Dynamic Scheduling of a Two-Server Parallel Server System with Complete Resource Pooling and Reneging in Heavy Traffic: Asymptotic Optimality of a Two-Threshold Policy

Mathematics of Operations Research, Forthcoming

Posted: 3 Nov 2013

See all articles by Samim Ghamami

Samim Ghamami

Securities and Exchange Commission (SEC); New York University (NYU); University of California, Berkeley - Center for Risk Management Research

Amy R. Ward

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE)

Date Written: October 29, 2013

Abstract

We consider a dynamic control problem for a parallel server system commonly known as the N-system. An N-system is a two-server parallel server system with two job classes, one server that can serve both classes, and one server that can only serve one class. We assume that jobs within each class arrive according to a renewal process. The random service time of a job has a general distribution that may depend on both the job’s class and the server providing the service. Each job independently reneges, or abandons the queue without receiving service, if service does not begin within an exponentially distributed amount of time. The objective is to minimize the expected infinite horizon discounted cost of holding jobs in the system and having customers abandon, by dynamically scheduling waiting jobs to available servers. It is not possible to solve this control problem exactly, and so, we consider an asymptotic regime in which the system satisfies both a heavy traffic and a resource pooling condition. Then, we solve the limiting Brownian control problem, and interpret its solution as a policy in the original N-system. We label the servers and job classes so that server 1 can only serve class 1 and server 2 can serve both classes. The policy we propose has two thresholds. There is one threshold on the total number of jobs in the system, and one threshold on the number of class 1 jobs in the system. These thresholds are used to determine which job class server 2 should serve. We show that this proposed policy is asymptotically optimal in the heavy traffic limit, and has the same limiting cost as the Brownian control problem solution.

Keywords: Asymptotically optimal stochastic control, approximating diffusion control, threshold control, parallel server system

JEL Classification: C6

Suggested Citation

Ghamami, Samim and Ward, Amy R., Dynamic Scheduling of a Two-Server Parallel Server System with Complete Resource Pooling and Reneging in Heavy Traffic: Asymptotic Optimality of a Two-Threshold Policy (October 29, 2013). Mathematics of Operations Research, Forthcoming, Available at SSRN: https://ssrn.com/abstract=2346932

Samim Ghamami (Contact Author)

Securities and Exchange Commission (SEC) ( email )

450 Fifth Street, NW
Washington, DC 20549-1105
United States

New York University (NYU) ( email )

Bobst Library, E-resource Acquisitions
20 Cooper Square 3rd Floor
New York, NY 10003-711
United States

University of California, Berkeley - Center for Risk Management Research ( email )

581 Evans Hall
Berkely, CA 94720
United States

Amy R. Ward

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE) ( email )

765 Ferst Drive
Atlanta, GA 30332-0205
United States

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