Relaxed asynchronous ¯ow-control algorithms for multiclass service networks CHRISTOPHER M. RUMP 1 and SHALER STIDHAM JR 2 1 Department of Industrial Engineering, State University of New York at Bualo, 342 Bell Hall, Bualo, NY 14260-2050, USA E-mail: crump@eng.bualo.edu 2 Department of Operations Research, University of North Carolina at Chapel Hill, CB #3180, 210 Smith Building, Chapel Hill, NC 27599-3180, USA E-mail: sandy@email.unc.edu Received January 1998 and accepted May 1999 This paper addresses resource sharing in a multiclass service center. The ¯ow of service requests from each class of customers are regulated by a class manager who attempts to maximize the class net bene®t (throughput reward less delay cost). Relaxed asynchronous algorithms are proposed for obtaining a Nash equilibrium among these competing classes in which each manager iteratively updates his throughput strategy in response to local (and possibly delayed) information on the strategies of the others. The novelty of these ¯ow-control algorithms are in the relaxation, whereby each manager employs a strategy update that is a convex combination of his previous strategy and his best-reply strategy to current information. Alternatively, this relaxation can be viewed as an exponential smoothing of all previous best-reply strategies. For a particular number of classes with speci®ed cost/ reward parameters, relaxation conditions for asymptotic convergence to the unique interior Nash equilibrium are presented. For more than two classes, it is shown that relaxation not only accelerates convergence (at least in a special case), but also is a necessary condition for convergence. Due to an equivalent functional form, these results can be directly translated to a network of users employing a power criterion, a common objective in the communications literature. 1. Introduction We consider the management of service requests to a service center. Examples of such service centers include a shared computer or printer, communication link, road- way, etc. Oftentimes such a service center is a component of a larger service network. In this setting, we assume that the management of service requests is decentralized, i.e., ¯ow control is performed at each service center using only local information on that center's server utilization. In this setting there is no need for a centralized network controller, and hence we may consider a service center separately from the rest of the network. We suppose that a service center is shared by m com- peting customer classes. In this multiclass environment, we investigate the self-regulation of the input of service requests. The input ¯ow rate from each class i; i  1; ... ; m, is regulated by a class manager. For exam- ple, a class manager may be an individual user of a communication network such as the Internet who con- trols the submission of several requests for information. These customer classes are distinguished by dierences in their performance objective, which includes the cost as- sociated with service delay and the value placed on re- ceiving the service. The class manager weighs the tradeo between the service value received and the associated quality of service as re¯ected by the service delay. In a game theoretic context, we view the managers as competing players in a ``game'', the ®ght for access to a shared resource. The performance payo to the players is the net bene®t that they receive from service, i.e., the service value less the cost of delay. In a similar context, Yao (1995) established S -modularity properties among general performance payos that are sucient to ensure convergence of ¯ow-control algorithms. Unfortunately, with m  3 or more competing players, submodularity of the payo between each pair of players prohibits appli- cation of this general theory in our setting. Several other studies have investigated rate-based al- gorithms for implementing decentralized ¯ow control in communication networks. (For a survey see Korilis and Lazar (1992)). Whereas these studies involve maximizing network power as the performance objective, this paper investigates maximizing the net bene®t (throughput re- ward less delay cost) experienced by the class managers.  Corresponding author 0740-817X Ó 2000 ``IIE'' IIE Transactions (2000) 32, 873±880