Some asymptotic results for the transient distribution of the Halfin-Whitt diffusion process
Document Type
Article
Publication Date
6-28-2015
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 x 0 > 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.
Publication Title
European Journal of Applied Mathematics
Volume
26
Issue
3
First Page
245
Last Page
295
Digital Object Identifier (DOI)
10.1017/S0956792515000030
ISSN
09567925
E-ISSN
14694425
Citation Information
ZHEN, & KNESSL, C. (2015). Some asymptotic results for the transient distribution of the Halfin–Whitt diffusion process. European Journal of Applied Mathematics, 26(3), 245–295. https://doi.org/10.1017/S0956792515000030