On the analysis of field failure data for repairable systems

Reliability assessments of repairable (electronic) equipment are often based on failure data recorded under field conditions. The main objective in the analyses is to provide information that can be used in improving the reliability through design changes. For this purpose it is of particular interest to be able to locate ‘trouble-makers’, i.e. components that are particular likely to fail. In the present context, reliability is measured in terms of the mean cumulative number of failures as a function of time. This function may be considered for the system as a whole, or for stratified data. The stratification is obtained by sorting data according to different factors, such as component positions, production series, etc. The mean cumulative number of failures can then be estimated either nonparametrically as an average of the observed failures, or parametrically, if a certain model for the lifetimes of the components involved is assumed. As an example we here consider a simple component lifetime model based on the assumption that components are ‘drawn’ randomly from a heterogeneous population, where a small proportion of the components are weak (with a small mean lifetime), and the remaining are standard components (with a large mean lifetime). This model enables formulation of an analytical expression for the mean cumulative number of failures. In both the nonparametric and the parametric case the uncertainty of the estimation may be assessed by computing a confidence interval for the estimated values (a confidence band for the estimated time functions). The determination of confidence bands provides a basis for assessing the significance of the factors underlying the stratification. The methods are illustrated through an industrial case study using field failure data.     

A non-parametric monotone maximum likelihood estimator of time trend for repairable system data

The trend-renewal process (TRP) is defined to be a time-transformed renewal process, where the time transformation is given by a trend function λ(·) which is similar to the intensity of a non-homogeneous Poisson process (NHPP). A non-parametric maximum likelihood estimator of the trend function of a TRP is obtained under the often natural condition that λ(·) is monotone. An algorithm for computing the estimate is suggested and examined in detail in the case where the renewal distribution of the TRP is a Weibull distribution. The case where one has data from several systems is also briefly studied.     

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.     

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.     

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.     

Additive hazards models in repairable systems reliability

The paper is concerned with the analysis of a repairable system's failure behaviour. In a brief survey the shortcomings of commonly used probabilistic models are mentioned. Appearing to be more appropriate, a regression type of analysis is followed, with explanatory variables acting additively on the hazard function. As repairable systems may experience multiple failures, the hazard function modelling is continued beyond a system's first failure to second and subsequent failures. In this way a kind of additive variant of the model developed by Prentice, Williams and Peterson (which is an extension of Cox's proportional hazards model) is formulated. The techniques are illustrated in an example.     

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.     

Closed loop two-echelon repairable item systems

In this paper we consider closed loop two-echelon repairable item systems with repair facilities both at a number of local service centers (called bases) and at a central location (the depot). The goal of the system is to maintain a number of production facilities (one at each base) in optimal operational condition. Each production facility consists of a number of identical machines which may fail incidentally. Each repair facility may be considered to be a multi-server station, while any transport from the depot to the bases is modeled as an ample server. At all bases as well as at the depot, ready-for-use spare parts (machines) are kept in stock. Once a machine in the production cell of a certain base fails, it is replaced by a ready-for-use machine from that bases stock, if available. The failed machine is either repaired at the base or repaired at the central repair facility. In the case of local repair, the machine is added to the local spare parts stock as a ready-for-use machine after repair. If a repair at the depot is needed, the base orders a machine from the central spare parts stock to replenish its local stock, while the failed machine is added to the central stock after repair. Orders are satisfied on a first-come-first-served basis while any requirement that cannot be satisfied immediately either at the bases or at the depot is backlogged. In case of a backlog at a certain base, that bases production cell performs worse.To determine the steady state probabilities of the system, we develop a slightly aggregated system model and propose a special near-product-form solution that provides excellent approximations of relevant performance measures. The depot repair shop is modeled as a server with state-dependent service rates, of which the parameters follow from an application of Nortons theorem for Closed Queuing Networks. A special adaptation to a general Multi-Class Marginal Distribution Analysis (MDA) algorithm is proposed, on which the approximations are based. All relevant performance measures can be calculated with errors which are generally less than one percent, when compared to simulation results. The approximations are used to find the stock levels which maximize the availibility given a fixed configuration of machines and servers and a certain budget for storing items. Correspondence to: W.H.M. Zijm     

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.     

Calibration tests for count data

Calibration, the statistical consistency of forecast distributions and observations, is a central requirement for probabilistic predictions. Calibration of continuous forecasts has been widely discussed, and significance tests are commonly used to detect whether a prediction model is miscalibrated. However, calibration tests for discrete forecasts are rare, especially for distributions with unlimited support. In this paper, we propose two types of calibration tests for count data: tests based on conditional exceedance probabilities and tests based on proper scoring rules. For the latter, three scoring rules are considered: the ranked probability score, the logarithmic score and the Dawid-Sebastiani score. Simulation studies show that all the different tests have good control of the type I error rate and sufficient power under miscalibration. As an illustration, we apply the methodology to weekly data on meningoccocal disease incidence in Germany, 2001–2006. The results show that the test approach is powerful in detecting miscalibrated forecasts.     

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.     

Availability, reliability and downtime of systems with repairable components

Closed-form expressions are derived for the steady-state availability, mean rate of failure, mean duration of downtime and lower bound reliability of a general system with randomly and independently failing repairable components. Component failures are assumed to be homogeneous Poisson events in time and repair durations are assumed to be exponentially distributed. The results are expressed in terms of the mean rates of failure and mean durations of repair of the individual components. Closed-form expressions are also derived for the rates of change of the various probabilistic system performance measures with respect to the mean rate of failure and the mean duration of repair of each component. These expressions provide a convenient framework for identifying important components within the system and for decision-making aimed at upgrading the system availability or reliability, or reducing the mean duration of system downtime. Example applications to an electrical substation system demonstrate the use of the formulas developed in the paper.     

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.     

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.     

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。     

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.     

A class of rank-based tests for doubly-truncated data

A class of rank-based tests is proposed for the two-sample problem with doubly-truncated data. We consider both nonparametric and semiparametric approaches, where the truncation distribution is parameterized, while the lifetime distribution is left unspecified. The asymptotic distribution theory of the test is presented. The small-sample performance of the test is investigated under a variety of situations by means of Monte Carlo simulations. The proposed tests are illustrated using the CDC AIDS Blood Transfusion Data.     

Analysing event data from a repairable machine subject to imperfect preventive maintenance

A machine is minimally repaired on failure and imperfect preventive maintenance (PM) is also carried out from time to time, not necessarily at regular intervals. A simple point process model is proposed for the sequence of corresponding failure times, and estimates of the machine lifetime parameters and the degree of age rejuvenation at PMs are obtained using maximum likelihood techniques. These results are then applied to real event data. © 1997 John Wiley & Sons, Ltd.     

Availability allocation to repairable systems with genetic algorithms: a multi-objective formulation

This paper describes a methodology based on genetic algorithms (GA) and experiments plan to optimize the availability and the cost of reparable parallel-series systems. It is a NP-hard problem of multi-objective combinatorial optimization, modeled with continuous and discrete variables. By using the weighting technique, the problem is transformed into a single-objective optimization problem whose constraints are then relaxed by the exterior penalty technique. We then propose a search of solution through GA, whose parameters are adjusted using experiments plan technique. A numerical example is used to assess the method.