## The Annals of Applied Probability

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

#### 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 $B^{\ast}\approx1.85772\mbox{ s.t.}$ when a certain excess parameter $B\in(0,B^{\ast}]$, the error in the steady-state approximation converges exponentially fast to zero at rate $\frac{B^{2}}{4}$. For $B>B^{\ast}$, 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 $B<B^{\ast}$. Our bounds scale independently of $n$ in the Halfin–Whitt regime, and do not follow from the weak-convergence theory.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 5 (2013), 1879-1912.

Dates
First available in Project Euclid: 28 August 2013

https://projecteuclid.org/euclid.aoap/1377696301

Digital Object Identifier
doi:10.1214/12-AAP889

Mathematical Reviews number (MathSciNet)
MR3134725

Zentralblatt MATH identifier
1287.60111

Subjects

#### Citation

Gamarnik, David; Goldberg, David A. On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23 (2013), no. 5, 1879--1912. doi:10.1214/12-AAP889. https://projecteuclid.org/euclid.aoap/1377696301

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