The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities $\bar{P}_k$ of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

中文翻译：

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基于平均失效时间函数的寿命分布类别

年龄替换策略的性能和有效性可以通过其平均故障时间 (MTTF) 函数来评估。我们基于 MTTF 函数在不同情况下为生命分布类别开发冲击模型理论，其中概率$\bar{P}_k$幸存的第一个ķ假设冲击具有离散的 DMTTF、IMTTF 和 IDMTTF 属性。在此背景下研究了 A-Hameed 和 Proschan [1] 的累积损伤模型，并建立了类似的结果。探讨了 IDMTTF 类生命分布中的弱收敛和矩收敛问题。还研究了在一些基本可靠性操作下 IDMTTF 属性的保留。最后，我们展示了 IDMRL 和 IDMTTF 类的交集包含 BFR 族，并建立了概述各种非单调老化类在层次结构中的位置的结果。