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Parametric Markov chains have been introduced as a model for families ofstochastic systems that rely on the same graph structure, but differ in theconcrete transition probabilities. The latter are specified by polynomialconstraints for the parameters. Among the tasks typically addressed in theanalysis of parametric Markov chains are (1) the computation of closed-formsolutions for reachabilty probabilities and other quantitative measures and (2)finding symbolic representations of the set of parameter valuations for which agiven temporal logical formula holds as well as (3) the decision variant of (2)that asks whether there exists a parameter valuation where a temporal logicalformula holds. Our contribution to (1) is to show that existing implementationsfor computing rational functions for reachability probabilities or expectedcosts in parametric Markov chains can be improved by using fraction-freeGaussian elimination, a long-known technique for linear equation systems withparametric coefficients. Our contribution to (2) and (3) is acomplexity-theoretic discussion of the model checking problem for parametricMarkov chains and probabilistic computation tree logic (PCTL) formulas. Wepresent an exponential-time algorithm for (2) and a PSPACE upper bound for (3).Moreover, we identify fragments of PCTL and subclasses of parametric Markovchains where (1) and (3) are solvable in polynomial time and establishNP-hardness for other PCTL fragments.

Parametric Markov Chains: PCTL Complexity and Fraction-free Gaussian Elimination