Transient behavior of the Halfin–Whitt diffusion

We consider the heavy-traffic approximation to the GI/M/sGI/M/s queueing system in the Halfin–Whitt regime, where both the number of servers ss and the arrival rate λλ grow large (taking the service rate as unity), with λ=s−βs and ββ some constant. In this asymptotic regime, the queue length process can be approximated by a diffusion process that behaves like a Brownian motion with drift above zero and like an Ornstein–Uhlenbeck process below zero. We analyze the transient behavior of this hybrid diffusion process, including the transient density, approach to equilibrium, and spectral properties. The transient behavior is shown to depend on whether ββ is smaller or larger than the critical value β∗≈1.85722β∗≈1.85722, which confirms the recent result of Gamarnik and Goldberg (2008) [8].