Markov network processes with string transitions
type: Item
instance of: scholarly article
title: Markov network processes with string transitions
zbMATH Open document ID: 0948.60085
author: Richard F. Serfozo
author: Bingyi Yang
published in: The Annals of Applied Probability
publication date: 2000-11-19T00:00:00Z
review text: The authors consider Markov network processes with string transitions. They assume that it is described by a Markov process \(\{X(t)\), \(t\geq 0\}\) that represents the numbers of units at \(m\) nodes. The state space consists of \(m\)-dimensional vectors \(x= (x_1, x_2,\dots, x_m)\) where \(x_j\) denotes the number of units at node \(j\). A transition of the network from \(x\) to \(y\) has a rate of the form \(\sum_{sa} \lambda_{sa} r_{sa} (x,y)\). \(\lambda_{sa}\) is the rate of selecting a string of vectors \(s\) and another add-on vector \(a\) as the increments in the state. \(r_{sa} (x,y)\) is a system-dependent transition-initiation rate that may represent service rates at the nodes plus other transition information. The authors give necessary and sufficient conditions for a string-net to have an invariant measure of the form \(\Phi(x) \prod_{j=1}^m w_j^{x_j}\). \(\Phi\) is determined by the transition-initiation rates \(r_{sa} (x,y)\), and the parameters \(w_j\) are solutions to certain polynomial equations involving the string-generation rates \(\lambda_{sa}\). They also give sufficient conditions for the existence of a solution to these so-called traffic equations and show that the equations are equivalent to equalities of certain average flows in the network (a partial balance property).
Mathematics Subject Classification ID: 60K25
zbMATH DE Number: 1319833
zbMATH Keywords: Markov queueing
zbMATH Keywords: network
reviewed by: Władysław Szczotka
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1028903451
OpenAlex ID: W2043590054
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DOI: 10.1214/AOAP/1028903451
description: scientific article; zbMATH DE number 1319833