This article is a study on the reliability characteristics of a system under a failure model called the generalized mixed $\delta$-shock model. We assume that the system is subject to shocks according to a stochastic process. Each shock may cause some damage to the system. The system fails either the magnitude of the damage caused by a shock exceeds a threshold ${\gamma }_2$ or the time between two consecutive shocks is less than a pre-specified threshold $\delta$ and simultaneously magnitude of the damage is bigger than a pre-specified critical threshold ${\gamma }_1$ (${\gamma }_1<{\gamma }_2$). The survival function and other characteristics of the system lifetime are investigated. By imposing a cost function, we arrive at an optimal replacement policy for the system based on the proposed failure model. Several examples are provided under which we illustrate the theoretical results numerically and graphically.

本文是对故障模型下的系统可靠性特性的研究，该模型称为广义混合 $\delta$冲击模型。我们假设系统是根据随机过程受到冲击的。每次电击都可能对系统造成一些损坏。系统因冲击而导致的损坏程度超过阈值时失败${\gamma }_{\mathrm{2个}}$ 或两次连续电击之间的时间小于预定阈值 $\delta$ 同时损坏程度大于预定的临界阈值 ${\gamma }_{\mathrm{1个}}$ （${\gamma }_{\mathrm{1个}}<{\gamma }_{\mathrm{2个}}$）。研究了系统寿命的生存功能和其他特征。通过施加成本函数，我们基于提出的故障模型得出了系统的最佳更换策略。提供了几个示例，在这些示例下我们以数字和图形方式说明了理论结果。