Abstract
Two different techniques for bounding the steady-state solution of large Markov chains are presented. The first one is a verification technique based on monotone iterative methods. The second one is a decomposition technique based on the concepts of eigen-vector polyhedron. The computation of the availability of repairable fault-tolerant systems is used to illustrate the performances and the numerical aspects of both methods.
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References
Berman A. and Plemmons R.J., Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
P.J. Courtois, Decomposability: Queueing and Computer System Application,Academic Press, N-Y, 1977.
P.J. Courtois, P. Semal, Bounds for the positive Eigenvectors of Nonnegative Matrices and for their Approximations by Decomposition, JACM, 31, 1984.
P.J. Courtois, P. Semal, On Polyhedra of Perron-Frobenius Eigenvectors, LAA, 65, 1985.
Koury J.R., McAllister D.F. and Stewart W.J., Iterative Methods for Computing Stationary Distributions of nearly Decomposable Markov Chains, SIAM Alg. Disc. Meth., Vol 5, 1984.
Linear Algebra and Its Applications, Special Issue on Iterative Methods, LAA, Vol 154–156, 1991.
Lui J.C.S. and Muntz R.R.,Evaluating bounds on Steady-State Availability of Repairable Systems from Markou Models, in [17], 1991.
Muntz R.R, E. De Souza Silva and A. Goyal, Bounding Availability of Repairable Systems, SIGMETRICS, 1989.
Philippe B., Saad Y. and Stewart W.J., Numerical Methods in Markov Chain Modeling, Op. Res., Vol. 40, 1156–1179, 1992.
Schneider H., Theorems on M-splittings of a singular M-matriz which depend on graph structure, LAA, 58, 1984.
Schweitzer P.J., A Survey of Aggregation-Disaggregation in Large Markov Chains, in [17], 1991.
Semal P., Iterative Algorithms for Large Stochastic Matrices, LAA, 154–156, 1991.
Semal P., Analysis of Large Markov Models: Bounding Techniques and Applications, Ph.Dissertation, Université Catholique de Louvain, Louvainla-Neuve, Belgium, 1992.
Semal P., Monotone Iterative Methods for Markov Chains, to appear in LAA, 1995.
Semal P., Bounding the steady-state solution of large Markov chains, submitted to IEEE trans. on computers, 1995.
Stewart G.W., Introduction to Matrix Computations, Academic Press, New York, 1973.
Stewart W.J., Numerical Solution of Markov Chains, Stewart W.J. (Edt.), Marcel Dekker Inc., New York, 1991.
Varga R.S., Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N.J., 1962.
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Semal, P. (1995). Two Bounding Schemes for the Steady-State Solution of Markov Chains. In: Stewart, W.J. (eds) Computations with Markov Chains. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2241-6_18
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DOI: https://doi.org/10.1007/978-1-4615-2241-6_18
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5943-2
Online ISBN: 978-1-4615-2241-6
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