On some reliability growth models with simple graphical interpretations

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.     

A new reliability growth model: its mathematical comparison to the Duane model

The Duane reliability growth model has been traditionally used to model electronic systems undergoing development testing. This paper proposes a new reliability growth model derived from variance stabilisation transformation theory which surpasses the Duane model in typical reliability growth situations. This new model is simpler to plot and fits the data more closely than the Duane model whenever the Duane slope is less than 0.5. This paper explores the mathematical relationships between these two models; and shows that at a Duane slope of 0.5, both models are mathematically equivalent in their capacity to fit the observed data. The instantaneous MTBF of the new model is also developed and compared to that of Duane. As the new model is influenced by the later failures, compared to early failures for the Duane model, it has the further advantage of leading to reduced test times for achieving a specified instantaneous MTBF. As the reliability of electronic systems increases, this has positive implications for testing.     

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.     

Some Aspects of Reliability Growth

Reliability growth is examined with the use of a variant of the Duane model. This variant is based on stepwise growth and avoids some of the mathematical and philosophic dificulties associated with the Duane model. Monte Carlo simulation, while confirming the noisy behavior of reliability growth data, indicates that central tendencies are sufficiently strong to justify the use of Gaussian approximations for analysis of some aspects of growth. Some exact methods and some approximations for parameter estimation are presented.     

Operational repairable equipments and the duane model

For many years, the assessment of the rate of reliability growth during the development of repairable equipments has been carried out using the Duane model. As part of a design-to-life-cycle cost study, it was required to determine whether the Duane model was also applicable to operational equipment data. Observation of data trends indicated that the majority of the 370 data sets tested possessed nonconstant failure rates. Further analysis revealed nearly 80% of the data sets accepted a Weibull repairable model (which is equivalent to Duane). However, fitting the model to the data using either or both the least squares and maximum likelihood regression methods produced problems which led to the conclusion that the Duane model was unsuitable for use on operational failure data.     

Rationale for a Modified Duane Model

The Duane model for reliability growth involves a rate function which is an inverse power law and has an ``infinite' value at t = 0. The model is usually motivated entirely empirically. Here a probabilistic rationale is proposed via a reliability growth model involving the removal of design faults. This rationale results in a modified power law rate, finite at the origin. A wider class of rate functions should be investigated for NHPP models of reliability growth.     

Starting & Limiting Values for Reliability Growth

Modifications of the Duane model for reliability growth which permit formulating test plans prior to test-data availability are presented herein. The modified model yields finite and nonzero MTBFs at the start and end of development testing. The unmodified Duane model is inadequate for formulating the Development Test plan for a Reliability Program because the MTBF is zero at the start of testing and is unbounded for long tests. The parameters of the modified model; limiting MTBF, initial MTBF, and Duane shape parameter (logarithmic growth rate) are estimatable from handbooks and previous experience prior to the start of development tests. An implicit expression for the Duane-model scale-parameter as a function of the limiting MTBF, initial MTBF, and shape parameter is given. The time required and/or the reliability improvement potential of development tests are obtained from the model to plan the reliability program. When the development tests are underway interval estimates for the modified Duane model parameters, needed to monitor the progress of the reliability program, can be computed.     

Reliability growth models for hardware and software systems based on nonhomogeneous Poisson processes: A survey

We summarize the reliability growth models for hardware and software systems described by a stochastic process, where the underlying stochastic process is assumed to be a nonhomogeneous Poisson process (NHPP). The background of reliability growth modelling based on an NHPP is surveyed. The Duane model, which was first postulated as a reliability growth model and is commonly used, is first explained. Secondly, the Weibull growth and modified Weibull growth models for hardware systems and the exponential type growth and gamma type growth models for error detection for software systems are discussed. The parameter estimates can be obtained by maximum likelihood estimation. Finally, the goodness-of-fit tests based on chi-square, Cramér-von Mises and Kolmogorov-Smirnov statistics are presented for the reliability growth models based on an NHPP.     

Software-reliability evaluation of the TROPICO-R switching system

The analysis and evaluation of the software of the TROPICO-R switching system are described. Failure data were collected over 27 months, including validation phase, field trial phase, and operational life. During this period 461 failure reports have been recorded. Two models (exponential and S-shaped) have been applied to these failure data, and have been analyzed according to their repetitive and predictive capabilities. It is found that it is difficult for the models to reproduce the observed failure data when changes in trend do not follow the model assumptions. The Laplace trend test is a major tool for guiding the partitioning of failure data according to the assumptions of model reliability growth. Only the S-shaped model has been considered for the analysis according to trend periods. Prediction yields good results over a time period of a few months, showing that reliability modeling is a major tool for test/maintenance planning and follow-up     

Software reliability growth supermodels

The aim of this paper is to present a method for obtaining a more accurate prediction for software reliability growth models (SRGMs). It is our belief that if we try to use a more general approach implying the building of a supermodel as a weighted sum of several SRGMs, it may be possible to obtain more accurate results in prediction. The weight factors will depend on the values of the prequential likelihood functions as calculated for each SRGM, the values varying each time a new error is observed. The basic models chosen are the Jelinski-Moranda, Goel-Okumoto, Duane, Littlewood-Verrall and Keiller-Littlewood models. Finally, we shall compare the SRGMs with the supermodels by using the median estimate and deciding if there are any benefits or constraints in applying this technique.     

Reliability Growth and Duane Learning Curves

This paper examines Duane learning curves for reliability improvement between successive reliability tests. A constant failure rate ratio between successive tests gives curved lines on a Duane plot. Gordon's observation that dominant failures appear exponentially with test time leads to exponential failure rate ratio models. These models, although not the only possible ones, give results that agree well with observations from practice.     

Reliability Investment and Life-Cycle Cost

The reliability of avionic equipment profoundly influences life-cycle cost; the level of reliability attained largely depends upon the investment in reliability programs during development. As more investment is made in reliability improvement, some cost elements increase and others decrease. These opposing cost trends yield a unique minimum life-cycle cost (LCC). In order to find the level of investment in a reliability improvement program that minimizes LCC, the Reliability Investment Optimization (RIO) model has been developed. It identifies, for a particular avionic system, the level of reliability investment that minimizes the LCC of the equipment. This model employs a reliability-growth relationship based on the Duane model. The RIO model uses this reliability growth pattern to compute LCC as a function of MTBF (mean time between failures) where LCC comprises: 1) research, development, test and evaluation (RDT&E), 2) procurement, and, 3) operations and support (O&S). The RIO model uses summary level data that are appropriate for the timeframe of its most advantageous use, i.e., prior to detail design of the system. The degree of accuracy for the input parameters need not be high because results are not very sensitive to data accuracy. The model's results thus are quite stable. The RIO model was designed with avionic systems in mind. However, the model applies to a wider range of systems. Certain assumptions should be particularly scrutinized in extending usage beyond avionics, e.g., Poisson demand assumption versus a wearout failure pattern (failure rate increases over time), scheduled maintenance, and LCC element breakdown.     

Failure correlation in software reliability models

Perhaps the most stringent restriction in most software reliability models is the assumption of statistical independence among successive software failures. The authors research was motivated by the fact that although there are practical situations in which this assumption could be easily violated, much of the published literature on software reliability modeling does not seriously address this issue. The research work in this paper is devoted to developing the software reliability modeling framework that can consider the phenomena of failure correlation and to study its effects on the software reliability measures. The important property of the developed Markov renewal modeling approach is its flexibility. It allows construction of the software reliability model in both discrete time and continuous time, and (depending on the goals) to base the analysis either on Markov chain theory or on renewal process theory. Thus, their modeling approach is an important step toward more consistent and realistic modeling of software reliability. It can be related to existing software reliability growth models. Many input-domain and time-domain models can be derived as special cases under the assumption of failure s-independence. This paper aims at showing that the classical software reliability theory can be extended to consider a sequence of possibly s-dependent software runs, viz, failure correlation. It does not deal with inference nor with predictions, per se. For the model to be fully specified and applied to estimations and predictions in real software development projects, we need to address many research issues, e.g., the detailed assumptions about the nature of the overall reliability growth, way modeling-parameters change as a result of the fault-removal attempts     

Accelerated Discrete Degradation Models for Leakage Current of Ultra-Thin Gate Oxides

Using degradation measurements is becoming more important in reliability studies because fewer failures are observed during short experiment times. Most of the literature discusses continuous degradation processes such as Wiener, gamma, linear, and nonlinear random effect processes. However, some types of degradation processes do not occur in a continuous pattern. Discrete degradations have been found in many practical problems, such as leakage current of thin gate oxides in nano-technology, crack growth of metal fatigue, and fatigue damage of laminates used for industrial specimens. In this research, we establish a procedure based on a likelihood approach to assess the reliability using a discrete degradation model. A non-homogeneous Weibull compound Poisson model with accelerated stress variables is considered. We provide a general maximum likelihood approach for the estimates of model parameters, and derive the breakdown time distributions. A data set measuring the leakage current of nanometer scale gate oxides is analyzed by using the procedure. Goodness-of-fit tests are considered to check the proposed models for the amount of degradation increment, and the rate of event occurrence. The estimated reliabilities are calculated at lower stress of the accelerated variable, and the approximate confidence intervals of quantiles for breakdown time distribution are given to quantify the uncertainty of the estimates. Finally, a simulation study based on the gate oxide data is built for the discrete degradation model to explore the finite sample properties of the proposed procedure.     

Approaches for reliability modeling of continuous-state devices

Three approaches for reliability modelling of continuous state devices are presented in this paper. One uses the random process to fit model parameters of a statistical distribution as functions of time. This approach allows the data set to be from any general distribution. The second approach uses the general path model to fit parameters of the model as functions of time. The relationship between the random process model and the general path model is illustrated. The third approach uses multiple linear regression to fit the distribution of lifetime directly. This approach has less restriction on the degradation data to be analyzed. All three approaches are illustrated with examples. Finally a mixture model is proposed which can be used to model both catastrophic failures and degradation failures. This mixture model also shows engineers how to design experiments to collect both hard failure data and soft failure data. Topics for further investigation in continuous device reliability modelling include further investigation of the mixture model, application of these models to practical situations, and using complex statistical distributions to fit degradation data     

Exploring Genetic Programming and Boosting Techniques to Model Software Reliability

Software reliability models are used to estimate the probability that a software fails at a given time. They are fundamental to plan test activities, and to ensure the quality of the software being developed. Each project has a different reliability growth behavior, and although several different models have been proposed to estimate the reliability growth, none has proven to perform well considering different project characteristics. Because of this, some authors have introduced the use of Machine Learning techniques, such as neural networks, to obtain software reliability models. Neural network-based models, however, are not easily interpreted, and other techniques could be explored. In this paper, we explore an approach based on genetic programming, and also propose the use of boosting techniques to improve performance. We conduct experiments with reliability models based on time, and on test coverage. The obtained results show some advantages of the introduced approach. The models adapt better to the reliability curve, and can be used in projects with different characteristics.     

基于BP神经网络的产品寿命分布类型选择

产品寿命分布类型选择问题,是可靠性工程中的一个重要问题。在分析了几种可靠性寿命分布类型识别方法的基础上,设计了一种基于BP神经网络的可靠性寿命分布类型选择方法,并进行了计算机模拟验证。结果表明,该方法原理简单,算法易以实现,便于工程应用。     

Extension of the Duane Plotting Technique

A Duane plot is a straight line relating the log of cumulative failure rate (or of cumulative MTBF) to the log of cumulative equipment operating hours in a series of equipment tests. The straight line provides an easy translation from the cumulative line to a line of instantaneous failure rate (or to instantaneous MTFB). This technique has also been applied to the early failure rates of operating integrated circuits (ICs); the cumulative plot has the same usefulness in data smoothing, and the same relationship to instantaneous failure rate applies. The cumulative failure rate plot produces a curved line instead of a straight line. The Duane relationship applies at any time point, however, and depends only on the slope of the line at that time. For either application, the cumulative data should not be forced to fit a straight line; such forcing causes the early reliability history to be lost, and the latest history may be improperly determined from too few points. This is not just of historical interest; it concerns whether reliability is still improving usefully, or whether a point of limited return is being approached. An additional point of concern is indicated by the example of rapidly reducing failure rates in early life of semiconductor devices. For equipments using many of these devices, the Duane plot assumption of a constant failure rate during each equipment test period is probably not true.     

Multi-state homogeneous Markov models in reliability analysis

In the standard Markov technique applied to reliability analysis, components are characterized by two states: an up state and a down state. The present paper explores the possibility of studying system reliability, by modelling each component with a multi-state homogeneous Markov model (MHMM). It is shown that this approach is of value both in approximating non-exponential probability distributions and in helping to build up suitable models for physical processes. Examples are presented which illustrate how the multi-state technique fits many practical situations. Finally some open problems on this topic are suggested.     

三点弯曲法的PBGA封装实验测试

任何电子产品在进入市场之前的可靠性测试是十分重要的环节。为此,在之前的几十年中开发了多种类型的测试方法。一种广泛使用的测试方法便是线路板弯曲测试。进行可靠性测试的一个主要问题就是需要大量的测试样品,这是因为这种测试必须重复进行直到获得一致并满意的测量结果。其测试样品的设计,组装工艺和测试程序需要花费长期的时间。如果可靠性测试结果不满意,测试样品需要重新设计、重新生产、重新测试,这从商业的观点来看成本过高且低效。为简化可靠性测试程序,提出了计算模拟的方法在前期设计阶段来预测印制板组装线的可靠性。在目前的研究中,可靠性测试采用了三点弯曲测试法。实现了有限元模式确认及关于该模式应用的2种情况研究。