# First passage times to congested states of many-server systems in the Halfin-Whitt regime

B.H. Fralix, C. Knessl, J.S.H. Leeuwaarden, van

We consider the heavy-traffic approximation to the GI/M/s queueing system in the Halfin–Whitt regime, where both the number of servers s and the arrival rate ¿ grow large (taking the service rate as unity), with $\lambda = s - \beta\sqrt s$ and ß some constant. In this asymptotic regime, the queue length process can be approximated by a diffusion process that behaves as a Brownian motion with drift above zero and as an Ornstein–Uhlenbeck process below zero. We analyze the first passage times of this hybrid diffusion process to levels in the state space that represent congested states in the original queueing system. Keywords: Asymptotic analysis, Diffusion process, First passage times, GI/M/s queue, Halfin–Whitt regime, Queues in heavy traffic.