Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-30T22:14:06.206Z Has data issue: false hasContentIssue false
  • English
  • Français

An Aggregation/Disaggregation Algorithm for Stochastic Automata Networks

Published online by Cambridge University Press:  27 July 2009

Peter Buchholz
Peter Buchholz
Affiliation:
Informatik IV, Universität Dortmund, D-44221 Dortmund, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Stochastic automata networks (SANs) have recently received much attention in the literature as a means to analyze complex Markov chains in an efficient way. The main advantage of SANs over most other paradigms is that they allow a very compact description of the generator matrix by means of much smaller matrices for single automata. This representation can be exploited in different iterative techniques to compute the stationary solution. However, the set of applicable solution methods for SANs is restricted, because a solution method has to respect the specific representation of the generator matrix to exploit the compact representation. In particular, aggregation/disaggregation (a/d) methods cannot be applied in their usual realization for SANs without losing the possibility to exploit the compact representation of the generator matrix.

In this paper, a new a/d algorithm for SANs is introduced. The algorithm differs significantly from standard a/d methods because the parts to be aggregated are defined in a completely different way, exploiting the structure of the generator matrix of a SAN. Aggregation is performed with respect to single automata or sets of automata, which are the basic parts generating a SAN. It is shown that the new algorithm is efficient even if the automata are not loosely coupled.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 11 , Issue 2 , April 1997 , pp. 229 - 253
Copyright
Copyright © Cambridge University Press 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Buchholz, P. (1994). Markovian process algebra: Composition and equivalence. In Herzog, U. & Rettelbach, M. (eds.) Proceedings of the 2nd Workshop on Process Algebras and Performance Modelling. Arbeitsberichte des IMMD, University of Erlangen, Erlangen, Germany, No. 27, pp. 11–30.Google Scholar
2.Buchholz, P. (1995). Equivalence relations for stochastic automata networks. In Stewart, W.J. (ed.), Computation with Markov chains. Dordrecht: Kluwer, pp. 197–216.CrossRefGoogle Scholar
3.Chatelin, F. & Miranker, W.L. (1982). Acceleration by aggregation successive approximation methods. Linear Algebra and Its Applications 43: 17–47.CrossRefGoogle Scholar
4.Courtois, P.J. (1977). Decomposability queueing and computer system applications. New York: Academic Press.Google Scholar
5.Davio, M. (1981). Kronecker products and shuffle algebra. IEEE Transactions on Computers 30: 116–125.CrossRefGoogle Scholar
6.Donatelli, S. (1993). Superposed stochastic automata: A class of stochastic Petri nets amenable to parallel solution. Performance Evaluation 18: 21–26.CrossRefGoogle Scholar
7.Fernandes, P., Plateau, B., & Stewart, W.J. (1994). Efficient descriptor–vector multiplications in stochastic automata networks. Rapport Apache (LGI, LMC) 12. Institut National Polytechnique de Grenoble, France.Google Scholar
8.Fourneau, J.M. & Quessette, F. (1995). Graphs and stochastic automata networks. In Stewart, W.J. (ed.), Computation with Markov chains. Dordrecht: Kluwer, pp. 217–235.CrossRefGoogle Scholar
9.Haviv, M. (1986). Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain. SIAM Journal on Numerical Analysis 24: 952–966.CrossRefGoogle Scholar
10.Heyman, D.P. & Goldsmith, M.J. (1995). Comparisons between aggregation/disaggregation and a direct algorithm for computing the stationary probabilities of a Markov chain. ORSA Journal on Computing 7: 101–108.CrossRefGoogle Scholar
11.Horton, G. & Leutenegger, S. (1995). On the utility of the multi-level algorithm for the solution of nearly completely decomposable Markov chains. In Stewart, W. J. (ed.), Computation with Markov chains. Dordrecht: Kluwer, pp. 425–442.Google Scholar
12.Kemper, P. (1995). Numerical analysis of superposed GSPNs. In Proceedings of the 6th International Workshop on Petri Nets and Performance Models. Los Alamitos: IEEE CS-Press, pp. 52–61.CrossRefGoogle Scholar
13.Plateau, B. (1985). On the stochastic structure of parallelism and synchronization models for distributed algorithms. Performance Evaluation Review 13: 142–154.CrossRefGoogle Scholar
14.Plateau, B. & Atif, K. (1991). Stochastic automata networks for modeling parallel systems. IEEE Transactions on Software Engineering 17: 1093–1108.CrossRefGoogle Scholar
15.Plateau, B. & Fourneau, J.M. (1991). A methodology for solving Markov models of parallel systems. Journal of Parallel and Distributed Computing 12: 370–387.CrossRefGoogle Scholar
16.Schweitzer, P.J. (1991). A survey of aggregation/disaggregation methods in large Markov chains. In Stewart, W.J. (ed.), Numerical solution of Markov chains. New York: Marcel Dekker, pp. 63–88.Google Scholar
17.Schweitzer, P.J. & Kindle, K.W. (1986). An iterative aggregation/disaggregation algorithm for solving linear equations. Applied Mathematics and Computation 18: 313–353.CrossRefGoogle Scholar
18.Siegle, M. (1994). Structured Markovian performance modelling with automatic symmetry exploitation. In Tools and Posters of the 7th International Conference on Modeling Techniques and Tools for Computer Performance Evaluation. Technical Report, Institut fur Angewandte Informatik und Informationssysteme, Universität Wien, Austria, pp. 77–83.Google Scholar
19.Stewart, W.J. (1994). Introduction to the numerical solution of Markov chains. Princeton, NJ: Princeton University Press.Google Scholar
20.Stewart, W.J., Atif, K., & Plateau, B. (1995). The numerical solution of stochastic automata networks. European Journal of Operations Research 86: 503–525.CrossRefGoogle Scholar