Efficient descriptor-vector multiplications in stochastic automata networks

This paper examines numerical issues in computing solutions to networks of stochastic automata. It is well-known that when the matrices that represent the automata contain only constant values, the cost of performing the operation basic to all iterative solution methods, that of matrix-vector multiply, is given by <display-equation> <fd> <fl><g>r</g><inf>N</inf>=<pr align="c"><ll>i=1</ll><ul>N</ul></pr> n<inf>i</inf>×<sum align="c"><ll>i=1</ll><ul>N</ul></sum> n<inf>i</inf></fl> </fd> </display-equation> where <?Pub Fmt italic>n<subscrpt>i</subscrpt><?Pub Fmt /italic> is the number of states in the <italic>i</italic>th automaton and <italic>N</italic> is the number of automata in the network. We introduce the concept of a generalized tensor product and prove a number of lemmas concerning this product. The result of these lemmas allows us to show that this relatively small number of operations is sufficient in many practical cases of interest in which the automata contain functional and not simply constant transitions. Furthermore, we show how the automata should be ordered to achieve this.

[1]  William Henderson,et al.  Aggregation and Disaggregation Through Insensitivity in Stochastic Petri Nets , 1993, Perform. Evaluation.

[2]  Giuliana Franceschinis,et al.  Computing bounds for the performance indices of quasi-lumpable stochastic well-formed nets , 1993, Proceedings of 5th International Workshop on Petri Nets and Performance Models.

[3]  Jean-Michel Fourneau,et al.  PEPS: A Package for Solving Complex Markov Models of Parallel Systems , 1989 .

[4]  Brigitte Plateau,et al.  Numerical Issues for Stochastic Automata Networks , 1996 .

[5]  Abderezak Touzene.  Résolution des modèles markoviens sur machines à mémoires distribuées. (Markovian models resolution on distributed memory machines) , 1992 .

[6]  Matteo Sereno,et al.  Arrival Theorems for Product-Form Stochastic Petri Nets , 1994, SIGMETRICS.

[7]  C. Berge.  Graphes et hypergraphes , 1970 .

[8]  Peter Kemper.  Closing the Gap Between Classical and Tensor Based Iteration Techniques , 1995 .

[9]  Brigitte Plateau,et al.  Stochastic Automata Networks: Product Forms and Iterative Solutions , 1996 .

[10]  Giovanni Chiola,et al.  Stochastic Well-Formed Colored Nets and Symmetric Modeling Applications , 1993, IEEE Trans. Computers.

[11]  H. Hermanns,et al.  Syntax , Semantics , Equivalences , and Axioms for MTIPP y , 1994 .

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

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

[14]  Wayne R. Dyksen,et al.  Efficient vector and parallel manipulation of tensor products , 1996, TOMS.

[15]  William H. Sanders,et al.  Reduced base model construction methods for stochastic activity networks , 1989, Proceedings of the Third International Workshop on Petri Nets and Performance Models, PNPM89.

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

[17]  François Baccelli,et al.  Quantitative Methods in Parallel Systems , 1995, Esprit Basic Research Series.

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

[19]  Richard J. Boucherie.  A Characterization of Independence for Competing Markov Chains with Applications to Stochastic Petri Nets , 1994, IEEE Trans. Software Eng..

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

[21]  Karim Atif.  Modélisation du parallélisme et de la synchronisation , 1992 .

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

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

[24]  Marco Ajmone Marsan,et al.  The Application of EB-Equivalence Rules to the Structural Reduciton of GSPN Models , 1992, J. Parallel Distributed Comput..

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

[26]  Giuliana Franceschinis,et al.  Computing Bounds for the Performance Indices of Quasi-Lumpable Stochastic Well-Formed Nets , 1994, IEEE Trans. Software Eng..

[27]  Peter Buchholz.  Aggregation and reduction techniques for hierarchical GCSPNs , 1993, Proceedings of 5th International Workshop on Petri Nets and Performance Models.

[28]  P. Buchholz.  Equivalence Relations for Stochastic Automata Networks , 1995 .

[29]  Jean-Michel Fourneau,et al.  Graphs and Stochastic Automata Networks , 1995 .