Fluid model for a data network with α-fair bandwidth sharing and general document size distributions: two examples of stability

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1 IMS Lecture Notes Monograph Series??? Vol. 0 (0000) 1 c Institute of Mathematical Statistics, 0000 Flui moel for a ata network with α-fair banwith sharing an general ocument size istributions: two examples of stability H. C. Gromoll 1, an R. J. Williams 2, University of Virginia an University of California, San Diego Abstract: The esign an analysis of congestion control mechanisms for moern ata networks such as the Internet is a challenging problem. Mathematical moels at various levels have been introuce in an effort to provie insight to some aspects of this problem. A moel introuce an stuie by Roberts an Massoulié [13] aims to capture the ynamics of ocument arrivals an epartures in a network where banwith is share fairly amongst flows that correspon to continuous transfers of iniviual elastic ocuments. With generally istribute interarrival times an ocument sizes, except for a few special cases, it is an open problem to establish stability of this stochastic flow level moel uner the nominal conition that the average loa on each resource is less than its capacity. As a step towars the stuy of this moel, in a separate work [8], we introuce a measure value process to escribe the ynamic evolution of the resiual ocument sizes an prove a flui limit result: uner mil assumptions, rescale measure value processes corresponing to a sequence of connection level moels (with fixe network structure) are tight, an any weak limit point of the sequence is almost surely a solution of a certain flui moel. The invariant states for the flui moel were also characterize in [8]. In this paper, we review the structure of the stochastic flow level moel, escribe our flui moel approximation an then give two interesting examples of network topologies for which stability of the flui moel can be establishe uner a nominal conition. The two types of networks are linear networks an tree networks. The result for tree networks is particularly interesting as there the istribution of the number of ocuments process in steay state is expecte to be sensitive to the (non-exponential) ocument size istribution [2]. Future work will be aime at further analysis of the flui moel an at using it for stuying stability an heavy traffic behavior of the stochastic flow level moel. 1 Department of Mathematics, University of Virginia, Charlottesville, VA 22903; gromoll@virginia.eu 2 Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA ; williams@math.ucs.eu Research supporte in part by an NSF Mathematical Sciences Postoctoral Research Fellowship, NSF Grant DMS FRG , a European Union Marie Curie Postoctoral Research Fellowship, an EURANDOM. Research supporte in part by NSF grants DMS an DMS AMS 2000 subject classifications: Primary 60K30; seconary 60F17, 90B15. Keywors an phrases: Banwith sharing, α-fair, flow level Internet moel, connection level moel, congestion control, measure value process, flui moel, workloa, Lyapunov function, simultaneous resource possession, Lagrange multipliers, stability, sensitivity. 1

2 2 H.C. Gromoll an R.J. Williams Contents 1 Introuction Notation Flow level moel Network structure Banwith sharing policy Stochastic moel Flui moel Flui stability for some network topologies Linear Network Tree network References Introuction The esign an analysis of congestion control mechanisms for moern ata networks such as the Internet is a challenging problem. Mathematical moels at various levels have been introuce in an effort to provie insight to some aspects of this problem. Roberts an Massoulié [13] have introuce an stuie a flow level moel of congestion control that represents the ranomly varying number of flows present in a ata network where banwith is share fairly between flows that correspon to continuous transfers of iniviual elastic ocuments. This moel assumes a separation of time scales such that the time scale of the flow ynamics (i.e., of ocument arrivals an epartures) is much longer than the time scale of the packet level ynamics on which rate control schemes such as TCP converge to equilibrium. Subsequent to the work of Roberts an Massoulié, assuming Poisson arrivals an exponentially istribute ocument sizes, e Veciana, Lee an Konstantopoulos [7] an Bonal an Massoulié [1] stuie the stability of the flow level moel operating uner various banwith sharing policies. Lyapunov functions constructe in [7] for weighte max-min fair an proportionally fair policies, an in [1] for weighte α-fair policies (α (0, )) [12], imply positive recurrence of the Markov chain associate with the moel when the average loa on each resource is less than its capacity. Srikant [14] an Lin an Shroff [10] have recently given sufficient conitions for stability of a Markov moel where the assumption of time scale separation is relaxe. Here we consier the moel of Roberts an Massoulié with generally istribute ocument sizes an interarrival times. We are intereste in the stability an heavy traffic behavior of this flow level moel operating uner a weighte α-fair banwith sharing policy (α (0, )) [12]. (Despite the claim in [1], the proof of sufficient conitions for stability given there oes not apply when ocument sizes are other than exponentially istribute. The reason for this is that the metho of Dai [5] quote there implicitly assumes (through the form of the moel equations) that the service iscipline is a hea-of-the-line iscipline an consequently the metho oes not apply in general to processor sharing types of isciplines, such as the banwith sharing policy consiere here.) There are a few results on sufficient conitions for stability of the flow level moel with general ocument size istributions. With Poisson arrivals an ocument sizes having a phase-type istribution, for α = 1, Lakshmikantha et al. [9] have establishe stability of some two resource linear networks an a 2 2 gri

3 Flui moel for banwith sharing 3 network when the average loa on each resource is less than its capacity. For generally istribute interarrival an ocument sizes, Bramson [3] has shown sufficiency of such a conition for stability uner a max-min fair policy (corresponing to α ). Uner proportional fair sharing, Massoulié [11] has recently establishe stability of a flui moel for the flow level moel with exponential interarrival an ocument sizes, an aitional routing. From this he infers stability when ocuments have phase type istributions. In contemporaneous work escribe in a very recent preprint, Chiang, Shah an Tang [4] have evelope a flui approximation for the flow level moel when the arrival rate an capacity are allowe to grow proportionally but the banwith per flow stays uniformly boune. Using their flui moel, they erive some conclusions concerning stability for general ocument size istributions when α (0, ) is sufficiently small. However, in general, it remains an open question whether, with renewal arrivals an arbitrarily (rather than exponentially) istribute ocument sizes, the flow level moel is stable uner an α-fair banwith sharing policy when the nominal loa place on each resource is less than its capacity. This paper reports on some first steps in our stuy of the flow level moel operating uner a weighte α-fair banwith sharing policy with general interarrival an ocument size istributions. Here we review the efinition of a measure value process that keeps track of the resiual sizes of all ocuments in the system at any given time. We escribe a flui moel (or formal functional law of large numbers approximation) for the flow level moel. In a separate work [8], we showe that uner mil conitions, appropriately rescale measure value processes corresponing to a sequence of flow level moels (with fixe network structure) are tight, an any weak limit point of the sequence is almost surely a flui moel solution. The invariant states for the flui moel were also characterize in [8]. Here, as an illustration of sufficient conitions for stability of the flui moel, we establish stability of flui moel solutions with finite initial workloa for linear networks an tree networks, uner the nominal conition that the average loa place on each resource is less than its capacity. The result for tree networks is particularly interesting as there the istribution of the number of ocuments process in steay state is expecte to be sensitive to the (non-exponential) ocument size istribution [2]. Future work will be aime at further analysis of the flui moel an at using it for stuying stability an heavy traffic behavior of the flow level moel. The paper is organize as follows. In Section 2, we efine the network structure, the weighte α-fair banwith sharing policy, the stochastic moel, an we introuce the measure value processes use to escribe the evolution of the system. The notion of a flui moel solution is efine in Section 3. In Section 4 we give sufficient conitions for stability of flui moel solutions with finite initial workloa for linear networks an tree networks Notation Let N = {1, 2,..., }, R = (, ), an let R enote -imensional Eucliean space for any 1. For x, y R, x y is the minimum of x an y, an x + is the positive part of x. For x, y R, let x = max i=1 x i, an interpret vector inequalities componentwise: x y means x i y i for all i = 1,...,. The positive -imensional orthant is enote R + = {x R : x 0}. To ease notation throughout the paper, efine c/0 to be zero for any real constant c, an efine a sum over an empty set of inices or of the form l k=j with j > l to be zero.

4 4 H.C. Gromoll an R.J. Williams For two functions f an g with the same omain, f g means f(x) = g(x) for all x in the omain. For a boune function f : R + R, let f = sup x R+ f(x). Let C b (R + ) be the set of boune continuous functions f : R + R, let C 1 (R + ) be the set of once continuously ifferentiable functions f : R + R, an let C 1 b (R +) be the set of functions f in C 1 (R + ) that together with the first erivative f are boune on R +. If w C 1 (R + ) is consiere as a function of time, its first erivative will be enote by ẇ. For a Polish space (i.e., a complete separable metrizable space) S, let D([0, ), S) enote the space of right continuous functions from [0, ) into S that have left limits in S on (0, ). We enow this space with the Skoroho J 1 -topology. For a finite non-negative Borel measure ξ on R + an a ξ-integrable function f : R + R, efine f, ξ = fξ. R + If ξ = (ξ 1,..., ξ ) is a vector of such measures, then we use f, ξ to enote the vector ( f, ξ 1,..., f, ξ ). All functions f : R + R are extene to be ientically zero on (, 0) so that f( x) is well efine on R + for all x > 0. Let χ : R + R + enote the ientity function χ(x) = x. Let M be the set of finite non-negative Borel measures on R +, enowe with the weak topology: ξ k w ξ in M if an only if f, ξ k f, ξ for all f C b (R + ) as k. For I N, let M I = {(ξ 1,..., ξ I ) : ξ i M for all i I}. The spaces M an M I are Polish spaces. Convergence in M I is also enote ξ k w ξ. The zero measure in M is enote Flow level moel 2.1. Network structure Consier a network with finitely many resources labelle by j = 1,..., J, an a finite set of routes labelle by i = 1,..., I. A route i is a non-empty subset of {1,..., J}, interprete as the set of resources use by the route. Let A be the J I incience matrix satisfying A ji = 1 if resource j is use by route i, an A ji = 0 otherwise. Since each route is a non-empty subset of {1,..., J}, no column of A is ientically zero. A flow on route i is the continuous transfer of a ocument through the resources use by the route. Assume that, while being transferre, a flow takes simultaneous possession of all resources on its route. The processing rate allocate to a flow is the rate at which the ocument associate with the flow is being transferre. There may be multiple flows on a route, an the banwith Λ i allocate to route i is the sum of the processing rates allocate to flows on route i. The banwith allocate through resource j is the sum of the banwiths allocate to routes using resource j. Assume that each resource j J has finite capacity C j > 0, interprete as the maximum banwith that can be allocate through it. Let C = (C 1,..., C J ) be the vector of capacities in R J +. Then any vector Λ = (Λ 1,..., Λ I ) of banwith allocations must satisfy AΛ C. We further assume that A has rank J, so that it has full row rank.

5 2.2. Banwith sharing policy Flui moel for banwith sharing 5 We consier the network operating uner a banwith sharing policy first introuce by Mo an Walran [12]. Banwith is ynamically allocate to routes as a function of the number of flows on all routes, an the resulting allocation is share equally among iniviual flows on each route. Let Z i (t) enote the number of flows on route i I at time t, an let Z(t) = (Z 1 (t),..., Z I (t)) be the corresponing vector in R I +. The banwith allocate to route i at time t is a function of the vector Z(t) an is enote Λ i (Z(t)). The ( corresponing vector of banwith allocations at time t is given by Λ(Z(t)) = Λ1 (Z(t)),..., Λ I (Z(t)) ). Although the coorinates of Z( ) are non-negative an integer value, the function Λ is efine on the entire orthant R I + to accommoate flui analogues of Z( ) later. Fix a parameter α (0, ) an a vector of strictly positive weights κ = (κ 1,..., κ I ). For z R I +, let I 0 (z) = {i I : z i = 0} an I + (z) = {i I : z i > 0}. Let O(z) = {λ R I + : λ i = 0 for all i I 0 (z)}. Define a function G z : R I + [, ) by (2.1) G z (λ) = i I +(z) i I +(z) κ i z α i λ 1 α i 1 α, α (0, ) \ {1}, κ i z i log λ i, α = 1, where the value of G z (λ) is taken to be if α [1, ) an λ i = 0 for some i I + (z). For each z R I +, efine Λ(z) as the unique vector λ R I + that solves the optimization problem: (2.2) (2.3) (2.4) maximize G z (λ), subject to Aλ C, over O(z). The resulting allocation is calle a weighte α-fair allocation, an the function Λ : R I + R I + is calle a weighte α-fair banwith sharing policy. Note that by (2.3), (2.5) sup Λ(z) C. z R I + Note also that for any z R I +, Λ i (z) = 0 for all i I 0 (z). This implies that no banwith is allocate to routes with no flows. The banwith Λ i (Z(t)) allocate to route i at time t is share equally by all flows on the route. That is, if there are Z i (t) > 0 flows on route i at time t, then each flow on route i is allocate a processing rate of Λ i (Z(t))/Z i (t) at time t. When κ i = 1 for all i I, the cases α 0, α 1, an α correspon respectively to a banwith allocation which achieves maximum throughput, is proportionally fair or is max-min fair [1, 12]. Weighte α-fair allocations provie a tractable theoretical abstraction of ecentralize packet-base congestion control algorithms such as TCP, the transmission control protocol of the Internet, particularly when α = 2 an κ i is the reciprocal of the square of the roun trip time on route i.

6 6 H.C. Gromoll an R.J. Williams 2.3. Stochastic moel Fix a network structure (A, C) an a weighte α-fair banwith sharing policy Λ with parameters (α, κ). Our stochastic moel of ocument flows consists of the following: a collection of stochastic primitives E 1,..., E I an {v 1k } k=1,...,{v Ik} k=1 escribing the arrivals of ocument flows (incluing their sizes) to the network, a ranom initial conition Z(0) M I specifying the state of the system at time zero, an a collection of performance processes escribing the time evolution of the system state. The performance processes are efine in terms of the primitives an initial conition through a set of escriptive equations. The ranom objects involve are efine on a common probability space (Ω, F, P), with expectation operator E. The stochastic primitives consist of an exogenous arrival process E i an a sequence of ocument sizes {v ik } k=1 for each route i I. The arrival process E i is a rate ν i > 0 elaye renewal process with kth jump time U ik. For t 0, E i (t) represents the number of flows that have arrive to route i uring the time interval (0, t]. The kth such arrival is calle flow k on route i an arrives at time U ik ; flows alreay on route i at time zero are calle initial flows. For each i I an k 1, the ranom variable v ik represents the initial size of the ocument associate with flow k on route i. This is the cumulative amount of processing that must be allocate to the flow to complete its transfer throught the network. Assume that the ranom variables {v ik } k=1 are strictly positive an form an inepenent an ientically istribute sequence with common istribution ϑ i on R +. Assume that the mean χ, ϑ i (0, ) an let µ i = χ, ϑ i 1. Define the traffic intensity on route i by ρ i = ν i /µ i. The initial conition specifies Z(0) = (Z 1 (0),..., Z I (0)), the number of initial flows on each route at time zero, as well as the initial sizes of the ocuments on these flows at time zero. Assume that the components of Z(0) are nonnegative, integer value ranom variables. The initial ocument sizes of the initial flows on route i I are the first Z i (0) elements of a sequence {ṽ il } l=1 of strictly positive ranom variables. The performance processes consist of a measure value process Z, taking values in D([0, ), M I ), an a collection of auxiliary processes (Z, T, U, W ). The process Z = (Z 1,..., Z I ) takes values in D([0, ), R I +). For i I an t 0, Z i (t) is the number of flows on route i at time t. Recall that at time t, the banwith allocate to route i is Λ i (Z(t)), an this banwith is share equally by all Z i (t) flows on route i; each such flow receives a processing rate of Λ i (Z(t))/Z i (t), which equals zero by convention if Z i (t) = 0. Thus, a flow that is active on route i uring a time interval [s, t] [0, ) receives cumulative service uring [s, t] equal to (2.6) S i (s, t) = t s Λ i (Z(u)) u. Z i (u) Consier the kth flow on route i. This flow arrives at time U ik an has initial ocument size v ik. At time t U ik, the cumulative service receive by this flow uring [U ik, t] equals S i (U ik, t) v ik. The amount of service still require therefore equals (v ik S i (U ik, t)) +. For t 0, k E i (t), an l Z i (0), efine the resiual ocument size at time t of the kth flow on route i, an the lth initial flow on route i, by (2.7) v ik (t) = ( v ik S i (U ik, t) ) +, ṽ il (t) = ( ṽ il S i (0, t) ) +.

7 Flui moel for banwith sharing 7 The measure value process Z = (Z 1,..., Z I ) is calle the state escriptor; it tracks the resiual ocument sizes of flows on all routes at any given time. Let δ x + M enote the Dirac measure at x if x (0, ), with δ 0 + = 0. For t 0 an i I, (2.8) Z i (t) = We can recover Z from Z by Z i(0) l=1 E i(t) δ + ṽ il (t) + k=1 δ + v ik (t). (2.9) Z i (t) = 1, Z i (t), for all t 0, i I. The process T takes values in D([0, ), R I +) an tracks the cumulative banwith allocate to each route. For t 0 an i I, (2.10) T i (t) = t 0 Λ i (Z(s))s. The process U takes values in D([0, ), R J +) an tracks the cumulative unuse banwith capacity of each resource. For t 0, (2.11) U(t) = Ct AT (t). The process W takes values in D([0, ), R I +) an tracks the immeiate amount of work still to be transferre on each route. For t 0, (2.12) W (t) = χ, Z(t). Recall that χ(x) = x an that integration against the vector of measures Z(t) is interprete componentwise. 3. Flui moel Fix a network structure (A, C) an a weighte α-fair banwith sharing policy Λ with parameters (α, κ). This section efines a flui analogue of the stochastic moel introuce in Section 2.3. In [8], uner mil assumptions, it was shown that this flui moel is a first orer approximation (uner functional law of large numbers scaling) to the stochastic moel. As in the stochastic moel, fix a vector of positive arrival rates ν = (ν 1,..., ν I ) an a vector of probability measures ϑ = (ϑ 1,..., ϑ I ) in M I, satisfying the assumptions of Section 2. Recall that µ i = χ, ϑ i 1 an ρ i = ν i /µ i for each i I. The flui moel consists of a eterministic measure value function of time, calle the flui moel solution, an a collection of auxiliary functions of time efine below. Definition 3.1 Given a continuous function ζ : [0, ) M I, efine the auxiliary functions (z, τ, u, w) of ζ, with respect to the ata (A, C, α, κ, ν, ϑ), by for all t 0. z(t) = 1, ζ(t), t ( ) τ i (t) = Λ i (z(s))1 (0, ) (z i (s)) + ρ i 1 {0} (z i (s)) s, i I, 0 u(t) = Ct Aτ(t), w(t) = χ, ζ(t),

8 8 H.C. Gromoll an R.J. Williams Note that z(t) an τ(t) take values in R I + an u(t) takes values in R J +. On the other han, w(t) takes values in [0, ] I, as the flui moel solution nee not have finite first moments. The function ζ is the flui analogue of the measure value process Z. The functions z, τ, u, an w, are flui analogues of the processes Z, T, U, an W, which keep track of queue length, cumulative banwith allocation, unuse capacity an workloa, respectively. The equation satisfie by τ i may seem counterintuitive at first. However, the presence of the term involving ρ i is accounte for by the fact that in passing to a flui limit of the stochastic moel, banwith allocations mae when a queue length is near zero in the stochastic moel are average with the zero banwith allocations mae when a queue length is zero. The fact that ρ i is the correct form here is relate to the fact that when the flui workloa function w is real-value, at a positive time where it is ifferentiable (which occurs a.e.) an at which the value of w is zero, the erivative of the workloa function must be zero (cf. (3.2) below). The notion of a flui moel solution is efine via projections against test functions in the class C = {f C 1 b(r + ) : f(0) = f (0) = 0}. Definition 3.2 A flui moel solution for the ata (A, C, α, κ, ν, ϑ) is a continuous function ζ : [0, ) M I that, together with its first three auxiliary functions (z, τ, u), satisfies (i) 1 {0}, ζ(t) = 0 for all t 0, (ii) u j is non-ecreasing for all j J, (iii) for each f C, i I, an t 0, (3.1) f, ζ i (t) = f, ζ i (0) t 0 f, ζ i (s) Λ i(z(s)) z i (s) s + ν i f, ϑ i t 0 1 (0, ) (z i (s)) s. Recall that in (3.1), the integran in the first integral term is efine to be zero when its enominator is zero. The first integral term in (3.1) relates to the movement to the left of the ranom measure ζ i at the processing rate of Λ i (z(s))/z i (s), an the secon integral term correspons to new infusion of mass ue to new arrivals coming at a rate of ν i with a istribution of ϑ i for route i. The appearance of the inicator function in the last term again relates to the fact that in the flui limit, mass that was near zero in the stochastic moel can be crushe to zero in the scaling limit. To iscern the correct form for this term, one uses the fact that at a time t > 0 for which z i (t) = 0 an f, ζ i ( ) is ifferentiable, we must have that the time erivative of f, ζ i ( ) is zero. When the initial flui workloa is finite, we have the following result which is prove in [8]. Lemma 3.3 Suppose ζ is a flui moel solution with finite initial workloa, i.e., w i (0) = χ, ζ i (0) < for all i I. Then, for each i I an t 0, (3.2) w i (t) = w i (0) + t = w i (0) + ρ i t τ i (t). 0 ( ρi Λ i (z(s)) ) 1 (0, ) (z i (s)) s In particular, the flui workloa w i (t) is finite for all t 0 an i I.

9 Flui moel for banwith sharing 9 Fig 1. A linear network with 3 resources (enote by circles) an 4 routes (enote by line segments) For later use, when ζ( ) is a flui moel solution with finite initial workloa an flui workloa function w, we efine υ : [0, ) R J + by (3.3) υ(t) = Aw(t), t 0, so that the jth component of υ(t) efines the flui workloa at resource j at time t. In other wors, υ is a resource level workloa, whereas w is a route level workloa. 4. Flui stability for some network topologies In this section, we use Lyapunov functions to show stability of flui moel solutions with finite initial workloa for two types of network topologies, linear networks an tree networks, uner the nominal conition: (4.1) A ji ρ i < C j for all j J, i I i.e., the average loa place on each resource is less than its capacity. (We note that it follows from the characterization of invariant states for the flui moel given in [8] that uner this nominal conition, the only invariant state is the zero state.) We assume that (4.1) hols henceforth. Let (4.2) ε = min j J C j A ji ρ i, i I so that ε > Linear Network A linear network consists of J resources an I = J + 1 routes where route j consists of resource j alone for j = 1,..., J an route J + 1 consists of all of the J resources. A schematic of such a network is shown in Figure 1 for J = 3. Consier a flui moel solution ζ with finite initial workloa w(0) = χ, ζ(0) an associate resource level workloa function υ as efine in (3.3). Consier the Lyapunov function H : R J + R + efine by (4.3) H(υ) = max j J υ j. A Lipschitz continuous function x : [0, ) R is absolutely continuous, hence it is ifferentiable almost everywhere an it can be recovere by integration from

10 10 H.C. Gromoll an R.J. Williams its a.e. efine erivative. We call a point at which such a Lipschitz continuous function is ifferentiable a regular point for the function. The auxiliary functions τ i : [0, ) R +, i I, are Lipschitz continuous, an hence so too are u j, j J, w i, i I an υ j, j J. The function H( ) is Lipschitz continuous an hence so too is H(υ( )). Let t > 0 be a regular point for H(υ( )), τ i, w i : i I, u j, υ j : j J, such that for all i I, (4.4) τ i (t) = Λ i (z(t))1 (0, ) (z i (t)) + ρ i 1 {0} (z i (t)), (such points occur a.e.). Suppose that H(υ(t)) > 0 an let Then, J t = {j J : H(υ(t)) = υ j (t)}. H(υ(t)) = υ j (t) for j J t, H(υ(t)) > υ j (t) for j / J t, an by the fact that t > 0 is a regular point for H(υ( )) an υ j, j J t, we have (cf. [6], Section 3), (4.5) t H(υ(t)) = υ j(t) for all j J t. Now, by Lemma 3.3 an (4.4), (4.6) υ j (t) = A ji w i (t) = A ji (ρ i Λ i (z(t)))1 (0, ) (z i (t)). i I i I We consier two cases. Case (a). Suppose z j (t) > 0 for some j J t. Then by the efinition of Λ( ) an the fact that route j just contains resource j, it follows that the full capacity of resource j will be use by Λ(z(t)), i.e., A ji Λ i (z(t))1 (0, ) (z i (t)) = C j. i I Thus, for this j J t, (4.6) becomes υ j (t) = i I A ji ρ i 1 (0, ) (z i (t)) C j i I A ji ρ i C j ε < 0, by the assumption (4.1). It follows that in Case (a), H(υ(t)) ε. t Case (b). Suppose z j (t) = 0 for all j J t. Then w j (t) = 0 for all j J t an since υ j (t) = w j (t) + w J+1 (t), j J,

11 we have Since Flui moel for banwith sharing 11 υ j (t) = w J+1 (t) for all j J t. w J+1 (t) υ l (t) < υ j (t) for all l / J t, j J t, it follows that J t = {1,..., J}, an υ j (t) = w J+1 (t) for all j J. By Lemma 3.3 an (4.4), since H(υ(t)) = w J+1 (t) > 0 an hence z J+1 (t) > 0, we have (4.7) ẇ J+1 (t) = ρ J+1 Λ J+1 (z(t)). Since z j (t) = 0 for all j J, Λ j (z(t)) = 0 for all j J, an it follows from the efinition of Λ(z(t)) as the solution of an optimization problem where at least one constraint must be bining, that there is at least one j J such that Λ J+1 (z(t)) = C j. Here C j > ρ j + ρ J+1 by (4.1). It follows that, for this j, ẇ J+1 (t) = ρ J+1 C j < ρ j + ρ J+1 C j ε < 0. Hence in Case (b), t H(υ(t)) = ẇ J+1(t) ε. Thus, in either Case (a) or (b), at the regular point t > 0, H(υ(t)) ε when H(υ(t)) > 0. t Since H(υ( )) is non-negative, it follows from Lemma 2.2 of Dai an Weiss [6] that H(υ(t)) = 0 for all t δ = H(υ(0))/ε. We summarize the above analysis as follows. Lemma 4.1 Consier a linear network satisfying the conition (4.1) an let ε > 0 be as efine in (4.2). Suppose that ζ is a flui moel solution with finite initial workloa w(0) = χ, ζ(0). Then where δ = max j J υ j (0)/ε. ζ(t) = 0 for all t δ, In the above sense, the flui moel for any linear network is stable uner the natural conition (4.1) Tree network As pointe out by Bonal an Proutière [2], tree networks, as illustrate in Figure 2, are practically interesting as they may represent an access network consisting of several multiplexing stages. Furthermore [2], they typically exhibit sensitivity to ocument size istributions.

12 12 H.C. Gromoll an R.J. Williams A tree network consists of J 2 resources an I = J 1 routes such that a single resource (labele J an referre to as the trunk) belongs to all routes an each of the other resources (labele by 1,..., J 1) belongs to a single route. Proceeing in a similar manner to that for the linear network, consier a flui moel solution ζ with finite initial workloa χ, ζ(0). We use the total workloa function H : R+ J 1 R + efine by J 1 (4.8) H(w) = as a Lyapunov function. Note that H(w( )) = υ J ( ), the resource level workloa for the trunk resource J. Suppose t > 0 is a regular point for τ i, i J 1, such that for all i J 1, i=1 w i τ i (t) = Λ i (z(t))1 (0, ) (z i (t)) + ρ i 1 {0} (z i (t)), (such points t occur a.e.) Then t is a regular point for all w i, i J 1. Suppose H(w(t)) > 0. Then by Lemma 3.3, (4.9) We consier two cases. Case (a). Suppose t H(w(t)) = i J 1 i J 1 (ρ i Λ i (z(t)))1 (0, ) (z i (t)). Λ i (z(t))1 (0, ) (z i (t)) = C J. Then by (4.9) an (4.1) with j = J, we have Case (b). Suppose t H(w(t)) i J 1 i J 1 ρ i C J ε. Λ i (z(t))1 (0, ) (z i (t)) < C J. Then, by the efinition of Λ(z(t)), we must have (4.10) Λ i (z(t)) = C i for those i J 1 satisfying z i (t) > 0. For if not, the value of Λ i (z(t)) coul be increase on some non-empty route i without exceeing the capacity of the resources i an J on that route. From (4.9) an (4.10), it follows that t H(w(t)) = i J 1 ε < 0, (ρ i C i )1 (0, ) (z i (t)) since ρ i < C i for all i J 1 by (4.1), an since z i (t) > 0 for some i J 1 as H(w(t)) > 0. Thus, in either Case (a) or (b), H(w(t)) ε < 0, when H(w(t)) > 0. t

13 Flui moel for banwith sharing 13 Fig 2. A tree network with 4 resources an 3 routes Since H(w( )) is non-negative, it follows from Lemma 2.2 of [6] that H(w(t)) = 0 for all t δ = H(w(0))/ε. We summarize the above analysis as follows. Lemma 4.2 Consier a tree network satisfying the conition (4.1) an let ε > 0 be as efine in (4.2). Suppose that ζ is a flui moel solution with finite initial workloa w(0) = χ, ζ(0). Then where δ = i J 1 w i(0)/ε. ζ(t) = 0 for all t δ, References [1] Bonal, T. an Massoulié, L. (2001). Impact of fairness on Internet performance. In Proceeings of ACM Sigmetrics [2] Bonal, T. an Proutiére, A. (2003). Insensitive banwith sharing in ata networks. Queueing Systems [3] Bramson, M. (2005). Stability of networks for max-min fair routing. Presentation at the 13th INFORMS Applie Probability Conference, Ottawa. [4] Chiang, M., Shah, D. an Tang, A. K. (2006). Stochastic stability uner network utility maximization: general file size istribution Preprint. [5] Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: a unifie approach via flui limit moels. Annals of Applie Probability [6] Dai, J. G. an Weiss, G. (1996). Stability an instability of flui moels for re-entrant lines. Mathematics of Operations Research [7] e Veciana, G., Konstantopoulos, T. an Lee, T.-J. (2001). Stability an performance analysis of networks supporting elastic services. IEEE/ACM Trans. Netw [8] Gromoll, H. C. an Williams, R. J. (2006). Flui limit of a network with fair banwith sharing an general ocument size istributions. Preprint. [9] Lakshmikantha, A., Beck, C. L. an Srikant, R. (2004). Connection level stability analysis of the internet using the sum of squares (sos) techniques. In Conference on Information Sciences an Systems, Princeton. [10] Lin, X. an Shroff, N. (2004). On the stability region of congestion control. In Proceeings of the Allerton Conference on Communications, Control an Computing. [11] Massoulié, L. (2005). Structural properties of proportional fairness: stability an insensitivity. Preprint.

14 14 H.C. Gromoll an R.J. Williams [12] Mo, J. an Walran, J. (2000). Fair en-to-en winow-base congestion control. IEEE/ACM Transactions on Networking [13] Roberts, J. an Massoulié, L. (2000). Banwith sharing an amission control for elastic traffic. Telecommunication Systems [14] Srikant, R. (2004). On the positive recurrence of a Markov chain escribing file arrivals an epartures in a congestion-controlle network. Presente at the IEEE Computer Communications Workshop.