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Performance Analysis of Stochastic Timed Petri Nets Using Linear Programming Approach
November 1998 (vol. 24 no. 11)
pp. 1014-1030

Abstract—Stochastic timed Petri nets are a useful tool in performance analysis of concurrent systems such as parallel computers, communication networks, and flexible manufacturing systems. In general, performance measures of stochastic timed Petri nets are difficult to obtain for practical problems due to their sizes. In this paper, we provide a method to compute efficiently upper and lower bounds for the throughputs and mean token numbers for a large class of stochastic timed Petri nets. Our approach is based on uniformization technique and linear programming.

[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] M.Ajmone Marsan,G. Balbo,, and G. Conte,“A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems,” ACM Trans. Computer Systems, pp. 93-122, vol. 2, no. 2, May 1984.
[3] M. Ajmone Marsan, G. Balbo, and G. Conte, Performance Models of Multiprocessor Systems.Cambridge, Mass.: MIT Press, 1986.
[4] F. Baccelli, "Ergodic Theory of Stochastic Petri Nets," The Annals of Probability, vol. 20, pp. 375-396, 1992.
[5] F. Baccelli, G. Balbo, R.J. Boucherie, J. Campos, and G. Chiola, "Annotated Bibliography on Stochastic Petri Nets," Performance Evaluation of Parallel and Distributed Systems—Solution Methods, O.J. Boxma and G.M. Koole, eds., CWI Tract 105&106. Amsterdam: CWI, 1994.
[6] F. Baccelli and P. Bremaud, Elements of Queueing Theory.Berlin: Springer-Verlag, 1994.
[7] F. Baccelli and M. Canales, "Parallel Simulation of Stochastic Petri Nets Using Recurrence Equations," ACM Trans. Modeling and Computer Simulation, vol. 3, no. 1, pp. 20-41, Jan. 1993.
[8] F. Baccelli, G. Cohen, and B. Gaujal, "Recursive Equations and Basic Properties of Timed Petri Nets," J. Discrete Event Systems, vol. 1, pp. 415-439, 1992.
[9] F. Baccelli, S. Foss, and B. Gaujal, "Free Choice Nets, An Algebraic Approach," IEEE Trans, Automatic Control, vol. 41, pp. 1,751-1,778, 1996.
[10] F. Baccelli, A. Jean-Marie, and Z. Liu, "A Survey on Solution Methods for Task Graph Models," Second QMIPS Workshop, N. Götz, U. Herzog, and M. Rettelbach, eds., vol. 26, no. 14, pp. 163-183, Arbeitsberichte der IMMD, Erlangen, 1993.
[11] Quantitative Methods in Parallel Systems, F. Baccelli, A. Jean-Marie, and I. Mitrani, eds. Springer-Verlag, 1995.
[12] F. Baccelli and P. Konstantopoulos, "Estimates of Cycle Times in Stochastic Petri Nets," Lecture Notes in Control and Information Sciences, vol. 177, pp. 1-21. Springer-Verlag, 1992.
[13] O.J. Boxma, G.M. Koole, and Z. Liu, "Queueing-Theoretic Solution Methods for Models of Parallel and Distributed Systems," Performance Evaluation of Parallel and Distributed Systems—Solution Methods, O.J. Boxma and G.M. Koole, eds., CWI Tract 105&106. Amsterdam: CWI, 1994.
[14] F. Baccelli and Z. Liu, "Comparison Properties of Stochastic Decision Free Petri Nets," IEEE Trans. Automatic Control, vol. 37, pp. 1,905-1,920, 1992.
[15] F. Baccelli, Z. Liu, and M. Silva, "Global and Local Monotonicities of Stochastic Petri Nets," in preparation.
[16] F. Baccelli and A.M. Makowski, "Queueing Models for Systems with Synchronization Constraints," Proc. IEEE, vol. 77, pp. 138-161, 1989.
[17] D. Bertsimas, I. Paschalidis, and J. Tsitsiklis, "Optimization of Multiclass Queueing Networks: Polyhedral and Nonlinear Characterizations of Achievable Performance," Annals of Applied Probability, vol. 4, pp. 43-73, 1994
[18] D. Bertsimas, I. Paschalidis, and J. Tsitsiklis, "Branching Bandits and Klimov's Problem: Achievable Region and Side Constraints," IEEE Trans. Automatic Control, vol. 40, pp. 2,063-2,075, 1995.
[19] R.J. Boucherie, "A Characterisation of Independence for Competing Markov Chains with Applications to Stochastic Petri Nets," Proc. Petri Nets and Performance Models, 1993.
[20] J.P. Buzen, “Computational Algorithms for Closed Queueing Networks with Exponential Servers,” Comm. ACM, vol. 16, no. 9, pp. 527-531, 1973.
[21] 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.
[22] 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.
[23] 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.
[24] J. Campos,B. Sanchez,, and M. Silva,“Throughput lower bounds for Markovian petri nets: Transformation techniques,” Fourth Int’l Workshop of Petri Nets and Performance Models, pp. 322-331,Melbourne, Australia, December2-5, 1991.
[25] J. Campos and M. Silva, "Embedded Product-Form Queueing Networks and the Improvement of Performance Bounds for Petri Net Systems," Performance Evaluation, vol. 18, pp. 3-19, 1993.
[26] K.M. Chandy and C.H. Sauer, "Computational Algorithms for Product Form Queueing Networks," Comm. ACM, vol. 23, pp. 573-583, 1980.
[27] G. Chiola, "GreatSPN 1.5 Software Architecture," Proc. Fifth Int'l Conf. Modeling Techniques and Tools for Computer Performance Evaluation,Torino, Italy, Feb. 1991.
[28] G. Chiola, M. Ajmone Marsan, G. Balbo, and G. Conte, "Generalized Stochastic Petri Nets: A Definition at the Net Level and Its Implications," IEEE Trans. Software Eng., vol. 19, no. 2, pp. 89-107, Feb. 1993.
[29] G. Chiola, C. Anglano, J. Campos, J.M. Colom, and M. Silva, "Operational Analysis of Timed Petri Nets and Application to the Computation of Performance Bounds," Proc. Fifth Int'l Workshop Petri Nets and Performance Models, pp. 128-137,Toulouse, France, Oct. 1993.
[30] G. Chiola, S. Donatelli, and G. Franceschinis, "GSPNs versus SPNs: What is the Actual Role of Immediate Transitions? Proc. Fourth Int'l Workshop Petri Nets and Performance Models, pp. 20-31,Melbourne, Australia, IEEE CS Press, Dec. 1991.
[31] G. Ciardo, J. Muppala, and K. Trivedi, SPNP: Stochastic Petri Net Package Proc. Third Int'l Workshop Petri Nets and Performance Models, pp. 142-151, 1989.
[32] Y. Dallery, Z. Liu, and D. Towsley, "Equivalence, Reversibility and Symmetry Properties in Assembly/Disassembly Networks," J. ACM., vol. 41, no. 5, pp. 903-942, 1994.
[33] Y. Dallery, Z. Liu, and D. Towsley, "Properties of Fork/Join Queueing Networks with Blocking Under Various Operating Mechanisms," IEEE Trans. Robotics and Automation, vol. 13, no. 4, pp. 503-518, Aug. 1997.
[34] J. Desel and J. Esparza, Free-Choice Petri Nets, Cambridge Tracts in Theoretical Computer Science, vol. 40. Cambridge Univ. Press, 1995.
[35] J. B. Dugan, K. S. Trivedi, R. M. Geist, and V. F. Nicola,“Extended stochastic Petri nets: Applications and analysis,”inPerformance '84, E. Gelenbe, Ed. Amsterdam, The Netherlands: North-Holland, 1984, pp. 507–519.
[36] G. Florin and S. Natkin, "Les réseaux de Petri stochastiques," Technique et Science Informatiques, vol. 4, Feb. 1985.
[37] B. Gaujal, "Liveness in Weighted Routed Nets," INRIA Research Report No. 2899, 1996.
[38] C. Hanen and A. Munier, "Cyclic Scheduling on Parallel Processors: An Overview," Scheduling Theory and Its Applications, P. Chretienne et al., eds., John Wiley&Sons, 1995.
[39] A. Jean-Marie, S. Lefebvre-Barbaroux, and Z. Liu, "An Analytical Approach to the Performance Evaluation of Master-Slave Computational Models," Parallel Computing, vol. 24, nos. 4-5, 1998.
[40] N.D. Jones, L.H. Landweber, and Y. E Lien, "Complexity of Some Problems in Petri Nets," Theoretical Computer Science, vol. 4, pp. 277-299, 1977.
[41] J. Keilson, Markov Chain Models|Rarity and Exponentiality. Springer-Verlag, 1979.
[42] S. Kumar and P.R. Kumar, "Performance Bounds for Queueing Networks and Scheduling Policies," IEEE Trans. Automatic Control, vol. 39, pp. 1,600-1,611, 1994.
[43] P.R. Kumar and S.P. Meyn, "Stability of Queueing Networks and Scheduling Policies," IEEE Trans. Automatic Control, vol. 40, pp. 251-260, 1995.
[44] Y. Li and C.M. Woodside, "Complete Decomposition of Stochastic Petri Nets Representing Generalized Service Networks," IEEE Trans. Computers, vol. 44, pp. 577-592, 1995.
[45] Z. Liu, "Performance Bounds for Stochastic Timed Petri Nets," Proc. 16th Int'l Conf. Applications and Theory of Petri Nets,Torino, Italy, 1995.
[46] Z. Liu, "Performance Analysis of Stochastic Timed Petri Nets using Linear Programming Approach," INRIA Research Report No. 2642, 1995.
[47] M.K. Molloy, "Performance Analysis using Stochastic Petri Nets," IEEE Trans. Computers, vol. 31, no. 9, pp. 913-917, Sept. 1982.
[48] T. Murata, “Petri Nets: Properties, Analysis and Application,” Proc. IEEE, vol. 77, no. 4, 1989.
[49] M. Neuts, Matrix-Geometric Solutions in Stochastic Models.Baltimore, Md.: Johns Hopkins, 1981.
[50] M. Reiser and H. Kobayashi, "Queueing Networks with Multiple Closed Chains: Theory and Computational Algorithms." IBM J. Research and Development, vol. 19, pp. 283-294, 1975.
[51] M. Reiser and S. Lavenberg, “Mean-Value Analysis of Closed Multichain Queueing Networks,” J. ACM, vol. 27, no. 2, pp. 313-322, 1980.
[52] S. Stidham, "A Last Word on L =λW," Operations Research, vol. 22, pp. 417-421, 1974.
[53] 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.
[54] E. Teruel and M. Silva, "Well-Formedness of Equal Conflict Systems," Application and Theory of Petri Nets, R. Valette ed., 1994.

Index Terms:
Stochastic timed Petri net, performance bound, throughput, mean token number, uniformization, linear programming.
Citation:
Zhen Liu, "Performance Analysis of Stochastic Timed Petri Nets Using Linear Programming Approach," IEEE Transactions on Software Engineering, vol. 24, no. 11, pp. 1014-1030, Nov. 1998, doi:10.1109/32.730548
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