Simple bounds on cumulative intensity functions of renewal and G-renewal processes with increasing failure rate...

The article considers point processes most commonly used in reliability and risk analysis. Short-term and long-term behavior for the point processes used as models for repairable systems(1) are introduced. As opposed to the long term, the term short term implies that a process is observed during an interval limited by a time close to the mean (or the median) of the respective underlying distribution. A new simple upper bound is proposed on the cumulative intensity function of the renewal process and G-renewal process with an increasing failure rate underlying distribution. The new bound is compared with some known bounds for the renewal process. Finally, a formal definition of "a boundary point" between the short-term repairable system behavior and long-term behavior is introduced. This point can also be used as a lower time limit beyond which the "long-term" Barlow and Proschan bound for the renewal process with NBUE underlying distribution could be effectively applied.