On the Existence and Computability of Long-Run Average Properties in Probabilistic VASS

We present recent results about the long-run average properties of probabilistic vector additions systems with states (pVASS). Interestingly, for probabilistic pVASS with two or more counters, long-run average properties may take several different values with positive probability even if the underlying state space is strongly connected. This contradics the previous results about stochastic Petri nets established in 80s. For pVASS with three or more counters, it may even happen that the long-run average properties are undefined (i.e., the corresponding limits do not exist) for almost all runs, and this phenomenon is stable under small perturbations in transition probabilities. On the other hand, one can effectively approximate eligible values of long-run average properties and the corresponding probabilities for some sublasses of pVASS. These results are based on new exponential tail bounds achieved by designing and analyzing appropriate martingales. The paper focuses on explaining the main underlying ideas.