Survival of systems with time redundancy runs

For some repairable systems, the ‘initial failures’, to be called defects, if repaired within the specified period of time, do not result in the final/functional failure. This situation can be also interpreted as some time redundancy that can be encountered in various applications. However, if there is a run of k≥2 interruptions in performance of less than specified duration that do not result in the final failure each, this run at some instances can also be considered as a fatal failure. Moreover, this run can be random. To obtain the corresponding survival probabilities for systems under the described criteria of failures, along with the conventional method of random sums, we employ and develop further the approach based on the corresponding integral equations of the Volterra type. The latter allows for more operation-wise interpretations and further generalizations on the more complex problems. Finally, our general solutions for survival probabilities are presented as the corresponding Laplace transforms that can be easily inverted numerically. The specific case of exponential distributions of times to defect and repair are considered, and the detailed numerical examples illustrate our findings.