Dynamic Scheduling of a System with Two Parallel Servers in Heavy Traac with Resource Pooling: Asymptotic Optimality of a Threshold Policy 1

This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to innnite capacity buuers and two (non-identical) servers working in parallel. One server can only process jobs from one buuer, whereas the other server can process jobs from either buuer. The service time distribution may depend on the buuer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisses both a heavy traac condition and a resource pooling condition. Our cost function is a mean cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traac scaling) queue length. We rst review the analytic solution of the Brownian control problem (formal heavy traac approximation) for this system. We \interpret" this solution by proposing a threshold control policy for use in the original parallel server system. We show that this policy is asymptotically optimal in the heavy traac limit and the limiting cost is the same as the optimal cost in the Brownian control problem. The techniques developed here are expected to be useful for analyzing the performance of threshold-type policies in more complex multiserver systems. Short title: Dynamic scheduling of two parallel servers.