Model-checking algorithms for continuous-time markov chains

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., 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 timebounded 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.

CSL Model Checking Algorithms for Infinite-State Structured Markov Chains

“…All the remaining depicted states are representative states. We have different types of representative states: the white ones represent a set of states that is infinite in one dimension, the grey ones represent a set that is infinite in two dimensions and the light grey one (3,3,3) represents a set that is infinite in three dimensions. …”

“…As for finite CTMCs, a probability measure on paths can now be defined depending on the starting state [3]. Starting from there, two different types of state probabilities can be distinguished.…”

“…We use the logic CSL [3] to express properties for LJQNs. The syntax and semantics are the same as for finite CTMCs, with the only difference that we now interpret the formulas over states and paths in LJQNs.…”

“…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.…”

“…Since logics like CSL have been shown to be extremely helpful in specifying similar properties for finite CTMCs [2,3], we also pursue a CSL model checking procedure for the CTMCs that underlie so-called Jackson queueing networks (JQNs). As we know from [3], for CSL model checking of CTMCs, we need to be able to compute both steady-state and transient state probabilities for all states, and for all possible starting states. The key issue lies in the fact that the CTMC underlying a JQN is infinite in as many dimensions as there are queues in the JQN.…”