Subadditivity and stability of a class of discrete-event systems

We investigate the stability of discrete-event systems modeled as generalized semi-Markov processes with event times that satisfy (max,+) recursions. We show that there exists for each event a cycle time, which is the long-run average time between event occurrences. We characterize the rate of convergence to this limit, bounding the error for finite horizons. The main tools we use are (max,+) matrix products, the subadditive ergodic theorem, and martingale inequalities. We discuss connections with these different fields, with the general theory of random matrix products, and with recent results for discrete-event systems modeled as Petri nets