# COMBINED MODELS (1-SCHEMES), Basic relations. - Modeling of systems

## COMBINED MODELS (1-SCHEMES)

The most famous general approach to the formal description of the processes of the functioning of systems is the approach proposed by Ya. Ya. Buslenko. This approach allows us to describe the behavior of continuous and discrete, deterministic and stochastic systems, that is, in comparison with the considered is generalized (universal) and is based on the notion of the aggregate system , which is a formal scheme of a general form, which we will call the A-scheme (4, 35].

## Basic relations.

Analysis of the existing means of modeling systems and tasks solved using the computer simulation method inevitably leads to the conclusion that a complex solution of the problems arising in the process of creation and machine realization of the model is possible only if the simulation systems are based on a single formal mathematical scheme, i.e., an A-scheme. Such a scheme must simultaneously perform several functions: to be an adequate mathematical description of the object of modeling, that is, of system 5, to serve as the basis for constructing the Algor tmov and program implementation in machine model L /, allowing a simplified version (for special cases) to conduct analytical studies.

These requirements are to some extent contradictory. Nevertheless, within the framework of the generalized approach on the basis of A-schemes , it is possible to find a certain compromise between them.

According to the tradition established in mathematics in general and in applied mathematics in particular, the aggregate approach first gives a formal definition of the modeling object - the aggregative system, which is a mathematical scheme that reflects the system character of the studied objects. In the aggregative description, a complex object (system) is divided into a finite number of parts (subsystems), while preserving the links that ensure their interaction. If some of the subsystems obtained turn out to be quite complicated in their turn, then the process

their decomposition continues until the subsystems that under the conditions of the considered modeling problem can be considered convenient for the mathematical description are formed. As a result of such decomposition, a complex system is represented as a multi-level design of interrelated elements, combined into subsystems of different levels [35].

As an element of the A-scheme is an aggregate, and the connection between aggregates (inside the system 5 and with the environment E) is performed using the conjugation operator H. Obviously, the aggregate itself can be considered as an A-scheme , ie it can be divided into elements (aggregates) of the next level.

Any aggregate is characterized by the following sets: time moments T, input X and output Y signals, states X at each time t. The state of the aggregate at a time instant/e is denoted as r (f) 6Z, and the input and output signals are as x (!) EX y (() e Y < 4].

We assume that the transition of the aggregate from the state r (* 2 ) to state r (/ < sub> 2 ) ^ 2 (* 1) occurs in a small time interval, ie, (j 1 ) in 2 (/ 2 ) are determined by the intrinsic (internal) parameters of the aggregate A itself (i) and the input signals x (/) e X

At the initial time t 0 , the states r have values ​​equal to r °, i.e. <2> = r 0 y given by the law of distribution of the process r (0 at the time/ 0 , namely b [г (/ 0 )]] Suppose that the process of the aggregate operation in the case of the input signal x " is described by the random operator V. Then when the input signal x n is received into the unit/ n eH, you can determine the state

We denote the half-time interval as (r A , and the half-interval/ 4 - as [/ A , r 2 ). If the time interval (/ ", r n + |) does not contain any moment of signal arrival, then the state of the aggregate for/e (/",/"+,) is determined by the random operator and in accordance with the relation

The set of random operators V and and is treated as an operator of aggregate transitions to new states. The process of aggregate functioning consists of states jumps at the moments of input of the input signals x (the operator V) and state changes between these moments and/ i + , (operator C). The and operators do not have any constraints, so state jumps bt are possible at times that are not the moments of input of the input signals x. In what follows, the moments of jumps b2 will be called the special moments of time r *,

Thus, by aggregate we mean any object determined by an ordered set of considered sets T y YY, Zy ZS ^ y H and random operators V , C /, W y G.

The sequence of input signals located in the order of their arrival in the A-scheme y will be called the input message or the x-message. The sequence of output signals, the output message or the y-message.

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