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.     

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.     

TTT-based tests for trend in repairable systems data

A major aspect of analysis of failure data for repairable systems is the testing for a possible trend in interfailure times. This paper reviews some important and popular graphical methods and tests for the nonhomogeneous Poisson process model. In particular, the total time on test (TTT) plot is considered, and trend tests based on the TTT-statistic are motivated and derived. In particular, a test based on the Anderson-Darling statistic is suggested. The tests are evaluated and compared in a simulation study, both with respect to the achievement of correct significance level and rejection power. The considered alternatives to ‘no trend’ are the log-linear, power law and a class of bathtub-shaped intensity functions. The simulation study involves single systems, as well as the case where several independent systems of the same kind are observed.     

Graphical weibull analysis of repairable systems

A graphical method is given for the analysis of repairable systems failing according to a Weilbull process. The integrated hazard rate is plotted in a logarithmic co-ordinate system. If the process is of Weibull type the observed data will be located around a straight line. The plot provides estimates of the parameters. The graphical procedure is completed by confidence bands and variation bands built either around the observed failure numbers at fixed time or vice versa.     

Practical aspects of modelling of repairable systems data using proportional hazards models

Cox's Proportional Hazards Model (PHM) has been widely applied in the analysis of lifetime data. The model is semi-parametric, so that weak assumptions are made about form of the hazard function. There have been medical developments of this model which have aided studies of repairable systems.A review of the practical use of this PHM model is given and particular attention is paid to the used of diagnostics statistics and graphs. Illustrations are given using field data from the semiconductor and electrical industries, and repairable data will be illustrated by data from the hydrocarbon industry.     

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.     

Simulation strategies to identify the failure parameters of repairable systems under the influence of general repair

With repairable systems, it is suspected than many of the post-failure repairs have the effect of partially resetting the failure rate of the system. Consequently, identifying the correct failure model and its associated parameters is confounded by the impact of these repair effects. The purpose of this paper is to compare the use of several simulation strategies involving mimicking experiments, to assist in this identification process. Data collected from a hydrelectric turbine unit is used to illustrate the approach.     

Proportional hazards modelling of repairable systems

The purpose of this paper is to illustrate some situations under which the proportional hazards model (PHM) and its extensions can be used for identification of the most important covariates influencing a repairable system. First of all an overview of the application of the PHM in engineering is presented. Then the concepts of the PHM and its extensions, such as stratified PHM, PHM in the case of non-homogeneous Poisson processes and PHM in the case of jumps in the hazard rate or different intensity function at failures of a large number of copies of a repairable system, are presented. Selection of a suitable extension of the PHM for given data on the basis of residual plots is also discussed. Finally applications of the PHM and its extensions are illustrated with a suitable example. Only the semi-parametric method has been considered. The assumptions made in the PHM for the analysis of repairable systems have been explained graphically as far as possible. Perfect, minimal or imperfect repairs carried out on repairable systems can be taken into consideration for the reliability analysis using the PHM.     

Testing for reliability improvement or deterioration in repairable systems

This paper reviews the derivation and application of the reverse arrangement test which is used to assess whether a system has improved or deteriorated in a reliability sense. Because the well-known and published table of statistics for this test is erroneous, the prime objective of this paper is to provide the correct statistical table. A secondary aim, however, is to demonstrate how this table can be applied when there is a paucity of failure data for each individual system and to show how a more meaningful reliability assessment can be obtained by pooling results, from the reverse arrangement test, for several systems.     

On calibration and validation of eigendeformation-based multiscale models for failure analysis of heterogeneous systems

We present a new strategy for calibration and validation of hierarchical multiscale models based on computational homogenization. The proposed strategy hinges on the concept of the experimental simulator repository (SIMEX) which provides the basis for a generic algorithmic framework in calibration and validation of multiscale models. Gradient-based and genetic algorithms are incorporated into SIMEX framework to investigate the validity of these algorithms in multiscale model calibration. The strategy is implemented using the eigendeformation-based reduced order homogenization (EHM) model and integrated into a commercial finite element package (Abaqus). Ceramic- and polymer- matrix composite problems are analyzed to study the capabilities of the proposed calibration and validation framework.     

Accelerated failure time models for reliability data analysis

Despite the popularity of the proportional hazards model (PHM) in analysing many kinds of reliability data, there are situations in which it is not appropriate. The accelerated failure time model (AFT) then provides an alternative. In this paper, a unified treatment of the accelerated failure time model is outlined for the standard reliability distributions (Weibull, log-normal, inverse Gaussian, gamma). The problem of choosing between the accelerated failure time models and proportional hazard models is discussed and effects of misspecification are reported. The techniques are illustrated in the analysis of data from a fatigue crack growth experiment.     

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.     

On the analysis of mixed-mode failure

The present paper is concerned with the problem of mixed-mode fracture, especially where such failure occurs along a plane of weakness at a bimaterial interface or by interlaminar failure in polymer fibre-composite materials. Firstly, two different schemes for analysing such mixed-mode failures are discussed. These schemes are: (i) a method based upon a consideration of the local singular field ahead of a crack tip, and (ii) a global method based upon a consideration of the applied energy release rates. Secondly, experimental data taken from the literature are reviewed and it is concluded that the latter scheme results in a more consistent interpretation of the data. Indeed, in several cases the global method clearly gives far better agreement between the theoretical predictions and the observed results. The reasons for this are suggested to be the very localised nature of the singular-dominated region ahead of a crack tip in many test specimens and the relatively large damage zone in the materials studied. Finally, the present work has led to the proposal of a general criterion for fracture under mixed-mode loading.     

多状态可修复k/n系统的随时间响应可靠性研究

利用离散时间的马尔科夫链和半马尔科夫链对复杂的多状态可修复k/n系统元件的多样性进行了分析,给出了元件状态变化以及在状态逗留时间的概率分布计算公式,然后给出了元件在状态变化、状态寿命变化的一步概率转移矩阵,最后根据对元件的分析,导出了系统的可靠度与可用度的预测模型。算例表明,得出的模型易行、有效。     

Gamma distribution parameter estimation for field reliability data with missing failure times

Maximum likelihood estimators have been developed for the gamma distribution when there is missing time-to-failure information. Data sets with missing time-to-failure data can arise from field data collection systems that rely on recorded observations of the system by the operators and maintenance personnel. In many regards, this type of data is highly desirable because it implicitly accounts for all actual usage and environmental stresses. Unfortunately the component times-to-failure are not always recorded for fielded systems because of a lack of elapsed time meters, unsatisfactory data reporting requirements, or incomplete or lost information. When only data of this type is available, it creates a non-standard form of da'ta censoring and it has generally not been possible to fit most common time-to-failure distributions. Reliability practitioners have sometimes made unsubstantiated simplifying assumptions so the data can be used. In this paper, a more rigorous approach is presented. Maximum likelihood estimators are derived and demonstrated for the gamma distribution based on merged data records where the individual failure times have not been recorded. These results are important because the gamma distribution can model diverse time-to-failure behavior. This provides a particularly useful tool for data sets that may otherwise not be satisfactorily analyzed.     

威布尔分布场合下无失效数据的Bayes分析

通过对分布函数进行变换，使变换后的函数成为凹函数，利用凹函数性质给出了各检测时刻失效概率的Bayes估计，进而得到了产品可靠性指标的估计。最后，通过对实际数据进行计算，验证了方法的稳定性。     

Generalized renewal process for repairable systems based on finite Weibull mixture

Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’ and ‘better than old but worse than new’. 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 one, which is more realistic in practice. In this paper, a probabilistic model that is applicable to all of the three after-repair states, called generalized renewal process (GRP), is applied. Simplistically, GRP addresses the repair assumption by introducing the concept of virtual age into the stochastic point processes to enable them to represent the full spectrum of repair assumptions. The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that a finite Weibull mixture, with positive component weights only, can be used as underlying distribution of the time to first failure (TTFF) of the GRP model, on condition that the unknown parameters can be estimated. To support the main idea, three examples are presented. In order to estimate the unknown parameters of the GRP model with m-fold Weibull mixture, the EM algorithm is applied. The GRP model with m mixture components distributions is compared to the standard GRP model based on two-parameter Weibull distribution by calculating the expected number of failures. It can be concluded that the suggested GRP model with Weibull mixture with an arbitrary but finite number of components is suitable for predicting failures based on the past performance of the system.     

A new approach to the cumulative operational time for semi-Markov models of repairable systems

Semi-Markovian reliability models of repairable systems are considered here, whose state space is partitioned into the set of up-, and the set of down-states. The cdf of the cumulative operational time over a finite time interval [0,t] is represented in terms of the work-mission-availability and a system of integral equations is shown to hold for the latter. The equations are of the convolution type which then allows closed form expressions to be established for both the work-mission-availability and the cdf of the cumulative operational time. The semi-Markov model of a two-unit system is examined numerically by solving the resulting integral equations with the two-point trapezoidal rule. The results are compared with those from simulation and an earlier solution scheme based on (non-convolution) integral equations for the cdf of the cumulative operational time.     

Structure field construction for nonlinear static systems based on measurement data processing

A method is proposed for the construction of the structure field for nonlinear static systems, including the sector condition, which should belong to nonlinearity, and the method of construction on the structure field. Definitions are given of the coherence concept for the system variables and the quantitative evaluation. Algorithms are considered for structural identification of nonlinearity for certain function classes.