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Evaluation model of the recovery processes of non-markovian systems, considering the elements unreliability under arbitrary distribution laws
(Національний технічний університет "Харківський політехнічний інститут", 2022) Raskin, Lev; Ivanchikhin, Yuriy; Sukhomlyn, Larysa; Svyatkin, Iaroslav; Korsun, Roman
The subject of the study is the reliability of recoverable non–Markovian systems, functioning of which is described by arbitrary distribution laws. The purpose of the article is to develop a mathematical model of the functioning of modern computer systems under arbitrary laws of the distribution of stay duration in each of the states, taking into account the recovery system and the provision of spare elements. The main task is to develop an adequate model of the system functioning process, taking into account the non-Markovian character of the processes occurring in the system, its possible large dimension, and the presence of a hierarchical recovery system. Based on this model, a method for calculating the density of the system recovery time distribution has been developed. At the same time, a universal four-parameter distribution is proposed to describe random processes occurring in the system. Using this approximation, the calculation of the desired parameter of the recovery flow is performed by solving the Volterra integral equation with a difference kernel.
Analysis of marсovian systems with a given set of selected states
(Національний технічний університет "Харківський політехнічний інститут", 2022) Raskin, Lev; Sukhomlyn, Larysa; Korsun, Roman
Analysis of stationary Marcovian systems is traditionally performed using systems of linear Kolmogorov differential equations. Such systems make it possible to determine the probability of the analyzed system being in each of its possible states at an arbitrary time. This standard task becomes more complicated if the set of possible states of systems is heterogeneous and some special subset can be distinguished from it, in accordance with the specifics of the system functioning. Subject of the study is technology development for such systems analysis. In accordance with this, the purpose of the work is to find the distribution law of the random duration of such a system's stay on a set of possible states until it falls into a selected subset of these states. Method for solving the problem is proposed based on splitting the entire set of possible states of the system into two subsets. The first of them contains a selected subset of states, and the second contains all the other states of the system. Now a subset of states is allocated from the second subset, from which a direct transition to the states of the first subset is possible. Next, a system of differential equations describing the transitions between the formed subsets is formed. The solution of this system of equations gives the desired result – distribution of the random duration of the system's stay until the moment of the first hit in the selected subset of states. The method allows solving a large number of practical problems, for example, in the theory of complex systems reliability with many different failure states. In particular, finding the law of the uptime duration distribution, calculating the average duration of uptime.
Analysis of multi-threaded markov systems
(Національний технічний університет "Харківський політехнічний інститут", 2021) Raskin, Lev; Sukhomlyn, Larysa; Sagaidachny, Dmytro; Korsun, Roman
Known technologies for analyzing Markov systems use a well-operating mathematical apparatus based on the computational implementation of the fundamental Markov property. Herewith the resulting systems of linear algebraic equations are easily solved numerically. Moreover, when solving lots of practical problems, this numerical solution is insufficient. For instance, both in problems of structural and parametric synthesis of systems, as well as in control problems. These problems require to obtain analytical relations describing the dependences of probability values of states of the analyzed system with the numerical values of its parameters. The complexity of the analytical solution of the related systems of linear algebraic equations increases rapidly along with the increase in the system dimensionality. This very phenomenon manifests itself especially demonstratively when analyzing multi-threaded queuing systems. Accordingly, the objective of this paper is to develop an effective computational method for obtaining analytical relations that allow to analyze high-dimensional Markov systems. To analyze such systems this paper provides for a decomposition method based on the idea of phase enlargement of system states. The proposed and substantiated method allows to obtain analytical relations for calculating the distribution of Markov system states. The method can be effectively applied to solve problems of analysis and management in high-dimensional Markov systems. An example has been considered.

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