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Switching processes: Averaging principle, diffusion approximation and applications

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Abstract

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 is 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.

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  1. Applied Statistics Department, Kiev University, Ukraine

    Vladimir V. Anisimov

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  1. Vladimir V. Anisimov
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Anisimov, V.V. Switching processes: Averaging principle, diffusion approximation and applications. Acta Appl Math 40, 95–141 (1995). https://doi.org/10.1007/BF00996931

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