Model Checking Infinite-State Markov Chains

In this paper algorithms for model checking CSL (continuous stochastic logic) against infinite-state continuous-time Markov chains of so-called quasi birth-death type are developed. In doing so we extend the applicability of CSL model checking beyond the recently proposed case for finite-state continuous-time Markov chains, to an important class of infinite-state Markov chains. We present syntax and semantics for CSL and develop efficient model checking algorithms for the steady-state operator and the time-bounded next and until operator. For the former, we rely on the so-called matrix-geometric solution of the steady-state probabilities of the infinite-state Markov chain. For the time-bounded until operator we develop a new algorithm for the transient analysis of infinite-state Markov chains, thereby exploiting the quasi birth-death structure. A case study shows the feasibility of our approach.

[1]  Robert K. Brayton,et al.  Model-checking continuous-time Markov chains , 2000, TOCL.

[2]  Alexander Ost.  Performance of communication systems: a model based approach with matrix geometric methods , 2001 .

[3]  Marta Z. Kwiatkowska,et al.  Probabilistic symbolic model checking with PRISM: a hybrid approach , 2004, International Journal on Software Tools for Technology Transfer.

[4]  Philippe Schnoebelen,et al.  The Verification of Probabilistic Lossy Channel Systems , 2004, Validation of Stochastic Systems.

[5]  Holger Hermanns,et al.  A tool for model-checking Markov chains , 2003, International Journal on Software Tools for Technology Transfer.

[6]  Boudewijn R. Haverkort,et al.  Performance of computer communication systems - a model-based approach , 1998 .

[7]  Christel Baier,et al.  On the Logical Characterisation of Performability Properties , 2000, ICALP.

[8]  A.K.I. Remke.  Model checking Quasi Birth Death processes , 2004 .

[9]  Alexander Ost.  Performance of Communication Systems , 2001 .

[10]  George S. Avrunin,et al.  Patterns in property specifications for finite-state verification , 1999, Proceedings of the 1999 International Conference on Software Engineering (IEEE Cat. No.99CB37002).

[11]  Alexander Bell.  Distributed Evaluation of Stochastic Petri nets , 2004, MMB.

[12]  Parosh Aziz Abdulla,et al.  A Survey of Regular Model Checking , 2004, CONCUR.

[13]  Christel Baier,et al.  Model-Checking Algorithms for Continuous-Time Markov Chains , 2002, IEEE Trans. Software Eng..

[14]  Isi Mitrani,et al.  TIPP and the Spectral Expansion Method , 1995 .

[15]  Joost-Pieter Katoen,et al.  On the use of model checking techniques for dependability evaluation , 2000, Proceedings 19th IEEE Symposium on Reliable Distributed Systems SRDS-2000.

[16]  Vaidyanathan Ramaswami,et al.  A logarithmic reduction algorithm for quasi-birth-death processes , 1993, Journal of Applied Probability.

[17]  M. Kwiatkowska,et al.  Solving Infinite Stochastic Process Algebra Models Through Matrix-Geometric Methods , 1999 .

[18]  Joost P. Katoen,et al.  Concepts, Algorithms, and Tools for Model Checking , 1999 .

[19]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[20]  Boudewijn R. Haverkort.  Performance of computer communication systems , 1998 .

[21]  Donald Gross,et al.  The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes , 1984, Oper. Res..

[22]  Edward J. Coyle,et al.  Transient analysis of quasi-birth-death processes , 1989 .