Imperfect repair and lifesaving in heterogeneous populations

In this theoretical paper we generalize the notion of minimal repair to the heterogeneous case, when the lifetime distribution function can be modeled by continuous or a discrete mixture of distributions. The statistical (black box) minimal repair and the minimal repair based on information just before the failure of an object are considered. The corresponding failure (intensity) rate processes are defined and analyzed. Demographic lifesaving model is also considered: each life is saved (cured) with some probability (or equivalently a proportion of individuals who would have died are now resuscitated and given another chance). Those who are saved experience the statistical minimal repair. Both of these models are based on the Poisson or non-homogeneous Poisson processes of underlying events, which allow for considering heterogeneity. We also consider the new model of imperfect repair in the homogeneous case and present generalizations to the heterogeneous setting.