Dynamic Scheduling of a System with Two Parallel Servers in Heavy Traffic with Complete Resource Pooling: Asymptotic Optimality of a Continuous Review Threshold Policy

S. L. Bell and R. J. Williams

This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (non-identical) servers working in parallel. One server can only process jobs from one buffer, whereas the other server can process jobs from either buffer. The service time distribution may depend on the buffer 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 satisfies both a heavy traffic 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 traffic scaling) queue length. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We ``interpret" this solution by proposing a ``continuous review" threshold control policy for use in the original parallel server system. We show that this policy is asymptotically optimal in the heavy traffic 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.

This paper appears in the Annals of Applied Probability, 11 (2001), 608-649. The final printed version of the article is freely available through open access on Project Euclid by clicking here.