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Iterative decomposition and aggregation of Stochastic marked graph Petri nets

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Advances in Petri Nets 1993 (ICATPN 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 674))

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Abstract

This paper addresses the problem of state explosion and computational complexity in solving marked graph Stochastic Petri Nets (SPNs). We first develop a simple and powerful Petri Nets (PNs) reduction technique that reduces the size of a marked graph PN. Then we propose an iterative delay equivalent reduction technique to solve the reduced SPN. This method iteratively tunes two or more partly aggregated auxiliary SPNs until they are approximately delay equivalent to the original SPN. It avoids the direct solution of the original SPN. In our experience the iteration usually converges and the accuracy is usually better than 5%, though there are exceptional cases. The technique is a step towards approximating more general SPNs.

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References

  1. Ammar, H.H., Y.F. Huang and R.W. Liu, “Hierarchical Models for Systems Reliability, Maintainability, and Availability,” IEEE Transactions on Circuits and Systems, Vol. CAS-34, No. 6, June 1987, pp. 629–638.

    Google Scholar 

  2. Ammar, H.H. and S.M.R. Islam, “Time Scale Decomposition of A Class of Generalized Stochastic Petri Net Models,” IEEE Transactions on Software Eng., Vol. 15, No. 6, June 1989, pp. 809–820.

    Google Scholar 

  3. Balsamo, S. and G. Iazeolla, “An Extension of Norton's Theorem for Queueing Networks,” IEEE Trans. on Software Eng., Vol. SE-8, No. 4, July 1982, pp. 298–305.

    Google Scholar 

  4. Berthelot, G. and G. Roucairol, “Reduction of Petri-Nets,” Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 45, 1976, pp. 202–209.

    Google Scholar 

  5. Berthelot, G. and G. Roucairol, “Reductions of Nets and Parallel Programs,” Net Theory and Applications, Lecture Notes in Computer Science, Vol. 84, 1979, pp. 277–290.

    Google Scholar 

  6. Berthelot, G. and Lri-Iie, “Checking Properties of Nets Using Transformations,” Advances in Petri Nets, Lecture Notes in Computer Science, Vol. 222, 1985, pp. 19–40.

    Google Scholar 

  7. Berthelot, G., Cham-Iie and Lri, “Transformations and Decompositions of Nets,” Advances in Petri Nets, Lecture Notes in Computer Science, Vol. 254, Part 1, 1986, pp. 359–376.

    Google Scholar 

  8. Best, E., “Structure Theory of Petri Nets: the Free Choice Hiatus,” Petri Nets: Central Models and Their Properties, Lecture Notes in Computer Science, Vol. 254, Springer-Verlag, 1986, pp. 165–205.

    Google Scholar 

  9. Campos, J., B. Sanchez, and M. Silva, “Throughput Lower Bounds for Markovian Petri Nets: Transformation Techniques,” The 4th International Workshop of Petri Nets and Performance Models, Melbourne, Australia, December 2–5, 1991, pp. 322–331.

    Google Scholar 

  10. Chandy, K.M., U. Herzog and L. Woo, “Parametric Analysis of Queueing Networks,” IBM Journal of the Research Development, Vol. 19, No. 9, Jan. 1975, pp. 36–42.

    Google Scholar 

  11. Ciardo, G. and K.S. Trivedi, “A Decomposition Approach for Stochastic Petri Net Models,” The 4th International Workshop of Petri Nets and Performance Models, Melbourne, Australia, December 2–5, 1991, pp. 74–83.

    Google Scholar 

  12. Courtois, P.J., “Decomposability: Queueing and Computer Applications,” Academic Press, New York, 1977.

    Google Scholar 

  13. Florin, G. and S. Natkin, “Generalization of Queueing Network Product Form Solutions to Stochastic Petri Nets,” IEEE Trans. on Software Eng., Vol. 17, No. 2, Feb. 1991, pp. 99–107.

    Google Scholar 

  14. Giglmayr, J., “Analysis of Stochastic Petri Nets by the Decomposition of the Transition Rate Matrix,” NTZ Archiv (Germany), Part 1: Vol. 9, No. 5, May 1987, pp. 115–120; Part 2: Vol. 9, No. 6, June 1987, pp. 147–152.

    Google Scholar 

  15. Hatono, I., K. Yamagata and H. Tamura, “Modeling and On-Line Scheduling of Flexible Manufacturing Systems Using Stochastic Petri Nets,” IEEE Trans. on Software Eng., Vol. 17, No. 2, Feb. 1991, pp. 126–132.

    Google Scholar 

  16. Jacobson, P.A. and E.D. Lazouska, “Analyzing Queueing Networks with Simultaneous Resource Possession,” Communications of the ACM, Vol. 25, No. 2, Feb., 1982, pp. 142–151.

    Google Scholar 

  17. Johnsonbaugh, R. and T. Murata, “Additional Methods for Reduction and Expansion of Marked Graphs,” IEEE Trans. on Circuits and Systems, Vol. CAS-28, No. 10, Oct. 1981, pp. 1009–1014.

    Google Scholar 

  18. Lazar, A.A. and T.G. Robertazzi, “Markovian Petri Net Protocols with Product Form Solution,” The 6th Annual Conference on Computer Communications, IEEE INFOCOM'87, San Francisco, California, March 31–April 2, 1987.

    Google Scholar 

  19. Lee, K.H. and J. Favrel, “Hierarchical Reduction Method for Analysis and Decomposition of Petri Nets,” IEEE Trans. on Systems, Man, and Cybernetics., Vol. SMC-15, No. 2, March/April 1985, pp. 272–280.

    Google Scholar 

  20. Lee, K.H., J. Favrel and P. Baptiste, “Generalized Petri Net Reduction Methods,” IEEE Trans. on Systems, Man, and Cybernetics., Vol. SMC-17, No. 2, March/April 1987, pp. 297–303.

    Google Scholar 

  21. Li, Y. and C.M. Woodside, “Product Form Stochastic Petri Nets and Norton's Aggregation,” Technical Report, 1989.

    Google Scholar 

  22. Li, Y. and C.M. Woodside, “Iterative Decomposition and Aggregation of Stochastic Marked Graph Petri Nets,” The 12th International Conference on Application and Theory of Petri Nets, Aarhus, Denmark, June 26–28, 1991, pp. 257–275.

    Google Scholar 

  23. Lien, Y.E., “Termination Properties of Generalized Petri Nets,” SIAM J. Comput., Vol. 5, No. 2, June 1976, pp. 251–265.

    Google Scholar 

  24. Marsan, M.A. and G. Balbo, “A Class of Generalized Stochastic Petri Nets for the Performance Evaluation of Multiprocessor Systems,” ACM Trans. on Computer Systems, Vol. 2, No. 2, May 1984, pp. 93–122.

    Google Scholar 

  25. Marsan, M.A., G. Balbo, G. Chiola and S. Donatelli, “On the Product-Form Solution of A Class of Multiple-Bus Multiprocessor System Models,” The Journal of Systems and Software, Vol. 1, No. 2, 1986, pp. 117–124.

    Google Scholar 

  26. Marsan, M.A., G. Balbo, G. Chiola and G. Conte, “Generalized Stochastic Petri Nets Revisited: Random Switches and Priorities,” International Workshop on Petri Nets and Performance Models, Madison, Wisconsin, August 24–26, 1987, pp. 44–53.

    Google Scholar 

  27. Marsan, M.A. and G. Chiola, “On Petri Nets with Deterministic and Exponentially Distributed Firing Times,” Advances in Petri Nets 1987, Lecture Notes in Computer Science 266, Springer-Verlag, 1987, pp. 132–145.

    Google Scholar 

  28. Molloy, M.K., “Performance Analysis Using Stochastic Petri Nets,” IEEE Trans. on Computers, Vol. C-31, No. 9, Sept. 1982, pp. 913–917.

    Google Scholar 

  29. Murata, T. and J.Y. Koh, “Reduction and Expansion of Live and Safe Marked Graphs,” IEEE Trans. on Circuits and Systems, Vol. CAS-27, No. 1, Jan. 1980, pp. 68–70.

    Google Scholar 

  30. Murata, T., “Synthesis of Decision-Free Concurrent Systems for Prescribed Resources and Performance,” IEEE Trans. on Software Eng., Vol. SE-6, No. 6, Nov. 1980, pp. 525–529.

    Google Scholar 

  31. Murata, T., “Petri Nets: Properties, Analysis and Applications,” Proceedings of the IEEE, Vol. 77, No. 4, April 1989, 541–580.

    Google Scholar 

  32. Peterson, J.L., “Petri Net Theory and the Modeling of Systems,” Prentice-Hall, Inc., Englewood Cliffs, N.J., 1981.

    Google Scholar 

  33. Silva, M. and R. Valette, “Petri Nets and Flexible Manufacturing,” Advances in Petri Nets 1989, Lecture Notes in Computer Science, Vol. 424, Springer-Verlag, 1989, pp. 374–417.

    Google Scholar 

  34. Song, J-S., S. Satoh and C.V. Ramamoorthy, “The Abstraction of Petri Net,” Telcon 87, Vol. 2 of 3, August 25–28, 1987, Seoul, Korea, pp. 467–471.

    Google Scholar 

  35. Suzuki, T., S.M. Shatz and T. Murata, “A Protocol Modeling and Verification Approach Based on a Specification Language and Petri Nets,” IEEE Trans. on Software Eng., Vol. 16, No. 5, May 1990, pp. 523–536.

    Google Scholar 

  36. Woodside, C.M., J.E. Neilson, J.W. Miernik, D.C. Petriu and R. Constantin, “Performance of Concurrent Rendezvous Systems with Complex Pipeline Structures,” Technical Report, Dec. 1988.

    Google Scholar 

  37. Woodside, C.M., “Throughput Calculation for Basic Stochastic Rendezvous Networks,” Performance Evaluation, No. 9, 1988/89, pp. 143–160.

    Google Scholar 

  38. Woodside, C.M. and Y. Li, “Performance Petri Net Analysis of Communications Protocol Software by Delay-Equivalent Aggregation,” The 4th International Workshop of Petri Nets and Performance Models, Melbourne, Australia, December 2–5, 1991, pp. 64–73.

    Google Scholar 

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Authors and Affiliations

  1. Telecommunications Research Institute of Ontario Department of Systems and Computer Engineering, Carleton University, K1S 5B6, Ottawa, Ontario, Canada

    Yao Li & C. Murray Woodside

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  1. Yao Li
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  2. C. Murray Woodside
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Grzegorz Rozenberg

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© 1993 Springer-Verlag Berlin Heidelberg

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Li, Y., Woodside, C.M. (1993). Iterative decomposition and aggregation of Stochastic marked graph Petri nets. In: Rozenberg, G. (eds) Advances in Petri Nets 1993. ICATPN 1991. Lecture Notes in Computer Science, vol 674. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56689-9_50

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  • DOI: https://doi.org/10.1007/3-540-56689-9_50

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