Asymptotic expansions for a stochastic model of queue storage
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 \(m/\rho= X\in (1,\infty)\), \[ P(m)\sim \sqrt{\rho} L(X)\exp\{\rho\Psi(X)\},\;\Psi(X)= -1+ X-X\log X,\;L(X)= {1\over\sqrt{2\pi}} {X-1\over X} \log\Biggl({X\over X- 1}\Biggr). \]
scientific article; zbMATH DE number 2110917