Solution of the integral equation for the average cost of restoration in the theory of reliability of technical systems

Failures of elements during the operation of technical and many other systems are, as a rule, random in nature. This leads to various models of the recovery process, studied in probability theory and mathematical reliability theory. During the restoration process, failed elements are restored or replaced with new ones, and there is often a change in the costs and quality of the restored elements (time-to-failure distribution functions). The work examines the cost function (average cost of restoration) in the process of restoration of order in which, according to a certain rule, the costs of each restoration and the distribution functions of operating time change. Considering, that the recovery function (average number of failures) is well studied in reliability theory, a solution to the integral equation for the cost function is obtained through the recovery function of the model under consideration. For the order restoration process a formula is obtained for calculating the cost function through the restoration function of a simple process formed by the convolution of all distribution functions of the periodic part. For practical application, explicit formulas are obtained for the cost function during the restoration process, in which the periodic part is distributed according to an exponential law or an Erlang law of order with the same exponent α. The resulting formulas can be used to study the properties of the cost function and solve optimization problems in strategies for carrying out the restoration process in terms of price, quality, risk, if, for example, the average number of failures is taken as quality, the average cost of restorations as price, the dispersion of the number of failures as the risk, or cost of restoration.