The analysis of multiple dependent degradation processes is a challenging research problem in the reliability field, especially for complex degradation processes with random jumps. To integrally handle the jumps with uncertainties and the dependence among degradation processes, we construct general multi-dimensional Lévy processes to describe multiple dependent degradation processes in engineering systems. The evolution of each degradation process can be modeled by a one-dimensional Lévy subordinator with a marginal Lévy measure. The dependence among all dimensions is described by Lévy copulas and the associated multiple-dimensional Lévy measure. The multi-dimensional Lévy measure is obtained from one-dimensional marginal Lévy measures and the Lévy copula. We develop the Fokker-Planck equations to describe the time evolution of the probability density for stochastic processes. The Laplace transforms of both reliability function and lifetime moments are then derived. Numerical examples are used to demonstrate our models in lifetime analysis. The results of this research are expected to provide a precise reliability prediction, help to avoid failures caused by multiple dependent degradation processes, and maintain the long-term operation of a system.

在可靠性领域，尤其是对于具有随机跳跃的复杂退化过程而言，对多个相关退化过程的分析是一个具有挑战性的研究问题。为了整体处理不确定性和退化过程之间的依存关系的跳跃，我们构造了通用的多维Lévy过程来描述工程系统中的多个因果退化过程。每个降解过程的演化可以由一维Lévy从属者以边际Lévy度量建模。Lévycopulas和相关的多维Lévy度量描述了所有维度之间的依赖性。多维Lévy度量是从一维边际Lévy度量和Lévycopula获得的。我们开发了Fokker-Planck方程来描述随机过程的概率密度的时间演化。然后导出可靠性函数和寿命矩的拉普拉斯变换。数值示例用于说明我们的寿命分析模型。预期该研究的结果将提供精确的可靠性预测，有助于避免由多个相关的退化过程引起的故障，并维持系统的长期运行。