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.
Citation
David Gamarnik. David A. Goldberg. "On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime." Ann. Appl. Probab. 23 (5) 1879 - 1912, October 2013. https://doi.org/10.1214/12-AAP889
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