On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime
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Gamarnik-2013-On the rate.pdf
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Author(s)
Gamarnik, David
Goldberg, David A.
Date Issued
October 2013
Journal
The Annals of Applied Probability
Publisher
Institute of Mathematical Statistics
Citation
Gamarnik, David, and David A. Goldberg. “On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime.” The Annals of Applied Probability 23, no. 5 (October 2013): 1879-1912. © Institute of Mathematical Statistics
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Final published version
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[superscript ∗] ≈ 1.85772 s.t. when a certain excess parameter B ∈ (0,B[superscript ∗]], the error in the steady-state approximation converges exponentially fast to zero at rate B[superscript 2 over 4]. For B > B[superscript ∗], 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[superscript ∗]. Our bounds scale independently of n in the Halfin–Whitt regime, and do not follow from the weak-convergence theory.
MIT Department
Sloan School of Management
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DOI of Published Version
https://doi.org/10.1214/12-aap889
