Bayes prediction for the number of failures of a repairable system

After observing a repairable system for some time, we wish to predict the number of failures of the system in some fixed future interval. Such a prediction depends on the: 1) assumed model for the failure process, and 2) length of the interval. We use a Bayes approach to obtain point & interval predictions for the number of failures in a future interval. Two situations are discussed: 1) the power law process (PLP) governs failure times during the period of observation, but in the future interval the homogeneous Poisson Process (HPP) governs the failure times, and 2) the failure process is the PLP. A rationale and an example of each situation is presented. We discuss the use of informative & noninformative priors for the parameters of the failure process. The Bayes approach can incorporate both sources of uncertainty: 1) the number of failures in the future interval is random, so even if we knew the parameters of the failure process, we would still not predict with certainty the number of failures that would occur in a future interval, and 2) the parameters of the failure process are not known and must be estimated from the observed data.