Multi-dimensional Lévy Processes and Lévy Copulas For Dependent Degradation Processes in Reliability Analysis




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Cumulative degradation is one of unavoidable failure mechanisms that occur in many engineering systems in chemical, civil, mechanical and other fields. These deterioration phenomena are caused by internal structures and dynamic external conditions. In critical systems (e.g., aircrafts, power systems, and railways), such degradation can lead to operational failure of a system, loss of economic profit, and even endangered human lives. Moreover, multiple dependent degradation processes can happen in a system simultaneously. In order to avoid failures in engineering systems, it becomes critical and urgent in reliability studies to develop new multi-dimensional dependent degradation models using appropriate stochastic processes. This dissertation aims to develop a framework to integrally handle the complicated degradation with uncertain jumps in multi-dimensional dependent degradation processes based on multi-dimensional Lévy processes and various Lévy copulas for reliability and lifetime analyses in various industries. To model the common degradation that is non-decreasing over time, we use Lévy subordinators that are a class of Lévy processes with non-decreasing paths. Random jumps are described by special Lévy measures for Lévy subordinators of interest. The relationship between high-dimensional Lévy copulas and the associated multi-dimensional Lévy measures is introduced in Chapter 3, which is the foundation to model the internal structures for multi-dimensional degradation processes. Based on multi-dimensional Lévy measures and high-dimensional Fokker-Planck equations, we derive the Laplace transform expressions for reliability function and lifetime moments. In Chapter 4, we study the reliability and lifetime through characteristic functions of multi-dimensional stochastic processes. We derive the reliability function for a two-dimensional degradation process modeled by two-dimensional Lévy subordinators. In Chapter 5, we consider the degradation and random jump with time dependence and extend the Lévy subordinators to non-homogeneous subordinators. The marginal/joint reliability functions and probability density functions of lifetime are derived for different types of Lévy measures. To illustrate our proposed models, we simulate multi-dimensional Lévy subordinators and conduct numerical analysis for these simulated processes under Lévy copulas, respectively. The results demonstrate that our multi-dimensional Lévy subordinator and non-homogeneous subordinator models perform well and provide a new methodology to analyze degradation, reliability and lifetime for a system that degrades over time.



Multi-dimensional Lévy processes, Multiple dependent degradation processes