Cycle time of stochastic max-plus linear systems

Abstract

We analyze the asymptotic behavior of sequences of random variables (x(n))n ∈ N defined by an initial condition and the induction formula xi(n + 1) = maxj(Aij(n) + xj (n)), where (A(n))n ∈ N is a stationary and ergodic sequence of random matrices with entries in R ⋃ ( − ∞ ). 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 (1/nx(n))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/nx(n))n ∈ N does not converge almost-surely. © 2008 Applied Probability Trust.