Age replacement policy is concerned with finding an optional time tominimize the cost, at which time the unit is replaced even if itdoes not fail. So far, age replacement policy involving random agehas been proposed. This paper will assume the age of the unit is anuncertain variable, and find the optimal time to replace the unit.

The alternating renewal process is one of the renewal processes in renewal theory. It is widely used in inventory systems and queueing systems. In classical alternating renewal processes, one of the assumptions is that the on times and off times are independent random variables. The notion of a fuzzy set was introduced by Zadeh [1] who defined it as a generalized characteristic function. Recent...

This article describes the Key renewal theorem and the Blackwell’s renewaltheorem. These two limit theorems for renewal processes are equivalent but of different forms.They are particularly useful for characterizing the asymptotic behavior of a probabilisticquantity of interest in a renewal process. We present two applications of these limit theorems:the limiting distributions o...