Modeling Failure Process and Quantifying the Effects of Multiple Types of Preventive Maintenance for a Repairable System

In this paper, we consider a repairable system whose failures follow a non‐homogenous Poisson process with the power law intensity function. The system is subject to corrective and multiple types of preventive maintenance. A corrective maintenance has a minimal effect on the system; however, a preventive maintenance may reduce the system's age. We assume the effects of different preventive maintenance on the system are not identical and derive the likelihood function to estimate the parameters of the failure process as well as the effects of preventive maintenance. Moreover, we derive the conditional reliability and the expected number of failures between two consecutive preventive maintenance types. The proposed methods are applied to a real case study of four trucks used in a mining site in Canada. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of repairable systems with severe left censoring or truncation

Left censoring or left truncation occurs when specific failure information on machines is not available before a certain age. If only the number of failures but not the actual failure times before a certain age is known, we have left censoring. If neither the number of failures nor the times of failure are known, we have left truncation. A datacenter will typically include servers and storage equipment installed on different dates. However, data collection on failures and repairs may not begin on the installation date. Often, the capture of reliability data starts only after the initiation of a service contract on a particular date. Thus, such data may exhibit severe left censoring or truncation, since machines may have operated for considerable time periods without any reliability history being recorded. This situation is quite different from the notion of left censoring in non-repairable systems, which has been dealt with extensively in the literature. Parametric modeling methods are less intuitive when the data has severe left censoring. In contrast, non-parametric methods based on the Mean Cumulative Function (MCF), recurrence rate plots, and calendar time analysis are simple to use and can provide valuable insights into the reliability of repairable systems, even under severe left censoring or truncation. The techniques shown have been successfully applied at a large server manufacturer to quantify the reliability of computer servers at customer sites. In this discussion, the techniques will be illustrated with actual field examples.     

Uptime of Systems Subject to Repairable and Nonrepayable Failures

Expressions are derived for the distribution and expected value of uptime for systems subject to both repairable and nonrepairable failures. The results are applicable to a wide range of situations, including the analysis of systems subject to major structural failure, product or process obsolescence, or failure of critical nonrepairable subsystems. Several examples are investigated, including multiple component series systems, systems containing standby redundant nonrepairable subsystems and systems containing standby redundant repairable subsystems.     

A Simple Method for Regression Analysis With Censored Data

Problems requiring regression analysis of censored data arise frequently in practice. For example, in accelerated testing one wishes to relate stress and average time to failure from data including unfailed units, i.e., censored observations.

Maximum likelihood is one method for obtaining the desired estimates; in this paper, we propose an alternative approach. An initial least squares fit is obtained treating the censored values as failures. Then, based upon this initial fit, the expected failure time for each censored observation is estimated. These estimates are then used, instead of the censoring times, to obtain a revised least squares fit and new expected failure times are estimated for the censored values. These are then used in a further least squares fit. The procedure is iterated until convergence is achieved. This method is simpler to implement and explain to non-statisticians than maximum likelihood and appears to have good statistical and convergence properties.

The method is illustrated by an example, and some simulation results are described. Variations and areas for further study also are discussed.     

Reliability Modelling of Multiple Repairable Units

This paper proposes a model selection framework for analysing the failure data of multiple repairable units when they are working in different operational and environmental conditions. The paper provides an approach for splitting the non‐homogeneous failure data set into homogeneous groups, based on their failure patterns and statistical trend tests. In addition, when the population includes units with an inadequate amount of failure data, the analysts tend to exclude those units from the analysis. A procedure is presented for modelling the reliability of a multiple repairable units under the influence of such a group to prevent parameter estimation error. We illustrate the implementation of the proposed model by applying it on 12 frequency converters in the Swedish railway system. The results of the case study show that the reliability model of multiple repairable units within a large fleet may consist of a mixture of different stochastic models, that is, the homogeneous Poisson process/renewal process, trend renewal process, non‐homogeneous Poisson process and branching Poisson processes. Therefore, relying only on a single model to represent the behaviour of the whole fleet may not be valid and may lead to wrong parameter estimation. Copyright © 2015 John Wiley & Sons, Ltd.     

Statistical Inference for Power-Law Process With Competing Risks

The focus of this article is on failure history of a repairable system for which the relevant data comprise successive event times for a recurrent phenomenon along with an event-count indicator. We undertake an investigation for analyzing failures from repairable systems that are subject to multiple failure modes. Failure data representing a cluster of recurrent events from a single system are studied under the parametric framework of a power-law process, a model that has found considerable attention in industrial applications. Some interesting and nonstandard asymptotic results ensue in this context that are discussed in detail. Extensive simulation has been carried out that supplements the theoretical findings. An extension to the case where the specific cause of failure may be missing is investigated in detail. The methodology has been implemented on recurrent failure data obtained from a warranty claim database for a fleet of automobiles. Supplementary material for this article is available online.     

Modeling the failure data of a repairable equipment with bathtub type failure intensity

The paper deals with the reliability modeling of the failure process of large and complex repairable equipment whose failure intensity shows a bathtub type non-monotonic behavior. A non-homogeneous Poisson process arising from the superposition of two power law processes is proposed, and the characteristics and mathematical details of the proposed model are illustrated. A graphical approach is also presented, which allows to determine whether the proposed model can adequately describe a given failure data. A graphical method for obtaining crude but easy estimates of the model parameters is then illustrated, as well as more accurate estimates based on the maximum likelihood method are provided. Finally, two numerical applications are given to illustrate the proposed model and the estimation procedures.     

A stochastic regime switching model for the failure process of a repairable system

This paper presents a stochastic model and estimation procedure for analyzing the failure process of a repairable system. We consider repairable systems whose successive interfailure times reveal a significant dependence while showing an insignificant trend. Neither the renewal process nor the non-homogeneous Poisson process are adequate for modeling such failure processes. Especially when the interfailure times show a cyclic pattern, we may consider a switching of the regimes (states) governing the lifetime distribution of the system. We propose a Markov switching model describing the failure process for such a case. The model postulates that a finite number of states governs the distinct lifetime distributions, and the state makes transitions according to a discrete-time Markov chain. Each of the distinct lifetime distributions represents a failure type that may change after successive repairs. Our model generalizes the mixture model by allowing the mixture probabilities to change during the transient period of the system. The model can capture the transient behavior of the system. The interfailure times constitute a set of incomplete data because the states are not explicitly identified. For the incomplete data, we propose a procedure for finding the maximum likelihood estimates of the model parameters by adopting the expectation and maximization principle. We also suggest a statistical method to determine the number of significant states. A Monte Carlo study is performed with two-parameter Weibull lifetime distributions. The results show consistency and good properties of the estimates. Some sets of Proschan's air conditioning unit data [Technometrics, 1963, 5′ 375–383] are also analyzed and the results are discussed with respect to the number of significant states and the performance of the prediction.     

Bayesian analysis of repairable systems showing a bounded failure intensity

The failure pattern of repairable mechanical equipment subject to deterioration phenomena sometimes shows a finite bound for the increasing failure intensity. A non-homogeneous Poisson process with bounded increasing failure intensity is then illustrated and its characteristics are discussed. A Bayesian procedure, based on prior information on model-free quantities, is developed in order to allow technical information on the failure process to be incorporated into the inferential procedure and to improve the inference accuracy. Posterior estimation of the model-free quantities and of other quantities of interest (such as the optimal replacement interval) is provided, as well as prediction on the waiting time to the next failure and on the number of failures in a future time interval is given. Finally, numerical examples are given to illustrate the proposed inferential procedure.     

Balancing burn-in and mission times in environments with catastrophic and repairable failures

In a system subject to both repairable and catastrophic (i.e., nonrepairable) failures, ‘mission success’ can be defined as operating for a specified time without a catastrophic failure. We examine the effect of a burn-in process of duration τ on the mission time x, and also on the probability of mission success, by introducing several functions and surfaces on the (τ,x)-plane whose extrema represent suitable choices for the best burn-in time, and the best burn-in time for a desired mission time. The corresponding curvature functions and surfaces provide information about probabilities and expectations related to these burn-in and mission times. Theoretical considerations are illustrated with both parametric and, separating the failures by failure mode, nonparametric analyses of a data set, and graphical visualization of results.     

Generalized renewal process for analysis of repairable systems with limited failure experience

Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’, ‘better than old but worse than new’, ‘better than new’, and ‘worse than old’. The probabilistic models traditionally used to estimate the expected number of failures account for the first two states, but they do not properly apply to the last three, which are more realistic in practice. In this paper, a robust solution to a probabilistic model that is applicable to all of the five after repair states, called generalized renewal process (GRP), is presented. This research demonstrates that the GRP offers a general approach to modeling repairable systems and discusses application of the classical maximum likelihood and Bayesian approaches to estimation of the GRP parameters. This paper also presents a review of the traditional approaches to the analysis of repairable systems as well as some applications of the GRP and shows that they are subsets of the GRP approach. It is shown that the proposed GRP solution accurately describes the failure data, even when a small amount of failure data is available.Recent emphasis in the use of performance-based analysis in operation and regulation of complex engineering systems (such as those in space and process industries) require use of sound models for predicting failures based on the past performance of the systems. The GRP solution in this paper is a promising and efficient approach for such performance-based applications.     

Robustness evaluation of a semi-parametric proportional intensity model

The Prentice, Williams and Peterson (PWP) family of proportional intensity reliability models has been proposed for application to repairable systems. This paper reports results of a study on the robustness of one PWP model over early failure history. The assessment of robustness was based on the semi-parametric PWP model's ability to predict the successive times of occurrence of failure events when the underlying process actually is parametric (specifically a nonhomogeneous Poisson process having power-law proportional intensity function with one covariate). A parametric method was also used to obtain maximum likelihood estimates of the power-law parameters, for purposes of validation and as a reference for comparison.     

Linear consecutive-k-out-of-n: F system reliability with common-mode forced outages

This paper presents a model of consecutive-k-out-of-n: F system subject to common-mode forced outages, whose interarrival times are independent and exponentially distributed. The objective is to analytically derive the mean operating time between failures for a non-repairable component system. The average system failure time and the system availability are also considered. Then, the model is extended to a system with repairable components and unrestricted repair, in which service times are exponentially distributed.     

Modeling repair events under intermittent failures and failures subject to unsuccessful repair

Intermittent failures are sometimes observed for a repairable system, such as an automobile. Such failures often lead to a series of unsuccessful repair attempts before the source of the failure has been identified and removed. Unsuccessful repair is also observed in cases where the failures are not intermittent. In order to model data involving repeated repair attempts related to the same failure, repair times cannot usually be assumed negligible and the model must be designed or modified to account for these. In this paper, we propose a modified version of the branching Poisson process model introduced by Lewis (1964). We discuss statistical inference procedures for this model and demonstrate these procedures using the service history of a new automobile. Copyright © 2002 John Wiley & Sons, Ltd.     

An improved decomposition method for the analysis of production lines with unreliable machines and finite buffers

We consider production lines consisting of a series of machines separated by finite buffers. The processing time of each machine is deterministic and all the machines have the same processing time. All machines are subject to failures. As is usually the case for production systems we assume that the failures are operation-dependent (Buzacott and Hanifin 1978, Dallery and Gershwin 1992). Moreover, we assume that the times to failure and the times to repair are exponentially distributed. To analyze such systems, a decomposition method was proposed by Gershwin (1987). The computational efficiency of this method was later significantly improved by the introduction of the so-called DDX algorithm \[Dallery et al. 1988, 1989). In general, this method provides fairly accurate results. There are however cases for which the accuracy of this decomposition method may not be so good. This is the case when the reliability parameters (mean times to failure and mean times to repair) of the different machines have different orders of magnitude. Such a situation may be encountered in real production lines. The purpose of this paper is to propose an improvement of Gershwin's original decomposition method that provides accurate results even in the above mentioned situation. The basic difference between the decomposition method presented in this paper with that of Gershwin is that the times to repair of the equivalent machines are modeled as generalized exponential distributions (Kouvatsos 1986) instead of exponential distributions. This allows us to use a two-moment approximation instead of a one- moment approximation of the repair time distributions of these equivalent machines.     

A decision model for deteriorating repairable systems

Complex systems are generally repaired rather than replaced after failures. If deterioration in a repairable system is detected, i.e., the successive inter-failure times become stochastically smaller and smaller, then the decision of when to overhaul or discard the system is of fundamental importance. However, such a decision involves many uncertainties, such as the initial status of the system, the degree of deterioration, expected system lifetime, repair cost, accidental cost, etc. which are important factors and need to be evaluated carefully. In this paper, a Bayesian decision theoretic approach is developed. A nonhomogeneous Poisson process with a power law failure intensity function is used to describe the behavior of the deteriorating repairable system. Also, a proposed natural conjugate prior distribution is applied to make the Bayesian decision-making process more effective and efficient.     

Analysis of equivalent dynamic reliability with repairs under partial information

For repairable system it is usual to evaluate the effectiveness of repair by considering the life extended between failures. Traditional definition of dynamic reliability is not suitable for the evaluation due to that the population diminishes gradually as failed events may induce non-repairable situations. More often the failure record presents sometimes only the sequence of failure number, it does not specify the system in a repairable or non-repairable condition as it fails. This kind of information is dim in describing the whole picture of failures after repair that should be accounted for estimating the repaired system dynamic reliability. In this paper the cumulative failure data set with repairs, which depicts the failure number in the successive operating ranges, is constructed from such incomplete information. It is developed by fuzzy consideration which (1) distinguishes repairable vs. non-repairable cases in the failure sequence data, and (2) identifies the system failed again after repair in the next time failure sequence data. The membership values in the fuzzy treatment about the data are incorporated with physical sense of cumulative damage. Thus, an equivalent dynamic reliability with repairs (EDRWR) is obtained in comparison with that by jumps representation at the time taking repairs. Finally, an example with different times of repairs for about 191 bus motors is used to demonstrate the suggested methodology. The fittings of EDRWR are quite well accepted in Weibull distribution.     

Reward optimization of a repairable system

This paper analyzes a system subject to repairable and non-repairable failures. Non-repairable failures lead to replacement of the system. Repairable failures, first lead to repair but they lead to replacement after a fixed number of repairs. Operating and repair times follow phase type distributions (PH-distributions) and the pattern of the operating times is modelled by a geometric process. In this context, the problem is to find the optimal number of repairs, which maximizes the long-run average reward per unit time. To this end, the optimal number is determined and it is obtained by efficient numerical procedures.     

Bayesian reliability assessment of repairable systems during multi-stage development programs

New repairable systems are generally subjected to development programs in order to improve system reliability before starting mass production. This paper proposes a Bayesian approach to analyze failure data from repairable systems undergoing a Test-Find-Test program. The system failure process in each testing stage is modeled using a Power-Law Process (PLP). Information on the effect of design modifications introduced into the system before starting a new testing stage is used, together with the posterior density of the PLP parameters at the current stage, to formalize the prior density at the beginning of the new stage. Contrary to the usual assumption, in this paper the PLP parameters are assumed to be dependent random variables. The system reliability is measured in terms of the number of failures that will occur in a batch of new units in a given time interval, for example the warranty period. A numerical example is presented to illustrate the proposed procedure.     

Adaptive progressive Type-II censoring

Extending the model of progressive Type-II censoring, we introduce an adaption process. It allows us to choose the next censoring number taking into account both the previous censoring numbers and previous failure times. After deriving some distributional results, we show that maximum likelihood estimators coincide with those in deterministic progressive Type-II censoring. Finally, we establish inferential results for the one- and two-parameter exponential distribution. Using the independence of normalized spacings, we present the distributions of the maximum likelihood estimators. Moreover, explicit confidence bounds and tests of hypotheses can be established.