On the function of time distribution of a complex computing system uptime

Any space computing complex is a complicated system. A complicated system is understood as a set of functionally related heterogeneous devices designed to perform certain functions and solve problems facing the system. One of the important characteristics of a system is its uptime. This characteristic is often considered to be a random variable. However, such a mathematical model is quite limited, since the uptime depends on many characteristics (parameters) that describe a system. Therefore, the uptime can be assumed to be a continuous random field (that is, a random function of many variables). It is this approach that is used in this work. If there are certain restrictions on the uptime of a computing system, upper estimates are found for the distributions of a random number of system failures. Therefore, the problem of estimating Gaussian field distribution in Hilbert space arises.

Two theorems that allow calculating the probability of a Gaussian vector falling into a sphere of a given radius are proved in the paper.

The paper is devoted to the reliability of a computing system. The random number of a computing system failures v(r) is a characteristic of its reliability.  The v(r) distribution is the distribution of the sum of a computing system random uptime. It is impossible to write down the distribution v (r) explicitly. Therefore, one has to look for an estimate of these distributions from above. Assuming that the uptime of a computing system is the sum of many variables, the authors of the paper obtained the following results: it is shown that the problem of estimating the distributions of a random number of system failures can be considered as the problem of estimating the convergence rate  in the central limit theorem in Banach spaces; if there are certain restrictions on the uptime of a computing system, upper estimates are found for the distributions of a random number of system failures. The estimates obtained can be used for further research in the theory of computing systems reliability. Knowing these upper estimates, it is possible to predict the level of average costs for computer systems restoration, as well as for the development of special mathematical and algorithmic support for analysis systems, for management, decision-making and information processing tasks.