Feinberg_YangG - Optimality of Trunk Reservation for an...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 = for n = 0 , 1 , . . . , k - 1, and μ n = 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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern