Optimal backward and backup policies in reliability theory

We are sometimes faced with the problem of searching for the actual failure time of a unit when it has failed until time t(0). This is called a backward time problem of how much time we go back o search for the failure time from time t(0), and is solved by using the reversed failure rate. This paper proposes an extended model of multiple backward trials in which we find the actual failure time from time t(0). Another problem is that when an operating unit fails, we have to go back to the newest checking time and reconstruct it as recovery techniques, which is called a backup policy. This paper takes up three backup policies where the unit is checked at periodic, successive and random working times, and discusses analytically their optimal policies. Finally, it would be important to keep a record of operational behavior of a unit when its failure is detected, which is called traceability. This paper compares two models with traceability and without traceability, and proposes three traceability models where the unit operates for a finite interval, and is checked at periodic and successive times. Optimal policies for each model are discussed by using backup techniques.