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Nonergodic Jackson networks with infinite supply–local stabilization and local equilibrium analysis

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  09 December 2016

Jennifer Sommer ,
Hans Daduna  and
Bernd Heidergott
Jennifer Sommer*
Affiliation:
HPC-Hamburg Port Consulting GmbH
Hans Daduna*
Affiliation:
University of Hamburg
Bernd Heidergott*
Affiliation:
Tinbergen Institute and Vrije University Amsterdam
*
* Postal address: HPC-Hamburg Port Consulting GmbH, PO Box 11 14 06, 20414 Hamburg, Germany. Email address: j.sommer@hpc-hamburg.de
** Postal address: Center of Mathematical Statistics and Stochastic Processes, Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany. Email address: daduna@math.uni-hamburg.de
*** Postal address: Department of Econometrics and Operations Research, Vrije University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Email address: b.f.heidergott@vu.nl
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Abstract

Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

Keywords

Jackson networkstabilityinstabilityproduct-form solutionbottleneck analysisshortest path

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B15: Network models, stochastic
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1125 - 1142
Copyright
Copyright © Applied Probability Trust 2016 

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