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ON THE FLUID LIMITS OF A RESOURCE SHARING ALGORITHM WITH LOGARITHMIC WEIGHTS 2 1 PHILIPPE ROBERTANDAMANDINEVE´BER 0 2 Abstract. The paper investigates the properties of a class of resource allo- v cation algorithms for communication networks: if anode of this network has o xrequeststotransmit,thenitreceivesafractionofthecapacityproportional N tolog(1+x),thelogarithmofitscurrentload. Afluidscalinganalysisofsuch anetwork ispresented. Itisshownthat theinteraction ofseveral timescales 6 playsanimportantroleintheevolutionofsuchasystem,inparticularitsco- 2 ordinates mayliveonverydifferenttimeandspacescales. Asaconsequence, theassociatedstochasticprocessesturnouttohaveunusualscalingbehaviors ] R whichgiveaninterestingfairnesspropertytothisclassofalgorithms. Aheavy traffic limittheorem for the invariant distribution is also proved. Finally, we P present a generalization to the resource sharing algorithm for which the log . h functionisreplacedbyanincreasingfunction. t a m [ Contents 1 1. Introduction 1 v 2. Presentation of the Results 6 8 6 3. The Stochastic Model 9 9 4. The Initial Phase 10 5 5. A Local Equilibrium 13 . 1 6. The Fluid Time Scale 20 1 7. Heavy Traffic Regime 26 2 8. General Functions for Resource Sharing 28 1 9. The Network with J Nodes 32 : v References 34 i X r 1. Introduction a TheresourceallocationproblemconsideredinthispaperinvolvesJ nodeswhich have access to a common shared resource, for example a communication channel or a processing unit. The latter is assumed to have a fixed capacity, say 1. The resource is shared among nodes in the following way: for 1 j J, if node j has ≤ ≤ n requests pending, it receives the instantaneous fraction of capacity j f(n ) j (1) f(n )+f(n )+ +f(n ) 1 2 J ··· from the resource. The algorithm is thus defined by the function x f(x). There 7→ are several situations where the capacity is allocated in this way. Date:November27,2012. 1991 Mathematics Subject Classification. Primary:60K25, 60K30, 60F05; Secondary:68M20, 90B22. 1 2 PHILIPPEROBERTANDAMANDINEVE´BER 1.1. Saturated Node of the Internet. In this context, the nodes correspond to TCP flows with different sources and destinations. The resource here is the processing time of a fixed router on the path of these flows. The packets of a flow are queued in the buffer of the router until they are routed to the next stage of their path. Congestionis simply the situation when the buffer is full and therefore incomingpacketsarelost. BecauseoftheTCPprotocol,agivenflowwillincreaseor decreasetherateatwhichitsendsthepackets,dependingonthelevelofcongestion of the routers on its path. There are several ways to represent this phenomenon. It should be kept in mind that the following descriptions are mathematical models of the way TCP is thought to allocate the bandwidth, not of the TCP algorithm itself. See Massoulié and Roberts [18]. a) Processor-Sharing disciplines. A popular, simplified, stochastic model of this situation consists in consid- ering that the router allocates its processing power to each flow according to a slight generalization of the allocation policy given by Relation (1) with a function f depending on the node j and of the form w n, where 1/w can j j be the round trip time between the source and the destination. This allocation algorithmcorresponds to the Discriminatory Processor-Sharing Policy. Node j has an instantaneous fraction of capacity given by w n j j (2) . w n +w n + +w n 1 1 2 2 J J ··· See Altman et al. [2] and references therein for a survey. When all the w ’s are 1, we obtain the classical processor-sharing policy: node j receives j the fraction of capacity n /(n + +n ), and the bandwidth is equally j 1 J ··· divided among the current requests. In the last ten years, different classes ofstochasticmodels ofprocessor-sharingpolicieshavebeenextensivelyused to describe the congestion in IP networks. b) Alpha-fair disciplines. These policies have also been introduced to describe the allocation of band- widthinIPnetworks(seeMoandWalrand[20]),intermsofanoptimization problem (cf. Kelly et al. [13]). In our context, a related policy would correspond to the case f(n) = nα, n 0, so that a non-empty node j has an ≥ instantaneous fraction of capacity given by nα j (3) . nα+nα+ +nα 1 2 ··· J The case α=1 is the processor-sharingdiscipline presented above. In the wireless section below, the situation is quite different since the bandwidth allocation algorithm is defined explicitly by relations similar to (1). 1.2. Wireless Networks. This is again a simplified, but meaningful stochastic model of bandwidth allocation, this time in wireless networks. The resource here is a radio channel in a region where there are J stations/mobiles. At a given time, because of interferences, only one station can transmit successfully in this region. Astationwithn messageswaitingfortransmissioncandetectifthereisacommu- j nication going on or not. If not, a classical backoff mechanism is used: the station starts transmitting after an exponentially distributed amount of time with parameter f(n ). If another station starts a transmission before that time, the attempt j BANDWIDTH SHARING ALGORITHM 3 oftransmissioniscanceled. Consequently,ifinitiallythereisnotransmission,node j willaccessthe channelsuccessfully withprobabilityf(n )/(f(n )+ +f(n )). j 1 J ··· SeeAbramson[1]andMetcalfandBoggs[19]forhistoricalreferences,andTassiulas and Ephremides [27]. If a small quantum δ is transmitted at each access, it is not difficult to see that, provided that backoff times are small (which is the case if one ofthecomponentsislarge),asδ goesto0theeffectivecapacityallocatedtostation j is indeed given by f(n ) j (4) . f(n )+ +f(n ) 1 J ··· Thisistheanalogueoftheapproximationoftheroundrobinpolicybytheprocessor- sharing discipline. Fair Access to Resource: The Choice of the function x f(x). 7→ The function f should clearly be increasing, so that the fraction of the capacity allocated may grow with the number of requests. This is the case if f(x) = xα whichcorrespondsto the Alpha-fair disciplines alreadymentioned. However,these policies may have a serious drawback. Indeed, if a station j has a large number of requests pending while the other stations are lightly loaded, the latter will receive a negligible fraction of the bandwidth. The station j will therefore capture the channelforitsownbenefit,untiltheinstantwhensomeoftheotherstationsreacha comparable level of congestion. This is a highly undesirable property for a network where fairness issues (for nodes, not requests) are of primary importance. See Bonald and Massoulié [5]. Apossiblewayofsolvingthisproblemistoconsiderincreasingfunctionsf which grow slowly to infinity like, for example, the concave function x log(1+x), or 7→ x loglog(e+x). Inthisway,onecanexpecttoreducesignificantlytheimpactof 7→ saturatednodes evenifthey stillreceiveasizablefractionofthe availablecapacity. Related algorithms have been considered in the context of wireless networks, see Shah and Wischik [26] and references therein. Bouman et al. [6] and Ghaderi et al. [12] investigate the impact of the “agressivity” of the function f on the stability andonthe delays for severalwireless networkarchitectureswith a related bandwidth allocation scheme. Inthispaper,wemainlyinvestigatethecasef(x)=log(1+x). Thegeneralcase is sketched in Section 8. The instantaneous fraction of capacity of the jth node is therefore given by log(1+n ) j (5) . log(1+n )+log(1+n )+ +log(1+n ) 1 2 J ··· Thelogfunctionmoderatestherateatwhichasaturatedstationtriestoaccessthe resource, which is a desirable property in an heterogeneous network where traffics mayhaveverydifferentcharacteristics. Inthecontextofwirelessnetworks,arelated algorithm was used to show that an optimal stability region is possible in a quite generalnetwork. The growth properties of the log function play an important role in the proof of the result. Basically, the log of the states of the saturated stations beingquitestableonsomelargetimeintervals,theschedule(thesetofstationsthat can transmit at some time) quickly reaches some equilibrium and stays around it. ALyapounovfunctionargumentcanthenbeusedtoproveergodicity(seeShahand Shin [25]). Up to now, apart from these stability results, little is known about the 4 PHILIPPEROBERTANDAMANDINEVE´BER quantitative and qualitative properties of these algorithms. As we shall see below, themathematicalanalysisofthis classofalgorithmspresentssomechallengingand unusual problems (see also Wischik [28]). We first achieve a fluid limit scaling analysis, which gives a very precise description of the qualitative behavior of these algorithms. Additionally, we derive a heavy traffic limit theorem result for the invariant distribution of the associated Markov process. Before presenting our main results, we briefly recall the main definitions of the fluid limit scaling. The interestedreaderwillfind anextendedpresentationinBramson[8]orinChapter8 of Robert [23]. Throughoutthe paper itwill be assumedthat, for every1 j J, the requests ≤ ≤ arriving at the jth node form a Poisson process with rate λ >0. Each request at j nodej leavesthenetworkwhenithasreceivedanexponentiallydistributedamount oftime with parameterµ fromthe commonresource. The averageloadof the jth j node is denoted by ρ =λ /µ . j j j Fluid Limits. The fluid limit scaling of a stochastic process(Z(t)) in RJ consists in speeding up time and space in proportion to the norm of its initial state: Z(Nt) Z (t)= , with N = Z(0) . N N k k A possible limit in distribution of the sequence of processes (Z (t)) is called as a N fluid limit of the process (Z(t)). Hence, in some sense fluid limits give a first order description of (Z(t)). This is a convenient tool to investigate multi-dimensional processes for which general results are scarce. In the context of Markov processes, there is an additional interest since the ergodicity of the process can be connected to the fact that fluid limits (whose initial states lie on the unit sphere) return to the origin. See Rybko and Stolyar [24] and Dai [9]. Note however that the fluid limit scaling is wellsuited for processesthat behave locally like randomwalks. For other processes, different scalings may have to be considered. In the case of the generalized processor-sharing policy defined by Relation (2), for every 1 j J and t 0, let X (t) denote the number of jobs waiting at the j ≤ ≤ ≥ jth node. The evolution of this process can be represented as t w X (s) j j X (t)=X (0)+M (t)+λ t µ ds, j j j j j − w X (s)+w X (s)+ +w X (s) Z0 1 1 2 2 ··· J J where (M (t)) is a martingale. The scaled process with N = X(0) is thus given j k k by N N Mj(Nt) (6) X (t)=X (0)+ j j N t w XN(s) +λ t µ j j ds. j − j N N N Z0 w1X1 (s)+w2X2 (s)+···+wJXJ (s) A standard tightness criterion and the fact that the martingale ((M (Nt)/N),1 j J) j ≤ ≤ convergesindistribution to 0 imply that any fluid limit ((x (t)),1 j J) should j ≤ ≤ satisfy the ordinary differential equations dx w x (t) j j j (7) (t)=λ µ , 1 j J, j j dt − w x (t)+w x (t)+ +w x (t) ≤ ≤ 1 1 2 2 J J ··· BANDWIDTH SHARING ALGORITHM 5 on the interval [0,t ], provided that the vector (x (t),1 j J) does not hit 0 j ≤ ≤ 0 before t . See Ben Tahar and Jean-Marie [3], Ramanan and Reiman [22], and 0 references therein. Similarly, for alpha-fair disciplines the corresponding fluid model (x (t)) is the j solution to the ODE dx x (t)α j j (8) (t)=λ µ , 1 j J. dt j − jx (t)α+x (t)α+ +x (t)α ≤ ≤ 1 2 J ··· Observe that for these two choices of function f the scaled process satisfies an autonomous ODE, like (8), with a stochastic noise component that vanishes as N gets large. Secondly, a remarkable feature of these convergence results is that all coordinatesof the scaledprocess are of order N. That is, as long as x(t) is not the vector 0, one has x (t)>0 for all 1 j J. j ≤ ≤ Problem of Fluid Limits for Algorithms with Logarithmic Weights. Let (L (t),1 j J) denote the Markov process associated to the policy with loga- j ≤ ≤ rithmic weights, i.e. associatedto Relation(5). Due to the log function, the scaled process (L (t)) does not have an autonomous representation analogous to (6). In j fact, it is easy to see that the corresponding stochastic equations involve both L (Nt) and log(1+L (Nt)), two quantities which evolve on very different scales. j j For this reason, there is no way of guessing a system of plausible “fluid equations” correspondingtoSystem(7)[resp. to(8)]fordiscriminatoryprocessor-sharingpolicy [resp. Alpha-fair policies]. See Wischik [28] and Ghaderi et al. [12]. Note that the question of stability of the system is not an issue here. Indeed, because of the work conserving property of these policies, a necessary and sufficient condition for the ergodicity of (L (t),1 j J) is simply given by j ≤ ≤ λ j ρ + +ρ <1, with ρ = , 1 j J. 1 J j ··· µ ≤ ≤ j Tothebestofourknowledge,thereisnoexplicitexpressionknownfortheinvariant distribution. The fluid scaling gives a first order description of the behavior of this policy. As we shall see, an interesting convergence result for the invariant distribution just below saturation can also be derived from these results. Outlineofthe Paper. Section2presentsthemainresultsofthepaper. Section3 introduces the notation and the stochastic differential equations associated to the Markov process (L (t),1 j J). Sections 4, 5 and 6 are respectively devoted j to the scaling properties o≤f the≤time scales t Nt, t Nα∗1logNt and t Nt. 7→ 7→ 7→ Thereweprovideaprecisedescriptionoftheevolutionofthenetwork,togetherwith some estimates of hitting times. The key results on the fluid limits are presented in Section 6. In Section 7, we prove a heavy traffic limit theorem for the invariant distribution. The case of a two node network with a general f is discussed in Section8. The correspondingtime scalesareidentified inthis case. Section9 gives a brief sketch of the case of a network of J nodes. Acknowledgments. The first authorwould like to thank DamonWischik, whose presentationatthe ICMS workshopinEdinburghin2010is one ofthe motivations at the origin of this work. 6 PHILIPPEROBERTANDAMANDINEVE´BER 2. Presentation of the Results The General Picture. The main result of the paper is that the states of the nodes may live on different space and time scales depending on the set of parameters (λ ,µ ). Indeed, there j j exists a set of real numbers (α ,1 j J) with values in (0,1] such that, if j ≤ ≤ αj < 1, then on the fluid scale Lj is of the order of L(0) αj and the process k k t Lj( L(0) αjlog L(0) t) has a local equilibrium which can be expressed in 7→ k k k k terms of an Ornstein-Uhlenbeck process. In the next paragraph we give a more precise description of this phenomenon in the case J = 2. An illustration of the different scales is given in Figure 1, where the y-axis is on a log /logN scale. · The case of Two Nodes. Even in the case where there are only two classes of customers, J = 2, giving an asymptotic picture of this queueing system is already a challenging problem. To concentrate on the interesting case, let us assume that the parameters satisfy the following conditions: ρ < 1/2 and ρ > 1/2 and the initial state of the process is 1 2 (LN(0),LN(0))=(0,N). 1 2 a) The time scale t Nt. → A convergence result, Proposition 2, shows the equivalence in distribution t ρ LN(Nt),,0<t<α∗ λ µ Nt,0<t<α∗ for α∗ d=ef. 1 . 1 1 ∼ 1− 1t+1 1 1 1 ρ (cid:18)(cid:18) (cid:19) (cid:19) − 1 (cid:0) (cid:1) The process LN stays at N on this time scale. The condition ρ < 1/2 2 1 implies that α∗ < 1. Note that the pre-factor of Nt vanishes at t = α∗. 1 1 For this reason, this convergence result does not prove that values of the order of Nα∗1 can be reached. In fact, an extra (logN) factor is required and Proposition 3 shows that if δ <1, the average hitting time of the value o⌊δthNeαr∗1h⌋abnyd,(rLeN1ac(ht)in)gisthbeouvnadlueed bNyαK∗1 1Nisαs∗1liglohgtlNy lfoonrgseor:mteheKa1v>era0g.eOhintttinhge time of Nα∗1 is bounded by C⌊Nα∗1(⌋logN)2loglog(N) for some C > 0, see ⌊ ⌋ Relation (28) in Section 5. b) The time scale t Nα∗1(logN)t. We now assume→that LN1 (0) = ⌊Nα∗1⌋ and L2(0) = N. Theorem 1 then proves the convergence of the processes LN Nα∗1(logN)t Nα∗1 (9) 1 − ,t 0 (cid:18) (cid:0) Nα∗1logN(cid:1) ≥ (cid:19) to an Ornstein-Uhlenbeckpprocess. In other words, on this time scale LN is 1 stabilized around the value Nα∗1. Again, the process LN2 stays at N on this time scale. c) The fluid time scale t Nt. → Thistimescaleisnatural,sincethenetworkemptiesonit. Theorem3shows the convergence, (10) N→lim+∞ LN1N(αN∗1t),LN2N(Nt) = γ(t)α∗1,γ(t) (cid:18) (cid:19) (cid:0) (cid:1) BANDWIDTH SHARING ALGORITHM 7 for the convergence in distribution of processes, with γ(t)=(1+(λ µ (1 ρ ))t)+. 2 2 1 − − Consequently, as long as the fluid limit of (L (t)) is not 0, the process LN 2 1 lives on the space scale Nα∗1. t Nt 7→ t Nα∗1(logN)t 7→ N t Nt 7→ L 2 Nα∗1 L 1 Figure 1. A first order picture of the network with ρ +ρ < 1, 1 2 ρ <1/2 and (L (0),L (0))=(0,N). 1 1 2 Properties of algorithms with logarithmic weights. The bandwidth allocation algorithm with logarithmic weights exhibits an interest- ingproperty. Forexample,whenJ =2andthe systemis overloaded(ρ +ρ >1), 1 2 if ρ <1/2 the size of the queue of class 1 requests growsat a rate proportionalto 1 tα∗1, with α∗ < 1. This implies that queue 1 is stable at the fluid level, i.e. that 1 L (t)/t goes to 0 in distribution as t becomes large. Hence, without any priority 1 mechanism among nodes, if a node has a light load, ρ < 1/2, then most of its 1 messages will be transmitted with success even in the case where the system is globallysaturated. Recallthatinthe transientcaseofthe processor-sharingpolicy or even with the Alpha-fair disciplines, this is not true at all: the states of the nodes diverge to infinity at the same speed, linearly in time. This is an interesting featurefromthe pointofviewoffairnessissues. Itcanbe shownthatananalogous property is valid for the network with J nodes, see Section 9. Thisstudyalsosuggeststhat,forgeneralnetworksusingsuchalgorithms,several spaceandtime scalesshouldplayanimportantroleinthe dynamicsofthe system. Forthetwonodenetworkdescribedabove,whentheinitialstateis(0,N)weprove that the fluid limit is given by (0,1+(λ µ (1 ρ ))t)+ . 2 2 1 − − Thisshowsthatnode2re(cid:16)ceivesthe capacity1 ρ ,wh(cid:17)ichisanotherwayofsaying 1 − that node 1 is stable at the fluid level. The simplicity of this expression somewhat hides the complexity of the situation, since the quantity α∗ does not show up. 1 Yet, as we have seen α∗ has a crucial impact in the transient case, as well as in 1 the equilibrium situation: if the pair of variables (L ,L ) has the equilibrium 1,ρ 2,ρ 8 PHILIPPEROBERTANDAMANDINEVE´BER distributionofthe Markovprocess(L (t),L (t))andifρ <1/2isfixed,the heavy 1 2 1 traffic limit-Theorem 4 shows the convergence in distribution lim (1 ρ1 ρ2)α∗1L1,ρ,(1 ρ1 ρ2)L2,ρ =(Xα∗1,X), ρ2ր1−ρ1 − − − − (cid:16) (cid:17) whereX isanexponentialrandomvariable. Henceatequilibriumandinthe heavy α∗ traffic regime, the relation L L 1 also holds. 1 ∼ 2 Itis unlikelythatastandardfluidanalysis,i.e. derivingdirectlysomeequations similar to Relation (7) for example, can be done to investigate the qualitative behaviorof more complex networks. This is where the considerationof the various time scales is useful. It gives a tool to explain, via a dynamic picture, the multiple orders of magnitude of the state variables at equilibrium. An Interaction of Time Scales. Thereis anunconventionalpropertyforthe fluidscalingofa queueingsystem. For most of the queueing networks investigated up to now, the classic general scheme for the fluid scaling of the associated Markov process (X(t)) is as follows: there is a subset of the coordinates whose values are of the order of N = X(0) and the k k other coordinatesform anergodic Markovprocess whose invariantdistribution de- terminestheevolutionofthelargecoordinatesonthefluidscale. SeeMalyshev[17] and Bramson [8] for some examples. Here the situation is different. If the initial state is (0,N) and if ρ < 1/2, 1 then the large coordinate L is of the order of N but L is an order of magnitude 2 1 smaller, Nα∗1 with 0 < α∗ < 1. There is indeed an underlying ergodic Markov 1 process but at the second order, namely an Ornstein-Uhlenbeck process scaled by a factor Nα∗1logN. See Relation (9). The associated stochastic model exhibits a stochastic averaging principle at the p origin of the second expansion in Relation (10). The key technical result of the paper,Theorem2,statesthatwhenρ <1/2,onthe fluidtime scalet Nt,LN is uniformly of the order of (LN2 )α∗1 on a1ny finite time interval with high7→probabi1lity. Recall that a) LwIfN1eLhN2caa(vn0e)bL=eN2rNe∼praeLnsdN2en(Lt0eN1)d(a0bn)yd=aLnNN1Oαir∗1sn,sottfheietnhn-eUoonhrldetnhebreeotcifkm(peLrN2socc(a0els)es)αta∗1r7→.ouANnddαd∗1Nit(ilαoo∗1ng,aNlsle)yet, point b) in the previous paragraph. b) Onthefluidtimescale,LN(Nt)isoftheorderofγ(t)LN(0)withγ(t)defined 2 2 above by Relation (10). The problem lies in proving that on the fluid time scale LN adapts sufficiently 1 quicklytopreservetherelationLN1 ∼(LN2 )α∗1. Acentrallimitresult,Proposition5, suggeststhatthisisnotthecaseonthetimescaleoftheOrnstein-Uhlenbeckprocess, at least for a second order description. On the other hand, on the fluid time scale Theorem 2 shows that this separation of time scales holds. Its proof uses several estimates related to average hitting times of reflected random walks and some coupling arguments. One of the problems encountered is that the potential natural stochastic fluctuations of the fluid time scale, of the order of √N, can be largecomparedtoNα∗1 (ifα∗1 <1/2forinstance). Inparticular,standardstochastic calculuscannotbeusedassuchtoprovetheresult. Itturnsoutthatthepotentially largefluctuations are reducedbythe strongergodicitypropertiesofthe underlying Ornstein-Uhlenbeckprocess. Thus,itdoesnotseemthattheclassicaltechniquesto BANDWIDTH SHARING ALGORITHM 9 prove stochastic averaging results can be used here. See Khasminskii [14], Freidlin andWentzell [11] andPapanicolauet al.[21] for a generalpresentationofmethods to prove stochastic averagingprinciples. 3. The Stochastic Model In this section, we introduce the main stochastic processes and some notation. If h is a nonnegative Borelian function on R , we let denote a Poisson process + h with rate x h(x) on R . This process can be dNefined as follows. If is a + homogeneous7→Poisson point process on R2 with rate 1 and f is some nonnePgative + Borelian function on R , then (f) is defined by + h N f(u) (du)= f(u) ([0,h(u)] du). h(u) N R2 P × Z Z + Forξ 0, denotes the Poissonprocesswith rate ξ on R , i.e. correspondingto ξ + ≥ N the constant function equal to ξ. In addition, for any 0 a b, ([a,b]) stands ξ ≤ ≤ N for the number of points of in the interval [a,b]. Throughout the paper, the ξ N various Poisson processes used will be assumed to be independent. We consider two classes of customers. The arrival process of class j customers is a Poisson process with rate λ , the distribution of the duration of the required j service is exponential with rate µ , and ρ denotes the ratio λ /µ . Each class of j j j j customershasa dedicatedqueue andthere isa singleserverworkingatunitspeed. If the state of the system is (x ,x ) N2, where x is the number of jobs in queue 1 2 j ∈ j, then customers of class j receive the fraction of service log(1+x ) def. i (11) W (x ,x ) = , i=1,2, i 1 2 log(1+x )+log(1+x ) 1 2 from the server, with the convention that 0/0 is 0. The process of the number of jobs in queue j 1,2 is denoted by (L (t)). Since we are only interested in j ∈ { } the total number of customers of each class, there is no need to specify the service disciplineforeachqueue. ItcanbeProcessor-SharingorFIFO(FirstInFirstOut), for example. Stochastic Differential Equation. The stochastic process (L (t),L (t)) can be 1 2 expressed as the solution to the following stochastic differential equation (SDE): (12) dL (t)= (dt) (dt), i=1,2, i Nλi −NµiWi(L1(t−),L2(t−)) where L (t ) denotes the left limit of L at t and W is the function defined by i i i − Relation (11). A Saturated System. For N N, λ, µ > 0, it will be convenient to introduce ∈ a one-dimensional process (X (t)) describing the evolution of the number of cus- N tomersinagivenqueue whenthe numberofjobsinthe otherqueueis “large”,i.e., of the order of N. The process (X (t)) is thus defined as the solution to the SDE N (13) dX (t)= (dt) (dt). N Nλ −NµW1(x,N−1) From a Markov process point of view, (X (t)) is simply a birth and death process N on N whose Q-matrix (Q(x,y)) is defined by q(x,x+1) =λ log(1+x) q(x,x 1) =µ , x>0.  − log(1+x)+logN)  10 PHILIPPEROBERTANDAMANDINEVE´BER As we shallsee, when (L (0),L (0))=(0,N 1), X (0)=0, λ=λ , and µ=µ , 1 2 N 1 1 − the two processes (L (t)) and (X (t)) are close enough (for our purposes). Note 1 N that this is not completely clear since the process (L (t)) may drift away from N 2 and therefore change the service rate received by each class. It turns out that, because of the slow increase of the log function, this property will hold at least at the beginning of the sample paths. By integrating the SDE (13) one obtains that, for any t 0, ≥ t (14) X (t)=X (0)+ ([0,t]) (du) N N Nλ − NµW1(XN(u−),N−1) Z0 t log(1+X (u)) N =X (0)+λt µ du+M (t), N N − log(1+X (u))+logN Z0 N where (M (t)) is the martingale N t M (t)= ([0,t]) λt+ (du) µW (X (u),N 1)du , N Nλ − NµW1(XN(u−),N−1) − 1 N − Z0 whose increasing process is give(cid:2)n by (cid:3) t log(1+X (u)) N (15) M (t)=λt+µ du. N h i log(1+X (u))+logN Z0 N 4. The Initial Phase This section is devoted to the very beginning of the evolution of the first component (LN(t)), when it starts from 0 while LN(0) = N. To start with, we have 1 2 the following asymptotic result on the initial growth rate of the process (X (t)) N defined by Equation(13). Here andlater, we write a b for the quantity min(a,b). ∧ Proposition 1. If X (0)=0 and N α∗ d=ef. ρ , where ρ= λ, 1 ρ µ − then the convergence in distribution of stochastic processes X (Nt) t lim N , 0<t<α∗ 1 = λ µ , 0<t<α∗ 1 N→+∞ Nt ∧ − t+1 ∧ (cid:18) (cid:19) (cid:18) (cid:19) holds for the uniform topology on compact sets of (0,α∗ 1). ∧ Proof. The evolutionequation(14)anda changeofvariablesgiveusthatfor every t 0, ≥ X (Nt) X (1) λ M (1) M (Nt) N N N N = − − + Nt Nt Nt t log(1+X (Nu)) +λ µ N (logN)Nu−tdu. − log(1+X (Nu))+logN Z0 N Letting Z (t)d=ef.(1+X (Nt))/Nt, we thus have N N X (1)+1 λ M (1) M (Nt) N N N (16) Z (t)= − − + N Nt Nt t log(Z (t v))+(t v)logN +λ µ N − − (logN)N−vdv. − log(Z (t v))+(t v+1)logN Z0 N − −