Numerical Analysis of Superposed GSPNs

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 (GSPNs), 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 million states, where other exact numerical methods become impracticable on a common workstation.

[1]  Marc Davio,et al.  Kronecker products and shuffle algebra , 1981, IEEE Transactions on Computers.

[2]  J. Hillston Compositional Markovian Modelling Using a Process Algebra , 1995 .

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

[4]  Giovanni Chiola,et al.  GSPNs versus SPNs: what is the actual role of immediate transitions? , 1991, Proceedings of the Fourth International Workshop on Petri Nets and Performance Models PNPM91.

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

[6]  G. De Micheli,et al.  Computer-Oriented Formulation of Transition-Rate Matrices via Kronecker Algebra , 1981, IEEE Transactions on Reliability.

[7]  Falko Bause,et al.  QPN -Tool for Qualitative and Quantitative Analysis of Queueing Petri Nets , 1994, Computer Performance Evaluation.

[8]  Peter Buchholz,et al.  A Hierarchical View of GCSPNs and Its Impact on Qualitative and Quantitative Analysis , 1992, J. Parallel Distributed Comput..

[9]  Laure Petrucci,et al.  Modular State Space Analysis of Coloured Petri Nets , 1995, Application and Theory of Petri Nets.

[10]  Giovanni Chiola Compiling Techniques for the Analysis of Stochastic Petri Nets , 1989 .

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

[12]  Brigitte Plateau,et al.  Stochastic Automata Network For Modeling Parallel Systems , 1991, IEEE Trans. Software Eng..

[13]  W. Stewart,et al.  The numerical solution of stochastic automata networks , 1995 .

[14]  Marco Ajmone Marsan,et al.  Generalized Stochastic Petri Nets: A Definition at the Net Level and Its Implications , 1993, IEEE Trans. Software Eng..

[15]  Christoph Lindemann,et al.  DSPNexpress: A Software Package for the Efficient Solution of Deterministic and Stochastic Petri Nets , 1993, Perform. Evaluation.

[16]  Kishor S. Trivedi,et al.  SPNP: stochastic Petri net package , 1989, Proceedings of the Third International Workshop on Petri Nets and Performance Models, PNPM89.

[17]  Jean-Michel Fourneau,et al.  A Methodology for Solving Markov Models of Parallel Systems , 1991, J. Parallel Distributed Comput..

[18]  Susanna Donatelli,et al.  Superposed Generalized Stochastic Petri Nets: Definition and Efficient Solution , 1994, Application and Theory of Petri Nets.

[19]  Manuel Silva,et al.  A Simple and Fast Algorithm to Obtain All Invariants of a Generalized Petri Net , 1980, Selected Papers from the First and the Second European Workshop on Application and Theory of Petri Nets.

[20]  Tadao Murata,et al.  Petri nets: Properties, analysis and applications , 1989, Proc. IEEE.

[21]  Kishor S. Trivedi,et al.  A decomposition approach for stochastic Petri net models , 1991, Proceedings of the Fourth International Workshop on Petri Nets and Performance Models PNPM91.

[22]  Susanna Donatelli,et al.  A comparison of performance evaluation process algebra and generalized stochastic Petri nets , 1995, Proceedings 6th International Workshop on Petri Nets and Performance Models.

[23]  Giovanni Chiola,et al.  On the Efficient Construction of the Tangible Reachability Graph of Generalized Stochastic Petri Nets , 1987, PNPM.

[24]  Marco Ajmone Marsan,et al.  A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems , 1984, TOCS.

[25]  Yao Li,et al.  Performance Petri net analysis of communications protocol software by delay-equivalent aggregation , 1991, Proceedings of the Fourth International Workshop on Petri Nets and Performance Models PNPM91.

[26]  Ajmone MarsanMarco,et al.  A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems , 1984 .

[27]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..