## The Annals of Applied Probability

### Steady-state $GI/G/n$ queue in the Halfin–Whitt regime

#### 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].

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2382-2419.

Dates
First available in Project Euclid: 22 October 2013

https://projecteuclid.org/euclid.aoap/1382447692

Digital Object Identifier
doi:10.1214/12-AAP905

Mathematical Reviews number (MathSciNet)
MR3127939

Zentralblatt MATH identifier
1285.60090

Subjects

#### Citation

Gamarnik, David; Goldberg, David A. Steady-state $GI/G/n$ queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23 (2013), no. 6, 2382--2419. doi:10.1214/12-AAP905. https://projecteuclid.org/euclid.aoap/1382447692

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