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Jackson networks in nonautonomous random environments

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  10 June 2016

Ruslan Krenzler ,
Hans Daduna  and
Sonja Otten
Ruslan Krenzler*
Affiliation:
University of Hamburg
Hans Daduna*
Affiliation:
University of Hamburg
Sonja Otten*
Affiliation:
University of Hamburg
*
* Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.
* Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.
* Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.
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Abstract

We investigate queueing networks in a random environment. The impact of the evolving environment on the network is by changing service capacities (upgrading and/or degrading, breakdown, repair) when the environment changes its state. On the other side, customers departing from the network may enforce the environment to jump immediately. This means that the environment is nonautonomous and therefore results in a rather complex two-way interaction, especially if the environment is not itself Markov. To react to the changes of the capacities we implement randomised versions of the well-known deterministic rerouteing schemes 'skipping' (jump-over protocol) and `reflection' (repeated service, random direction). Our main result is an explicit expression for the joint stationary distribution of the queue-lengths vector and the environment which is of product form.

Keywords

Randomised random walkJackson networkskippingprocesses in random environmentreflectionproduct form steady statebreakdown of nodesdegrading servicespeed-up of service

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 60K25: Queueing theory 90B22: Queues and service 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 315 - 331
Copyright
Copyright © Applied Probability Trust 2016 

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