Some asymptotic results for the transient distribution of the Halfin–Whitt diffusion process. (June 2015)

• Record Type: Journal Article
• Authors: ZHEN, QIANG; KNESSL, CHARLES
• Is Part Of: European journal of applied mathematics. Volume 26:Number 3(2015:Jun.)
• Page Start: 245
• Page End: 295
• Publication Date: 2015-06
• Subjects: Mathematics -- Periodicals; 519
• DOI: 10.1017/S0956792515000030
• Languages: English
• ISSNs: 0956-7925

Abstract

<abstract abstract-type="normal"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>We consider the Halfin–Whitt diffusion process <italic>X<sub>d</sub></italic>(<italic>t</italic>), which is used, for example, as an approximation to the <italic>m</italic>-server <italic>M/M/m</italic> queue. We use recently obtained integral representations for the transient density <italic>p</italic>(<italic>x, t</italic>) 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 <italic>x</italic> and the initial condition <italic>x</italic><sub>0</sub> (with <italic>X<sub>d</sub></italic>(0) = <italic>x</italic><sub>0</sub> > 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 <italic>x</italic><sub>0</sub> > 0 the probability mass migrates from <italic>X<sub>d</sub></italic>(<italic>t</italic>) > 0 to the range <italic>X<sub>d</sub></italic>(<italic>t</italic>) < 0, which is where it concentrates as <italic>t</italic> → ∞, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.</p> </abstract>