Perturbation analysis of stochastic systems with empirical distribution parameters

Probabilistic model checking is a quantitative verification technology for computer systems and has been the focus of intense research for over a decade. While in many circumstances of probabilistic model checking it is reasonable to anticipate a possible discrepancy between a stochastic model and a real-world system it represents, the state-of-the-art provides little account for the effects of this discrepancy on verification results. To address this problem, we present a perturbation approach in which quantities such as transition probabilities in the stochastic model are allowed to be perturbed from their measured values. We present a rigorous mathematical characterization for variations that can occur to verification results in the presence of model perturbations. The formal treatment is based on the analysis of a parametric variant of discrete-time Markov chains, called parametric Markov chains (PMCs), which are equipped with a metric to measure their perturbed vector variables. We employ an asymptotic method from perturbation theory to compute two forms of perturbation bounds, namely condition numbers and quadratic bounds, for automata-based verification of PMCs. We also evaluate our approach with case studies on variant models for three widely studied systems, the Zeroconf protocol, the Leader Election Protocol and the NAND Multiplexer.