We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the queue in the Halfin–Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant when a certain excess parameter , the error in the steady-state approximation converges exponentially fast to zero at rate . For , the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer].

We also prove explicit bounds on the distance to stationarity for the queue in the Halfin–Whitt regime, when . Our bounds scale independently of in the Halfin–Whitt regime, and do not follow from the weak-convergence theory.