Model checking of performance measures using bounding aggregations

This paper presents an algorithm based on stochastic comparisons in order to check formulas with rewards on multidimensional Continuous Time Markov Chains (CTMC). These formulas are expressed in Continuous Stochastic Logic (CSL) which includes means to express transient, steady-state and path performance measures. However, using simulations or analytical methods, computation of transient and steady state distribution are limited to relatively small sizes because of the state space explosion problem. We propose a model checking algorithm based on aggregated bounding Markov processes in order to perform the verification on the bounds values instead of the exact one. The stochastic comparison has been largely applied in performance evaluation however the state space is generally assumed to be totally ordered which induces less accurate bounds for multidimensional Markov processes. We use the increasing set theory and the comparison by mapping functions in order to derive performance measures bounds on reduced state spaces. The relevance of the proposed checking algorithm is the possibility of a parametric aggregation scheme in order to improve the accuracy of the bounds and in the same time the precision of the checking, but in return with an increasing of the complexity. We apply the algorithm to the performance evaluation of a tandem queueing network in order to verify if loss probabilities are included or not in an interval.