The IFR/DFR property of the forward recurrence-time distribution

When a failure or replacement process is modeled as a renewal process, the residual life of the unit in use at a given time is generally referred to as the forward recurrence time (RT). Distributional properties of this random variable are critically important in many applications. This paper investigates the extent to which the failure-rate function monotonicity of a life distribution is inherited by the forward RT distribution at time s of its renewal process. For DFR life distributions, the forward RT distribution is also DFR for every s⩾0. However, the corresponding property does not necessarily hold for IFR life distributions. The forward RT distribution is IFR in the limit as s→0 and as s→∞. For IFR Weibull life distributions, we demonstrate numerically that the forward RT distribution is IFR for small s. As s is increased, it alternates between being IFR and non-IFR in an interesting cyclical pattern, and remains IFR beyond a large enough s