Abstract
This paper presents a framework to decompose a single GSPN model into a set of small interacting models. This decomposition technique can be applied to any GSPN model with a finite set of tangible markings and a generalized tensor algebra (Kronecker) representation can be produced automatically. The numerical impact of all the possible decompositions obtained by our technique is discussed. To do so we draw the comparison of the results for some practical examples. Finally, we present all the computational gains achieved by our technique, as well as the future extensions of this concept for other structured formalisms.
This work was partially funded by CNPq/Brazil.
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Ajmone-Marsan, M., Balbo, G., Chiola, G., Conte, G., Donatelli, S., Franceschinis, G.: An Introduction to Generalized Stochastic Petri Nets. Microelectronics and Reliability 31(4), 699–725 (1991)
Ajmone-Marsan, M., Conte, G., Balbo, G.: A Class of Generalized Stochastic Petri Nets for the Performance Evaluation of Multiprocessor Systems. ACM Transactions on Computer Systems 2(2), 93–122 (1984)
Amoia, V., De Micheli, G., Santomauro, M.: Computer-Oriented Formulation of Transition-Rate Matrices via Kronecker Algebra. IEEE Transactions on Reliability R-30(2), 123–132 (1981)
Bellman, R.: Introduction to Matrix Analysis. McGraw-Hill, New York (1960)
Benoit, A., Brenner, L., Fernandes, P., Plateau, B.: Aggregation of Stochastic Automata Networks with replicas. Linear Algebra and its Applications 386, 111–136 (2004)
Benoit, A., Brenner, L., Fernandes, P., Plateau, B., Stewart, W.J.: The PEPS Software Tool. In: Kemper, P., Sanders, W.H. (eds.) TOOLS 2003. LNCS, vol. 2794, pp. 98–115. Springer, Heidelberg (2003)
Brenner, L., Fernandes, P., Sales, A.: The Need for and the Advantages of Generalized Tensor Algebra for Kronecker Structured Representations. International Journal of Simulation: Systems, Science & Technology 6(3-4), 52–60 (2005)
Brewer, J.W.: Kronecker Products and Matrix Calculus in System Theory. IEEE Transactions on Circuits and Systems CAS-25(9), 772–780 (1978)
Buchholz, P., Ciardo, G., Donatelli, S., Kemper, P.: Complexity of memory-efficient Kronecker operations with applications to the solution of Markov models. INFORMS Journal on Computing 13(3), 203–222 (2000)
Buchholz, P., Dayar, T.: Block SOR for Kronecker structured representations. Linear Algebra and its Applications 386, 83–109 (2004)
Buchholz, P., Kemper, P.: Hierarchical reachability graph generation for Petri nets. Formal Methods in Systems Design 21(3), 281–315 (2002)
Ciardo, G., Forno, M., Grieco, P.L.E., Miner, A.S.: Comparing implicit representations of large CTMCs. In: 4th International Conference on the Numerical Solution of Markov Chains, Urbana, IL, USA, September 2003, pp. 323–327 (2003)
Ciardo, G., Jones, R.L., Miner, A.S., Siminiceanu, R.: SMART: Stochastic Model Analyzer for Reliability and Timing. In: Tools of Aachen 2001 International Multiconference on Measurement, Modelling and Evaluation of Computer-Communication Systems, Aachen, Germany, September 2001, pp. 29–34 (2001)
Ciardo, G., Trivedi, K.S.: A Decomposition Approach for Stochastic Petri Nets Models. In: Proceedings of the 4th International Workshop Petri Nets and Performance Models, Melbourne, Australia, December 1991, pp. 74–83. IEEE Computer Society Press, Los Alamitos (1991)
Davio, M.: Kronecker Products and Shuffle Algebra. IEEE Transactions on Computers C-30(2), 116–125 (1981)
Donatelli, S.: Superposed stochastic automata: a class of stochastic Petri nets with parallel solution and distributed state space. Performance Evaluation 18, 21–36 (1993)
Donatelli, S.: Superposed generalized stochastic Petri nets: definition and efficient solution. In: Valette, R. (ed.) Proceedings of the 15th International Conference on Applications and Theory of Petri Nets, pp. 258–277. Springer, Heidelberg (1994)
Fernandes, P., Plateau, B., Stewart, W.J.: Efficient descriptor - Vector multiplication in Stochastic Automata Networks. Journal of the ACM 45(3), 381–414 (1998)
Hillston, J., Kloul, L.: An Efficient Kronecker Representation for PEPA models. In: de Alfaro, L., Gilmore, S. (eds.) Proceedings of the first joint PAPM-PROBMIV Workshop), Aachen, Germany, September 2001, pp. 120–135. Springer, Heidelberg (2001)
Miner, A.S.: Data Structures for the Analysis of Large Structured Markov Models. PhD thesis, The College of William and Mary, Williamsburg, VA (2000)
Miner, A.S.: Efficient solution of GSPNs using Canonical Matrix Diagrams. In: 9th International Workshop on Petri Nets and Performance Models (PNPM 2001), Aachen, Germany, September 2001, pp. 101–110. IEEE Computer Society Press, Los Alamitos (2001)
Miner, A.S., Ciardo, G.: Efficient Reachability Set Generation and Storage Using Decision Diagrams. In: Donatelli, S., Kleijn, J. (eds.) ICATPN 1999. LNCS, vol. 1639, pp. 6–25. Springer, Heidelberg (1999)
Miner, A.S., Ciardo, G., Donatelli, S.: Using the exact state space of a Markov model to compute approximate stationary measures. In: Proceedings of the 2000 ACM SIGMETRICS Conference on Measurements and Modeling of Computer Systems, Santa Clara, California, USA, June 2000, pp. 207–216. ACM Press, New York (2000)
Murata, T.: Petri nets: Properties, analysis and applications. Proceedings of the IEEE 77(4), 541–580 (1989)
Plateau, B.: On the stochastic structure of parallelism and synchronization models for distributed algorithms. In: Proceedings of the 1985 ACM SIGMETRICS conference on Measurements and Modeling of Computer Systems, Austin, Texas, USA, pp. 147–154. ACM Press, New York (1985)
Plateau, B., Atif, K.: Stochastic Automata Networks for modelling parallel systems. IEEE Transactions on Software Engineering 17(10), 1093–1108 (1991)
Reisig, W.: Petri nets: an introduction. Springer, Heidelberg (1985)
Stewart, W.J.: Introduction to the numerical solution of Markov chains. Princeton University Press, Princeton (1994)
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Brenner, L., Fernandes, P., Sales, A., Webber, T. (2005). A Framework to Decompose GSPN Models. In: Ciardo, G., Darondeau, P. (eds) Applications and Theory of Petri Nets 2005. ICATPN 2005. Lecture Notes in Computer Science, vol 3536. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494744_9
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DOI: https://doi.org/10.1007/11494744_9
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