Abstract: The problem on replacement policies for a random checking repairable system is studied in this paper.In order to improve systems safety and reliability,a random overhauling before system failures is considered,and the repairable models are studies in this case.Under the assumption that the random over-hauling is as good as old,the successive survival time of the system constitutes a decreasing geometric process stochastically and the consecutive repair time of the system constitutes an increasing geometric process,and the repair after system failures is not as good as new,we consider replacement policies based on the failure number of the system.By using the monot one geometric process,the explicitex pression of the long-run expected profit per unit time is derived.Finally,we analyze the result through numerical example too.

Key words: expected profit, random overhauling, failure repair, renewal process, geometric process