# Transient analysis of non-Markovian models using stochastic state classes

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

**Abstract:** 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.

Stochastic ProcessesTheory

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