Degradation modeling based on the gamma process with random initial degradation level and random threshold

Abstract

The stochastic modeling of performance characteristics is an important approach for the quality assessment of products. As part of the stochastic modeling, it is possible to consider different variation sources such as temporal, unit-to-unit heterogeneity and measurement error. These sources have been incorporated into different stochastic processes in the literature. Considering the unit-to-unit heterogeneity, two sources of variation can be found for certain products, particularly when the initial level and the critical threshold vary from unit-to-unit. Although, these sources have been studied separately, they have not been studied simultaneously. In this paper, a modeling approach is proposed which considers the gamma process to take into account the temporal uncertainty and the initial degradation level and the critical threshold level as two additional sources of variation. The modeling consists in first obtaining the distribution of the subtraction of the critical threshold and the initial level via a deconvolution operation. Then, the cumulative distribution function of the first passage time distribution is obtained through numerical integration. A simulation study is conducted to evaluate the performance of the proposed method, where different scenarios are analyzed. Furthermore, this approach is applied to a fatigue-crack propagation case study, from which a reliability assessment is performed.