Products of Irreducible Random Matrices in the ( max , + ) Algebra 1

We consider the recursive equation \x(n + 1) = A(n) x(n)" where x(n + 1) and x(n) are R k-valued vectors and A(n) is an irreducible random matrix of size k k. The matrix-vector multiplication in the (max,+) algebra is deened by (A(n) x(n)) i = max j (A ij (n) + x j (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 fA(n); n 2 Ng is i.i.d or more generally stationary and ergodic. The main result of the paper states that the system couples in nite time with a unique stationary regime if and only if there exists a set of matrices C such that PfA(0) 2 Cg > 0 and the matrices C 2 C have a unique periodic regime.