Model Checking Infinite-State Markov Chains

Abstract

Abstract. In this paper algorithms for model checking CSL (continuous stochastic logic) against infinite-state continuous-time Markov chains of so-called quasi birth-death type are developed. In doing so we extend the applicability of CSL model checking beyond the recently proposed case for finite-state continuous-time Markov chains, to an important class of infinite-state Markov chains. We present syntax and semantics for CSL and develop efficient model checking algorithms for the steady-state operator and the time-bounded next and until operator. For the former, we rely on the so-called matrix-geometric solution of the steady-state probabilities of the infinite-state Markov chain. For the time-bounded until operator we develop a new algorithm for the transient analysis of infinite-state Markov chains, thereby exploiting the quasi birth-death structure. A case study shows the feasibility of our approach.

CSL Model Checking Algorithms for Infinite-State Structured Markov Chains

“…From [15] we know that CSL formulas are not level independent on QBDs in general, even if the atomic propositions are level independent. However, their validity does not change arbitrarily between levels.…”

“…For model checking P ⊲⊳p (Φ U I Ψ ) we adopt the general approach for finite CTMCs [3] and QBDs [15]. Recall that the CSL path formula ϕ = Φ U I Ψ is valid if a Ψ -state is reached on a path during the time interval [t 1 , t 2 ] via only Φ-states.…”

“…In [15] we have shown how transient probabilities can be computed for QBDs using uniformization [9] by considering only a finite fraction of the infinite state space. As standard property of uniformization, the finite time bound t is transformed to a finite number of steps l [9].…”

“…Once the accumulated probabilities are calculated, similar inequalities as presented in [15], can be used to decide the validity of P ⊲⊳p (Φ U I Ψ ) on LJQNs. The accumulated probability is always an underestimation of the actual probability.…”

“…For the steady-state probabilities we can rely on the seminal work of Jackson, however, for the transient state probabilities, no results are readily available. Inspired by our recent work on CSL model checking for quasi-birth-death processes (QBDs), which form a different class of infinitestate CTMCs [15], we use an uniformization-based approach to compute just those transient state probabilities in JQNs that are needed to verify the validity of CSL properties. The highly structured state space allows us to conclude the validity of CSL properties for groups of states on the basis of the validity for a so-called representative state in such a group.…”