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A Characterization of the Stochastic Process Underlying a Stochastic Petri Net
July 1994 (vol. 20 no. 7)
pp. 506-515

Stochastic Petri nets (SPN's) with generally distributed firing times can model a large class of systems, but simulation is the only feasible approach for their solution. We explore a hierarchy of SPN classes where modeling power is reduced in exchange for an increasingly efficient solution. Generalized stochastic Petri nets (GSPN's), deterministic and stochastic Petri nets (DSPN's), semi-Markovian stochastic Petri nets (SM-SPN's), timed Petri nets (TPN's), and generalized timed Petri nets (GTPN's) are particular entries in our hierarchy. Additional classes of SPN's for which we show how to compute an analytical solution are obtained by the method of the embedded Markov chain (DSPN's are just one example in this class) and state discretization, which we apply not only to the continuous-time case (PH-type distributions), but also to the discrete case.

[1] M. K. Molloy, "Performance analysis using stochastic Petri nets,"IEEE Trans. Comp., vol. 31, no. 9, pp. 913-917, Sept. 1982.
[2] S. Natkin, "Reseaux de Petri stochastiques," These de docteur ingeneur, CNAM-Paris, Paris, France, June 1980.
[3] F. J. W. Symons, "Modeling and analysis of communication protocols using numerical Petri nets," Ph.D. thesis, Univ. of Essex, Great Britain, 1978.
[4] M. Ajmone Marsan, G. Balbo, and G. Conte, "A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems,"ACM Trans. Comput. Syst., vol. 2, pp. 93-122, May 1984.
[5] J. F. Meyer, A. Movaghar, and W. H. Sanders, "Stochastic activity networks: Structure, behavior, and application," inProc. Int. Workshop Timed Petri Nets, Torino, Italy, July 1985, pp. 106-115.
[6] G. Ciardo, A. Blakemore, P. F. Jr. Chimento, J. K. Muppala, and K. S. Trivedi, "Automated generation and analysis of Markov reward models using Stochastic reward nets," inLinear Algebra, Markov Chains, and Queueing Models, C. Meyer and R. J. Plemmons, Eds., vol. 48 ofIMA Volumes in Mathematics and its Applications. New York: Springer-Verlag, 1993, pp. 145-191.
[7] J. B. Dugan, K. S. Trivedi, R. M. Geist, and V. F. Nicola, "Extended stochastic petri nets: Applications and analysis," inProc. 10th Int. Symp. Comput. Performance (PERFORMANCE 84), Dec. 1984, pp. 507-520.
[8] G. Ciardo, "Analysis of large stochastic Petri net models," Ph.D. thesis, Duke Univ., Durham, NC, 1989.
[9] M. Ajmone Marsan and G. Chiola, "On Petri nets with deterministic and exponentially distributed firing times," inAdvances in Petri Nets 1987, Lecture Notes in Computer Science 266, G. Rozenberg, Ed. New York: Springer, 1987, pp. 132-145.
[10] C. Lindemann, "DSPNexpress: A software package for the efficient solution of deterministic and stochastic Petri nets,"Perform. Evul., vol. 21, Feb. 1995 (to appear).
[11] W. L. Zuberek, "Timed Petri nets definitions, properties, and applications,"Microelectronics and Reliability, vol. 31, pp. 627-644, 1991.
[12] M. A. Holliday and M. K. Vernon, "A generalized timed Petri net model for performance analysis,"IEEE Trans. Software Eng., vol. SE-13, no. 12, pp. 1297-1310, Dec. 1987.
[13] R. German and C. Lindemann, "Analysis of stochastic Petri nets by the method of supplementary variables," inPerformance '93, G. Iazeolla and S. S. Lavenberg, Eds. Amsterdam: North-Holland, Sept. 1993.
[14] H. Choi, V. G. Kulkarni, and K. S. Trivedi, "Markov regenerative stochastic Petri nets," in16th IFIP W.G. 7.3 Int. Symp. on Computer Performance Modelling, Measurement and Evaluation (Performance'93), (Rome, Italy), Sept. 1993.
[15] H. Choi, V. G. Kulkarni, and K. S. Trivedi, "Transient analysis of deterministic and stochastic Petri nets," inApplication and Theory of Petri Nets 1993, Lecture Notes in Computer Science, M. Ajmone Marsan, Ed. New York: Springer-Verlag, 1993, vol. 691, pp. 166-185.
[16] P. J. Haas and G. S. Shedler, "Regenerative stochastic Petri nets,"Perf. Eval., vol. 6, pp. 189-204, 1986.
[17] W. Whitt, "Continuity of generalized semi-Markov processes,"Mathematics of Operat. Res., vol. 5, no. 4, pp. 494-501, Nov. 1980.
[18] P. J. Haas and G. S. Shedler, "Stochastic Petri net representation of discrete event simulations,"IEEE Trans. Software Eng., vol. 15, no. 4, pp. 381-393, Apr. 1989.
[19] M. K. Molloy, "Discrete time stochastic Petri nets,"IEEE Trans. Software Eng., vol. SE-11, Apr. 1985.
[20] 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 Trans. Software Eng., vol. 15, no. 7, pp. 832-846, July 1989.
[21] G. Ciardo and C. Lindemann. "Analysis of deterministic and stochastic Petri nets," inProc. 5th Int. Workshop on Petri Nets and Perform. Models, Toulouse, France, 1993, pp. 160-169.
[22] D. Gross and D. Miller, "The randomization technique as a modeling tool and solution procedure for transient Markov processes,"Oper. Res., vol. 32, no. 2, pp. 926-944, Mar.-Apr. 1984.
[23] E.Çinlar,Introduction to Stochastic Processes. Englewood Cliffs. NJ: Prentice-Hall, 1975.
[24] R. German, "Analysis of stochastic Petri nets with nonexponentially distributed firing times," Ph.D. thesis, Technical Univ. of Berlin, Berlin, Germany, 1994.
[25] C. Lindemann and R. German, "Modeling discrete event systems with state-dependent deterministic service times,"Discrete Event Dynamic Syst.: Theory and Applicat., vol. 3, pp. 249-270, July 1993.

Index Terms:
Petri nets; stochastic processes; Markov processes; stochastic process; stochastic Petri net; distributed firing times; simulation; SPN classes; modeling power; generalized stochastic Petri nets; stochastic Petri nets; deterministic Petri nets; semiMarkovian stochastic Petri nets; timed Petri nets; generalized timed Petri nets; embedded Markov chain; state discretization; continuous-time case; PH-type distributions
Citation:
G. Ciardo, R. German, C. Lindemann, "A Characterization of the Stochastic Process Underlying a Stochastic Petri Net," IEEE Transactions on Software Engineering, vol. 20, no. 7, pp. 506-515, July 1994, doi:10.1109/32.297939
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