**A. Horváth, M. Paolieri, L. Ridi, E. Vicario**

The method of stochastic state classes approaches the analysis of Generalised Semi Markov
Processes (GSMPs) through the symbolic derivation of probability density functions over
supports described by Difference Bounds Matrix (DBM) zones. This makes steady state
analysis viable, provided that at least one regeneration point is visited by every cyclic
behaviour of the model.

We extend the approach providing a way to derive transient probabilities. To this end, stochastic state classes are extended with a supplementary timer that enables the symbolic derivation of the distribution of time at which a class can be entered. The approach is amenable to efficient implementation when model timings are given by expolynomial distributions, and it can be applied to perform transient analysis of GSMPs within any given time bound. In the special case of models underlying a Markov Regenerative Process (MRGP), the method can also be applied to the symbolic derivation of local and global kernels, which in turn provide transient probabilities through numerical integration of generalised 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.

horvath 2012-04-05