Skip to main content
SpringerLink
Log in

Introduction to Stochastic Petri Nets

  • Chapter
  • First Online:
Lectures on Formal Methods and PerformanceAnalysis (EEF School 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2090))

Included in the following conference series:

Abstract

Stochastic Petri Nets are a modelling formalism that can be conveniently used for the analysis of complex models of Discrete Event Dynami Systems (DEDS) and for their performance and reliability evaluation. The automatic construction of the probabilistic models that underly the dynamic behaviours of these nets rely on a set of results that derive from the theory of untimed Petri nets. The paper introduces the basic motivations for modelling DEDS and briefly overviews the basic results of net theory that are useful for the definition of Stochastic Petri Nets and Generalized Stochastic Petri Nets. The different approaches that have been used for introducing the concept of time in these models are discussed in order to provide the basis for the definition of SPNs and GSPNs as well. Details on the solution techniques and on ntheir computational aspects are provided. A brief overview of more advanced material is included at the end of the paper to highlight the state of the art in this field and to give pointers to relevant results published in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Agerwala. Putting Petri nets to work. IEEE Computer, pages 85–94, December 1979.

    Google Scholar 

  2. M. Ajmone Marsan, G. Balbo, A. Bobbio, G. Chiola, G. Conte, and A. Cumani. The effect of execution policies on the semantics and analysis of stochastic Petri nets. IEEE Transactions on Software Engineering, 15(7):832–846, July 1989.

    Article  MathSciNet  Google Scholar 

  3. M. Ajmone Marsan, G. Balbo, G. Chiola, and G. Conte. Generalized stochastic Petri nets revisited: Random switches and priorities. In Proc. Intern. Workshop on Petri Nets and Performance Models, pages 44–53, Madison, WI, USA, August 1987. IEEE-CS Press.

    Google Scholar 

  4. M. Ajmone Marsan, G. Balbo, and G. Conte. A class of generalized stochastic Petri nets for the performance analysis of multiprocessor systems. ACM Transactions on Computer Systems, 2(1), May 1984.

    Google Scholar 

  5. M. Ajmone Marsan, G. Balbo, and G. Conte. Performance Models of Multiprocessor Systems. MIT Press, Cambridge, USA, 1986.

    Google Scholar 

  6. M. Ajmone Marsan, G. Balbo, G. Conte, S. Donatelli, and G. Franceschinis. Modelling with Generalized Stochastic Petri Nets. J. Wiley, 1995.

    Google Scholar 

  7. M. Ajmone Marsan, G. Balbo, and K.S. Trivedi, editors. Proc. Intern. Workshop on Timed Petri Nets, Torino, Italy, July 1985. IEEE-CS Press.

    Google Scholar 

  8. M. Ajmone Marsan and G. Chiola. On Petri nets with deterministic and exponential transition firing times. In Proc. 7 th European Workshop on Application and Theory of Petri Nets, pages 151–165, Oxford, England, June 1986.

    Google Scholar 

  9. H. Alaiwan and G. Memmi. Algorithmes de recherche des solutions entieres positives d’un systeme d’equations lineaires homogeneus en nombres entieres. Revue Technique Thomson-CSF, 14(1):125–135, March 1982. in French.

    Google Scholar 

  10. H. Alaiwan and J.M. Toudic. Research des semiflots, des verrous et des trappes dans le reseaux de Petri. Technique et Science Informatiques, 4(1), February 1985.

    Google Scholar 

  11. H.H. Ammar and S.M. Rezaul Islam. Time scale decomposition of a class of generalized stochastic Petri net models. IEEE Transactions on Software Engineering, 15(6):809–820, June 1989.

    Article  Google Scholar 

  12. G. Balbo, M. Silva, editors. Performance Models for Discrete Event Systems with Synchronizations: Formalisms and Analysis Techniques. KRONOS, Zaragoza, Spain, 1998.

    Google Scholar 

  13. G. Berthelot. Transformations and decompositions of nets. In W. Brauer, W. Reisig, and G. Rozenberg, editors, Advances in Petri Nets’ 86-Part I, volume 254 of LNCS, pages 359–376. Springer Verlag, Bad Honnef, West Germany, February 1987.

    Google Scholar 

  14. E. Best, R. Devillers, and J. Hall. The Petri box calculus: a new causal algebra with multilabel communication. In G. Rozenberg, editor, Advances in Petri Nets, volume 609 of LNCS, pages 21–69. Springer Verlag, 1992.

    Google Scholar 

  15. A. Blakemore. The cost of eliminating vanishing markings from generalized stochastic Petri nets. In Proc. 3 rd Intern. Workshop on Petri Nets and Performance Models, Kyoto, Japan, December 1989. IEEE-CS Press.

    Google Scholar 

  16. A. Blakemore and S.K. Tripathi. Automated time scale decomposition and analysis of stochastic Petri nets. In Proc. 5 th Intern. Workshop on Petri Nets and Performance Models, pages 248–257, Toulouse, France, October 1993. IEEE-CS Press.

    Google Scholar 

  17. R.J. Boucherie. A characterization of independence for competing Markov chains with applications to stochastic Petri nets. In Proc. 5 th Intern. Workshop on Petri Nets and Performance Models, pages 117–126, Toulouse, France, October 1993. IEEE-CS Press.

    Google Scholar 

  18. P. Buchholz. Hierarchical high level Petri nets for complex system analysis. In Proc. 15 th Intern. Conference on Applications and Theory of Petri Nets, number 185 in LNCS, Zaragoza, Spain, 1994. Springer-Verlag.

    Google Scholar 

  19. Moler C. and C. Van Loan. Nineteen doubious ways to compute the exponential of a matrix. SIAM Review, 20(4):801–836, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Chen, S.C. Bruell, and G. Balbo. Alternative methods for incorporating nonexponential distributions into stochastic Petri nets. In Proc. 3 rd Intern. Workshop on Petri Nets and Performance Models, pages 187–197, Kyoto, Japan, December 1989. IEEE-CS Press.

    Google Scholar 

  21. G. Chiola, M. Ajmone Marsan, G. Balbo, and G. Conte. Generalized stochastic Petri nets: A definition at the net level and its implications. IEEE Transactions on Software Engineering, 19(2):89–107, February 1993.

    Article  Google Scholar 

  22. G. Chiola, S. Donatelli, and G. Franceschinis. GSPN versus SPN: what is the actual role of immediate transitions? In Proc. 4 th Intern. Workshop on Petri Nets and Performance Models, pages 20–31, Melbourne, Australia, December 1991. IEEECS Press.

    Google Scholar 

  23. G. Chiola, S. Donatelli, and G. Franceschinis. Priorities, inhibitor arcs, and concurrency in P/T nets. In Proc. 12 th Intern. Conference on Application and Theory of Petri Nets, Aarhus, Denmark, June 1991.

    Google Scholar 

  24. G. Chiola, C. Dutheillet, G. Franceschinis, and S. Haddad. Stochastic Well-Formed coloured nets for symmetric modelling applications. IEEE Transactions on Computers, 42(11), November 1993.

    Google Scholar 

  25. G. Chiola and G. Franceschinis. Colored GSPN models and automatic symmetry detection. In Proc. 3 rd Intern. Workshop on Petri Nets and Performance Models, Kyoto, Japan, December 1989. IEEE-CS Press.

    Google Scholar 

  26. G. Chiola, G. Franceschinis, R. Gaeta, and M. Ribaudo. GreatSPN1.7: GRaphical Editor and Analyzer for Timed and Stochastic Petri Nets. Performance Evaluation, 24, 1995. Special Issues on Performance Modelling Tools.

    Google Scholar 

  27. H. Choi, G. Kulkarni, and K.S. Trivedi. Transient analysis of deterministic and stochastic petri nets. In Proc. 14 th Intern. Conference on Application and Theory of Petri Nets, Chicago, Illinois, June 1993. Springer Verlag.

    Google Scholar 

  28. H. Choi, V.S. Kulkarni, and K.S. Trivedi. Markov regenerative stochastic Petri nets. In Proc. of Performance 93, Rome, Italy, September 1993.

    Google Scholar 

  29. G. Ciardo. Analysis of large Petri net models. PhD thesis, Department of Computer Science, Duke University, Durham, NC, USA, 1989. Ph. D. Thesis.

    Google Scholar 

  30. G. Ciardo, R. German, and C. Lindemann. A characterization of the stochastic process underlying a stochastic Petri net. In Proc. 5 th Intern. Workshop on Petri Nets and Performance Models, pages 170–179, Toulouse, France, October 1993. IEEE-CS Press.

    Google Scholar 

  31. G. Ciardo and C. Lindemann. Analysis of deterministic and stochastic Petri nets. In Proc. 5 th Intern. Workshop on Petri Nets and Performance Models, pages 160–169, Toulouse, France, October 1993. IEEE-CS Press.

    Google Scholar 

  32. J.E. Coolhan and N. Roussopoulos. Timing requirements for time-driven systems using augmented Petri nets. In Proc. Intern. Workshop on Timed Petri Nets, Torino, Italy, July 1985. IEEE-CS Press.

    Google Scholar 

  33. P.J. Courtois. Decomposability: Queueing and Computer Systems Applications. Academic Press, New York, 1977.

    Google Scholar 

  34. S. Donatelli. L’uso delle reti di Petri per la valutazione e la validazione di sistemi di grandi dimensioni. PhD thesis, Dipartimento di Informatica, Università di Torino, Corso Svizzera 185, 10149, Torino, Italy, February 1990. (in Italian).

    Google Scholar 

  35. J.B. Dugan, K.S. Trivedi, R.M. Geist, and V.F. Nicola. Extended stochastic Petri nets: Applications and analysis. In Proc. PERFORMANCE’ 84, Paris, France, December 1984.

    Google Scholar 

  36. C. Dutheillet and S. Haddad. Aggregation and disaggregation of states in colored stochastic Petri nets: Application to a multiprocessor architecture. In Proc. 3 rd> Intern. Workshop on Petri Nets and Performance Models, Kyoto, Japan, December 1989. IEEE-CS Press.

    Google Scholar 

  37. G. Florin and S. Natkin. Les reseaux de Petri stochastiques. Technique et Science Informatiques, 4(1), February 1985.

    Google Scholar 

  38. G. Florin and S. Natkin. Matrix product form solution for closed synchronized queueing networks. In Proc. 3 rd Intern. Workshop on Petri Nets and Performance Models, pages 29–39, Kyoto, Japan, December 1989. IEEE-CS Press.

    Google Scholar 

  39. R. German and C. Lindemann. Analysis of stochastic Petri nets by the method of supplementary variables. In Proc. of Performance 93, Rome, Italy, September 1993.

    Google Scholar 

  40. P.J. Haas and G.S. Shedler. Regenerative stochastic Petri nets. Performance Evaluation, 6(3):189–204, September 1986.

    Article  MathSciNet  MATH  Google Scholar 

  41. P.J. Haas and G.S. Shedler. Regenerative simulation of stochastic Petri nets. In Proc. Intern. Workshop on Timed Petri Nets, Torino, Italy, July 1985. IEEE-CS Press.

    Google Scholar 

  42. W. Henderson and D. Lucic. Exact results in the aggregation and disaggregation of stochastic Petri nets. In Proc. 4 th Intern. Workshop on Petri Nets and Performance Models, pages 166–175, Melbourne, Australia, December 1991. IEEE-CS Press.

    Google Scholar 

  43. W. Henderson and P.G. Taylor. Aggregation methods in exact performance analysis of stochastic Petri nets. In Proc. 3 rd Intern. Workshop on Petri Nets and Performance Models, pages 12–18, Kyoto, Japan, December 1989. IEEE-CS Press.

    Google Scholar 

  44. M.A. Holliday and M.K. Vernon. A generalized timed Petri net model for performance analysis. In Proc. Intern. Workshop on Timed Petri Nets, Torino, Italy, July 1985. IEEE-CS Press.

    Google Scholar 

  45. K. Lautenbach. Linear algebraic technique for place/transition nets. In W. Brawer, W. Reisig, and G. Rozenberg, editors, Advances on Petri Nets’ 86-Part I, volume 254 of LNCS, pages 142–167. Springer Verlag, Bad Honnef, West Germany, February 1987.

    Google Scholar 

  46. A.A. Lazar and T.G. Robertazzi. Markovian Petri net protocols with product form solution. Performance Evaluation, 12:67–77, 1991.

    Article  MathSciNet  Google Scholar 

  47. C. Lin and D.C. Marinescu. On stochastic high level Petri nets. In Proc. Intern. Workshop on Petri Nets and Performance Models, Madison, WI, USA, August 1987. IEEE-CS Press.

    Google Scholar 

  48. V. Mainkar, H. Choi, and K. Trivedi. Sensitivity analysis of Markov regenerative stochastic Petri nets. In Proc. 5 th Intern. Workshop on Petri Nets and Performance Models, Toulouse, France, October 1993. IEEE-CS Press.

    Google Scholar 

  49. J. Martinez and M. Silva. A simple and fast algorithm to obtain all invariants of a generalized Petri net. In Proc. 2 nd European Workshop on Application and Theory of Petri Nets, Bad Honnef, West Germany, September 1981. Springer Verlag.

    Google Scholar 

  50. P.M. Merlin and D.J. Farber. Recoverability of communication protocols: Implications of a theoretical study. IEEE Transactions on Communications, 24(9):1036–1043, September 1976.

    Article  MATH  MathSciNet  Google Scholar 

  51. J.F. Meyer, A. Movaghar, and W.H. Sanders. Stochastic activity networks: Structure, behavior, and application. In Proc. Intern. Workshop on Timed Petri Nets, pages 106–115, Torino,Italy, July 1985.

    Google Scholar 

  52. R. Milner. Communication and concurrency. Prentice Hall, 1989.

    Google Scholar 

  53. M.K. Molloy. On the Integration of Delay and Throughput Measures in Distributed Processing Models. PhD thesis, UCLA, Los Angeles, CA, 1981. Ph.D. Thesis.

    Google Scholar 

  54. M.K. Molloy, T. Murata, and M.K. Vernon, editors. Proc. Intern. Workshop on Petri Nets and Performance Models, Madison, Wisconsin, August 1987. IEEE-CS Press.

    Google Scholar 

  55. T. Murata. Petri nets: properties, analysis, and applications. Proceedings of the IEEE, 77(4):541–580, April 1989.

    Article  Google Scholar 

  56. S. Natkin. Les Reseaux de Petri Stochastiques et leur Application a l’Evaluation des Systemes Informatiques. PhD thesis, CNAM, Paris, France, 1980. These de Docteur Ingegneur.

    Google Scholar 

  57. M.F. Neuts. Matrix Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD, 1981.

    MATH  Google Scholar 

  58. J.D. Noe and G.J. Nutt. Macro e-nets representation of parallel systems. IEEE Transactions on Computers, 31(9):718–727, August 1973.

    Article  Google Scholar 

  59. J.L. Peterson. Petri Net Theory and the Modeling of Systems. Prentice-Hall, Englewood Cliffs, NJ, 1981.

    Google Scholar 

  60. C.A. Petri. Communication with automata. Technical Report RADC-TR-65-377, Rome Air Dev. Center, New York, NY, 1966.

    Google Scholar 

  61. C.V. Ramamoorthy and G.S. Ho. Performance evaluation of asynchronous concurrent systems using Petri nets. IEEE Transactions on Software Engineering, 6(5):440–449, September 1980.

    Article  MathSciNet  Google Scholar 

  62. C. Ramchandani. Analysis of Asynchronous Concurrent Systems by Timed Petri Nets. PhD thesis, MIT, Cambridge, MA, 1974. Ph.D. Thesis.

    Google Scholar 

  63. W. Reisig. Petri Nets: an Introduction. Springer Verlag, 1985.

    Google Scholar 

  64. T.G. Robertazzi. Computer Networks and Systems: Queueing Theory and Performance Evaluation. Springer Verlag, 1991.

    Google Scholar 

  65. J. Sifakis. Petri nets for performance evaluation. In H. Beilner and E. Gelenbe, editors, Proc. 3 rd Intern. Symp. IFIP, pages 75–93, 1978.

    Google Scholar 

  66. H.A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29, 1961.

    Google Scholar 

  67. W.J. Stewart. Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, New Jersey, USA, 1994.

    MATH  Google Scholar 

  68. F.J.W. Symons. Introduction to numerical Petri nets, a general graphical model of concurrent processing systems. Australian Telecommunications Research, 14(1):28–33, January 1980.

    Google Scholar 

  69. R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1962.

    Google Scholar 

  70. C.Y. Wong, T.S. Dillon, and K.E. Forward. Timed places Petri nets with stochastic representation of place time. In Proc. Intern. Workshop on Timed Petri Nets, Torino, Italy, July 1985. IEEE-CS Press.

    Google Scholar 

  71. W.M. Zuberek. Timed Petri nets and preliminary performance evaluation. In Proc. 7 th Annual Symposium on Computer Architecture, pages 88–96, La Baule, France, May 1980.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università di Torino, Torino, Italy

    Gianfranco Balbo

Authors
  1. Gianfranco Balbo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Computer Science, Formal Methods and Tools Group, University of Twente, 217, 7500, AE, Enschede, The Netherlands

    Ed Brinksma , Holger Hermanns  & Joost-Pieter Katoen ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Balbo, G. (2001). Introduction to Stochastic Petri Nets. In: Brinksma, E., Hermanns, H., Katoen, JP. (eds) Lectures on Formal Methods and PerformanceAnalysis. EEF School 2000. Lecture Notes in Computer Science, vol 2090. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44667-2_3

Download citation

  • DOI: https://doi.org/10.1007/3-540-44667-2_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42479-6

  • Online ISBN: 978-3-540-44667-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation