Open Access
December 2013 Steady-state $GI/G/n$ queue in the Halfin–Whitt regime
David Gamarnik, David A. Goldberg
Ann. Appl. Probab. 23(6): 2382-2419 (December 2013). DOI: 10.1214/12-AAP905

Abstract

We consider the FCFS $GI/G/n$ queue in the so-called Halfin–Whitt heavy traffic regime. We prove that under minor technical conditions the associated sequence of steady-state queue length distributions, normalized by $n^{1/2}$, is tight. We derive an upper bound on the large deviation exponent of the limiting steady-state queue length matching that conjectured by Gamarnik and Momcilovic [Adv. in Appl. Probab. 40 (2008) 548–577]. We also prove a matching lower bound when the arrival process is Poisson.

Our main proof technique is the derivation of new and simple bounds for the FCFS $GI/G/n$ queue. Our bounds are of a structural nature, hold for all $n$ and all times $t\geq0$, and have intuitive closed-form representations as the suprema of certain natural processes which converge weakly to Gaussian processes. We further illustrate the utility of this methodology by deriving the first nontrivial bounds for the weak limit process studied in [Ann. Appl. Probab. 19 (2009) 2211–2269].

Citation

Download Citation

David Gamarnik. David A. Goldberg. "Steady-state $GI/G/n$ queue in the Halfin–Whitt regime." Ann. Appl. Probab. 23 (6) 2382 - 2419, December 2013. https://doi.org/10.1214/12-AAP905

Information

Published: December 2013
First available in Project Euclid: 22 October 2013

zbMATH: 1285.60090
MathSciNet: MR3127939
Digital Object Identifier: 10.1214/12-AAP905

Subjects:
Primary: 60K25

Keywords: Gaussian process , large deviations , Many-server queues , Stochastic comparison , weak convergence

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 6 • December 2013
Back to Top