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Products of Irreducible Random Matrices in the (Max, +) Algebra

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Jean Mairesse
Jean Mairesse*
Affiliation:
INRIA
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Abstract

We consider the recursive equation x(n + 1)= A(n)⊗x(n), where x(n + 1) and x(n) are ℝk-valued vectors and A(n) is an irreducible random matrix of size k × k. The matrix-vector multiplication in the (max, +) algebra is defined by (A(n)⊗x(n))= maxj (Aij (n) + xj(n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices such that and the matrices have a unique periodic regime.

Keywords

RANDOM MATRICESCYCLIC JACKSON NETWORK

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 444 - 477
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Supported by the European Grant BRA-QMIPS of CEC DG XIII.

The research of the author is supported by the Direction des Recherches Etudes et Techniques (DRET) under contract no 91 815.

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