×

zbMATH — the first resource for mathematics

Nonlinearly perturbed stochastic processes and systems. (English) Zbl 1404.60130
Rykov, Vladimir V. (ed.) et al., Mathematical and statistical models and methods in reliability. Applications to medicine, finance, and quality control. Invited papers based on the presentation at the 6th international conference (MMR 2009), Moscow, Russia, June 22–26, 2009. Boston, MA: Birkhäuser (ISBN 978-0-8176-4970-8/hbk; 978-0-8176-4971-5/ebook). Statistics for Industry and Technology, 19-37 (2010).
Summary: This paper is a survey of results presented in the recent book M. Gyllenberg and D. S. Silvestrov [Quasi-stationary phenomena in nonlinearity perturbed stochastic systems. Berlin: de Gruyter (2008; Zbl 1175.60002)]. This book is devoted to studies of quasi-stationary phenomena for nonlinearly perturbed stochastic processes and systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on asymptotic expansions for perturbed renewal equation and recurrence algorithms for construction of asymptotic expansions for Markov type processes with absorption are presented. Asymptotic expansions are given in mixed ergodic (for processes) and large deviation theorems (for absorption times) for nonlinearly perturbed regenerative processes, semi-Markov processes, and Markov chains. Applications to analysis of quasi-stationary phenomena in nonlinearly perturbed queueing systems, population dynamics and epidemic models, and for risk processes are presented. The book also contains an extended bibliography of works in the area.
For the entire collection see [Zbl 1203.60007].
MSC:
60K05 Renewal theory
60K15 Markov renewal processes, semi-Markov processes
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anisimov, V.V.: Switching Processes in Queuing Models. Applied Stochstic Models Series, ISTE and Wiley, London (2008) · Zbl 1156.60002
[2] Asmussen, S.: Applied Probability and Queues. Wiley Series in Probability and Mathematical Statistics, Wiley, New York and Stochastic Modelling and Applied Probability,51, Springer, New York (1987, 2003) · Zbl 0624.60098
[3] Asmussen, S.: Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability,2, World Scientific, Singapore (2000) · Zbl 0960.60003
[4] Bening, V.E., Korolev, V.Yu.: Generalized Poisson Models and their Applications in Insurance and Finance. Modern Probability and Statistics, VSP, Utrecht (2002) · Zbl 1041.60004
[5] Borovkov, A.A.: Ergodicity and Stability of Stochastic Processes. Wiley Series in Probability and Statistics, Wiley, New York (1998) · Zbl 0917.60005
[6] Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Applications of Mathematics,33, Springer, Berlin (1997) · Zbl 0873.62116
[7] Englund, E.: Perturbed renewal equations with application to M/M queueing systems. 1, Teor. Ĭmovirn. Math. Stat., 60, 31-37 (1999a) (Also in Theory Probab. Math. Stat., 60, 35-42) · Zbl 0955.60080
[8] Englund, E.: Perturbed renewal equations with application to M/M queueing systems. 2. Teor. Ĭmovirn. Math. Stat.,61, 21-32 (1999b) (Also in Theory Probab. Math. Stat., 61, 21-32)
[9] Englund, E.: Nonlinearly perturbed renewal equations with applications to a random walk. Theory Stoch. Process., 6(22), no. 3-4, 33-60 (2000) · Zbl 0973.60026
[10] Englund, E.: Nonlinearly perturbed renewal equations with applications. Doctoral Dissertation, Umeå University (2001)
[11] Englund, E., Silvestrov, D.S.: Mixed large deviation and ergodic theorems for regenerative processes with discrete time. Theory Stoch. Process.,3(19), no. 1-2, 164-176 (1997) · Zbl 0946.60079
[12] Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II. Wiley, New York (1966) · Zbl 0138.10207
[13] Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol.,33, 35-70 (1994) · Zbl 0816.92016
[14] Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary phenomena for semi-Markov processes. In: Janssen, J., Limnios, N. (eds) Semi-Markov Models and Applications. Kluwer, Dordrecht, 33-60 (1999a) · Zbl 0966.60086
[15] Gyllenberg, M., Silvestrov, D.S.: Cramér-Lundberg and diffusion approximations for nonlinearly perturbed risk processes including numerical computation of ruin probabilities. Theory Stoch. Process.,5(21), no. 1-2, 6-21 (1999b) · Zbl 0952.60080
[16] Gyllenberg, M., Silvestrov, D.S.: Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems. Stoch. Process. Appl., 86, 1-27 (2000a) · Zbl 1028.60067
[17] Gyllenberg, M., Silvestrov, D.S.: Cramér-Lundberg approximation for nonlinearly perturbed risk processes. Insur. Math. Econom.,26, no. 1, 75-90 (2000b) · Zbl 0956.91044
[18] Gyllenberg, M., Silvestrov, D.S.: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter Expositions in Mathematics,44, Walter de Gruyter, Berlin (2008) · Zbl 1175.60002
[19] Hanen, A.: Théorèmes limites pour une suite de chaînes de Markov. Ann. Inst. H. Poincaré,18, 197-301 (1963) · Zbl 0202.48206
[20] Ho, Y.C., Cao, X.R.: Perturbation Analysis of Discrete Event Dynamic Systems. Kluwer International Series in Engineering and Computer Science, Kluwer, Boston (1991) · Zbl 0744.90036
[21] Kalashnikov, V.V.: Mathematical Methods in Queuing Theory. Mathematics and its Applications, 271, Kluwer, Dordrecht (1994) · Zbl 0836.60098
[22] Kalashnikov, V.V.: Geometric Sums: Bounds for Rare Events with Applications. Mathematics and its Applications,413, Kluwer, Dordrecht (1997) · Zbl 0881.60043
[23] Kalashnikov, V.V., Rachev, S.T.: Mathematical Methods for Construction of Queueing Models. Nauka, Moscow (1988) (English edition: The Wadsworth & Brooks/Cole Operations Research Series, Wadsworth & Brooks/Cole, Pacific Crove, CA (1990)). · Zbl 0714.60083
[24] Kartashov, M.V.: Strong Stable Markov Chains. VSP, Utrecht and TBiMC, Kiev (1996) · Zbl 0874.60082
[25] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966, 1976, 1995) · Zbl 0148.12601
[26] Kijima, M.: Markov Processes for Stochastic Modelling. Stochastic Modeling Series, Chapman & Hall, London (1997) · Zbl 0866.60056
[27] Kingman, J.F.: The exponential decay of Markovian transition probabilities. Proc. Lond. Math. Soc.,13, no. 3, 337-358 (1963) · Zbl 0154.43003
[28] Korolyuk, V.S., Korolyuk, V.V.: Stochastic Models of Systems. Mathematics and its Applications,469, Kluwer, Dordrecht (1999) · Zbl 0960.60004
[29] Koroliuk, V.S., Limnios, N.: Stochastic Systems in Merging Phase Space. World Scientific, Singapore (2005) · Zbl 1101.60003
[30] Korolyuk, V.S., Turbin, A.F.: Semi-Markov Processes and Its Applications. Naukova Dumka, Kiev (1976) · Zbl 0371.60106
[31] Korolyuk, V.S., Turbin, A.F.: Mathematical Foundations of the State Lumping of Large Systems. Naukova Dumka, Kiev (1978) (English edition: Mathematics and its Applications, 264, Kluwer, Dordrecht (1993))
[32] Kovalenko, I.N.: Rare events in queuing theory - a survey. Queuing Syst. Theory Appl.,16, no. 1-2, 1-49 (1994) · Zbl 0804.90059
[33] Kovalenko, I.N., Kuznetsov, N.Yu., Pegg, P.A.: Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications. Wiley Series in Probability and Statistics, Wiley, New York (1997) · Zbl 0899.60074
[34] Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA and American Statistical Association, Alexandria, VA (1999) · Zbl 0922.60001
[35] Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Communications and Control Engineering Series, Springer, London (1993) · Zbl 0925.60001
[36] Ni, Y., Silvetrov, D., Malyarenko, A.: Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations. J. Numer. Appl. Math.1(96), 173-197 (2008) · Zbl 1164.60062
[37] Seneta, E.: Non-negative Matrices and Markov Chains. Springer Series in Statistics, Springer, New-York (1981, 2006) · Zbl 0471.60001
[38] Shurenkov, V.M.: Transition phenomena of the renewal theory in asymptotical problems of theory of random processes. I. Mat. Sbornik,112(154), no. 1 (5), 115-132 (1980a) (English translation in Math. USSR-Sbornik, 40, no. 1, 107-123).
[39] Shurenkov, V.M.: Transition phenomena of the renewal theory in asymptotical problems of theory of random processes. II. Mat. Sbornik,112(154), no. 2(6), 226-241 (1980b) (English translation in Math. USSR-Sbornik, 40, no. 2, 211-225). · Zbl 0444.60071
[40] Silvestrov, D.S.: A generalization of the renewal theorem. Dokl. Akad. Nauk. Ukr. SSR, Ser. A, no. 11, 978-982 (1976)
[41] Silvestrov, D.S.: The renewal theorem in a series scheme 1. Teor. Veroyatn. Mat. Stat.,18, 144-161 (1978) (English translation in Theory Probab. Math. Statist. 18, 155-172)
[42] Silvestrov, D.S.: The renewal theorem in a series scheme 2. Teor. Veroyatn. Mat. Stat.,20, 97-116 (1979) (English translation in Theory Probab. Math. Statist.20, 113-130).
[43] Silvestrov, D.S.: Semi-Markov Processes with a Discrete State Space. Library for an Engineer in Reliability, Sovetskoe Radio, Moscow (1980)
[44] Silvestrov, D.S.: Exponential asymptotic for perturbed renewal equations. Teor. Ĭmovirn. Mat. Stat.,52, 143-153 (1995) (English translation in Theory Probab. Math. Statist., 52, 153-162). · Zbl 0946.60080
[45] Silvestrov, D.S.: Perturbed renewal equation and diffusion type approximation for risk processes. Teor. Ĭmovirn. Mat. Stat.,62, 134-144 (2000a) (English translation in Theory Probab. Math. Statist., 62, 145-156). · Zbl 1004.60086
[46] Silvestrov, D.S.: Nonlinearly perturbed Markov chains and large deviations for lifetime functionals. In: Limnios, N., Nikulin, M. (eds) Recent Advances in Reliability Theory: Methodology, Practice and Inference. Birkhauser, Boston, 135-144 (2000b) · Zbl 0961.60072
[47] Silvestrov, D.S.: Limit Theorems for Randomly Stopped Stochastic Processes. Probability and Its Applications, Springer, London (2004) · Zbl 1057.60021
[48] Silvestrov, D.S.: Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes. Theory Stoch. Process.,13(29), no. 1-2, 267-271 (2007a) · Zbl 1142.60397
[49] Silvestrov, D.S.: Asymptotic expansions for distributions of the surplus prior and at the time of ruin. Theory Stoch. Process., 13(29), no. 4, 183-188 (2007b) · Zbl 1164.60068
[50] Silvestrov, D.S.: Nonlinearly perturbed stochstic processes. Theory Stoch. Process.,14(30), no. 3-4, 129-164 (2008) · Zbl 1224.60001
[51] Solov’ev, A.D.: Analytical methods for computing and estimating reliability. In: Gnedenko, B.V. (ed) Problems of Mathematical Theory of Reliability. Radio i Svyaz’, Moscow, 9-112 (1983)
[52] Stewart, G.W.: Matrix Algorithms. Vol. I. Basic Decompositions. Society for Industrial and Applied Mathematics, Philadelphia, PA (1998)
[53] Stewart, G.W.: Matrix Algorithms. Vol. II. Eigensystems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2001)
[54] Stewart, G.W., Sun, Ji Guang: Matrix Perturbation Theory. Computer Science and Scientific Computing, Academic Press, Boston (1990) · Zbl 0706.65013
[55] Vere-Jones, D.: Geometric ergodicity in denumerable Markov chains. Quart. J. Math.,13, 7-28 (1962) · Zbl 0104.11805
[56] Wentzell, A.D., Freidlin, M.I.: Fluctuations in Dynamical Systems Subject to Small Random Perturbations. Probability Theory and Mathematical Statistics, Nauka, Moscow (1979) (English edition: Random Perturbations of Dynamical Systems. Fundamental Principles of Mathematical Sciences,260, Springer, New York (1998)).
[57] Whitt, W.: Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and their Application to Queues. Springer Series in Operations Research, Springer, New York (2002) · Zbl 0993.60001
[58] Yin, G.G., Zhang, Q.: Continuous-time Markov Chains and Applications. A Singular Perturbation Approach. Applications of Mathematics, 37, Springer, New York (1998) · Zbl 0896.60039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.