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Dynamic Control of a Queue with Adjustable Service Rate

Melbourne Business School, 200 Leicester St, Carlton, Victoria 3053, Australia
Graduate School of Business, Stanford University, Stanford, California 94305
j.george{at}mbs.unimelb.edu.au
harrison_michael{at}gsb.stanford.edu

We consider a single-server queue with Poisson arrivals, where holding costs are continuously incurred as a nondecreasing function of the queue length. The queue length evolves as a birth-and-death process with constant arrival rate = 1 and with state-dependent service rates µ_n that can be chosen from a fixed subset A of [0, ). Finally, there is a nondecreasing cost-of-effort function c(·) on A, and service costs are incurred at rate c(µ_n) when the queue length is n. The objective is to minimize average cost per time unit over an infinite planning horizon. The standard optimality equation of average-cost dynamic programming allows one to write out the optimal service rates in terms of the minimum achievable average cost *. Here we present a method for computing * that is so fast and so transparent it may be reasonably described as an explicit solution for the problem of service rate control. The optimal service rates are nondecreasing as a function of queue length and are bounded if the holding cost function is bounded. From a managerial standpoint it is natural to compare *, the minimum average cost achievable with state-dependent service rates, against the minimum average cost achievable with a single fixed service rate. The difference between those two minima represents the economic value of a responsive service mechanism, and numerical examples are presented that show it can be substantial.

Subject classifications: Queues: dynamic control; Dynamic programming: service rate control in queues.
History: Received January 1999; revision received January 2000; revision received May 2000; accepted June 2000.

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