Calculation of the mean restoration time, of an item based on the characteristics of the restoration process

The well-known formulas for calculating the mean restoration time (as a mathematical expectation of the restoration time of an item–after-failure operational state) assume that the characteristics of the restoration process are set on the entire time axis. This circumstance makes it difficult to use these formulas in solving engineering problems, since, on the one hand, a specific restoration process is limited by the duration of repair works, and, on the other hand, the approximation of the restoration process over an adjacent infinite period of time after the completion of the repair works is not adequate to the process as observed. Then a problem to solve is to calculate the mean restoration time of an item through the characteristics of the restoration process, knowing the ways they work for a limited period of time, the length of which is determined by (i), the duration of repair works, and (ii), their specific conditions. In this paper, such conditions are considered as some specified level of the gamma-percentile restoration time of an item, as well as the restoration rate of an item, which results from the repair works. The selected characteristics of the restoration process have turned out to be sufficient to solve the task. In this paper, the formulas are proved for calculating the indicator “mean restoration time” based on the functional manifestations of the restoration process characteristics (taking into account the specified gamma-percentile restoration level, and the restoration rate) during the repair works to restore the operational state of an item after failure. The proved formulas are valid for any law of the restorative process. The paper also provides two examples of calculating the mean restoration time. In the first example, the uniform law of the restoration process is considered for the time interval of the item being repaired after failure, and in the second example, its exponential law is under study.