Martingale relations for the M/GI/1 queue with Markov modulated Poisson input
type: Item
instance of: scholarly article
title: Martingale relations for the M/GI/1 queue with Markov modulated Poisson input
zbMATH Open document ID: 0734.60092
DOI: 10.1016/0304-4149(91)90074-M
author: Francois Baccelli
author: Armand M. Makowski
published in: Stochastic Processes and their Applications
publication date: 1991-01-01T00:00:00Z
review text: A single server queueing system with renewal service process and Poisson arrivals modulated by a finite-state Markov process \((y(t),\quad t\geq 0)\) with rate \(\lambda\) (i) when the environment is in state i is considered. The arrival process \((A(t),\quad t\geq 0)\) is a counting process with \(A(0)=0\) such that the process \((\alpha (t),\quad t\geq 0)\) defined by \[ \alpha (t)=A(t)-\int^{t}_{0}\lambda (Y(s^-))ds,\quad t\geq 0, \] is an \({\mathcal F}_ t\)-martingale, where \({\mathcal F}_ t=\sigma \{Y(s),A(s)\), \(0\leq s\leq t\}\). It is shown that martingales are a useful tool for analyzing the dynamic, transient and stationary behaviors of the system. Several exponential martingales which are associated with a chain embedded at service completion epochs in the process describing the joint evolution of the queue-size and the state of the environment are investigated. New conservation laws in the form of a system of linear relations satisfied by the joint distributions and the absolute continuity properties of certain queues are obtained. Some computational problems are discussed.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 4215115
zbMATH Keywords: stability condition
zbMATH Keywords: single server queueing system
zbMATH Keywords: stationary behaviors of the system
zbMATH Keywords: exponential martingales
zbMATH Keywords: absolute continuity properties
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4215115