Cycle time of stochastic max-plus linear systems

Glenn Merlet (1)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

We analyze the asymptotic behavior of sequences of random variables defined by an initial condition, a stationary and ergodic sequence of random matrices, and an induction formula involving multiplication is the so-called max-plus algebra. This type of recursive sequences are frequently used in applied probability as they model many systems as some queueing networks, train and computer networks, and production systems. We give a necessary condition for the recursive sequences to satisfy a strong law of large numbers, which proves to be sufficient when the matrices are i.i.d. Moreover, we construct a new example, in which the sequence of matrices is strongly mixing, that condition is satisfied, but the recursive sequence do not converges almost surely.

Mots clés: Law of Large Numbers; products of random matrices; max-plus; discrete event systems; stochastic recurrent sequences

Domaines: Probabilités [math.PR]; Mathématique discrète [cs.DM]

Glenn Merlet. Cycle time of stochastic max-plus linear systems. Electronic Journal of Probability, 2008, 13 (2008), Paper 12, 322-340. ⟨hal-00197850v2⟩