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Model Checking CSL until Formulae with Random Time Bounds

  • Marta Kwiatkowska
  • Gethin Norman
  • António Pacheco
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2399)

Abstract

Continuous Time Markov Chains (CTMCs) are widely used as the underlying stochastic process in performance and dependability analysis. Model checking of CTMCs against Continuous Stochastic Logic (CSL) has been investigated previously by a number of authors [2,4,13]. CSL contains a time-bounded until operator that allows one to express properties such as “the probability of 3 servers becoming faulty within 7.01 seconds is at most 0.1”. In this paper we extend CSL with a random time-bounded until operator, where the time bound is given by a random variable instead of a fixed real-valued time (or interval). With the help of such an operator we can state that the probability of reaching a set of goal states within some generally distributed delay while passing only through states that satisfy a certain property is at most (at least) some probability threshold. In addition, certain transient properties of systems which contain general distributions can be expressed with the extended logic. We extend the efficient model checking of CTMCs against the logic CSL developed in [13] to cater for the new operator. Our method involves precomputing a family of coefficients for a range of random variables which includes Pareto, uniform and gamma distributions, but otherwise carries the same computational cost as that for ordinary time-bounded until in [13]. We implement the algorithms in Matlab and evaluate them by means of a queueing system example.

Keywords

Model Check Random Time Pareto Distribution Queueing System Continuous Time Markov Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Aziz, K. Sanwal, V. Singhal, and R. Brayton. Verifying continuous time Markov chains. In Proc. CAV’96, volume 1102 of LNCS, pages 269–276. Springer, 1996.Google Scholar
  2. 2.
    A. Aziz, K. Sanwal, V. Singhal, and R. Brayton. Model checking continuous time Markov chains. ACM Transactions on Computational Logic, 1(1):162–170, 2000.CrossRefMathSciNetGoogle Scholar
  3. 3.
    C. Baier, B. Haverkort, H. Hermanns, and J.-P. Katoen. Model checking continuous-time Markov chains by transient analysis. In Proc. CAV 2000, volume 1855 of LNCS, pages 358–372, 2000.Google Scholar
  4. 4.
    C. Baier, J.-P. Katoen, and H. Hermanns. Approximative symbolic model checking of continuous-time Markov chains. In Proc. CONCUR’99, volume 1664 of LNCS, pages 146–162. Springer, 1999.Google Scholar
  5. 5.
    E. Clarke, E. Emerson, and A. Sistla. Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Transactions on Programming Languages and Systems, 8(2):244–263, 1986.MATHCrossRefGoogle Scholar
  6. 6.
    M. Crovella and A. Bestavros. Self-similarity in world wide Web traffic: evidence and possible causes. IEEE/ACM Transactions on Networking, 5(6):835–846, 1997.CrossRefGoogle Scholar
  7. 7.
    B. Fox and P. Glynn. Computing Poisson probabilities. Communications of the ACM, 31(4):440–445, 1988.CrossRefMathSciNetGoogle Scholar
  8. 8.
    R. German. Performance Analysis of Communication Systems: Modeling with Non-Markovian Stochastic Petri Nets. John Wiley and Sons, 2000.Google Scholar
  9. 9.
    J. Grandell. Mixed Poisson Processes. Chapman & Hall, 1997.Google Scholar
  10. 10.
    D. Gross and D. Miller. The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operations Research, 32(2):343–361, 1984.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    H. Hansson and B. Jonsson. A logic for reasoning about time and probability. Formal Aspects of Computing, 6:512–535, 1994.MATHCrossRefGoogle Scholar
  12. 12.
    A. Jensen. Markov chains as an aid in the study of Markov processes. Skandinavisk Aktuarietidsskrift, Marts, pages 87–91, 1953.Google Scholar
  13. 13.
    J.-P. Katoen, M. Kwiatkowska, G. Norman, and D. Parker. Faster and symbolic CTMC model checking. In Proc. PAPM-PROBMIV 2001, volume 2165 of LNCS, pages 23–38. Springer, 2001.Google Scholar
  14. 14.
    M. Kwiatkowska, G. Norman, and D. Parker. Probabilistic symbolic model checking with PRISM: A hybrid approach. In Proc. TACAS 2002, volume 2280 of LNCS, pages 52–66. Springer, 2002.Google Scholar
  15. 15.
    G.I. Lópes, H. Hermanns, and J.-P. Katoen. Beyond memoryless distributions. In Proc PAPM-PROBMIV 2001, volume 2165 of LNCS, pages 57–70. Springer, 2001.Google Scholar
  16. 16.
    S. Molnár and I. Maricza, editors. Source characterization in broadband networks. COST 257 Mid-term seminar interim report on source characterization, 2000.Google Scholar
  17. 17.
    J. Muppala and K. Trivedi. Queueing Systems, Queueing and Related Models, chapter Numerical Transient Solution of Finite Markovian Queueing Systems, pages 262–284. Oxford University Press, 1992.Google Scholar
  18. 18.
    H. Panjer. Recursive evaluation of a family of compound distributions. Astin Bulletin, 12(1):22–26, 1982.MathSciNetGoogle Scholar
  19. 19.
  20. 20.
    W. J. Stewart. Introduction to the Numerical Solution of Markov Chains. Princeton, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Marta Kwiatkowska
    • 1
  • Gethin Norman
    • 1
  • António Pacheco
    • 2
  1. 1.School of Computer ScienceUniversity of BirminghamEdgbaston BirminghamUK
  2. 2.Department of Mathematics and CEMATInstituto Superior TécnicoLisboaPortugal

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