Modeling and reliability analysis of systems subject to multiple sources of degradation based on Lévy processes

Abstract

In this paper we present a framework to model the effect of multiple sources of degradation on deteriorating systems, and to find easy-to-evaluate expressions for important reliability quantities. By considering the system deterioration as a general increasing Lévy process (i.e., a subordinator), which is a process with independent, stationary and non-negative increments, the proposed methodology allows the modeling of shock-based degradation (in the form of compound Poisson processes) with different distributions of shock sizes, progressive degradation as deterministic linear drift plus a Gamma process, and multiple sources of degradation by the superposition of any of these models into the same mathematical formalism. In addition, we obtain expressions for the reliability quantities (reliability function, and probability density and moments of the lifetime L), and the mean and central moments of the deterioration process Xt. Moreover, we present efficient and accurate numerical methods to compute the reliability quantities and to simulate sample paths. Several deterioration models are compared in terms of their reliability quantities, the simulated sample paths and the feasible moments of the deterioration Xt. Furthermore, we propose a method to mix different Lévy models that extend the available moments with possible applications to data fitting. The results demonstrate the generality, versatility, efficiency and accuracy of the proposed Lévy degradation framework, which can open a new productive research field in the area of probabilistic degradation models.