Power Law Process Model Research Articles

Repairable System Analysis Using the Discrete Weibull Distribution

In many practical circumstances, a repair can be performed when a system fails to restore to a condition before a failure. This type of repair is known as minimal repair and one of the most used models is the power law process (PLP). It is common to consider equipment failure time as a continuous variable in this model, assuming a high degree of accuracy in the measurement tool. However, in practical situations, failures are usually observed and recorded as integer numbers of time units, such as the number of days and hours, indicating a discrete process. In this study, we used a discrete Weibull distribution instead of a continuous Weibull distribution. Since both Weibull models have similar complexities, some benefits are observed in the usage of the discrete model, such as a lower standard deviation of the parameter related to the system deterioration and also a lower Akaike's information criterion. For illustrative purposes, the use of a discrete Weibull distribution was applied to a dataset related to the failures of concrete mixer trucks. Moreover, an approach using a Markov chain with discrete states to obtain the average number of failures and the respective optimum maintenance policy was included. Additionally, six cases (found in the literature) fitted by the PLP model were reanalyzed and compared under the discrete model perspective.

수리 가능한 체계에 대한 Upside-Down 욕조 곡선 형태의 삼분할된 고장 강도 모형

Purpose: This study proposes an upside-down bathtub-shaped three-segmented failure intensity model for repairable systems. The procedure for estimating the parameters for the proposed model is presented under certain assumptions regarding the change points of the model.BRMethods: The proposed model comprises three failure intensity functions - one for each of the three model segments. The model is therefore a P-H-P model, with the power law process model, the homogeneous Poisson process model, and the power law process model that follows the non-homogeneous Poisson process as the failure intensity functions. The change points and the parameters of each failure intensity function were determined using maximum likelihood estimation and least squares estimation.BRResults: The case study shows that compared with other extant models, the proposed P-H-P model is more suitable in terms of log-likelihood function value, Akaike information criterion corrected, and mean squared error.BRConclusion: The proposed P-H-P model is expected to better explain the failure intensities of similar systems and to allow for more accurate reliability analysis.

Non-homogeneous Poisson and linear regression models as approaches to study time series with change-points

In this study we consider some stochastic models and a Bayesian approach to analyze count time series data in the presence of one or more change-points. When the observed count data are all different of zero, it is possible to perform an analysis using linear regression models with normal distribution errors for the log-transformed data. When we have the presence of zero counts at different times, the statistical model based on the log-transformation is not suitable. Hence, in this case, it is possible to consider non-homogeneous Poisson processes (NHPP) models with a suitable rate function. In the present work we consider both models (linear regression and Poisson) to analyze three data sets. These sets are data recording the monthly visitors to New Zealand with the purpose of education, yearly tuberculosis incidence data from New York City, and monthly measles incidence data from Brazil. When the NHPP model is used a PLP (power law process) model is assumed for the intensity function. Additionally, in both models, the presence of change-points will be allowed.

Statistical modeling and reliability analysis of multiple repairable systems with dependent failure times under perfect repair

In repairable systems, a fundamental aspect to be considered is to predict the reliability of the systems under study. Establishing reliability models for multiple repairable systems, however, is still a challenging problem when considering the dependency or unobserved heterogeneity of component failures. This paper focuses on a special repair assumption, called perfect repair, for repairable systems with dependent failures, where only the failed component is restored to as good as new condition. A frailty model is proposed to capture the statistical dependency and unobserved heterogeneity. The classical inferential method for estimating parameters and the reliability predictors will be shown for models in repairable systems under the assumption of perfect repair. An extensive simulation study is conducted under different scenarios to verify the suitability of the model and the asymptotic consistency and efficiency properties of the maximum likelihood estimators. The proposed methodology can be especially useful for industries that operate with repairable systems subjected to the replacement of parts after failures and to non-quantifiable factors that can interfere with the failure times of these parts. We illustrate the practical relevance of the proposed model on two real data sets. The first deals with a set of 9 sugarcane harvesters, observed during a fixed period of time, whose cutting blades failed several times in this interval. The other deals with a set of 5 dump trucks whose diesel engines also had recurring failures during the observation period. Parametric inference is carried out under the power-law process model that has found significant attention in industrial applications.

Bayesian non-parametric frailty model for dependent competing risks in a repairable systems framework

The aim of this article is to analyze multiple repairable systems data under the presence of dependent competing risks. It is known that the dependence effect in this scenario influences the estimates of the model parameters. Hence, under the assumption that the cause-specific intensities follow a power law process (PLP), we propose a frailty-induced dependence approach to incorporate the dependence among the cause-specific recurrent processes. Moreover, the misspecification of the frailty distribution may lead to errors when estimating the parameters of interest. Because of this, we considered a nonparametric approach to model the frailty density using a Dirichlet process mixture prior, which offers more flexibility to provide consistent estimates for the PLP model, as well as insights about heterogeneity among the systems. We proposed an orthogonal parametrization for the PLP model parameters that allowed us to specify a joint prior distribution for the parameters that returned closed-form estimators for the posterior mean. Additionally, a hybrid MCMC sampler algorithm composed by Hamiltonian Monte Carlo and Gibbs sampling was built for computing the posterior estimates with respect to the frailty distribution. A simulation study was conducted to evaluate the efficiency of our estimates. This method was used to analyze a real dataset. Algorithms, code, and data are provided in supplementary material available online.

Reliability analysis assessment of the wind turbines system under multi-dimensions

Aiming at the frequent occurrence of the wind turbines failures, a set of analytical methods was developed to carry out the reliability assessment from multi-dimensions, considering the fault characteristics of the wind turbines subsystems, the variation of its failure process, and the wind turbines reliability indexes. Through classification processing of failure data, the Pareto diagram was applied to search the weak subsystem. A fault tree model is constructed, which can figure out the logical relationship between its failure events by the Fussell–Vesely algorithm. According to various characteristics of the bathtub curve, the failure process based on power law process (PLP) model was proposed, it has been discussed that the change criteria of the wind turbines failure with the running time. At last, reliability indexes such as availability were solved and compared to judge the wind turbine performance status. A case study was given in which the failure data are from a wind farm in China. The results indicate that electrical control and pitch subsystems are weak subsystems, and the minimum bottom event in their fault tree model may lead to system failure. Besides, the PLP model can describe the failure process of wind turbines.

Parametric inference for multiple repairable systems under dependent competing risks

The focus of this article is on the analysis of repairable systems that are subject to multiple sources of recurrence. The event of interest at the system level is assumed to be caused by the earliest occurrence of a source, thereby conforming to a series system competing risks framework. Parametric inference is carried out under the power law process model that has found significant attention in industrial applications. Dependence among the cause‐specific recurrent processes is induced via a shared frailty structure. The theoretical inference results are implemented to a warranty database for a fleet of automobiles, for which the warranty repair is triggered by the failure of one of many components. Extensive finite‐sample simulation is carried out to supplement the asymptotic findings. Copyright © 2014 John Wiley & Sons, Ltd.

Testing the Performance of the Power Law Process Model Considering the Use of Regression Estimation Approach

Testing the Performance of the Power Law Process Model Considering the Use of Regression Estimation Approach

Cost-effective maintenance scheduling of cutting arms of drum shearer machine

Cutting arms are the main productive units of coal shearer machine that contact directly with coal seam. More importantly, the cutting arms operate under a high-stressed environment. In this paper, in order to suggest an optimum maintenance plan for the subsystem of shearer machine, firstly, we conducted a failure and reliability analysis using the data acquired from the Parvade coal mine, Iran over the two years. The analysis shows that the Power Law Process model provides a good fit to the failure data. Also, we applied estimated distribution functions and calculated cost parameters to identify the cost-based optimum preventive maintenance interval. Furthermore, we conducted a sensitivity analysis of the studied machine and identified that if we consider the variation of the world price of the raw coal then the maintenance plan is sensitive to the price. The analysis showed that, big changes in world price of raw coal are not so effective on maintenance plan but have a considerable effect on maintenance costs of the cutting arms of the shearer machine. However, the effects of this change on reliability improvement are almost zero.

New Reliability Model in the Development Phase of a System

HeReliability growth models have been widely applied to track the system failure intensity during the development phase. The power law process (PLP) model is the most commonly model use in reliability growth analyses. However, the PLP model has two drawbacks that the failure intensities at the start and end of the development test-fix-test program are infinite and zero, respectively. In this paper, we propose a new model based on the Burr XII distribution to overcome the drawbacks. The estimates of the model parameters can be easily obtained using the maximum likelihood estimation method. Through three numerical examples, the results show that our proposed model outperforms other existing models regarding log-likelihood value and mean of squared error (MSE).

Reliability analysis of the cable system of drum shearer using the power law process model

In this article, the reliability of the cable system of a drum shearer machine was studied and analysed using operation and maintenance data from an Iranian mine for a period of two years. The tests for trend and serial correlation show that the times between successive failures for the cable system are not independent and identically distributed. A goodness-of-fit test shows that the power law process model provides a good fit to the failure data of this system. After parameter estimation for the power law model, reliability and failure rate plots were obtained. Based on analysis and results, a period of 125 h was defined as the reliability-based maintenance interval for the cable system of the shearer. The analysis shows that, by using this strategy, the reliability of the system will improve by at least 50%.

Reliability Assessment Test Design for Numerical Control Machine Tools

According to the required interval estimation precision of mean time between failures (MTBF) and reliability for numerical control (NC) machine tools, where failure processes can be described by the power law process model, the minimal testing truncated time and sample size of the reliability assessment test are given using Fisher information matrix method. The results show that there are two main factors which affect the truncated time and sample size. They are the log ratio of upper bound to lower bound and confidence of interval estimation for reliability indices, respectively. Among them, the minimal sample size increases significantly as the improvement of required precision of interval estimation for reliability indices and confidence increases, but the minimal testing truncated time is less affected by these two factors.

A Bayesian Hierarchical Power Law Process Model for Multiple Repairable Systems with an Application to Supercomputer Reliability

Los Alamos National Laboratory was home to the Blue Mountain supercomputer, which at one point was the world's fastest computer. This paper presents and analyzes hardware failure data from Blue Mountain. Nonhomogeneous Poisson process models are fit to the data within a hierarchical Bayesian framework using Markov chain Monte Carlo methods. The implementation of these methods is convenient and flexible. Simulations are used to demonstrate strong frequentist properties and provide comparisons between time-truncated and failure-count designs and demonstrate the benefits of hierarchical modeling of multiple repairable systems over the modeling of such systems separately.

Reliability Prediction During Development Phase of a System

Reliability Growth analysis is a process of collecting, modeling, analyzing and interpreting data from the reliability development test program to achieve reliability goal. Many mathematical models have been developed for reliability growth in the literature. In general, it is not possible to fix a priori the model to be used in a given situation. A physical motivation of the problem, based on failure or defect causes/mechanisms, can help in such a choice. The Power Law Process (PLP) is vastly used for monitoring reliability growth during the development test phase. However, in the reliability growth context, the intensity function of the PLP model has two unrealistic situations, viz., it tends to infinity as t tends to zero and tends to zero as t tends to infinity. In this paper, we have proposed Modified PLP model to overcome the second drawback and Doubly Bounded PLP model to overcome both the drawbacks. For the proposed models, the estimates of the model parameters and other quantities of interest have been obtained using Maximum Likelihood approach, for both failure truncated and time truncated data. Reliability Prediction is done using system reliability measures of intensity function and mean time between failures (MTBF). Further confidence intervals of the relevant parameters have been obtained. The procedures developed have been illustrated using numerical examples.

Failure Forecasting of Aircraft Air Conditioning/Cooling Pack with Field Data

This paper presents methods for modeling the failure of air-conditioning/cooling packs for a particular type of aircraft with field data. In many regards, field data are highly desirable for more accurate failure prediction by aircraft operators, because the data implicitly account for all actual usage and environmental stresses. It is not always possible to accurately anticipate or simulate these stresses in a laboratory or even in a field test. Field data, in a larger extent, are also important to the manufacturer, because the data identify product deficiencies and areas of improvement. In this study, the failure of the aircraft air-conditioning/cooling pack under a customer-use environment is first modeled at the component level by using the Weibull distribution and its extensions. These include the two-parameter Weibull model, three-parameter Weibull model, mixture model, and phased bi-Weibull model. The number of failures over time is estimated by a renewal process. The failure of the air-conditioning/cooling pack at the system level is then modeled by using the power law process model. The failure trend is tested by the Laplace test. The results give an insight into the reliability and quality of the air-conditioning/cooling pack under actual operating conditions. The models presented here can be used by aircraft operators for assessing system and component failures and customizing the maintenance programs recommended by the manufacturer.

INFERENCES ON THE PARAMETERS AND SYSTEM RELIABILITY FOR A FAILURE-TRUNCATED POWER LAW PROCESS: A BAYESIAN APPROACH USING A CHANGE-POINT

The reliability of a repairable system that is either improving or deteriorating depends on the system's chronological age. If such a system undergoes "minimal repair" at the occurrence of each failure so that the rate of system failures is not disturbed by the repair, then a nonhomogeneous Poisson process (NHPP) may be used to model the "age-dependent" reliability of the system. The power law process (PLP) is a model within the class of NHPP models and is commonly used as a model for describing the failure times of a repairable system. We introduce a new model that is an extension of the PLP model: the power law process change-point model. This model is capable of describing the failure times of particular types of repairable systems that experience a single change in their rates of occurrence of failures. Bayesian inference procedures for this model are developed.

A simple goodness-of-fit test for the power-law process, based on the duane plot

The PLP (power-law process) or the Duane model is a simple model that can be used for both reliability growth and reliability deterioration. GOF (goodness-of-fit) tests for the PLP have attracted much attention. However, the practical use of the PLP model is its graphical analysis or the Duane plot, which is a log-log plot of the cumulative number of failures versus time. This has been commonly used for model validation and parameter estimation. When a plot is made, and the coefficient of determination, R/sup 2/, of the regression line is computed, the model can be tested based on this value. This paper introduces a statistical test, based on this simple procedure. The distribution of R/sup 2/ under the PLP hypothesis is shown not to depend on the true model parameters. Hence, it is possible to build a statistical GOF test for the PLP. The critical values of the test depend only on the sample size. Simulations show that this test is reasonably powerful compared with the usual PLP GOF tests. It is sometimes more powerful, especially for deteriorating systems. Implementing this test needs only the computation of a coefficient of determination. It is much easier than, for example, computing an Anderson-Darling statistic. Further study is needed to compare more precisely this new test with the existing ones. But the R/sup 2/ test provides a very simple and useful objective approach for decision making with regard to model validation.

Data-based modeling of the failure rate of repairable equipment.

A database of failures of many types of medical equipment was analysed, to study the dependence of failure rate on equipment age and on time since repair. The intention was to use this large dataset to assess the validity of some widely-used models of failure rate, such as the power-law and loglinear Poisson processes, and so to recommend simple and adequate models to those practitioners having little data to discriminate between rival models. The aim is also to illustrate a methodology for computing policy costs from failure databases. The power-law process model was found to fit slightly better overall than did the loglinear and linear processes. Some related models were created to fit an observed peaking of failure rate. The data showed a decreasing hazard of (first) failure after repair for some equipment types. This can be due to imperfect or hazardous repair, and also to differing failure rates among a population of machines. Two simple models of imperfect repair were used to fit the data, and an Empirical Bayes method was used to fit a model of variable failure rate between machines. Neglect of such variation can lead to an over-estimate of the hazardousness of repair.

Estimation of Current Reliability in a Duane-Based Reliability Growth Model

In this article, I investigate the statistical inference concerning the current (field-stage) reliability of a reliability growth model. The model, assuming a step-intensity structure, evolves from the physical consideration of the Duane learning-curve property and incorporates the effect of a testanalyze- and-fix program that is typically undertaken in a developmental testing program. Both exact and large-sample distributional results are derived for the maximum likelihood and the least squares estimators of the current intensity. Under the assumption that the step-intensity model represents the reality, I provide an assessment of the extent of “misspecification” when the widely used power law process model is fit to the failure data of a system experiencing recurrent failures. Extensive simulation results are carried out to supplement the theoretical findings. An illustration with a dataset is provided as a demonstration of the application of the inference results.

Estimation of Current Reliability in a Duane-Based Reliability Growth Model

In this article, I investigate the statistical inference concerning the current (field-stage) reliability of a reliability growth model. The model, assuming a step-intensity structure, evolves from the physical consideration of the Duane learning-curve property and incorporates the effect of a testanalyze- and-fix program that is typically undertaken in a developmental testing program. Both exact and large-sample distributional results are derived for the maximum likelihood and the least squares estimators of the current intensity. Under the assumption that the step-intensity model represents the reality, I provide an assessment of the extent of “misspecification” when the widely used power law process model is fit to the failure data of a system experiencing recurrent failures. Extensive simulation results are carried out to supplement the theoretical findings. An illustration with a dataset is provided as a demonstration of the application of the inference results.