Some Asymptotic Results for the Transient Distribution of the Halfin-Whitt Diffusion Process

TL;DR

This paper derives asymptotic expressions for the transient density of the Halfin-Whitt diffusion process, revealing how probability mass shifts over time and providing insights into the process's convergence to steady state.

Contribution

The paper introduces new asymptotic formulas for the transient density of the Halfin-Whitt diffusion, using advanced integral and asymptotic analysis techniques.

Findings

Density approximations via saddle point and Laplace methods

Insights into probability mass migration from positive to negative states

Alternative asymptotic approaches using geometrical optics

Abstract

We consider the Halfin-Whitt diffusion process $X_d(t)$, 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 $\beta$ in the model is large, and the state variable $x$ and the initial condition $x_0$ (with $X_d(0)=x_0>0$) 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 $X_d(t)>0$ to the range $X_d(t)<0$, which is where it concentrates as $t\to\infty$, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.