Efficient solution of GSPNs using canonical matrix diagrams

The solution of a generalized stochastic Petri net (GSPN) is severely restricted by the size of its underlying continuous-time Markov chain. In recent work (G. Ciardo and A.S. Miner, 1999), matrix diagrams built from a Kronecker expression for the transition rate matrix of certain types of GSPNs were shown to allow for more efficient solution; however, the GSPN model requires a special form, so that the transition rate matrix has a Kronecker expression. In this paper, we extend the earlier results to GSPN models with partitioned sets of places. Specifically, we give a more restrictive definition for matrix diagrams and show that the new form is canonical. We then present an algorithm that builds a canonical matrix diagram representation for an arbitrary non-negative matrix, given encodings for the sets of rows and columns. Using this algorithm, a Kronecker expression is not required to construct the matrix diagram. The efficient matrix diagram algorithms for numerical solution presented earlier are still applicable. We apply our technique to several example GSPNs.

[1]  William H. Sanders,et al.  An Efficient Disk-Based Tool for Solving Very Large Markov Models , 1997, Computer Performance Evaluation.

[2]  Gianfranco Ciardo,et al.  Storage Alternatives for Large Structured State Spaces , 1997, Computer Performance Evaluation.

[3]  S TrivediKishor,et al.  A decomposition approach for stochastic reward net models , 1993 .

[4]  Gianfranco Ciardo,et al.  SMART: simulation and Markovian analyzer for reliability and timing , 1996, Proceedings of IEEE International Computer Performance and Dependability Symposium.

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

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

[7]  Brigitte Plateau,et al.  On the stochastic structure of parallelism and synchronization models for distributed algorithms , 1985, SIGMETRICS '85.

[8]  Peter Buchholz,et al.  Numerical analysis of stochastic marked graph nets , 1995, Proceedings 6th 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]  G. Ciardo,et al.  ON THE USE OF KRONECKER OPERATORS FOR THE SOLUTION OF GENERALIZED STOCHASTIC PETRI NETS , 1996 .

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

[12]  Gianfranco Ciardo,et al.  Efficient Reachability Set Generation and Storage Using Decision Diagrams , 1999, ICATPN.

[13]  Paulo Fernandes,et al.  Efficient descriptor-vector multiplications in stochastic automata networks , 1998, JACM.

[14]  William H. Sanders,et al.  An Efficient Disk-Based Tool for Solving Large Markov Models , 1998, Perform. Evaluation.

[15]  William H. Sanders,et al.  Stochastic Activity Networks: Structure, Behavior, and Application , 1985, PNPM.

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

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

[18]  Gianfranco Ciardo,et al.  Saturation: An Efficient Iteration Strategy for Symbolic State-Space Generation , 2001, TACAS.

[19]  M. Siegle,et al.  Multi Terminal Binary Decision Diagrams to Represent and Analyse Continuous Time Markov Chains , 1999 .

[20]  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).

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

[22]  Kishor S. Trivedi,et al.  Modeling Using Stochastic Reward Nets , 1993, MASCOTS.

[23]  Robert K. Brayton,et al.  Algorithms for discrete function manipulation , 1990, 1990 IEEE International Conference on Computer-Aided Design. Digest of Technical Papers.