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This paper addresses the issues of charging, rate control and routing for a communication network carrying elastic traffic, such as an ATM network offering an available bit rate service. A model is described from which max-min fairness of rates emerges as a limiting special case; more generally, the charges users are prepared to pay influence their allocated rates. In the preferred version of the model, a user chooses the charge per unit time that the user will pay; thereafter the user's rate is determined by the network according to a proportional fairness criterion applied to the rate per unit charge. A system optimum is achieved when users' choices of charges and the network's choice of allocated rates are in equilibrium.

This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system de®ned by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Advances in Applied Probability. AbstractThis paper is concerned with blocking and loss probabilities in circuitswitched networks. We show that when the capacity of links and the offered traffic are increased together, a limiting regime emerges in which loss probabilities are as if links block independently, with blocking probabilities given by the solution of a simple convex programming problem. We then show that an approximate procedure, based on solving Erlang's formula under the assumption of independent blocking, produces a unique solution when routes are fixed, and that under the limiting regime the estimates of loss probabilities obtained from the procedure converge to the correct values. LOSS PROBABILITIES; NETWORK FLOW; LOCAL LIMIT THEOREMS; ERLANG'S FORMULA; FIXED ROUTING; DYNAMIC ROUTING; CELLULAR RADIO R = ({1}, {2}, {1, 2}, {3, 5}, {4, 5}, {1, 3, 5}) for the network of Figure 1. Assume that calls requesting route r arrive as a Poisson process of rate Vr, and that as r varies over R it indexes independent Poisson streams. A call requesting route r is blocked and lost if on any link j, j = 1, 2, ..., J, there are less than Air circuits free. Otherwise the call is connected and simultaneously holds Air circuits from link j, j = 1, 2, ---, J, for the holding period of the call. The call holding period is randomly distributed j jnrl l The stationary probability that a call requesting route r is accepted is then G(C)G(C -Aer)-1 where er is the unit vector from 91(C) describing just one call in progress on route r. These rather simple explicit forms might be thought to provide the complete solution. However, as Harvey and Hills [15] have emphasized, this is far from the case. For all but the smallest networks it is impractical to compute G directly-observe that the number of routes I|I may grow as fast as exponentially with the number of nodes J, and that in the (otherwise trivial) case when I = J and A = I the size of the state space 19I = HrI1 C, grows rapidly with the capacity limitations C1, CZ, ***, C,. The product form (1.2) can be used as the basis for Monte Carlo estimation of loss probabilities, and Harvey and Hills [15] describe a method based on a refinement of acceptancerejection sampling. This method extends the range of networks for which loss probabilities can be accurately calculated, but still requires an effort which This content downloaded from 195.34.78.161 on Tue, 10 Jun 2014 14:05:04 PM All use subject to JSTOR Terms and Conditions Blocking probabilities in large circuit-switched networks 475 grows quickly with the complexity of the network or the capacitie...

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