The settling time of a class of discrete event systems

The linear (in the (max, +)-algebra) discrete-event system model is considered. Under the assumption that the coefficients and the input are stationary and ergodic random sequences, a cutoff phenomenon is proved for the convergence of the transient process to its steady state. This cutoff phenomenon works as follows: if d_N(n ) is the total variation distance between the transient process at time n and the steady state, when the initial state is proportional to N, then d_N(Nn) converges, as N→∞, to 1 or 0, depending on whether n is smaller or larger than a certain constant α that is identified in the paper. The settling time of the system can then be taken to be Nα when N is large. The meaning of this is that, roughly speaking, the system is far from stationarity before time Nα and reaches stationarity after Nα