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This paper presents various semantics in the branching-time spectrum of discrete-time and continuous-time Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation pre-orders are covered and are logically characterised in terms of the temporal logics PCTL and CSL. Apart from presenting various existing branching-time relations in a uniform manner, our contributions are: (i) weak simulation for DTMCs is defined, (ii) weak bisimulation equivalence is shown to coincide with weak simulation equivalence, (iii) logical characterisation of weak (bi)simulations are provided, and (iv) a classification of branching-time relations is presented, elucidating the semantics of DTMCs, CTMCs and their interrelation.

This paper presents a simulation preorder for continuous-time Markov chains (CTMCs). The simulation preorder is a conservative extension of a weak variant of probabilistic simulation on fully probabilistic systems, i.e., discrete-time Markov chains. The main result of the paper is that the simulation preorder preserves safety and liveness properties expressed in continuous stochastic logic (CSL), a stochastic branching-time temporal logic interpreted over CTMCs.

Invited Contributions.- Failure Detection vs Group Membership in Fault-Tolerant Distributed Systems: Hidden Trade-Offs.- Probability and Timing: Challenges for Secure Programming.- Selected Papers.- Security Analysis of a Probabilistic Non-repudiation Protocol.- The Mean Value of the Maximum.- Reduction and Refinement Strategies for Probabilistic Analysis.- Action Refinement for Probabilistic Processes with True Concurrency Models.- Probabilistic Unfoldings and Partial Order Fairness in Petri Nets.- Possibilistic and Probabilistic Abstraction-Based Model Checking.- Out-of-Core Solution of Large Linear Systems of Equations Arising from Stochastic Modelling.- Model Checking CSL until Formulae with Random Time Bounds.- Probabilistic Model Checking of the IEEE 802.11 Wireless Local Area Network Protocol.- Deriving Symbolic Representations from Stochastic Process Algebras.- Short Abstracts.- A Generalization of Equational Proof Theory?.- An Integrated Approach for the Specification and Analysis of Stochastic Real-Time Systems.- Probabilistic Abstract Interpretation and Statistical Testing.- Approximate Verification of Probabilistic Systems.

Markov chains are widely used in practice to determine system performance and reliability characteristics. The vast majority of applications considers continuous-time Markov chains (CTMCs). This tutorial paper shows how successful model specification and analysis techniques from concurrency theory can be applied to performance evaluation. The specification of CTMCs is supported by a stochastic process algebra, while the quantitative analysis of these models is tackled by means of model checking. Process algebra provides: (i) a high-level specification formalism for describing CTMCs in a precise, modular and constraint-oriented way, and (ii) means for the automated generation and aggregation of CTMCs. Temporal logic model checking provides: (i) a formalism to specify complex measures-of-in terest in a lucid, compact and flexible way, (ii) automated means to quantify these measures over CTMCs, and (iii) automated measure-driven aggregation (lumping) of CTMCs. Combining process algebra and model checking constitutes a coherent framework for performance evaluation based on CTMCs.

Markov-reward models, as extensions of continuous-time Markov chains, have received increased attention for the specification and evaluation of performance and dependability properties of systems. Until now, however, the specification of reward-based performance and dependability measures has been done manually and informally. In this paper, we change this undesirable situation by the introduction of a continuous-time, reward-based stochastic logic. We argue that this logic is adequate for expressing performability measures of a large variety. We isolate two important sub-logics, the logic CSL [#!ASS+96!#,#!BKH99!#], and the novel logic CRL that allows one to express reward-based properties. These logics turn out to be complementary, which is formally established in our main duality theorem. This result implies that reward-based properties expressed in CRL for a particular Markov reward model can be interpreted as CSL properties over a derived continuo us-time Markov chain, so that model checking procedures for CSL [#!BKH99!#,#!BHHK00!#] can be employed.