Steady-state GI/GI/n Queue in the Halfin-Whitt Regime

Details Steady-state GI/GI/n Queue in the Halfin-Whitt Regime Steady-state GI/GI/n Queue in the Halfin-Whitt Regime

2011-03-09

arxiv.org/abs/1103.1709v1

Mathematics

Probability

Scientific paper

We consider the FCFS $GI/GI/n$ 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 in \cite{GM.08}. 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 $GI/GI/n$ queue. Our bounds are of a structural nature, hold for all $n$ and all times $t \geq 0$, 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 non-trivial bounds for the diffusion process studied in \cite{R.09}.

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