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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 6, JUNE 2002 917 Nash Equilibria for Combined Flow Control and Routing in Networks: Asymptotic Behavior for a Large Number of Users Eitan Altman , Senior Member, IEEE , Tamer Bas ¸ar , Fellow, IEEE , and R. Srikant , Senior Member, IEEE Abstract— We consider a noncooperative game framework for combined routing and flow control in a network of parallel links, where the number of users (players) is arbitrarily large. The utility function of each user is related to the power criterion, and is taken as the ratio of some positive power of the total throughput of that user to the average delay seen by the user. The utility function is nonconcave in the flow rates of the user, for which we introduce a scaling to make it well defined as the number of users, , becomes arbitrarily large. In spite of the lack of concavity, we obtain ex- plicit expressions for the flow rates of the users and their associated routing decisions, which are in @I A Nash equilibrium. This @I A equilibrium solution, which is symmetric across different users and could be multiple in some cases, exhibits a delay-equal- izing feature among the links which carry positive flow. The paper also provides the complete optimal solution to the single-user case, and includes several numerical examples to illustrate different fea- tures of the solutions in the single- as well as -user cases, as becomes arbitrarily large. Index Terms— Asymptotic Nash equilibria, flow control, net- works, noncooperative equilibria, nonzero-sum games, routing. I. INTRODUCTION F LOW CONTROL and routing are two components of re- source and traffic management in today’s high-speed net- works, such as the Internet and the ATM. Flow control is used by best-effort type traffic in order to adjust the input transmis- sion rates (the instantaneous throughput of a connection) to the available bandwidth in the network. Routing decisions are taken to select paths with certain desirable properties, for example, minimum delays. In many cases, both flow control and routing decisions can be made by the users (rather than by the network) so as to meet some performance criteria. The appropriate frame- work for modeling this situation is that of noncooperative game theory. Manuscript received June 10, 2001; revised January 12, 2002. Recom- mended by Associate Editor L. Dai. This work was supported in part by the National Science Foundation under Grants ANI-9813710, NCR 97-01525, CCR 00-85917 ITR, and INT-9804950, in part by the Air Force Office of Scientific Research under MURI Grant AF DC 5-36128, an EPRI/ARO Grant, and in part by DARPA under Grant F30602–00–2–0542. E. Altman is with INRIA, 06902 Sophia Antipolis Cedex, France (e-mail:, and also with Centro de Simulacion y Modelos, Universidad de Los Andes, Facultad de Ingenieria, Venezuela (e-mail: T. Bas
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