1 Rate control for communication networks: shaow prices, proportional fairness an stability F Kelly, AK Maulloo an DKH Tan University of Cambrige, UK This paper analyses the stability an fairness of two classes of rate control algorithm for communication networks. The algorithms provie natural generalisations to large-scale networks of simple aitive increase/multiplicative ecrease schemes, an are shown to be stable about a system optimum characterise by a proportional fairness criterion. Stability is establishe by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provies a Lyapunov function for the ynamical system e ne by the rate control algorithm. The network's optimisation problem may be cast in primal or ual form: this leas naturally to two classes of algorithm, which may be interprete in terms of either congestion inication feeback signals or explicit rates base on shaow prices. Both classes of algorithm may be generalise to inclue routing control, an provie natural implementations of proportionally fair pricing. Keywors: ATM network; congestion inication; elastic traf c; Internet; Lyapunov function; proportionally fair pricing; queues; routing Introuction The esign an control of moern communication networks raises several issues well suite to stuy using techniques of operational research such as optimisation, network programming an stochastic moelling. In this paper we illustrate this theme, through the presentation an analysis of a mathematical moel that arises in connection with the evelopment an eployment of large-scale broaban networks. In future communication networks there are expecte to be applications that are able to moify their ata transfer rates accoring to the available banwith within the network. Traf c from such applications is terme elastic; 1 a typical current example is TC traf c over the Internet, 2 an future examples may inclue the controlle-loa service of the Internet Engineering Task Force 3 an the Available Bit Rate transfer capability of ATM (asynchronous transfer moe) networks. 4 The key issue we aress in this paper concerns how the available banwith within the network shoul be share between competing streams of elastic traf c; in particular, we present a tractable mathematical moel an use it to analyse the stability an fairness of a class of rate control algorithms. Traitionally stability has been consiere an engineering issue, requiring an analysis of ranomness an feeback operating on fast time-scales, while fairness has been consiere an economic issue, involving static Corresponence: Frank Kelly, Statistical Laboratory, University of Cambrige, 16 Mill Lane, Cambrige CB2 1SB, UK. f.p.kellystatslab.cam.ac.uk comparisons of utility. In future networks the intelligence embee in en-systems, acting on behalf of human users, is likely to lessen the istinction between engineering an economic issues an increase the importance of an interisciplinary view. (This general theme was the subject of the 1996 Blackett Memorial Lecture; further aspects are evelope elsewhere, see Reference 5). There is a substantial literature on rate control algorithms, recently reviewe by Hernanez-Valencia et al. 6 Key early papers of Jacobson 2 an Chiu an Jain 7 ienti e the avantages of aaptive schemes that either increase ows linearly or ecrease ows multiplicatively, epening on the absence or presence of congestion. Important recent papers of Bolot an Shankar, 8 Fenick et al 9 an Bonomi et al 1 have analyse the stability of networks with a single bottleneck resource, where congestion is signalle by the buil-up of a queue at the bottleneck's buffer, an where propagation elays are signi cant. (In wie-area networks propagation times may be signi cant in comparison with queueing times: for a transatlantic link of 6 Megabits per secon, ten million bits may be in ight between queues.) The framework we aopt in this paper is simpler than that analyse by these authors in that we irectly moel only rates an not queue lengths, but more complex in that we moel a network with an arbitrary number of bottleneck resources. Theoretical work 11,12 on queues serving the superposition of a large number of streams inicates circumstances when the busy perio preceing a buffer over ow may be relatively short, an several authors have argue the avantages of preventing queue buil-up through the bouning of rates (see Charny et al ). 13
2 Any iscussion of the performance of a rate control scheme must aress the issue of fairness, since there exist situations where a given scheme might maximise network throughput, for example, while enying access to some users. The most commonly iscusse fairness criterion is that of max±min fairness: loosely, a set of rates is max±min fair if no rate may be increase without simultaneously ecreasing another rate which is alreay smaller. In a network with a single bottleneck resource max±min fairness implies an equal share of the resource for each ow through it. Mazumar et al 14 have pointe out that from a game-theoretic stanpoint such an allocation is not special, an have avocate instea the Nash bargaining solution, from cooperative game theory, as capturing natural assumptions as to what constitutes fairness. The nee for networks to operate in a public (an therefore potentially non-cooperative) environment has stimulate work on charging schemes for broaban networks: see Kelly 15 for a scheme base on time an volume measurements for non-elastic traf c, MacKie- Mason an Varian 16 for a escription of a `smart market' base on a per-packet charge when the network is congeste, an the collection eite by McKnight an Bailey 17 for several further papers an references. Kelly 18 escribes a moel for elastic traf c in which a user chooses the charge per unit time that the user is willing to pay; thereafter the user's rate is etermine by the network accoring to a proportional fairness criterion applie to the rate per unit charge. It was shown that a system optimum is achieve when users' choices of charges an the network's choice of allocate rates are in equilibrium. There remaine the question of how the proportional fairness criterion coul be implemente in a large-scale network. In this paper we show that simple rate control algorithms, using aitive increase/multiplicative ecrease rules or explicit rates base on resource shaow prices, can provie stable convergence to proportional fairness per unit charge, even in the presence of ranom effects an elays. Mechanisms by which supply an eman reach equilibrium have, of course, long been a central concern of economists, an there exists a substantial boy of theory on the stability of what are terme tatonnement processes. 19±21 From this viewpoint the rate control algorithms escribe in this paper are particular emboiments of a `Walrasian auctioneer', searching for market clearing prices. The `Walrasian auctioneer' of tatonnement theory is usually consiere a rather implausible construct; we show that the structure of a communication network provies a natural context within which to investigate the consequences for a tatonnement process of stochastic perturbations an time lags. The organisation of the paper is as follows. In the next section we escribe our basic moel of a network, escribe two classes of rate control algorithm, an provie an outline of our results. Detaile proofs are provie in the next two sections, following which we illustrate our theoretical results through a iscussion of some numerical examples. We then consier user aaptation an routing, an nally conclue with some remarks on open issues. Outline of results The basic moel Consier a network with a set J of resources, an let C j be the nite capacity of resource j, for j 2 J. Let a route r be a non-empty subset of J, an write R for the set of possible routes. Set A jr ˆ 1ifj 2 r, so that resource j lies on route r, an set A jr ˆ otherwise. This e nes a ±1 matrix A ˆ A jr ; j 2 J; r 2 R. Associate a route r with a user, an suppose that if a rate x r is allocate to user r then this has utility U r x r to the user. Assume that the utility U r x r is an increasing, strictly concave an continuously ifferentiable function of x r over the range x r 5 (following Shenker, 1 we call traf c that leas to such a utility function elastic traf c). Assume further that utilities are aitive, so that the aggregate utility of rates x ˆ x r ; r 2 R is r2r U r x r. Let U ˆ U r ; r 2 R an C ˆ C j ; j 2 J. Uner this moel the system optimal rates solve the following problem. SYSTEM(U, A, C): subject to over max r2r U r x r Ax 4 C x 5 : While this optimisation problem is mathematically fairly tractable (with a strictly concave objective function an a convex feasible region), it involves utilities U that are unlikely to be known by the network. We are thus le to consier two simpler problems. Suppose that user r may choose an amount to pay per unit time, w r, an receives in return a ow x r proportional to w r,sayx r ˆ w r =l r, where l r coul be regare as a charge per unit ow for user r. Then the utility maximisation problem for user r is as follows. USER r U r ; l r : over max U r w r l r w r 5 : w r Suppose next that the network knows the vector w ˆ w r ; r 2 R, an attempts to maximize the function
3 r w r log x r. The network's optimisation problem is then as follows. NETWORK(A, C; w): subject to over max r2r w r log x r Ax 4 C x 5 : It is known 18 that there always exist vectors l ˆ l r ; r 2 R, w ˆ w r ; r 2 R an x ˆ x r ; r 2 R, satisfying w r ˆ l r x r for r 2 R, such that w r solves USER r U r ; l r for r 2 R an x solves NETWORK(A, C; w); further, the vector x is then the unique solution to SYSTEM(U, A, C). A vector of rates x ˆ x r ; r 2 R is proportionally fair if it is feasible, that is x 5 an Ax 4 C, an if for any other feasible vector x*, the aggregate of proportional changes is zero or negative: x* r x r 4 : 1 r2r x r If w r ˆ 1; r 2 R, then a vector of rates x solves NETWORK A; C; w if an only if it is proportionally fair. Such a vector is also the Nash bargaining solution (satisfying certain axioms of fairness 22 ), an, as such, has been avocate in the context of telecommunications by Mazumar et al. 14 A vector x is such that the rates per unit charge are proportionally fair if x is feasible, an if for any other feasible vector x* x* w r x r r 4 : 2 r2r x r The relationship between the conitions (1) an (2) is well illustrate when w r ; r 2 R, are all integral. For each r 2 R, replace the single user r by w r ientical sub-users, construct the proportionally fair allocation over the resulting r w r users, an provie to user r the aggregate rate allocate to its w r sub-users; then the resulting rates per unit charge are proportionally fair. This construction also illustrates the nee to aapt the notion of fairness to a non-cooperative context, where it is possible for a single user to represent itself as several istinct users. It is straightforwar to check 18 that a vector of rates x solves NETWORK A; C; w if an only if the rates per unit charge are proportionally fair. We note in passing that if, for a xe set of users an arbitrary parameters w ˆ w r ; r 2 R, the network solves NETWORK A; C; w, then the resulting rates x ˆ x r ; r 2 R solve a variant of the problem SYSTEM U; A; C, with a weighte objective function r a ru r x r where a r ˆ w r = x r Ur x r for r 2 R. Thus a choice of the parameters w ˆ w r ; r 2 R by the network (rather than by users) correspons to an implicit weighting by the network of the relative utilities of ifferent users, with weights relate to the users' various marginal utilities. Uner the ecomposition of the problem SYSTEM U; A; C into the problems NETWORK A; C; w an USER r U r ; l r ; r 2 R, the utility function U r x r is not require by the network, an only appears in the optimisation problem face by user r. The Lagrangian 23 for the problem NETWORK A; C; w is L x; z; m ˆ w r log x r m T C Ax z r2r where z 5 is a vector of slack variables an m is a vector of Lagrange multipliers (or shaow prices). Then L ˆ wr m x r x j ; r an so the unique optimum to the primal problem is given by wr x r ˆ S m j where x r ; r 2 R, m j ; j 2 J solve 3 m5; Ax 4 C; m T C Ax ˆ 4 an relation (3). Furthermore the associate ual problem quickly reuces, after elision of terms not epenent on the shaow prices m, to the following problem. DUAL A; C; w : max w r log m j m j C j r2r j2j j2j over m5: While the problems NETWORK A; C; w an DUAL A; C; w are mathematically tractable, it woul be if cult to implement a solution in any centralise manner. A centralise processor, even if it were itself completely reliable an coul cope with the complexity of the computational task involve, woul have its lines of communication through the network vulnerable to elays an failures. Rather, interest focuses on algorithms which are ecentralize an of a simple form: the challenge is to unerstan how such algorithms can be esigne so that the network as a whole reacts intelligently to perturbations. Next we escribe two simple classes of ecentralise algorithm, esigne to implement solutions to relaxations of the problems NETWORK A; C; w an DUAL A; C; w.
4 A primal algorithm Consier the system of ifferential equations t x r t ˆk w r x r t m j t where m j t ˆp j s:j2s 5 x s t : 6 (Here an throughout we assume that, unless otherwise speci e, r ranges over the set R an j ranges over the set J.) We may motivate the relations (5)±(6) in several ways. For example, suppose that p j y is a price charge by resource j, per unit ow through resource j, when the total ow through resource j is y. Then by ajusting the ow on route r; x r t, in accorance with (5)±(6), the network attempts to equalise the aggregate cost of this ow, x r t m j t, with a target value w r, for every r 2 R. (For an enlightening escription of the technological implementation of such algorithms in an ATM network, see Courcoubetis et al 24 ). For an alternative motivation, suppose that resource j generates a continuous stream of feeback signals at rate p j y when the total ow through resource j is y. Suppose further that when resource j generates a feeback signal, a copy is sent to each user r whose route passes through resource j, where it is interprete as a congestion inicator requiring some reuction in the ow x r. Then (5) correspons to a response by user r that comprise two components: a steay increase at rate proportional to w r, an a multiplicative ecrease at rate proportional to the stream of feeback signals receive. (For early iscussions of algorithms with aitive increase an multiplicative ecrease see Chiu an Jain 7 an Jacobson 2 ; Hernanez-Valencia et al 6 review several algorithms base on congestion inication feeback.) Later we establish that uner mil regularity conitions on the functions p j ; j 2 J, the expression u x ˆ r2r w r log x r j2j Ss:j2s x s p j y y provies a Lyapunov function for the system of ifferential equations (5)±(6), an we euce that the vector x maximising u x is a stable point of the system, to which all trajectories converge. The functions p j ; j 2 J, may be chosen so that the maximisation of the Lyapunov function u x arbitrarily closely approximates the optimisation problem NETWORK A; C; w, an, in this sense, is a relaxation of the network problem. In our penultimate section we shall see that certain relaxations correspon naturally to a system objective which takes into account loss or elays, as well as ow rates. 7 The Lyapunov function (7) thus provies an enlightening analysis of the global stability of the system (5)±(6), an of the relationship between this system an the problem NETWORK A; C; w. However, the system (5)±(6) has omitte to moel two important aspects of ecentralise systems, namely stochastic perturbations, an time lags. We analyse these aspects by consiering small perturbations to the stable point x. Stochastic perturbations within the network may well arise from a resource's metho of sensing its loa. Equations (6) represents the response m j t of resource j as a continuous function of a loa, y ˆ s:j2s x s, which is assume known. In practice a resource may assess its loa by an error-prone measurement mechanism, an then choose a feeback signal from a small set of possible signals. (See Hernanez-Valencia et al 6 an Bonomi et al 1 for more etaile escriptions of binary feeback an congestion inication rate control algorithms.) In the next section we escribe how such mechanisms motivate various stochastic moels of the network. One particular moel takes the form x r t ˆk w r t x r t m j t t m j t 1=2 e 1=2 j B j t where B j t is a stanar Brownian motion, representing stochastic effects at resource j, an e j is a scaling parameter for these effects. If the scaling parameters e j ; j 2 J, are small then the stochastic ifferential equation (8) has, as solution, a multiimensional Ornstein±Uhlenbeck process, centre on the stable point x of the ifferential equations (5)±(6). The stationary istributions for x r t ; r 2 R is a multivariate normal istribution, with covariance matrix that can be explicitly calculate in terms of the parameters of the network. Similarly we shall escribe a moel incorporating time lags that generalises (5)±(6), an shall analyse its behaviour close to the stable point x. Our moels of stochastic effects an of time-lags provie important insights into the behaviour of the network, an allows us to quantify the various relationships an trae-offs between spee of convergence, the magnitue of uctuations about the equilibrium point, an the stability of the network. A ual algorithm The equations (5)±(6) represent a system where rates vary graually, an shaow prices are given as functions of the rates. Next we consier a system where shaow prices vary graually, with rates given as functions of the shaow prices. Let t m j t ˆk x r t q j m j t 9 8
5 where x r t ˆ w r k2r m k t : 1 The relationship between the algorithm (9)±(1) an the problem DUAL A; C; w parallels that between the primal algorithms (5)±(6) an the problem NETWORK A; C; w, an, again, we may motivate the algorithm in several ways. For example, suppose that q j Z is the ow through resource j which generates a price at resource j of Z. Then an economist woul escribe the right han sie of (9) as the vector of excess eman at prices m j t ; j 2 J, an woul recognise (9)±(1) as a tatonnement process by which prices ajust accoring to supply an eman (Varian, 21 Chapter 21). Later we establish that uner mil regularity conitions on the functions q j, j 2 J, the expression v m ˆ w r log m j mj q j Z Z 11 r2r j2j provies a Lyapunov function for the system of ifferential equations (9)±(1), an we euce that the vector m maximising v m is a stable point of the system, to which all trajectories converge. Further, by appropriate choice of the functions q j ; j 2 J, the maximisation of the function v m can arbitrarily approximate the problem DUAL A; C; w. We consier stochastic perturbations of system (9)±(1), with a typical example taking the form m j t ˆk x r t t x r t 1=2 e 1=2 B r t q j m j t t r 12 where B r t is a stanar Brownian motion, representing stochastic effects associate with the ow on route r. If the scaling parameters e r ; r 2 R, are small then the stationary istribution for m j t ; j 2 J is centre on the stable point m of the ifferential equations (9)±(1), with a covariance matrix that can be explicitly calculate in terms of the parameters of the network. Also it is possible to analyse the stability of the moel (9)±(1) when time-lags are introuce. User aaptation Our analyses of the primal algorithm (5)±(6) an the ual algorithm (9)±(1) assume that the parameters w r ; r 2 R chosen by the users are xe, at least on the time scales concerne in the analyses. With increasing intelligence embee in en-systems, users may in the future be able to vary the parameters w r ; r 2 R even within these short time scales. Both the algorithms may be extene to this situation. Suppose that user r is able to monitor its rate x r t continuously, an to vary smoothly the parameter w r t so as to track accurately the optimum to USER r U r ; l r t, where l r t ˆw r t =x r t is the charge per unit ow to user r at time t. Then, using revise Lyapunov functions, stability of both the primal an ual algorithms may again be establishe. Our next sections provie etaile proofs of the various results outline above together with some numerical illustrations. In our penultimate section we shall look again at the system ecomposition relating the problems SYSTEM U; A; C an NETWORK A; C; w, an exten the iscussion to inclue routing control. A primal algorithm In this section we establish the global stability of the primal algorithm (5)±(6), etermine the rate of convergence, an, by consiering perturbations about the stable point, moel stochastic effects an time lags. Global stability Let the function u x be e ne by (7) where w r > ; r 2 R, an suppose that, for j 2 J, the function p j y ; y 5, is a non-negative, continuous, increasing function of y, not ientically zero. Theorem 1 The strictly concave function u x is a Lyapunov function for the system of ifferential equations (5)±(6). The unique value x maximising u x is a stable point of the system, to which all trajectories converge. roof. The assumptions on w r > ; r 2 R, an p j ; j 2 J, ensure that u x is strictly concave on x 5 with an interior maximum; the maximising value of x is thus unique. Observe that u x ˆwr p x r x j r x s ; 13 s: j2s setting these erivatives to zero ienti es the maximum. Further t u x t ˆ u x r t x r t r2r ˆ k 1 r2r x r t w r x r t p j 2 x x t ; s: j2s establishing that u x t is strictly increasing with t, unless x t ˆx, the unique x maximising u x. The function u x is thus a Lyapunov function for the system (5)±(6), an the theorem follows (see Reference 25, Chapter 5). u De ne the continuous functions p j y ˆ y C j e =e 2 for j 2 J. Then, as e, the maximisation of the Lyapunov
6 function u x approximates arbitrarily closely the primal problem NETWORK A; C; w ; in particular the vector x maximizing u x approaches the solution x to relations (3) an (4). Note, however, that the erivative p j y may become arbitrarily large as the approximation improves. Rate of convergence We have seen, in Theorem 1, that the system (5)±(6) converges to a unique stable point: next we investigate the rate of convergence, by linearisation about the stable point. Let x ientify the unique vector maximising u x, let m j ˆ p j s: j2s x s, an suppose p j is ifferentiable at this point, with erivative p j. Let x r t ˆx r x 1=2 r y r t. Then, linearising the system (5)±(6) about x, we obtain t y r t ˆ k y r t m j x 1=2 r p j x 1=2 s y s t s: j2s ˆ k w r y x r t x 1=2 r p ja jr A js x 1=2 s y s t : r We may write this in matrix form as t y t ˆ k WX 1 X 1=2 A T AX 1=2 y t 14 where X ˆ iag x r ; r 2 R, W ˆ iag w r ; r 2 R an ˆ iag p j; j 2 J. Let G T FG ˆ WX 1 X 1=2 A T AX 1=2 j s 15 where G is an orthogonal matrix, G T G ˆ I, an F ˆ iag f r ; r 2 R is the matrix of eigenvalues, necessarily positive, of the real, symmetric, positive e nite matrix (15). Then t y t ˆ kgt FGy t ; 16 an thus the rate of convergence to the stable point is etermine by the smallest eigenvalue, f r ; r 2 R, of the matrix (15). Note that spee of convergence increases both with the gain parameter k an with the magnitue of the erivatives ; we shall see that this conclusion requires quali cation in the presence of either stochastic effects or of time-lags. Our early assumption that p j y ; j 2 J, are increasing functions is convenient an often natural: it implies that u is strictly concave with an interior maximum. If the functions p j y ; j 2 J, are not increasing, then u x may not have an interior maximum or it may have multiple stationary points: we escribe an example later. rovie p j y ; j 2 J; are ifferentiable at a stationary point, the local behaviour near the stationary point is escribe by (16). Stochastic analysis Next we consier a stochastic perturbation of the linearise equation (16). Let y t ˆ k G T FGy t t FB t 17 where F is an arbitrary jrjji j matrix an B t ˆ B i t ; i2 I is a collection of inepenent stanar Brownian motions, extene to 1 < t < 1. (Later we escribe how the moelling of ifferent sources of ranomness may lea to various explicit forms for the matrix F.) The stationary solution to the system (17) is t y t ˆ k e k t t GT FG FB t ; 1 18 as can be checke by ifferentiating both sies of (18) with respect to t. The solution (18) is a linear transformation of the Gaussian process B t ; t < t ; hence y t has a multivariate normal istribution, y t N ; S, where S ˆ E y t y t T Š ˆ k 2 e ktgt FG FF T e ktgt FG t 1 ˆ kg T e tf GFF T G T e tf t G: 1 De ne the symmetric matrix GF; FŠ by GF; FŠ rs ˆ e tf GFF T G T e tf t Then 1 ˆ GFFT G T Š rs f r f s : S ˆ kg T GF; FŠG: rs 19 Note that the covariance matrix increases linearly with the gain parameter k; ask increases, the faster convergence to equilibrium escribe by relation (16) is at the cost of a greater sprea at equilibrium. Varying the erivatives has a more subtle effect, through relation (15) an the construction (19), on the covariance matrix; broaly, as increases, not only is convergence to equilibrium faster, but also the sprea at equilibrium ecreases. However, we shall see later that, in the presence of time-lags, increasing may compromise stability. We next illustrate how various sources of ranomness may lea to ifferent covariance structures. Congestion inication with joint feeback. Consier the following stochastic version of the system (5)±(6). Let
7 N j t ; t5, for j 2 J, be a collection of inepenent unit rate oisson processes, an let x r t ˆk w r t x r t e j N j e 1 j t m j t t ; 2 where the functions m j, for j 2 J, are given by (6). Equation (2) woul escribe the following moel: resource j generates feeback signals inicating congestion as a time-epenent oisson process at rate e 1 j m j t ; when resource j generates a feeback signal, a copy is sent to each user r whose route passes through resource j; an user r reacts to such a feeback signal by reucing x r t by an amount ke j x r t. Now as e, the normalise oisson process en j t=e t e 1=2 ; t5 converges in istribution to a stanar Brownian motion. This motivates the approximation, vali when e j are small, of the oisson riving equation (2) by its Brownian version x r t ˆk w r t x r t m j t t e 1=2 j m j t 1=2 B j t where B j t ; t 5, for j 2 J, are a collection of inepenent stanar Brownian motions. The corresponing Brownian version of the linearise system (16) is just (17) where B t ˆ B j t ; j 2 J, an F is an jrjjjj matrix with elements Then F rj ˆ e 1=2 j m 1=2 j A jr x 1=2 r : 21 FF T ˆ X 1=2 A T EAX 1=2 ; x r t ˆk w r t e j N jr e 1 j 22 where E ˆ iag e j ; j 2 J an ˆ iag m j ; j 2 J, an hence the stationary covariance matrix S may be calculate from expression (19). Congestion inication with iniviual feeback. Consier next the oisson riving equation t x r t m j t t 23 where N jr t ; t5, for j 2 J; r 2 R, are a collection of inepenent unit rate oisson processes. This woul escribe the following moel: feeback signals from resource j to user r arise at rate e 1 j x r t m j t ; an user r reacts to such a feeback signal by reucing x r t by an amount ke j. The Brownian approximation, vali when e j are small, becomes x r t ˆk w r t x r t m j t t e 1=2 j x r t 1=2 m j t 1=2 B jr t ; whose linearisation is (17) where the jrjjj jjrj matrix F is given by F r; j;s ˆ e 1=2 j m 1=2 j A jr I r ˆ sš: 24 Thus FF T is the matrix iag m je j ; r 2 R, an the stationary covariance matrix S may be calculate from expression (19). Later we provie a numerical illustration of this calculation, an contrast the results erive from the forms (21) an (24). Source uctuations. Consier the Brownian riving equation x r t ˆk w r t x r t m j t t e 1=2 r x r t 1=2 B r t ; which might correspon to uctuations arising at sources, rather than within the network. For this system the jrjjrj matrix F is the iagonal matrix iag e 1=2 r ; r 2 R. Time lags Consier next the lagge, iscrete time system x r t 1Š ˆx r tš k w r x r tš m j t j; r Š where m j tš ˆp j s: j2s 25 x s t j; s Š ; 26 an j; r ; j 2 J; r 2 R, are non-negative integers. This might correspon to a moel of congestion inication with joint feeback, where there is a elay of j; r between resource j generating a feeback signal an user r receiving it, an another elay of j; r between user r changing its rate an the altere ow reaching resource j. Say that a vector x is an equilibrium point of the system (25)±(26) if x r tš ˆx r, for t ˆ...; ; 1; 2;...; satis es these equations. Theorem 2 The vector x maximising the strictly concave function u x is the unique equilibrium point of the system (25)±(26). roof. The vector x is an equilibrium point if an only if it solves w r ˆ x r x s : p j s: j2s
8 But this is precisely the stationarity conition implie by the partial erivatives (13) of the function u x, a strictly concave function with a unique maximum. u For small enough values of k the equilibrium point will be asymptotically stable, since if we replace k by k in (25) an let x r t ˆx r t=š, then as we may approximate arbitrarily closely a solution to (5)±(6). But for small values of k convergence to the equilibrium point is slow, an so it is of interest to investigate the local stability of the equilibrium point for general values of k. Let m j ˆ p j s: j2s x sš, an suppose p j is ifferentiable at this point, with erivative p j. Let x r tš ˆx r x 1=2 r y r tš. Then, linearising the system (25)±(26) about x, we obtain y r t 1Š ˆy r tš k y r tš m j x 1=2 r p j s: j2s y s t j; r j; s Š x 1=2 s effect on the convergence matrix (19) as ecreasing the gain parameter k; in contrast the estabilising effect on the matrix (29) of increasing is broaly the same as increasing k. For simplicity of notation we have use the same gain parameter k for each r 2 R. Ifk is replace by k r in (25), then we again obtain relations (28)±(29), but now with k interprete as the matrix iag k r ; r 2 R. An interesting topic concerns how the time elays within a network affect the choice of gain parameters; we might for example stuy the problem of choosing iag k r ; r 2 R in orer to minimize the spectral raius of the matrix L. There exist other natural iscrete time versions of the equation (5)±(6), an these too may be analyse in a similar manner. For example, consier the metho of repeate substitution x r t 1Š ˆw r = m j t j; r Š or its ampe version x r t 1Š ˆ 1 k x r tš k m j t j; r Š w r ˆ y r tš k w r x r y r tš j y s t j; r j; s Š s p ja jr A js x 1=2 r x 1=2 s 27 where m j tš is again given by (26). Then the linearise relations (28)±(29) are altere in that the top row of the matrix (29) becomes I k I XW 1 L Š ; kxw 1 L 1Š;...; kxw 1 L DŠ : De ne the jrjjrj matrices L Š; ˆ ; 1;...; D by L Š rs ˆ p ja jr A js x 1=2 I j; r j; s ˆŠ j r x 1=2 s where D ˆ max j;r;s f j; r j; s g. Thus D ˆ L Š ˆX 1=2 A T AX 1=2 ; the secon term of the key matrix (15). De ne the vector y tš ˆ y r tš; r 2 R. Then we can rewrite (28) in the matrix form 1 1 y t 1Š y tš y tš B. C A ˆ L y t 1Š B. C 28 A y t D 1Š y t DŠ where L ˆ I k WX 1 L Š kl 1Š kl 2Š... 1 kl DŠ I... I : B..... C A The equilibrium point x of the system (25)±(26) is stable if an only if the spectral raius of the matrix L is less than unity. Recall that in our moel of stochastic effects, increasing the erivatives ha broaly the same reuctive A ual algorithm In this section we investigate the stability of the ual algorithm (9)±(1), incluing a perturbation analysis of stochastic effects an time lags. Finally we note that the system (9)±(1) is just one example of a ual algorithm, an consier variants that share the Lyapunov function (11). Global stability Let the function v m be e ne by (11), where w r > ; r 2 R, an suppose that, for j 2 J; q j ˆ an q j Z ; Z5, is a continuous, strictly increasing function of Z. Theorem 3 The strictly concave function v m is a Lyapunov function for the system of ifferential equations (9)±(1). The unique value m maximising v m is a stable point of the system, to which all trajectories converge. roof. The assumptions on w r > ; r 2 R, an on q j ; j 2 J, ensure that v m is strictly concave on m5 with an interior maximum; the maximising value of m is thus unique, an is etermine by setting the erivatives v m ˆ w r m j k2r m q j m j 3 k
9 to zero. Also, t v m t ˆ v m j t m j t j2j ˆ k j2j 2 w r k2r m k t q j m j t ; establishing that v m t is strictly increasing with t, unless m t ˆm, the unique value m maximising v m. The function v m is thus a Lyapunov function for the system (9)± (1), an the theorem follows. 25 u The maximisation of the Lyapunov function v m becomes the ual problem if, for j 2 J; Z > ; q j Z ˆC j. These functions violate the assumption that q j Z is continuous at Z ˆ, but they may be arbitrarily closely approximate, for example by the functions q j Z ˆC j Z= Z e for small positive e. Note, however, that the erivative q j Z may become arbitrarily large as the approximation improves. Rate of convergence Let m ientify the unique vector maximising v m, let x r ˆ w r = k2r m k, an suppose q j y ifferentiable at the point y ˆ m j, with erivative q j. Let m j t ˆm j x j t. Then, linearising the system (9)±(1) about m, we obtain, after some reuction, t x t ˆ k AXW 1 XA T Q x t where W ˆ iag w r ; r 2 R an Q ˆ iag q j; j 2 J. Let Y T CY ˆ AXW 1 XA T Q 31 where Y is an orthogonal matrix, Y T Y ˆ I, an C ˆ iag c j ; j 2 J is the matrix of eigenvalues, necessarily non-negative, of the real, symmetric, positive semie nite matrix (31). Then t x t ˆ kyt CYx t ; 32 an thus the rate of convergence to the stable point is etermine by the smallest eigenvalue of the matrix (31). Note that the spee of convergence increases both with the gain parameter k an with the magnitue of the erivatives Q. Stochastic analysis Next consier a stochastic perturbation of the linearize equation (32). Let x t ˆ k Y T CYx t t GB t 33 where B t ˆ B i t ; I 2 I is a collection of inepenent stanar Brownian motions, an G is a jjjjij matrix. A similar analysis to that of the last section etermines the stationary covariance matrix S of x t. De ne the symmetrix matrix YG; CŠ by Then YG; CŠ jk ˆ YGGT Y T Š jk c j c k : S ˆ ky T YG; CŠY: 34 Note that the covariance matrix increases linearly with the gain parameter k; ask increases, the faster convergence to equilibrium escribe by relation (32) is at the cost of a greater sprea at equilibrium. Next we escribe an example illustrating how a moel of the form (33) might arise. Shaow prices inferre from uctuating ow rates. Consier the oisson riving equation m j t ˆk e r N r e 1 r t x r t t q j m j t t where N r t ; t5, for r 2 R, are a collection of inepenent unit rate oisson processes. This woul escribe a moel where, on a very ne time-scale, the ow on route r takes the form of a time-epenent oisson process of rate x r t =e r, with each point of the process containing a workloa of size e r. The Brownian approximation, vali when e r are small, becomes m j t ˆk x r t t e 1=2 x r t 1=2 B r t q j m j t t r whose linearisation is (33) where G is a jjjjrj matrix with elements G jr ˆ e 1=2 r x 1=2 r A jr ; thus GG T ˆ AXEA T where E ˆ iag e r ; r 2 R. Time lags Consier next the system m j t 1Š ˆm j tš k x r t j; r Š q j m j tš where w r x r tš k2r m k t k; r Š : A vector m is an equilibrium point of the system (35)±(36) if m j tš ˆm j, for t ˆ...; ; 1; 2;...; satis es these equations. Theorem 4 The vector m maximising the strictly concave function v m is the unique equilibrium point of the system (35)±(36).
10 roof. solves The vector m is an equilibrium point if an only if w r k2r m ˆ q j m j : k But this is precisely the stationarity conition implie by the partial erivatives (3) of the function v m, a strictly concave function with a unique maximum. The result follows. u Next we investigate the stability of the equilibrium point. Let x r ˆ w r = k2r m k, an suppose q j is ifferentiable at the point y ˆ m j, with erivative q j. Let m j tš ˆm j x j tš. Then, linearising the system (35)±(36) about m, we obtain x j t 1Š ˆx j tš k x 2 r w 1 r x k t j; r k; r Š k2r q jx j tš : 37 De ne the jjjjjj matrices M Š; ˆ ; 1;...; D by M Š jk ˆ x r A jr A kr I j; r k; r ˆŠ r where now D ˆ max j;k;r f j; r k; r g. Thus D ˆ M Š ˆAXW 1 XA T : De ne the vector x tš ˆ x j tš; j 2 J. Then we can rewrite (37) in the matrix form 1 1 x t 1Š x tš x tš B. C A ˆ M x t 1Š B. C A x t D 1Š x t DŠ where M ˆ I k M Š Q km 1Š km 2Š... 1 km DŠ I... I... B.... : C A... The equilibrium point m of the system (35)±(36) is stable if the spectral raius of the matrix M is less than unity. With stochastic effects, increasing the erivatives Q has broaly the same reuctive effect on the covariance matrix (34) as ecreasing the gain parameter k; in constrast the estabilising effect on the matrix (38) of increasing Q is broaly the same as increasing k. Variants Several variants of the primal algorithm (5)±(6) an the ual algorithm (9)±(1) allow a similar analysis. For example, if the right han sie of (5) is multiplie by a postive function f r x t ; m t then Theorem 1 remains vali. Similarly, if the right han sie of (9) is multiplie by a positive function f j x t ; m t then Theorem 3 remains vali. As a simple example, we coul ivie the right han sie of (5) by w r, or of (9) by q j m j t. A more subtle variation is obtaine if (9) is replace by t m j t ˆk p j x r t m j t ; 39 where p j is the inverse function of q j, an x r t is again given by (1). Note that the expression (39) is of the same sign as expression (9), an so the proof of Theorem 3 goes through as before. Suppose p j y is ifferentiable at the stable point with erivative p j, an let m j t ˆm j x j t. Then, linearising the system (39) about the equilibrium point, we obtain t x t ˆ k I AXW 1 XA T x t where ˆ iag p j; j 2 J, allowing the local convergence properties of the algorithm (39) to be stuie. Examples In this section we illustrate the results of the last two sections through a iscussion of some examples. The rst sub-section illustrates how the functions p j ; j 2 J, maybe etermine by the etaile stochastic behavior of resource j; a simple four noe network is use to facilitate comparisons between feeback mechanisms. The results of this paper are, of course, intene to apply to large-scale networks, an our secon sub-section iscusses the behaviour of a ual algorithm in a ranom network with 1 resources an 1 routes. Congestion inication in a four noe network Suppose that the total loa y on a resource takes the form, on a very ne time-scale, of a oisson stream of cells at rate y=e. Suppose that the time-axis is ivie into non-overlapping slots each of length te, an that a feeback signal is generate for a slot if the total number of cells arriving in that slot excees a threshol N. (While there may well be a queue at a resource, we suppose for the moment that the feeback signals are generate by the process just escribe, rather than, for example, by the queue size exceeing a threshol.) Suppose that when a feeback signal is generate, it is sent to each user r whose route passes through resource j, where it is interprete as a congestion inicator requiring a reuction in the rate x r t
11 of size kex r t. If the probability that a signal is generate in any single slot is small, then this moel correspons to (2), with e j ˆ e an yt yt n p j y ˆ1 e : 4 t n>n n Consier the network illustrate in Figure 1, where jjj ˆjRj ˆ4. Let w r ˆ :2; r 2 R, suppose p j y is given by (4), an choose N ˆ 128; t ˆ 5, so that the equilibrium point is x r ˆ 1:; r 2 R; m j ˆ :1, j 2 J. Then, from relations (19) an (22), the covariance matrix of the rates x r t ; r 2 R can be calculate to be 1 2:4 :8 :8 :8 S 1 ˆ ke1 2 :8 2:4 :8 :8 B C :8 :8 2:4 :8 A ; :8 :8 :8 2:4 a matrix whose form we shall iscuss shortly. For a secon example, suppose again that a feeback signal is generate by a slot when the total number of cells arriving in that slot excees a threshol, N. But now suppose that when a feeback signal is generate at resource j, it is irecte at a ranom route r with probability x r =y (for example, the signal might be sent to the route responsible for the last cell arriving uring the slot that generate the feeback signal). If user r receives a feeback signal, then the rate x r t is reuce by an amount ke. This moel correspons to (23), with e j ˆ e an p j y ˆ 1 yt yt n e : 41 ty n>n n Again let w r ˆ :2; r 2 R, an now choose N ˆ 125, t ˆ 5 so that, now using (41), x r ˆ 1:, r 2 R; m j ˆ :1, j 2 J, is once again an equilibrium point. Then, from relations (19) an (24), the covariance matrix of the rates x r t ; r 2 R can be calculate to be 1 1:5 1:2 1:1 1:2 S 2 ˆ ke1 1 1:2 1:5 1:2 1:1 B C 1:1 1:2 1:5 1:2 A : 1:2 1:1 1:2 1:5 Figure 1 A four noe network. (The function (41) is not an increasing function of y for all values of y. But it is increasing at the point y ˆ 2, hence the matrix (15) is positive e nite an this implies that the equilibrium point is locally stable. We note in passing that function (41) provies an example where, if the parameters w r ; r 2 R, are set too large, the function u x has no interior maximum an the system (5)±(6) has no equilibrium point). It is interesting to compare the magnitues, an structures of the matrices S 1 an S 2. That S 2 is larger in magnitue is expecte, since with iniviual feeback there are aitional sources of variation in the ranom choice of which rate is to be reuce by a feeback signal. Note also that rates on routes sharing a noe are positively correate for the joint feeback moel. The explanation is that, with joint feeback, congestion inication at a noe causes both routes through that noe to ecrease their rates simultaneously. However, for iniviual feeback, routes sharing a noe are negatively correlate. In this case, a ecrease in the ow on a route will allow increases on routes sharing a common noe with it. We have simulate (2) an (23), with e ˆ 1:; k ˆ :1, with results that agree well with the matrices S 1 an S 2. Suppose next that when the number of cells within a slot excees N, each of the cells within the slot causes a feeback signal to be sent to the user responsible for that cell. Suppose a user respons to each feeback signal by reucing its rate by ke. Then the expecte number of feeback signals generate per slot is an so n>n yt yt n ne ˆ yt yt yt n e n n5n n p j y ˆ yt yt n e ; 42 n 5 N n an expression rather similar to the form (4). The covariance structure of the moel epens on the size of N relative to the number of routes through a resource. If N is large an most or all routes through a resource receive a feeback signal when overloa occurs, then the covariance structure resembles that of the joint feeback moel; if N is small, an an essentially ranom set of routes receives a feeback signal, the covariance structure resembles more closely that of the iniviual feeback moel. Many other mechanisms for signal generation are possible. For example, suppose that cells pass through a buffer which acts as a single server queue, an signals are generate whenever a cell arrives to n the buffer above a threshol level. Uner oisson arrival assumptions the rate of signal generation, an hence p j y, may be etermine from the analysis of an M/D/1 queue. Note that the buffer may well be a virtual buffer, with service rate lower than that of a real buffer at the resource, in orer to signal congestion before the onset of cell loss. 26 ;
12 oisson streams an non-overlapping slots allow simple calculations, an are suggestive of the results that may be obtaine with more complex moels. 12 It is important to note, however, that the main results of earlier sections o not epen upon oisson assumptions; the erivation of the covariance structure (19) was base on a more general central limit approximation, while Theorems 1 an 3 rely only upon rather weak properties of the functions p j ; q j ; u or v. A ranom network Next we consier a network where the elements of the matrix A are inepenent ranom variables, each taking the value 1 with probability p an the value otherwise, an where the elements of the matrix are inepenent ranom variables uniformly istribute over the set f; 1;...; Dg. Let w r ˆ j A j; r, an let q j Z ˆZ s A j; s, so that the unique stable point for the system (9)±(1) is x r ˆ 1; r 2 R; m j ˆ 1; j 2 J. Consier the system m j t 1Š ˆm j tš k N r t j; r Š q j m j tš 43 where N r tš; t ˆ 1; 2;..., for r 2 R, are a collection of inepenent oisson ranom variables, an N r tš has mean x r tš as e ne by (36). The system (43) is thus a iscrete time version of the ual algorithm that combines both stochastic uctuations an time lags. Figures 2 an 3 illustrate the behaviour of ve ranomly chosen routes an ve ranomly chosen resources for the following parameter choices: J ˆ 1; R ˆ 1; p ˆ :1; D ˆ 1; k ˆ :5. For these parameter choices the average length of a route is 1, the average number of routes through a resource is 1, an the largest time elay between a source an a resource is 1 time units. Note that Figure 3 Shaow prices for three ranomly chosen resources. for this example rates oscillate within a narrower ban than shaow prices, an both are relatively well controlle. In Figure 4, the curve labelle a ˆ 1 recors the effect of the gain parameter k on the mean square eviation of shaow prices, s 2, e ne as the expecte value of m j t 1 2 average over all resources, j 2 J. For small values of k, the relationship is approximately linear, with a slope in goo agreement with that preicte by relation (34). However as k increases, the mean square eviation iverges, with an asymptote at the value of k (approximately.11) at which the spectral raius of the matrix (38) reaches unity an the eterministic time-lagge system becomes unstable. Finally, let us consier brie y the effect of more general choices for the functions q j ; j 2 J, escribing the relationship between ow rates an shaow prices at resources. Suppose that q j Z ˆ a Z 1 1 s A j; s, so that the unique stable point for the system (9)±(1) is again x r ˆ 1; r 2 R, m j ˆ 1; j 2 J. The case a ˆ 1 is that iscusse so far; the case a ˆ 2, also illustrate in Figure Figure 2 Rates on three ranomly chosen routes. Figure 4 Relation between the gain parameter k an the mean square eviation of shaow prices, s 2, for the resources of the ranom network. The parameter a labels the sensitivity of the relationship between ow rates an shaow prices at resources.
13 4, correspons to a oubling of the matrix of erivatives Q. As preicte by our earlier analysis, the effect of increasing a is to reuce variability for smaller values of k, but also to lower the critical value of k at which the system becomes unstable. User aaptation In this section we consier the stability of systems where users are able to aapt very quickly to their experience of congestion, an illustrate brie y how our methos exten to this situation. Suppose that user r is able to monitor its rate x r t continuously, an to vary smoothly the parameter w r t so as to track accurately the optimum to USER r U r ; l r t, where l r t ˆw r t =x r t is the charge per unit ow to user r at time t. A simple ifferentiation establishes that the solution to the problem USER r U r ; l r has w r ˆ x r Ur x r, where x r ˆ w r =l r. Thus, uner accurate tracking by user r of the optimum to USER r U r ; l r t, the parameter w r t will satisfy w r t ˆx r t U r x r t ; 44 while, for the primal algorithm, x r t evolves accoring to the revise ifferential equation t x r t ˆk w r t x r t m j t where m j t is given by (6). We shall establish stability of the revise system, by using a revision of the argument leaing to Theorem 1. Consier the revise expression Note that u x ˆ r2r U r x r j2j Ss: j2s x s p j y y: again be use to investigate behaviour near the stable point: for example, the revise form of (14) becomes t y t ˆ kx 1=2 A T A U X 1=2 y t where U ˆ iag Ur x r ; r 2 R. A similar analysis is possible for the ual algorithm. Uner accurate tracking by user r of the optimum to USER r U r ; l r t the parameter w r t will be given by (44), where, for the ual algorithm, x r t ˆ wr t k2r m k t an m j t evolves accoring to the ifferential equation (9). To n a Lyapunov function for this system, it is helpful to rst construct the ual to problem SYSTEM U; A; C. Let D r l ˆx r, where x r is the solution to l ˆ Ur x r, with D r l ˆ if l5ur an D r l ˆ1 if l4ur 1. Then, after elision of a constant term, the ual of the problem SYSTEM U; A; C becomes max lr Dr z z m j C j r2r j2j subject to over l4m T A m5 where the lower limit in the integral of the function D r can be chosen to be any xe value in the range Ur 1 ; Ur. We may interpret D r l as the eman of user r when confronte with a price per unit ow of l; uner accurate tracking by user r x r t ˆD r m j t : 45 Consier now the revise Lyapunov function u x ˆU x r x r p j r x s ; s: j2s S m j v m ˆ D r z z mj r2r j2j q j Z Z: an thus t u x t ˆ u x r t x r t r2r ˆ k 1 r2r x r t w r t x r t 2 p j x s t ; s: j2s using relation (44) to substitute for U r x r t. Hence u x provies a Lyapunov function for the revise system, an the unique value maximising u x is a stable point of the system, to which all trajectories converge. Linearisation may Note that v m ˆ m j j2j D r m k k2r an thus t v m t ˆ v m j t m j t ˆ k j2j q j m j 2 x r t q j m j t ;
14 using (9) an relation (45). Hence v m provies a Lyapunov function for the revise system, an the unique value maximising v m is a stable point of the system, to which all trajectories converge. The moels consiere in this section assume very fast aaptation of the users, inee so rapi that user r is essentially varying its rate x r t optimally in response to the resource shaow prices m j t ; j 2 J. Interesting questions remain concerning the stability of the system uner more general assumptions on users' spee of aaptation. A more general optimisation problem The optimisation problem implicitly solve by the primal algorithm (5)±(6) is not our initial network problem NETWORK A; C; w, but rather the maximisation of the Lyapunov function (7). We begin this section by iscussing a possible interpretation of this relaxation of the network problem, where the constraint Ax 4 C is replace by penalties, perhaps expresse in terms of elay or loss, that increase as the capacity of a resource is approache. Following this we inicate how the system problem SYSTEM U; A; C may be recast both to motivate the relaxation of the network problem, an to allow routing choices. Delay or loss Suppose that when a resource is heavily loae the network incurs some cost, perhaps expresse in terms of elay or loss. Then the optimisation of the Lyapunov function (7) might be interprete in terms of a penalty function C j y ˆ y p j Z Z 46 that escribes the rate at which cost is incurre at resource j when the loa through it is y. For example, suppose the rate at which cost is incurrre at resource j is yt yt n C j y ˆ1 n N e t n n>n when the loa is y, that is, e times the expecte number of cells per unit time that excee a threshol N in the slotte oisson moel of the previous section. Then a simple ifferentiation establishes that p j y, etermine by (46), is given by (42). Routing Next we exten the basic moel to allow routing choices to be mae. Let s 2 S now label a user, an suppose s is ienti e with a subset of R, the routes available to serve the user s. Set H sr ˆ 1ifr 2 s, so that route r serves user s, an set H sr ˆ otherwise. This e nes a ±1 matrix H ˆ H sr ; s 2 S; r 2 R. For each r 2 R let s r ientify a value s 2 S such that H sr ˆ 1, an suppose this value is unique; view s r as the user serve by route r. Now let y r be the ow on route r, an suppose that resource j incurs a cost C j y r epenent on the ow through that resource, where C j is a strictly convex function. Consier the following optimisation problems. SYSTEM U; H; A; C : max U s x s C j s2s j2j y r subject to Hy ˆ x over x; y 5 : NETWORK H; A; C; w : max w s log y r C j s2s r2s j2j over y 5 : y r Then, following the approach of Kelly, 18 it is possible to show that there exist vectors l ˆ l s ; s 2 S, w ˆ w s ; s 2 S an x ˆ x s ; s 2 S satisfying w s ˆ l s x s for s 2 S, such that w s solves USER s U s ; l s for s 2 S an x solves NETWORK H; A; C; w ; further x is then the unique vector with the property that there exists a vector y such that x; y solves SYSTEM U; H; A; C. Thus the relaxation of the network problem may be motivate by a similar relaxation of the overall system problem an both problems may be generalise to inclue routing choices. Finally we sketch the natural generalisations of the primal an ual algorithms, an their corresponing Lyapunov functions. Suppose that p j an C j are relate by (46). Generalise the primal algorithm (5)±(6) to become t y r t ˆk w s r y a t m j t 47 a2s r (or zero if this expression is negative an y r t ˆ where m j t ˆp j y r t ; 48 an let u y ˆ s2s w s log r2s y r S y r p j y y: j2j
15 Then the ynamical system (47)±(48) has the property that t u y t > unless y solves NETWORK H; A; C; w. Similarly the ual algorithm (9)±(1) may be generalize to incorporate a form of least cost routing. For s 2 S let w y r t ˆx s t ˆ s r2s min r2s m j t ; an suppose y r t is only positive on routes r that attain the minimum in the enominator. Then the ynamical system t m j t ˆk y r t q j m j t has the property that v m t is an increasing function of t, where v m ˆ w s log min m j mj q j Z Z: s2s r2s j2j Thus routing, as well as rate control, may be naturally integrate with proportionally fair pricing. Concluing remarks In this paper we have aresse the issue of how available banwith within a large-scale broaban network shoul be share between competing streams of elastic traf c. An optimisation framework leas to a ecomposition of the overall system problem into a separate problem for each user, in which the user chooses a charge per unit time that the user is willing to pay, an one for the network; we have shown that two classes of rate control algorithm are naturally associate with the objective functions appearing in, respectively, the primal an ual formulation of the network's problem. In consequence the algorithms provie natural implementations of proportionally fair pricing. We have stuie the stability of the algorithms in the presence of stochastic perturbations an time lags, an have illustrate our results with a stuy of ranom network with a hunre resources an a thousan routes. Interesting an challenging questions remain concerning the stability of the entire system uner more general assumptions on users' reactions to the rates allocate to them by the network, an when the numbers of users an the amounts of capacity available for elastic traf c vary ranomly. An outstaning practical issue concerns how protocols, such as TC in the Internet or the Available Bit Rate transfer capability of an ATM network, can be aapte to be charge sensitive. AcknowlegementsÐAman Maulloo was supporte by the Staff Development rogramme, University of Mauritius, an Davi Tan by St John's College, Cambrige; computing equipment was provie by the ESRC uner grant GR/J31896, an further support was provie by the Commission of the European Communities ACTS project AC39, entitle Charging an Accounting Schemes in Multiservice ATM Networks (CA$hMAN). The authors are grateful to Martin Biiscombe, Costas Courcoubetis, Jon Crowcroft, Richar Gibbens, James Giles, Meena Lakshmanan, Ravi Mazumar, Anrew Olyzko, George D. Stamoulis an Richar Weber for valuable iscussions an suggestions. References 1 Shenker S (1995). Funamental esign issues for the future Internet. IEEE J Selecte Areas Commun 13: 1176± Jacobson V (1988). Congestion avoiance an control. In: roc. ACM SIGCOMM '88, pp 314±329. An upate version is available via ftp://ftp.ee.lbl.gov/papers/congavoi.ps.z. 3 Wroclawski J (1996). Speci ciation of the controlle-loa network element service. Internet-Draft. (Internet-Drafts are working ocuments of the Internet Engineering Task ForceÐ their current status is available by ftp from nic.noru.net.) 4 International Telecommunications Union ((1996). Recommenation I371: Traf c control an congestion control in B-ISBN. Geneva. 5 Kelly F (1996). 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