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Time Scale Decomposition of a Class of Generalized Stochastic Petri Net Models
June 1989 (vol. 15 no. 6)
pp. 809-820

A time-scale decomposition (TSD) algorithm of a class of generalized stochastic Petri net (GSPN) models of systems comprising activities whose duration differ by orders of magnitude is presented. The GSPN model of a system can be decomposed into a hierarchical sequence of aggregated subnets, each of which is valid at a certain time scale. These smaller subnets are solved in isolation and their solutions are combined to get the solution of the whole system. A degradable multiprocessor system which would be intractable using conventional techniques, is analyzed using TSD. The complexity of the TSD algorithm can be orders of magnitude smaller without any significant loss in the accuracy of the result. In general, the error due to aggregation is proportional to the maximum degree of coupling between aggregates. An expression of the error due to aggregation is also given in terms of the ratio of fast and slow transitions in the GSPN model. The algorithm is easy to use and can be easily automated.

[1] 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.
[2] M. A. Marsan, G. Chiola, and G. Conte, "Performance models of task synchronization in computer systems," inProc. Int. Conf. Computers and Applications, IEEE, Beijing, China, June 1984.
[3] S. Natkin, "Timed and stochastic Petri nets: From the validation to the performance of synchronization schemes," inProc. Int. Workshop Timed Petri Nets, Torino, Italy, July 1985.
[4] E. Gressier, "A stochastic Petric net model for ethernet," inProc. Int. Workshop on Timed Petri Nets, Torino, Italy, July 1985.
[5] M. K. Molloy, "On the integration of delay and throughput measures in distributed processing models," Ph.D. dissertation, Dep. Comput. Sci., Univ. of California, Los Angles, 1981.
[6] M. Ajmone Marsan, G. Balbo, G. Chiola, and G. Conte, "Generalized stochastic Petri nets revisited: Random switches and priorities," inProc. Int. Workshop Petri Nets and Performance Models, Madison, WI, IEEE Computer Society Press, Aug. 1987, pp. 44-53.
[7] H. H. Ammar, Y. F. Huang, and R. W. Liu, "Hierarchical models for systems reliability maintainability, and availability,"IEEE Trans. Circuits Syst., vol. CAS-34, no. 6, June 1987.
[8] H. H. Ammar and R. W. Liu, "Analysis of the generalized stochastic Petri nets by state aggregation," inProc. Int. Workshop Timed Petri Nets, Torino, Italy, July 1985.
[9] H. H. Ammar and R. W. Liu, "Hierarchical models for parallel processing systems using the generalized stochastic Petri nets," inProc. 1984 Int. Conf. Parallel Processing, IEEE, 1984.
[10] H. H. Ammar, Y. F. Huang, and R. W. Liu, "Analysis by aggregation of the generalized stochastic petri nets with application to reliability/maintainability and fault diagnosis," inProc. Int. Symp. Circuits and Systems (ISCAS 85), Kyoto, Japan, June 1985.
[11] C. Petri, "Communication with automata," RADC, Griffis Air Force Base, NY, Tech. Rep. RADC-TR-65-377, vol. 1, Jan. 1966.
[12] J. L. Peterson,Petri Net Theory and the Modeling of Systems. Englewood Cliffs, NJ: Prentice-Hall, 1981.
[13] J. L. Peterson, "Petri nets,"ACM Comput. Surveys, vol. 9, no. 3, pp. 223-252, Sept. 1977.
[14] M. K. Molloy, "Performance analysis using stochastic Petri nets,"IEEE Trans. Comput., vol. C-31, no. 9, Sept. 1982.
[15] H. H. Ammar, "Analytical models for parallel processing," Ph.D. dissertation, Univ. Notre Dame, 1985.
[16] P. J. Courtois,Decomposability: Queueing and Computer Systems Applications. New York: Academic, 1977.
[17] P. J. Courtois, "On time and space decomposition of complex structures,"Commun. ACM, vol. 28, pp. 590-603, 1985.
[18] M. Goderch, A. S. Willsky, S. S. Sastry, and D. A. Cantanon, "Hierarchical aggregation of singularly perturbed finite state Markov pro cess,"Stochastic, vol. 8, 1983.
[19] A. Brandwajan, "Equivalence and decomposition in queuing systems a unified approach,"Perform. Eval., vol. 5, 1985.
[20] A. Bobbio and K. S. Trivedi, "An aggregation technique for the transient analysis of stiff Markov chains,"IEEE Trans. Comput., vol. C-35, pp. 803-814, Sept. 1986.
[21] 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.
[22] M. Ajmone Marsan, G. Balbao, and G. Conte,Performance Modeling of Multiprocessor Systems. Cambridge, MA: MIT Press, 1986.
[23] H. A. Simon and A. Ando, "Aggregation of variables in dynamic systems,"Econometrica, vol. 29, 1961.
[24] K. S. Trivedi, and J. B. Dugan, "Hybrid reliability modeling of fault-tolerant computer systems,"Comput. Elec. Eng., vol. 11, no. 2-3, 1984.
[25] K. S. Trivedi, and R. M. Geist, "Decomposition in reliability analysis of fault-tolerant systems,"IEEE Trans. Rel., vol. R-32, no. 5, Dec. 1983.
[26] A. Bobbio, A. Cumani, and R. Del Bello, "Reduced Markovian representation of stochastic Petri net models,"Syst. Sci., vol. 10, no. 2, 1984.
[27] P. J. Courtois and P. Semal, "Computable bounds on conditional steady-state probabilities in large markov chains and queueing models,"IEEE J. Select. Areas Commun., vol. 4, no. 6, pp. 926-937, Sept. 1986.
[28] M. A. Marsan, A. Bobbio, G. Conte, and A. Cumani, "Performance analysis of degradable multiprocessor systems using generalized stochastic Petri nets,"IEEE Comput. Soc. Distrib. Processing T. C. Newslett., vol. 6, no. SI-1, 1984.

Index Terms:
generalized stochastic Petri net models; time-scale decomposition; GSPN model; hierarchical sequence; aggregated subnets; degradable multiprocessor system; complexity; TSD algorithm; aggregation; coupling; slow transitions; multiprocessing systems; performance evaluation; Petri nets; stochastic processes
H.H. Ammar, S.M.R. Islam, "Time Scale Decomposition of a Class of Generalized Stochastic Petri Net Models," IEEE Transactions on Software Engineering, vol. 15, no. 6, pp. 809-820, June 1989, doi:10.1109/32.24734
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