Best-effort networks: modeling and performance analysis via large networks asymptotics

Transcription

1 IEEE INFOCOM 200 Best-effort networs: modeing and performance anaysis via arge networs asymptotics Guy Fayoe Arnaud de La Fortee Jean-Marc Lasgouttes Laurent Massouié James Roberts Abstract In this paper we introduce a cass of Marov modes termed best-effort networs designed to capture performance indices such as mean transfer times in data networs with best-effort service. We introduce the so-caed bandwidth sharing poicy as a conservative approximation to the cassica max- poicy. We estabish necessary and sufficient ergodicity conditions for best-effort networs under the poicy. We then resort to the mean fied technique of statistica physics to anayze networ performance deriving fixed point equations for the stationary distribution of arge symmetrica best-effort networs. A specific instance of such networs is the star-shaped networ which constitutes a pausibe mode of a networ with an overprovisioned bacbone. Numerica and anaytica study of the equations aows us to state a number of quaitative concusions on the impact of traffic parameters (in oads) and topoogy parameters (route engths) on mean document transfer time. Keywords best-effort service max- fairness poicy mean fied star-shaped networ. I. INTRODUCTION Consider a networ handing data fows from severa users and assume no quaity of service commitments (such as imum bandwidth aocations) have been made by the networ to the users. Such a situation has been prevaent in the Internet unti now and is iey to remain so for another few years. The preferred service mode in that situation nown as best effort service consists in aocating a fair proportion of bandwidth to contending users; see e.g. Bertseas and Gaager. There are actuay severa possibe notions of fairness avaiabe for this bandwidth aocation probem (see e.g. Mo and Warand 2 for a parametric famiy of fairness criteria covering a other notions proposed so far) athough the cassica notion proposed in is the so-caed max- fairness. Recent wor has ed to a reativey good understanding of how bandwidth is shared between networ users when a given congestion contro agorithm is used; see e.g. Massouié and Roberts 3 and references therein. The question of what type of fairness is achieved in the current Internet where Jacobson s congestion avoidance agorithm as impemented in TCP is responsibe for congestion contro has been studied in depth by Hurey et a. 4. These studies a assume the number of fows remains fixed. In comparison there is itte wor accounting for the random nature of traffic and its impact on user perceived quaity of service. Consider for instance the transfer of digita documents (Web pages fies emais... ) using a transport protoco ie TCP. This constitutes the bu of Internet traffic today. The performance criterion reevant to such transfers is the overa This wor has been party supported by a grant from France Téécom R&D. G. Fayoe A. de La Fortee and J.-M. Lasgouttes are with INRIA Domaine de Vouceau BP 05 Rocquencourt 7853 Le Chesnay CEDEX France. L. Massouié is with Microsoft Research Saint George House Guidha Street CB2 3NH Cambridge United Kingdom. J. Roberts is with France Téécom R&D rue du Généra Lecerc Issy es Mouineaux CEDEX 9 France. document transfer time. This time is ceary highy dependent on the number of ongoing transfers on the ins shared by the considered connection. This number varies as a random process as new connections are estabished and existing ones terate in a way which depends on how bandwidth is aocated as we as on the underying traffic parameters. In the case of a singe bottenec resource shared perfecty fairy simpe traffic assumptions of Poisson arrivas and identicay and independenty distributed document size ead to a processor sharing queueing mode 5. This fuid fow mode provides usefu resuts on expected response times as a function of the oad of an access in or a Web server for instance. It aso shows how a form of congestion coapse can occur when demand (arriva rate mean document size) exceeds capacity. The processor sharing queue is then no onger ergodic eading to unbounded response times. To understand the impact of mutipe bottenecs and to investigate the effect of different sharing strategies one woud ie to dispose of simiar anaytica resuts for mutipe resource systems. To the best of our nowedge the ony anaytica resuts avaiabe so far are in Massouié and Roberts 5 where the so-caed inear networ topoogy is investigated. Simuation resuts for the inear networ can be found both in 5 and in de Veciana et a. 6. The main motivation for the present paper is to study the performance of best-effort networs with aternative topoogies the utimate obective being the derivation of heuristics enabing the performance evauation of bandwidth sharing in a genera networ. In the present paper we report the resuts of our preiary investigations. These incude an anaysis of the stabiity conditions under which the expected response time remains finite in a genera networ. We aso appy mean fied techniques to evauate the performance of arge symmetrica networs. Numerica resuts derived from the mode iustrate how response times depend on the number of bottenec ins and their utiization. These resuts are of some practica interest and aide our understanding of the behavior of best effort networs. A further significant contribution is the insight provided into the inherent difficuty of deriving performance estimates when more than one bottenec imits throughput. Section II introduces a genera cass of Marov modes for best-effort networs which is intended to capture the impact of networ topoogy traffic parameters and bandwidth sharing (fairness) criteria on document transfer times. A brief account of the resuts obtained in 5 is given and the so-caed bandwidth aocation is introduced as a conservative approximation to max- fairness. Section III then estabishes the necessary and sufficient ergodicity criteria for best-effort networs under the poicy. Section IV introduces the so-caed star topoogy. Its reevance as a mode of rea networs is discussed

2 2 IEEE INFOCOM 200 route 0 route in route 2 in 2 Fig.. The inear networ route L in L and a mean fied heuristic is proposed. This heuristic is expected to be accurate in the asymptotic regime where the number of star branches is arge. The derived fixed point equations are investigated numericay in Section V. Simuation is used to verify the accuracy of the heuristics. Extensions to the star-shaped networ are aso considered in Sections IV and V which notaby aow an evauation of the impact of the number of bottenecs on the mean transfer time. II. BEST-EFFORT NETWORKS Consider the foowing networ mode: a set L of ins is given where each in L has an associated capacity or bandwidth C > 0. A set R of routes is given each route being identified with a subset of ins. Fig. iustrates the so-caed inear networ: it consists of L ins with equa capacity route 0 which crosses each in and routes r =...L which cross a singe in. To each route r are associated two parameters: λ r is the arriva rate of new transfer requests aong route r and σ r is the average document size. We mae the foowing standard simpifying assumptions: requests for document transfers aong route r arrive at the instants of a Poisson process with intensity λ r whie the corresponding document sizes are mutuay independent independent of the arriva times and drawn from an exponentia distribution of mean σ r. These traffic assumptions mae the process specifying the number of transfers in progress on different routes Marovian (see beow) and thus greaty simpify anaysis. The Poisson arrivas assumption is not unreasonabe in a arge networ. In view of the insensitivity of performance resuts for an isoated in to the exact document size distribution we do not expect divergence of the rea distribution from the exponentia size assumption to invaidate the derived concusions. However the main reason for assug an exponentia distribution is ceary one of anaytica tractabiity. The networ state is summarized by the variabes X = {x r r R} where x r denotes the number of transfers in progress aong route r. It remains to specify at what speed documents are transmitted in any given state X in order to turn X into a Marov process with we ined dynamics. Indeed given the rate ζ r (X) at which documents aong route r are transferred when the networ state is X X is a Marov process with non-zero transition rates given by { xr x r + : rate λ r x r x r : rate x r ζ r (X)/σ r. A natura assumption woud be to consider that each document receives its fair share of bandwidth. For instance if as in fairness is understood as max- fairness each transfer aong route r receives a bandwidth share ζr mm where x r ζr mm C L () r and for every route r there is at east one in r such that x r ζr mm = C mm and ζr = max ζ r r r mm. (2) These two conditions uniquey detere the bandwidth shares. Having specified the Marov process X one can then attempt to study its steady state properties identifying the conditions on the oad parameters ζr mm ρ = C λ r σ r under which it is ergodic and when it is detering the stationary distribution. Mean transfer times T r aong each route r can then be computed using Litte s aw: T r = IE x r /λ r. It turns out that expicit formuas for steady state distributions are typicay beyond reach. A notabe exception is the inear networ with bandwidth shares being aocated to reaize proportiona rather than max- fairness; see 5. In order to obtain formuas in other cases one therefore has to resort to asymptotics on various parameters. For instance for the inear networ with max- fair rate sharing the regime where the arriva rate λ 0 aong route 0 goes to zero (essentiay a form of ight traffic anaysis) is considered in 5; this eads to approximate formuas for T 0. It can be shown in particuar that T 0 increases as the ogarithm of the number of ins L when L increases. This is in contrast to the case of proportionay fair sharing where it increases ineary in L. The main purpose of this paper is to investigate an aternative asymptotic regime where it is the networ topoogy which evoves. The precise description of this imiting regime wi be given in Section IV. In the foowing sections we consider bandwidth aocations according to the foowing poicy: given the networ state X each transfer aong route r receives a bandwidth share ζr given by ζ r r C = (3) r X where we have introduced the notation X = r x r to represent the tota number of transfers maing use of in. It is easy to chec that this poicy satisfies the capacity constraints (). Moreover it is sub-optima with respect to the max fairness poicy as shown in the next theorem. Theorem : Under the same initia conditions the vector X mm (t) for the system under the ζ mm aocation poicy is stochasticay smaer than X (t) corresponding to the ζ aocation. Proof: Assume that for some t X mm (t) X (t). Then with the notation of (2) ζr mm = max ζ r r mm X mm C X (t) xr mm r r. (t) ζ mm (t)ζr = C X mm (t)

3 FAYOLLE ET AL: BEST-EFFORT NETWORKS: MODELING AND PERFORMANCE ANALYSIS VIA LARGE NETWORKS ASYMPTOTICS 3 Thus using a couping argument one can ine the processes X mm and X (t) in such a way that X mm (t) X (t) for a t > 0. The previous theorem motivates the study of the poicy as it impies for instance that mean transfer times T r under the poicy provide upper bounds on the corresponding transfer times under the max- poicy. III. ERGODICITY CONDITIONS In the foowing we demonstrate that and max- bandwidth sharing poicies have a stationary regime under the usua conditions i.e. when the oad on each in is ess than : Theorem 2: Under the aocation poicies ζ mm and ζ the networ is (i) ergodic if max L ρ < ; (ii) transient if max L ρ >. This resut has aready been proven for the max- poicy in 6; we note that by Theorem ergodicity under the poicy impies ergodicity under the max- poicy thus the treatment of the poicy given beow provides an aternative proof to that of 6. However we fee that since the proof beow is simper and uses ony eementary Lyapunov functions resuts it shoud be easier to adapt to a more compicated situation. Transience under condition (ii) is in fact vaid for any aocation poicy which meets the capacity constraints (). Proof: Consider the discrete time chain ( ˆX(n) n IN) describing the sequence of states visited by the continuous time ump process X. Transitions from a given state ˆX = ( ˆx r r R ) satisfy IP ˆx r (n)= ˆX(n)= ˆX = λ r D IP ˆx r (n)= ˆX(n)= ˆX = ˆx r D where ˆx r (n) = ˆx r (n + ) ˆx r (n) and D ( = λ r + ˆx r σ r r ( R max λ r + max = D. C ˆX ) σ r r ) max σ C r L C ˆX Ergodicity of the continuous time process X wi foow from that of ˆX and from the fact that the mean soourn times in each state X are bounded from above uniformy in X (or equivaenty that the ump rates out of each state X are bounded away from zero) a property which is easiy verified. Sufficient condition. Assume ρ M = max L ρ < and ine the Lyapunov function f ( ˆX) = γr ˆx r where γ r > wi be chosen ater. The structure of the function which may seem unnatura has been chosen for the sae of computation; it is in fact of the genera form β r γ ˆx r r + K for appropriate constants β r and K. In order to express the transition rates in terms of ˆx r and ρ remar that ˆX = r ˆx r = r Then using the notation we have r R Thus ˆx r = ˆx r λ r σ r ˆx r ˆx r λ r σ r ρ λ r σ C max. r r λ r σ r ˆx M = max ˆx r λ r IP ˆx r (n)= ˆX(n)= ˆX Dρ M IE f ( ˆX(n + )) f ( ˆX(n)) ˆX(n)= ˆX ˆx r ˆx M ( = γ ˆx r+ r IP ˆx r (n)= ˆX(n)= ˆX γ ˆx r IP ˆx r (n)= ˆX(n)= ˆX ) λ r ρ M D γ ˆx r r ρ M γ r ˆx r ˆx M Let γ r = γ λ r σ r where γ is such that ρ M γ r = ρ M γ λ r σ r < θ < r R for some rea number θ satisfying ρ M < θ <. The foowing inequaity sums up what we have so far: IE f ( ˆX(n + )) f ( ˆX(n)) ˆX(n)= ˆX λ r γ ˆx r θ ˆx r ρ M D ˆx M. Let α be a rea number such that θ < α <. The foowing quantities wi be evauated separatey Σ Σ 2 λ r γ = ˆx r r:ˆx r >α ˆx M ρ M D = r:ˆx r α ˆx M λ r γ ˆx r ρ M D θ ˆx r ˆx M θ ˆx r ˆx M Since Σ is a sum of negative terms the foowing bound hods for any r 0 is such that ˆx r 0 = ˆx M Σ λ r 0 γ ˆx M ρ M D Bounding Σ 2 is straightforward: Σ 2 r:ˆx r α ˆx M. ˆx γ M (θ α) (θ α) ρ M D λ r < 0. λ r γ α ˆx M ρ M D θ γ ˆx M ρ M D γ(α ) ˆx M R θ max λ r..

4 4 IEEE INFOCOM 200 connection in Fig. 2. A star-shaped networ Now if C > 0 and ε > 0 are chosen to satisfy the inequaity (θ α) λ r + γ (α )C R θ max λ r ε we have ˆX {ˆx M > C} IE f ( ˆX(n + )) f ( ˆX(n)) ˆX(n)= ˆX γc = Σ + Σ 2 ε γ ˆx M ρ M D ε ρ M D < 0. Since { ˆx M C} is a compact set Foster s theorem appies (see e.g. 7) and the Marov chain is ergodic. Necessary condition. Assume now that there exists 0 such that ρ 0 >. Defining g( ˆX) = σ r ˆx r r 0 we immediatey have IE g( ˆX(n + )) g( ˆX(n)) ˆX(n)= ˆX = σ r (IP ˆx r (n)= ˆX(n)= ˆX r 0 IP ˆx r (n)= ˆX(n)= ˆX ) D C 0 ρ 0 C 0 > 0. Since the umps are bounded the chain is transient. IV. MEAN FIELD ANALYSIS OF LARGE NETWORKS It does not appear possibe to obtain cosed form expressions for the stationary distribution of the best-effort networ state under the poicy. We therefore turn to the study of these stationary distributions under a imiting regime on networ size and topoogy. A simiar approach has previousy been successfuy appied to oss networs (see 8 9 and references therein) and to queueing networs in 0. It is inspired by the so-caed mean fied modes of statistica physics. Mean fied anaysis in the present context is best iustrated by the star-shaped networ of Fig. 2. This networ has N/2 branches each consisting of one inbound and one outbound in (thus impicity N is an even number) and a ins have unit capacity. Each route r connects the endpoints of two branches via the center node. It has an associated arriva rate 2λ/N and mean message ength σ. The factor 2/N is introduced to mae the tota oad on each in ρ = λσ independent of the number of ins N. As discussed beow when N goes to infinity the number of ongoing transfers on any in becomes independent of the number of ongoing transfers on any finite coection of the other ins (this was termed the chaos propagation property in 9). This aows us to derive fixed point equations for the probabiity distribution of the number of transfers in progress on any in. A. Symmetrica star-shaped networs Athough amenabiity to a mean fied anaysis is a significant motivation for considering the star-shaped topoogy it shoud be noted that it is aso reevant to the study of rea networs. Any overprovisioned ins in a rea networ are argey transparent to the throughput of eastic best effort fows. Ony bottenec resources typicay ocated in the access networ and within Web servers need to be incuded in the networ mode. The star-shaped networ may thus be considered to represent any networ with a we provisioned bacbone where throughput is imited by bottenecs at the source and destination edges. For exampe inbound ins might represent Web server CPU whie outbound ins correspond to the ast hop of an ISP s interconnection networ. This discussion not ony motivates the consideration of such a topoogy but aso suggests that etting N go to infinity might indeed be reaistic if N represents the number of Web servers over the Internet. Of course there woud be no reason in practice to assume symmetry. This assumption is introduced soey for reasons of tractabiity. Athough our focus is on the star-shaped topoogy the mean fied approach can be appied to other symmetrica topoogies. It thus aows one to consider routes with more than two hops. The corresponding extended mode is described in detai in Section IV-B beow where the corresponding fixed point equations are derived. Section IV-C then presents anaytica resuts for the star-shaped networ. Resuts of numerica investigation of the fixed point equations are reported in Section V. B. Fixed point equations for arge symmetrica networs We use the foowing notation in the seque. N: tota number of ins; L: ength of a route through the networ; R (N) : number of routes going through a given in; X (N) : number of active connections on in L in stationary state; x r (N) : number of active connections on route r going through ins r()...r(l); λ: arriva rate on a in; σ: mean message ength; ρ = λσ: oad of a in. We have impicity assumed here that the number of routes going through a in R (N) is the same for a ins. We sha in fact assume further that the networ topoogy is the same as seen from any route. We do not attempt to give a forma

5 FAYOLLE ET AL: BEST-EFFORT NETWORKS: MODELING AND PERFORMANCE ANALYSIS VIA LARGE NETWORKS ASYMPTOTICS 5 The chaos propagation assumption impies that aw of arge numbers: obeys a (0000) (0000) (0000) (00000) (0000) (0000) Fig. 3. A hypercube-shaped networ with dimension 5. inition of this symmetry assumption here. The reader is referred to 9 for a thorough discussion on the ima symmetry assumptions required. Symmetry impies notaby that each route has the same number of hops and the same traffic parameters. The star-shaped networ discussed above constitutes an exampe of such a symmetrica networ when L = 2 (with R (N) = N/2). It is more difficut to come up with meaningfu exampes of symmetrica networs supporting routes with L > 2: in particuar the networ shoud not be fuy connected since routes onger than one hop woud then be pointess. One reasonabe mode is a hypercube (Fig. 3) of arge dimension in which each edge contains two one-way ins. The hypercube is a cassica structure with many symmetries. It is characterized by its dimension d. Its vertices are represented by d-tupes of 0s and s (e.g. (00)) and its edges connect two vertices differing in ony one coordinate. The tota number of ins in such a networ is N = d2 d. The number of routes going through any in is (d )! R (N) = L (d L)! where the ony routes considered are the shortest paths between two vertices which differ in exacty L coordinates. Note that the resuts beow do not depend on the precise topoogy of the networ. We now derive the fixed point equations. It shoud be stressed that this derivation is heuristic. We ceary mention which steps need further ustification in the course of the derivation. We do beieve that the equations are very good approximations however especiay in view of the numerica and simuation resuts presented in the foowing section. Assume now that ρ < and that the system is in stationary state X (N) =(x r (N) r R). For any 0 the proportion of ins in state is = N L By symmetry it hods that IP(X (N). {X (N) =} = )=IE L. α = im (N) = im IP(X = ) = IP(X = ). N N It appears that the dynamics of the system are driven by traditionay referred to as the mean fied. The foowing notation wi aso be usefu: (N) ᾱ = >0 ᾱ = IE X = α. In order to derive the equation satisfied by the imit stationary distribution α we must first describe the possibe transitions for. The two cases of interest are arriva on a in with connections: N > N departure from a in with > 0 connections: N + N The transition corresponding to a new connection arriva has rate λ. The main probem is to compute the departure rate from a in given that it has X (N) = ongoing transfers. This can be written as σ IE x r (N) X (N) =. (4) r r X (N) r( ) Since the tota number of routes is much arger than the size N of the networ we assume that the probabiity of having more than one connection on a route r is negigibe 2 and that the ins on route r are independent conditioned on {x r (N) = } 3. The first property aows to rewrite (4) as σ IEx r (N) X (N) = IE r IP X (N) = x (N) r = = r X (N) r( ) x r (N) =. (5) Let be a given in and et r be one route using in i.e. r. The distribution of X (N) conditioned on there being one connection on r is then IE {X (N) = } {x r (N) =} IP X (N) = x (N) r = = IE {x (N) r =} By symmetry it is possibe to sum both sides of the above fraction over a the routes going through in IE X (N) {X (N) IE X (N) = } = IE IE ᾱ (N). We have not proven that this assumption hods. It seems however that the techniques deveoped in 9 coud be appied to prove that this is the case provided the parameter R (N) goes to infinity with N. 2 This fact is easy to prove in finite time but requires more wor for the stationary regime. 3 This is the point where the heuristic is not competey exact; it is however iey to be true when ρ tends either to 0 or..

6 6 IEEE INFOCOM 200 Departure rate (5) then becomes in view of the assumed independence property between the X r(i) given x r = σ 2... L = ( 2 L ) L i=2 i IE i IE ᾱ (N). Taing the imit N the invariant measure equations foow. We have: Inbound ins Outbound ins and λ(α α )+ σ for where u = u λα 0 + σ α = 0 (6) ᾱl 2... L = ( ( u α + + ᾱ L α u ) ᾱ L = 0 (7) ) L 2 i α i. (8) L i=2 Equations (6) and (7) can be rewritten in a more concise form as α + u + = ρᾱ L α 0. (9) The two sets of equations (8) and (9) together constitute the fixed point equations we require. As noted in the introduction of this section these equations do not depend on the topoogy of the networ. The expression for u can be simpified. Let Y 2...Y L be random variabes with distribution IP(Y i = y)= yα y ᾱ 2 i L and et Y = max(y 2...Y L ). Then (8) reads u = ᾱ L IE. (0) Y Straightforward cacuations yied IE Y = = y= IP(Y = y) y y= IP(Y y). y(y + ) The simpified form for u is thus from the basic properties of the imum of independent random variabes: y L. u = y= y(y + ) mα m () m=0 Note that in the case L = 2 (i.e. for the star-shaped networ) the origina equation (8) is perhaps simper than the equivaent expression (). It yieds the foowing form for the fixed point equations: α + u + = ρᾱα (2) u = ( y)α y. (3) y>0 Fig. 4. The asymmetrica star-shaped networ Remar : When considering the star-shaped networ as a mode for Web transfers over the Internet as suggested in Section IV-A inbound ins coud be seen as the CPU of Web servers and outbound ins as the ast hop between the ISP s bacbone and the end customers. It thus maes sense to reax the symmetry assumption we had made between inbound and outbound ins as the two types of bottenecs are of a different nature. We might thus consider a star-shaped networ with N in inbound ins N out outbound ins inbound (resp. outbound) ins having capacity C in (resp. C out ) see Fig. 4. Assume the mean message ength σ is the same for each two-hop route and the in capacities C out are fixed. The arriva rate on each route has the form λ r = λ/n in and the oad ρ out = λσ/c out is ess that. The capacity C in of a bacbone in is chosen to ensure ρ in = λσn out /C in N in < is fixed when the size of the system grows. Then as N in and N out increase with N in /N out sma the inbound ins have many active connections and a arge capacity whie the outbound ins remain in norma utiization. The same approach as above can then be appied to yied the set of fixed point equations where α+ in + = ρ in ᾱ out α in α+ out + = ρ in ᾱ out α out u in = αy in Cout y>0 C y in C out = αy out C y in u out y>0 and α in out (resp. α ) represents the proportion of inbound (resp. outbound) ins with ongoing transfers. C. Anaytica resuts for L = 2 Whie equation (9) oos superficiay ie a birth and death process equation it is in fact non-inear due to the fact that u and ᾱ both depend on the α y. From (0) one ceary sees that u is increasing in and tends to ᾱ L when. Therefore α is increasing as ong as u < ρᾱ L and decreasing after that. This means that the

7 FAYOLLE ET AL: BEST-EFFORT NETWORKS: MODELING AND PERFORMANCE ANALYSIS VIA LARGE NETWORKS ASYMPTOTICS 7 α form a moda distribution which maxima vaue is attained at 0 = max{ > 0 u < ρᾱ L }. We now present anaytica resuts on the soution of the fixed point equations for L = 2. The proof of these resuts can be found in 2. It reies heaviy on functiona anaysis. Equations (2) (3) have an unique soution for ρ < and when ρ the foowing asymptotic expansions hod: IE X ( ρ) 2 A im ρ IP(X = ) ( ρ)bexp ( ρ)a where A and B are non-negative constants. Thus under the poicy any in in the star-shaped networ has a mean queue ength which is one order of magnitude arger than for a singe server queue with the same oad (ρ/( ρ)). Its tai distribution is sti geometrica with factor ρ. It is possibe to give an expression for the constant A: ifc and v are soutions of the foowing system of differentia equations zc (z)+c(z)v(z)=0 zv (z)+v (z)=c(z) v(0)=0 v (0)= c(0)= then A can be written as foows: A = 0 c(z)dz = im z zv (z).30. Since this system is numericay highy unstabe it has proven difficut (with the Livermore stiff ODE sover from MAPLE) to derive a better estimate for A. It is interesting to note that the function w(y) = v(e y )+ satisfies the so-caed Basius 3 equation w (y)+w(y)w (y)=0 which describes a aar boundary ayer aong a fat pate (see e.g. 4). V. NUMERICAL ANALYSIS AND SIMULATIONS Whie the anaytica resuts of Section IV-C give some good estimates they are ony vaid in the heavy traffic regime ρ. In addition simiar resuts for L > 2 are not avaiabe. We thus resort to numerica resoution of the equations to gain a better understanding of the performance of transfers across arge symmetrica networs. The very form of the equations suggests the use of a fixed-point method for this numerica resoution: starting from a priori vaues (α (0) 0) the agorithm computes the corresponding u (0) from (8) and then new vaues α () from (9). Provided specia care is taen to avoid instabiities the iteration of this process converges rapidy (ess than 00 steps). Sampe resuts are shown in Fig. 5 corresponding to a arge symmetrica networ with routes of ength L = 20. distribution number of connections ρ = 0.70 ρ = 0.80 ρ = 0.90 ρ = 0.92 ρ = 0.94 ρ = 0.95 Fig. 5. Distribution of the mean-fied probabiities IP(X = ) for L = 20 and different vaues of the oad ρ. distribution L = 2 L = 3 L = 4 L = 5 L = 0 L = 20 L = 50 L = number of connections Fig. 6. Distribution of the mean-fied probabiities IP(X = ) for different vaues of the route ength L and oad ρ = 0.9. As ceary seen in the figure the distributions are very different from what woud be obtained for routes of ength L =. In this case the system consists of a coection of independent M/M/ queues and the associated distribution {α } is geometric. For L > the {α } distributions are maredy moda and the positions of the pea vaues are roughy proportiona to ( ρ) 2 a fact has ony been proven in Section IV-C for the case L = 2. Moreover since the shape of the distribution is rather narrow this position roughy coincides with the mean number of active connection (as can be seen from the raw data). The impact of route ength is iustrated in Fig. 6. It seems that the mean number of active connections (which is again approximatey the pea vaue of the distribution) is roughy proportiona to og L. Note that a ogarithmic growth rate is very sow suggesting that beyond 2 or 3 the number of bottenecs does not have a significant impact on mean transfer times. They depend much more on the oad ρ. The resuts presented so far ony concern the soution of the fixed point equations. As mentioned earier there are gaps in the derivation of these equations. To assess their quaity and to investigate the accuracy of the asymptotic approximation for finite size networs we ran a number of simuations of the starshaped networ. Fig. 7 dispays the corresponding resuts when

8 8 IEEE INFOCOM 200 cumuative distribution number of connections N= 3 N= 0 N= 30 N=00 N=300 mean fied Fig. 7. Cumuative distribution IP(X ) for different vaues of N (simuation) and infinite size (fixed point) for a star-shaped networ with oad ρ = 0.9. the oad on each in is set to ρ = 0.9 for a varying number of ins. The agreement between the simuation resuts and the fixed point equation resuts is exceent for N = 00 ins and improves as N increases. VI. CONCLUSIONS In this paper we have considered a cass of Marov processes caed best-effort networs which constitutes a natura probabiistic mode for evauating the performance of document transfers over data networs such as the Internet. Unie amost a previous wor this mode accounts for the random nature of traffic: document transfers begin at the epochs of a certain arriva process and the size of each document is drawn from a given probabiity distribution. In the interests of tractabiity we assumed Poisson arrivas and exponentiay distributed sizes. We introduced the bandwidth sharing poicy as a conservative approximation to the more cassica max- poicy. Necessary and sufficient ergodicity conditions for best effort networs under the and max- poicies have been estabished. In order to pursue the anaysis of the stationary distributions of the number of transfers in progress we have resorted to arge networ asymptotics appying the mean fied approach of statistica physics. This enabed us to derive fixed point equations for the probabiity distribution of the number of ongoing transfers on a given networ in. The vaidity of these equations has been estabished by comparing their soution with the resuts of simuations. Anaytica and numerica resuts show how the mean transfer time depends on the number of bottenec ins and their oad. The steady state distribution in networs where routes have severa bottenecs (L > ) has a mared moda behavior. This is significanty different to the geometric distribution which hods when routes have a singe bottenec (L = ). Performance is aso much more sensitive to in oad ρ for mutipe bottenec routes: as ρ mean transfer time increases ie /( ρ) 2 in the case L = 2 whereas the dependence is in /( ρ) when L =. Finay the impact of the number of hops per route L appears sma (given that L > ) compared to that of parameter ρ. This suggests that the star-shaped networ is perhaps a sufficienty compex mode and that the study of symmetrica networs with L > 2 is ess reevant. The wor presented here can be pursued in severa directions. On the theoretica side the anaytica resuts presented in Section IV-C constitute a first step to understanding the soution of the fixed point equations which coud be taen further. Another chaenging theoretica question is to improve the fixed point equations in a rigorous way. On a more practica side the fixed point equations might be simpified so as to find simpe approximate formuas for mean transfer times as a function of ey parameters (such as ρ in ρ out in the case of the asymmetrica star-shaped networ described in Remar ). Such approximate formuas coud then ead to engineering rues for capacity panning. We view the present study as a preiary investigation into the performance of best effort networs with mutipe bottenec ins. A significant resut of this investigation is the discovery that the extension of the processor sharing mode vaid for a singe bottenec proves to be very hard. There appears to be no simpe parae to the famiiar fixed point techniques used in oss networs. The probem is however of considerabe practica importance for providers seeing to engineer their networ to ensure adequate throughput for document transfers. We hope therefore that this paper wi incite further wor and the deveopment of aternative heuristic approaches. REFERENCES D. 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