Decomposition of the M/M/1 transition function
type: Item
instance of: scholarly article
title: Decomposition of the M/M/1 transition function
zbMATH Open document ID: 0731.60086
DOI: 10.1007/BF01158470
author: Joseph Abate
author: Masaaki Kijima
author: Ward Whitt
published in: Queueing Systems
publication date: 1991-01-01T00:00:00Z
review text: The authors investigate transition probabilities of the queue length process in the classical M/M/1 queue. With this goal two auxiliary stochastic processes are introduced. The first process corresponds to the presence of the absorbing barrier at the origin (equivalently, it describes the queue length process during a busy period). The second process corresponds to the absence of barrier at the origin at all (equivalently, it describes the difference between two Poisson processes). By simple probabilistic arguments some relations between transition probabilities of these processes are established. They imply several inequalities for the transition probabilities and asymptotic behaviour as \(t\to \infty\). An asymptotic formula for the transition probabilities of the queue length process in the case of a geometric initial distribution should be noted. This formula describes dependence of the relaxation time on the parameter of the initial geometric distribution.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 4209263
zbMATH Keywords: transient behaviour
zbMATH Keywords: queue length process
zbMATH Keywords: absorbing barrier
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: dependence of the relaxation time on the parameter of the initial geometric distribution
reviewed by: G. I. Falin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4209263