Multiphase until formulas over Markov reward models: An algebraic approach

We consider the probabilistic model checking problem of continuous-time Markov chains with rewards. We first extend multiphase until formulas in continuous stochastic logic (CSL) with reward constraints. Then we present an effective integral-style algorithm to compute the probability under the assumption of harmony, and give upper and lower bounds of the probability without this assumption. Furthermore, the resulting probability value (or its upper and lower bounds) is shown to be a real number of a well-formed structure, with which we can successfully (or partially) decide whether the constraints in the CSL formula are satisfied. Our method is entirely based on algebraic manipulations and number theory. Finally, to show the practical usefulness, we apply the results to evaluate the performance of a small multi-processor system.