Fair Internet traffic integration: network flow models and analysis

Transcription

1 Fai Intenet taffic integation: netwok flow models and analysis Pete Key, Lauent Massoulié Micosoft Reseach, 7 J J Thompson Avenue, Cambidge CB3 0FB, UK Alan Bain, Fank Kelly Statistical Laboatoy, Univesity of Cambidge, Cambidge CB3 0WB, UK Abstact We use flow-level models to study the integation of two types of Intenet taffic, elastic file tansfes and steaming taffic. Pevious studies have concentated on just one type of taffic, such as the flow level models of Intenet congestion contol, whee netwok capacity is dynamically shaed between elastic file tansfes, with a andomly vaying numbe of such flows. We conside the addition of steaming taffic in two cases, unde a fainess assumption that includes TCP-fiendliness as a special case, and unde cetain admission contol schemes. We establish sufficient conditions fo stability, using a fluid model of the system. We also assess the impact of each taffic type on the othe: file tansfes ae seen by steaming taffic as educing the available capacity, wheeas fo file tansfes the pesence of steaming taffic amounts to eplacing shap capacity constaints by elaxed constaints. Simulation esults suggest that the integation of steaming taffic and file tansfes has a stabilizing effect on the vaiability of the numbe of flows pesent in the system. Keywods: Intenet, Quality of Sevice, Fainess, Elastic Taffic, Steaming Taffic, Fluid Models, Flow Level 1

2 Intégation équitable du tafic dans l Intenet: modèles fluides de flots et leu analyse. Résumé Nous poposons des modèles de flots epésentant l intégation de deux types de tafic Intenet: les tansfets de données, ou tafic élastique, et le tafic temps-éel. Les tavaux antéieus ont pincipalement taité un seul type de tafic, comme les modèles de flots du contôle de congestion dans l Intenet, où la capacité du éseau est patagée dynamiquement ente les tansfets de données en cous, dont le nombe évolue dans le temps. Nous considéons deux scénaios d intégation, l un eposant su une hypothèse d équité généale, dont un cas paticulie est la compatibilité avec TCP ou TCP-fiendliness ), et l aute eposant su une politique de contôle d admission des flots temps-éel. Nous considéons une enomalisation des pocessus décivant l état du éseau. Nous donnons des conditions suffisantes de stabilité pou ces pocessus enomalisés. Nous évaluons aussi l impact qu a chaque type de tafic su l aute: le tafic élastique a pou seul effet de éduie la capacité offete au tafic temps-éel, alos que l effet du tafic temps-éel su le tafic élastique est de change des containtes stictes de capacité en des containtes pénalisées. Des ésultats de simulation suggèent que l intégation des deux types de tafic éduit la vaiabilité du nombe de flots pésents dans le système. Mots clés: Intenet, Qualité de Sevice, Equité, Tafic élastique, Tafic temps-éel, Modèles fluides, modèles de flots. I INTRODUCTION The motivation fo this pape aises fom the need to undestand and model the integation of diffeent types of taffic within the Intenet. At the tanspot laye, the cuent Intenet is dominated by flows which use TCP. The pecentage of TCP taffic is vaiable, and may depend on time of day and the paticula oute chosen; howeve typical measuements on a backbone [17] show that upwads of 70% of flows use TCP, ising to ove 90% by volume, with UDP the main altenative potocol up to 20% of packets, o 10% of bytes). Pevailing applications can change apidly: wheeas Web taffic used to be the dominant application type fo TCP taffic, at the time of witing file-shaing applications can dominate and may account fo 40% of the taffic on backbone links. The cuent volume of steaming taffic caied by UDP is small less than 10%), but the apid incease in pee-to-pee taffic illustates how quickly the status-quo can change, and we would like to pedict behaviou in diffeent scenaios. How TCP and UDP should co-exist is a vexed question, and many egad UDP-elated taffic as inheently poblematic. Some authos have poposed that steaming taffic should be TCP-fiendly, so that it can shae netwok esouces faily with the dominant fom of existing taffic [11]. Indeed some steaming applications use TCP as the tanspot potocol. Applications that use UDP often need some fom of quality of sevice to function adequately, which has led some eseaches to conside distibuted o end-point admission contol [14, 6, 3, 16]. 2

3 The need to model such situations equies modelling the heteogeneous taffic steams, with thei diffeent chaacteistics. Pevious wok in this aea has focused on analysing occupancy distibutions of single esouce systems, via eithe exact o appoximate techniques, see e.g. [2],[1],[21] and [23]. In contast we look at abitay netwok topologies. We conside two types of taffic, which we label file tansfes and steaming taffic. A flow caying a file tansfe must tansfe a given volume: the volume may be andom, but is independent of the level of congestion expeienced. An admitted flow caying steaming taffic emains pesent fo a holding time: the holding time may be andom, but is independent of the level of congestion expeienced. Ou analysis is a flow-level analysis, that genealises the flow level model of Intenet congestion contol of Massoulié and Robets [18] by incopoating steaming taffic. They consideed a netwok whee a andomly vaying numbe of flows is pesent, and capacity is dynamically shaed between elastic file tansfes using diffeent shaing mechanisms. The analysis of steaming taffic on its own gives ise to a poduct-fom solution unde cetain easonable assumptions, a fom which is peseved unde cetain types of call admission contol [14]. Moeove the limiting behaviou as the size of the system gows leads natually to a non-degeneate limit fo the scaled) numbe of connections. In contast, a simila scaling applied to just file tansfe taffic esults in numbes of competing flows eithe inceasing to infinity o deceasing to zeo; it has been suggested [8] that such a model is flawed, lacking any self-limiting behaviou. We shall see that this citicism is avoided when the two types of taffic ae mixed, and that the pesence of even a small amount of steaming taffic has a stabilising effect. The oganization of this pape is as follows. In Section 2 we descibe the shaing policy between flows whee we assume a genealized fom of TCP-fiendliness fo the steaming taffic. The genealisation is based on the so-called α-fai allocations [20]. In Section 3 we descibe the flow level stochastic model of a netwok, a genealization of [18]. File tansfes ae chaacteized by a andom Poisson aival pocess, with exponentially distibuted file sizes, wheeas steaming taffic has Poisson aival ates but an exponentially distibuted holding time. In Section 4 we establish appopiate stability conditions, fo a fluid model of the system, though the constuction of an appopiate Lyapunov function. We also chaacteize and intepet the netwok state in equilibium. In Section 5 we conside extensions whee we elax the shaing assumptions between the two types of taffic. in paticula, we discuss admission contol stategies fo the case whee the steaming taffic entes at a fixed ate, and show how a paticula admission contol stategy peviously consideed in [13] can be consideed TCP-fai. In Section 6 we discuss simulations of the flow level model fo a sta netwok, and exploe the impact of steaming taffic on the vaiability of flow alloctions. We conclude in Section 7. II FAIRNESS ASSUMPTIONS Conside a netwok with esouces labelled by j J. Let a oute identify a non-empty subset of J intepeted as the set of esouces used by a flow on oute ). Wite R fo the set of possible outes. Set A j = 1 if esouce j lies on oute i.e. j ), and set A j = 0 othewise. We 3

4 assume positive finite capacities C j,j J). Let N be the numbe of flows on oute. Given a fixed paamete α 0, ) and stictly positive weights w, R), we suppose that the bandwidth allocation to each of the N flows on oute is x, whee x = x, R) is a solution to the following optimization poblem: maximize subject to R x 1 α w N 1 α 1) A j N x C j, j J 2) ove x 0, R. 3) Call the esulting allocation a weighted α-fai allocation [20]. The fom of a solution to the poblem 1 3) can be given in tems of Lagange multiplies p j,j J), one fo each of the capacity constaints 2), as ) 1/α w x = j p, 4) ja j whee p j 0, p j C j ) A j N x = 0 j J. 5) The stict concavity of the objective function 1) as a function of x, : N > 0) ensues that the component x is unique if N is positive. When w = 1, R, the cases α 0, α 1 and α coespond espectively to an allocation which achieves maximum thoughput, is popotionally fai o is max-min fai [5, 20]. Weighted α-fai allocations povide a tactable theoetical abstaction of decentalized packet-based congestion contol algoithms such as TCP. If α = 2 and w is the ecipocal of the squae of the ound tip time on oute, then the fomula 4) is a vesion of the invese squae oot law familia fom studies of the thoughput of TCP connections [10, 19, 22]. A flow caying steaming taffic is temed TCP-fiendly if, inte alia, it adapts its ate to coespond with the steady-state ate of a TCP connection, usually chaacteized in tems of a vesion of the invese squae oot law [11]. The elations 2 5), and moe efined vesions of these elations, can be solved to give pedictions of thoughput, given the numbes of flows N pesent [7, 12, 25]. Given N, netwok pefomance along diffeent outes can be pedicted. But what detemines the behaviou of N? One aim of this pape is to bette undestand how the behaviou of N is influenced by the mix of taffic types pesent. III FLOW LEVEL STOCHASTIC MODEL We now descibe ou model of how flows aive and depat. Ou aim is to genealize the stochastic model fo file tansfes intoduced in [18] to include steaming flows. 4

5 Let N be the numbe of document tansfes on oute, and let M be the numbe of steaming flows on oute. Define the indicato function I[ = s] = 1 if = s, I[ = s] = 0 othewise. Let T s N = N + I[ = s], R), with invese Ts 1 N = N I[ = s], R). We suppose that N,M) = N, R;M, R) is a Makov pocess, with state space Z J + ZJ + and non-tivial tansition ates qn,m),t N,M)) = ν, qn,m),t 1 N,M)) = µ N x N + M), R qn,m),n,t M)) = κ, qn,m),n,t 1 M)) = M η, R fo N,M) Z J + Z J +, whee xn) is a solution to the optimization poblem 1 3). This coesponds to a model whee new file tansfes aive on oute as a Poisson pocess of ate ν, new steaming flows aive on oute as a Poisson pocess of ate κ, and x N + M) is the bandwidth allocated to each flow on oute, whethe it is a file tansfe o steaming flow. A file tansfe on oute tansfes a file whose size is exponentially distibuted with paamete µ, and a steaming flow on oute has an exponentially distibuted holding time with paamete η. If κ = 0, R, then this model educes to the model intoduced by Robets and Massoulié [18], in which thee ae no steaming flows, only file tansfes. Fo this case, De Veciana, Lee and Konstantopoulos [9] and Bonald and Massoulié [5] have shown that a sufficient condition fo the Makov chain Nt), t 0) to be positive ecuent is that A j ρ < C j, j J, 6) whee ρ = ν /µ ; this condition is also necessay [15]. The condition is natual: ρ is the load on oute, and we can identify the atio of the two sides of the inequality 6) as T the taffic intensity at esouce j. Kelly and Williams [15] have exploed the behaviou of a fluid model fo this case in heavy taffic, when the inequalities 6) ae close to being tight, which is a key step towads poving state space collapse. The papes [5, 9, 15] all make use of a fluid model of the Makov pocess, an appoach which we shall use fo ou analysis of the extended model. We shall hencefoth assume that κ > 0, R, and that condition 6) is satisfied. Define the educed capacities C j = C j A j ρ, j J. 7) Thus the educed capacity C j on esouce j is just the amount by which inequality 6) fails to be tight. The educed capacities will detemine the capacity available to steaming flows in a sense that will be made pecise in the next section. IV STABILITY OF FLUID MODELS Next we descibe a fluid model, which can be thought of as a fomal law of lage numbes appoximation unde the scaling Nc t) n,m)t) =, M ) ct) c, c c 5

6 whee N c t),m c t)) is the model of the pevious Section but with C j,j J, and ν,κ, R, eplaced by cc j,j J, and cν,cκ, R, espectively. The fluid model is an appoximation appopiate fo the case whee C j,j J, and ν,κ, R, ae all lage, an impotant case in applications. The fluid model fo the Makov pocess of the last Section takes the fom d dt n t) = ν µ n t)x nt) + mt)), R 8) d dt m t) = κ η m t), R. 9) Note that ou assumption that κ > 0, R, implies that m t) > 0, R,t > 0. Poposition 1. Povided the condition 6) is satisfied, the diffeential equations 8,9) have a unique invaiant point, ˆn, ˆm ). It takes the fom ˆm = κ /η and ˆn = ν j J p ) 1/α ja j R, 10) µ w fo some p R J +. At the invaiant point the bandwidth allocation to each flow on oute is x = w j p ja j ) 1/α. 11) The pai x,p) foms a solution of equation 11) and the conditions p j 0, p j C j ) A j ˆm x = 0 j J, 12) and togethe these elations detemine x uniquely. Poof. At an invaiant point m t) = ˆm, fom equation 9). Futhe, fom equation 8). Now at any time t, x nt) + mt)) = ˆn x ˆn + ˆm) = ρ, 13) w j p jt)a j ) 1/α whee ) p j t) 0, p j t) C j A j n t) + m t))x nt) + mt)) = 0 j J, 6

7 fom the chaacteization of x as a solution to an optimization poblem of the fom 1 3). Thus, at an invaiant point, p j 0, p j Cj ) 1/α w A j ˆm j p = 0 j J, ja j using equation 13) and the definition 7). Thus x, given by 11), is the unique optimum to a poblem of the fom 1 3), with C eplaced by C and N eplaced by ˆm. Equation 10) descibes the vecto ˆn, of dimension R, in tems of p, a vecto which may have a much smalle dimension, J, a phenomenon fist noted in the balanced fluid model of [15]. Remak 2. The invaiant point can be intepeted as follows. File tansfes place an ieducible load A jρ on esouce j fo each j J. The educed capacities C j,j J) that emain afte this load is satisfied ae available to be shaed amongst steaming taffic, and detemine the bandwidth allocation to flows on oute fo both types of taffic. When κ = 0, R, the unique invaiant point of the fluid model is ˆn = 0 [9, 5]. It is notable that the inclusion of steaming taffic within the fluid model foces the components of ˆn to be positive. We now discuss convegence to the equilibium point of the above dynamics. In ode to do so, it is convenient to intoduce a modification fo the dynamics of file tansfes. This is natually descibed in tems of the quantities λ, which epesent the total capacity allocated to type file tansfes, and thus with the pevious notation, λ = n x. Let the function ψλ) be a penalty function. Then the modified dynamics ae as follows: d dt n t) = ν µ λ nt)), R, 14) whee the vecto λ of sevice ates λ is defined as the solution to the optimisation poblem maximize φλ) := w n α λ 1 α + ψλ) 15) 1 α R subject to A j λ C j, j J 16) ove λ 0, R. 17) In the case whee ψ is identically zeo, this educes to the pevious dynamics fo the file tansfes in the absence of steaming taffic. The function ψ is assumed to be concave and stictly monotonic deceasing in each coodinate on the domain of the optimisation poblem. This latte condition implies that the ate λ goes to zeo as n goes to zeo, and hence the tajectoies n stay away fom the bounday of the othant R R +. Let us pove stability of the above dynamics. 7

8 Theoem 3. Unde the stability conditions 6), the function Ln) defined by Ln) = 1 µ {w n 1+α 1 + α)ρ α } + n ψ ρ), 18) whee ψ ψ ρ) stands fo the -th patial deivative λ evaluated at the vecto of loads ρ, is a Lyapunov function fo the dynamics 14 17). Hence these dynamics convege to the unique minimise of L on the othant R R +, that is ) ψ ˆn = ρ ρ) 1/α. 19) w Poof. Unde the condition 6), the vecto ρ = ρ, R) lies in the inteio of the domain 16 17) of the optimisation poblem defining the vecto λ. The function φ is stictly concave on this domain, since both tems in its definition 15) ae concave, with stict concavity of the fist tem. Hence φ ρ)ρ λ ) 0, and this inequality is stict unless λ = ρ. The left-hand side also eads { ) α } n w + ψ ρ ρ) ρ λ ), and is thus equal to L n nt)) d dt n t) = d dt Lnt)). Thus the value of Ln) deceases stictly along the tajectoies of the system, except at the equilibium point specified by 19), which is the only point fo which the coesponding ate vecto λ equals the load vecto ρ. Remak 4. If the concave function ψ fails to be diffeentiable at ρ, by adapting the above poof it can be shown that the dynamics 14 17) convege to the set of points ˆn satisfying 19), whee the vecto ψ ρ), R) spans the set of sub-gadients of the convex function ψ at ρ. We efe the eade to [24], p.214 fo a definition and basic popeties of sub-gadients of convex functions. We now apply this esult to establish stability of the dynamics 8 9). Coollay 5. Unde the stability condition 6), the dynamics 8 9) ae asymptotically stable. Poof. We shall only teat the special case whee the m have aleady conveged to thei equilibium values, ˆm. As the convegence of mt) to ˆm does not depend on the evolution of nt), the geneal case can be deduced by continuity aguments. We now show that the n evolve accoding 8

9 to 14 17) fo some suitable choice of a penalty function ψ. Indeed, 14) holds, with the sevice ates λ solving maximize φλ,γ) := w {n α λ 1 α 1 α + γ 1 α } ˆmα 1 α R subject to A j λ + γ ) C j, j J ove λ,γ 0, R. Pefoming the optimisation ove the γ fist, this is of the fom 15 17), with { } ψλ) := sup w ˆm α γ 1 α, 20) 1 α { } ove γ Sλ) := γ R R +, A j γ C j A j λ, j J It is eadily seen that ψ is deceasing in each coodinate: given λ, λ, such that λ λ fo all, the inequality being stict fo some, any vecto γ in Sλ) is such that γ := γ + λ λ ) is in Sλ), so that ψλ) < ψλ ). Concavity of ψ also holds: given λ, λ and ǫ in [0,1], denote by γ and γ the maximising vectos in the definition of ψλ), ψλ ) espectively. Then ǫγ + 1 ǫ)γ lies in Sǫλ + 1 ǫ)λ ), and hence ψǫλ + 1 ǫ)λ ) w ˆm α ǫγ + 1 ǫ)γ ) 1 α ǫψλ) + 1 ǫ)ψλ ), 1 α whee concavity of the function maximised in the definition of ψ gives the second inequality. Remak 6. Unde the paticula choice 20) of penalty function, and compaing equations 10) and 19), we deduce that j J p ja j = ψ ρ). Notice the identification between the sensitivity of the penalty function ψ with espect to the load ρ and the sum of the Lagange multiplies along oute.. V EXTENSIONS: PACKET MODELS AND ADMISSION CONTROL V.1 Constaint elaxation The fomulation 14 17) is also useful to model situations whee the had capacity constaints descibed by the intesection of half-spaces 2) ae elaxed. If the optimization poblem 1 3) is eplaced by x 1 α maximize w N 1 α ) C j A j N x j ove x 0, 9

10 whee C j ),j J, ae convex, stictly inceasing, diffeentiable functions, then an optimum is again given by equation 4), but whee now p j,j J, satisfy ) p j = C j A j N x. This fomulation aises natually fom packet level models, with x the mean ate of a stochastic packet geneation pocess. Fo example, if the esouces j coespond to output pots of outes, then thee is a limited amount of buffeing available, and packets will be dopped if the capacity is exceeded, o moe geneally maked accoding to some active queue management technique. We may intepet p j y j ) as the pobability of dopping o making) a packet at esouce j when the load on the esouce is y j. Stability of the coesponding fluid model can be deduced fom the fomulation 14 17), by setting ψλ) = ) C j A j λ. j V.2 Admission contolled taffic Steaming may need some minimal non-zeo ate fo the application to function adequately. Fo example in the case of steaming multimedia, even with adaptive codecs, some minimal tansmission ate is often equied fo acceptable pefomance. Suppose that type steaming taffic only entes if x x min κ I[x x min x x min : then in both the flow level stochastic model and in the fluid limit, κ is eplaced by ]. At an invaiant point, eithe m > 0 and x x min o m = 0. The condition is equivalent to p j A j w x min ) α R. 21) j Thus an invaiant point ˆn, ˆm ) is descibed in tems of a vecto p of dimension J which lies in the polyhedal egion defined by the intesection of the positive othant with the R halfspaces 21). If the paametes p j,j J, satisfy the linea constaints 21) with stict inequality, then the fluid model pedicts thee will be no call admission blocking. A moe exteme case is when eal-time steaming taffic cannot adapt its ate at all. We now descibe a shaing model elevant fo such a scenaio, accoding to which steaming flows eithe poceed at thei taget ate, o ae ejected, in such a way that the equilibium points ae the same as fo the pevious model at least in a situation of inteest). Moe specifically, type steaming flows have a taget ate π. At a given time t, they measue the cuent TCP-fiendly ate they would get unde the pevious shaing model, say λ ; they then poceed at full taget ate π with pobability min1,λ /π ), and ae ejected with the complementay pobability. Once stated, they no longe adapt to the netwok state. Such an appoach has been poposed by Kalsson [13]. 10

11 Keeping the same notations as in the pevious sections, the fluid equations descibing the evolution of the numbes of flows ae now d dt n = ν µ n x n;c Aπm), R 22) d dt m = κ min 1, x ) n;c Aπm) η m, R. 23) π In the above we denote by x n,c) the solution to the optimisation poblem 1 3), whee we have made explicit both the numbes of flows n and the capacity constaints C. Note that the capacity allocations ae now defined based on the numbes of file tansfes n, and the educed capacities C Aπm, whee A is the link-flow incidence matix as befoe), π is the diagonal matix with diagonal enties π, and m = m ). We now chaacteize the equilibium points unde these dynamics. Poposition 7. Povided the condition 6) is satisfied, any invaiant point ˆn, ˆm ) of the diffeential equations 22,23) takes the fom ˆm = κ /η min1,x /π ), ˆn = ν /µ )x 1, whee x satisfies 11), fo some fo some p R J +, and is the equilibium bandwidth allocation to each flow on oute. The pai x,p) foms a solution of equation 11) and the conditions p j 0, p j C j ) A j ˆm π = 0 j J. 24) The quantities y := minπ,x ) solve the optimisation poblem maximize subject to κ y 1 α w η 1 α R 25) A j κ η y C j, j J, y π, R, 26) ove y 0, R, 27) and the p j s constitute a set of Lagange multiplies associated with the capacity constaint C j in the above. The y s ae thus uniquely detemined. The x s ae not necessaily uniquely detemined. Poof. The expessions of the quantities of inteest at any invaiant point ae obtained exactly as in the poof of Poposition 1. Rewiting 24) as p j 0, p j C j ) κ A j minπ,x ) = 0 j J, η 11

12 we can eadily intepet the quantities y := minπ,x ) as the solutions to the optimisation poblem 25 27). By stict concavity of the function 25) being maximised, y is indeed uniquely defined. That x is in geneal not uniquely defined can be on the following counte-example: Conside a netwok with two links {1,2} of equal capacities C, and thee outes {1}, {2}, {1,2}. Routes {1} and {2} cay taffic with exactly the same chaacteistics. It is then easy to select paametes such that steaming taffic along oute {1, 2} expeiences admission contol, while steaming taffic along the two othe outes is always accepted. In that case, only the sum of multiplies p 1 + p 2 is detemined, not the individual multiplies. As a esult the coesponding ates x {1}, x {2} ae not uniquely defined eithe. We can again povide an intepetation fo invaiant points, in paticula when fo all it holds that y < π. This is the case whee admission contol is active along each oute. Note that in this specific situation, x is now uniquely detemined and coincides with y. Remak 7. In the case the admission contol is active along each oute, i.e. fo all it holds that y < π, the equilibium allocation x is the same unde the pesent admission contol mechanism as unde the pevious ate adaptation scheme, the invaiant numbes of file tansfes ˆn ae also unchanged, and the load used up by steaming taffic, hee ˆm π, is also unchanged. In this sense, the pesent admission contol mechanism may be deemed TCP-fai at the level of detail captued by the pesent fluid flow models. Remak 8. Conside the following modified admission ule: instead of choosing to poceed with pobability min1,x /π ), type steaming flows will instead chose to poceed with pobability min1,ǫx /π ), whee ǫ > 0 is some fixed paamete. Denote by x ǫ, m ǫ and p ǫ j the coesponding equilibium vaiables. It is eadily seen that a valid solution is povided by chosing ǫx ǫ = x1, m ǫ = m1, and pǫ j = ǫα p 1 j. The intepetation is as follows: by basing thei admission decision on the scaled down TCP-fiendly ate ǫx athe than x, the eal-time flows ensue that the ate obtained by file tansfes in equilibium is scaled up by ǫ 1, but this comes at no cost fo them, as they have the same admission pobability in equilibium. This stems fom the fact that in the cuent model, file tansfes contibute an incompessible load on the system, independent of thei pefomance. If we wee to conside an Engset-like model fo file tansfes, this popety would no longe hold. We expect the dynamical system 22 23) to be stable unde the stability condition 6), howeve we have not poven this yet. VI EXAMPLE: A STAR NETWORK As a concete example, conside a sta netwok of 10 links connected to a coe. This example is motivated by the cuent Intenet, whee the back-bone is elatively uncongested, and congestion occus mainly on the access links. Flows use two links, with taffic spead andomly acoss links. 12

13 n m n1) n2) Numbe Time s) a) balanced taffic mix Numbe Time s) b) file tansfe dominated taffic mix Figue 1: Impact of steaming taffic on file tansfes. The substantial amount of steaming taffic pesent in mix a) elative to mix b) has a stabilizing effect on the numbe of flows in pogess. Impact du tafic temps-éel su les tansfets de fichies. La quantité impotante de tafic temps-éel dans a) elativement à b) un effet stabilisant su le nombe de flots en cous. Fo the example, J = {1,2,...,10},R = {i,j) : i < j,i,j J}. The capacity of each link C j was chosen equivalent to a T3 link 45 Mbit/s), fo j J. The mean holding time of steaming taffic 1/η, R) was taken to be 200 seconds, coesponding to voice taffic, with the mean file size 1/µ, R) taken to be 600kB. The aival ates fo the two types of taffic ν, and κ ) wee chosen to be identical, giving a file-tansfe taffic intensity of 0.5 on each link, and such that in equilibium each flow has ate 25kbit/s x ). Unde this egime the equilibium numbe of flows of each type is 100 ˆm = ˆn = 100) pe oute, giving 900 flows of each type on each link j. Figue 1a) shows the evolution of the numbe of each type A jn and A jm on a typical link, obtained by simulation of the Makov chain of Section 3. Note that the two cuves look vey simila and have a mean of 900. The numbe of steaming flows in pogess has a standad deviation of 30, while the numbe of file tansfes has a slightly highe standad deviation of just ove 40. We now alte the offeed load of each type of taffic, to keep the nominal quality x ) seen by the flows fixed while significantly alteing the popotions of the two types of taffic. We make the file-tansfe taffic intensity on each link, with a vey small amount of steaming taffic. The load was such that the equilibium of the fluid model has ˆn = 199, ˆm = 1. With so little steaming taffic we do not expect ou fluid model to be a good appoximation; as the amount of steaming taffic deceases to zeo, we expect the behaviou of the system to be bette descibed by the Bownian model of [15].) In Figue 1b) we plot the behaviou of A jn on two typical links: obseve the diffeent vetical scale in this figue, and the maked vaiability of the numbe of flows in pogess. Compaing the two figues, we see that the substantial popotion of steaming taffic pesent, in Figue 1a), has the effect of educing the vaiability of the numbe of flows in 13

14 pogess. Of couse Figue 1b) concens a faily exteme case whee thee is a vey small amount of steaming taffic. Moe geneally, the lage the popotion of steaming taffic fo a given nominal quality) the lowe the vaiability of the numbe of flows in pogess, and hence the lowe the vaiability of the bandwidth eceived by flows. VII CONCLUSION We have studied a flow level model of Intenet congestion contol, that epesents the andomly vaying numbe of flows pesent in a netwok. Bandwidth was assumed to be dynamically shaed between file tansfes and steaming taffic, accoding to a fainess citeion that includes TCP fiendliness as a special case. Though the constuction of an appopiate Lyapunov function we have established stability, unde conditions, fo a fluid model of the system. The pesence of faishaing steaming taffic esults in a non-degeneate fluid model. Analysis of the model suggests that file tansfes ae seen by steaming taffic as educing the available capacity, wheeas fo file tansfes the pesence of steaming taffic amounts to eplacing shap capacity constaints by elaxed constaints. Simulations show that the integation of steaming taffic and file tansfes has a stabilizing effect on the vaiability of the numbe of flows pesent in the system. Acknowledgement Alan Bain and Fank Kelly wee patially suppoted by EPSRC gant GR/S86266/01. Refeences [1] E. Altman, D. Atiges and K. Taoe, On the Integation of Best-Effot and Guaanteed Pefomance Sevices, INRIA Reseach Repot No. 3222, July [2] Andesen S.), Blaabjeg S.), Fodo G.), Telek M.), A Patially-blocking queueuing system with CBR/VBR and ABR/UBR aival steams. Tel. Syst. J. 19 1) 2002) [3] Bain A.), Key P. B.), Modelling the pefomance of distibuted admission contol fo adaptive applications. Pefomance Evaluation Review, Decembe [4] Ben Fedj S.), Bonald T.), Poutièe A.), Regnié G.), Robets J.), Statistical bandwidth shaing: a study of congestion at flow level. In Poceedings of SIGCOMM [5] Bonald T.), Massoulié L.), Impact of fainess on Intenet pefomance. In Poceedings of ACM SIGMETRICS [6] Beslau L.), Knightly E. W.), Shenke S.), Stoica I.), Zhang H.), Endpoint admission contol: Achitectual issues and pefomance. In Poceedings of SIGCOMM 2000, [7] Bu T.), Towsley D.), Fixed Point Appoximation fo TCP behavio in an AQM Netwok. In Poceedings of ACM SIGMETRICS [8] Coucoubetis C.), Dimakis A.), Reiman M. I.), Poviding bandwidth guaantees ove a best-effot netwok: call admission and picing. IEEE INFOCOM 2001,

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