Model Checking of Continuous-Time Markov Chains Against Timed Automata Specifications

TL;DR

This paper presents a method for verifying continuous-time Markov chains against timed automata specifications by translating the problem into reachability analysis in piecewise deterministic Markov processes, enabling the use of standard CTMC algorithms.

Contribution

It introduces a novel approach to model checking CTMCs against timed automata using PDPs and integral equations, extending verification techniques to real-time probabilistic systems.

Findings

Verification reduces to reachability in PDPs.

Qualitative analysis involves graph analysis of PDPs.

Single-clock DTA verification uses standard CTMC algorithms.

Abstract

We study the verification of a finite continuous-time Markov chain (CTMC) C against a linear real-time specification given as a deterministic timed automaton (DTA) A with finite or Muller acceptance conditions. The central question that we address is: what is the probability of the set of paths of C that are accepted by A, i.e., the likelihood that C satisfies A? It is shown that under finite acceptance criteria this equals the reachability probability in a finite piecewise deterministic Markov process (PDP), whereas for Muller acceptance criteria it coincides with the reachability probability of terminal strongly connected components in such a PDP. Qualitative verification is shown to amount to a graph analysis of the PDP. Reachability probabilities in our PDPs are then characterized as the least solution of a system of Volterra integral equations of the second type and are shown to be approximated by the solution of a system of partial differential equations. For single-clock DTA, this integral equation system can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations. As the coefficients are in fact transient probabilities in CTMCs, this result implies that standard algorithms for CTMC analysis suffice to verify single-clock DTA specifications.