Computer Science – Other Computer Science
Scientific paper
2007-07-25
Advances in Applied Probability 29, 2 (1997) 444-477
Computer Science
Other Computer Science
Scientific paper
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.
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