Modeling Repairs of Systems with a Bathtub-Shaped Failure Rate Function
  Most of the reliability literature on modeling the effect of repairs on systems assumes the failure rate functions are monotonically increasing. For systems with non-monotonic failure rate functions, most models deal with minimal repairs (which do not affect the working condition of the system) or replacements (which return the working condition to that of a new and identical system). We explore a new approach to model repairs of a system with a non-monotonic failure rate function; in particular, we consider systems with a bathtub-shaped failure rate function. We propose a repair model specified in terms of modifications to the virtual age function of the system, while preserving the usual definitions of the types of repair (minimal, imperfect and perfect repairs) and distinguishing between perfect repair and replacement. In addition, we provide a numerical illustration of the proposed repair model. 
Sima Varnosafaderani and Stefanka Chukova
stat.AP