Ergodicity of a polling network
type: Item
instance of: scholarly article
title: Ergodicity of a polling network
zbMATH Open document ID: 0802.60084
DOI: 10.1016/0304-4149(94)90122-8
published in: Stochastic Processes and their Applications
publication date: 1994-12-12T00:00:00Z
review text: A queueing system with a single server and with \(N\) service stations is considered which are successively visited by the server who is polling them according to a Markovian routing scheme. For every \(i\in \{1,\dots, N\}\), the server can serve no more than a random maximum number \(D_ i\) of customers during each visit at station \(i\). The \(D_ i\) and the service times at this station are drawn independently from general distributions which may depend on \(i\). For switching from station \(i\) to station \(j\), the server needs a random switching time whose distribution may depend on the pair \((i,j)\). At each station, groups of customers arrive according to a homogeneous Poisson process whose intensity and group-size distribution may be station-dependent, too. Arrival processes, routing, switching times, and service times are mutually independent. The temporal evolution of the system is described by the Markov chain \((S, {\mathcal X})= \{(S(n), X_ 1(n),\) \(\dots, X_ i (n))\); \(n\geq 0\}\) where \(S(n)\) denotes the \(n\)th station polled by the server, and \(X_ i(n)\) the queue length at station \(i\) at the moment when the \(n\)th switching time is terminated. Conditions are derived under which \((S,{\mathcal X})\) is positive recurrent.
reviewed by: Volker Schmidt
Mathematics Subject Classification ID: 60K25
zbMATH DE Number: 590988
zbMATH Keywords: polling server
zbMATH Keywords: Poisson arrivals
zbMATH Keywords: queue length
zbMATH Keywords: stability
zbMATH Keywords: single server
zbMATH Keywords: Markovian routing scheme
zbMATH Keywords: random switching
zbMATH Keywords: homogeneous Poisson process
author: A. A. Borovkov
author: R. Schaßberger
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)90122-8
OpenAlex ID: W2090268469
cites work: Stability, monotonicity and invariant quantities in general polling systems
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cites work: Queueing systems on a circle
cites work: Queuing analysis of polling models
description: scientific article; zbMATH DE number 590988