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# Optimal Control of Service Rates in Networks of Queues

Richard R. Weber and Shaler Stidham Jr.
Vol. 19, No. 1 (Mar., 1987), pp. 202-218
DOI: 10.2307/1427380
Stable URL: http://www.jstor.org/stable/1427380
Page Count: 17
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## Abstract

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of · /M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi(xi) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval $[0,\overline{\mu}]$, where cost ci(μ) is charged for each unit of time that the service rate μ is in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are extended to an average-cost measure, and an example is included to show that in general the assumption of convex holding costs may not be relaxed. A further example shows that the optimal policy may not be monotone unless the choice of possible service rates at each queue includes 0.

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