This Article 
 Bibliographic References 
 Add to: 
Approximate Mean Value Analysis for Stochastic Marked Graphs
September 1996 (vol. 22 no. 9)
pp. 654-664

Abstract—An iterative technique for the computation of approximate performance indices of a class of stochastic Petri nets models is presented. The proposed technique is derived from the mean value analysis algorithm for product form solution stochastic Petri nets. In this paper, we apply the approximation technique to stochastic marked graphs. In principle, the proposed technique can be used for other stochastic Petri net subclasses in the paper, some of these possible applications are presented. Several examples are presented in order to validate the approximate results.

[1] M.A. 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, pp. 832-846, 1989.
[2] G. Balbo, S.C. Bruell, and M. Sereno, "Arrival Theorems for Product-Form Stochastic Petri Nets," Proc. 1994 SIGMETRICS Conf.,Nashville, Tenn., May 1994.
[3] Y. Bard, "Some Extensions to Multiclass Queuing Network Analysis," Performance of Computers, M. Arato, A. Butrimenko, and E. Gelenbe, eds. North-Holland, 1979.
[4] F. Baskett, K.M. Chandy, R.R. Muntz, and R. Palacios, “Open, Closed and Mixed Networks of Queues with Different Classes of Customers,” J. ACM, vol. 22, no. 2, pp. 248-260, 1975.
[5] R.J. Boucherie and M. Sereno, "On the Traffic Equations for Batch Routing Queuing Networks and Stochastic Petri Nets," Technical Report ERCIM-04/94-R032, European Research Consortium for Informatics and Mathematics, 1994.
[6] P. Buchholtz and P. Kemper, "Numerical Analysis of Stochastic Marked Graph Nets," Proc. Sixth Int'l Workshop on Petri Nets and Performance Models,Durham, N.C., Oct., 1995.
[7] J. Campos, G. Chiola, and M. Silva, "Ergodicity and Throughput Bounds of Petri Nets with Unique Consistent Firing Count Vector," IEEE Trans. Software Eng., vol. 17, no. 2, pp. 117-125, Feb. 1991.
[8] J. Campos, G. Chiola, and M. Silva, "Properties and Performance Bounds for Closed Free Choice Synchronized Monocles Queueing Networks," IEEE Trans. Automatic Control, vol. 36, no. 12, pp. 1,368-1,382, Dec. 1991.
[9] J. Campos,J.M. Colom,H. Jungnitz,, and M. Silva,“A general iterative technique for approximate throughput computation of stochastic marked graphs,” Fifth Int’l Workshop of Petri Nets and Performance Models, , pp. 138-147,Toulouse, France, Oct.19-22, 1993.
[10] K.M. Chandy, U. Herzog, and L.S. Woo, "Approximate Analysis of General Queuing Networks," IBM J. Research and Development, vol. 19, no. 1, pp. 43-49, Jan. 1975.
[11] K.M. Chandy and D. Neuse, “Linearizer: A Heuristic Algorithm for Queueing Network Models of Computing Systems,” Comm. ACM, vol. 25, no. 2, pp. 126-134, 1982.
[12] G. Chiola, G. Franceschinis, R. Gaeta, and M. Ribaudo, “GreatSPN 1.7: Graphical Editor and Analyzer for Timed and Stochastic Petri Nets,” Performance Evaluation, vol. 24, nos. 1-2, pp. 47-68, Nov. 1995.
[13] J.L. Coleman, W. Henderson, and P.G. Taylor, "Product Form Equilibrium Distributions and an Algorithm for Classes of Batch Movement Queuing Networks and Stochastic Petri Nets," Technical report, Univ. of Adelaide, 1992.
[14] A.J. Coyle, W. Henderson, C.E.M. Pearce, and P.G. Taylor, "A General Formulation for the Mean-Value Analysis in Product Form Batch-Movement Queuing Networks," Queuing Systems, vol. 16, pp. 363-372, 1994.
[15] J. Desel and J. Esparza, "Reachability in Reversible Free-Choice Systems, Technical Report TUM-I9023, Inst. Fur Informatik, 1990.
[16] M. Di Mascolo, M.Y. Frein, Y. Dallery, and R. David, "Modeling of Kanban Systems Using Petri Nets," Proc. Third ORSA/TIMS Conf. on Flexible Manufacturing Systems, NorthHolland: Elsevier Science 1989.
[17] G. Florin and S. Natkin, "Les reseaux de Petri stochastiques," Technique et Science Informatiques, vol. 4, no. 1, Feb. 1985.
[18] W.J. Gordon and G.F. Newell, "Closed Queuing Systems With Exponential Servers," Operations Research, vol. 15, pp. 254-265, 1967.
[19] W. Henderson, D. Lucic, and P.G. Taylor, "A Net Level Performance Analysis of Stochastic Petri Nets," J. Australian Math. Soc. Series B, vol. 31, pp. 176-187, 1989.
[20] W. Henderson and P.G. Taylor, “Embedded Processes in Stochastic Petri Nets,” IEEE Trans. Software Eng., vol. 17, pp. 108–116, Feb. 1991.
[21] J.R. Jackson, "Jobshop-Like Queuing Systems," Management Science, vol. 10, no. 1, pp. 131-142, Oct. 1963.
[22] A. Jungnitz, H. Desrochers, and M. Silva, "Approximation Techniques for Stochastic Macroplace/Macrotransition Nets," Proc. QMIPS Workshop Stochastic Petri Nets, pp. 118-146, Sophia Antipolis, France, Nov. 1992.
[23] Y. Li and C.M. Woodside,“Iterative decomposition and aggregation of stochastic marked graph Petri nets,” 12th Int’l Conf. Application and Theory of Petri Nets, pp. 257-275,Aarhus, Denmark, June26-28, 1991.
[24] J.D.C. Little, "A Proof of the Queuing Formula l =λw," Operations Research, vol. 9, pp. 383-387, 1961.
[25] M.K. Molloy, "Performance Analysis Using Stochastic Petri Nets," IEEE Trans. Computers, vol. 31, no. 9, pp. 913-917, Sept. 1982.
[26] T. Murata, “Petri Nets: Properties, Analysis and Application,” Proc. IEEE, vol. 77, no. 4, 1989.
[27] K.R. Pattipati, M.M. Kostreva, and J.L. Teele, "Approximate Mean Value Analysis Algorithms of Queuing Networks: Existence, Uniqueness and Convergence Results," J. ACM, vol. 3, pp. 643-673, Jan. 1990.
[28] P.J. Schweitzer, "A Survey of Mean Value Analysis, Its Generalizations, and Applications, for Networks of Queues," Technical Report, Univ. of Rochester, Rochester, N.Y., Dec. 1990.
[29] M. Sereno and G. Balbo, "Mean Value Analysis of Stochastic Petri Nets," Performance Evaluation, vol. 29, no. 1, pp. 35-62, 1997.
[30] M. Sereno and G. Balbo, “Computational Algorithms for Product form Solution Stochastic Petri Nets,” Proc. Fifth Int'l Workshop Petri Nets and Performance Models, pp. 98–107, Oct. 1993.
[31] E. Teruel, P. Chrzastowski-Watchel, J.M. Colom, and M. Silva, "On Weighted T-Systems," Proc. Application and Theory of Petri Nets, K. Jensen, ed., 1992.

Index Terms:
Stochastic Petri nets, marked graphs, computational algorithms, approximate mean value analysis algorithm, approximation techniques.
Matteo Sereno, "Approximate Mean Value Analysis for Stochastic Marked Graphs," IEEE Transactions on Software Engineering, vol. 22, no. 9, pp. 654-664, Sept. 1996, doi:10.1109/32.541436
Usage of this product signifies your acceptance of the Terms of Use.