General Repair Model Research Articles

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.

Mining repair model for exception-related bug

It has long been a hot research topic to detect and to repair bugs automatically. As a common practice, researchers propose approaches for specific bugs, and their approaches typically are limited in handling the variety among bugs. Recently, researchers start to explore automatic program repair. With predefined repair operators and test cases, test-based repair approaches use search algorithms to generate patches for a bug, until a patch passes all the test cases. To improve the effectiveness to generate patches, Martinez and Monperrus (2013b) proposed an approach that mines repair models from past fixes. Although their approach produces positive results, we argue that it can be feasible to further improve their approach, if we mine repair models for bug categories, instead of all bugs. However, the benefits are still unclear, since existing benchmarks do not classify bugs into categories and existing approaches cannot mine repair models for bug categories. In this paper, we implement a tool, called ExFi, that classifies bugs into categories based on their related exceptions. With its support, we construct a benchmark, in which bug categories are marked. Furthermore, we propose an approach, called MiMo, that mines a repair model for each exception. We compared the general repair model with our mined repair models. Our results show that our mined models are all significantly different from the general model. Outside of the projects where our models are mined, we selected 59 real bugs. For each bug, we used our models and the general model to generate correct repair shapes for these bugs. The results show that for 43 out of 59 bugs, our models found faster a correct shape than the general repair model (Martinez and Monperrus, 2013b), and for 5 bugs, our models were able to find correct shapes that were not found by the compared model.

Warranty claims forecasting based on a general imperfect repair model considering usage rate

Because manufacturers of products sold with any type of warranty incur additional costs from servicing customer warranty claims, accurately forecasting the number of warranty claims has become increasingly important to manufacturers. Under the assumption of Weibull distributed time to failure, we propose two models based on a generalised renewal process that consider product usage rate to forecast the number of failures of products sold with a two-dimensional warranty policy. The accelerated failure time model is used to investigate the effects of the product usage rate on system reliability. The maximum likelihood estimation combined with a nonlinear constrained programming method is used to estimate the parameters of the proposed models. We conduct data experiments based on both simulation and real data collected from an excavator manufacturer in China to test the performance of the proposed models. The results indicate that the models incorporating a variable usage rate more accurately forecast the number of failures than those models based on a nominal usage rate.

Comparative Analysis of Optimal Maintenance Policies Under General Repair With Underlying Weibull Distributions

Various replacement policies under Kijima's general repair model with the underlying Weibull distribution function are studied via two efficient methods. The first one is based on our previously derived approximate formula for the g-renewal function; the second is an improved Monte Carlo method. These methods enable an in-depth, comparative analysis of the maintenance polices in question. An efficient algorithm is suggested for finding optimal preventive replacement times. The influence of restoration factor, including the deviation from a minimal repair assumption, on the optimal solution is studied. A practical study illustration is provided.

General Repair Models for Maintained Systems

Any system when put on operation deteriorates over a period of time and fails on crossing a threshold. Maintenance is performed either on failure or for mitigating the effects of this deterioration. In general the system is either partially maintained or renewed to bring it back to its operating state. The general behaviour of the failure process of a maintained system is modelled by means of stochastic processes called general repair processes. In these processes the effect of the maintenance carried out on the system is captured by means of a variable called the maintenance indicator which indicates the degree to which the system has been maintained. Such a maintenance indicator acting on the system’s failure age or intensity post maintenance generalizes the modelling of the failure processes. The general repair process models based on the type of the maintenance indicator can be broadly classified into two categories based on age or intensity loss or recovery and two sub-categories based on whether these take place additively or multiplicatively. This paper focuses on such models, their inference, their uses, the physical reality they depict and their shortcomings when modelling the failure processes of maintained systems. DOI: http://dx.doi.org/10.4038/sljastats.v12i0.4971 Sri Lankan Journal of Applied Statistics Vol.12 2011 pp.117-143

Modeling and Analysis of the Exponential Unit with General Repair

As to the fact that the common model of repair does not suit for the immemorial exponential unit, it first proposes the general repair model based on the Repair Degree. This model can describe the repair effect of the exponential unit exactly. Then it studies the classical estimation method of the parameters for the Repair Degree as well as the Failure Rate in the condition of general repair, that’s Moment Estimation and Maximum Likelihood Estimation. On this foundation, it compares the two methods by large amount of simulative data. Further, it figures out the estimation value of the Failure Rate on the assumption of ‘As Good As new after repair’. There exists apparent difference from the exact value. So it shows that the assumption of ‘As Good As new after repair’ is not appropriate.

A General Imperfect Repair Model Considering Time-Dependent Repair Effectiveness

Kijima I and Kijima II models are two important imperfect repair models in literature. These two models use one constant parameter to represent the degree of repair, which is called Repair Effectiveness (RE) in this paper. We developed a more general imperfect repair model by extending the constant RE to a time-dependent function based on the virtual age process, where the Kijima models are special cases of the new model. A simulation method is developed to estimate the cumulative number of failures for the new model, and a Bayesian inference method is proposed to select the best imperfect repair model. Finally, to demonstrate the new model, a numerical example is provided. From this example, the new model shows a more accurate mean and a narrower confidence interval than that of the Non-Homogeneous Poisson Processes, and Kijima I and Kijima II models.

Optimal maintenance policy for a multi-state deteriorating system with two types of failures under general repair

We present a general repair model for a multi-state deteriorating system subject to major and minor failures. The process is modeled as a semi-Markov decision process with the optimality criterion being the minimization of the long-run expected average cost per unit time. A modified policy-iteration algorithm using the embedded technique is developed as the computational approach used to find the optimal maintenance policy. The advantage of using the embedded technique is that it reduces the size of the linear system in the value-determination step of the algorithm resulting in reduced computational efforts for large state spaces. Numerical examples are given which illustrate the implementation of the computational approach.

Nonparametric Bayes estimation in repair models

Consider a sequence of dependent random variables X 1 , X 2 , … , X n , where X 1 has distribution F (or probability measure P), and the distribution of X i + 1 given X 1 , … , X i and other covariates and environmental factors depends on F and the previous data, i = 1 , … , n - 1 . General repair models give rise to such random variables as the failure times of an item subject to repair. There exist nonparametric non-Bayes methods of estimating F in the literature, for instance, Whitaker and Samaniego [1989. Estimating the reliability of systems subject to imperfect repair. J. Amer. Statist. Assoc. 84, 301–309], Hollander et al. [1992. Nonparametric methods for imperfect repair models. Ann. Statist. 20, 879–896] and Dorado et al. [1997. Nonparametric estimation for a general repair model. Ann. Statist. 25, 1140–1160], etc. Typically these methods apply only to special repair models and also require repair data on N independent items until exactly only one item is left awaiting a “perfect repair”. In this paper, we define a general model for dependent random variables taking values in a general space, which includes most of the repair models in the literature. We describe nonparametric Bayesian methods to estimate P, without making any assumptions on when we stop collecting data. To do this we introduce a new class of priors called partition-based (PB) priors and show that it is a conjugate class to a large class of our general repair models. We also define a subclass of such priors called partition-based Dirichlet (PBD) priors which also forms a conjugate family of priors. For a special case of the repair model called the aging repair model, we obtain an easily computable Bayes estimate of P under a Dirichlet prior. The Bayes estimates are smoother than Whitaker and Samaniego non-Bayes estimates. Graphical comparisons show that the Bayes and non-Bayes estimates tend to be close.

A New Stochastic Model for Systems Under General Repairs

Numerous stochastic models for repairable systems have been developed by assuming different time trends, and repair effects. In this paper, a new general repair model based on the repair history is presented. Unlike the existing models, the closed-form solutions of the reliability metrics can be derived analytically by solving a set of differential equations. Consequently, the confidence bounds of these metrics can be easily estimated. The proposed model, as well as the estimation approach, overcomes the drawbacks of the existing models. The practical use of the proposed model is demonstrated by a much-discussed set of data. Compared to the existing models, the new model is convenient, and provides accurate estimation results

Determining optimal warranty periods from the seller's perspective and optimal out-of-warranty replacement age from the buyer's perspective

This paper considers a general repairable product sold under a failure-free renewing warranty agreement. In the case of a general repair model, there can be two types of failure: Type I failure (a minor failure), which can be rectified by minimal repairs; and type II failure (a catastrophic failure), which can be rectified only by replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. The aim of this paper is to determine the optimal warranty period and the optimal out-of-warranty replacement age, from the perspective of the seller (manufacturer) and the buyer (consumer), respectively, while minimizing the corresponding cost functions. We prove under mild conditions, that the optimal solution of minimizing the cost function exists and is finite. Further, a concise numerical example is demonstrated, and the sensitivity analysis of some of the parameters related to costs is carried out as well. Finally, some practical aspects of renewing the warranty policy for the practitioner and reader are addressed.

Estimation in a General Repair Model Based on Left‐truncated Data

We consider a repairable system with general repairs introduced by Last & Szekli (1998a). Apart from simple special cases this model leads to a strong dependency among the observed failure times. Our aim is to estimate the underlying failure time distribution and its cumulative hazard given that the failure process has been observed up to the nth failure. We use non‐parametric estimators of Kaplan–Meier and Nelson–Aalen type. We prove strong uniform consistency of the estimators as n tends to infinity. Further results on weak convergence are derived. Neither stationarity nor mixing conditions are required.

Optimal repair/replacement policy for a general repair model

In this paper, we study a maintenance model with general repair and two types of replacement: failure and preventive replacement. When the system fails a decision is made whether to replace or repair it. The repair degree that affects the virtual age of the system is assumed to be a random function of the repair-cost and the virtual age at failure time. The system can be preventively replaced at any time before failure. The objective is to find the repair/replacement policy minimizing the long-run expected average cost per unit time. It is shown that a generalized repair-cost-limit policy is optimal and the preventive replacement time depends on the virtual age of the system and on the length of the operating time since the last repair. Computational procedures for finding the optimal repair-cost limit and the optimal average cost are developed. This model includes many well-known models as special cases and the approach provides a unified treatment of a wide class of maintenance models.

Optimal repair/replacement policy for a general repair model

In this paper, we study a maintenance model with general repair and two types of replacement: failure and preventive replacement. When the system fails a decision is made whether to replace or repair it. The repair degree that affects the virtual age of the system is assumed to be a random function of the repair-cost and the virtual age at failure time. The system can be preventively replaced at any time before failure. The objective is to find the repair/replacement policy minimizing the long-run expected average cost per unit time. It is shown that a generalized repair-cost-limit policy is optimal and the preventive replacement time depends on the virtual age of the system and on the length of the operating time since the last repair. Computational procedures for finding the optimal repair-cost limit and the optimal average cost are developed. This model includes many well-known models as special cases and the approach provides a unified treatment of a wide class of maintenance models.

A general minimal repair model

Minimal repairs have been given considerable attention in the reliability literature. Instead of replacing a failed system by a new one, such a minimal repair restores the system to the state it had just before failure. But the state just before failure depends on the information which is available about the system. Different information levels are possible. This paper gives a general definition characterizing point processes which describe time points of minimal repairs with respect to a certain information level. Some examples demonstrate the wide range of applications.

A general minimal repair model

Minimal repairs have been given considerable attention in the reliability literature. Instead of replacing a failed system by a new one, such a minimal repair restores the system to the state it had just before failure. But the state just before failure depends on the information which is available about the system. Different information levels are possible. This paper gives a general definition characterizing point processes which describe time points of minimal repairs with respect to a certain information level. Some examples demonstrate the wide range of applications.

Nonparametric estimation for a general repair model

The construction and analysis of repair models is an important area in reliability. A commonly used model is the minimal repair model. Under this model, repair restores the state of the system to its level prior to failure. Kijima introduced repair models that could be classified as better-than-minimal. Under Kijima's models, the system, upon repair, is functionally the same as a working system of lesser age which has never experienced failure. In this paper, we present a new approach to the modeling of better-than-minimal repair models. Using this approach, we construct a general repair model that contains Kijima's models as special cases. We also study the problem of estimating the distribution of the time to first failure of a system maintained by general repair. We make use of counting processes to show strong consistency of the estimator and prove results on weak convergence. Finally, we derive a Hall-Wellner type asymptotic confidence band for the distribution of the time to first failure of the system.

Modelling of repairable systems with various degrees of repair

The availability of a stochastic repairable system depends on the failure behaviour and on repair strategies. In this paper, we deal with a general repair model for a system using auxiliary counting processes and corresponding intensities which include various degrees of repair (between minimal repair and perfect repair). For determining the model parameters we need estimators depending on failure times and repair times: maximum likelihood (ML) estimator and Bayes estimators are considered. Special results are obtained by the use of Weibull-type intensities and random observation times.

Modeling of repairable systems

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes. Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.

Modeling of repairable systems

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes.Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.