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

TL;DR

This paper investigates the convergence rate to stationarity of the M/M/n queue in the Halfin-Whitt regime, revealing an asymptotic phase transition and providing explicit bounds that are independent of the number of servers.

Contribution

It identifies the limiting spectral gap and describes an asymptotic phase transition in the convergence rate, with explicit bounds that do not depend on n.

Findings

Existence of a critical constant B* ≈ 1.85772 for convergence rate transition.

For B ≤ B*, convergence error decays exponentially at rate B²/4.

For B > B*, the decay rate is given by an explicit equation involving special functions.

Abstract

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 $B^*\approx1.85772$ s.t. when a certain excess parameter $B\in(0,B^*]$, the error in the steady-state approximation converges exponentially fast to zero at rate $\frac{B^2}{4}$. 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.