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Approximating the GI/GI/1+GI Queue with a Nonlinear Drift Diffusion: Hazard Rate Scaling in Heavy Traffic

Department of Information, Operations, and Management Sciences, Leonard N. Stern School of Business, New York University, New York, New York 10012
Information and Operations Management Department, Marshall School of Business, University of Southern California, Los Angeles, California 90089
jreed{at}stern.nyu.edu, http://www.stern.nyu.edu/jreed
amyward{at}marshall.usc.edu, http://www-rcf.usc.edu/amyward/

We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy-traffic regime in which the resulting diffusion approximation for both the offered waiting time process (the process that tracks the amount of time an infinitely patient arriving customer would wait for service) and the queue-length process contain the entire abandonment distribution. To use a continuous mapping approach to establish our weak convergence results, we additionally develop existence, uniqueness, and continuity results for nonlinear generalized regulator mappings that are of independent interest. We further perform a simulation study to evaluate the quality of the proposed approximations for the steady-state mean queue length and the steady-state probability of abandonment suggested by the limiting diffusion process.

Key Words: abandonment; deadlines; reneging; customer impatience; queueing theory; diffusion approximations; generalized regulator mappings
History: Received: May 25, 2005; revision received: July 6, 2007;