Model-checking algorithms for continuous-time markov chains

TLDR

Continuous‑time Markov chains are widely used to analyze steady‑state and transient probabilities, and the logic introduced here extends the continuous stochastic logic CSL. The paper introduces a branching temporal logic for expressing real‑time probabilistic properties on CTMCs and proposes approximate model‑checking algorithms for it. The logic includes a time‑bounded until operator and a steady‑state operator, and model checking reduces to solving linear equations or Volterra integral equations, which can be handled by transient analysis techniques such as uniformization, while a lumping equivalence preserves formula validity. The authors demonstrate that model checking reduces to linear and Volterra equations, that time‑bounded until properties can be verified via transient analysis such as uniformization, and that a lumping equivalence preserves all formulas.

Abstract

Continuous-time Markov chains (CTMCs) have been widely used to determine system performance and dependability characteristics. Their analysis most often concerns the computation of steady-state and transient-state probabilities. This paper introduces a branching temporal logic for expressing real-time probabilistic properties on CTMCs and presents approximate model checking algorithms for this logic. The logic, an extension of the continuous stochastic logic CSL of Aziz et al. (1995, 2000), contains a time-bounded until operator to express probabilistic timing properties over paths as well as an operator to express steady-state probabilities. We show that the model checking problem for this logic reduces to a system of linear equations (for unbounded until and the steady-state operator) and a Volterra integral equation system (for time-bounded until). We then show that the problem of model-checking time-bounded until properties can be reduced to the problem of computing transient state probabilities for CTMCs. This allows the verification of probabilistic timing properties by efficient techniques for transient analysis for CTMCs such as uniformization. Finally, we show that a variant of lumping equivalence (bisimulation), a well-known notion for aggregating CTMCs, preserves the validity of all formulas in the logic.