A New Approach to the Evaluation of Non Markovian Stochastic Petri Nets

  • Serge Haddad
  • Lynda Mokdad
  • Patrice Moreaux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4024)


In this work, we address the problem of transient and steady-state analysis of a stochastic Petri net which includes non Markovian distributions with a finite support but without any additional constraint. Rather than computing an approximate distribution of the model (as done in previous methods), we develop an exact analysis of an approximate model. The design of this method leads to a uniform handling of the computation of the transient and steady state behaviour of the model. This method is an adaptation of a former one developed by the same authors for general stochastic processes (which was shown to be more robust than alternative techniques). Using Petri nets as the modelling formalism enables us to express the behaviour of the approximate process by tensorial expressions. Such a representation yields significant savings w.r.t. time and space complexity.


Probability Vector Probabilistic Choice Continuous Time Markov Chain Discrete Time Markov Chain Reachability Graph 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cox, D.R.: A use of complex probabilities in the theory of stochastic processes. Proc. Cambridge Philosophical Society, 313–319 (1955)Google Scholar
  2. 2.
    German, R., Logothesis, D., Trivedi, K.: Transient analysis of Markov regenerative stochastic Petri nets: A comparison of approaches. In: Proc. of the 6th International Workshop on Petri Nets and Performance Models, Durham, NC, USA, pp. 103–112. IEEE Computer Society Press, Los Alamitos (1995)CrossRefGoogle Scholar
  3. 3.
    Cox, D.R.: The analysis of non-Markov stochastic processes by the inclusion of supplementary variables. Proc. Cambridge Philosophical Society (Math. and Phys. Sciences) 51, 433–441 (1955)MATHCrossRefGoogle Scholar
  4. 4.
    German, R., Lindemann, C.: Analysis of stochastic Petri nets by the method of supplementary variables. Performance Evaluation 20(1–3), 317–335 (1994); special issue: Peformance 1993CrossRefMathSciNetGoogle Scholar
  5. 5.
    Ajmone Marsan, M., Chiola, G.: On Petri nets with deterministic and exponentially distributed firing times. In: Rozenberg, G. (ed.) APN 1987. LNCS, vol. 266, pp. 132–145. Springer, Heidelberg (1987)Google Scholar
  6. 6.
    Lindemann, C., Schedler, G.: Numerical analysis of deterministic and stochastic Petri nets with concurrent deterministic transitions. Performance Evaluation 27–28, 576–582 (1996); special issue: Proc. of PERFORMANCE 1996Google Scholar
  7. 7.
    Lindemann, C., Reuys, A., Thümmler, A.: DSPNexpress 2.000 performance and dependability modeling environment. In: Proc. of the 29th Int. Symp. on Fault Tolerant Computing, Madison, Wisconsin (1999)Google Scholar
  8. 8.
    German, R.: Cascaded deterministic and stochastic petri nets. In: Plateau, B., Stewart, W.J., Silva, M. (eds.) Proc. of the third Int. Workshop on Numerical Solution of Markov Chains, Zaragoza, Spain, Prensas Universitarias de Zaragoza, pp. 111–130 (1999)Google Scholar
  9. 9.
    Puliafito, A., Scarpa, M., Trivedi, K.: K-simultaneously enable generally distributed timed transitions. Performance Evaluation 32(1), 1–34 (1998)MATHCrossRefGoogle Scholar
  10. 10.
    Bobbio, A., Telek, A.H.M.: The scale factor: A new degree of freedom in phase type approximation. In: International Conference on Dependable Systems and Networks (DSN 2002) - IPDS 2002, Washington, DC, USA, pp. 627–636. IEEE C.S. Press, Los Alamitos (2002)CrossRefGoogle Scholar
  11. 11.
    Jones, R.L., Ciardo, G.: On phased delay stochastic petri nets: Definition and an application. In: Proc. of the 9th Int. Workshop on Petri nets and performance models (PNPM 2001), Aachen, Germany, pp. 165–174. IEEE Comp. Soc. Press, Los Alamitos (2001)CrossRefGoogle Scholar
  12. 12.
    Horváth, A., Puliafito, A., Scarpa, M., Telek, M.: A discrete time approach to the analysis of non-markovian stochastic Petri nets. In: Haverkort, B.R., Bohnenkamp, H.C., Smith, C.U. (eds.) TOOLS 2000. LNCS, vol. 1786, pp. 171–187. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Haddad, S., Mokdad, L., Moreaux, P.: Performance evaluation of non Markovian stochastic discrete event systems - a new approach. In: Proc. of the 7th IFAC Workshop on Discrete Event Systems (WODES 2004), Reims, France, IFAC (2004)Google Scholar
  14. 14.
    Gross, D., Miller, D.: The randomization technique as a modeling tool an solution procedure for transient markov processes. Operations Research 32(2), 343–361 (1984)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lindemann, C.: DSPNexpress: A software package for the efficient solution of deterministic and stochastic Petri nets. In: Proc. of the Sixth International Conference on Modelling Techniques and Tools for Computer Performance Evaluation, Edinburgh, Scotland, UK, pp. 9–20. Edinburgh University Press (1992)Google Scholar
  16. 16.
    Donatelli, S., Haddad, S., Moreaux, P.: Structured characterization of the Markov chains of phase-type SPN. In: Puigjaner, R., Savino, N.N., Serra, B. (eds.) TOOLS 1998. LNCS, vol. 1469, pp. 243–254. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  17. 17.
    Sidje, R., Stewart, W.: A survey of methods for computing large sparse matrix exponentials arising in Markov chains. Computational Statistics and Data Analysis 29, 345–368 (1999)MATHCrossRefGoogle Scholar
  18. 18.
    Stewart, W.J.: Introduction to the numerical solution of Markov chains. Princeton University Press, USA (1994)MATHGoogle Scholar
  19. 19.
    German, R.: Iterative analysis of Markov regenerative models. Performance Evaluation 44, 51–72 (2001)MATHCrossRefGoogle Scholar
  20. 20.
    Ciardo, G., Zijal, R.: Well defined stochastic Petri nets. In: Proc. of the 4th Int. Workshop on Modeling, Ananlysis and Simulation of Computer and Telecommunication Systems (MASCOTS 1996), San Jose, CA, USA, pp. 278–284. IEEE Comp. Soc. Press, Los Alamitos (1996)CrossRefGoogle Scholar
  21. 21.
    Scarpa, M., Bobbio, A.: Kronecker representation of stochastic Petri nets with discrete PH distributions. In: International Computer Performance and Dependability Symposium - IPDS 1998, Duke University, Durham, NC. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  22. 22.
    Python team: Python home page (2004), http://www.python.org
  23. 23.
    Dubois, P.: Numeric Python home page: And the Numpy community (2004), http://www.pfdubois.com/numpy/
  24. 24.
    Geus, R.: PySparse home page (2004), http://www.geus.ch
  25. 25.
    Buchholz, P., Ciardo, G., Kemper, P., Donatelli, S.: Complexity of memory-efficient kronecker operations with applications to the solution of markov models. INFORMS Journal on Computing 13(3), 203–222 (2000)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Haddad, S., Moreaux, P.: Approximate analysis of non-markovian stochastic systems with multiple time scale delays. In: Proc. of the 12th Int. Workshop on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (MASCOTS 2004), Volendam, The Netherlands, pp. 23–30 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Serge Haddad
    • 1
  • Lynda Mokdad
    • 1
  • Patrice Moreaux
    • 2
  1. 1.LAMSADE, UMR CNRS 7024Universit Paris DauphinePARISFrance
  2. 2.LISTIC, ESIAUniversit de SavoieANNECYFrance

Personalised recommendations