Transient Analysis of Generalised Semi-Markov Processes Using Transient Stochastic State Classes

The method of stochastic state classes approaches the analysis of Generalised Semi Markov Processes (GSMP) through symbolic derivation of probability density functions over Difference Bounds Matrix (DBM) zones. This makes viable steady state analysis in both discrete and continuous time, provided that each cyclic behavior that changes the enabling status of generally distributed transitions visits at least one regeneration point. However, transient analysis is supported only in discrete time. We extend the approach providing a way to derive continuous time transient probabilities. To this end, stochastic state classes are extended with a supplementary age clock that enables symbolic derivation of the distribution of times at which the states of a zone can be reached. The approach is amenable to efficient implementation when model timings are given by expolynomial distributions, and it can in principle be applied to transient analysis with any given time bound for any GSMP. In the special case of models underlying a Markov Regenerative Process (MRP), the method can also be applied to symbolic derivation of local and global kernels, which in turn provide transient probabilities through numerical integration of generalized renewal equations. Since much of the complexity of this analysis is due to the local kernel, we propose a selective derivation of its entries depending on the specific transient measure targeted by the analysis.