Efficient Quantile Computation in Markov Chains via Counting Problems   for Parikh Images
  A cost Markov chain is a Markov chain whose transitions are labelled with non-negative integer costs. A fundamental problem on this model, with applications in the verification of stochastic systems, is to compute information about the distribution of the total cost accumulated in a run. This includes the probability of large total costs, the median cost, and other quantiles. While expectations can be computed in polynomial time, previous work has demonstrated that the computation of cost quantiles is harder but can be done in PSPACE. In this paper we show that cost quantiles in cost Markov chains can be computed in the counting hierarchy, thus providing evidence that computing those quantiles is likely not PSPACE-hard. We obtain this result by exhibiting a tight link to a problem in formal language theory: counting the number of words that are both accepted by a given automaton and have a given Parikh image. Motivated by this link, we comprehensively investigate the complexity of the latter problem. Among other techniques, we rely on the so-called BEST theorem for efficiently computing the number of Eulerian circuits in a directed graph. 
Christoph Haase and Stefan Kiefer and Markus Lohrey
cs.FL cs.CC cs.DM cs.LO