Preventive maintenance with limited historical data

We consider a piece of equipment that is subject to breakdown. If breakdowns are very costly, it may be attractive to undertake preventive maintenance on a regular basis. We model the time until breakdown as a discrete random variable whose probability distribution is not exactly known (due to limited historical data). An optimal Bayesian strategy for selecting the preventive maintenance interval (T) is developed. In some circumstances the optimal strategy turns out to involve hedging due to the uncertainty in the probabilities, namely using a T-value 1 period less than what would be selected under the assumption that the probabilities are exactly known at their current (updated) values. Two simple heuristic methods are presented. On numerous tests one of them is shown to give very low percentage cost penalties.