On the Rate of Convergence to Stationarity of the M/M/N Queue in the Halfin-Whitt Regime

Details On the Rate of Convergence to Stationarity of the M/M/N Queue in the Halfin-Whitt Regime On the Rate of Convergence to Stationarity of the M/M/N Queue in the Halfin-Whitt Regime

2010-03-10

arxiv.org/abs/1003.2004v1

Mathematics

Probability

Scientific paper

We study the rate of convergence to stationarity of the $M/M/n$ queue in the Halfin-Whitt regime. We prove that there is an interesting \emph{phase transition} in the system's behavior, occurring when the excess parameter $B$ reaches $B^* \approx 1.85772$. For $B < B^*$, the exponential rate of convergence is $\frac{B^2}{4}$; above $B^*$ it is the solution to an equation involving the parabolic cylinder functions. We also bound the prefactor governing the rate of convergence uniformly over $n$ when $B < B^*$, and use our bounds to derive a rule-of-thumb for determining the time it takes a severely overloaded (underloaded) queueing system to return (probabilistically) to the steady-state.

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