ON THE RATE OF CONVERGENCE TO STATIONARITY OF THE M/M/N QUEUE IN THE HALFIN-WHITT REGIME

• 成果类型: Article
• 署名作者: Gamarnik, David; Goldberg, David A.
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP889
• 发表日期: 2013
• 页码: 1879-1912
• 关键词: asymptotic analysis; transient-behavior; decay parameter; erlang; birth; POLYNOMIALS; krawtchouk; bounds

摘要

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* approximate to 1.85772 s.t. when a certain excess parameter B is an element of (0, B*], the error in the steady-state approximation converges exponentially fast to zero at rate B-2/4. or B > B*, 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 Doom [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*. Our bounds scale independently of n in the Halfin-Whitt regime, and do not follow from the weak-convergence theory.