Multi-dimensional Lévy processes with Lévy copulas for multiple dependent degradation processes in lifetime analysis

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.