Individual equilibrium and learning in processor sharing systems

We consider a processor-sharing service system, where the service rate to individual customers decreases as the load increases. Each arriving customer may observe the current load and should then choose whether to join the shared system. The alternative is a constant-cost option, modeled here for concreteness as a private server (e.g., a personal computer that serves as an alternative to a central mainframe computer). The customers wish to minimize their individual service times (or an increasing function thereof). However, the optimal choice for each customer depends on the decisions of subsequent ones, through their effect on the future load in the shared server. This decision problem is analyzed as a noncooperative dynamic game among the customers. We first show that any Nash equilibrium point consists of threshold decision rules and establish the existence and uniqueness of a symmetric equilibrium point. Computation of the equilibrium threshold is demonstrated for the case of Poisson arrivals, and some of its properties are delineated. We next consider a reasonable dynamic learning scheme, which converges to the symmetric Nash equilibrium point. In this model customers simply choose the better option based an available performance history. Convergence of this scheme is illustrated here via a simulation example and is established analytically in subsequent work.