Suppose that a system is subject to a sequence of shocks which occur with probability p in any period of time $t=1,2,\dots$, and suppose that ${\delta }_1$ and ${\delta }_2$ are two critical levels ($1\le {\delta }_1<{\delta }_2$). The system fails when the time interval between two consecutive shocks is less than ${\delta }_1$, and the time interval bigger than ${\delta }_2$ has no effect on the system activity. In addition, the system fails with a probability of, say, $\theta$, when the time interval varies between ${\delta }_1$ and ${\delta }_2$. Therefore, this model can be regarded as an extension of discrete time version of $\delta$-shock model, and such an idea can be also applied in the extension of other shock models. The present study obtains the reliability function and the probability generating function of the system’s lifetime under this model. The present study offers some properties of the system and refers to a generalization of the new model. In addition, the mean time of the system’s failure is obtained under reduced efficiency which is created when the time between two consecutive shocks varies between ${\delta }_1$ and ${\delta }_2$ for the first time.

假设系统在任何时间段内遭受一系列以概率p发生的冲击$Ť=\mathrm{1个}，2，\dots$，并假设 ${\delta }_{\mathrm{1个}}$ 和 ${\delta }_2$ 有两个关键级别（$\mathrm{1个}\le {\delta }_{\mathrm{1个}}<{\delta }_2$）。当两次连续电击之间的时间间隔小于${\delta }_{\mathrm{1个}}$，且时间间隔大于 ${\delta }_2$对系统活动没有影响。此外，系统发生故障的可能性为$\theta$，当时间间隔在 ${\delta }_{\mathrm{1个}}$ 和 ${\delta }_2$。因此，该模型可以看作是离散时间版本的扩展。$\delta$-shock模型，这种思想也可以应用于其他冲击模型的扩展。在此模型下，本研究获得了系统寿命的可靠性函数和概率生成函数。本研究提供了系统的一些属性，并引用了新模型的一般化。此外，系统故障的平均时间是在效率降低的情况下获得的，效率降低是在两次连续电击之间的时间间隔变化时产生的。${\delta }_{\mathrm{1个}}$ 和 ${\delta }_2$ 首次。