Reference:

Tomáš Brázdil, Stefan Kiefer, Antonín Kučera, and Petr Novotný. Long-run average behaviour of probabilistic vector addition systems. Technical report, arXiv.org, 2015. Available at http://arxiv.org/abs/1505.02655.

Abstract:

We study the pattern frequency vector for runs in probabilistic Vector Addition Systems with States (pVASS). Intuitively, each configuration of a given pVASS is assigned one of finitely many patterns, and every run can thus be seen as an infinite sequence of these patterns. The pattern frequency vector assigns to each run the limit of pattern frequencies computed for longer and longer prefixes of the run. If the limit does not exist, then the vector is undefined. We show that for one-counter pVASS, the pattern frequency vector is defined and takes only finitely many values for almost all runs. Further, these values and their associated probabilities can be approximated up to an arbitrarily small relative error in polynomial time. For stable two-counter pVASS, we show the same result, but we do not provide any upper complexity bound. As a byproduct of our study, we discover counterexamples falsifying some classical results about stochastic Petri nets published in the 80s.

Suggested BibTeX entry:

@techreport{15BKKN-LICS-TR,
    author = {Tom\'a\v{s} Br\'azdil and Stefan Kiefer and Anton\'in Ku\v{c}era and Petr Novotn\'y},
    institution = {arXiv.org},
    note = {Available at http://arxiv.org/abs/1505.02655},
    title = {Long-Run Average Behaviour of Probabilistic Vector Addition Systems},
    year = {2015}
}