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.

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This paper surveys the theoretical developments in the ÿeld of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems -like large-scale computers, clientserver architectures, networks -can accurately be described using such stochastic speciÿcation formalisms. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for di erent types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also brie y indicate how one can proÿt from the separation of time and actions when incorporating more general, non-Markovian distributions.

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This paper presents MODEST (MOdeling and DEscription language for Stochastic Timed systems), a formalism that is intended to support 1) the modular description of reactive systems' behavior while covering both 2) functional and 3) nonfunctional system aspects such as timing and quality-of-service constraints in a single specification. The language contains features such as simple and structured data types, structuring mechanisms like parallel composition and abstraction, means to control the granularity of assignments, exception handling, and nondeterministic and random branching and timing. MODEST can be viewed as an overarching notation for a wide spectrum of models, ranging from labeled transition systems to timed automata (and probabilistic variants thereof), as well as prominent stochastic processes such as (generalized semi-)Markov chains and decision processes. The paper describes the design rationales and details of the syntax and semantics.

The Markov Reward Model Checker (MRMC) is a software tool for verifying properties over probabilistic models. It supports PCTL and CSL model checking, and their reward extensions. Distinguishing features of MRMC are its support for computing time-and reward-bounded reachability probabilities, (property-driven) bisimulation minimization, and precise on-the-fly steady-state detection. Recent tool features include time-bounded reachability analysis for uniform CTMDPs and CSL model checking by discrete-event simulation. This paper presents the tool's current status and its implementation details. This research was performed as part of the MC=MC project financed by the Netherlands Organization for Scientific Research (NWO) and the DFG Research Training Group 623 on Leistungsgarantien für Rechnersysteme. We thank Maneesh Khattri (Oxford Univ.), Christina Jansen (RWTH Aachen), and Tim Kemna (Univ. Twente) for their implementation efforts.