Asymptotic expansions for a stochastic model of queue storage
type: Item
instance of: scholarly article
title: Asymptotic expansions for a stochastic model of queue storage
zbMATH Open document ID: 1056.60095
author: Charles Knessl
published in: The Annals of Applied Probability
publication date: 2004-10-27T00:00:00Z
review text: Let be given an M/M/\(\infty\) queue with a Poisson arrival rate \(\lambda\) and identical servers working at rate \(\mu\). The servers are ranked, and an arriving customer takes the lowest ranked available server. Let \(S\) be the set of occupied servers, \(\max(S)\) be the highest ranked occupied server, and \(P(m) =\text{Pr}[\max(S)> m]\) be its steady-state distribution. The paper studies asymptotic expansions of \(P(m)\) as \(\rho= \lambda/\mu\to \infty\). The following result is proved: Theorem 1. For \(\rho\to\infty\), the distribution of \(\max(S)\) has the following asymptotic behaviors, listed in increasing size of \(m/\rho\): a) \(\rho\to\infty\), \(m= O(1)\), \[ 1-P(m)\sim\exp\{-\rho\}\rho^{m/2} {1\over\sqrt{2\pi i}} \int_{\text{Br}} {\exp\{\xi^2/2\}\over \text{He}_m(\xi)}\,d\xi, \] where He\(_m(\cdot)\) is the \(m\)th Hermite polynomial and the contour Br is a vertical contour in the \(\xi\)-plane, which lies to the right of all the zeroes of He\(_m(\cdot)\); b) \(m,\rho\to\infty\) with \(m/\rho= X\in(0, 1)\), \[ \begin{gathered} 1-P(m)\sim\rho^{-1/3} K(X)\exp\{\rho\Phi(X)\}\exp\{\rho^{1/3} \Phi_1(X)\},\\ \Phi(X)=- 1+ \sqrt{X}- \textstyle{{1\over 2}} X\log X+(1- \sqrt{X})^2\log(1- \sqrt{X}),\\ \Phi_1(x)= -r_0 X^{-1/6}(1- \sqrt{X})^{2/3}\log(1- \sqrt{X}),\\ K(X)= [\text{Ai}'(r_0)]^{-1} X^{-1/3}(1- \sqrt{X})^{-2/3} \exp\Biggl\{- {1\over 2\sqrt{X}} \log(1- \sqrt{X})\Biggr\}.\end{gathered} \] Here \(r_0= \max\{z: \text{Ai}(z)= 0\}= -2.33810741\dots\) is the largest (negative) root of the Airy function Ai\((\cdot)\); c) \(m,\rho\to\infty\) with \(m= \rho+ \beta\sqrt{\rho}\) and \(\beta\) fixed, \[ 1- P(m)\sim- \rho^{-z_0/2}\exp\{- \beta^2/4\} \Gamma(z_0)/\Delta(\beta), \] where \(z_0= z_0(\beta)= \min\{z> 0: D_z(- \beta)= 0\}\) is the smallest positive root of the parabolic cylinder function \(D_z(-\beta)\) (viewed as a function of the index) and \(\Delta(\beta)= -{d\over dz} D_z(-\beta)|_{z= z_0(\beta)}\); d) \(m,\rho\to\infty\) with \(m/\rho= X\in (1,\inft…
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60K30
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 2110917
reviewed by: Oleg K. Zakusilo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1019487357
OpenAlex ID: W2028323177
cites work: Q5765718
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DOI: 10.1214/AOAP/1019487357
description: scientific article; zbMATH DE number 2110917