Feinberg_YangG

# Feinberg_YangG - Optimality of Trunk Reservation for an...

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

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

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

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern