Properties of the Reflected Ornstein–Uhlenbeck Process

Authors:
Amy R. Ward;Peter W. Glynn
Affiliations:
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA amy@isye.gatech.edu;Department of Management Science & Engineering, Stanford University, Stanford, CA 94305, USA glynn@stanford.edu
Venue:
Queueing Systems: Theory and Applications
Year:
2003

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Abstract

Consider an Ornstein–Uhlenbeck process with reflection at the origin. Such a process arises as an approximating process both for queueing systems with reneging or state-dependent balking and for multi-server loss models. Consequently, it becomes important to understand its basic properties. In this paper, we show that both the steady-state and transient behavior of the reflected Ornstein–Uhlenbeck process is reasonably tractable. Specifically, we (1) provide an approximation for its transient moments, (2) compute a perturbation expansion for its transition density, (3) give an approximation for the distribution of level crossing times, and (4) establish the growth rate of the maximum process.