Cumulative Number Of Failures Research Articles

Development of software reliability models using a hybrid approach and validation of the proposed models using big data

This article proposes three software reliability models based on a hybrid approach combining NHPP models, Weibull model, and exponential model. The software failure is first categorised into three categories, namely pure software failures, hardware-induced software failures, and user-induced software failures. Based on the failure behaviour, NHPP models were adapted for pure software failures, Weibull model for hardware-induced failures, and exponential model for user-induced software failures. The failure intensity function, mean value function, and reliability function were determined. The proposed models are validated using big data analysis. From the data collected during the testing phase, the optimal values of parameters were estimated using maximum likelihood estimation and genetic algorithm. The expected number of failures and the cumulative number of failures were calculated, and comparison was made between the observed values to show the performance of the proposed models. A comparison criterion was also proposed to confirm the estimation accuracy. Finally, a t test was conducted to test the significance of the difference between the observed and estimated values. Experimental results confirm the better estimation accuracy of the proposed models.

Toward lean satellites reliability improvement using HORYU-IV project as case study

Lean satellite programs are programs in which the satellite development philosophy is driven by fast delivery and low cost. Though this concept offers the possibility to develop and fly risky missions without jeopardizing a space program, most of these satellites suffer infant mortality and fail to achieve their mission minimum success. Lean satellites with high infant mortality rate indicate that testing prior to launch is insufficient. In this study, the authors monitored failures occurring during the development of the lean satellite HORYU-IV to identify the evolution of the cumulative number of failures against cumulative testing time. Moreover, the sub-systems driving the failures depending on the different development phases were identified. The results showed that half to 2/3 of the failures are discovered during the early stage of testing. Moreover, when the mean time before failure was calculated, it appeared that for any development phase considered, a new failure appears on average every 20h of testing. Simulations were also performed and it showed that for an initial testing time of 50h, reliability after 1 month launch can be improved by nearly 6 times as compared to an initial testing time of 20h. Through this work, the authors aim at providing a qualitative reference for lean satellites developers to better help them manage resources to develop lean satellites following a fast delivery and low cost philosophy while ensuring sufficient reliability to achieve mission minimum success.

Non-parameter Estimation of Failures Intensity of Tractor Based on Bootstrap

The failures data of tractors operating in field conditions were collected by failure tracking tests. A mathematical model is established with non-homogenous Poisson process to get the tractor failure intensity function. Bootstrap method is presented for constructing confidence regions for the failure intensity of a repairable system. Combined with fault data in tracking experiment, the early period failure intensity curves are drawn, revealing the failure regular in the initial stages of tractors. A confidence interval can be estimated by standard likelihood asymptotic theory in the parametric estimation, however in the non-parametric case it can be obtained by using the bootstrap. Comprising the MSE of cumulative intensity and cumulative number of failures between the two estimations, it is seen that the curve which was estimated by non-parameter estimation method is more realistic to the actual experiment process. Results provide the reference to improve tractor reliability, maintainability and maintenance strategy.

Kernel-based nonparametric estimation methods for a periodic replacement problem with minimal repair

In this article, we consider nonparametric estimation methods for a periodic replacement problem with minimal repair, where the expected cumulative number of failures (minimal repairs) is unknown. To construct the confidence interval of an estimator of the optimal periodic replacement time which minimizes the long-run average cost per unit time, we apply two kernel-based bootstrap estimation methods and three replication techniques for bootstrap samples, to estimate the optimal periodic replacement time under incomplete knowledge on the failure time distribution. In simulation experiments, we compare those results with the well-known constrained nonparametric maximum likelihood estimate and some parametric models. We also conduct the field data analysis based on an actual minimal repair data and refer to an applicability of our methods.

Log-linear process modeling for repairable systems with time trends and its applications in reliability assessment of numerically controlled machine tools

Two non-homogeneous Poisson processes including the power law process and the log-linear process with reliability improvement or deterioration are analyzed. Based on Akaike information criterion and Bayesian information criterion, the best model of failure data is presented. The point maximum likelihood and interval estimators of the parameters, as well as seven reliability indices of the log-linear process model, such as cumulative mean time between failures, cumulative number of failures, reliability at a given time, and warranty time given reliability are given. In tests for failure time trends, both the graphical methods, including the cumulative failures versus time plot and the total-time-on-test plot, and the analytical methods including the Laplace, the Military Handbook, and the Lewis–Robinson tests are used. Three real cases for failure data with failure truncation and time truncation of multiple numerically controlled machine tools are given to illustrate the use of the proposed models.

Software Reliability Prediction using Neural Network with Encoded Input

neural network based software reliability model to predict the cumulative number of failures based on Feed Forward architecture is proposed in this paper. Depending upon the available software failure count data, the execution time is encoded using Exponential and Logarithmic function in order to provide the encoded value as the input to the neural network. The effect of encoding and the effect of different encoding parameter on prediction accuracy have been studied. The effect of architecture of the neural network in terms of hidden nodes has also been studied. The performance of the proposed approach has been tested using eighteen software failure data sets. Numerical results show that the proposed approach is giving acceptable results across different software projects. The performance of the approach has been compared with some statistical models and statistical models with change point considering three datasets. The comparison results show that the proposed model has a good prediction capability. Software engineers can measure or forecast software reliability using software reliability models. In real software development process, cost and time are limited. The models are used to develop reliable software under given time and cost constraints. The software managers also determine the release time of the software with the help these models. In past few years, a large number of software reliability models/statistical models have been developed. Every model is based on certain assumptions. So the predictive capability of different models is different for different data sets. There exist no single models that can best suit for all cases. The statistical models are also influenced by different external parameters. To overcome these problems, non-parametric models like neural network and support vector machines have been used for last few years. The non-parametric models are not influenced by any external parameter and also not based on assumptions which are unrealistic in real situation. Neural network has become an alternative method in software reliability modeling, evaluation and prediction. In the literature, many neural network based software reliability prediction models exist and their prediction capability is proven better than some statistical models (4-6). All the neural network based software reliability models are basically experimental in nature. Anyone who wants to apply the models on real software data set, they have to first find out the architecture of the network in hit and trial method. In real cases, people lose their confidence to apply the models with high level of confidence. In this work, neural network models with Exponential and Logarithmic encoding scheme have been proposed to build non-parametric models for software reliability prediction. It has been shown in (14) that neural network models with the above two encoding scheme has better prediction for cumulative number of failures than some statistical models. But in (14), the value of the encoding parameter is determined by doing repeated experiments with hit and trial method. The purpose of this paper is to provide a guideline for the selection of encoding parameter which gives consistent results for different datasets. The proposed approach is applied using eighteen different data sets and found to provide consistent result for all datasets. The approach has been compared using three data sets with existing statistical models with change points. The rest of the paper has been organized as follows: In Section 2, some works related to software reliability prediction are introduced. Section 3 presents the proposed method to predict cumulative number of failures in software using neural network with two different encoding schemes such as Exponential and Logarithmic encoding. Section 4 shows the experimental result. Finally, Section 5 concludes the paper.

Reliability analysis and joint assessment of Brown–Proschan preventive maintenance efficiency and intrinsic wear-out

An imperfect maintenance model for repairable systems is considered. Corrective Maintenances (CM) are assumed to be minimal, i.e. As Bad As Old (ABAO). They are considered to be left and right censored. Preventive Maintenances (PM) are assumed to be done at predetermined planned times and to follow a Brown–Proschan (BP) model, i.e. they renew (As Good As New, AGAN) the system with probability p, and they are minimal (ABAO) with probability 1−p. BP PM effects (AGAN or ABAO) are assumed to be not observed. In this context, different methods (maximum likelihood, moment estimation, and expectation–maximization algorithm) are considered in order to estimate jointly the PM efficiency parameter p and the parameters of the first time to failure distribution corresponding to the new unmaintained system. A method to individually assess the effect of each PM is also proposed. Finally, some reliability characteristics are computed: failure and cumulative failure intensities, reliability and expected cumulative number of failures. All the corresponding algorithms are detailed and applied to a real maintenance data set from an electricity power plant.

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.

Fitting software reliability growth curves using nonparametric regression methods

A simple and effective method of assessing the reliability of a piece of software is to plot the cumulative number of failures observed during testing, N ( x ) , against time x . Since no software is ever completely free of errors, be it careless minor oversights or the results of serious design problems, it is therefore expected that with prolonged and systematic testing, N ( x ) will increase with x . Since the 1970s, there have been many models, aptly named Software Reliability Growth Models (SRGMs) which have been proposed to fit software failure data to the curve m ( x ) = E ( N ( x ) ) . Unfortunately, due to the complexity of the software development processes, which include the possibility of imperfect debugging and introduction of new faults into the system, many of these SRGMs are very complex and standard estimation procedures such as Maximum Likelihood estimation (MLE) fails to estimate correctly, if at all, the parameters of these models. In this paper, we investigate the potential benefits of using Nonparametric Regression (NPR) methods to fit SRGMs. In addition, we will also develop methods based on Stein two-stage and modified two-stage sequential procedures to find a fixed-width confidence interval for the estimator of m ( x ) . Finally, numerical examples based on real software failure data will be presented to illustrate the techniques developed and compare the results with some parametric SRGMs.

Prediction of remaining life of power transformers based on left truncated and right censored lifetime data

Prediction of the remaining life of high-voltage power transformers is an important issue for energy companies because of the need for planning maintenance and capital expenditures. Lifetime data for such transformers are complicated because transformer lifetimes can extend over many decades and transformer designs and manufacturing practices have evolved. We were asked to develop statistically-based predictions for the lifetimes of an energy company’s fleet of high-voltage transmission and distribution transformers. The company’s data records begin in 1980, providing information on installation and failure dates of transformers. Although the dataset contains many units that were installed before 1980, there is no information about units that were installed and failed before 1980. Thus, the data are left truncated and right censored. We use a parametric lifetime model to describe the lifetime distribution of individual transformers. We develop a statistical procedure, based on age-adjusted life distributions, for computing a prediction interval for remaining life for individual transformers now in service. We then extend these ideas to provide predictions and prediction intervals for the cumulative number of failures, over a range of time, for the overall fleet of transformers.

A Poisson Limit Theorem for Reliability Models Based on Markov Chains

In this article, we deal with a class of discrete-time reliability models. The failures are assumed to be generated by an underlying time inhomogeneous Markov chain. The multivariate point process of failures is proved to converge to a Poisson-type process when the failures are rare. As a result, we obtain a Compound Poisson approximation of the cumulative number of failures. A rate of convergence is provided.

Non-standard asymptotics in an inhomogeneous gamma process

Nonhomogeneous Poisson process (NHPP) is a commonly used stochastic model that is utilized to describe the pattern of repeated occurrence of certain events or conditions. Aninhomogeneous gamma process evolves as a generalization to NHPP, where the observed failure epochs correspond to every successive κ-th event of the underlying Poisson process, κ being an unknown parameter to be estimated from the data. This article focuses on a special class of inhomogeneous gamma process, calledmodulated power law process (MPLP) that assumes the Weibull form of the intensity function. The traditional power law process is a popular stochastic formulation of certain empirical relationships between the time to failure and the cumulative number of failures, often observed in industrial experiments. The MPLP retains this underlying physical basis and provides a more flexible modeling environment potentially leading to a better fit to the failure data at hand. In this paper, we investigate inference issues related to MPLP. The maximum likelihood estimators (MLE’s) of the model parameters are not in closed form and enjoy the curious property that they are asymptotically normal with a singular variance-covariance matrix. Consequently, the derivation of the large-sample results requires non-standard modifications of the usual arguments. We also propose a set of simple closed-form estimators that are asymptotically equivalent to the MLE’s. Extensive simulation results are carried out to supplement the theoretical findings. Finally, we implement our inference results to a failure dataset arising from a repairable system.

Simple Plots Improve Software Reliability Prediction Models

Software reliability prediction is accomplished by fitting a nonhomogeneous Poisson process (NHPP) model to data from software testing. The data consist of the cumulative time and the cumulative number of failures found in software testing. The NHPP model can be used to predict the reliability of the software product at the time of release or to determine how much further testing must be done to reach a specified failure rate. Models are normally fitted to software testing data using Poisson regression by the method of maximum likelihood. We encountered difficulties fitting models when numerical algorithms failed to converge or when we were unable to discriminate among several models with the same number of parameters. These difficulties were the result of having no likelihood ratio test to compare models with the same number of parameters and anomalies in the data that caused numerical algorithms to fail. We found that a simple cumulative plot of the data (cumulative failures on the vertical axis vs. cumulative test time on the horizontal axis) helped in spotting anomalies in the data and selecting an appropriate model to fit. A second plot of running products of ratios of the probability densities for the predictions made from competing models, called the prequential likelihood ratio, helped in discriminating between models. Use of these plots helped resolve the difficulties we experienced in fitting models to the software testing data.

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.

Weibull prediction of a future number of failures

This paper presents simple new prediction limits for the number of failures that will be observed in a future inspection of a sample of units. The past data consist of the cumulative number of failures in a previous inspection of the same sample of units. Life of such units is modelled with a Weibull distribution with a given shape parameter value. Copyright © 2000 John Wiley & Sons, Ltd.

A new modified discrete preventive maintenance policy and its application to hard disk management

This study proposes a new modified discrete preventive maintenance policy where failures of a system can be detected only by inspections and be fixed by minimal repairs. The maximum number of minimal repairs allowed to the system is, however, a prespecified integer, N. Under the proposed policy, inspections are conducted at times T, 2T, ...(T>0) to detect and fix failures by minimal repairs, and the whole system is preventively replaced by a new one when the cumulative number of failures detected becomes equal to an integer, k or more. The expected cost per unit time is formulated under the proposed maintenance policy. The existence of an optimal integer k=k* which minimizes the expected cost is then shown. Numerical examples are also presented to illustrate the theoretical underpinnings of the proposed formulation.

On some reliability growth models with simple graphical interpretations

Abstract Some useful reliability growth models, which have simple graphical interpretations, are studied in this paper. The proposed models are inspired by the Duane model. For each of the models, the plot of the cumulative number of failures against the running time, when a suitable scale is used, will tend to be on a straight line if the model is valid. Otherwise the model should be rejected. The slope of the fitted line and its intercept on the vertical axis will give us the estimates of the parameters. Hence, it provides us with a simple graphical model validation and parameter estimation tool. In particular, we propose a “first-model-validation-then-parameter-estimation” approach which will simplify the model validation and parameter estimation problem in software reliability analysis. Numerical analysis of several sets of software failure data are also provided to enlighten the ideas.

On the analysis of field failure data for repairable systems

Abstract 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.

Applications of crude incidence curves.

Crude incidence curves display the cumulative number of failures of interest as a function of time. With competing causes of failure, they are distinct from cause-specific incidence curves that treat secondary types of failures as censored observations. After briefly reviewing their definition and estimation, we present five applications of crude incidence curves to show their utility in a broad range of studies. In some of these applications it is helpful to model survival-time distributions with use of two different time metameters, for example, time from diagnosis and age of the patient. We describe how one can incorporate published vital statistics into the models when secondary types of failure correspond to common causes of death.

Estimation of the mean cumulative number of failures in a repairable system with mixed exponential component lifetimes

AbstractComponents in electronic systems are often observed to be likely to fail at an early age, an effect well known as the ‘infant mortality’ effect. Similarly, for systems composed of such components, a decrease in the rate of occurrence of failures, usually called the ‘reliability improvement’ effect, is seen in the early operating period. In this paper we show that such improvement may be considered as a simple consequence of the heterogeneity among the components in the population. We present a simple three‐parameter model for the distribution of the component lifetime, which has a simple physical interpretation, and from which we obtain methods for statistical inference, suitable for implementation on even small computers. The methods have been applied successfully to field failure data, collected from an industrial company.