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

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 \u3e 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 \u3e 0 the probability mass migrates from Xd (t) \u3e 0 to the range Xd (t) \u3c 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