Optimal Allocation in a Queueing System with Shared Resources

In this paper, we study the problem of dynamic allocation of heterogeneous processors to parallel heterogeneous job traffic flows. Each traffic flow is a stationary ergodic random marked point process, with a parameter (traffic intensity rate) that depends on the job class. The service rates of the various job flows depend on both the job class and the processor class. This model captures the essential features of several practical systems, including flexible manufacturing ones, packet switches, distribution systems, etc. We first specify precisely the necessary and sufficient condition for stability of the system. We then identify a family of policies that achieves maximum throughput; i.e. they stabilize the system under the maximum possible input rates. The approach taken introduces the concept of virtual queueing, which proves powerful in establishing strong probabilistic results (convergence in distribution to a finite stationary regime) for queueing systems with complex dynamics due to sharing resources, under a very general stationary ergodic probabilistic structure.