On the analysis of the virtual waiting time in open queueing networks
type: Item
instance of: scholarly article
title: On the analysis of the virtual waiting time in open queueing networks
zbMATH Open document ID: 1171.60021
published in: Acta Applicandae Mathematicae
publication date: 2009-04-29T00:00:00Z
review text: An open network of single server stations with waiting rooms of unlimited capacities and operating under first come first serve (FCFS) discipline is considered. The external arrival streams of customers at the stations are independent renewal processes and the service times at the nodes are generally i.i.d. distributed and independent of the external arrival processes. The routing of the customers after finishing service at the stations is Markovian and governed by a transition matrix. The main result of the paper is that under proper overload of all stations and further moment conditions the vector of the virtual waiting time processes of the stations -- associated to a considered sequence of arrival and service processes as well as routing processes -- converges under an appropriate scaling to the vector process of independent standard Wiener processes. The proof uses results from \textit{L. L. Sakalauskas} and \textit{S. Minkevičius} [Eur. J. Oper. Res. 120, No. 3, 632--640 (2000; Zbl 0981.60093)]. Further an application to a reliability model is given. Further the authors state that the law of iterated logarithm for the virtual waiting time of the stations can be proved similarly as their main theorem of the paper.
reviewed by: Andreas Brandt
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 68M20
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 5549560
zbMATH Keywords: open network of GI/GI/1 queues
zbMATH Keywords: overload regime
zbMATH Keywords: virtual waiting time
zbMATH Keywords: limit theorem
zbMATH Keywords: Wiener process
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: reliability model
author: Saulius Minkevičius
MaRDI profile type: Publication
full work available at URL: s10440-008-9256-9
OpenAlex ID: W2042042241
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DOI: 10.1007/S10440-008-9256-9
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description: scientific article; zbMATH DE number 5549560