Abstract
New algorithms for construction of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces are presented. These algorithms are based on a special technique of sequential phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces. Asymptotic expansions are given in two forms, without and with explicit upper bounds for remainders.
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Avrachenkov KE, Filar JA, Howlett PG (2013) Analytic perturbation theory and its applications. SIAM, Philadelphia, p xii+ 372
Bini DA, Latouche G, Meini B (2005) Numerical methods for structured markov chains. Numerical mathematics and scientific computation. Oxford Science Publications, Oxford University Press, New York, p xii+ 327
Courtois PJ (1977) Decomposability. Queueing and computer system applications. ACM monograph series. Academic Press, New York, p xiii+ 201
Gyllenberg M, Silvestrov DS (2008) Quasi-stationary phenomena in nonlinearly perturbed stochastic systems. De Gruyter expositions in mathematics, vol 44. Walter de Gruyter, Berlin, p ix+ 579
Hanen A (1963) Théorèmes limites pour une suite de chaînes de Markov. Ann Inst H Poincaré 18:197–301
Kartashov MV (1996) Strong stable Markov chains. VSP, Utrecht and TBiMC, Kiev, p 138
Konstantinov M, Gu DW, Mehrmann V, Petkov P (2003) Perturbation theory for matrix equations. Studies in computational mathematics, vol 9. North-Holland, Amsterdam, p xii+ 429
Korolyuk VS (1969) On asymptotical estimate for time of a semi-Markov process being in the set of states. Ukr Mat Zh 21:842–845. (English translation in Ukr Math J 21:705–710)
Korolyuk VS, Korolyuk VV (1999) Stochastic models of systems. Mathematics and its applications, vol 469. Kluwer, Dordrecht, p xii+ 185
Koroliuk VS, Limnios N (2005) Stochastic systems in merging phase space. World Scientific, Singapore, p xv+ 331
Korolyuk V, Swishchuk A (1995) Semi-markov random evolutions. Mathematics and its applications, vol 308. Kluwer, Dordrecht, p x + 310. (English revised edition of Semi-Markov Random Evolutions. Naukova Dumka, Kiev, 1992, 254 pp)
Korolyuk VS, Turbin AF (1976) Semi-markov processes and its applications. Naukova Dumka, Kiev, p 184
Korolyuk VS, Turbin AF (1978) Mathematical foundations of the state lumping of large systems. Naukova Dumka, Kiev, p 218. (English edition: Mathematics and its Applications, 264, Kluwer, Dordrecht, 1993, x + 278 pp)
Meshalkin LD (1958) Limit theorems for Markov chains with a finite number of states. Teor Veroyatn Primen 3:361–385. (English translation in Theory Probab Appl 3:335–357)
Schweitzer PJ (1968) Perturbation theory and finite Markov chains. J. Appl. Probab. 5:401–413
Seneta E (1967) Finite approximations to infinite non-negative matrices. Proc Cambridge Philos Soc 63:983–992
Seneta E (2006) Non-negative matrices and markov chains. Springer series in statistics. Springer, New York, p xvi+ 287. (A revised reprint of the second (1981) edition)
Silvestrov D, Manca R (2015) Reward algorithms for semi-markov processes. Research report 2015:16, department of mathematics, stockholm university, pp 23 and arXiv:1603.05693. (to appear in method. Comp Appl Probab)
Silvestrov D, Silvestrov S (2016a) Asymptotic expansions for stationary distributions of perturbed semi-Markov processes. In: Silvestrov S, Ranc̆ić M (eds) Engineering mathematics II. Algebraic, stochastic and analysis structures for networks, data classification and optimization, Chapter 10. Springer proceedings in mathematics & statistics, vol 179. Springer, Cham, pp 151–222
Silvestrov D, Silvestrov S (2016b) Asymptotic expansions for stationary distributions of nonlinearly perturbed semi-markov processes. I, II. Part I: arXiv:1603.04734, pp 30 , Part II: arXiv:1603.04743, pp 33
Simon HA, Ando A (1961) Aggregation of variables in dynamic systems. Econometrica 29:111–138
Stewart GW (1998) Matrix algorithms, vol I. Basic decompositions. SIAM, Philadelphia, p xx+ 458
Stewart GW (2001) Matrix algorithms, vol II. Eigensystems. SIAM, Philadelphia, p xx+ 469
Stewart GW, Sun JG (1990) Matrix perturbation theory. Computer science and scientific computing. Academic Press, Boston, p xvi+ 365
Yin GG, Zhang Q (2005) Discrete-time markov chains. Two-time-scale methods and applications. Stochastic modelling and applied probability, vol 55. Springer, New York, p xix+ 348
Yin GG, Zhang Q (2013) Continuous-time markov chains and applications. a two-time-scale approach, stochastic modelling and applied probability, vol 37, 2nd edn. Springer, New York, p xxii+ 427. (An extended variant of the first (1998) edition)
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Silvestrov, D., Silvestrov, S. Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes. 1. Methodol Comput Appl Probab 21, 945–964 (2019). https://doi.org/10.1007/s11009-017-9605-0
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DOI: https://doi.org/10.1007/s11009-017-9605-0
Keywords
- Markov chain
- Semi-Markov process
- Nonlinear perturbation
- Stationary distribution
- Expected hitting time
- Laurent asymptotic expansion