Exploiting Non-deterministic Analysis in the Integration of Transient Solution Techniques for Markov Regenerative Processes

  • Marco Biagi
  • Laura Carnevali
  • Marco Paolieri
  • Tommaso Papini
  • Enrico Vicario
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10503)


Transient analysis of Markov Regenerative Processes (MRPs) can be performed through the solution of Markov renewal equations defined by global and local kernels, which respectively characterize the occurrence of regenerations and transient probabilities between them. To derive kernels from stochastic models (e.g., stochastic Petri nets), existing methods exclusively address the case where at most one generally-distributed timer is enabled in each state, or where regenerations occur in a bounded number of events. In this work, we analyze the state space of the underlying timed model to identify epochs between regenerations and apply distinct methods to each epoch depending on the satisfied conditions. For epochs not amenable to existing methods, we propose an adaptive approximation of kernel entries based on partial exploration of the state space, leveraging heuristics that permit to reduce the error on transient probabilities. The case study of a polling system with generally-distributed service times illustrates the effect of these heuristics and how the approach extends the class of models that can be analyzed.


Non-markovian Petri Nets Markov Regenerative Process Enabling restriction Stochastic state class Non-deterministic analysis 


  1. 1.
    Amparore, E.G., Buchholz, P., Donatelli, S.: A structured solution approach for Markov regenerative processes. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 9–24. Springer, Cham (2014). doi: 10.1007/978-3-319-10696-0_3 Google Scholar
  2. 2.
    Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., Van der Vorst, H.: Templates for the Solutions of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berthomieu, B., Diaz, M.: Modeling and verification of time dependent systems using time Petri nets. IEEE Trans. Softw. Eng. 17(3), 259–273 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bucci, G., Carnevali, L., Ridi, L., Vicario, E.: Oris: a tool for modeling, verification and evaluation of real-time systems. STTT 12(5), 391–403 (2010)CrossRefGoogle Scholar
  5. 5.
    Çinlar, E.: Markov renewal theory: a survey. Manag. Sci. 21(7), 727–752 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choi, H., Kulkarni, V.G., Trivedi, K.S.: Markov regenerative stochastic Petri nets. Perform. Eval. 20(1–3), 337–357 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ciardo, G., German, R., Lindemann, C.: A characterization of the stochastic process underlying a stochastic Petri net. IEEE Trans. Softw. Eng. 20(7), 506–515 (1994)CrossRefGoogle Scholar
  8. 8.
    German, R., Lindemann, C.: Analysis of stochastic Petri nets by the method of supplementary variables. Perform. Eval. 20(1), 317–335 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    German, R., Logothetis, D., Trivedi, K.S.: Transient analysis of Markov regenerative stochastic Petri nets: a comparison of approaches. In: International Workshop on Petri Nets and Performance Models, pp. 103–112. IEEE (1995)Google Scholar
  10. 10.
    Horváth, A., Paolieri, M., Ridi, L., Vicario, E.: Transient analysis of non-Markovian models using stochastic state classes. Perform. Eval. 69(7–8), 315–335 (2012)CrossRefGoogle Scholar
  11. 11.
    Ibe, O.C., Trivedi, K.S.: Stochastic Petri net models of polling systems. IEEE J. Sel. Areas Commun. 8(9), 1649–1657 (1990)CrossRefGoogle Scholar
  12. 12.
    Kulkarni, V.: Modeling and Analysis of Stochastic Systems. Chapman & Hall, London (1995)zbMATHGoogle Scholar
  13. 13.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM: probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, pp. 200–204. Springer, Heidelberg (2002). doi: 10.1007/3-540-46029-2_13 CrossRefGoogle Scholar
  14. 14.
    Lime, D., Roux, O.H.: Expressiveness and analysis of scheduling extended time Petri nets. In: IFAC Conference on Fieldbus and their Applications. Elsevier Science (2003)Google Scholar
  15. 15.
    Lindemann, C., Thümmler, A.: Transient analysis of deterministic and stochastic Petri nets with concurrent deterministic transitions. Perform. Eval. 36–37(1–4), 35–54 (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Paolieri, M., Horváth, A., Vicario, E.: Probabilistic model checking of regenerative concurrent systems. IEEE Trans. Softw. Eng. 42(2), 153–169 (2016)CrossRefGoogle Scholar
  17. 17.
    Telek, M., Horváth, A.: Transient analysis of Age-MRSPNs by the method of supplementary variables. Perform. Eval. 45(4), 205–221 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Zimmermann, A: Modeling and evaluation of stochastic Petri nets with TimeNET 4.1. In: International ICST Conference on Performance Evaluation Methodologies and Tools, pp. 54–63 (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Marco Biagi
    • 1
  • Laura Carnevali
    • 1
  • Marco Paolieri
    • 2
  • Tommaso Papini
    • 1
  • Enrico Vicario
    • 1
  1. 1.Department of Information EngineeringUniversity of FlorenceFlorenceItaly
  2. 2.Department of Computer ScienceUniversity of Southern CaliforniaLos AngelesUSA

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