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Some asymptotic results for the transient distribution of the Halfin–Whitt diffusion process

Published online by Cambridge University Press:  20 February 2015

QIANG ZHEN  and
CHARLES KNESSL
QIANG ZHEN
Affiliation:
Department of Mathematics and Statistics, University of North Florida, 1 UNF DR, Jacksonville, FL 32224, USA email: [email protected]
CHARLES KNESSL
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan St.(M/C 249), Chicago, IL 60607, USA email: [email protected]
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Abstract

We consider the Halfin–Whitt diffusion process Xd(t), which is used, for example, as an approximation to the m-server M/M/m queue. We use recently obtained integral representations for the transient density p(x,t) of this diffusion process, and obtain various asymptotic results for the density. The asymptotic limit assumes that a drift parameter β in the model is large, and the state variable x and the initial condition x0 (with Xd(0) = x0 > 0) are also large. We obtain some alternate representations for the density, which involve sums and/or contour integrals, and expand these using a combination of the saddle point method, Laplace method and singularity analysis. The results give some insight into how steady state is achieved, and how if x0 > 0 the probability mass migrates from Xd(t) > 0 to the range Xd(t) < 0, which is where it concentrates as t → ∞, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.

Keywords

diffusion processesasymptotic expansionssingular perturbation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 3 , June 2015 , pp. 245 - 295
Copyright
Copyright © Cambridge University Press 2015 

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