The actual problem of systems theory is the development of mathematically justified methods of simplification of complicated systems whose mathematical analysis is difficult to perform even with help of modern computers. The main difficulties are caused by the complexity of the phase (state) space of the system, which leads to virtually boundedless mathematical models.A simplified model for a system must satisfy the following conditions: (i)The local characteristics of the simplified model are determined by rather simple functions of the local characteristics of original model. (ii) The global characteristics describing the behavior of the stochastic system can be effectively calculated on large enough time intervals. (iii) The simplified model has an effective mathematical analysis, and the global characteristics of the simplified model are close enough to the corresponding characteristics of the original model for application.Stochastic systems considered in the present book are evolutionary systems in random medium, that is, dynamical systems whose state space is subject to random variations. From a mathematical point of view, such systems are naturally described by operator-valued stochastic processes on Banach spaces and are nowadays known as Random evolution.This book gives recent results on stochastic approximation of systems by weak convergence techniques. General and particular schemes of proofs for average, diffusion, diffusion with equilibrium, and Poisson approximations of stochastic systems are presented. The particular systems studied here ['(t), t 2 0 , describes the system evolution, and, in general, is a stochastic functional of a third process. The switching process z'(t), t 2 0, also called the driving or modulation process, is the perturbing process or the random medium, and can represent the environment, the technical structure, or any perturbation factor. The two modes of switching considered here are Markovian and semiMarkovian. Of course, we could present only the semi-Markov case since the Markov is a special case. But we present both mainly for two reasons: the first is that proofs are simpler for the Markov case, and the second that most of the readers are mainly interested by the Markov case.The switching processes are considered in phase split and merging scheme. The phase merging scheme is based on the split of the phase space into disjoint classesand further merging these classes Ek, k E v, into distinct states k E V .So the merged phase space of the simplified model of system is E = V (see Figure 4. l). ' The transitions (connections) between the states of the original system S are merged to yield the transitions between merged states of the merged system S . The analysis of the merged system is thus significantly simplified.It is important to note that the additional supporting system So with the same phase space E but without connections between classes of states Ek is used. Split the phase space (0.1) just means introducing a new supporting system consisting of isolated subsy...

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'Figures, theorems, lemmas, etc. are numbered by x.y, where x is the number of chapter, and y is the number of figure, theorem, etc. into the chapter.

The biological process of cooperative interaction with equilibrium state will be described as a model of binary statistical experiments with Wright-Fisher normalization, which sets the concentration of a certain characteristic. Such mathematical model is supposed to have a property of persistent regression which means that all current elementary transitions depend on the mean concentration of the said characteristics in the previous state. Equilibrium state of the model is expressed in the terms of the regression function, given by a cubic parabola with three real roots. We construct stochastic approximation of the model by autoregressive process with normal disturbances. Such approach was developed for effective and calculable mathematical description of dynamic concentration for experiment planning, parameters evaluation and hypotheses verification of mechanism of action.

The authors propose a mathematical model for a multi-channel queueing system with feedback in which one part of calls instantaneously enters the system for repeated service and the other part either retries in some random time or finally leaves the system. The behavior of the serviced calls is randomized. Both exact and asymptotic methods are developed to calculate the characteristics of the proposed model. The results of numerical experiments are presented.

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Regular and singular parts of asymptotic expansions of semi-Markov random evolutions are given. Regularity of boundary conditions is shown. An algorithm for calculation of initial conditions is proposed.

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