On the Optimal Control of Two Queues with Server Setup Times and Its Analysis

Abstract

Two queues are fed by independent, time-homogeneous Poisson arrival processes. One server is available to handle both. All service durations, in both queues, are drawn independently from the same distribution. A setup time is incurred whenever the server moves (switches) from one queue to the other. We prove that in order to minimize the sum of discounted setup charges and holdings costs, assumed linear in queue length and having the same rate at the two queues, the service at each queue should be exhaustive, A “threshold policy” is defined as a policy under which the server switches (from an empty queue) only when the other reaches a critical size. It is shown to be a likely candidate for the optimal policy, both for the discounted version and for the long-time average criterion. The steady-state performance of this policy (under somewhat more general distributional assumptions) and the optimal thresholds are determined for a number of cases.