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## Model-checking algorithms for continuous-time Markov chains (2003)

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Venue: | IEEE TRANSACTIONS ON SOFTWARE ENGINEERING |

Citations: | 228 - 44 self |

### Citations

3194 | Model Checking
- Clarke, Grumberg, et al.
- 1999
(Show Context)
Citation Context ...a special kind of a transition system. The validity of CTL-formulas over finite-state automata can be established by fully automated techniques such as model checking [32], [65]; for an overview, see =-=[25]-=-. The basis of model checking CTL is a systematic, usually exhaustive, state-space exploration to check whether a property is satisfied in each state of the model, thereby using effective methods to c... |

2914 |
An Introduction to Probability Theory and Its Applications
- Feller
- 1971
(Show Context)
Citation Context ...gic. Index Terms—Continuous-time Markov chain, lumping, model checking, temporal logic, steady-state analysis, transient analysis, uniformization. 1 INTRODUCTION CONTINUOUS-TIME Markov chains (CTMCs) =-=[34]-=-, [46], [50], [53], [69] are an important class of stochastic processes that have been widely used in practice to determine system performance and dependability characteristics. To mention just a few ... |

2869 |
The design and analysis of computer algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ... whether s ƒS / pð Þ: We first recursively determine the set of states that satisfy . Then, the BSCCs are computed using (a slight variant of) an algorithm for computing strongly connected components =-=[1]-=-, [70]. For each BSCC B that contains a state, the steady-state probabilities are determined using standard means for solving the linear equation system (2). Formally, Bðs0Þ1 if B fs0g; otherwise, B... |

2619 | A theory of timed automata
- Alur, Dill
- 1994
(Show Context)
Citation Context ...ate sk in the interval I0 , where the probability density function of the residence time of sk equals EðskÞ e EðskÞ t (for time instant t). As opposed to the traditional approach in real-time systems =-=[6]-=-, we do not assume time divergence for infinite paths s0! t0 s1! t1 .... Although P j 0 tj might converge, in which case represents an “unrealistic” computation where infinitely many transitions are... |

1849 |
Markov Decision Processes: Discrete Stochastic Dynamic Programming
- Puterman
- 1994
(Show Context)
Citation Context ...me temporal operators like next step and until. This will be exploited when we address the semantics of the logic CSL in Section 3.2. CTMCs are also treated in this way in, among others, the textbook =-=[64]-=-. A state s is called absorbing iff Rðs; s0Þ0 for all states s0 . Whenever appropriate, we assume that, for any state s, AP contains an atomic proposition ats which is characteristic for s, i.e., ats... |

1493 | The model checker SPIN
- Holzmann
- 1997
(Show Context)
Citation Context ... problem. Model checking has been successfully used to validate, among others, hardware and software systems, security protocols, and e-commerce systems. With appropriate tools such as SMV [22], SPIN =-=[45]-=-, and Mur’ [31], systems of several millions of states have been analyzed. In this paper, we present the branching-time logic CSL (Continuous Stochastic Logic) that provides us ample means to specify ... |

1373 | Automatic verification of finite-state concurrent systems using temporal logic specifications
- Clarke, Emerson, et al.
- 1986
(Show Context)
Citation Context ...aracterizations provide the theoretical basis for model checking CSL over CTMCs in the same way as the fixedpoint characterizations for CTL provide the basis for the model checking algorithms for CTL =-=[23]-=-. We show that model checking time-bounded untilformulas can be reduced to the problem of computing transient-state probabilities for CTMCs. In particular, our result states that, for a given CTMC M a... |

1369 | Depth-first search and linear graph algorithms
- Tarjan
- 1972
(Show Context)
Citation Context ...es in the parse tree of the form tt, : , or ^ is straightforward and takes OðNÞ time. Steady-state operator. First, a graph analysis is carried out to determine the BSCCs of M. This takes OðNþMÞ time =-=[70]-=-. In the worst case, for each identified BSCC B, a linear system of jBj equations needs to be solved once. Ranging over all BSCCs, this leads to at most N equations since each state belongs to at most... |

1089 |
Finite Markov Chains
- Kemeny, Snell
- 1960
(Show Context)
Citation Context ...or denumerable and continuous state spaces. The presentation is focused on the concepts needed for the understanding of the rest of this paper; for a more elaborate treatment, we refer to [34], [46], =-=[49]-=-, [53], [69]. 2.1 Labeled CTMCs To ease the definition of the semantics of the logic CSL, we slightly depart from the standard notations for CTMCs andsBAIER ET AL.: MODEL-CHECKING ALGORITHMS FOR CONTI... |

744 | A compositional approach to performance modelling
- Hillston
- 1996
(Show Context)
Citation Context ...cification techniques have been developed, most notably those based on queuing networks [26], stochastic Petri nets [2], stochastic activity networks [57], [61], and stochastic process algebras [41], =-=[44]-=-. With appropriate software tools supporting these specification methods, such as those provided by MACOM [52], SPNP [21], UltraSAN [68], or TIPPtool [42], it is relatively comfortable to specify perf... |

737 |
Introduction to the Numerical Solution of Markov Chains
- Stewart
- 1994
(Show Context)
Citation Context ...ous-time Markov chain, lumping, model checking, temporal logic, steady-state analysis, transient analysis, uniformization. 1 INTRODUCTION CONTINUOUS-TIME Markov chains (CTMCs) [34], [46], [50], [53], =-=[69]-=- are an important class of stochastic processes that have been widely used in practice to determine system performance and dependability characteristics. To mention just a few practical applications, ... |

521 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...ion of the CTMC. In addition, we show that lumping—an equivalence notion on Markov chains to aggregate state spaces [18], [44] that can be viewed as a continuous variant of probabilistic bisimulation =-=[55]-=-—preserves the validity of all CSL-formulas. This allows us to switch from the original state space to the (possibly much smaller) quotient space under lumping prior to carrying out the model checking... |

456 |
Numerical Recipes: The art of scientific computing. Cambridge Univ
- Press, Teukolsky, et al.
- 1992
(Show Context)
Citation Context ...m: Z tX Fkþ1ðs; tÞ Rðs; s 0 Þ e EðsÞ x Fkðs 0 ;t xÞ dx; 0 s 0 2S if s ƒ ^: . These integrals can be solved numerically using integration methods such as trapezoidal, Simpson and Romberg integration =-=[63]-=-. Experiments have shown that this approach is rather time consuming and that numerical stability is hard to achieve [43]. Alternatively, the recursive integral formula equation of Theorem 1 can be re... |

455 | Modelling with generalized stochastic Petri nets
- Marsan, Balbo, et al.
- 1995
(Show Context)
Citation Context ...o avoid the specification of CTMCs directly at the state level, high-level model specification techniques have been developed, most notably those based on queuing networks [26], stochastic Petri nets =-=[2]-=-, stochastic activity networks [57], [61], and stochastic process algebras [41], [44]. With appropriate software tools supporting these specification methods, such as those provided by MACOM [52], SPN... |

421 | Nineteen dubious ways to compute the exponential of a matrix, twenty-fiveyearslater
- Moler, Loan
- 1994
(Show Context)
Citation Context ...Þ e Q t X1 i0 with Q R diagðEÞ; ðQ tÞ I where we recall that ð ; tÞ denotes the vector of state probabilities at time t. This characterization is not attractive for a numerical algorithm since =-=[58]-=-, [69] 1) it suffers from numerical instability as Q contains both positive and negative entries and 2) it is difficult to find a proper truncation criterion for the infinite summation. Therefore, oth... |

359 | B.: A logic for reasoning about time and reliability
- Hansson, Jonsson
- 1994
(Show Context)
Citation Context ...s logic is basically a probabilistic timed extension of CTL and is strongly based on the (equally named) logic by Aziz et al. [9] and on PCTL, a variant of CTL for discrete-time Markov chains (DTMCs) =-=[38]-=-. Besides the standard steady-state and transient measures, the logic allows for the specification of (constraints over) probabilistic measures over paths through CTMCs. For instance, the probability ... |

310 |
Automatic verification of probabilistic concurrent finite-state programs
- Vardi
- 1985
(Show Context)
Citation Context ...hs in the CTMC M, and P athMðsÞ the set of paths in M that start in s. The superscript M is omitted whenever convenient. 2.3 Borel Space Our definition of a Borel space on paths through CTMCs follows =-=[71]-=-, [38]. An initial distribution yields a probability measure Pr on paths as follows: Let s0; ...;sk 2 S with Rðsi;siþ1Þ > 0 (0 i<k) and I0; ...;Ik 1 nonempty intervals in IR 0. Then, Cðs0;I0; ...;Ik 1... |

284 | Model checking of probabilistic and nondeterministic systems
- Bianco, Alfaro
- 1995
(Show Context)
Citation Context ... the CSL operators that do not refer to the real-time behavior of the CTMC, the PCTL algorithms can be exploited. The logic PCTL* contains both LTL and PCTL. Its verification is studied in [8], [15], =-=[16]-=-. Its basic idea is the reduction to the verification of quantitative LTL properties. Branching-time model checking of Markov decision processes is considered in [4], [14], [15], [16]. Here, nondeterm... |

278 |
Modeling and Analysis of Stochastic Systems
- Kulkarni
- 1995
(Show Context)
Citation Context ...ontinuous-time Markov chain, lumping, model checking, temporal logic, steady-state analysis, transient analysis, uniformization. 1 INTRODUCTION CONTINUOUS-TIME Markov chains (CTMCs) [34], [46], [50], =-=[53]-=-, [69] are an important class of stochastic processes that have been widely used in practice to determine system performance and dependability characteristics. To mention just a few practical applicat... |

228 |
Using branching time temporal logic to synthesize synchronization skeletons
- Emerson, Clarke
- 1982
(Show Context)
Citation Context ...nsition systems consisting of a finite set of states and a set of transitions that describe how the system evolves from one state to another. Branchingtime logics such as CTL (Computation Tree Logic) =-=[32]-=- allow one to express state-based properties as well as properties over paths, i.e., state sequences through transition systems. Typical properties expressible in CTL are that along all (or some) path... |

216 |
The complexity of probabilistic verification
- Courcoubetis, Yannakakis
- 1995
(Show Context)
Citation Context ...k has concentrated on discrete-time models. Methods to verify a DTMC or a Markov decision process against a linear-time temporal logic (LTL) formula (or a Büchi automaton) have been considered, e.g., =-=[28]-=-, [62], [71]. The basis of these works is the nontrivial reduction of the model-checking problem to the computation of the probabilities to reach certain sets of states (mostly, BSCCs). Courcoubetis a... |

184 | Multiterminal binary decision diagrams: an efficient data structure for matrix representation - Fujita, McGeer, et al. - 1997 |

169 | Formal Verification of Probabilistic Systems - Alfaro - 1997 |

166 | NuSMV: a new symbolic model checker
- Cimatti, Clarke, et al.
- 2000
(Show Context)
Citation Context ...e explosion problem. Model checking has been successfully used to validate, among others, hardware and software systems, security protocols, and e-commerce systems. With appropriate tools such as SMV =-=[22]-=-, SPIN [45], and Mur’ [31], systems of several millions of states have been analyzed. In this paper, we present the branching-time logic CSL (Continuous Stochastic Logic) that provides us ample means ... |

156 | Approximate symbolic model checking of continuous-time Markov chains
- Baier, Katoen, et al.
- 1999
(Show Context)
Citation Context ...mputation of probability measures for time-bounded until formulas by transient analysis; . the preservation of the validity of CSL formulas under lumping. This paper is based on the extended abstract =-=[11]-=- and the paper [12]. Organization of the paper. Section 2 introduces the basic concepts of CTMCs. Section 3 presents the logic CSL and provides fixed-point characterizations of CSL-formulas that form ... |

144 |
Specification and Verification of Concurrent Systems
- Queille, Sifakis
- 1982
(Show Context)
Citation Context ...iew a finite-state CTMC as a special kind of a transition system. The validity of CTL-formulas over finite-state automata can be established by fully automated techniques such as model checking [32], =-=[65]-=-; for an overview, see [25]. The basis of model checking CTL is a systematic, usually exhaustive, state-space exploration to check whether a property is satisfied in each state of the model, thereby u... |

131 | Model checking for a probabilistic branching time logic with fairness
- Baier, Kwiatkowska
- 1998
(Show Context)
Citation Context ...]. For the CSL operators that do not refer to the real-time behavior of the CTMC, the PCTL algorithms can be exploited. The logic PCTL* contains both LTL and PCTL. Its verification is studied in [8], =-=[15]-=-, [16]. Its basic idea is the reduction to the verification of quantitative LTL properties. Branching-time model checking of Markov decision processes is considered in [4], [14], [15], [16]. Here, non... |

123 | Stochastic automata network of modeling parallel systems
- Plateau, Atif
- 1991
(Show Context)
Citation Context ...tly at the state level, high-level model specification techniques have been developed, most notably those based on queuing networks [26], stochastic Petri nets [2], stochastic activity networks [57], =-=[61]-=-, and stochastic process algebras [41], [44]. With appropriate software tools supporting these specification methods, such as those provided by MACOM [52], SPNP [21], UltraSAN [68], or TIPPtool [42], ... |

123 | Probabilistic Verification of Discrete Event Systems Using Acceptance Sampling
- Younes, Simmons
- 2002
(Show Context)
Citation Context ...product of the model and the specification automaton. Other recent works on CSL are the extension to continuousspace Markov processes [30], the use of discrete-event simulation and hypothesis testing =-=[72]-=-, and the use of Kronecker algebra to exploit the structure of the CTMC [20]. 8 CONCLUDING REMARKS This paper proposed the use of the temporal logic CSL to specify performance and reliability measures... |

117 | Automatic verification of real-time systems with discrete probability distributions. Theoretical Computer Science 282
- Kwiatkowska, Norman, et al.
- 2002
(Show Context)
Citation Context ...d on regions, finite partitions of the infinite continuous-time domain tailored to the property and model under consideration. Recently, this approach has been adopted for quantitative model checking =-=[54]-=-. Model-checking continuous-time Markov chains. A stochastic extension of CTL, also called CSL, was initially proposed in [9]. Using transcendental number theory, the elementary result that the model-... |

111 | Exact and ordinary lumpability of finite Markov chains
- Buchholz
- 1994
(Show Context)
Citation Context ... integration of the Volterra equation system—it employs a measure-driven transformation of the CTMC. In addition, we show that lumping—an equivalence notion on Markov chains to aggregate state spaces =-=[18]-=-, [44] that can be viewed as a continuous variant of probabilistic bisimulation [55]—preserves the validity of all CSL-formulas. This allows us to switch from the original state space to the (possibly... |

110 |
The randomization technique as a modeling tool and solution procedure for transient Markov processes
- Gross, Miller
- 1984
(Show Context)
Citation Context ...nother CTMC M 0 , which can easily be derived from M using ’. This allows us to adopt efficient and numerically stable techniques for performing transient analysis of CTMCs, like uniformization [36], =-=[37]-=-, [47], for model checking time-bounded until-formulas. The reduction of the model checking problem for the time-bounded until-operator to the transient analysis of a CTMC has the advantage that—besid... |

101 | It usually works: The temporal logic of stochastic systems
- Aziz, Singhal, et al.
(Show Context)
Citation Context ...wn as lumping on CTMCs, preserves all CSL formulas. The result presented below is similar to that for relating bisimulation and CTL (and CTL* equivalence) [17] and probabilistic bisimulation and PCTL =-=[8]-=-. 5.1 Bisimulation (Lumping) Equivalence Lumpability is an important notion on CTMCs that allows their aggregation without affecting performance properties [18], [49]. We adapt the standard notion in ... |

93 |
Modalities for Model Checking: Branching Time Logic Strikes Back
- Emerson, Lei
- 1987
(Show Context)
Citation Context ...a path for which ’ holds—corresponds to P>0ð’Þ, and 8’—for all paths ’ holds—corresponds to P 1ð’Þ. For instance, P>0 ð aÞ is equivalent to 9 a, and P 1ð aÞ stands for 8 a given a fair interpretation =-=[33]-=- of the CTL formula 8 a. Ina fair interpretation of CTL, paths that do not satisfy certain fairness constraints, like “visit a set of states infinitely often,” are ruled out. Satisfaction of formulae ... |

89 | Model checking continuous-time markov chains by transient analyisis
- Baier, Havekort, et al.
- 2000
(Show Context)
Citation Context ...ility measures for time-bounded until formulas by transient analysis; . the preservation of the validity of CSL formulas under lumping. This paper is based on the extended abstract [11] and the paper =-=[12]-=-. Organization of the paper. Section 2 introduces the basic concepts of CTMCs. Section 3 presents the logic CSL and provides fixed-point characterizations of CSL-formulas that form the basis for a mod... |

86 |
Numerical transient analysis of Markov models
- Reibman, Trivedi
- 1988
(Show Context)
Citation Context ...], [47] is currently regarded as the most attractive. Under special conditions, e.g., when the rates in R differ a large number of magnitudes, Runge-Kutta-like methods might perform better, see [66], =-=[67]-=-. For the sake of completeness, we briefly discuss the main aspects of uniformization here. These details will play a significant role in discussing the efficiency of our model-checking algorithms, cf... |

83 | Modelchecking continuous time Markov chains
- Aziz, Sanwal, et al.
- 2000
(Show Context)
Citation Context ...and dependability measures for CTMCs in a compact and unambiguous way. This logic is basically a probabilistic timed extension of CTL and is strongly based on the (equally named) logic by Aziz et al. =-=[9]-=- and on PCTL, a variant of CTL for discrete-time Markov chains (DTMCs) [38]. Besides the standard steady-state and transient measures, the logic allows for the specification of (constraints over) prob... |

83 |
Computing Poisson probabilities
- Fox, Glynn
- 1988
(Show Context)
Citation Context ...bution vector after i epochs in the DTMC with transition matrix U determined recursively by i i 1 U with 0 . The Poisson probabilities can be computed in a stable way with the Fox-Glynn algorithm =-=[35]-=-. The number of terms k in (7) to be taken given a required accuracy is the smallest value satisfying: i; ð7ÞsBAIER ET AL.: MODEL-CHECKING ALGORITHMS FOR CONTINUOUS-TIME MARKOV CHAINS 13 Fig. 5. (a) A... |

71 | Process algebra for performance evaluation
- Hermanns, Herzog, et al.
- 2002
(Show Context)
Citation Context ...el specification techniques have been developed, most notably those based on queuing networks [26], stochastic Petri nets [2], stochastic activity networks [57], [61], and stochastic process algebras =-=[41]-=-, [44]. With appropriate software tools supporting these specification methods, such as those provided by MACOM [52], SPNP [21], UltraSAN [68], or TIPPtool [42], it is relatively comfortable to specif... |

67 | On algorithmic verification methods for probabilistic systems
- Baier
- 1998
(Show Context)
Citation Context ...racterizations for the timed-next and timed-until operators and show that the characterizations for their untimed variants coincide with those for model checking PCTL over discrete-time Markov chains =-=[10]-=-, [27], [38]. We first observe that it suffices to consider time bounds specified by closed intervals since: P robðs; U I ÞP robðs; U clðIÞ P robðs; X I ÞP robðs; X clðIÞ Þ; Þ and where clðIÞ denote... |

65 |
Performance of Computer Communication Systems: A Model-Based Approach
- Haverkort
- 1998
(Show Context)
Citation Context ...tegrated in the computational procedure, see [59]. This steady-state truncation point is often smaller than k , making the trailing matrix-vector multiplications superfluous. For further details, see =-=[39]-=-, [69]. Example 6. Consider the TMR example and P 0:99ðtu 0;3Š ðup3 _ up2ÞÞ. As explained in Example 5, it suffices to verify on the CTMC of Fig. 4. This CTMC is equivalent to the CTMC depicted in F... |

61 | How to specify and verify the long-run average behavior of probabilistic systems
- Alfaro
- 1998
(Show Context)
Citation Context ...or of CTMCs. The first work on logics and model-checking algorithms for studying the stationary behavior of stochastic systems, in particular semi-Markov decision processes, has been reported in [4], =-=[5]-=-. Semi-Markov decision processes extend CTMCs with nondeterminism and nonexponential distributions. Apart from the fact that we are considering a more specific model, our approach differs in several a... |

53 | Compositional performance modelling with TIPPTool
- Hermanns, Herzog, et al.
- 1998
(Show Context)
Citation Context ..., [61], and stochastic process algebras [41], [44]. With appropriate software tools supporting these specification methods, such as those provided by MACOM [52], SPNP [21], UltraSAN [68], or TIPPtool =-=[42]-=-, it is relatively comfortable to specify performance and dependability models of which the underlying CTMCs have millions of states, cf. [69]. In combination with state-of-the art numer. C. Baier is ... |

47 |
Probabilistic Verification
- Pnueli, Zuck
- 1993
(Show Context)
Citation Context ...concentrated on discrete-time models. Methods to verify a DTMC or a Markov decision process against a linear-time temporal logic (LTL) formula (or a Büchi automaton) have been considered, e.g., [28], =-=[62]-=-, [71]. The basis of these works is the nontrivial reduction of the model-checking problem to the computation of the probabilities to reach certain sets of states (mostly, BSCCs). Courcoubetis and Yan... |

44 |
Finding transient solutions in Markovian event systems through randomization
- Grassmann
- 1990
(Show Context)
Citation Context ...s of another CTMC M 0 , which can easily be derived from M using ’. This allows us to adopt efficient and numerically stable techniques for performing transient analysis of CTMCs, like uniformization =-=[36]-=-, [37], [47], for model checking time-bounded until-formulas. The reduction of the model checking problem for the time-bounded until-operator to the transient analysis of a CTMC has the advantage that... |

42 | Temporal logics for the specification of performance and reliability
- Alfaro
(Show Context)
Citation Context ...fication is studied in [8], [15], [16]. Its basic idea is the reduction to the verification of quantitative LTL properties. Branching-time model checking of Markov decision processes is considered in =-=[4]-=-, [14], [15], [16]. Here, nondeterminism is resolved by adversaries. The model checking of until formulas reduces to the computation of a minimum (or maximum) probability, depending whether one quanti... |

42 |
Markov and Markov reward model transient analysis: an overview of numerical approaches
- Reibman, Smith, et al.
- 1989
(Show Context)
Citation Context ...], [37], [47] is currently regarded as the most attractive. Under special conditions, e.g., when the rates in R differ a large number of magnitudes, Runge-Kutta-like methods might perform better, see =-=[66]-=-, [67]. For the sake of completeness, we briefly discuss the main aspects of uniformization here. These details will play a significant role in discussing the efficiency of our model-checking algorith... |

39 |
On markovian process algebra
- Buchholz
- 1994
(Show Context)
Citation Context ... to F) and the cumulative rate of moving from these states to any equivalence class C is equal. The notion of F bisimulation is a slight variant of lumping equivalence [18] and Markovian bisimulation =-=[19]-=-, [44]. For s 2 S, let sŠR denote the equivalence class of s under R. For MðS; R;LÞ, the CTMC M=R is defined by M=R ðS=R; RR;LRÞ with RRðsŠR ;CÞRðs; CÞ and LRðsŠRÞ LF ðsÞ. The performance mea... |

39 |
Verifying temporal properties of finite-state probabilistic programs
- Courcoubetis, Yannakakis
- 1988
(Show Context)
Citation Context ...izations for the timed-next and timed-until operators and show that the characterizations for their untimed variants coincide with those for model checking PCTL over discrete-time Markov chains [10], =-=[27]-=-, [38]. We first observe that it suffices to consider time bounds specified by closed intervals since: P robðs; U I ÞP robðs; U clðIÞ P robðs; X I ÞP robðs; X clðIÞ Þ; Þ and where clðIÞ denotes the ... |

36 |
Modeling and analysis of communication systems based on computational methods for Markov chains
- Krieger, Miiller-CIoscermaun, et al.
- 1990
(Show Context)
Citation Context ... nets [2], stochastic activity networks [57], [61], and stochastic process algebras [41], [44]. With appropriate software tools supporting these specification methods, such as those provided by MACOM =-=[52]-=-, SPNP [21], UltraSAN [68], or TIPPtool [42], it is relatively comfortable to specify performance and dependability models of which the underlying CTMCs have millions of states, cf. [69]. In combinati... |

35 |
Optimal State-Space Lumping
- Derisavi, Hermanns, et al.
- 2003
(Show Context)
Citation Context ... consider the coarsest AP bisimulation R on M and construct the quotient Markov model M=R prior to model checking M. R can be computed by a modification of the standard partition refinement algorithm =-=[29]-=-. It now follows from Theorem 5 that any CSL formula can be equally well checked on M=R instead of on M. Furthermore, we can add a formula-specific aggregation. Let APð Þ denote the set of atomic prop... |

31 | Continous Stochastic Logic Characterizes Bisimulation of Continuous-time Markov Processes. The Journal of Logic and Algebraic Programming
- Desharnais, Panangaden
- 2003
(Show Context)
Citation Context ...simulation R. Theorem 5. s ƒM iff s0 ƒM for all CSL formulas iff s and s0 are AP bisimilar. Proof. The “if” part of the proof follows directly from Theorem 4. The “only if” part was recently shown in =-=[30]-=-.tu We can exploit the above result and apply it to the transformations of the previous section by using the following observation: From 1 of Theorem 4, it follows: X X M 0 M=R ðs; s ;tÞ ðsŠR ;S 0 ;... |

27 |
Queuing Networks - Exact Computational Algorithms: A Unified Theory Based on Decomposition and Aggregation
- Conway, Georganas
- 1989
(Show Context)
Citation Context ...umbersome and error-prone. To avoid the specification of CTMCs directly at the state level, high-level model specification techniques have been developed, most notably those based on queuing networks =-=[26]-=-, stochastic Petri nets [2], stochastic activity networks [57], [61], and stochastic process algebras [41], [44]. With appropriate software tools supporting these specification methods, such as those ... |

27 | State-Space Support for Path-Based Reward Variables
- Obal, Sanders
- 1999
(Show Context)
Citation Context ...ted differently, we would also like to be able to express measures that address the probability on paths through the CTMC obeying particular properties. Except for the recent work by Obal and Sanders =-=[60]-=-, suitable mechanisms to express such measures have not been considered. In Section 3, we will introduce a logicbased approach that allows us to express such path-based measures. 3 THE CONTINUOUS STOC... |

18 |
Characterizing Finite Kripke Structures
- Browne, Clarke, et al.
- 1988
(Show Context)
Citation Context ...out the model checking. Using this property, we indicate how the state space for checking probabilistic timing properties on the derived CTMC M 0 can be obtained. This result is in the same spirit as =-=[17]-=- where bisimulation is shown to agree with CTL and CTL* equivalence. Summarizing, the main contributions of this paper are: . the definition of a stochastic branching-time logic that facilitates the f... |

10 |
Dynamic Probabilistic System: Markov Models. Volume 1
- Howard
- 1971
(Show Context)
Citation Context ...ndex Terms—Continuous-time Markov chain, lumping, model checking, temporal logic, steady-state analysis, transient analysis, uniformization. 1 INTRODUCTION CONTINUOUS-TIME Markov chains (CTMCs) [34], =-=[46]-=-, [50], [53], [69] are an important class of stochastic processes that have been widely used in practice to determine system performance and dependability characteristics. To mention just a few practi... |

8 |
Numerical transient solution of finite Markovian queueing systems
- Trivedi
- 1992
(Show Context)
Citation Context ...e other hand, if q t is large, the DTMC described by U might have reached steady state along the way. Such an “on-the-fly” steady-state detection can be integrated in the computational procedure, see =-=[59]-=-. This steady-state truncation point is often smaller than k , making the trailing matrix-vector multiplications superfluous. For further details, see [39], [69]. Example 6. Consider the TMR example a... |

7 |
J.-P.: On the use of model checking techniques for quantitative dependability evaluation
- Haverkort, Hermanns, et al.
- 2000
(Show Context)
Citation Context ...t time units, a set of goal-states is reached, thereby avoiding or deliberately visiting particular intermediate states before. This is a useful feature for dependability analysis, as demonstrated in =-=[40]-=-, and goes beyond the standard measures in performance and dependability analysis. The model checking problem for CSL is known to be decidable [9] (for rational time bounds), but, to the best of our k... |

7 |
Markov Chains as an Aid
- Jensen
- 1953
(Show Context)
Citation Context ... CTMC M 0 , which can easily be derived from M using ’. This allows us to adopt efficient and numerically stable techniques for performing transient analysis of CTMCs, like uniformization [36], [37], =-=[47]-=-, for model checking time-bounded until-formulas. The reduction of the model checking problem for the time-bounded until-operator to the transient analysis of a CTMC has the advantage that—besides avo... |

6 |
Stochastic activity networks
- Meyer, Movaghar, et al.
- 1985
(Show Context)
Citation Context ... directly at the state level, high-level model specification techniques have been developed, most notably those based on queuing networks [26], stochastic Petri nets [2], stochastic activity networks =-=[57]-=-, [61], and stochastic process algebras [41], [44]. With appropriate software tools supporting these specification methods, such as those provided by MACOM [52], SPNP [21], UltraSAN [68], or TIPPtool ... |

3 |
The Murϕ verification system. Computer-Aided Verification
- Dill
- 1996
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Citation Context ... checking has been successfully used to validate, among others, hardware and software systems, security protocols, and e-commerce systems. With appropriate tools such as SMV [22], SPIN [45], and Mur’ =-=[31]-=-, systems of several millions of states have been analyzed. In this paper, we present the branching-time logic CSL (Continuous Stochastic Logic) that provides us ample means to specify state as well a... |

3 |
Investigation of an important case of dependent trials
- Markov
- 1907
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Citation Context ...ed research. Finally, Section 8 concludes the paper. 2 CONTINUOUS-TIME MARKOV CHAINS This section recalls the basic concepts of continuous-time Markov chains (CTMCs) as originally developed by Markov =-=[56]-=- for finite state spaces and Kolmogorov [51] for denumerable and continuous state spaces. The presentation is focused on the concepts needed for the understanding of the rest of this paper; for a more... |

2 |
D.Dill, Model-checking for probabilistic real-time
- Alur, Courcoubetis
- 1991
(Show Context)
Citation Context ...or some adversaries, respectively. Model-checking real-time probabilistic systems. A qualitative model-checking algorithm for a continuous probabilistic variant of timed automata has been proposed in =-=[7]-=-. This technique is based on regions, finite partitions of the infinite continuous-time domain tailored to the property and model under consideration. Recently, this approach has been adopted for quan... |

2 | On the Verification of Qualitative
- Baier, Kwiatkowska
- 1998
(Show Context)
Citation Context ... is only with respect to the remaining fair paths. An elaborate discussion about the relation between fairness and probabilities goes beyond the scope of this paper; we refer the interested reader to =-=[14]-=-. The temporal operator XI is the timed variant of the standard next operator in CTL; the path formula XI asserts that a transition is made to a state at some time point t 2 I. Operator U I is the tim... |

2 |
Über die analytische Methoden in der Wahrscheinlichkeitsrechnung
- Kolmogorov
- 1931
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Citation Context ...erms—Continuous-time Markov chain, lumping, model checking, temporal logic, steady-state analysis, transient analysis, uniformization. 1 INTRODUCTION CONTINUOUS-TIME Markov chains (CTMCs) [34], [46], =-=[50]-=-, [53], [69] are an important class of stochastic processes that have been widely used in practice to determine system performance and dependability characteristics. To mention just a few practical ap... |

1 |
Anfangsgründe der Theorie der Markoffschen Ketten mit unendlichen vielen möglichen Zuständen
- Kolmogorov
- 1936
(Show Context)
Citation Context ...e paper. 2 CONTINUOUS-TIME MARKOV CHAINS This section recalls the basic concepts of continuous-time Markov chains (CTMCs) as originally developed by Markov [56] for finite state spaces and Kolmogorov =-=[51]-=- for denumerable and continuous state spaces. The presentation is focused on the concepts needed for the understanding of the rest of this paper; for a more elaborate treatment, we refer to [34], [46]... |