Load balancing in processor sharing systems

We investigate optimal load balancing strategies for a multi-class multi-server processor-sharing system with a Poisson input stream, heterogeneous service rates, and a server-dependent holding cost per unit time. Specifically, we study (i) the centralized setting in which a dispatcher routes incoming jobs based on their service time requirements so as to minimize the weighted mean sojourn time in the system; and (ii) the decentralized, distributed non-cooperative setting in which each job, aware of its service time, selects a server with the objective of minimizing its weighted mean sojourn time in the system. For the decentralized setting we show the existence of a potential function, which allows us to transform the non-cooperative game into a standard convex optimization problem.