics literatureas the partitionfunction. In order to simplify the calculation of the stationarydistribution, we introduce another subgraphTm, which we refer to as a rooted (q, m) Cayley tree. The graph Tmis a tree of size m which comprises a distinguishednode or root O, internalnodes, and terminalnodes that are a distance m from the root O. The distinguishednode has q edges emanatingfrom it, the terminalnodes have just one, and all other nodes are incident to q + 1 edges (see Figure 1). (We will also use Tmto denote the set of nodes of this finite subgraph.) In orderto derive a recursionrelation which characterizesthe stationarydistribution, we exploit the propertythat the removal of any internaledge splits the tree Tminto two disjoint trees. Let 2m := 2Tm denote the set of feasible configurationson the tree Tmas definedin (2.1). For i e Z, we define Zm(i) to be the weighted sum of all feasible configurationsin Tmthat have i multicastcalls at the root node O. More precisely,

nt V Xnne := Zm(i) n!tn n e nEam:no=i tETm eEE(Tm) " e-

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Then the configurationvector (Zm+l(i), i E Z) for the tree Tm+lcan be expressedin termsof that for Tmas follows:

C-i C-i-j q i Xj Zm(k) . (2.4) Zm+1(i)-i! LJ j j=0 k=O • Since a sphericaltree of radiusL can be decomposedinto a centralnode with q + 1 edges, and q + 1 rooted trees each of size L - 1, the normalizationconstant (or partitionfunction) Z, for the tree SL is given by C

Z, = ZrZ(i), (2.5) i=0 where jC-i-j q+1 1= iC-i ;,- ZL-1(k)) (2.6) Zr j=0 k=0 Thus for a finite sphericalr(i)-•'tree SL, given boundary=0 conditions for the externalnodes, the partitionfunction Z, can be calculatedfrom the iteration(2.4) and expressions(2.5) and (2.6). Free boundaryconditions correspond to setting

Vi Zo(i) = , for i = 0, ..., C. The problemof computingthe stationarydistribution 7r on a finite tree T is thus reducedto that of analysingthe recursionrelation (2.4). 2.2. Blocking probabilities In this section we derive expressions for the stationaryblocking probabilitiesin SL, the sphericaltree of radiusL. We begin with the stationaryunicast blocking probabilities.Let s be the centralnode, s' a neighbouringnode and aL be the probabilitythat a unicast call arrivingat the internaledge e = ss' is blocked. Removal of the internaledge ss' partitionsthe tree into two rooted trees, one of size L, with root s, and anotherof size L - 1, with root s' (see Figure 2). Then the weight of a configurationwith i multicastcalls at node s, j calls at node s' and k unicast calls on ss' is

-ZL(i)ZL-1IQ)k! providedi + j + k < C. The probabilityof call acceptanceon the edge ss' is the weighted sum of configurationsthat have one unit of free capacity on the edge ss' divided by the weighted sum of all configurations.Therefore, 1 Y ki = (Xk/k!) 01-i=k 01j= ZL(i)ZL-1(j) 1 e -orL - =0(Xk/k!) IC-k O-k-i ZL(i)ZL-1(j) (2.7)

Now considerthe stationarymulticast blocking probability,fit say, for a call arrivingat the centralnode s. Since the node s is central,the sphericalCayley tree can be partitionedinto the centralnode s with q + 1 edges emanatingfrom it, and q + 1 rootedCayley trees of size L - 1,

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I SI

FIGURE2: A sphericalCayley tree with radius 4 (q = 2).

each having as its root one of the nodes adjacentto the node s. Note thatif thereare i multicast calls at node s, then there can be at most C - i unicast calls on each edge emanatingfrom it; andif thereare then j unicastcalls on the edge ss', then therecan be at most C - i - j multicast calls at node s'. Hence, the weight of a configurationwith i multicastcalls at node s is

C-i C-i-J q+1 ZL-l(k)) i k-ci-j j=0 k=0

Consequently,

1 - L i/0i!) (=O- (XJ/j!) ZL-ZL-1(k))q+l (k))+ E- LCi (2.8) C=0 --k=0 /-i=O (pi/i!)(LC=l :,l ?jEO/ J (k))q+l . ZL-1 /j/-.k=0 Related expressionsfor othernetworks can be found in [22].

3. Large networks In this section, we examine call distributionsand blocking probabilitiesfor the collection of nodes that are located deep within the core of a large network. This collection of deep nodes in a large sphericalCayley tree is often referredto as the Bethe lattice. In Sections 3.1 and 3.2, we study the limit of the recursionand blocking probabilitiesdescribed in Sections 2.1 and 2.2 respectively. In Sections 3.3 and 3.4, we consider the special cases of C = 1 and C = 2, where some analyticalresults can be obtained. We present numericalresults for higher C in Section 3.5.

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3.1. Limit of the finite tree stationary distributions In Section 2.1, we showed thatgiven a boundarycondition Zo, we can use the recursion(2.4) to calculate the stationarydistribution r on any sphericalnetwork. Insight into the stationary distributionfor large networkscan be obtainedby analysing propertiesof the recursion(2.4) in the asymptoticlimit as m oo. For i = 1, C, and m > 0, define .... Zm(i) (i := and (m(1) m(C)). Zm(0) m ..

(Note that?m is well defined since Zm(0) is always strictlypositive for any m.) We can then express the recursion(2.4) in the more convenientform

m+1 = (3.1) 4V'1WX(m), where for e the two of + I v, > 0, is Rc+, parameterfamily mappings v, : c definedby (, , such that for k+ = 1, ..., C, 4v,(kI(=) = gv4(•)I4(.)), ( ... \C-k (/i!)[l k-i vk Sv,h E, =0 + - .(3.2) k k! =c /i!)[1i + EC- (j)](3.2)

We will referto the mappings4v,k as randomfield maps, to differentiatethem from the Erlang fixed point maps f introducedin Section 4. For the case X = 0, relatedmaps associatedwith a general homogeneousMarkov specification can be found in [20, p. 899], [21, (2.11)] and [22]. Suppose RC is endowed with the usual Euclidean metric and define the compact subset K C IRCto be the Cartesian product of the sets [0, vi/i!], i = 1,..., C. Observe that Svx(RC) C (Kb) and the functions 4)v, have continuous partial derivativeswith respect to ?(j), j = 1,..., C. This shows thatthere exists a continuousfunction g(v, X) : 2 2 that is nondecreasingin v with g(0, 0) = 0, which satisfies IV4vx(o)- )l g(v, )OII- 111. _<( Let S C IR2be the open set defined by {x e R2 : g(x) < 11. This implies the following theorem,which is relatedto resultsof Zachary[20, Theorem4.2], who showed thatthere exists at least one fixed point to the relatedmap in [20], which is equivalentto the case when X = 0 here. Theorem 3.1. Let : > be the Iv'? IRC-- JcR, v, X 0, family of mappings defined in (3.2), and let S C R2 be the associated set defined above. Thenfor any (v, X) E S, the equation 4Iv,X(?)= = has a solution E Kv. Moreover, initial condition?o E unique xv'k given any RC, the sequence definediteratively by In:= IvX(?n-1) convergesto xv,' and

- x' |II< g(V 1I|n - i1 - g(v, ) n+1II. lne+l Proof The metric space JRCis complete, and for every v > 0, K( is compact. Moreover, the fact that the mappings CV'h are Lipschitz continuouswith constantg(v, X) shows that the 4 is contractive S. is a mapping v,. : K( K strictly on The result then simple consequence of the Banach fixed-pointtheorem [23, Theorem 1.A].

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Theorem 3.1 indicates that for small enough v and k, the sequence of iterates of the map (Dv,Xhas a unique limit that is independentof the initial condition. Consequently,for such values of v and X the marginalsof the stationarydistribution at nodes in the core of sufficiently large networkswould be almost identical. However,increasing v or k can give rise to phase transitions,defined here to be the existence of v, k for which the sequences of iteratesof v,'X startingat differentinitial conditionshave multiple limit points. Phase transitionsdestroy the spatial homogeneity of the stationarydistribution at the core. For a given k, we denote the lowest value of v at which a phase transitionoccurs by vpt(k). As we see from the discussion below, for C = 1 the greaterthe value of k, the greaterthe value of vpt(k). These phenomena areillustrated in more detailfor the cases C = 1 and C = 2 in Sections 3.3 and 3.4 respectively.

Remark 3.1. It is possible to establisha rigourouscorrespondence between the fixed points of the randomfield maps Iv,k and Markovchains on the infinite tree T that are invariantto tree isomorphisms(as definedin [20, p. 895]) or, equivalently,simple invariantGibbs measureson Q2T,the set of feasible configurationson T (see [3, Definition 3.7]). The case k = 0 can be deducedfrom [3], [20]. The proof for the case X > 0, however,is more involved and requires the introductionof notationand concepts not centralto this work. Hence, we defer the details of the proof to a subsequentpaper.

3.2. Limit of the blocking probabilities L Recall the expression(2.7) thatcharacterizes c the stationaryunicast blocking probability of an edge comprisinga centralnode andits neighbourin SL, the sphericalCayley tree of radius L. Define ae to be the limiting probabilityas L -+ 0o0, so that ae provides an approximation for the blocking probabilityof edges thatare deep in the interiorof a largeCayley tree network. recall from (2.8) that is the that a multicastcall is blocked from the Similarly, sL probability centralnode s of and let ?s be the limit of as L SL, fsL -+ o0. Lemma 3.1. For (v, k) such that v < vpt(k), let (* = (*(v, k) be the uniquefixed point of the map 4Iv,x. Then

E + 2 C-l-k + C--k V C--k-i - C-1(0.k/k!)[1 1 a= = *() (3.3) 1-e C-k =o0(k/k!)[1 + 2 EC-k *(i) + CC-k-i 3(i3) and

=Ol(/i!) Lj=iO (/j!)(1 + -k= *(k) 1)5S -s = (3.4) )[ + *(k))]+ - o(vi/i =0 ('J/j!)(1 Proof. Using the definitionof 4m,we can rewrite(2.7) •-k=las

1 - aL= 1~C-l01 + C -1-kC 1 L kL(i)+ L-1(j) + -k (i)(L-1(j)] S---(.k/k!)[1 -i _-k -kl C..,Ck L C-k-i L i) ) k-• L(i) 1•- - () L- oC(k/k!)[1+ C=0- - (3.5) The expressionfor 1 - ae follows directlyfrom (3.5) and the fact that >* as L tL - - oc. Analogous calculationsyield the expressionfor 1 - s.s

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3.3. The case C = 1 In this section, we analyse the performanceof the model for the case C = 1. As mentioned earlier,the limiting stationarydistribution for C = 1 with k = 0 was obtainedin [12], [13]. Here we are also interestedin studyingthe trade-offbetween unicastblocking and the capacity to supportmulticast calls. 3.3.1. The recursionfor C = 1. For C = 1, (3.2) reduces to the one-dimensionalrecursion

(m+l - (1 + , + ?m)q

The critical curve Lcr in (v, X)-space of bifurcationpoints of the two-parameterfamily of mappings4v,k (see e.g. [7]) is definedby the relation

(*) --=1,

8n (v,,k)ELcr where, as in the previous section, =*= *(v, k) is the fixed point for the mapping4v,). Since

S(1 + + - )+-)q+1 and * (3.6) = + + , (1 X ?*)q simple algebraicmanipulations show that, at criticality,the fixed point is 1+' q-1 and hence the critical curve Lcr is describedby the set of points (vcr(X), X3)where

= vcr(P) q)q+l. (1 + (3.7) (q - 1)q+1 The value of the critical points of bifurcationfor the family of mappings for C = 1 was obtainedin the absence of unicast calls (,k = 0) in [12], [13]. We define thev,,' subcritical region to be the set of points (v, 3) that satisfy v < vcr(,k)and the supercriticalregion to be the set of points that satisfy v > vcr(X().To see that bistable behaviourcannot exist in the subcritical region, consider the iteratedmap 4v,a o v,rx(?). Then it is straightforwardto check that (a) )v,'1ovX(0) > 0; (b) 4V'' o4v,' (.) is increasingin ; (c) v,' o v,'(.) has apoint of inflection (i.e. its second derivativevanishes and changes sign) at ? = [v(q - 1)/(1 + X)]l/q - (1 + )); and finally (d) for v < vcr, vd, o0 < 1. From (d) we can conclude that, for 4v,'x()/Iaj v < vcr, o 'v,'(?)/?j < 1 for all ?, and hence thatthe iteratedmap has a uniquefixed laDv,' point for v < This yields the following result. vcr. Theorem 3.2. For C = 1, the region where there is no phase transition is precisely the 2 subcritical region {(v, X) e )2 v < vcr()}, where vcr(() is given by (3.7). In Section 4, we will show thatthe criticalpoint for the Erlangfixed-point approximation is 2vcr, which providesevidence of the inadequacyof this approximationin this context, although we note that for v < vcrthe approximationis still very good.

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Note that in the asymptoticregime as q - +c, for the case of a pure multicastnetwork = 0) we have (k e lim vcr" lim - = 0. q--oo q-4ooq This suggests that in pure multicastnetworks with a higher degree of connectivity,heterogeneous multicastblocking probabilitiesin the networkwill arise at lower arrivalrates. Beyond the criticalpoint, an orbitof period 2 appears,and thereis no longer a uniquelimit point for the sequenceof iteratesof the map Iv,x. Let the points of this orbitbe denotedby and *. Then we observe that

? = ,)( 1 ) 2 and (3.8) 'I~,(*) =_ and so, for i = 1, 2, 4IV,Xk0 cIVX(*)=

Spitzer [18, Theorem 9] and Zachary [20, Theorems 4.1 and 4.3] showed that given any homogeneousMarkov specification on a regularCayley tree, there exists an associatedrecursion map such thatthe existence of a 2-periodorbit (that is not a fixed point) for the mapis equivalent to the existence of a Markovchain thatis not invariantunder isomorphisms of the tree, but that odd onto themselves. is invariantunder isomorphisms that map the even and lattices Thus, vcr denotesthe point at which thereappears a Markovchain on the infinitetree that is not translation invariant,but has an alternatingpattern on the even and odd lattices. 3.3.2. Blockingv. capacitytrade-offsfor C = 1. We now calculatethe trade-offin the subcritical region between the volume of multicastand unicastcalls that can be supportedin the network for a given blocking probabilityfor unicastcalls. For simplicity of notation,we let a = ae be the unicastblocking probability.Then for (v, X) in the subcriticalregion, from the expression (3.3) it follows that 1 1-a= 1 + + 2* where '* is the unique fixed point of the map 4v,X. Rearrangingthe above equationyields

a * 2(1 - a) 2

Using the fact that '* satisfies the fixed point equation(3.6), for (v, k) in the subcriticalregion we obtain - S= 1 + + (3.9) ( - a) 2)( 2(1 - a) 2+ 2(1 The above equationis plotted for differentvalues of a for the case q = 2 in Figure 3. In the subcriticalregion, the curves characterizethe trade-offbetween multicast and unicast arrival rates for fixed a and are markedas solid curves, whereasin the supercriticalregion the curves no longer have this physical interpretation,and are thus markedwith dashedlines. It is intuitivelyclear that as the unicastarrival rate decreases,the multicastarrival rate must increasein orderfor the unicastblocking probability to be maintainedat a. Indeed,as illustrated in Figure 3, the v-X trade-offcurves are monotone. In additionthey are also convex, which means that, for every fixed a, the rate of increase of the multicastarrival rate decreases with decreasein the unicast arrivalrate. Let X = X(a) be the point at which becomes zero, v/8a•

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Lcr 30

20 -a

10 - -- -

2 3 0

0 1 2 ,k 3 4 5

FIGURE 3: Trade-offbetween unicast and multicast calls (C = 1, q = 2).

- and let = Vi(a) be the correspondingvalue of v at which this happens. Let 0 = a/(1 - a), and note that

a 1 ' k q qq-1 2 + -1 ( 1+2+ +)(+-4 ) a- 2 2 2 4 1 2 2

Then by the above equationwe conclude that

S (q - 1) 2 q+1 q+l' and substitutingthis back in (3.9), we see that

(9 + 1)q+l (q + 1)q+1l Since - (q-1)( +1) 1+ q+1 it follows thatVi satisfies - qq (q 1)q+1

a E Comparingthis with (3.7), we see thatthe locus of points (k(a), for (0, 1) coincides with the critical phase transition curve Thus we note the remarkablei(a)) feature that (at least for Lcr. the case C = 1) phase transitionsoccur precisely at points (k, Vi)at which the rate of increase of the multicastarrival rate with decreasein the unicastarrival rate (for a fixed unicastblocking probability)is zero.

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Now, for (v, X) beyond the criticalcurve, let •, , be the two period orbit definedin (3.8). TakingL - o in (3.5), it can be seen thatthe blockingprobability a in the supercriticalregion is given by 1 I-a=- 1 + X + + *22

Thus, althoughthere is a bifurcationof solutions for large enough values of (v, X), the unicast blocking probabilitiesare still homogeneous throughoutthe tree. Moreover,we observe that since the two points ?* and?2* move in a continuousfashion away from the fixed point as the critical curve is crossed, the unicast blocking probabilitieschange continuouslythrough the phase transition.As we show later,this is not the case when the link capacityC = 2 (and,more generally, for even link capacities C), where we could have an abruptchange in the unicast blocking probabilitiesat certainvalues (v, X) in the subcriticalregion of the parameterspace (i.e. where the fixed point is still locally stable). We now consider the multicastblocking probability. For C = 1, from (3.4) we have, for (v, k) in the subcriticalregion,

1 1 1-B= v 1+ [1 + k + ,*]q+1 = v[1 + (1 + + (*)/(*]' where the last equalityfollows from the fixed-pointequation (3.6). Rearranging,we find that

v(A + 1)(1 - P) 1 - 2v(1 -) ' and therefore + (1 (1 v(1 -P)) 1X) - 2v(1- - P) Note that the above two equationsimply that, in the subcriticalregion, 1 - 2v(1 - P) > 0. Substitutingthe last two displays into the fixed point equation(3.6), we see that - - S- ( -)-1 2v(1 P) [1 - v(1 - f)]q/(q+1)

Analogousto the trade-offcurves for fixed unicastblocking probabilities, here too it is clearthat with a decreasein the multicastarrival rate, to maintainthe same multicastblocking probability, the unicastarrival rate should increase. In this case, however,beyond the phasetransition curve, the multicastblocking probabilities bifurcate, and take on differentvalues depending on whether the node lies on the 'even' or 'odd' lattice. 3.4. The case C = 2 In this section, we analyse the performanceof the model for the case where the capacity of each edge (link) is C = 2. As we will see below, this map displays behaviourthat is not found for the C = 1 case. In particular,it is possible to observe a discontinuouschange in the blocking probabilitieswith a change in parametervalues. As discussed in Section 3.5, maps with even link capacity C seem to behave like C = 2, while maps with odd link capacity C behave like C = 1.

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3.4.1. The recursion for C = 2. For C = 2 the recursion defined in (3.1) yields the following two-dimensionalmap: 1 [ 1+X+((1) , + +1) 2 + (1 +k)X(1) (+ (2) v2 q 2 2 1+ + 12 + (1+ )((1)

Setting ? (1) = and ? (2) = qr,and computingpartial derivatives of the map, we see that the Jacobianat (1, 7j)is given by

_ _ _ -qv__ (1?+ + + 1j)2 + (1 (I + + 1 ? (1 + + [(1-++-)q-l(+ (l 1)2-k) . . +1.)r qvl Now we know from Theorem3.1 that the map has a unique fixed point * (? 9*) =(=*(*v,)), 0 *(V,X)) for all sufficientlysmall v and X. At the fixed point, the Jacobianhas the value

S*() -q 1d'22 - r*) J(t*, n*) = -+q1?+X* -+- ) j)2 s 17) + + + (12 )* + + J(, +• 1 7*(1+,) 7* "1 ) Thus the eigenvaluesat the fixed point (1*, 7*) are given by q 2[11 'X +( + X)* + •++ 2 1 17*] F + X) + + ) -*(1 (*X(1 ) S1 +X) + *

+_) + _*(1_ + ))2 4[?*r*2 + ?*r* + *21r*+ ?*r1*X(1+ ~*+ 1))] (17*(l + 4?2. (1 + + (*)2 1 + + •* In the pure multicast. case, when X = 0 the eigenvaluesare given. by

4 q_ + *7*2 + 17__** 17*2 4?*r* . 2(1+ + 17*) 1+* -, (1+ *)2 1 * We know from Theorem 3.1 that for small v the fixed point (1*, rl*) is globally stable in the sense that,starting from any initialcondition, the iteratesconverge to thatfixed point. In orderto determinewhen this fixed pointloses its local stability,it is necessaryto calculatethe bifurcation K point Vcr,which is the point at which IRe(K)l= 1, where is the eigenvalue of the Jacobian. Since in this case the eigenvaluesare real, this correspondsto findingthe point at which K I = 1. If $*, 9* could be expressedas a function of v, then by setting the absolutevalue of the above expressionequal to 1, we could determineVcr. However, since such an explicit expressiondoes not seem feasible, we use numerics and approximationsto discern the behaviourof the map.

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222/V2

globally stable fixedpoint

252/lV2 22Iv/2 stable two-cycle

marginallyall stable unstable stable locally two-cyle two-cycle locally two-cycle stable two-cycle stable fixedpoint fixedpoint

v=Vf V>Vf

FIGURE4: First-orderphase transitions (C = 2, k = 0).

For large v, it is easy to see that the fixed point is approximatelygiven by (v, lv2-q). Thus, as v -* oc, the absolute value of both eigenvalues tends to zero-implying that locally the fixed point becomes extremelystable, andthus thatthere is no finite vcr. However,although the fixed point maintainsits local stabilityfor arbitrarylarge values of v, it loses global stabilityat a point vf when a marginallystable two-periodorbit appearsin additionto the locally stable fixed point. As v becomes large, the marginaltwo-cycle splits into a stable and an unstable cycle. The domain of attractionof the unstable two-cycle defines the boundarybetween the basins of attractionof the fixed point andthe stabletwo-cycle. This phenomenonis reminiscent of what is referredto as a first order transitionin statisticalphysics, in which (at v = vf) a new stable state appearswhose orbits are far away from the priorstable state describedby the fixed point (referto Figure 4). Hence for C = 2, the phase transitionpoint vptcoincides with vf. In contrast,recall that for the case C = 1 the phase transitionoccurs at Vcr,the value at which the stable fixed point loses even its local stabilityand bifurcatesinto a stable two-cycle whose orbitslie close to the fixed point. Such gradualtransitions are referred to as second-order transitionsin the statisticalmechanics literature.

3.5. Higher link capacity We have examined the cases C = 1 and C = 2 with some care. Clearly,in a realistically sized network,C will be considerablylarger. However, we conjecturethat in some respectsthe

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00 -

vpt ?

-10

1 S x C

FIGURE5: Plotof vptfor varying q andC values(k = 0). qualitativebehaviour will be similar. In particularwe conjecturethat for odd C, entry to the phase transitionregion will be with a second orderphase transition(vpt = Vcr),as with C = 1; while for even C, the entryto the phase transitionregion will be accomplishedvia a firstorder phase transition(vpt = vf). (Recall that vpt is defined to be the lowest value of v at which a phase transitionoccurs.) Numericalresults seem to indicate that for higher C there will be a numberof phase transitions,corresponding to modes of the distributionwhere, say, nodes in the even lattice carry i calls and nodes in the odd lattice carry j calls, with i + j = C. In Figure 5, we plot numericallyobtained values for Vptfor X = 0 and varying C and q. We see that these numericalresults indicate that, as q increases, the phase transitionpoint decreases, and that vpt is of a lower orderof magnitudefor odd C, when comparedto even C, particularly for small q. The latterbehaviour may be linked with our conjectureabout the differingnature of the phase transitionsfor odd and even C.

4. The Erlang fixed-point approximation We have observedin previoussections that,for (v, X) in the region priorto phase transition, the blocking probabilityin the centre of SL converges as L -- oo to the fixed point of a map, and that link blocking probabilitiesin the centre of the tree are homogeneous. This suggests that the naturalformulation of the Erlangfixed-point (EFP) approximation,at least initially,is one where the blocking probabilityon every link is the same, assumingthat the tree is infinite. This then gives the blocking probability,B, for a single link as a fixed point of the map - = E(X + 2v(1 B)q, C), (4.1) fv',(B) where E(v, C) := 'i/i!)-'vc/C! is the Erlang-Bblocking formula [13], [17]. Thus, each link sees a reduced(E•o load of X + 2v(1 - B)q offered to it. Recall that X is the arrivalrate or load of unicast calls at every link, and there are also two streamsof multicastcalls offered to every link, each centredat one end of the link, each arrivingat rate v and thinnedby a factor of (1 - B)q.

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It would, of course, also be possible to consider the Erlang fixed-point approximationto the blocking probabilitiesat the centre of finite trees, as they become large, without a priori assumingthat they are the same on every link. For the tree SL, this involves findingthe fixed point of an L-dimensionalmap. Numerical results indicate that, for (v, X) in the subcritical region of the map (4.1), the approximationto the blocking probabilityof a centrallink in SL obtainedfrom the full L-dimensionalmap convergesto the fixed point of (4.1). We therefore confine our attentionto (4.1) for the rest of this section. We note, however,that our numerical resultsindicate that beyond the criticalpoint of the EFPmap, the single-linkblocking probability correspondingto that of the centrallink for the L-dimensionalEFP approximationconverges to the unstablefixed point of the map (4.1). Let B*(v, ,) be the unique fixed point of the mapping (4.1), which is guaranteedto exist for small enough X and v (by analogy with Theorem3.1). As in Section 3.3, by definitionthe values (v, X) on the criticalcurve Mcr of bifurcationpoints of the family of mappingsfvx (see e.g. [7]) must satisfy

(B*) = 1. aB (VX)EMcr When C = 1, - (B)+ 2v(1 B)q fv,1(B) - =- 1 + X + 2v(1 B)q so that _ - afV), 2qv(1 B)q-1 aB (1 + X + 2v(1 - B)q)2' and at the fixed point B* = B*(v, X) we obtain

1 - B* =1 1 + X + 2v(1 - B*)q Togetherthese show, after simple algebraicmanipulations, that at criticality

+ Aq cr B* 1 q(1 + X))' andthe criticalcurve LEFPfor the EFP mapis describedby the set of points (vEFP(X), X), where

1 -1- + (1 .)q+l VEFP( 2q q)1\-(q+ which is equalto exactly half the expression(3.7) for vcr(W) of the randomfield map(Dv, found in Section 3. Figure6 illustratesthe attractingsolutions to the variousmaps for the case C = 1, withX = 0, q = 2 and v rangingbetween 0 and 6. Figure 6(a) gives the multicastblocking probabilities 6 for the randomfield maps Qv,O(which when C = 1 can be found in Section 3.3), as well as exact blocking probabilitiesat the centre of SL, when the radius L = 10, 11 (o10 and f11) and when the radius L = 40, 41 (/40 and~41). These all assume free boundary conditions. We note that as the radiusL increases,the blocking probabilitiesat the centre of SL approach those of the random field map, following either the upper or lower branch, depending on whether L, the radius of the tree, is odd or even. Figure 6(b) gives the multicast blocking

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(a) (b) 1.0- 1.0- 0.8 - 0.8 - 0.6- 0.6- 0.4 I 0.4- 0.2 - 0.2- 0- 04-- I 1I I I I I I I 0 1 2 3 4 5 6 0 1 2 3 4 5 6 v v

Po10 and f11 ...... 1 - (1 - B*)3 Poand ------

FIGURE 6: Comparisonof blockingprobabilities for the randomfield andEFP maps (k = 0, C = 1, q = 2): (a)multicast blocking probability, P, obtainedfrom the random field map, (b) multicast blocking probability,1 - (1 - B*)3, obtainedfrom the EFP approximation.

30 Vpt 20

0- 2 4 6 8 10 C vpt for ) _ for f vErFP FIGURE7: Comparisonof phasetransitions for the random field and EFP maps (X = 0, q = 2). probabilitiesobtained from the Erlangfixed-point approximation below criticalityof that map (for comparisonpurposes the blocking probabilitiesobtained for the maps Qv,ohave also been plotted in Figure 6(b)). Note that if B* is a fixed point of the EFP map, then the multicast blocking probabilityis given by 1 - (1 - B*)q+ . We observe that the family of randomfield maps Qv,o bifurcatesat v = 4, while the family of EFP maps fv,o bifurcatesat v = 2, in agreementwith the results of this section and those of Section 3. For higher C too, there are inadequaciesin the EFP approximation.The most striking is that the bifurcationpoint for the EFP approximationoccurs at much lower values than for the randomfield map tv,O, particularlyfor even C. Figure 7 shows Vpt,the lowest value of v for which a phase transitionoccurs, for both the randomfield maps Dv,oand the EFP maps fv,o

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 76 K. RAMANANETAL. when q = 2, and for C rangingfrom 1 to 10. We note that,unlike the case of the randomfield map, for the EFP map this increasesmonotonically with C, and no differenceis apparentin the behaviourfor odd and even values of the link capacity C. In addition,for values of v past the bifurcationpoint for the EFP maps (i.e. v > vEFP),the stable solution to the map fvx in (4.1) is a two-cycle. However,we have shown in Section 3.3 that single-linkblocking probabilities are homogeneous in the centre of the tree even past criticality,and do not oscillate. It appears that beyond the criticalregion, approximationsto the single-linkblocking probabilitiesshould be obtainedfrom the unstablefixed point of the EFP approximation,but it is no longer clear how this can be used to approximatethe multicastblocking probabilities.

5. Some generalizations A simple symmetricnetwork such as the one we have studiedabove is, of course,unrealistic. The model could be generalizedin many ways. Two of these we study in greaterdetail below. The first is to allow longer multihopcalls into the system. This is a naturalextension of the model, and essentially only complicates the formulationof normalizingconstants, blocking probabilitiesetc. The second generalizationthat we studybreaks the symmetryof the model by allowing heterogeneousv. The interesthere is in seeing whetherphase transitionsstill occur. Van den Berg and Steif [19] have conjecturedthat if arrivalrates at odd and even nodes on the cubic lattice Zd are not equal, then the model has a unique Gibbs measure, and they give a partialproof of this conjecture.However, we find that on a tree network,phase transitionscan still occur with heterogeneousv. 5.1. Multihop multicast calls In this section, we indicate how to generalize this model to multicasts beyond nearest neighbours on the tree. For simplicity, we consider a model where there are no unicast calls (X = 0), but only multicast calls with the propertythat a multicast call originatingat a node t is broadcastto all its nearest neighbours t' e N(t) and next-nearestneighbours t"/ E := Ut'EV(t) NV(t')\ t. The numberof calls originatingat a node t is denotednt, as usual,,N2(t) and the total numberof types of calls is MT = ITI, the numberof nodes in the tree T. The set of feasible configurationsis then

T nE : for each (t1t2) nt C = IMTr E&(T), < tE.V(tl)U. (t2)

Once more, the stationarydistribution 7r has the truncatedproduct form 1 vnt

ZI teT nt! for n 2Tr,where Z, is the normalizingconstant

tTvnt nE~?T teT As before, we calculate Zz in terms of some partialsums, to be defined below, and then deriverecursion relations for these partialsums. To show how to derivethe recursionrelations for this model, we restrictourselves to the case C = 1 and q = 2. Let to be the centralnode in a sphericaltree, and let t-1, tl and t2 be its neighbours,with t-2 and t-3 being neighboursof t_1

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t-2 t-1 /

(a) (b)

o emptynode * nodewith 1 call o nodeeither empty or full

FIGURE8: Multihopmulticasting on S3 (C = 1, q = 2): (a) augmentedtrees of size 4 and 3 and configurationwith weight AL-1AL; (b) AL-IFL; (c) ALFL-1; (d) ALBL-11v.

(see for example Figure 8). There are only two possible types of configurationsof calls on these sites: either there is no call on any site, or there is only one call originatingfrom exactly one of these six sites. These possibilities give us seven differentconfigurations. We can find a rooted tree Tmfor some m, such that Tmis rooted at to. Without loss of generality,we could assume that tl, t2 E Tm. Now we define

Zm(i-1,io, i2) 1 i1, : nE?m:nta=ia, tETm a=-1,...,2 so that Zm is the weighted sum of all configurationsin the graphcomprising the rootedtree Tm with root to, augmentedby the node t_1. As shown in Figure 8(a), any sphericaltree of radius L with central node to can be decomposed into two overlappingaugmented trees of sizes L and L - 1 with roots to and t_1 respectively. Then Z, can be expressed as a weighted sum of the seven on the six nodes to the configurations t-3, t-2, t-l, to, ti, t2, corresponding empty configurationwith no calls on any of the nodes, andthe six configurationsassociated with a call on one of the nodes. Introducingthe notationAm = Zm(0, 0, 0, 0), Bm = Zm(1, 0, 0, 0), Im = Zm(0, 0, 1, 0) = Zm(0, 0, 0, 1), Am = Zm(0, 1, 0, 0), it is clear that ALAL-1 representsthe weight for the empty configurationshown in Figure 8(a), AL-11L and ALFL-1 representthe weights for the configurationsgiven in Figures8(b) and8(c) respectively.Likewise, ALBL L- 1/

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 78 K. RAMANANETAL. correspondsto the weight of the configurationin Figure8(d), where the dividingfactor v arises due to the fact thatthe occupied node to lies in both augmentedtrees. Arguingsimilarly for the remainingconfigurations, it follows that the normalizationconstant Z, is given by

= Z, AL-1AL + 2(FL-1AL + AL-1FL) + V-1(BL-1AL + AL-1BL) for a sphericaltree of radiusL. It is easy to see that the recursionrelations are:

Am+1 = (Am + 2Fm)2, = Bm+l vAm, Fm+l = AmAm, B2 Am+1 =

The free boundarycondition is imposed by taking A0 = 1, Fo = 0, B0 = A0 = v. As usual, the recursionrelations are homogeneous in Am, Bm, Fm, Am, andwe could instead obtain recursions on the three ratios im = Bm/Am, Ym= Im/Am, Sm = Am/Am as follows:

' (1 + 2ym)2 sm Ym+I - (1 + 2Ym)2'

Sm+l = v(1 + 2Ym)2

Once more, for small v, these recursionrelations lead to a unique fixed point. At large v, there is a four-cycleleading to a phase transition.This correspondsto an orderedstate which breaks homogeneity (i.e. translationalsymmetry). 5.2. Heterogeneous v In this section, we considerthe model with heterogeneousv. We assumethat multicast calls arriveat eitherrate vl or v2 at each node, andthat these arrivalrates alternate across the tree. In particular,we assumethat in the sphericaltree, nodes at a distancem from the nearestterminal node have arrivalrate vl if m is odd and v2 if m is even, and that terminalnodes have arrival rate v2. Thus, each link has multicaststreams of rate vl and of rate v2 offered to it. The expressionsobtained in Section 3 then need to be modifiedin the obvious fashion. The = recursions (3.1) now become ?m+1 = (IVl'X($m) if m + 1 is odd and ?m+1 (IV2,'(m) if m + 1 is even. Let 4* and * be fixed for the recursions o i'2,X and ~v2,' 0vI,X points ~lv,' respectively. (Note that due to Theorem 3.1, they certainly exist and are globally stable for vl = v2 = v sufficiently small.) Then in the subcriticalregion the limiting link blocking probabilityin (3.3) satisfies

1-a• C (0 1 1-k 1l-k 1-k-i 1 1C-- k/k!)[1- /1-k' k a C k C k!)[1 + 1 k-i 1 Ec=Olk ( k/ - - ) -lk (

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* We note that this is symmetricin and?2*, thus implying that even with these heterogeneous multicastarrival rates, the link blocking probabilitiesare homogeneous. The limiting multicast call acceptanceprobabilities in the subcriticalregion are now given by - -i=O i!)[ + E = (vI// -i/j=O 'k=l 2(k))]q+l 1 - il (.J/j!)(1 + k=1 ( /i!)[-j=O-i (XJ/j!)(1!) 2C-i-j (k))]q+l /=0=o1/_i!j)[ •EZ= for odd nodes, and - C (V/i!)[C-l-iJ (/j!)(1 + 1 - 82 = 1C--i-' *(k))]q+l 1!)[E + *,(k))]q+l S=o(/i CC-i-j(J/j!)(1 EZ(7kk=l ))] for even nodes. When C = 1, these equationsare simple. We use c1 and (D2 to denote particularly )Vli,O and Q>v2,0respectively, and define QD:= (1 o (2. Then note that by the definition given above, 4j*is a fixed point of Q. The recursionsreduce to

Vl v2 = and 1(() (1 + 1)+ a)q 4I2(4•) (1 + +_)q-+ which togetherthen yield V1 (1 + v2/(1 + )q4) The limiting acceptanceprobabilities are then given by 1 1-a= 1 + ? + M*(1) + (1)' 1 1 - = ll 1 1 - f12 = I (1))q+1 V292 + (+1 + l* )+1' .+ One question of interesthere is whetherthe phase transition(i.e. the existence of multiple limit points for a sequence of iterates of D) that we saw with homogeneous v can also occur when vl :A v2. The following theoremshows that indeed it can.

Theorem 5.1. For C = 1 there exists an open set 0 such that, for (vl, v2) E 0, there exists a iterates the (D = that has limit sequence of of map Co1 '0 0o2,0o multiple points.

Proof Assume that vl, v2 > 0. Note first that D (.) has first and second derivatives

d q2vlv12 1 d( (1 + v2/(1 + (1 + )q+1 ' 4)q)q+l and d2 _-q2(q + 1)vPv2 v2(q - 1) 4 d =2 (1 + + v2/(1 + 4)q-1)q+2 (1 + ')q

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(a) (b) 1.00 - - - 1.00 0.95 0.95 0.90 L0.90 0.85 -- 0.85 0.80 -.- 0.80 0.75 0.75 0 2 4 6 8 0 2 4 6 8 V2 V2

812

FIGURE9: Blocking probabilitieswith heterogeneousv (k = 0, C = 1, q = 2, vl = 6): (a) multicast (node)blocking probability, (b) unicast(link) blocking probability.

Thus ((0) = vi/(l + > 0, (4) = and V'(4) > 0 for 4 0. When v2 v2)q lim,,oo vm > < 1/(q - 1), V"(4) < 0 for 4 > 0, and so there is a unique fixed point for the map (. When > - > and has a of inflection at the Sv2 1/(q 1), however, ("(0) 0, Q(.) single point point = (v2(q - 1))l/q - 1, which is increasing in v2. (Note that the point of inflection 4 is such that ( is convex for < and concave for4 > 4.) At the point of inflection 4(D) = vi (1 - l/q)q, which does not depend on v2, and 1'(4) = vi(q - If V'(4) < 1, then a 1)q-1/q/(v/4qqq-1). single fixed point is guaranteed,since the derivativeis greatestat the point of inflection4. If 4 is also a fixed point of ((-), then simple algebra shows that 4 = [v2(q - 1)]1/q - - = - = - - 1 = vi(q 1)/q, and V'(4) q/(1 + qq/(vI(q 1)q)) q q/[v2(q 1)]1/q. Moreover, - - V'(?) > 1 if and only if > qq/((q 1)q+m) (or, equivalently, v2 > qq/((q 1)q+1)), and in this case 4 is clearlyan unstablevm fixed point. The propertiesof ( describedabove then imply that there also exist two stable fixed points 4*:< < . Thus, given any vi > qq/(q - 1)q+1, by continuity it is clear there exists an open interval (al, a2) around v2 = (q - 1)/q + 1)/(q - 1) (vm (i.e. the value at which 4 is a fixed point) such thatthere are two stablepoints *: <4 < (~ and an unstablefixed point e (, ). In additio, due to the form of the function 4, it is clear that for v2 < al the uniquefixed point <*< $, and for v2 > a2, the uniquefixed point 4* > 4.

We illustratethe above theoremwith an example. Figure 9(a) gives the limiting blocking probabilitiesfor multicastcalls at odd nodes and even nodes (f1i and 82 respectively)for fixed = 6.0, and v2 varying from 0.0 to 8.0, with C = 1 and = 0. Figure 9(b) gives the vm X. link blocking probabilityfor the same range of parameters.Again, all of these were obtained numericallyunder the assumptionof free boundaryconditions. We note that, as v2 increases, the blocking probabilityfor multicastcalls at odd nodes, ,1, increases. However,the blocking probabilityfor multicast calls at even nodes, f2, is very close to 1.0 when v2 is 0 and then decreases initially as v2 increases. The explanationfor this is that increasingv2 from 0 breaks the packing symmetryfor the odd calls, thus giving a slightly lower blocking probabilityfor the even calls. We note thatthis initial decreasedoes not occur for small vl (e.g. less than0.1). There is evidence of bistable behaviour(i.e. multiple solutions for 01,92 and a) over a range of values of v2, includingthe point v2 = 6.0.

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6. Conclusions

In this paper we developed a frameworkin which to analyse the effect of multicast calls in large unicast tree networks. Although most of our analysis is carriedout for the case of single-hop multicast calls in a regulartree networkwith uniform arrivalrates throughoutthe network,we showed in Section 5 that the qualitativephenomena observed persist even in the case of multihopcalls and asymmetricalarrival rates, and are thus of relevanceto more general networks.

6.1. Phase transitions, fairness and capacity Ourmain findingis that, at high enough multicastcall arrivalrates, the stationarycall distribution at the core of large networksloses its spatialhomogeneity, even though the call arrival patternis homogeneous. For any finite network,the distributionof calls near the boundaryof the networkwill clearly be influencedby the boundaryconditions (i.e. the distributionof calls at the edge of the network). However, as our results show, for small values of the multicast arrivalrate, v, the stationarydistribution at nodes in the core of a large networkis insensitive to the boundaryconditions. In addition,for the core of sufficientlylarge networks,this unique stationarydistribution approaches homogeneity, in the sense that the marginaldistributions of calls at differentnodes in the core network approacha common distribution. Higher values of v, however, lead to a phase transitionor symmetrybreaking in the sense that the marginal call distributionat a core node now depends on whether the distance of the node from the boundaryis odd or even. (Thus, for high v, the particulardistribution realized in the core of the networkdepends on the networkboundary conditions.) This leads to unfairnessin the sense that even in a symmetricnetwork with uniformloading, calls arrivingat even and odd nodes in the core may experience differentblocking probabilities. This phenomenonis particularly relevantin networks with a higher degree of connectivity (i.e. largerq), which manifest this phase transitionat lower arrivalrates, possibly well below the capacity of the network. The presence of uncorrelatedunicast calls in the networkseems to somewhatmitigate this effect as it raises the multicastarrival rate at which the phase transitiontakes place (at least for C = 1). Thus, if an operatorwould like to avoid 'unfairness',it may be desirableto work away from the region of phase transition. 6.2. Networks with tree-like structure v. lattices Markovrandom fields on a Cayley treebehave quite differently from those definedon regular lattices, andthus it is instructiveto contrastthe two structures.A boundedsquare lattice Zn x Zn has n2 points, of which 4n -4 lie on the boundary.A sphericaltree SLwith coordinationnumber q has qL + qL-l boundary nodes and (qL+l + qL - 2)/(q - 1) nodes in total. Note that the ratio of the numberof boundarynodes to the total numberof nodes, in the limit as L -+ o, converges to zero for the lattice, but stays finite in the case of the tree. In other words, in the thermodynamiclimit, regularlattices have negligibly few boundarynodes comparedto the total numberof nodes, whereasthe Cayley tree has a nonzerofraction of nodes on the boundary.As a result,the Cayley tree system is much more susceptibleto boundaryconditions than a regular lattice. To make this more concrete, we consider the particularquestion of whether there exist symmetrybroken (i.e. non-translationallyinvariant) stationary distributions when C = 2 and X = 0. The results of Section 3.4 indicate that the answer is affirmativefor Cayley trees, whereas the results of [11, p. 5] seem to suggest that the answeris negativefor lattice models. We argue heuristicallyto provide some intuition as to why this may be the case. For both

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 82 K. RAMANANETAL. graphs, we define the odd lattice to be the collection of Vi nodes that lie an odd distance away from the terminalnodes and the even lattice to be the collection of V2nodes that lie an even distance from the terminalnodes. Then for large v, it is reasonableto expect that the stationarydistribution would be dominatedby those configurationsthat sustainthe largesttotal number of calls in the network, subject to the capacity constraint. Consider two candidate configurations:(i) homogeneous,with one call per site, and (ii) symmetrybroken, two calls on one sublattice,and no calls on the other. Then the homogeneous configurationhas a weight VI v +V2, while the symmetrybroken one (say with calls supportedonly on the even lattice)has a weight (I v2)V2.Let V = V1+ V2be the total numberof nodes in the network. On the regular lattice, V1 V2 mV, and thus both configurationshave weights with the same power of v. However,for all values of v, the weight of the homogeneousconfiguration is greaterbecause it has a largerprefactor. The same configurationson the sphericalCayley tree have very different weights. For a tree with q = 3 (which is equal to the coordinationnumber of a regularsquare 3 lattice), since V2 includes the boundaries,it is easy to see that V2r 3 VI V. Hence, for small v the homogeneous configurationhas largerweight, while for v large enough such that (pI2)3/4 > v, the symmetrybroken configuration has greaterweight. Consequently,we would expect the Cayley tree model (for C = 2) to show a phase transitionfor large enough v. Anotherdifference between lattice and tree models was pointed out in Section 5.2, where it was shown that phase transitionscan occur on the tree even when the arrivalrates on the odd and even lattices differ. In contrast,this is not expected to happen in the case of the regular squarelattice [19]. In general,when we analysethe recursionrelations for the Cayley tree model, we often find fixed points of fixed cycles with particularbasins of attraction(as illustrated,for example, in Figure4). These basins of attractioncorrespond to initial conditionsfor the recursionrelations, which in turncorrespond to specific boundaryconditions. The fact that we findmore phases in the Cayley tree model thanthe regularlattice is not surprising.It is due to the stronginfluence of the boundary. In physics models of spin systems, this is often an artefactof the model. However, in actual telecommunicationnetworks the connectivity structureis often tree-like, especially in the outlying or 'access' regions of the networks,and this may significantlyaffect the performancein the 'core' network. Hence a stronginfluence of the boundaryon the core may be a welcome element in a networkmodel. 6.3. Dependence on parity of network capacity C An interestingby-product of our analysis is the dependenceof the natureand value of the phase transitionon the parityof the capacityC. The phasetransitions seem to be fundamentally differentfor odd and even C-with odd C showing a second ordertransition and lower phase transitionvalues, and even C showing a firstorder transition that occurs at highervalues. The resultsof [11, p. 5] suggest thatin latticemodels too the natureof phase transitionsdepends on the parityof the capacity. However,in thatcase, for even C thereis no phasetransition, as there is a unique Gibbs distributionfor all values of v, while for odd C there is a phase transition leading to multiple Gibbs distributionsfor large enough arrivalrates v. 6.4. Problems with traditional approximations for multicast networks Ouranalysis indicates that traditional techniques such as the EFP approximationfor estimat- ing link blockingprobabilities may not be very effective in networkssupporting multicast calls beyond the criticalpoint of the EFP map. This is probablydue to the fact that as we approach phase transition,the correlationsintroduced by multicastingseem to seriously underminethe independenceassumption underpinning such approximations.It would thus be of interestto

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 models of multicastingin tree networks 83 develop simple and effective approximationmethods that can be effectively applied to the multicast setting near and beyond the phase transition. Zachary and Ziedins [22] outline a generalframework for refiningthe EFP approximation,which could be appliedin the multicast setting for more generalnetworks provided no controls, such as trunkreservation, are in force. 6.5. Implications for transient behaviour in finite networks We discussed earlierhow phase transitionsgive rise to qualitativedifferences in stationary measures in the core of large networks for different regions of the parameterspace. This phenomenonof differentlimiting quantitiesfor differentparameter values often also appliesto transientbehaviour. Analysis of dynamicphenomena, even for such a simple model, is rather hard. However,the existence of a phase transitionsuggests that finite networkswould exhibit transientbehaviour that spends large amounts of time in these different phases and decays slowly to the equilibriumdistribution. This kind of behaviouris reasonablywell understood for lattice systems, and thus it would be worthwhile to carry out a similar analysis for tree networks. 6.6. Control of the network and future directions It would be interestingto study the effect of controls akin to trunkreservation on the phase transitioneffects-in the pasttrunk reservation has been used to controlbistable behaviour (see, for instance, [9], [13]), and we believe that it or a relatedcontrol could be used here to similar effect. We would also like to consider more realistic settings in which the arrivalrate at each node is not uniform(or picked from a set of two), but is insteadpicked from a distribution.

Acknowledgements Ilze Ziedins thanks Bell Labs, Lucent Technologies, for its generous hospitality while preparingmuch of this paper.Thanks are also due to Stan Zacharyfor his comments.

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