Optimal service-rate selection in an $M| G |\hat 1$Queue

Abstract

This paper considers a single server, Poisson arrival, general service queuing system in which the service rate may be varied continuously between fixed limits. The problem is to find a policy for selecting the service rate which minimizes the expected average service plus holding cost per unit time. We first consider, as a Markov decision process, the approximating model in which the service rate can be changed only at equally spaced points in time. We prove that if (i) the service cost rate is a convex function of the service rate, and (ii) the holding cost rate is a polynomial approximation to a convex function of the work remaining in the system, then there exists a stationary deterministic optimal policy in which the service rate is a nondecreasing function of the work remaining in the system. Finally, we show that the continuous-time problem, in which the service rate may be changed at any point in time, is in a natural sense the limit of a sequence of discrete-time problems as the transition interval approaches zero.

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