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Switching processes: Averaging principle, diffusion approximation and applications. (English) Zbl 0827.60022

Summary: A special class of processes with discrete interference of chance- switching processes (SP), is introduced. The limit theorems for these processes in the case of ‘fast’ switching (averaging principle and diffusion approximation) are proved for the models with simple and semi- Markov switchings. A new approach which is based on the investigation of the asymptotic properties of the special subclass of SP-recurrent process of the semi-Markov type (RPSM), theorems about the convergence of recurrent sequences to the solutions of stochastic differential equations and the convergence of superpositions of random functions are given.
The paper consists of seven sections. A description of SP and RPSM is given in Section 1. In Section 2 the models of stochastic systems described in the terms of SP are investigated. In Sections 3 and 4 the averaging principle and diffusion approximation for RPSM with simple and semi-Markov switchings are proved. In Section 5 these results are extended on SP with simple and semi-Markov switchings. Section 6 is devoted to the application of results obtained for asymptotic analysis of dynamic systems with fast semi-Markov switchings and Section 7 is devoted to the analysis of switching queueing systems and networks in the transient conditions and with large number of requirements.

MSC:

60G05 Foundations of stochastic processes
60F17 Functional limit theorems; invariance principles
60K15 Markov renewal processes, semi-Markov processes
60K25 Queueing theory (aspects of probability theory)
60J60 Diffusion processes
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[1] Anisimov, V. V., (1973): Asymptotic consolidation of the states of random processes,Cybernetics 9(3), 494–504. · doi:10.1007/BF01069207
[2] Anisimov, V. V., (1975): Limit theorems for random processes and their application to discrete summation schemes,Theory Probab. Appl. 20, 692–694.
[3] Anisimov, V. V., (1977): Switching processes,Cybernetics 13(4), 590–595. · doi:10.1007/BF01069653
[4] Anisimov, V. V., (1978): Limit theorems for switching processes and their application,Cybernetics 14(6), 917–929. · Zbl 0447.60075 · doi:10.1007/BF01070290
[5] Anisimov, V. V., (1984):Asymptotic Methods in Analysing Stochastic Systems, Metsniereba, Tbilisi (in Russian). · Zbl 0551.62060
[6] Anisimov, V. V., (1988a):Random Processes with Discrete Component. Limit Theorems, Publ. Kiev Univ. (in Russian). · Zbl 0686.60092
[7] Anisimov, V. V., (1988b): Limit theorems for switching processes,Theory Probab. Math. Statist. 37, 1–5. · Zbl 0663.60030
[8] Anisimov, V. V., (1992a): Limit theorems for switching processes,Functional Analysis III, Proc. Postgrad. School and Conf., Dubrovnik, Yugoslavia, 1989; Var. publ. ser., Aarhus Univ., No. 40. pp. 235–262.
[9] Anisimov, V. V., (1992b): Averaging principle for switching processes,Theory Probab. Math. Statist. 46, 1–10. · Zbl 0835.60018
[10] Anisimov, V. V., (1993): Averaging principle for the processes with fast switchings,Random Operators Stochas. Equations 1(2) 151–160. · Zbl 0842.60057 · doi:10.1515/rose.1993.1.2.151
[11] Anisimov, The relative compactness of sets of probability measures in D0X),Litovsk. Mat. Sb. 13(4), 83–96 (in Russian); English transl. inMath. Trans. Acad. Sci. Lithuanian SSR,13.
[12] Hersh, R. (1974): Random evolutions: survey of results and problems,Rocky Mount. J. Math. 4(3), 443–475. · Zbl 0366.60005 · doi:10.1216/RMJ-1974-4-3-443
[13] Kac, M. (1956): A stochastic model related to the telegrapher’s equation, Magnolia Petroleum Co. Colloq. Lectures repr. inRocky Mount.J. Math. 4(4) (1974), 497–510.
[14] Kertz, R. (1978a): Limit theorems for semigroups with perturbed generators with applications to multi-scaled random evolutions,TAMPS 27, 215–233. · Zbl 0372.47021
[15] Kertz, R. (1978b): Random evolutions with underlying semi-Markov processes,Publ. Res. Inst. Math. Sci. 14, 589–614. · Zbl 0416.60089 · doi:10.2977/prims/1195188829
[16] Khas’minskii, R. Z. (1968): About the averaging principle for ITO stochastic differential equations,Kybernetika 4(3), 260–279.
[17] Korolyuk, V. S. and Swishchuk, A. V. (1986): Central limit theorem for semi-Markov random evolutions,Ukrainian Math. J. 38, 330–335.
[18] Korolyuk, V. S. and Swishchuk, A. V. (1988): Semi-Markov random evolutions,Phase Aggregations and Applications, Internat. Summer. School Probab. Theory Math. Statist., Varna, Gold Sands, 1985, Sophia, pp. 116–131.
[19] Korolyuk, V. S. and Swishchuk, A. V. (1992):Random Evolutions, Naukova Dumka, Kiev (in Russian). English transl., Kluwer Acad. Publ., 1994. · Zbl 0813.60083
[20] Korolyuk, V. S. and Turbin, A. F. (1978):Mathematical Foundations of Phase Consolidations of Complex Systems, Naukova Dumka, Kiev (in Russian). · Zbl 0411.60003
[21] Korolyuk, V. S. and Turbin, A. F. (1983): Limit theorems for Markov random evolutions in the scheme of asymptotic phase lumping,Lecture Notes in Math. 1021, Springer, New York, pp. 327–332. · Zbl 0529.60065
[22] Kurtz, T. (1972): A random Trotter product formula,Proc. Amer. Math. Soc. 35, 147–154. · Zbl 0255.47051 · doi:10.1090/S0002-9939-1972-0303347-5
[23] Kurtz, T. (1973): A limit theorem for perturbed operator semigroups with applications to random evolutions,J. Funct. Anal. 12, 55–67. · Zbl 0246.47053 · doi:10.1016/0022-1236(73)90089-X
[24] Liptser, R. S. and Shiryaev, A. N. (1986):Martingale Theory, Nauka, Moscow.
[25] Papanicolaou, G. and Hersh, R. (1972): Some limit theorems for stochastic equations and applications,Indiana Univ. Math. J. 21, 815–840. · Zbl 0222.60037 · doi:10.1512/iumj.1972.21.21065
[26] Pinsky, M. (1971): Multiplicative operator functionals of Markov processes,BAMS 77, 377–380. · Zbl 0255.60053 · doi:10.1090/S0002-9904-1971-12703-9
[27] Pinsky, M. (1975a): Multiplicative operator functionals and their asymptotical properties,Adv. in Probab. 3, 1–100.
[28] Pinsky, M. (1975b): Random evolutions,Lecture Notes in Math. 451, Springer, New York, 89–100. · Zbl 0331.60051
[29] Skorokhod, A. V. (1956): Limit theorems for random processes,Theory Probab. Appl. 1, 289–319. · Zbl 0074.33802 · doi:10.1137/1101022
[30] Skorokhod, A. V. (1987):Asymptotic Methods in the Theory of Stochastic Differential Equations, Naukova Dumka, Kiev. English transl., Amer. Math. Soc., Providence, RI, 1989. · Zbl 0709.60057
[31] Skorokhod, A. V. (1992): Dynamic systems with random influences,Functional Analysis III, Proc. Postgrad. School and Conf., Dubrovnik, Yugoslavia, 1989; Var. publ. ser., Aarhus Univ., No. 40. pp. 193–234.
[32] Tsarkov, Je. (1993): Averaging and stability of impulse systems with rapid Markov switchings,Proc. Latv. Prob. Sem. 2, Riga, pp. 49–63.
[33] Tsarkov, Je. (1993): Limit theorems for impulse systems with rapid Markov switchings,Proc. Latv. Prob. Sem. 2, Riga, pp. 74–96.
[34] Watkins, J. C. (1984): A central limit problem in random evolutions,Ann. Prob. 12(2), 480–513. · Zbl 0547.60040 · doi:10.1214/aop/1176993302
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