An adaptive decomposition approach for the analysis of stochastic Petri nets

We present a new approximate solution technique for the numerical analysis of superposed generalized stochastic Petri nets (SGSPNs) and related models. The approach combines numerical iterative solution techniques and fixed point computations using the complete knowledge of state space and generator matrix. In contrast to other approximation methods, the proposed method is adaptive by considering states with a high probability in detail and aggregating states with small probabilities. Probabilities are approximated by the results derived during the iterative solution. Thus, a maximum number of states can be predefined and the presented method automatically aggregates states such that the solution is computed using a vector of a size smaller or equal to the maximum. By means of a non-trivial example it is shown that the approach computes good approximations with a low effort for many models.

[1]  William J. Stewart,et al.  Introduction to the numerical solution of Markov Chains , 1994 .

[2]  Peter Kemper.  Numerical Analysis of Superposed GSPNs , 1996, IEEE Trans. Software Eng..

[3]  B. R. Haverkort.  Matrix-geometric solution of infinite stochastic Petri nets , 1995, Proceedings of 1995 IEEE International Computer Performance and Dependability Symposium.

[4]  Gianfranco Ciardo,et al.  A data structure for the efficient Kronecker solution of GSPNs , 1999, Proceedings 8th International Workshop on Petri Nets and Performance Models (Cat. No.PR00331).

[5]  Brigitte Plateau.  On the stochastic structure of parallelism and synchronization models for distributed algorithms , 1985, SIGMETRICS 1985.

[6]  Manuel Silva Suárez,et al.  Structured Solution of Asynchronously Communicating Stochastic Modules , 1999, IEEE Trans. Software Eng..

[7]  Peter Buchholz.  Iterative Decomposition and Aggregation of Labeled GSPNs , 1998, ICATPN.

[8]  Peter Buchholz,et al.  Hierarchical structuring of superposed GSPNs , 1997, Proceedings of the Seventh International Workshop on Petri Nets and Performance Models.

[9]  Peter Buchholz,et al.  Complexity of Memory-Efficient Kronecker Operations with Applications to the Solution of Markov Models , 2000, INFORMS J. Comput..

[10]  Susanna Donatelli.  Superposed stochastic automata: a class of stochastic Petri nets amenable to parallel solution , 1991, Proceedings of the Fourth International Workshop on Petri Nets and Performance Models PNPM91.

[11]  Edmundo de Souza e Silva,et al.  State space exploration in Markov models , 1992, SIGMETRICS '92/PERFORMANCE '92.

[12]  Peter Buchholz,et al.  On generating a hierarchy for GSPN analysis , 1998, PERV.

[13]  Susanna Donatelli,et al.  Superposed Stochastic Automata: A Class of Stochastic Petri Nets with Parallel Solution and Distributed State Space , 1993, Perform. Evaluation.

[14]  Peter Buchholz,et al.  Structured analysis approaches for large Markov chains , 1999 .

[15]  Kishor S. Trivedi,et al.  A Decomposition Approach for Stochastic Reward Net Models , 1993, Perform. Evaluation.

[16]  Gianfranco Ciardo,et al.  Using the exact state space of a Markov model to compute approximate stationary measures , 2000, SIGMETRICS '00.

[17]  Yao Li,et al.  Complete Decomposition of Stochastic Petri Nets Representing Generalized Service Networks , 1995, IEEE Trans. Computers.

[18]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[19]  Matteo Sereno,et al.  Mean Value Analysis of Stochastic Petri Nets , 1997, Perform. Evaluation.

[20]  Marco Ajmone Marsan,et al.  Modelling with Generalized Stochastic Petri Nets , 1995, PERV.

[21]  P. Buchholz,et al.  Complexity of Kronecker Operations on Sparse Matrices with Applications to the Solution of Markov Models , 1997 .

[22]  Tugrul Dayar,et al.  Comparison of Partitioning Techniques for Two-Level Iterative Solvers on Large, Sparse Markov Chains , 1999, SIAM J. Sci. Comput..