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Unformatted text preview: Optimality of Trunk Reservation for an M/M/k/N Queue with Several Customer Types and Holding Costs Eugene A. Feinberg * and Fenghsu Yang † Abstract We study optimal admission to an M/M/k/N queue with several customer types. The reward structure consists of revenues collected from admitted customers and hold ing costs, both of which depend on customer types. This paper studies average rewards per unit time and describes the structures of stationary optimal, canonical, bias op timal, and Blackwell optimal policies. Similar to the case without holding costs, bias optimal and Blackwell optimal policies are unique, coincide, and have a trunk reserva tion form with the largest optimal control level for each customer type. Problems with one holding cost rate have been studied previously in the literature. 1 Introduction We consider an M/M/k/N queue with m customer types, where m ≥ 1. Customers of type j , j = 1 , 2 ,...,m , arrive at the system according to independent Poisson processes with rates λ j , where 0 < λ j < ∞ . When a customer arrives at the system, its type * Eugene A. Feinberg is with Faculty of Department of Applied Mathematics and Statistics, State Uni versity of New York at Stony Brook, NY 117943600, USA. Email:[email protected] † Fenghsu Yang is with the Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA. Email:[email protected] 1 becomes known. There are k identical servers in the system, where k ≥ 1. The service times are independent, do not depend on the customer types, and are exponentially distributed with rate μ , where 0 < μ < ∞ . When there are n customers in the system, the total service rate of the system is μ n , where μ n = nμ for n = 0 , 1 ,...,k 1, and μ n = kμ for n = k,k + 1 ,...,N . Moreover, there is no preemption for customers. The queue follows the firstinfirstout (FIFO) rule. At the arrival epochs, the system manager decides whether an arrival can enter the system or not. If a customer sees less than k customers in the system and is admitted, the customer goes to a free server immediately. If a customer sees at least k customers in the system and is admitted, the customer waits in the queue for service. If there are N customers in the system, the system is full and all the arrivals are rejected. Upon admitting a customer, the system collects a positive reward which depends on the customer type and incurs a nonnegative random holding cost which depends on the customer type and the number of customers in the system the admitted customer sees. Let r j ( n ) be the net reward collected by the system if a customer of type j sees n customers in the system and is admitted. The net rewards r j ( n ) are positive constants for n = 0 , 1 ,...,k 1, and are nonincreasing in n = k 1 ,k,...,N 1. The dependence of r j ( n ) on n reflects the fact that net rewards depend on waiting times....
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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.
 Fall '11
 EugeneA.Feinberg

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