Performance evaluation of marked graphs by linear programming

Performance evaluation is an important issue for discrete event dynamic systems, which in many cases can be described in terms of the language of Petri nets. In particular, for marked graphs, a subclass of Petri nets, a formula for performance evaluation is widely known. That is, the ratio of the total execution time to the number of tokens in a cycle gives a lower bound for the cycle time of the system, and the maximum of these ratios over all cycles determines the overall system performance. However, we need to enumerate all directed cycles in a graph to apply this formula of performance evaluation. Carrying out this enumeration for an actual system is often impractical, since the number of cycles in a graph usually grows exponentially with the size of a system. We give a linear algebraic characterization for directed cycles, and based on this result, transform the problem of performance evaluation into a simple linear programming problem. A few explanatory examples are also given.