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Transforming stochastic matricesfor stochastic comparison with the st-order

Published online by Cambridge University Press:  15 November 2003

Tuğrul Dayar ,
Jean-Michel Fourneau  and
Nihal Pekergin
Tuğrul Dayar
Affiliation:
Department of Computer Engineering, Bilkent University, 06533 Bilkent, Ankara, Turkey.
Jean-Michel Fourneau
Affiliation:
PRSM, Université de Versailles-St.Quentin, 45 avenue des États-Unis, 78035 France; jmf@prism.uvsq.fr.
Nihal Pekergin
Affiliation:
PRSM, Université de Versailles-St.Quentin, 45 avenue des États-Unis, 78035 France; jmf@prism.uvsq.fr.
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Abstract

We present a transformation for stochastic matrices and analyze theeffects of using it in stochastic comparison with the strong stochastic(st) order. We show that unless the given stochastic matrix is row diagonallydominant, the transformed matrix provides better st bounds on the steady state probability distribution.

Keywords

Markov processesprobability distributionsstochastic orderingst-order.
Type
Research Article
Information
RAIRO - Operations Research , Volume 37 , Issue 2 , April 2003 , pp. 85 - 97
Copyright
© EDP Sciences, 2003

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References

O. Abu-Amsha and J.-M. Vincent, An algorithm to bound functionals of Markov chains with large state space, in 4th INFORMS Conference on Telecommunications. Boca Raton, Florida (1998). Available as Rapport de recherche MAI No. 25. IMAG, Grenoble, France (1996).
M. Benmammoun, J.M. Fourneau, N. Pekergin and A. Troubnikoff, An algorithmic and numerical approach to bound the performance of high speed networks, IEEE MASCOTS 2002. Fort Worth, USA (2002) 375-382.
Keilson, J. and Kester, A., Monotone matrices and monotone Markov processes. Stochastic Process. Appl. 5 (1977) 231-241. CrossRef
J.M. Fourneau and N. Pekergin, An algorithmic approach to stochastic bounds, Performance evaluation of complex systems: Techniques and Tools. Springer, Lecture Notes in Comput. Sci. 2459 (2002) 64-88.
J.M. Fourneau, M. Le Coz, N. Pekergin and F. Quessette, An open tool to compute stochastic bounds on steady-state distributions and rewards, IEEE Mascots 03. USA (2003).
J.M. Fourneau, M. Le Coz and F. Quessette, Algorithms for an irreducible and lumpable strong stochastic bound, Numerical Solution of Markov Chains. USA (2003).
M. Kijima, Markov Processes for stochastic modeling. Chapman & Hall (1997).
Massey, W.A., Stochastic orderings for Markov processes on partially ordered spaces. Math. Oper. Res. 12 (1987) 350-367. CrossRef
N. Pekergin, Stochastic delay bounds on fair queueing algorithms, in Proc. of INFOCOM'99. New York (1999) 1212-1220.
Pekergin, N., Stochastic performance bounds by state reduction. Performance Evaluation 36-37 (1999) 1-17. CrossRef
M. Shaked and J.G. Shantikumar, Stochastic Orders and their Applications. Academic Press, California (1994).
D. Stoyan, Comparison Methods for Queues and Other Stochastic Models. John Wiley & Sons, Berlin, Germany (1983).
H.M. Taylor and S. Karlin, An Introduction to Stochastic Modeling. Academic Press, Florida (1984).
M. Tremolieres, J.-M. Vincent and B. Plateau, Determination of the optimal upper bound of a Markovian generator, Technical Report 106. LGI-IMAG, Grenoble, France (1992).
Truffet, L., Near complete decomposability: bounding the error by stochastic comparison method. Adv. Appl. Probab. 29 (1997) 830-855. CrossRef