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On the Generalized Drift Skorokhod Problem in One Dimension

Part of: Markov processes Stochastic processes Operations research and management science

Published online by Cambridge University Press:  30 January 2018

Josh Reed ,
Amy Ward  and
Dongyuan Zhan
Josh Reed*
Affiliation:
New York University
Amy Ward*
Affiliation:
University of Southern California
Dongyuan Zhan*
Affiliation:
University of Southern California
*
∗ Postal address: Stern School of Business, New York University, New York, NY 10012, USA. Email address: jreed@stern.nyu.edu
∗∗ Postal address: Marshall School of Business, University of Southern California, Los Angeles, CA 90089-0809, USA.
∗∗ Postal address: Marshall School of Business, University of Southern California, Los Angeles, CA 90089-0809, USA.
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Abstract

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We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime.

Keywords

Skorokhod mapreflected Ornstein–Uhlenbeck processabandonmentqueueing

MSC classification

Primary: 90B22: Queues and service
Secondary: 90B15: Network models, stochastic 60G17: Sample path properties 60J60: Diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 16 - 28
Copyright
Copyright © Applied Probability Trust 2013 

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