# Mathematical models of processes. - Modeling of systems

## Mathematical models of processes.

After moving from the description of the simulated system 5 to its model N *, constructed according to the block principle, it is necessary to construct mathematical models of processes occurring in different blocks. The mathematical model is a set of relationships (for example, equations, logical conditions, operators) that determine the characteristics of the process of the system 5 functioning depending on the structure of the system, behavior algorithms, system parameters, environmental conditions E, of the initial conditions and time. Mate -

Fig. 3.2. The model of the system: a - zhogaetstuzhlylaya; b is a block

The mathematical model is the result of the formalization of the process of functioning of the system under study, that is, the construction of a formal (mathematical) description of the process with the degree of approximation to reality necessary in the framework of the study [4, 35, 37].

To illustrate the possibilities of formalization, let us consider the process of functioning of some hypothetical system 5, which can be divided into t subsystems with the characteristics y 1 (/), in 2 ( 0 - parameters A x , L 2 , A " I with input effects x and x 2 , ... x nx and environmental impacts 1e i 2 , ... ь pu . Then the mathematical model of the process can be a system of relations of the form

If the functions/ 1 # / 2 , / t were known, then relations (3.1) would turn out to be an ideal mathematical model of the process of the functioning of system 5. However, in practice, it is usually impossible to obtain a model for a large system in large systems, therefore, the process of functioning of system 5 is usually divided into a number of elementary subprocesses. At the same time, it is necessary to subdivide into subprocesses so that building models of individual subprocesses was elementary and did not cause difficulties in formalization. Thus, at this stage, the essence of the formalization of subprocesses will consist in the selection of typical mathematical schemes. For example, for stochastic processes, these can be schemes of probability automata (P- schema), queuing schemes of the d-scheme , etc., which describe the basic features of real phenomena that make up subprocesses, from the point of view of solved applied problems.

Thus, the formalization of the functioning of any system 5 must precede the study of its constituent phenomena. As a result, a meaningful description of the process appears, which represents the first attempt to clearly state the patterns characteristic of the process under investigation and the formulation of the applied problem. The content description is the starting material for the subsequent stages of formalization: the construction of a formalized scheme of the process of the functioning of the system and the mathematical model of this process. To model the process of computer operation on a computer, it is necessary to transform the mathematical model of the process into the corresponding modeling algorithm and the machine program.

Sub-stages of the first stage of modeling. Let's consider in more detail the main sub-stages of constructing the conceptual model of the system and its formalization (see Figure 3.1).

1.1. Statement of the problem of computer simulation of the system. We give a clear formulation of the problem of studying a concrete system, 5 and focuses on such issues as: a) recognition of the existence of the problem and the need for machine modeling; b) choice of methods for solving the problem, taking into account available resources; c) determining the scale of the problem and the possibility of splitting it into sub-tasks.

It is also necessary to answer the question of the priority of solving various subtasks, to evaluate the effectiveness of possible mathematical methods and software and hardware for their solution. Careful study of these issues allows us to formulate the research task and proceed to its implementation. It is possible to revise the initial statement of the problem in the modeling process.

1.2. Analysis of the problem of modeling the system. Carrying out an analysis of the problem helps overcome the difficulties that arise in the future when solving it by the modeling method. In the second stage under consideration, the main work is reduced to the analysis, i