METHODS MODELING SYSTEMS FOR THE IMPROVEMENT OF THEIR RELIABILITY

The method of Markov’s processes for the analysis of systems with constant bounce and recovery intensities considered. The article presents calculations of the failure probability of the system for describing the various cases of redundancy of its components using Markov’s models. Expressions obtained for calculating the approximate value of the failure probability of the system and analyzed of failures to improve the reliability of the system. The Markov’s graph of transitions in the reservation of the system, which reflects its behavior, described. Analysis of the results of numerical solution of systems shows that when loaded with redundancy, the probability of failure is higher than with partially loaded, and with partially loaded higher than with unloaded backup. A tree of errors for the system of cooling and clearing of flue gas at the reservation made by replacing "2 of 3", which has seven minimum bounce cross sections. Calculated the probability of system failure. The obtained calculations allow to analyze failures of technical systems in order to increase the reliability of their functioning. KEYWORDS

Due to the comparative simplicity and clarity of the mathematical apparatus, the high probability and accuracy of the obtained decisions, Markov processes are of particular interest in risk assessment and design of decision support systems.
Standard approaches for reliability are based on a probabilistic model, which is often inappropriate for tasks of this kind [1,2]. Probability theory is often a complex and not intuitive approach, the result of which is difficult to analyze. Similarly, probabilistic analysis usually requires more information about the system than it is known about, for example, the distribution of failure rates [3]. Typically, this leads to false assumptions about the raw data. The probabilistic paradigm also has many limitations when applied to small-volume samples [4].
Analysis of the development of technical systems allows us to conclude that, despite the rapid development of such areas as systems theory, including theory of automatic control, theory of reliability, theory of security, to describe the behavior of complex systems of existing mathematical models and methods is not enough. This position is clearly reflected in the research of A.V. Akimova, M.A. Yastrebenetsʹkoho, H.M. Druzhynina [5][6][7]. Also, the risk of technical systems at different times was addressed by such researchers as M. Rasmussen, O.Renn, B.V. Hnedenko, I.A. Ryabinin et al. They noted that it was impossible to ignore this area of study of the security of technical systems.
Approaches using failure trees and their varieties are well adapted to analyze the reliability of technical systems, but they are somewhat limited in application to real complex systems.
Partial failures, coverage, system serviceability, and other important reliability issues are well covered by the failure tree analysis method [8]. An alternative to this approach is to use Markov processes. However, a review of their models showed that they were not sufficiently investigated in the problems of reliability of technical systems.
Purpose of the study is to perform system failure calculations to describe the various cases of redundancy of its components using Markov models. Obtain expressions to calculate the approximate value of the system failure probability and analyze the failures to improve the reliability of the system.
Research results. Consider the Markov process method for analyzing systems with constant failure rates and recoveries (  − conditional failure flow rate,  − conditional recovery flow rate).
The following expression system can be used to determine the conditional failure rate:  The probability of a system failure is the probability that ( ) 1. x t t +  = This probability, in turn, can be expressed in terms of two possible states () xt and corresponding transitions to the state ( ) 1: The last equation can be rewritten as: From where do we find: with the following initial conditions (0) 0. Q =  Special cases of partially loaded redundancy (0 <  <  ) are unloaded redundancy (  = 0) and unloaded redundancy (  =  ). The recovery rate of all components in the system is the same and equal  . For all types of redundancy considered above, the system is considered to have failed if it went to state 5.
Denote by () i Pt the probability that the system is in a state i at time t . The derivative of this probability is as follows: The use of the expression given for the system under consideration makes it possible to construct the following system of differential equations: The first equation in (3)  ( ). P P t P t P P t P t P P t = + = + = The system of differential equations (5) describes a system whose transition graph contains three states -(0), (1), and (2) (Fig. 3). The intensity of the transition stream coming out of state (0) is equal  + , and the intensity of the input stream is  . shows that the probability of failures is higher in the case of loaded redundancy than in the case of partially loaded and higher in the case of unloaded redundancy.

Fig. 4. Dependencies of the probabilities of failure of elements A and B on time
The numerical solution of problem (5) gives the following values of failure probabilities (Table 1): which describes a Markov graph of the states of the inability and inability of a component at constant values of the intensities of failures and recoveries [12].
Applying the Laplace transform, we have: The probability of system failure is generally calculated as: upper limit for rejection Let us now consider a system where pumps of a cooling device are switched on in the scheme "2 of 3". Suppose that the failure rate of each of the cooling unit pumps is equal to  when the corresponding pump is in operation  and when it is in reserve.
The transition graph for this system is shown in Fig. 5. Condition (0) corresponds to the situation when two pumps are in operation and one is in reserve. State (0) corresponds to three substations (1, 2, 3), each of which can go to state (1), and the intensities of the respective transitions are the same and make up 2 + . The intensity of the transition from state (0) to state (1) is given by: As a result of solving this system of differential equations, one can determine the probabilities of states.
In order to be able to work, it is necessary that at least two of the three pumps of the cooling unit available are functional. Thus, the parameter value () r Qt for the cooling system of the system under consideration, which is equal to the probability that "less than two cooling system pumps are operable", is given by the following expression: The numerical solution of problem (11) gives the following values of failure probabilities (   (11) The error tree for the system of cooling and purification of fugitive gas during redundancy reservation according to the scheme "2 of 3" (Fig. 7) has seven minimum failure sections: , , , , , , , , , .

C E H A B B D D A F G
The upper and lower bounds of the system failure probability are: Error tree for the cooling and purification system of the fugitive gas during redundancy reservation according to the scheme "2 of 3" Following the above methodology, we calculate the probability of failure of the system ( Table 2). The probability of a system failure for a time of 1000 hours lies within 0,0744064 ( ) 0,076274. s Qt  3. No more than r link components may be updated at any one time. The circuit of the scheme "m with n" is described by the following system of differential equations:

Conclusions.
1. No complex system can have absolute security. However, society cannot allow the possibility of serious accidents when operating such systems. Therefore, one of the main tasks of science is the justification of quantitative security requirements and the creation of methods for calculating security systems with risk.
2. The Markov Process Model is an adequate method for analyzing the fault tolerance of systems. This method works well with bounce trees -a well known tool for reliability.
3. The obtained calculations of the approximate probability of failure of the system allow to analyze the failures of technical systems in order to improve the reliability of their functioning.