The Mean-field Behavior of Processor Sharing Systems with General Job Lengths Under the SQ(d) Policy

Abstract

In this paper, we derive the mean-field behavior of empirical distributions of large systems that consist of N (large) identical parallel processor sharing servers with Poisson arrival process having intensity Nλ and generally distributed job lengths under the randomized SQ(d) load balancing policy. Under this policy, an arrival is routed to the server with the least number of progressing jobs among d randomly chosen servers. The mean-field is then used to approximate the statistical properties of the system. In particular, we show 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. This has important engineering relevance for the robustness of such routing policies that are often used in web server farms. We use a measure-valued process approach and martingale techniques to obtain our results. We also provide numerical results to support our analysis.