Abstract
We show how some graph theoretical arguments may be used to reduce the complexity of the computation of the steady-state distribution of Markov chain. We consider the directed graph associated to a Markov chain derived from a Stochastic Automata Network (SAN). The structural properties of the automata are used to establish new various results. First, we establish the complexity of the resolution for Stochastic Automata Networks with a sparse matrix representation of the automata. This results are used to compare simple SAN (i.e. without functions) with methods which generates a sparse representation of Markov chains (i.e. Markovian Petri Nets for instance) on some examples.
Then, we show how to apply state reduction techniques on a chain associated to a SAN. We present an algorithm to solve the steady-state equations and we prove its complexity. Finally, we extend our algorithm to allow the semi-parametric analysis of Stochastic Automata Networks.
1This work is partially supported by a grant from CNRS, project Aξ.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Buchholz P., “Numerical solution methods based on structural descriptions of Markovian models,” Balbo G., Serazzi G. (Eds.), Computer performance evaluation - modelling techniques and tools, Elsevier, 1992.
Donatelli S., “Superposed generalized stochastic Petri nets: Definition and efficient solution”, Valette R. (Ed.), Application and Theory of Petri Nets, LNCS, Vol. 815, pp. 258–277.
Feinberg B. N. and Chiu, S. S. “A method to calculate steady-state distributions of large Markov chains by aggregating states,” Oper. Res. Vol. 35, pp. 282–290, 1987.
Fourneau J.-M., Kloul L., Mokdad L. and Quessette F., “A new tool to model parallel systems and protocols,” European Simulation Symposium, Istanbul, Turkey, 1994.
Fourneau J.-M., Kloul L., Quessette F. and and Vèque V., “Performance evaluation of a simple congestion control at burst level for ATM networks,” 2nd International Conference on Telecommunication Systems Modelling and Analysis, Mars 1994, Nashville, USA.
Fourneau J.-M., Maisonniaux H., Pekergin N. and Vèque V., “Performance evaluation of buffer policies with stochastic automata networks,” Workshop IFIP on Modelling and Performance Evaluation of ATM Technology, pp. 433–451, La Martinique, 1993.
Lazar A., “Markovian petri net protocol with product form solution,” Performance Evaluation, Vol. 12, pp. 67–77, 1991.
Lal R. and Bhat U. N., “Reduced systems in Markov chains and their applications in queueing theory,” Queueing Systems, Vol. 2, pp. 147–172, 1987.
Li M. and Georganas N.D., “Parametric analysis of stochastic Petri nets,” 5th International Conference on Modelling Techniques and Tools for Computational Performance Evaluation, Torino, Italy, 1991.
Meyer C.D. “Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems,” SIAM review, Vol. 31(2), pp. 240–272, June 1989.
Plateau B. and Atif K., “Stochastic automata network for modeling parallel systems,” IEEE Transactions on Software Engineering, 17, 1991.
Plateau B. and Fourneau J.-M., “A methodology for solving complex Markov models of parallel systems,” Journal of Parallel and Distributed Computing, Vol. 12, pp. 370–387, 1991.
Plateau B., Fourneau J.-M. and Lee K.H., “Peps: A package for solving complex Markov models of parallel systems,” Proceedings of the 4th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, Palma de Majorca, Spain, 1988.
Robertazzi T., “Recursive solution of a class of non-product form protocol models,” INFOCOM, 1989.
Shamir A., “A linear time algorithm for finding minimum cutsets in reducible graph,” Siam J. Comput., Vol. 8, No. 4, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media New York
About this paper
Cite this paper
Fourneau, JM., Quessette, F. (1995). Graphs and Stochastic Automata Networks. In: Stewart, W.J. (eds) Computations with Markov Chains. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2241-6_14
Download citation
DOI: https://doi.org/10.1007/978-1-4615-2241-6_14
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5943-2
Online ISBN: 978-1-4615-2241-6
eBook Packages: Springer Book Archive