The mean-field behavior of processor sharing systems with general job lengths under the SQ([formula omitted]) policy

Abstract

This paper addresses the mean-field behavior of large-scale systems of parallel servers with a processor sharing service discipline when arrivals are Poisson and jobs have general service time distributions when an SQ(d) routing policy is used. Under this policy, an arrival is routed to the server with the least number of progressing jobs among d randomly chosen servers. The limit of the empirical distribution is then used to study the statistical properties of the system. In particular, this shows that in the limit as N grows, individual servers are statistically independent of others (propagation of chaos) and more importantly, the equilibrium point of the mean-field is insensitive to the job length distributions that has important engineering relevance for the robustness of such routing policies used in web server farms. We use a framework of measure-valued processes and martingale techniques to obtain our results. We also provide numerical results to support our analysis.