When do stochastic max-plus linear systems have a cycle time ?
Résumé
We analyze the asymptotic behavior of the sequence of random variables (x(n, x0 ))n∈N defined by x(0, x0 ) = x0 and x(n+1, x0 ) = A(n)x(n, x0 ), where (A(n))n∈N is a stationary and ergodic sequence of random matrices with entries in the semiring (R ∪ {−∞}, max, +). Such sequences model a large class of discrete event systems, among which timed event graphs, 1-bounded Petri nets, some queuing networks, train or computer networks. We give a necessary condition for 1/n x(n, x0 ) n∈N to converge almost-surely, which proves to be sufficient when the A(n) are i.i.d. Moreover, we construct a new example, in which (A(n))n∈N is strongly mixing, that condition is satisfied, but 1/n x(n, x0 ) n∈N do not converge almost-surely.
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