Decline and repair, and covariate effects

The failure processes of repairable systems may be impacted by operational and environmental stress factors. To accommodate such factors, reliability can be modelled using a multiplicative intensity function. In the proportional intensity model, the failure intensity is the product of the failure intensity function of the baseline system that quantifies intrinsic factors and a function of covariates that quantify extrinsic factors. The existing literature has extensively studied the failure processes of repairable systems using general repair concepts such as age-reduction when no covariate effects are considered. This paper investigates different approaches for modelling the failure and repair process of repairable systems in the presence of time-dependent covariates. We derive statistical properties of the failure processes for such systems

Optimal replacement policy under a general failure and repair model: Minimal versus worse than old repair

We analyze the optimal replacement policy for a system subject to a general failure and repair model. Failures can be of one of two types: catastrophic or minor. The former leads to the replacement of the system, whereas minor failures are followed by repairs. The novelty of the proposed model is that, after repair, the system recovers the operational state but its condition is worse than that just prior to failure (worse than old). Undertrained operators or low quality spare parts explain this deficient maintenance. The corresponding failure process is based on the Generalized Pólya Process which presents both the minimal repair and the perfect repair as special cases. The system is replaced by a new one after the first catastrophic failure, and also undergoes two sorts of preventive maintenance based on age and after a predetermined number of minor failures whichever comes first. We derive the long-run average cost rate and study the optimal replacement policy. Some numerical examples illustrate the comparison between the as bad-as-old and the worse than old conditions

Virtual series-system models of imperfect repair

Novel models of imperfect repair are fitted to classic reliability datasets. The models suppose that a virtual system comprises a component and a remainder in series. On failure of the component, the component is renewed, and on failure of the remainder, the component is renewed and the remainder is minimally repaired. It follows that the repair process is a counting process that is the superposition of a renewal process and a Poisson process. The repair effect, that is, the extent to the system is repaired by renewal of the component, depends on the relative intensities of the superposed processes. The repair effect may be negative, when the intensity of the part that is a renewal process is a decreasing function. Other special cases of the model exist (renewal process, Poisson process, superposed renewal process and homogeneous Poisson process). Model fit is important because the nature of the model and corresponding parameter values determine the effectiveness of maintenance, which we also consider. A cost-minimizing repair policy may be determined provided the cost of preventive-repair is less than the cost of corrective-repair and the repairable part is ageing. If the remainder is ageing, then policy needs to be adapted as it ages

The semi-geometric process and some properties

The geometric process has been widely applied in reliability engineering and other areas since its introduction. One of its assumptions is that the times between occurrences of events are independent. This assumption is rather restrictive and can limit its application in the real world. This paper extends the geometric process to a new process, which we call the semi-geometric process, by relaxing this assumption. Some probabilistic properties of the process are derived and parameter estimation is described. A numerical example, based on a real-world dataset, is used to illustrate the model and validate the estimation methodology

Doubly geometric processes and applications

The geometric process has attracted extensive research attention from authors in reliability mathematics since its introduction. However, it possesses some limitations, which include that: (1) it can merely model stochastically increasing or decreasing inter-arrival times of recurrent event processes, and (2) it cannot model recurrent event processes where the inter-arrival time distributions have varying shape parameters. Those limitations may prevent it from a wider application in the real world. In this paper, we extend the geometric process to a new process, the doubly geometric process, which overcomes the above two limitations. Probability properties are derived and two methods of parameter estimation are given. Application of the proposed model is presented: one is on fitting warranty claim data and the other is to compare the performance of the doubly geometric process with the performance of other widely used models in fitting real world datasets, based on the corrected Akaike information criterion

Two new stochastic models of the failure process of a series system

Consider a series system consisting of sockets into each of which a component is inserted: if a component fails, it is replaced with a new identical one immediately and system operation resumes. An interesting question is: how to model the failure process of the system as a whole when the lifetime distribution of each component is unknown? This paper attempts to answer this question by developing two new models, for the cases of a specified and an unspecified number of sockets, respectively. It introduces the concept of a virtual component, and in this sense, we suppose that the effect of repair corresponds to replacement of the most reliable component in the system. It then discusses the probabilistic properties of the models and methods for parameter estimation. Based on six datasets of artificially generated system failures and a real-world dataset, the paper compares the performance of the proposed models with four other commonly used models: the renewal process, the geometric process, Kijima's generalised renewal process, and the power law process. The results show that the proposed models outperform these comparators on the datasets, based on the Akaike information criterion