Cybernetics and Systems Analysis, Vol. 34, No. 2, 1998 DIFFUSION APPROXIMATION OF NONHOMOGENEOUS SWITCHING PROCESSES AND ITS APPLICATION TO RATE OF CONVERGENCE ANALYSIS OF COMPUTATIONAL PROCEDURES V. V. Anisimov and A. V. Naidenova UDC 519.21 The development of a broad class of stochastic systems can be described in terms of stochastic processes whose behavior spontaneously changes (switches) at certain time moments. These switching points are random functionals of the previous path. Such processes arise in the theory of queueing systems and networks, in branching and migration phenomena, in the analysis of stochastic dynamical systems with random errors, and in other applications. The description of these models relies on a special class of discrete-event stochastic processes introduced in [1-3], where they are called switching processes. A switching process is a two-component process (x(t), ~(t)), t > 0, with values in the space (X, Rr) for which there is a sequence of time moments t t < t 2 < ... such that x(t) = x(tk) on each time interval Itk, tk+ l) and the behavior of the process ~t) on this time interval depends only on the values (x(tk), ~(tk)). The moments tk are called switching points; x(t) is the discrete switching component. Such processes are describable in terms of constructive characteristics [ 1-3]. They are useful for the analysis of the asymptotic behavior of stochastic systems with "fast" and "rare" switching events [2-7]. Switching processes are a natural generalization of classes of Markov processes homogeneous in the second component [8], processes with independent increments and semi-Markov switchings [2, 3, 9], Markov aggregates [8], Markov processes with semi-Markovian random interventions [ 10], Markovian and semi-Markovian evolutions [ 11-14]. Limit theorems on the convergence of one switching process to another (in the class of switching processes) have been proved in [2, 3]. These results have led to a theory of asymptotic state aggregation of nonhomogeneous Markov and semi-Markov processes [3]. In this article we consider limit theorems on the convergence of the path of a switching process to the solution of some ordinary differential equation (the averaging principle) and the convergence of the normalized deviation to some diffusion process (the diffusion approximation) for an important subclass of switching processes - the so-called semi-Markov recurrent processes (SMRP) with additional dependence on the current switching point. The approach to the study of recursive algorithms with random response time is based on representation of the original process as a superposition of a recurrent embedded process and an accumulated-time counter process followed by application of limit theorems for discrete-time recursive stochastic algorithms and superpositions of random functions. We first prove some generalizations of theorems that constitute the averaging principle and the diffusion approximation for SMRP [6]. Given are independent families of random vectors {(~k(s, t), ~-k(s, t), s E R m, t >__ 0}, k >_ 0, with values in R m • [0, ~) whose characteristic functions are Bern-measurable (Bern is the Borel a-algebra in Rm), and the initial value s 0. Define the sequences ~ to = O, S k + t, h = S,. h + h.~(S,, h' t~. h)' t~ + ~. h = t,. h + h'r~,(Sk, h' t~ h)' ~ ;' O. Consider the process Sh(t) of the form Sh(O) = s o, Sh(t) = Sk, h for tk, h <_ t < tk+ l,h, t >__ O. Translated from Kibernetika i Sistemnyi Analiz No. 2, pp. 97-104, March-April, 1998. Original article submitted June 3, 1997. 238 1060-0396/98/3402-0238520.00 9 Plenum Publishing Corporation