Stochastic marked graphs

Stochastic marked graphs (SMGs) are marked graphs in which transmission delays of tokens, and firing durations are random variables. A technique is presented to study the performance of SMGs. The main performance measure is the rate of computation, i.e., the average number of firings of a vertex, per time unit. The effect of the topology and the probability of the random variables on the rate is investigated. For deterministic random variables, the rate is maximized, while for exponential random variables the rate is minimized (among a natural class of distributions). For random variables with exponential distribution several bounds on the rate are provided. The bounds depend on the degrees of the vertices and on the average number of tokens in a cycle, but not on the number of vertices itself. In particular, it is shown that the rate is at least the optimal (deterministic) rate, divided by a logarithmic factor of the vertex degrees. Thus, for some graphs the rate does not diminish below a bound, regardless of the number of vertices