Finding the variance and moments of the number of failures in models of restoration processes of technical and information systems

This article examines models of restoration processes with variable distribution functions of the mean time to failure of elements. Analytical expressions and integral equations are derived for calculating the variance and initial moments of the number of failures, enabling the formulation of new optimization problems in the management of technical and information systems. This enables the formulation of new optimization problems for technical and information system operation strategies related to the average number of failures, the variance of the number of failures, and the average cost of restoration. Here, the average cost of restoration and the cost intensity can be assigned the meaning of "price", the average number of failures and the availability factor the meaning of "quality", and the variance of the number of failures and the cost of restoration the meaning of "risk". For example, consider the problem of minimizing the variance of the number of failures under given constraints on the average number of failures or the average cost of restoration. As an example, an expression for the variance of the number of failures of a second-order periodic restoration process with an exponential distribution of the mean time and its asymptotic behavior is obtained. The paper formulates the problem of asymptotic behavior of the variance of the number of failures, investigated in the mathematical theory of reliability for the recovery function of the considered models of recovery processes.

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