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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 11794-3600, 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 first-in-first-out (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