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 > 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.
European Journal of Applied Mathematics
Digital Object Identifier (DOI)
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