Products of irreducible random matrices in the (Max,+) Algebra

TL;DR

This paper studies the behavior of recursive equations involving products of random matrices in the (max,+) algebra, showing conditions for convergence to a unique stationary regime in stochastic models like Jackson networks.

Contribution

It establishes necessary and sufficient conditions for the system to couple in finite time with a unique stationary regime based on matrix properties.

Findings

System couples in finite time with a unique stationary regime under certain matrix conditions.

Conditions involve existence of a set of matrices with a positive probability and a unique periodic regime.

Results apply to models like Jackson Networks and manufacturing systems with blocking.

Abstract

We consider the recursive equation ``x(n+1)=A(n)x(n)'' where x(n+1) and x(n) are column vectors of size k and where A(n) is an irreducible random matrix of size k x k. The matrix-vector multiplication in the (max,+) algebra is defined by (A(n)x(n))_i= max_j [ A(n)_{ij} +x(n)_j ]. 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 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 C such that P {A(0) in C} > 0, and the matrices in C have a unique periodic regime.