Rate of convergence to stationarity of the system \( M / M / N / N + R \)
type: Item
instance of: scholarly article
title: Rate of convergence to stationarity of the system \( M / M / N / N + R \)
zbMATH Open document ID: 1263.60081
published in: Top
publication date: 2013-02-26T00:00:00Z
review text: The paper under review considers the \(M/M/N/N+R\) queueing system, in which the rate of arrival is \(\lambda\) and the reciprocal of the mean service time in each server is \(\mu\). Let \(\lambda_j=\lambda\) and \(\mu_j=\mu\min\{j,N\}\), and let \(p_j(t)=\operatorname{P}\{X(t)=j\}\), where \(X(t)\) is a birth-and-death process with the birth and death rates \(\lambda_j\) and \(\mu_j\), respectively, \(0<j\leq N+R\). Denote \(a=\frac{\lambda}{\mu}\). Then, for \(0\leq j\leq N\), the stationary probabilities are \[ \pi_j=c\frac{a^j}{j!}, \] and, for \(N<j\leq N+R\), they are \[ \pi_j=c\frac{a^j}{N!N^{j-N}}, \] where \(c\) is the normalisation constant. The paper studies bounds for the rate of convergence, as \(t\to\infty\), of the probabilities \(p_j(t)\) to the stationary probabilities \(\pi_j\), and study the behaviour of \(p_j(t)\) as a function of \(R\), \(N\) and arrival rate \(\lambda\), allowing the parameter \(\lambda\) to be dependent on \(N\).
reviewed by: Vyacheslav Abramov
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 6139235
zbMATH Keywords: decay rate
zbMATH Keywords: delay and loss system
zbMATH Keywords: many-server queue
zbMATH Keywords: orthogonal polynomials
author: Erik A. van Doorn
MaRDI profile type: Publication
full work available at URL: s11750-011-0173-0
OpenAlex ID: W2132143048
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DOI: 10.1007/S11750-011-0173-0
description: scientific article; zbMATH DE number 6139235