A Block Replacement Policy for Systems Subject to Non-homogeneous Pure Birth Shocks

This note studies the block replacement policy with general repairs for an operating system subject to shocks occurring according to a non-homogeneous pure birth process. A shock causes the system to fail. There are two types of failures: a type-I failure (minor failure) is fixed by a general repair, whereas a type-II failure (catastrophic failure) is removed by an unplanned (or unscheduled) replacement. The failure type probabilities depend on the number of type-I failure shocks that occurred since the last replacement. Under the block replacement policy, the operating system is replaced every T time units to reduce the chances of more expensive unplanned replacements due to type-II failures. The aim of this note is to determine the optimal block interval T*, which minimizes the expected cost rate and the expected total discounted cost rate of the proposed policy. As the shock process is a more general non-homogeneous pure birth process, several previous models become the special cases of our model.