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12
Modelchecking algorithms for continuoustime Markov chains
 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING
, 2003
"... Continuoustime Markov chains (CTMCs) have been widely used to determine system performance and dependability characteristics. Their analysis most often concerns the computation of steadystate and transientstate probabilities. This paper introduces a branching temporal logic for expressing realt ..."
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Cited by 128 (26 self)
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Continuoustime Markov chains (CTMCs) have been widely used to determine system performance and dependability characteristics. Their analysis most often concerns the computation of steadystate and transientstate probabilities. This paper introduces a branching temporal logic for expressing realtime 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 timebounded until operator to express probabilistic timing properties over paths as well as an operator to express steadystate probabilities. We show that the model checking problem for this logic reduces to a system of linear equations (for unbounded until and the steadystate operator) and a Volterra integral equation system (for timebounded until). We then show that the problem of modelchecking 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 wellknown notion for aggregating CTMCs, preserves the validity of all formulas in the logic.
Comparative branchingtime semantics for Markov chains
 Information and Computation
, 2003
"... This paper presents various semantics in the branchingtime spectrum of discretetime and continuoustime Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation preorders are covered and are logically characterised in terms of the temporal logics PCTL (Probabilisti ..."
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Cited by 33 (12 self)
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This paper presents various semantics in the branchingtime spectrum of discretetime and continuoustime Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation preorders are covered and are logically characterised in terms of the temporal logics PCTL (Probabilistic Computation Tree Logic) and CSL (Continuous Stochastic Logic). Apart from presenting various existing branchingtime relations in a uniform manner, this paper presents the following new results: (i) strong simulation for CTMCs, (ii) weak simulation for CTMCs and DTMCs, (iii) logical characterizations thereof (including weak bisimulation for DTMCs), (iv) a relation between weak bisimulation and weak simulation equivalence, and (v) various connections between equivalences and preorders in the continuous and discretetime setting. The results are summarized in a branchingtime spectrum for DTMCs and CTMCs elucidating their semantics as well as their relationship. Key Words: comparative semantics, Markov chain, (weak) simulation, (weak) bisimulation, temporal logic
Weak Bisimulation is Sound and Complete for PCTL
, 2002
"... We investigate weak bisimulation of probabilistic systems in the presence of nondeterminism, i.e. labelled concurrent Markov chains (LCMC) with silent transitions. We build on the work of Philippou, Lee and Sokolsky [1] for finite state LCMCs. Their denition of weak bisimulation destroys the additiv ..."
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Cited by 14 (0 self)
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We investigate weak bisimulation of probabilistic systems in the presence of nondeterminism, i.e. labelled concurrent Markov chains (LCMC) with silent transitions. We build on the work of Philippou, Lee and Sokolsky [1] for finite state LCMCs. Their denition of weak bisimulation destroys the additivity property of the probability distributions, yielding instead capacities. The mathematics behind capacities naturally captures the intuition that when we deal with nondeterminism we must work with estimates on the possible probabilities. Our analysis leads to three...
Automated performance and dependability evaluation using model checking
 In Performance Evaluation of Complex Systems: Techniques and Tools, Performance 2002, Tutorial Lectures
, 2002
"... Abstract. Markov chains (and their extensions with rewards) have been widely used to determine performance, dependability and performability characteristics of computer communication systems, such as throughput, delay, mean time to failure, or the probability to accumulate at least a certain amount ..."
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Cited by 14 (1 self)
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Abstract. Markov chains (and their extensions with rewards) have been widely used to determine performance, dependability and performability characteristics of computer communication systems, such as throughput, delay, mean time to failure, or the probability to accumulate at least a certain amount of reward in a given time. Due to the rapidly increasing size and complexity of systems, Markov chains and Markov reward models are difficult and cumbersome to specify by hand at the statespace level. Therefore, various specification formalisms, such as stochastic Petri nets and stochastic process algebras, have been developed to facilitate the specification of these models at a higher level of abstraction. Uptill now, however, the specification of the measureofinterest is often done in an informal and relatively unstructured way. Furthermore, some measuresofinterest can not be expressed conveniently at all. In this tutorial paper, we present a logicbased specification technique to specify performance, dependability and performability measuresofinterest and show how for a given finite Markov chain (or Markov reward model) such measures can be evaluated in a fully automated way. Particular emphasis will be given to socalled pathbased measures and hierarchicallyspecified measures. For this purpose, we extend socalled model checking techniques to reason about discrete and continuoustime Markov chains and their rewards. We also report on the use of techniques such as (compositional) model reduction and measuredriven statespace generation to combat the infamous state space explosion problem. 1
Model checking action and statelabelled Markov chains
 DSN’04, Proceedings of International Conference on Dependable Systems and Networks
, 2004
"... In this paper we introduce the logic asCSL, an extension of continuous stochastic logic (CSL), which provides powerful means to characterise execution paths of action and statelabelled Markov chains. In asCSL, path properties are characterised by regular expressions over actions and stateformulas ..."
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Cited by 14 (2 self)
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In this paper we introduce the logic asCSL, an extension of continuous stochastic logic (CSL), which provides powerful means to characterise execution paths of action and statelabelled Markov chains. In asCSL, path properties are characterised by regular expressions over actions and stateformulas. Thus, the executability of a path not only depends on the available actions but also on the validity of certain state formulas in intermediate states. Our main result is that the model checking problem for asCSL can be reduced to CSL model checking on a modified Markov chain, which is obtained through a product automaton construction. We provide a case study of a scalable cellular phone system which shows how the logic asCSL and the model checking procedure can be applied in practice. 1.
Simulation for ContinuousTime Markov Chains
, 2002
"... This paper presents a simulation preorder for continuoustime Markov chains (CTMCs). The simulation preorder is a conservative extension of a weak variant of probabilistic simulation on fully probabilistic systems, i.e., discretetime Markov chains. The main result of the paper is that the simula ..."
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Cited by 8 (2 self)
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This paper presents a simulation preorder for continuoustime Markov chains (CTMCs). The simulation preorder is a conservative extension of a weak variant of probabilistic simulation on fully probabilistic systems, i.e., discretetime 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 branchingtime temporal logic interpreted over CTMCs.
Model Checking Meets Performance Evaluation
"... Markov chains are one of the most popular models for the evaluation of performance and dependability of information processing systems. To obtain performance measures, typically longrun or transient state probabilities of Markov chains are determined. Sometimes the Markov chain at hand is equipped ..."
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Cited by 5 (1 self)
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Markov chains are one of the most popular models for the evaluation of performance and dependability of information processing systems. To obtain performance measures, typically longrun or transient state probabilities of Markov chains are determined. Sometimes the Markov chain at hand is equipped with rewards and computations involve determining longrun or instantaneous reward probabilities.
Backward bisimulation in Markov chain model checking
 IEEE TSE
, 2006
"... Equivalence relations can be used to reduce the state space of a system model, thereby permitting more efficient analysis. We study backward stochastic bisimulation in the context of model checking continuoustime Markov chains against Continuous Stochastic Logic (CSL) properties. While there are s ..."
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Cited by 5 (0 self)
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Equivalence relations can be used to reduce the state space of a system model, thereby permitting more efficient analysis. We study backward stochastic bisimulation in the context of model checking continuoustime Markov chains against Continuous Stochastic Logic (CSL) properties. While there are simple CSL properties that are not preserved when reducing the state space of a continuoustime Markov chain using backward stochastic bisimulation, we show that the equivalence can nevertheless be used in the verification of a practically significant class of CSL properties. We consider an extension of these results to Markov reward models and Continuous Stochastic Reward Logic. Furthermore, we identify the logical properties for which the requirement on the equality of statelabeling sets (normally imposed on state equivalences in a modelchecking context) can be omitted from the definition of the equivalence, resulting in a better statespace reduction.
CONTINUOUS MARKOVIAN LOGICS AXIOMATIZATION AND QUANTIFIED METATHEORY
, 2011
"... Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuoustime labelled Markov processes with arbitrary (analytic) statespaces, henceforth called continuous Markov processes (CMPs). The modalities of CML evaluate the rates of the exp ..."
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Cited by 2 (2 self)
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Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuoustime labelled Markov processes with arbitrary (analytic) statespaces, henceforth called continuous Markov processes (CMPs). The modalities of CML evaluate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions of a CMP. In this paper we present weak and strong complete axiomatizations for CML and prove a series of metaproperties, including the finite model property and the construction of canonical models. CML characterizes stochastic bisimilarity and it supports the definition of a quantified extension of the satisfiability relation that measures the “compatibility ” between a model and a property. In this context, the metaproperties allows us to prove two robustness theorems for the logic stating that one can perturb formulas and maintain “approximate satisfaction”.
Under consideration for publication in Math. Struct. in Comp. Science Equilibrium and Termination II: the case of
, 2011
"... This paper is concerned with the asymptotic properties of a restricted class of Petri nets equipped with stochastic mass action semantics. We establish a simple algebraic criterion for the existence of an equilibrium, that is to say an invariant probability that satisfies the detailed balance condit ..."
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This paper is concerned with the asymptotic properties of a restricted class of Petri nets equipped with stochastic mass action semantics. We establish a simple algebraic criterion for the existence of an equilibrium, that is to say an invariant probability that satisfies the detailed balance condition familiar from the thermodynamics of reaction networks. We also find that when such a probability exists, it can be described by a free energy function which combines an internal energy term and an entropy one. Under strong additional conditions, we show how the entropy term can be deconstructed using the finergrained individualtoken semantics of Petri nets. 1.