In this paper, the life distribution behavior of a generalization of the mixed \(\delta\)-shock models in the multi-state systems is studied. In this model, the k out of interarrival times between two successive shocks with a magnitude less than \(\delta\) have a disaster result on the system which causes a complete failure. In addition to this event, another factor called the magnitude of shock causes the failure of the system, such that if the magnitude of a shock is greater than another critical threshold \(\gamma\), then the system fails. Such model create a multi-state system with a number of different states. The survival functions of the lifetime, the time spent by the system in a complete working state, and the total time spent by the system in partially working states are derived and the corresponding first two moments are also computed. An application in industry is analyzed to illustrate the proposed methodology. A simulation study is also presented to illustrate the behavior of the survival functions.

中文翻译：

---

多状态系统广义混合$$ / delta $$δ-Shock模型的寿命分布行为

本文研究了多状态系统中混合\（\ delta \）-休克模型的推广的寿命分布行为。在该模型中，两次连续冲击之间的间隔时间之外的k个幅度小于\（\ delta \）会对系统造成灾难性后果，从而导致完全故障。除此事件外，另一个称为震级的因素会导致系统故障，因此，如果震级大于另一个临界阈值\（\ gamma \），则系统发生故障。这种模型创建了具有许多不同状态的多状态系统。得出寿命的生存函数，系统在完全工作状态下花费的时间以及系统在部分工作状态下花费的总时间，并计算出相应的前两个时刻。分析了工业应用，以说明所提出的方法。还提供了一个仿真研究来说明生存功能的行为。