Abstract. Perturbation analysis in probabilistic verification addresses the robustness and sensitivity problem for verification of stochastic models against qualitative and quantitative properties. We identify two types of perturbation bounds, namely non-asymptotic bounds and asymptotic bounds. Non-asymptotic bounds are exact, pointwise bounds that quantify the upper and lower bounds of the verification result subject to a given perturbation of the model, whereas asymptotic bounds are closed-form bounds that approximate non-asymptotic bounds by assuming that the given perturbation is sufficiently small. We perform perturbation analysis in the setting of Discrete-time Markov Chains. We consider three basic matrix norms to capture the perturbation distance, and focus on the computational aspect. Our main contributions include algorithms and tight complexity bounds for calculating both non-asymptotic bounds and asymptotic bounds with respect to the three perturbation distances.

The majority of existing probabilistic model checking case studies are based on well understood theoretical models and distributions. However, real-life probabilistic systems usually involve distribution parameters whose values are obtained by empirical measurements and thus are subject to small perturbations. In this paper, we consider perturbation analysis of reachability in the parametric models of these systems (i.e., parametric Markov chains) equipped with the norm of absolute distance. Our main contribution is a method to compute the asymptotic bounds in the form of condition numbers for constrained reachability probabilities against perturbations of the distribution parameters of the system. The adequacy of the method is demonstrated through experiments with the Zeroconf protocol and the hopping frog problem.

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.

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Abstract-Probabilistic model checking is a verification technique that has been the focus of intensive research for over a decade. One important issue with probabilistic model checking, which is crucial for its practical significance but is overlooked by the state-of-the-art largely, is the potential discrepancy between a stochastic model and the real-world system it represents when the model is built from statistical data. In the worst case, a tiny but nontrivial change to some model quantities might lead to misleading or even invalid verification results. To address this issue, in this paper, we present a mathematical characterization of the consequences of model perturbations on the verification distance. The formal model that we adopt is a parametric variant of discrete-time Markov chains equipped with a vector norm to measure the perturbation. Our main technical contributions include a closed-form formulation of asymptotic perturbation bounds, and computational methods for two arguably most useful forms of those bounds, namely linear bounds and quadratic bounds. We focus on verification of reachability properties but also address automata-based verification of omega-regular properties. We present the results of a selection of case studies that demonstrate that asymptotic perturbation bounds can accurately estimate maximum variations of verification results induced by model perturbations.