Poisson's equation for queues driven by a Markovian marked point process
type: Item
instance of: scholarly article
title: Poisson's equation for queues driven by a Markovian marked point process
zbMATH Open document ID: 0809.60093
DOI: 10.1007/BF01158696
author: Mogens Bladt
author: Søren Asmussen
published in: Queueing Systems
publication date: 1995-03-28T00:00:00Z
review text: Let \(\{J_ t\}_{t>0}\) be a finite Markov process, generating a marked point process input, and denote by \(V_ t\) the virtual waiting time at time \(t\) in the queue. This paper dals with Poisson's equation which in the setting \(\{(V_ t, J_ t)\}\) has the form \((*)\) \({\mathcal A}_ g = - f\), where \({\mathcal A}\) is the infinitesimal generator of \(\{(V_ t,J_ t)\}\). The explicit form of a suitable kernel \(K\) is calculated, so that the solution of \((*)\) has the form \(Kf\). Further, these results are applied to the service times having phase-type distributions and for evaluating of variance constants. Though of complicated matrix-analytical form, the results are explicit and computational tractable.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 698842
zbMATH Keywords: virtual waiting time
zbMATH Keywords: queue
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: phase-type distributions
reviewed by: Elisaveta Pancheva
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01158696
OpenAlex ID: W2024989604
description: scientific article; zbMATH DE number 698842