Calculating Probabilities of Real-Time Test Cases

  • Marcin Jurdziński
  • Doron Peled
  • Hongyang Qu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3997)


When testing a system, it is often necessary to execute a suspicious trace in a realistic environment. Due to nondeterministic choices existing in concurrent systems, such a particular trace may not be scheduled for execution. Thus it is useful to compute the probability of executing the trace. Our probabilistic model of real-time systems requires that for each transition, the period from the time when its enabling condition becomes satisfied to the time when it is fired is bounded and the length of the period obeys a probabilistic distribution. This model is not Markovian if the distribution is not exponential. Therefore it cannot be analyzed by Markov processes. We propose to use integration to calculate the probability for a path. Then we discuss the possibility to optimize the calculation.


Partial Order Transition System Probabilistic Choice Process Algebra Global Clock 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur, R.: Timed Automata, NATO-ASI 1998 Summer School on Verification of Digital and Hybrid SystemsGoogle Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking for probabilistic real-time systems. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 115–136. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  3. 3.
    Borovkov, A.A.: Probability theory, ch. 3, Gordon and Breach (1998)Google Scholar
  4. 4.
    Bravetti, M., D’Argenio, P.R.: Tutte le algebre insieme: Concepts, Discussions and Relations of Stochastic Process Algebras with General Distributions. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 44–88. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Bryans, J., Bowman, H., Derrick, J.: Model checking stochastic automata. ACM Transactions on Computational Logic 4(4), 452–492 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P.: Model-checking algorithms for continuous-time markov chains. IEEE Transactions on Software Engineering 29(6), 524–541 (2003)CrossRefMATHGoogle Scholar
  7. 7.
    Dill, D.L.: Timing assumptions and verification of finite-state concurrent systems. In: Sifakis, J. (ed.) CAV 1989. LNCS, vol. 407, pp. 197–212. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  8. 8.
    Dyer, M.E., Frieze, A.M.: On the complexity of computing the volume of a polyhedron. SIAM Journal on Computing 17(5), 967–974 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gunter, E., Peled, D.: Path Exploration Tool. In: Cleaveland, W.R. (ed.) ETAPS 1999 and TACAS 1999. LNCS, vol. 1579, pp. 405–419. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. 10.
    Katoen, J.-P., D’Argenio, P.R.: General distributions in process algebra. In: Brinksma, E., Hermanns, H., Katoen, J.-P. (eds.) EEF School 2000 and FMPA 2000. LNCS, vol. 2090, pp. 375–430. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Kwiatkowska, M., Norman, G., Segala, R., Sproston, J.: Verifying Quantitative Properties of Continuous Probabilistic Timed Automata. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 123–137. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. 12.
    Kwiatkowska, M., Norman, G., Segala, R., Sproston, J.: Automatic Verification of Real-time Systems with Discrete Probability Distributions. Theoretical Computer Science 282, 101–150 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lovász, L., Vempala, S.: Simulated Annealing in Convex Bodies and an O *(n 4) Volume Algorithm. In: 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 650–659 (2003)Google Scholar
  14. 14.
    Infante-López, G.G., Hermanns, H., Katoen, J.-P.: Beyond memoryless distributions: Model checking semi-markov chains. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, pp. 57–70. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Manna, Z., Pnueli, A.: How to cook a temporal proof system for your pet language. In: Proceedings of the ACM Symposium on Principles on Programming Languages, pp. 141–151 (1983)Google Scholar
  16. 16.
    Peled, D., Qu, H.: Enforcing concurrent temporal behaviors. Electronic Notes in Theoretical Computer Science 113, 65–83 (2005)CrossRefMATHGoogle Scholar
  17. 17.
    Ruys, T.C., Langerak, R., Katoen, J.-P., Latella, D., Massink, M.: First passage time analysis of stochastic process algebra using partial orders. In: Margaria, T., Yi, W. (eds.) ETAPS 2001 and TACAS 2001. LNCS, vol. 2031, pp. 220–235. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Schechter, M.: Integration Over a Polyhedron: an Application of the Fourier-Motzkin Elimination Method. The American Mathematical Monthly 105(3), 246–251 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Segala, R.: Modelling and Verification of Randomized Distributed Real-Time Systems, Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1995)Google Scholar
  20. 20.
    Sproston, J.: Model checking for probabilistic timed systems. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 189–229. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Younes, H.L.S., Simmons, R.G.: Solving Generalized Semi-Markov Decision Processes using Continuous Phase-Type Distributions. In: Proc. Nineteenth National Conference on Artificial Intelligence, pp. 742–748. AAAI Press, Menlo Park (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Marcin Jurdziński
    • 1
  • Doron Peled
    • 1
  • Hongyang Qu
    • 1
  1. 1.Department of Computer ScienceUniversity of WarwickCoventryUK

Personalised recommendations