On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime
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...
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| Format: | Article |
| Language: | en_US |
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Institute of Mathematical Statistics
2013
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| Online Access: | http://hdl.handle.net/1721.1/83256 https://orcid.org/0000-0001-8898-8778 |
| Summary: | 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. |
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