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Networks and dynamical systems

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

V. A. Malyshev
V. A. Malyshev*
Affiliation:
INRIA
*
∗Postal address: INRIA—Domaine de Voluceau, Rocquencourt, BP105, 78153 Le Chesnay, France. On leave from Laboratory of Large Random Systems, Moscow State University, Moscow, 119899, Russia.
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Abstract

A new approach to the problem of classification of (deflected) random walks in or Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energy M < 0 corresponds to ergodicity, the Lyapounov exponent L < 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.

Keywords

CLASSIFICATION OF MARKOV PROCESSESSINGLE-CUSTOMER QUEUEING NETWORKSSCALING LIMIT FOR DEFLECTED RANDOM WALKSRANDOM WALKS IN AN ORTHANT

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 60K25: Queueing theory 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 140 - 175
Copyright
Copyright © Applied Probability Trust 1993 

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References

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