Statistical Approximation of Optimal Schedulers for Probabilistic Timed Automata

  • Pedro R. D’Argenio
  • Arnd Hartmanns
  • Axel Legay
  • Sean Sedwards
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9681)


The verification of probabilistic timed automata involves finding schedulers that optimise their nondeterministic choices with respect to the probability of a property. In practice, approaches based on model checking fail due to state-space explosion, while simulation-based techniques like statistical model checking are not applicable due to the nondeterminism. We present a new lightweight on-the-fly algorithm to find near-optimal schedulers for probabilistic timed automata. We make use of the classical region and zone abstractions from timed automata model checking, coupled with a recently developed smart sampling technique for statistical verification of Markov decision processes. Our algorithm provides estimates for both maximum and minimum probabilities. We compare our new approach with alternative techniques, first using tractable examples from the literature, then motivate its scalability using case studies that are intractable to numerical model checking and challenging for existing statistical techniques.


  1. 1.
    Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bohnenkamp, H., D’Argenio, P., Hermanns, H., Katoen, J.P.: MoDeST: a compositional modeling formalism for real-time and stochastic systems. IEEE Trans. Softw. Eng. 32(10), 812–830 (2006)CrossRefGoogle Scholar
  3. 3.
    Boyer, B., Corre, K., Legay, A., Sedwards, S.: PLASMA-lab: a flexible, distributable statistical model checking library. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 160–164. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Brázdil, T., Chatterjee, K., Chmelík, M., Forejt, V., Křetínský, J., Kwiatkowska, M., Parker, D., Ujma, M.: Verification of Markov decision processes using learning algorithms. In: Cassez, F., Raskin, J.-F. (eds.) ATVA 2014. LNCS, vol. 8837, pp. 98–114. Springer, Heidelberg (2014)Google Scholar
  5. 5.
    D’Argenio, P., Legay, A., Sedwards, S., Traonouez, L.M.: Smart sampling for lightweight verification of Markov decision processes. STTT 17(4), 469–484 (2015)CrossRefGoogle Scholar
  6. 6.
    David, A., Jensen, P.G., Larsen, K.G., Mikučionis, M., Taankvist, J.H.: Uppaal stratego. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 206–211. Springer, Heidelberg (2015)Google Scholar
  7. 7.
    David, A., Larsen, K.G., Legay, A., Mikucionis, M., Poulsen, D.B.: Uppaal SMC tutorial. STTT 17(4), 397–415 (2015)CrossRefGoogle Scholar
  8. 8.
    Hahn, E.M., Hartmanns, A., Hermanns, H.: Reachability and reward checking for stochastic timed automata. ECEASST, 70 (2014)Google Scholar
  9. 9.
    Henriques, D., Martins, J.G., Zuliani, P., Platzer, A., Clarke, E.M.: Statistical model checking for Markov decision processes. In: 2012 Ninth International Conference on Quantitative Evaluation of Systems, pp. 84–93. IEEE (2012)Google Scholar
  10. 10.
    Hérault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic model checking. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 73–84. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Jegourel, C., Legay, A., Sedwards, S.: A platform for high performance statistical model checking – PLASMA. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 498–503. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Knuth, D.E.: The Art of Computer Programming: Sorting and Searching, 2nd edn. Addison-Wesley, Redwood (1998)zbMATHGoogle Scholar
  13. 13.
    Kwiatkowska, M., Norman, G., Parker, D.: The PRISM benchmark suite. In: Proceedings of the 9th International Conference on Quantitative Evaluation of SysTems (QEST 2012). pp. 203–204. IEEE CS Press, September 2012Google Scholar
  14. 14.
    Kwiatkowska, M.Z., Norman, G., Parker, D.: Stochastic games for verification of probabilistic timed automata. In: Ouaknine, J., Vaandrager, F.W. (eds.) FORMATS 2009. LNCS, vol. 5813, pp. 212–227. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Kwiatkowska, M.Z., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Kwiatkowska, M.Z., Norman, G., Parker, D., Sproston, J.: Performance analysis of probabilistic timed automata using digital clocks. FMSD 29(1), 33–78 (2006)zbMATHGoogle Scholar
  17. 17.
    Kwiatkowska, M.Z., Norman, G., Segala, R., Sproston, J.: Automatic verification of real-time systems with discrete probability distributions. Theor. Comput. Sci. 282(1), 101–150 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kwiatkowska, M.Z., Norman, G., Sproston, J., Wang, F.: Symbolic model checking for probabilistic timed automata. Inf. Comput. 205(7), 1027–1077 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Legay, A., Sedwards, S., Traonouez, L.: Estimating rewards & rare events in nondeterministic systems. ECEASST, 72 (2015)Google Scholar
  20. 20.
    Younes, H.L.S., Simmons, R.G.: Probabilistic verification of discrete event systems using acceptance sampling. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 223–235. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Pedro R. D’Argenio
    • 1
  • Arnd Hartmanns
    • 2
  • Axel Legay
    • 3
  • Sean Sedwards
    • 3
  1. 1.Universidad Nacional de CórdobaCórdobaArgentina
  2. 2.University of TwenteEnschedeThe Netherlands
  3. 3.INRIA Rennes – Bretagne AtlantiqueRennesFrance

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