Aggregation of States in Colored Stochastic Petri Nets: Application to a Multiprocessor Architecture

In this paper, we present a lumping method based on a particular class of colored Petri nets. We prove on an example that the method can be usefully applied to the performance evaluation of symmetric systems. Stochastic Petri nets [Nat 80, Mol 81, Zub 85] are more and more widely used for the performance evaluation of synchronized systems. Some extensions of the original models have been proposed, either including more general timing constraints, such as immediate firing [Ajm 84], deterministic delays [Ajm 87a], general timing [Dug 84], or extending the underlying model by including simultaneous transition firings [Haa 87], or unbounded places [Flo 86]. Other models have been presented, that are particularly well suited to the study of some specific problems, such as fault tolerance and degradable performance [Mey 85], resource sharing [Hol 85], or communication protocol performance [Raz 85]. However, for any kind of model, obtaining performance results is not easy when complex systems are analyzed. Even when limiting to markovian models, the resolution is often quite difficult because of the size of the generated Markov process. Simulation techniques can be used, but they are very expensive and the accuracy of the result is not always guaranteed. Therefore, research has been done to provide more simple and exact analytic methods of resolution. Lumping techniques have been presented in [Kem 60]. They are based on an aggregation of states into classes that is performed once the Markov process has been constructed. The solution is then computed on the classes. The drawbacks of this method are twofold. On the one hand, the lumping phase generally depends on the ability of the analyst, as there is no algorithm that defines the partition of states into classes. On the other hand, it is not always possible to derive back the state probabilities from the class probabilities.