Evaluation of Iterative Methods on Large Markov Chains Generated by GSPN Models

GSPN models often generate high cardinality state spaces, whose analysis requires the solution of very large and sparse nonsymmetric linear systems for the associated Markov chain. In this paper we report the results of an empirical investigation of three well-known iterative methods for linear systems of equations: Gauss-Seidel, GMRES, and Bi-CGstab. We evaluate these methods on several large Markov chains generated by GSPN models proposed in the literature. Issues addressed include state space characterization, problem conditioning, numerical accuracy and stability, and computation time. Results show that increased attention should be paid to the numerical issues underlying performance and reliability analyses when dealing with large state spaces.

[1]  Gianni Conte,et al.  GSPN models of concurrent architectures with mesh topology , 1991, Proceedings of the Fourth International Workshop on Petri Nets and Performance Models PNPM91.

[2]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[3]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[4]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[5]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

[6]  Yousef Saad,et al.  Numerical Methods in Markov Chain Modelling , 1996 .

[7]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

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

[9]  Gianni Conte,et al.  Analysis of large GSPN models: a distributed solution tool , 1997, Proceedings of the Seventh International Workshop on Petri Nets and Performance Models.

[10]  G. Conte,et al.  Parallel State Space Exploration for GSPN Models , 1995, Application and Theory of Petri Nets.

[11]  Souheib Baarir,et al.  The GreatSPN tool: recent enhancements , 2009, PERV.

[12]  William H. Sanders,et al.  "On-the-Fly'' Solution Techniques for Stochastic Petri Nets and Extensions , 1998, IEEE Trans. Software Eng..

[13]  Peter Kemper,et al.  Transient analysis of superposed GSPNs , 1997, Proceedings of the Seventh International Workshop on Petri Nets and Performance Models.

[14]  Marco Ajmone Marsan,et al.  GSPN Models of Markovian Multiserver Multiqueue Systems , 1990, Perform. Evaluation.

[15]  Gianni Conte,et al.  A Distributed Algorithm for GSPN Reachability Graph Generation , 2001, J. Parallel Distributed Comput..

[16]  Giovanni Chiola,et al.  GreatSPN 1.7: Graphical Editor and Analyzer for Timed and Stochastic Petri Nets , 1995, Perform. Evaluation.

[17]  G. Ciardo,et al.  ON THE USE OF KRONECKER OPERATORS FOR THE SOLUTION OF GENERALIZED STOCHASTIC PETRI NETS , 1996 .

[18]  Hiromu Tanimoto.  Factory Automation: An Automatic Assembly Line for the Manufacture of Printers , 1984, Computer.

[19]  Per Brinch Hansen,et al.  Special Feature A Keynote Address on Concurrent Programming , 1979, Computer.