d-policy - Queueing Systems 42, 355–376, 2002  2002...

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Unformatted text preview: Queueing Systems 42, 355–376, 2002  2002 Kluwer Academic Publishers. Manufactured in The Netherlands. Optimality of D-Policies for an M/G/ 1 Queue with a Removable Server EUGENE A. FEINBERG ∗ [email protected] Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794-3600, USA OFFER KELLA ∗∗ [email protected] Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel Received 11 February 2002; Revised 20 May 2002 Abstract. We consider an M/G/ 1 queue with a removable server. When a customer arrives, the workload becomes known. The cost structure consists of switching costs, running costs, and holding costs per unit time which is a nonnegative nondecreasing right-continuous function of a current workload in the system. We prove an old conjecture that D-policies are optimal for the average cost per unit time criterion. It means that for this criterion there is an optimal policy that either runs the server all the time or switches the server off when the system becomes empty and switches it on when the workload reaches or exceeds some threshold D . Keywords: M/G/ 1 queue, removable server, D-policy, average cost criterion AMS subject classification: 60K25, 90C40 1. Introduction We consider an M/G/ 1 queue. The customers arrive according to a Poisson process F t with intensity λ . The i th arriving customer increases the total workload by its service time Y i . The assumption is that this service time becomes known upon arrival, rather than upon initiation of service. Following standard assumptions, { Y i | i > 1 } are i.i.d. and are independent of the arrival process. As usual, the service rate is 1. At any time the server can be switched on and off. If the server is off or the queue is empty, the service rate is 0. It costs K to switch the server on and K 1 to switch the server off. We assume that K and K 1 are nonnegative numbers and at least one of them is positive. If the server is on (off), the cost is r 1 ( r 2 ) per unit time, where r 1 > r 2 . Setting r = r 1 − r 2 it is clear that without loss of generality we may assume that r 1 = r and r 2 = 0. We make this assumption throughout this paper. ∗ Supported in part by grant DMI-9908258 from the National Science Foundation. ∗∗ Supported in part by grant 935/0 from the Israel Science Foundation. 356 E.A. FEINBERG AND O. KELLA If the workload is x , the holding cost per unit time is h(x) . We assume that h : [ , ∞ ) → [ , ∞ ) is a nonnegative nondecreasing right-continuous function with h(w) → ∞ as x → ∞ . Without loss of generality, we set h( ) = 0. Indeed, if h( ) = 0 we define a new holding cost function equal to h(x) − h( ) . For any policy, average costs per unit time will change by the same constant h( ) . The new holding cost function equals zero when x = 0....
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This note was uploaded on 12/06/2011 for the course MATH 101 taught by Professor Eugenea.feinberg during the Fall '11 term at State University of New York.

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d-policy - Queueing Systems 42, 355–376, 2002  2002...

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