Dynamic Scheduling of a Two-Class Queue: Small Interest Rates

Abstract

We consider a single server queuing system with two classes of customers who arrive according to independent Poisson processes. The two service time distributions are arbitrary, and we assume a linear holding cost and fixed service reward for each class. The problem is to decide, at the completion of each service and given the state of the system, which class (if any) to admit next into service. We seek a policy, called Blackwell optimal, which will maximize for all sufficiently small interest rates the expected net present value of service rewards received minus holding costs incurred over an infinite planning horizon. For a variety of different cases, it is shown that there exists a Blackwell optimal policy which simply enforces a static priority ranking of the classes, choosing idleness only when the system is empty. Criteria for ranking the classes are derived, extending classical results on optimal priority rules. For other cases, involving one zero holding cost and/or an unstable system, it is shown that there exists a Blackwell optimal policy which serves only one class. Attention is largely restricted to nondegenerate cases. It is shown, however, that lexicographic ranking criteria can be derived to deal with degeneracies.