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We study a model of a two-stage degradation process in a dynamic environment. The two stages, the normal stage and the defective stage, are separated by the first hitting time of the alarm threshold by the degradation level. Wiener processes with piecewise linear drift are used in each stage to describe the degradation level in a dynamic environment. System failure is triggered in two ways: the system degradation level reaches the defect-based failure threshold; the duration in the defective stage is larger than the duration-based failure threshold. Explicit expressions for the system reliability for different duration-based failure thresholds are obtained. These include when the duration-based failure threshold is zero, when it is a positive constant and when it tends to infinity. A simulation procedure is described for the case in which the duration-based failure threshold is a random variable. Finally, some numerical examples are presented to illustrate the proposed reliability assessment method. The modelling method and the results can be used for reliability evaluation, residual life prediction and maintenance decision optimization of a system with two-stage degradation in a dynamic environment. [ABSTRACT FROM AUTHOR]

• A two-stage degradation system with two performance characterizes is considered. • A correlated bivariate Wiener process is adopted in modeling. • The explicit expressions of system reliabilities and availabilities are derived. • Two algorithms for the simulations of system reliabilities are given. It is still a challenge to study the degradation mechanisms of complex systems with multiple performance characteristics. This paper develops a two-stage stochastic degradation model. The degeneration processes of two correlated performance characteristics are described by a correlated bivariate Wiener process in the first stage, in which neither of the degradation levels of performance characteristics reaches their thresholds. When one of the degradation levels of performance characteristics reaches its threshold, the system operates defectively in the second stage, and the system degradation process is described by the other performance characteristic which is modeled by a univariate Wiener process. The system fails completely when both the degradation levels of performance characteristics reach their thresholds. Explicit expressions of the system reliability and availabilities are given based on different inspection and maintenance policies. Simulation is also presented, and it shows that the analytical results and the simulation results are in good agreement. Finally, a detailed case study of the rail track geometry degradation is given to demonstrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

Highlights • Three types of thresholds including alarm line and two different failure thresholds are used to model the system reliability. • Two degradation process models, Wiener-Gamma and Gamma-Gamma, are developed. • Explicit formulas for the system reliability in terms of duration thresholds for the degradation models are presented. Abstract Stochastic degradation process models are developed in terms of cumulative degradation signals of systems under three types of thresholds including alarm line and two different failure thresholds. One of the failure thresholds corresponds to the degradation amount, and the other corresponds to the duration. We assume that the degradation process can be separated into two distinctive stages by a change-point which is the first passage time of the degradation process with respect to the alarm line. For simplicity, the first stage is regarded as an age-dependent (or non-stationary) Wiener process or an age-dependent (or non-stationary) Gamma process, and the second stage is regarded as an age-dependent (or non-stationary) Gamma process; thus, two special degradation models, Wiener-Gamma and Gamma-Gamma, are constructed. Specially, two-stage stationary Wiener-Gamma and Gamma-Gamma models can be viewed as special cases. The system reliability is defined as the probability that the degradation signals do not exceed a failure threshold and the duration of exceeding the alarm line is less than the duration threshold. Some reliability results including two cases in terms of duration thresholds (constants and random variables) are derived. The moments of lifetime are also given based on the results. In addition, simulation is carried out to verify the given results. And some numerical examples are presented to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

• A fixed change point model with a jump based on a Wiener process is developed. • The fixed change point model is extended to a random change point model. • Two phase changing patterns for the random change point model are proposed. • Analytical results of the system reliability indexes are given for Model 1 and the HPP case of PCP 2 in Model 2. A two-phase degradation system with a change point is commonly seen in many real-world degradation systems due to influence of internal mechanisms and external environments, such as lithium-ion batteries, light-emitting diodes (LEDs), plasma display panels (PDPs), and vacuum fluorescent displays (VFDs). Random jumps exist at the change points of the degradation processes. Motivated by this phenomenon, this paper attempts to model such kinds of systems based on the Wiener process. Specifically, In Model 1, a constant change point model is first considered. The analytical results of the system reliability, lifetime distribution, and the remaining useful lifetime (RUL) are obtained. Then, based on Model 1, a random change point model, i.e., Model 2, is proposed to explain the change point phenomenon which is influenced by the accumulative effect of a shock process. Two phase-change patterns (PCPs) are considered in Model 2. For PCP I, the change point is triggered when the number of shocks reaches a predetermined threshold. For PCP II, the change point is triggered when the accumulative shock damage exceeds the corresponding threshold. The system reliability indexes are studied by using analytical and simulation methods. Finally, several numerical examples are given to illustrate the approaches and results. [ABSTRACT FROM AUTHOR]