Simple Bounds on Cumulative Intensity Functions of Renewal and G‐Renewal Processes with Increasing Failure Rate Underlying Distributions

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 systems1 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.