Discrete Event Dynamic Systems

, Volume 22, Issue 2, pp 179–196 | Cite as

Piecewise constant timed continuous PNs for the steady state estimation of stochastic PNs

Article

Abstract

Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. The computation of numerical approximations is also time consuming due to the slow convergence of stochastic simulations. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes. The contributions of this paper are to point out that timed continuous Petri nets may lead to biased estimators of the stochastic steady state and to introduce fluid Petri nets with piecewise-constant maximal firing speeds and sufficient conditions in order to obtain unbiased estimators.

Keywords

Stochastic Petri nets Continuous Petri nets Fluidification Steady state Reliability analysis  

References

  1. Ajmone Marsan M, Chiola G (1987) On Petri nets with deterministic and exponentially distributed firing times. Lect Notes Comput Sci 266:132–145MathSciNetCrossRefGoogle Scholar
  2. Bobbio A, Puliafito A, Telek M, Trivedi K (1998) Recent developments in Stochastic Petri Nets. J Circuits Syst Comput 8(1):119–158MathSciNetCrossRefGoogle Scholar
  3. Campos J, Chiola G, Silva M (1991) Ergodicity and throughut bounds of Petri nets with unique consistent firing count vector. IEEE Trans Softw Eng 17(2):117–125MathSciNetCrossRefGoogle Scholar
  4. David R, Alla H (1992) Petri nets and grafcet – tools for modelling discrete events systems. Prentice Hall, LondonGoogle Scholar
  5. Diaz M (2001) Les réseaux de Petri : modèles fondamentaux. Hermes, ParisGoogle Scholar
  6. Dotoli M, Fanti MP, Giua A, Seatzu C (2008) First-order hybrid Petri nets. An application to distributed manufacturing systems. NAHS 2(2):408–430MathSciNetMATHGoogle Scholar
  7. Gaujal B, Giua A (2004) Optimal stationary behavior for a class of timed continuous Petri nets. Automatica 40(9):1505–1516MathSciNetMATHCrossRefGoogle Scholar
  8. Júlvez G, Recalde L, Silva M (2005) Steady-state performance evaluation of continuous mono-T-semiflow Petri nets. Automatica 41(4):605–616MathSciNetMATHCrossRefGoogle Scholar
  9. Kara R, Loiseau JJ, Djennoune S (2008) Quantitative analysis of continuous weighted marked graphs. NAHS 2:1010–1020MathSciNetMATHGoogle Scholar
  10. Lefebvre D, Leclercq E (2011) Attractive regions with finite attraction time for contPNs. In: Proceeding IEEE-MED 2011, Corfu, GreeceGoogle Scholar
  11. Lefebvre D, Leclercq E, Khalij L, Souza de Cursi E, El Akchioui N (2009) Approximation of MTS stochastic Petri nets steady state by means of continuous Petri nets: a numerical approach. In: Proceeding IFAC ADHS, Zaragoza, Spain, pp 62–67Google Scholar
  12. Lefebvre D, Leclercq E, El Akchioui N, Khalij L, Souza de Cursi E (2010) A geometric approach for the homothetic approximation of stochastic Petri nets. In: Proc. IFAC WODES 2010, Berlin, Germany, pp 245–250Google Scholar
  13. Mahulea C, Giua A, Recalde L, Seatzu C, Silva M (2006) On sampling continuous timed Petri nets: reachability “equivalence” under infinite servers semantics. In: Proceeding IFAC - ADHS, Alghero, Italy, pp 37–43Google Scholar
  14. Mahulea C, Ramirez Trevino A, Recalde L, Silva M (2008) Steady state control reference and token conservation laws in continuous Petri nets. Trans IEEE – TASE 5(2):307–320Google Scholar
  15. Molloy MK (1981) On the integration of delay and troughput in distributed processing models. Ph. D, UCLA, Los Angeles, USAGoogle Scholar
  16. Molloy MK (1982) Performance analysis using stochastic Petri nets. IEEE Trans Comput C 31:913–917CrossRefGoogle Scholar
  17. Rausand M, Hoyland A (2004) System reliability theory: models, statistical methods, and applications. Wiley, HobokenMATHGoogle Scholar
  18. Recalde L, Teruel E, Silva M (1999) Autonomous continuous P/T systems. In: Kleijn J, Donatelli S (eds) Application and theory of Petri Nets. Lecture notes in computer science, vol 1639. Springer, pp 107–126Google Scholar
  19. Rotella F, Borne P (1995) Theorie et pratique du calcul matriciel. Technip, ParisMATHGoogle Scholar
  20. Silva M, Recalde L (2002) Petri nets and integrality relaxations: a view of continuous Petri nets. Trans IEEE – SMC part C 32(4):314–326Google Scholar
  21. Silva M, Recalde L (2004) On fluidification of Petri Nets: from discrete to hybrid and continuous models. Annu Rev Control 28(2):253–266CrossRefGoogle Scholar
  22. Trivedi K, Kulkarni V (1993) FSPN’s Fluid Stochastic Petri nets. In: Ajmone Marsan M (ed) Lecture notes in computer science 691Google Scholar
  23. Valette R (2002) Les réseaux de Petri, cours Université P. sabatier. Toulouse, FranceGoogle Scholar
  24. Vazquez R, Silva M (2009) Hybrid approximations of Markovian Petri Nets. In: Proceeding IFAC – ADHS, Zaragoza, Spain, pp 56–61Google Scholar
  25. Vazquez R, Recalde L, Silva M (2008) Stochastic continuous-state approximation of markovian Petri net systems. In: Proceeding IEEE – CDC08, Cancun, Mexico, pp 901–906Google Scholar
  26. Zerhouni N, Alla H (1990) Dynamic analysis of manufacturing systems using continuous Petri nets. In: Proc. IEEE ICRA, Cincinnati, USAGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.GREAH, University Le HavreLe HavreFrance

Personalised recommendations