## MATHEMATICAL SCHEMES OF SYSTEM MODELING

The greatest difficulties and the most serious mistakes in modeling arise when moving from the content to the formal description of research objects, which is explained by the participation in this creative process of teams of different specialties: specialists in the field of systems that are required to model (customers) and specialists in the field of computer modeling (executors). An effective tool for finding mutual understanding between these groups of specialists is the language of mathematical schemes, which makes it possible to focus on the adequacy of the transition from a meaningful description of the system to its mathematical scheme, and only then to decide the question of a specific method of obtaining results using a computer: analytical or imitational, and possibly, combined, i.e., analytical-imitative. Applied to a particular simulation object, that is, to a complex system, the model developer should be assisted by concrete, already tested for a given class of systems mathematical schemes that have shown their effectiveness in applied computer studies and received the name of typical mathematical schemes.

## BASIC APPROACHES TO THE CONSTRUCTION OF SYSTEM MATHEMATICAL MODELS

The initial information in the construction of mathematical models of the processes of the functioning of systems is given by the data about the purpose and conditions of work of the system being investigated (projected). This information defines the main goal of modeling the system & pound; and allows us to formulate the requirements for the mathematical model being developed. And the level of abstraction depends on the range of questions the researcher of the system wants to get a response with the help of the model, and to some extent determines the choice of the mathematical scheme [4, 13, 29, 37, 42, 48].

### Mathematical schemes.

Introduction of the concept mathematical scheme allows us to view mathematics not as a method of calculation, but as a method of thinking, as a means of formulating concepts, which is the most important in the transition from a verbal description of the system to a formal representation of the process of its functioning in the form of a mathematical model (analytical or imitative). When using the mathematical scheme of the researcher of the 5 * system, the question of the adequacy of mapping in the form of concrete schemes of real processes in the system under investigation, and not the possibility of obtaining a response (the result of the solution) on a particular question of research, should be of primary interest. For example, the representation of the process of the operation of the information-computing system of collective use in the form of a network of queuing schemes makes it possible to describe the processes occurring in the system well, but with the complex laws of distribution of incoming flows and service flows, it does not make it possible to obtain the results in explicit form [13, 21, 30, 33, 37, 41].

* The mathematical scheme * can be defined as a link in the transition from a meaningful to a formal description of the process of functioning of the system taking into account the influence of the external environment, that is, the chain "descriptive model - mathematical scheme - mathematical [ analytical or (and) imitation] model. "

Each concrete system A is characterized by a set of properties, by which are meant quantities reflecting the behavior of the modeled object (real system) and taking into account the conditions of its functioning in interaction with the external environment (system) * Е. * In constructing the mathematical model system it is necessary to solve the problem of its completeness. The completeness of the model is governed mainly by the choice of the boundary of the environment-environment system & gt; & gt; & gt ;. The task of simplifying the model should also be solved, which helps to isolate the main properties of the system, discarding secondary ones. And the attribution of the properties of the system to the main or secondary significantly depends on the purpose of modeling the system (for example, analysis of the probability-time characteristics of the process of the functioning of the system, synthesis of the structure of the system, etc.).

The formal model of the object. The model of the modeling object, that is, of the system S, can be represented in the form of a set of quantities describing the process of functioning of the real system and forming the following * subsets in the general case* *: the set of input actions * on the system

a collection of * environmental influences *

a set of * internal (proprietary) parameters * of the system

a set of * output characteristics * of the system

In the above subsets, you can select managed and unmanaged variables. In the general case, x "y, A *,

* y * y are elements of disjoint subsets and contain both deterministic and stochastic components.

In the simulation of the system 5, the input actions, the external environment effects * E * and the internal parameters of the system are * independent (exogenous) variables *, which in the vector form are respectively of the form x (/) = (*! (0, x 2 (0 & gt; - * * x (0) * *

* * (0 = ( 1 (0. 2 (0, ...) (0; n (/) = 2 (0, ..., -H (0), and the output characteristics of the system are * dependent (endogenous) variables * and in vector form they have the form * y (0 = (y 1 0), y 2 (* 0 <•••> * YHW *

The process of functioning of system 5 is described in time by the operator/* 5 , which in the general case transforms exogenous variables into endogenous variables in accordance with relations of the form

The set of dependences of the output characteristics of the system on the time y (i) for all types y = 1, * n y * is called the output trajectory y (t). The dependence (2.1) is called * the law of the functioning of the system B * and is denoted * 5 . * In general, the law of the functioning of the system * E 5 * can be specified in the form of a function, functional, logical conditions, in algorithmic and tabular forms, or as a verbal correspondence rule.

*of the algorithm of functioning 5 ,*which means the method of obtaining output characteristics taking into account the input actions

*x*(/), the effects of the external environment

*V*(r) and the eigenvalues of the system

*AND*(/). It is obvious that the same law of the functioning of system 5 can be realized in various ways, that is, using a variety of different functioning algorithms

*L $ .*

The relations (2.1) are a mathematical description of the behavior of the object (system) of modeling in time /, ie, reflect its dynamic properties. Therefore, mathematical models of this kind are usually called * dynamic models (systems) * [4, 11,43, 44].

For static models, the mathematical model (2.1) is a mapping between two subsets of the properties of the modeled object * V * and * {X, V *, H}, which in vector form can be written as

The relations (2.1) and (2.2) can be specified in various ways: analytically (using formulas), graphically, tabularly, etc. Such relations can be obtained in a number of cases

through the properties of system 5 at specific times, called * states. * The state of system 5 is characterized by vectors

where *; (0) at the instant/e (/ 0 , __ 7) (/ 0 ) at the instant t> 0 (/ 0 , 7), etc., & pound; = 1, n g . __

If we consider the process of the functioning of system 5 as a successive change of states (/), r 2 (/), * r * to (/), then they

can be interpreted as the coordinates of a point in the ^ -dimensional phase space, with each phase of the process corresponding to a certain phase trajectory. The set of all possible state values * {r} * is called the * state space * of the simulation object * Z t * where * r to * e * Z. *

The states of system 5 at the moment of time completely

are determined by the initial conditions 7 ° = (2 ° 1 , 2 2 °, * г * ° to ) [where

* ° 1 = * 1 (* o) * ° r = * 2 (^ o) - * ° * = ** (* o)] (i), the internal parameters * k * (/) and the actions of the external environment * V * (0, which occurred during the time interval -/ 0 , using two vector equations

The first equation for the initial state * r ° and * defines the exogenous variables * x, V, AND * defines the vector function (/), and the second by the obtained state value * r * (/) - endogenous variables at the output of the system * y * (/). Thus, the chain of equations of the object input-states-output allows * define * the characteristics of the system

In general, the time in the model of the system * H * can be considered in the simulation interval (0, * T) * as continuous or discrete, that is, the number of times A by the line A/time units each, when * T = mA1, * where * m - * 1, * m T * is the number of sampling intervals.

Thus, by * mathematical model of object * (real system) is meant a finite subset of variables * (x * (/), * ь * (/), * AND * (r)} together with the mathematical relationships between them and the characteristics * y * (/) [4, 9, 10, 35].

If the mathematical description of the modeling object does not contain elements of randomness or they are not taken into account, ie if

we can assume that in this case the stochastic effects of the external environment * V * (/) and the stochastic internal parameters * AND * (/) are absent, then the model is called * deterministic * in the sense that the characteristics are uniquely determined by deterministic input actions

Obviously, a deterministic model is a particular case of a stochastic model.

### Typical schemes.

The above mathematical relations are mathematical schemes of a general kind and allow us to describe a wide class of systems. However, in the practice of modeling objects in the field of system engineering and system analysis, it is more rational to use the standard mathematical schemes: differential equations, finite and probabilistic automata, queuing systems, Petri nets, etc., at the initial stages of system research.

Without such a degree of generality, as the models considered, typical mathematical schemes have the advantages of simplicity and clarity, but with a significant narrowing of the possibilities of application. As deterministic models, when random factors are not taken into account in the study, differential, integral, integro-differential and other equations are used to represent systems operating in continuous time, and for the representation of systems operating in discrete time, finite automata and finite-difference scheme. Stochastic models (with allowance for random factors) for the representation of systems with discrete time are used by probability automata, and for the representation of a system with continuous time, queuing systems, etc.

The above typical mathematical schemes, of course, can not claim the possibility of describing on their basis all the processes occurring in large information-control systems. For such systems, in a number of cases, the use of aggregative models is more promising [4, 37]. Aggregative models (systems) allow us to describe a wide range of objects of research with a display of the systemic nature of these objects. It is with the aggregative description of a complex object (the system) is divided into a finite number of parts (subsystems), while maintaining connections that ensure the interaction of parts.

Thus, in constructing mathematical models of the processes of functioning of systems, the following basic approaches can be singled out: continuously deterministic (for example, differential equations); discrete-deterministic (finite automata); discrete stochastic (probabilistic automata); continuous-stochastic (queuing systems); generalized, or universal (aggregative systems).

The mathematical schemes discussed in the following paragraphs of this chapter should help to operate different approaches in practical work when modeling specific systems.

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