On the rate of convergence to stationarity of the M/M/N queue in the Halfin.Whitt regime

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the M/M/n 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 B..1.85772 s.t. when a certain excess parameter B.(0,B.], the error in the steady-state approximation converges exponentially fast to zero at rate B24. For B>B., 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 n 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 M/M/n queue in the Halfin.Whitt regime, when B<B.. Our bounds scale independently of n in the Halfin.Whitt regime, and do not follow from the weak-convergence theory.