Mathematical and Statistical Methods in Reliability

The following sections are included:

• CONTENTS

• PREFACE

• CONTRIBUTORS

It is argued that the mathematical theory of reliability as a separate discipline began in 1961 with the publication of "Multi-component systems and their structures and their reliability" by Birnbaum, Esary and Saunders⁸. Prior to this time, mathematicians were just applying standard mathematical techniques such as queueing theory, statistics and probability to engineering reliability problems. We will describe how the 1965 book⁵ "Mathematical Theory of Reliability" came to be written. Some personal historical perspectives will follow on probabilistic concepts of aging. Finally, we will discuss more recent work on Schur functions and Bayesian implications for reliability research.

The sophistication of science and technology is growing almost exponentially. Government and industry are relying more and more on science's advanced methods to assess reliability coupled with performance, safety, surety, cost, schedule, etc. Unfortunately, policy, cost, schedule, and other constraints imposed by the real world inhibit the ability of researchers to calculate these metrics efficiently and accurately using traditional methods. Because of such constraints, reliability must undergo an evolutionary change. The first step in this evolution is to reinterpret the concepts and responsibilities of scientists responsible for reliability calculations to meet the new century's needs. The next step is to mount a multidisciplinary approach to the quantification of reliability and its associated metrics using both empirical methods and auxiliary data sources, such as expert knowledge, corporate memory, and mathematical modeling and simulation.

The description of uncertainty about the world through probability is the defining idea of the Bayesian thinking. Probability models and parameters are just tools for expressing these uncertainties. Nonetheless, the Bayesian literature is mainly focused on probability models and the problem of assessing uncertainties of the parameters of these models. In this chapter we argue for the use of a stronger predictive emphasis in a Bayesian analysis, where the key elements are observable quantities to be predicted, models linking observable quantities on different level of detail, and probability used to express uncertainty of these observable quantities. And for some of the observable quantities, the assignments of the probabilities can be done without introducing parameters and adopting the somewhat sophisticated procedure of specifying prior distributions of parameters. Furthermore, if parameters are introduced they should have a physical meaning, they should represent states of the world. In this setting, model uncertainty has no meaning. What we look for is the "goodness" of the model, to represent the world and to perform its function as a tool in the analysis.

We consider the problem of incomplete conditional probability tables in Bayesian nets, noting that marginal probabilities for an effect, given a single cause are usually easy to elicit and can serve as constraints on the full conditional probability table (CPT) for occurrence of an effect given all possible conditions of its causes. A form of maximum entropy principle, local to an effect node is developed and contrasted with existing global methods. Exact maximum-entropy CPTs are computed and a conjecture about the exact solution for effects with a general number N of causes is examined.

This chapter tackles the difficulties of specifying subjective prior distributions for parameters that arise in reliability modelling. We review strategies for selecting families of priors, and propose some simplifications and enhancements. Sampling distributions of particular interest here are binomial, negative binomial, Poisson, exponential, normal, log-normal, gamma and Weibull. We then consider the formulation of suitable priors for generalized linear models and stochastic processes.

Our research also investigates methods of predictive elicitation to determine values for hyperparameters encountered in priors for subjective reliability analysis, and illustrates the philosophical beauty and practical benefits of this approach. We briefly discuss numerical algorithms needed to resolve the computational difficulties incurred, as an aid to decision making. Finally, we demonstrate these procedures by applying them to a problem of quality control in the electronics industry.

This survey of copulas reviews some aspects of copulas, their properties, tries to stress their relevance for statistics and their connection with Markov processes and conditional expectations.

"Domination theory" and the notion of the "signature" of a network, and their respective roles in calculating the reliability of a network, are briefly reviewed. The computational advantages of the former, and the interpretive richness of the latter, raise the question: how are the two related? The exact functional relationship between the signature vector and the vector of signed dominations is obtained.

In this chapter we show how Monte Carlo methods can be improved significantly by conditioning on a suitable variable. In particular this principle is applied to system reliability evaluation. In relation to this an efficient algorithm for simulating a vector of independent Bernoulli variables given their sum is presented. By using this algorithm one can generate such a vector in O(n) time, where n is the number of variables. Thus, simulating from the conditional distribution can be done just as efficient as simulating from the unconditional distribution. The special case where the Bernoulli variables are i.i.d. is also considered. For this case the reliability evaluation can be improved even further. In particular, we present a simulation algorithm which enables us to estimate the entire system reliability polynomial expressed as a function of the common component reliability. If the component reliabilities are not too different from each other, a generalized version of the improved conditional method can be used in combination with importance sampling.

The following sections are included:

• Main Directions of Research in Multi-State System Reliability

• History of Ideas in MSS Reliability

• References

In the present chapter, we study the distribution of a waiting time statistic associated with the sum of success run lengths in a sequence of binary trials. More specifically, we start accumulating the exact run lengths of subsequences (strings) consisting of k or more consecutive successes (k is a given positive integer) and then we focus on the waiting time until the sum exceeds a predetermined level. The investigation of the waiting time distribution of the statistic of interest is achieved by the aid of appropriate Markov chain embedding techniques.

Let X₁, X₂, … be a sequence of independent and identically distributed random variables with absolutely continuous distribution function. For n ≥ 1, we denote the order statistics of X₁, X₂, …, X_n by X_1,n ≤ X_2,n ≤ … ≤ X_n,n. Define L(1) = 1, L(n + 1) = min {j : j > L(n), X_j > X_j-1,j-1}, and X(n) = X_{L(n), L(n)}, n ≥ 1. The sequence {X(n)} ({L(n)}) is called upper record statistics (times). In this chapter, we deal with maximum likelihood prediction of record statistics. In particular, having observed a sequence of record statistics from an absolutely continuous population, we consider the maximum likelihood prediction of a future record statistic. The conditional median predictor (CMP) is also considered. Some desirable properties of maximum likelihood predictor (MLP) such as unbiasedness, consistency, and efficiency are discussed. The properties of MLP are examined in detail for the exponential and uniform populations. The conditional distribution is used to derive the CMP for the exponential and uniform populations. Finally, we compare the CMP with median unbiased predictor (MUP), MLP, and best linear unbiased predictor (BLUP). The comparisons indicate that the CMP performs well in terms of mean squared prediction error.

This chapter deals with the optimal detection of a change point in the intensity of a nonhomogeneous Poisson process N. We consider different information levels, namely sequential observation of N, retrospective analysis after having observed N up to some fixed time t* and a combination of both observation schemes. The corresponding detection problems are viewed as optimal stopping problems which only differ in the filtration used, and they are solved simultaneously by means of a semimartingale approach.

A key quantity for the assessment of the reliability of a dynamic system subjected to a stochastic load process is the mean level crossing rate of the response. This chapter discusses a new method for calculating this quantity for a stochastic response process represented as a second order stochastic Volterra series. The derivation of this procedure consists of three stages. First the expression for the mean crossing rate is rewritten in terms of a joint characteristic function. Secondly, it is shown that a closed form expression for this joint characteristic function can be derived. Thirdly, it is then demonstrated how the method of steepest descent can be applied to the numerical calculation of the mean crossing rate. It is shown by an example that the numerical accuracy of this method is apparently very high.

The chapter develops models that allow the determination of optimum inspection and maintenance policies. The models are based on Lévy processes and allow the use of the Markov property to construct renewal type integral equations for costs and dynamic programmes for determining optimum policies. The extension to models with imperfect observation or covariates is indicated. The models provide a single framework and depend only on the determination of an appropriate transition density to "plug in" to the equation. Some numerical results and a comparison of different policies are reported.

In survival and event history analysis the focus is usually on the mere occurrence of events. Not much emphasis is placed on understanding the processes leading up to these events. The simple reason for this is that these processes are usually unobserved. However, one may consider the structure of possible underlying processes and draw some general conclusions from this. One important concept being of use here is quasi-stationary distributions. These arise as limiting distributions on transient spaces where probability mass is continuously being lost to some set of absorbing states. Due to this leaking of probability mass, the limiting distribution is just stationary in a conditional sense, that is, conditioned on non-absorption. Quasi-stationarity is a research theme in stochastic process theory, with several established results, although not too much work has been done. Quasi-stationary distributions act as attractors on the set of individual underlying processes, and can be a tool for understanding the shape of the hazard rate.

We shall explain the use of this concept in survival analysis. Stochastic models based on Markov chains, diffusion processes and Lévy processes will be mentioned.

For a pair of exchangeable lifetimes we analyze several properties of the joint distribution of the residual lifetimes, conditional on histories of survivals. Attention is focused on properties of dependence, univariate aging and bivariate aging. In particular, we deal with notions of PQD, NBU, bivariate NBU and their negative counterparts. The role of copulæ in this analysis is highlighted.

The case of time-transformed exponential lifetimes, where the copulae involved are Archimedean, is analyzed in detail. Several examples and remarks illustrate the main points of this note and may give hints for future research.

The chapter presents the utilization of the Generalized Stochastic Petri Nets to model the reliability of the power systems. It is known that a dependability analysis for this kind of systems is often very difficult due to the multiple dependencies on the specific maintenance policies, on the great number of operation conditions and the possibility to consider non-exponential distribution of the operation periods and the time for maintenance activities. The main idea to use the GSPN formalism for the reliability modelling was to benefit of the advantages offered to generate more exactly the system's states taking into account its dynamic evolution and to use the Monte Carlo simulation of the model in the special situations imposed by the power systems complexity. In the first part is presented the main ideas for building-up a GSPN model, and the possibilities of its evaluation using stochastic processes or Monte Carlo simulation. A test network is modelled and evaluated. In the second part a study case is presented in order to establish the configuration for the Medium Voltage Distribution Systems (main auxiliary services) for a nuclear power plant (Cernavoda Nuclear Power Plant, CANDU reactor).

This chapter is motivated by a real problem, involving the prediction of damage to an item under different conditions. Problems of this type are generic. What is noteworthy here is that information about damage comes from three sources: a physics based model, actual test data, and informed testimonies from specialists. The problem then is to fuse or integrate this information in a coherent manner. Our proposed approach is Bayesian and involves some subtle manoeuvres of the likelihood function.

Weibull distribution is very useful in reliability and survival analysis because of its ability in modelling increasing and decreasing failure rate functions. However, when the failure rate function is of a bathtub shape, a single Weibull distribution will not be sufficient. Bathtub failure rate is a common phenomenon in reliability and it is an important ageing property. The system lifetime cycle can usually be divided into three distinct stages: early life, useful life and wear out period. Models for bathtub shaped failure rate functions are needed while it is also useful for a model to be generalisation of the Weibull distribution. In this paper, some models extending the traditional two-parameter Weibull distribution are presented and discussed. We also review some related models that can be used for the modelling of bathtub-shaped failure rate function.

The paper considers degradation and failure time models with multiple failure modes. Dependence of traumatic failure intensities on the degradation level are included into the models. Nonparametric estimators of various reliability characteristics are proposed and their asymptotic properties are established.

Degradation data analysis is a powerful tool for reliability assessment. Useful reliability information is available from degradation data when there are few or even no failures. For some applications the degradation measurement process destroys or changes the physical/mechanical characteristics of test units. In such applications, only one meaningful measurement can be can be taken on each test unit. This is known as "destructive degradation." Degradation tests are often accelerated by testing at higher than usual levels of accelerating variables like temperature.

This chapter describes an important class of models for accelerated destructive degradation data. We use likelihood-based methods for inference on both the degradation and the induced failure-time distributions. The methods are illustrated with the results of an accelerated destructive degradation test for an adhesive bond.

We develop an inferential procedure based on hierarchical Bayesian models which can take into account different kinds of heterogeneity in reliability/survival data. This is illustrated by an analysis of failure data from several motor operated closing valves at two nuclear power plants.

The chapter considers competing risk models suitable for analysis of modern reliability databases. Commonly used models are reviewed and a new simple and attractive model is developed. This model grew out of a study of real failure data from an ammonia plant. The use of graphical methods as an aid in model choice is advocated in the chapter.

The aim of this chapter is to propose a general model of the joint effect of corrective and preventive maintenance on repairable systems. The model is denned by the failure intensity of a stochastic point process. It takes into account the possibility of dependent maintenance times with different effects. The modelling is based on a competing risks approach. The likelihood function is derived, so parameter estimation and assessment of maintenance efficiency are possible.

An operating system (machine) is observed to undergo failures. On failure, one of three actions was taken: failures were minimally repaired, given a minor repair or given a major repair. Furthermore, periodically the machine was stopped for either minor maintenance action or major maintenance action. In addition to the kind of maintenance action, the length of duration for each repair action is known. Either on failure or maintenance stoppage, both types of repairs are assumed to impact the intensity following a virtual age process of the general form proposed by Kijima. There are several possibilities for assumptions of the impact of repair: it can be assumed that a minor or major repair impact the virtual age of the item to an unknown fixed part. It is also possible to assume that the impact of repair depends on the repair time. The issue in this research is to identify not only the virtual aging process associated with repairs but also the form of the failure intensity associated with the system. A series of models appropriate for such an operating/maintenance environment are developed and estimated in order to identify the most appropriate statistical structure. Field data from an industrial setting are used to fit the models.

A class of statistical tests for trend in repairable systems data based on the general null hypothesis of a renewal process is proposed. This class does in particular include a test which is attractive for general use by having good power properties against both monotonic and nonmonotonic trends. Both the single system and the several systems cases are considered.

We investigate the mathematical modelling of maintenance and repair of components that can fail due to a variety of failure mechanisms. Our motivation is to build a model, which can be used to unveil aspects of the quality of the maintenance performed. The model we propose is motivated by imperfect repair models, but extended to model preventive maintenance as one of several "competing risks". This helps us to avoid problems of identifiability previously reported in connection with imperfect repair models.

We obtain estimates of a surface by first finding the coefficient vector β(x) that minimizes a distance between θ(z) and an approximating function θ(z - x; β) for z in a neighborhood of a given point , next expressing this β(x) as a functional β(x; G) of a surface that admits an empirical estimate , and then using the empirical plug-in approach with and . Examples of G's are the multivariate distribution function and the integrated hazard function. With this approach the random part of the estimation error can be expressed in terms of empirical processes of the form , which facilitates asymptotic analysis. This approach gives simple derivations of old and new curve and surface estimates as well as their properties. It is particularly useful for providing simple and highly efficient estimates in boundary regions. We apply the approach to density estimation, univariate and multivariate; to hazard estimation; and to nonparametric regression.

Engineers commonly use the ratio of means to compare the strengths of two materials or even the deterioration of the strength of one material undergoing aging or various stresses. However, wood engineers also use the ratio of the fifth or other lower percentiles when comparing lumber of two different grades, moisture content, or sorted under two different grading systems. Motivated by this application, we develop some exact, and some approximate, confidence regions for the ratio of percentiles from two different populations. This work generalizes earlier work on inferences concerning the ratio of two means.

Our exact sampling results are limited to a normal linear model case, a related log-normal case, and negative exponential distributions. Other general distributions are treated by large sample methods.

We also consider the exact nonparametric Bayesian approach to obtain regions for the ratio of percentiles.

We consider a semi-Markov process with a finite state space, through one observation on a time interval [0,T], for a fixed time T. The aim of this paper is to present empirical estimators of the semi-Markov kernel, reliability, availability, failure rate and rate of occurrence of failure (rocof) of a system described by the semi-Markov process. We also give asymptotic properties of the above estimators, as the uniform strong consistency and normality. This paper is an overview of our recent results.^19–16,11

Classes of hazard-odds based fixed-order and adaptive smooth goodness-of-fit tests for the composite hypothesis that an unknown discrete distribution belongs to a family of distributions using right-censored observations are presented. The proposed classes of tests generalize Neyman's³³ smooth class of tests. The class of fixed-order tests is the discrete analog of the hazard-based class of tests for continuous failure times studied in Peña³⁵. The class of adaptive tests employs a modified Schwartz⁴⁰ Bayesian information criterion for choosing the order of the embedding class, with the modification on the criterion accounting for the incompleteness mechanism.

In an effort to address the continuing demand for high quality software, an enormous multitude of software reliability growth models have been proposed in recent years. In spite of the diversity and elegance of many of these, there is still a need for models which can be more readily applied in practice. One aspect which should perhaps be taken more account of is the complex and challenging nature of the testing of software. A brief overview of software reliability modelling and testing is given, together with an indication of the challenges that lie ahead if advances in theory are to address the practical needs of software developers.

We consider a Markovian model, proposed by Littlewood, to assess the reliability of a modular software. Specifically, we are interested in the asymptotic properties of the corresponding failure point process. We focus on its time-stationary version and on its behavior when reliability growth takes place. We prove the convergence in distribution of the failure point process to a Poisson process. Additionally, we provide a convergence rate using the distance in variation. This is heavily based on a similar result of Kabanov, Liptser and Shiryayev, for a doubly-stochastic Poisson process where the intensity is governed by a Markov process.

An adaptive sequential procedure for testing software is developed for a data domain model. The approach is that of predictive finite population modeling, when each modular unit is associated with one or more covariates. For a given link function, expressing the probability that there are faults in a unit as a function of the covariates, and for given costs of testing and failures in the field, a rectifying sampling plan is developed. If the link function is not completely known, the rectifying sampling is adaptive sequential.