A continuous review preventative maintenance replacement model with a periodic preventative replacement policy

Let a discounted continuous review preventive-maintenance replacement model be such that its total discounted cost is given by means of two functional equations. The authors assume that downtime is caused by equipment breakdowns, and the length of a given downtime is the time necessary to repair the equipment and set it back in operation. The periodic preventive replacement policy is to replace equipment by new identical equipment when service age X is reached, or when the equipment fails. It is demonstrated that there exists only one value for X, which determines the optimal policy.