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Probabilistic Time Petri Nets

  • Yrvann EmzivatEmail author
  • Benoît Delahaye
  • Didier Lime
  • Olivier H. Roux
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9698)

Abstract

We introduce a new model for the design of concurrent stochastic real-time systems. Probabilistic time Petri nets (PTPN) are an extension of time Petri nets in which the output of tokens is randomised. Such a design allows us to elegantly solve the hard problem of combining probabilities and concurrency. This model further benefits from the concision and expressive power of Petri nets. Furthermore, the usual tools for the analysis of time Petri nets can easily be adapted to our probabilistic setting. More precisely, we show how a Markov decision process (MDP) can be derived from the classic atomic state class graph construction. We then establish that the schedulers of the PTPN and the adversaries of the MDP induce the same Markov chains. As a result, this construction notably preserves the lower and upper bounds on the probability of reaching a given target marking. We also prove that the simpler original state class graph construction cannot be adapted in a similar manner for this purpose.

Keywords

Time Petri nets Probabilistic systems State classes Markov decision processes 

References

  1. 1.
    Stewart, W.J.: Introduction to the Numerical Solutions of Markov Chains. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  2. 2.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, London (1994)CrossRefGoogle Scholar
  3. 3.
    Kwiatkowska, M., Norman, G., Segala, R., Sproston, J.: Automatic verification of real-time systems with discrete probability distributions. Theoret. Comput. Sci. 282(1), 101–150 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eisentraut, C., Hermanns, H., Zhang, L.: On probabilistic automata in continuous time. In: 25th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 342–351. IEEE (2010)Google Scholar
  5. 5.
    Bertrand, N., Bouyer, P., Brihaye, T., Menet, Q., Baier, C., Größer, M., Jurdziński, M.: Stochastic timed automata. Log. Meth. Comput. Sci. 10(4), 6 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kwiatkowska, M., Norman, G., Sproston, J.: Probabilistic model checking of deadline properties in the IEEE 1394 FireWire root contention protocol. Formal Aspects of Comput. 14(3), 295–318 (2003)CrossRefGoogle Scholar
  7. 7.
    Kwiatkowska, M., 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
  8. 8.
    Marsan, M.A., Conte, G., Balbo, G.: A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems. ACM Trans. Comput. Syst. 2(2), 93–122 (1984)CrossRefGoogle Scholar
  9. 9.
    Molloy, M.K.: Discrete time stochastic Petri nets. IEEE Trans. Softw. Eng. 11(4), 417–423 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vicario, E., Sassoli, L., Carnevali, L.: Using stochastic state classes in quantitative evaluation of dense-time reactive systems. IEEE Trans. Softw. Eng. 35(5), 703–719 (2009)CrossRefGoogle Scholar
  11. 11.
    Berthomieu, B., Diaz, M.: Modeling and verification of time dependent systems using time Petri nets. IEEE Trans. Soft. Eng. 3, 259–273 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berthomieu, B., Menasche, M.: An enumerative approach for analyzing time Petri nets: In: Mason, R.E.A. (ed.) Information Processing: Proceedings of the IFIP Congress, vol. 9 of IFIP Congress Series, pp. 41–46 (1983)Google Scholar
  13. 13.
    Dugan, J.B., Trivedi, K.S., Geist, R.M., Nicola, V.F.: Extended stochastic Petri nets: applications and analysis. Technical report, DTIC Document (1984)Google Scholar
  14. 14.
    Berthomieu, B., Vernadat, F.: State class constructions for branching analysis of time Petri nets. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 442–457. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Baier, C., Katoen, J.: Principles of Model Checking. MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  16. 16.
    Norman, G., Parker, D., Sproston, J.: Model checking for probabilistic timed automata. Formal Meth. Syst. Des. 43(2), 164–190 (2013)CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2016

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Authors and Affiliations

  • Yrvann Emzivat
    • 1
    • 3
    Email author
  • Benoît Delahaye
    • 2
  • Didier Lime
    • 1
  • Olivier H. Roux
    • 1
  1. 1.UMR CNRS 6597École Centrale de Nantes, IRCCyNNantesFrance
  2. 2.UMR CNRS 6241Université de Nantes, LINANantesFrance
  3. 3.Renault S.A.S., Technocentre RenaultGuyancourtFrance

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