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Correlation formulas for Markovian network processes in a random environment

  • Hans Daduna (a1) and Ryszard Szekli (a2)
Abstract

We consider Markov processes, which describe, e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady-state distribution of product form available in the literature, we note that this steady-state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one-step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for travelling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.

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Corresponding author
* Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany. Email address: daduna@math.uni-hamburg.de
** Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
References
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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