Survival of repairable systems with decreasing time redundancy

For some repairable systems, the “initial” failures or defects, if repaired within the specified time period, do not result in the final/functional failures. This can also be interpreted as some time redundancy in various applications. However, with each consecutive repair, this specified time period is usually decreasing, showing a certain deterioration in the quality of repair that can also be interpreted as some imperfectness of repair. On the other hand, the threshold can be fixed, whereas the repair times can be stochastically increasing with each repair. To obtain the corresponding survival probabilities for systems under the described criteria of failures, we employ integral equations of the Volterra type that are solved via the Laplace transforms. The specific cases of exponential distributions of times to defect and repair, and of the geometrically decreasing redundancy time with each repair are considered and analyzed. Provided numerical examples illustrate our findings.