A Framework to Decompose GSPN Models

  • Leonardo Brenner
  • Paulo Fernandes
  • Afonso Sales
  • Thais Webber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3536)


This paper presents a framework to decompose a single GSPN model into a set of small interacting models. This decomposition technique can be applied to any GSPN model with a finite set of tangible markings and a generalized tensor algebra (Kronecker) representation can be produced automatically. The numerical impact of all the possible decompositions obtained by our technique is discussed. To do so we draw the comparison of the results for some practical examples. Finally, we present all the computational gains achieved by our technique, as well as the future extensions of this concept for other structured formalisms.


Decomposition Technique Tensor Format Tensor Element Input Place Transition Superposition 
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.


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  1. 1.
    Ajmone-Marsan, M., Balbo, G., Chiola, G., Conte, G., Donatelli, S., Franceschinis, G.: An Introduction to Generalized Stochastic Petri Nets. Microelectronics and Reliability 31(4), 699–725 (1991)CrossRefGoogle Scholar
  2. 2.
    Ajmone-Marsan, M., Conte, G., Balbo, G.: A Class of Generalized Stochastic Petri Nets for the Performance Evaluation of Multiprocessor Systems. ACM Transactions on Computer Systems 2(2), 93–122 (1984)CrossRefGoogle Scholar
  3. 3.
    Amoia, V., De Micheli, G., Santomauro, M.: Computer-Oriented Formulation of Transition-Rate Matrices via Kronecker Algebra. IEEE Transactions on Reliability R-30(2), 123–132 (1981)CrossRefGoogle Scholar
  4. 4.
    Bellman, R.: Introduction to Matrix Analysis. McGraw-Hill, New York (1960)zbMATHGoogle Scholar
  5. 5.
    Benoit, A., Brenner, L., Fernandes, P., Plateau, B.: Aggregation of Stochastic Automata Networks with replicas. Linear Algebra and its Applications 386, 111–136 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Benoit, A., Brenner, L., Fernandes, P., Plateau, B., Stewart, W.J.: The PEPS Software Tool. In: Kemper, P., Sanders, W.H. (eds.) TOOLS 2003. LNCS, vol. 2794, pp. 98–115. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Brenner, L., Fernandes, P., Sales, A.: The Need for and the Advantages of Generalized Tensor Algebra for Kronecker Structured Representations. International Journal of Simulation: Systems, Science & Technology 6(3-4), 52–60 (2005)Google Scholar
  8. 8.
    Brewer, J.W.: Kronecker Products and Matrix Calculus in System Theory. IEEE Transactions on Circuits and Systems CAS-25(9), 772–780 (1978)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Buchholz, P., Ciardo, G., Donatelli, S., Kemper, P.: 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
  10. 10.
    Buchholz, P., Dayar, T.: Block SOR for Kronecker structured representations. Linear Algebra and its Applications 386, 83–109 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Buchholz, P., Kemper, P.: Hierarchical reachability graph generation for Petri nets. Formal Methods in Systems Design 21(3), 281–315 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ciardo, G., Forno, M., Grieco, P.L.E., Miner, A.S.: Comparing implicit representations of large CTMCs. In: 4th International Conference on the Numerical Solution of Markov Chains, Urbana, IL, USA, September 2003, pp. 323–327 (2003)Google Scholar
  13. 13.
    Ciardo, G., Jones, R.L., Miner, A.S., Siminiceanu, R.: SMART: Stochastic Model Analyzer for Reliability and Timing. In: Tools of Aachen 2001 International Multiconference on Measurement, Modelling and Evaluation of Computer-Communication Systems, Aachen, Germany, September 2001, pp. 29–34 (2001)Google Scholar
  14. 14.
    Ciardo, G., Trivedi, K.S.: A Decomposition Approach for Stochastic Petri Nets Models. In: Proceedings of the 4th International Workshop Petri Nets and Performance Models, Melbourne, Australia, December 1991, pp. 74–83. IEEE Computer Society Press, Los Alamitos (1991)CrossRefGoogle Scholar
  15. 15.
    Davio, M.: Kronecker Products and Shuffle Algebra. IEEE Transactions on Computers C-30(2), 116–125 (1981)MathSciNetGoogle Scholar
  16. 16.
    Donatelli, S.: Superposed stochastic automata: a class of stochastic Petri nets with parallel solution and distributed state space. Performance Evaluation 18, 21–36 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Donatelli, S.: Superposed generalized stochastic Petri nets: definition and efficient solution. In: Valette, R. (ed.) Proceedings of the 15th International Conference on Applications and Theory of Petri Nets, pp. 258–277. Springer, Heidelberg (1994)Google Scholar
  18. 18.
    Fernandes, P., Plateau, B., Stewart, W.J.: Efficient descriptor - Vector multiplication in Stochastic Automata Networks. Journal of the ACM 45(3), 381–414 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hillston, J., Kloul, L.: An Efficient Kronecker Representation for PEPA models. In: de Alfaro, L., Gilmore, S. (eds.) Proceedings of the first joint PAPM-PROBMIV Workshop), Aachen, Germany, September 2001, pp. 120–135. Springer, Heidelberg (2001)Google Scholar
  20. 20.
    Miner, A.S.: Data Structures for the Analysis of Large Structured Markov Models. PhD thesis, The College of William and Mary, Williamsburg, VA (2000)Google Scholar
  21. 21.
    Miner, A.S.: Efficient solution of GSPNs using Canonical Matrix Diagrams. In: 9th International Workshop on Petri Nets and Performance Models (PNPM 2001), Aachen, Germany, September 2001, pp. 101–110. IEEE Computer Society Press, Los Alamitos (2001)CrossRefGoogle Scholar
  22. 22.
    Miner, A.S., Ciardo, G.: Efficient Reachability Set Generation and Storage Using Decision Diagrams. In: Donatelli, S., Kleijn, J. (eds.) ICATPN 1999. LNCS, vol. 1639, pp. 6–25. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  23. 23.
    Miner, A.S., Ciardo, G., Donatelli, S.: Using the exact state space of a Markov model to compute approximate stationary measures. In: Proceedings of the 2000 ACM SIGMETRICS Conference on Measurements and Modeling of Computer Systems, Santa Clara, California, USA, June 2000, pp. 207–216. ACM Press, New York (2000)CrossRefGoogle Scholar
  24. 24.
    Murata, T.: Petri nets: Properties, analysis and applications. Proceedings of the IEEE 77(4), 541–580 (1989)CrossRefGoogle Scholar
  25. 25.
    Plateau, B.: On the stochastic structure of parallelism and synchronization models for distributed algorithms. In: Proceedings of the 1985 ACM SIGMETRICS conference on Measurements and Modeling of Computer Systems, Austin, Texas, USA, pp. 147–154. ACM Press, New York (1985)CrossRefGoogle Scholar
  26. 26.
    Plateau, B., Atif, K.: Stochastic Automata Networks for modelling parallel systems. IEEE Transactions on Software Engineering 17(10), 1093–1108 (1991)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Reisig, W.: Petri nets: an introduction. Springer, Heidelberg (1985)zbMATHGoogle Scholar
  28. 28.
    Stewart, W.J.: Introduction to the numerical solution of Markov chains. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Leonardo Brenner
    • 1
  • Paulo Fernandes
    • 1
  • Afonso Sales
    • 1
  • Thais Webber
    • 1
  1. 1.Pontifícia Universidade Católica do Rio Grande do SulPorto AlegreBrazil

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