Degradation is an important phenomenon for industrial products, which manifests as the gradually deterioration of some performance characteristics. The degradation process is often relevant to both time and usage, and indexing the degradation process merely by the time or usage cannot characterize the process accurately. Considering a stochastic usage process, this study proposes a degradation process model indexed by two scales, i.e., the time and the usage, where the degradation along the two scales are modeled as correlated nonlinear Wiener processes. We develop two simulation-based algorithms for reliability evaluation and study the model inference problems for the proposed model. The estimation procedure and the reliability assessment algorithms are validated by simulations. The performance of the proposed model is justified with an application to a real degradation dataset of outdoor coating materials, which shows that indexing the degradation process by two scales can considerably improve the degradation modeling performance.

Since the degradation process is dependent on the environmental stresses, degrading products operating under dynamic environments can have time-varying degradation rates. Existing studies generally exploit a random walk to model the time-varying degradation rate, considering the randomness of the environmental effects. The random walk is not stationary, while the real environments, although dynamic, are often stationary. The degradation process under a stationary environment would have a stationary degradation rate. Therefore, instead of the random walk, we propose to model the stochastic degradation rate by an autoregressive model. The autoregressive rate can accommodate the randomness and stationarity of the environmental effects. Conditional on the autoregressive degradation rate, a Wiener process is used to model the degradation process. We develop an Expectation-Maximization algorithm to perform maximum likelihood estimation of model parameters. Moreover, to facilitate remaining useful life prediction, we derive the explicit probability density function for the remaining useful life (RUL). We validate the proposed model by a simulation study and justify the applicability and performance of the proposed model by two real degradation datasets.

Most industrial products degrade with usage. To capture degradation patterns and schedule condition-based maintenance actions, it is necessary to model the degradation process and predict the system remaining useful life (RUL). In practice, the operating environments and loading conditions are dynamic and stochastic, which introduces extra randomness in the degradation process. To account for the dynamic environmental impacts, this article proposes a nonlinear Wiener process model with a random time-varying covariate. We model the dynamic covariate with an Ornstein–Uhlenbeck process, and link it to the time-varying degradation rate by an exponential form covariate-effect function. Therefore, the proposed Wiener process has a time-varying degradation rate impacted by the dynamic environment. We propose to estimate the model parameters by maximum likelihood estimation based on given observations of the degradation and covariate. We also propose a simulation-based procedure to obtain the RUL for the proposed model. The model is verified by Monte Carlo simulations. We apply the proposed model to a dust-cleaning fan degradation dataset and a lithium-ion battery degradation dataset, and the results show that the proposed model outperforms the existing degradation models in fitting the real data and RUL prediction.

Components in a system work under the same dynamic operating conditions, and their lifetimes are generally positively correlated. Ignorance of the correlation may lead to a significant bias in the evaluation of system reliability. Using the cumulative exposure principle, we model the equivalent operating time of the components, resulting from the cumulative effects of the dynamic environments, as a monotone increasing stochastic time scale. Commonly-used models, such as the compound Poisson, gamma, and the inverse Gaussian processes, are adopted for the stochastic time scale. Based on the above settings, reliability models for multicomponent systems are developed. We investigate how the stochastic time scale influences the system reliability and the correlations between component lifetimes. Under the stochastic time scale, the component lifetimes are shown to be positively quadrant dependent. Overlook of the correlation would overestimate the reliability of a parallel system but underestimate the reliability of a series system. When the stochastic time scale degenerates to a deterministic function of the calendar time, on the other hand, the system reliability becomes the reliability of the system where components work independently. The proposed models are successfully applied to lifetime data of brake pads in the automobile braking system.

This paper studies the defense strategy for a parallel system subject to an unintentional impact and an intentional impact. Existing research assumes that system damage is caused by either natural disasters (unintentional impacts) or strategic attackers (intentional impacts). However, in practice, the defender may encounter diverse scenarios, where these two kinds of impacts can happen in a sequential order. Without knowing the precise occurrence sequence, the defender must allocate its limited resources against two successive impacts. In each contest, the defender can construct redundant elements and protect them from damage, to minimize the expected loss from system destructions. Illustrative examples of the optimal strategy are presented in three cases: first, where the unintentional impact comes first; second, where the intentional impact comes first; and third, where the two impacts come in an uncertain order. Supplemental protection is considered in the model extension, where the defender can allocate additional protection resources to those elements that have survived after the first impact.

In degradation tests, it is common to see that the initial degradation levels of test units are heterogeneous. Moreover, the degradation rate of a path may also be correlated with the initial value of the degradation measure. Motivated by this observation, in this paper, we introduce a time shift to the traditional stochastic process models, which presumes that the product has experienced some degradation at the beginning of the test. Such a modeling technique can also capture the correlation between the initial degradation and the degradation rate, when the degradation rate of each path does vary from unit to unit. We apply this technique to the three popular stochastic process models, i.e., the Wiener process, the gamma process, and the inverse Gaussian process, and develop the corresponding parameter inference procedures. Monte Carlo simulations are implemented to validate the proposed models and the estimation procedures. Applications to the degradation analysis of block error rates data and GaAs laser data reveal good performance of the proposed models.

Degradation modeling plays an important role in system health diagnosis and remaining useful life (RUL) prediction. Recently, a class of Wiener process models with adaptive drift was proposed for degradation-based RUL prediction, which has been proven flexible and effective. However, the existing studies use an autoregressive model of order 1 for the adaptive drift, which can result in difficulties in both model estimation and RUL prediction. This paper proposes a new adaptive Wiener process model that utilizes a Brownian motion for the adaptive drift. The new model shares the flexibility of the existing models, but avoids the difficulties in model estimation and RUL prediction. A model estimation procedure based on maximum likelihood estimation is developed, and the RUL prediction based on the proposed model is formulated. The effectiveness of the model in RUL prediction is validated using simulation and through an application to the lithium-ion battery degradation data.

This paper models repairable computing systems performing a mission that is successful if the system can accomplish a specified amount of work within the allowed mission time or deadline. During the mission the system is subject to a sequence of full and incremental data backup procedures to facilitate an effective system recovery and avoid repeating the entire mission work from the very beginning when a system failure happens. The repair time is fixed while the system time-to-failure can follow any arbitrary type of distributions. This paper makes novel contributions by first developing a new numerical algorithm to evaluate mission success probability and expected completion time of the considered repairable real-time computing systems subject to mixed full and incremental backups. Correctness of the proposed evaluation algorithm is verified using Monte Carlo simulations. We make another new contribution by formulating and solving the backup schedule optimization problem that finds the full and incremental backup frequencies maximizing the mission success probability. Through illustrative examples, effects of different parameters (including the system time-to-failure distribution parameter, maximum allowed mission time, data backup and retrieval times, storage availability, repair time and efficiency) on the mission success probability and expected completion time as well as on the optimal backup schedule solution are investigated.

To improve the system reliability while conserving the limited system resources, cold standby sparing is often used. In computing tasks, because active components fail randomly, and the standby component has to pick up the mission task whenever required, scheduled backups are often implemented to save the completed portions of the task. The backups can facilitate an effective system recovery where the standby component can take over the mission task from the last backup point instead of resuming the mission task from the very beginning. This paper considers a k-out-of- n cold standby system subject to scheduled backups, where k components are online and operating, with the remaining components waiting in the unpowered, cold standby mode. Whenever an online component fails, a cold standby component is activated to take over the mission task from the last backup point. The backup intervals are deterministic, but can be even or uneven. As the component may fail due to an imperfect switching from the standby state to the fully powered up state, the switching failure is also considered in the system model. A multi-valued decision diagram (MDD)-based analytical approach is proposed to evaluate the reliability of the considered system, and its complexity is analyzed. The proposed method is applicable to systems with non-identical components following arbitrary lifetime distributions. Examples are given to illustrate the MDD-based method. The correctness and efficiency of the proposed method are verified using Monte Carlo simulations.

Optimal component order can effectively improve the system reliability for the design of a warm standby sparing (WSP) system. The optimal component order in the 1-out-of- n WSP system is studied in this paper. The optimal component order strategies in terms of the expected system lifetime are proposed, and proved for several cases. Furthermore, the 1-out-of- n WSP system with exponential components is investigated from the viewpoint of the system reliability; it is proved that the components should be activated according to the descending order of their failure rates. These results are very useful and applicable in the design of 1-out-of- n WSP systems.