Cycle time of stochastic max-plus linear systems

TL;DR

This paper studies the long-term behavior of stochastic max-plus linear systems, providing conditions for convergence and illustrating cases where convergence fails despite mixing properties.

Contribution

It establishes necessary and sufficient conditions for the strong law of large numbers in max-plus systems, and presents a novel example with mixing matrices where convergence does not occur.

Findings

Necessary condition for strong law of large numbers.

Sufficient condition when matrices are i.i.d.

Counterexample with mixing matrices where convergence fails.

Abstract

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.