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Numerical Analysis of Superposed GSPNs
September 1996 (vol. 22 no. 9)
pp. 615-628

Abstract—The numerical analysis of various modeling formalisms profits from a structured representation for the generator matrix Q of the underlying continuous time Markov chain, where Q is described by a sum of tensor (Kronecker) products of much smaller matrices. In this paper we describe such a representation for the class of superposed generalized stochastic Petri nets (SGSPNs), which is less restrictive than in previous work. Furthermore a new iterative analysis algorithm is proposed. It pays special attention to a memory efficient representation of iteration vectors as well as to a memory efficient structured representation of Q. In consequence the new algorithm is able to solve models which have state spaces with several millions of states, where other exact numerical methods become impracticable on a common workstation.

[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. Computer Systems, pp. 93-122, vol. 2, no. 2, May 1984.
[2] V. Amoia, G. De Micheli, and M. Santomauro, "Computer-Oriented Formulation of Transition-Rate Matrices via Kronecker Algebra," IEEE Trans. Reliability, vol. 30, no. 2, pp. 123-132, June 1981.
[3] G. Balbo, G. Chiola, G. Franceschinis, and G. Molinar-Roet, "On the Efficient Construction of the Tangible Reachability Graph of Generalized Stochastic Petri Nets," Proc. Workshop Petri Nets and Performance Models, pp. 136-145. IEEE Computer Soc., 1987.
[4] F. Bause and P. Kemper, "QPN-Tool for Qualitative and Quantitative Analysis of Queueing Petri Nets," G. Haring and G. Kotsis, eds., Computer Performance Evaluation, Modelling Techniques and Tools, Proc. Seventh Int'l Conf.,Vienna, Austria, LNCS 794, pp. 321-334. Springer-Verlag, 1994.
[5] P. Buchholz, "A Hierarchical View of GCSPNs and its Impact on Qualitative and Quantitative Analysis," J. Parallel and Distributed Computing, vol. 15, pp. 207-224, 1992.
[6] G. Chiola, "Compiling Techniques for the Analysis of Stochastic Petri Nets," Proc. Fourth Int'l Conf. Modeling Techniques and Tools, pp. 13-27, 1989.
[7] G. Chiola, "1.5 Software Architecture," G. Balbo and G. Serazzi, eds., Computer Performance Evaluation, pp. 121-136. NorthHolland, 1992.
[8] 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.
[9] 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.
[10] S. Christensen and L. Petrucci, “Modular State Space Analysis of Coloured Petri Nets,” Proc. 16th Int'l Conf. Application and Theory of Petri Nets, June 1995.
[11] 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.
[12] G. Ciardo and K.S. Trivedi,“A decomposition approach for stochastic Petri net models,” Fourth Int’l Workshop of Petri Nets and Performance Models, pp. 74-83,Melbourne, Australia, December2-5, 1991.
[13] M. Davio, "Kronecker Products and Shuffle Algebra," IEEE Trans. Computers, vol. 30, no. 2, pp. 116-125, Feb. 1981.
[14] S. Donatelli, "Superposed Stochastic Automata: A Class of Stochastic Petri Nets with Parallel Solution and Distributed State Space," Performance Evaluation, vol. 18, pp. 21-36, 1993.
[15] S. Donatelli, "Superposed Generalized Stochastic Petri Nets: Definition and Efficient Solution," R. Valette, ed., Proc. 15th Int'l Conf. Applications and Theory of Petri Nets, Lecture Notes in Computer Science 815, pp. 258-277.Berlin, Heidelberg: Springer-Verlag, 1994.
[16] S. Donatelli, M. Ribaudo, and J. Hillston, "A Comparison of Performance Evaluation Process Algebra and Generalized Stochastic Petri Nets," Proc. Sixth Int'l Workshop Petri Nets and Performance Models, pp. 158-168, IEEE CS Press, 1995.
[17] J. Hillston, "Compositional Markovian Modelling using a Process Algebra," W.J. Stewart, ed., Computations with Markov Chains: Proc. Second Int'l Work, Numerical Solution of Markov Chains, pp. 177-196. Kluwer Academic Publishers, 1995.
[18] N. Karmarkar,"A New Polynomial-Time Algorithm for Linear Programming," Combinatorica, vol. 4, pp. 373-395, 1984.
[19] C. Lindemann, “DSPN Express: A Software Package for the Efficient Solution of Deterministic and Stochastic Petri Nets, Performance Evaluation, vol. 22, pp. 3-21, 1995
[20] J. Martinez and M. Silva, "A Simple and Fast Algorithm to Obtain all Invariants of a Generalized Petri Net," C. Girault and W. Reisig, eds., Application and Theory of Petri Nets, Informatik Fachberichte, p. 52, 1982.
[21] T. Murata, “Petri Nets: Properties, Analysis and Application,” Proc. IEEE, vol. 77, no. 4, 1989.
[22] B. Plateau and K. Atif, Stochastic Automata Network for Modeling Parallel Systems IEEE Trans. Software Eng., vol. 17, no. 10, pp. 1093-1108, Oct. 1991.
[23] B. Plateau and J.M. Fourneau, "A Methodology for Solving Markov Models of Parallel Systems," J. Parallel and Distributed Computing, vol. 12, pp. 370-387, 1991.
[24] W.J. Stewart, Introduction to the Numerical Solution of Markov Chains. Princeton Univ. Press, 1994.
[25] W.J. Stewart, K. Atif, and B. Plateau, "The numerical solution of stochastic automata networks," European J. Operatonal Research, vol. 86, pp. 503-525, 1995.
[26] C.M. Woodside and Y. Li,“Performance Petri net analysis of communications protocol software by delay-equivalent aggregation,” Fourth Int’l Workshop Petri Nets and Performance Models, pp. 64-73,Melbourne, Australia, Dec.2-5, 1991.

Index Terms:
Stochastic Petri net, superposed GSPN, Markov process, numerical solution algorithm for steady-state analysis tensor/Kronecker algebra, decomposition, reachability analysis, structured representation.
Peter Kemper, "Numerical Analysis of Superposed GSPNs," IEEE Transactions on Software Engineering, vol. 22, no. 9, pp. 615-628, Sept. 1996, doi:10.1109/32.541433
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