2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editorial Board   Contacts   Template   Publication Ethics   Peer Review Process

Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)



Asymptotic Expansions for Power-Exponential Moments of Hitting Times for Nonlinearly Perturbed Semi-Markov Processes

D. S. Silvestrov, S. D.Silvestrov

Download PDF

Abstract: New algorithms for construction of asymptotic expansions for exponential and power-exponential moments of hitting times for nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction and the systematical use of operational calculus for Laurent asymptotic expansions applied to moments of hitting times for perturbed semi-Markov processes. These algorithms have an universal character. They can be applied to nonlinearly perturbed semi-Markov processes with an arbitrary asymptotic communicative structure of a phase space. Asymptotic expansions are given in two forms, without and with explicit bounds for remainders. The algorithms are computationally effective, due to a recurrent character of the corresponding computational procedures.

Keywords: Semi-Markov process, Nonlinear perturbation, Hitting time, Power-exponential moment, Laurent asymptotic expansion

Bibliography:
1. K. E. Avrachenkov, J. A. Filar, P. G. Howlett, Analytic Perturbation Theory and Its Applications, SIAM, Philadelphia, 2013.
2. D. A. Bini, G. Latouche, B. Meini, Numerical Methods for Structured Markov Chains, Numerical Mathematics and Scientific Computation, Oxford Science Publications, Oxford University Press, New York, 2005.
3. P. J. Courtois, Decomposability: Queueing and Computer System Applications, ACM Monograph Series, Academic Press, New York, 1977.
4. M. Gyllenberg, D. S. Silvestrov, Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems, De Gruyter Expositions in Mathematics, vol. 44, Walter de Gruyter, Berlin,2008.
5. Y. C. Ho, X. R. Cao, Perturbation Analysis of Discrete Event Dynamic Systems, Springer International Series in Engineering and Computer Science, Springer, New York, 1991.
6. M. V. Kartashov, Strong Stable Markov Chains, VSP, Utrecht and TBiMC, Kiev, 1996.
7. M. Konstantinov, D. W. Gu, V. Mehrmann, P. Petkov, Perturbation Theory for Matrix Equations, Studies in Computational Mathematics, vol. 9, North-Holland, Amsterdam, 2003.
8. V. S. Korolyuk, V. V. Korolyuk, Stochastic Models of Systems, Mathematics and its Applications, vol. 469, Kluwer, Dordrecht, 1999.
9. V. S. Koroliuk, N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, Singapore, 2005.
10. V. Korolyuk, A. Swishchuk, Semi-Markov Random Evolutions, Mathematics and its Applications, vol. 308, Kluwer, Dordrecht, 1995.
11. V. S. Korolyuk, A. F. Turbin, Semi-Markov Processes and its Applications, Naukova Dumka, Kiev, 1976.
12. V. S. Korolyuk, A. F. Turbin, Mathematical Foundations of the State Lumping of Large Systems, Mathematics and its Applications, vol. 264, Kluwer, Dordrecht, 1993.
13. D. S. Silvestrov, Upper bounds for exponential moments of hitting times for semi-Markov processes, Comm. Statist. Theory Methods, 33 (2005), no. 3, 533-544.
14. D. Silvestrov, S. Silvestrov, Asymptotic expansions for stationary distributions of perturbed semi-Markov processes, Engineering Mathematics II. Algebraic, Stochastic and Analysis Struc-
tures for Networks, Data Classification and Optimization (S. Silvestrov and M. Rancic, eds.), Springer Proceedings in Mathematics & Statistics, vol. 179, Springer, Cham, 2016, 151-222.
15. D. Silvestrov, S. Silvestrov, Nonlinearly Perturbed Semi-Markov Processes, SpringerBriefs Probab. Math. Stat., Springer, Cham, 2017.
16. G. W. Stewart, Matrix Algorithms. Vol. I. Basic Decompositions, SIAM, Philadelphia, 1998.
17. G. W. Stewart, Matrix Algorithms. Vol. II. Eigensystems, SIAM, Philadelphia, 2001.
18. G. W. Stewart, J. G. Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing, Academic Press, Boston, 1990.
19. G. G. Yin, Q. Zhang, Discrete-time Markov chains. Two-time-scale methods and applications, Stochastic Modelling and Applied Probability, Springer, New York, 2005.
20. G. G. Yin, Q. Zhang, Continuous-Time Markov Chains and Applications. A Two-Time-Scale Approach, Stochastic Modelling and Applied Probability, vol. 37, Springer, New York, 2013.