Series on Quality, Reliability and Engineering StatisticsMathematical and Statistical Methods in Reliability, pp. 3-13 (2003) No AccessMATHEMATICAL RELIABILITY THEORY: FROM THE BEGINNING TO THE PRESENT TIMERichard E. BarlowRichard E. BarlowCollege of Engineering, University of California, Berkeley, CA 94720, USAhttps://doi.org/10.1142/9789812795250_0001Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: 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. FiguresReferencesRelatedDetailsCited By 4Reliability and Safety in Offshore EngineeringJunkai Feng30 June 2022Reliability and Safety in Offshore EngineeringJunkai Feng28 October 2021Stochastic Case Problems for the Secondary Classroom with Reliability TheoryUsha Kotelawala11 June 2011Bounds on MTBF of Systems Subjected to Periodic MaintenanceS.V. Amari1 Sep 2006 | IEEE Transactions on Reliability, Vol. 55, No. 3 Mathematical and Statistical Methods in ReliabilityMetrics History PDF download

In this chapter we assume that populations are effectively unbounded; i.e., we are assuming the limiting population case (N ↑ ∞).2.1. Data analysis for the exponential model.In the previous chapter we derived the finite population exponential model for populations of size N. Only in the limit, as N ↑ ∞, did we obtain the usual exponential model, i.e., in the one-dimensional case where θ is the limiting average lifetimep(x|θ)=1θe−xθ2.1.1for x≥0 and θ>0 . We call (2.1.1) the “exponential model.”The first idea in Chapter 1 was that the derivation should be based on a judgment of indifference on submanifolds. The importance of this derivation was due to the fact that we focused on the average lifetime of the population. Were we to focus on other aspects of the lifetimes of the population, we would have derived a different conditional probability model.The second idea was that when we insert data in our conditional probability model, the corresponding function of θ, called the likelihood L(θ), provides the key tool in analyzing data. (In Chapter 1, θ was the population lifetime average.) Using the likelihood and a prior opinion concerning θ we can compute the posterior probability function for θ and from this answer any probability question we may have concerning θ or future lifetimes.The influence of failures on the posterior density. Suppose NO failures are observed but all items have survived for some time t. How can we analyze item lifetimes in this case? To make the situation clear we will consider a particular case. We will show that, although survivors tend to increase our estimate for the mean life θ, lack of failures also increases our uncertainty concerning θ. While the engineer naturally wants to see no failures, the statistician—analyst wants to see failures in order to decrease the uncertainty in the analysis.

The common approach to analyzing censored data utilizes "competing risk" models; a class of distribution is first chosen and then the sufficient statistics are identified! An "operational Bayesian" approach (Barlow 1993) for analyzing censored data would require a somewhat different methodology. In this approach, we first determine potentially observable parameters of interest. We then determine the data summaries (sufficient statistics) for these parameters. Tsai (1994) suggests that the observed sample frequency is sufficient for predicting the population frequency. Invariant probability measures (likelihoods), conditional on the parameters of interest, are then derived based on the principle of sufficiency and the principle of insufficient reason.

We emphasize the derivation of likelihood models starting from a well specified problem of interest and finite populations. Parameters are given operational meaning. In particular, parameters are specified in terms of different forms of energy. Examples relevant to reliability theory are used to illustrate ideas. Examples in engineering probability are given.

A four-cell model to analyze dependent lifetimes by using the generalized multivariate Gumbel distribution is described.

This chapter focuses on one of the most basic stochastic models in reliability theory, that is, the on-off process generated by failures and repairs of components in a series system. A series system of k components operates if and only if each of the k components operates. No component operates while the system is down. Repaired components are assumed to function like new components. There is a large literature dealing with availability, the probability that the system is functioning. However, most papers assume special repair or failure distributions, or both. As only failed components are replaced with new or like-new components, the age distribution of components in the system quickly becomes mathematically very complicated. The process {ξ(t); t≥ 0} has no generation points. The chapter focuses on the process {U(t); t ≥ 0} where U(t) is the system functioning time in [0, t].

Let $\Pi_1, \Pi_2, \cdots, \Pi_k$ be $k$ populations. The random variable $X_i$ associated with $\Pi_i$ has a continuous distribution $F_i, i = 1, 2, \cdots, k$. We are primarily interested in selecting a subset such that the probability is at least $P^\ast$ that the selected subset includes the population with the largest (smallest) quantile of a given order $\alpha (0 < \alpha < 1)$. We assume each $F_i$ has a unique $\alpha$-quantile, $\xi_{\alpha i}$. Let $F_{\lbrack i\rbrack}(x) = F_{\lbrack x\rbrack}$ denote the cumulative distribution function of the population with the $i$th smallest $\alpha$-quantile. In the following, we consider families of distributions ordered in a certain sense with respect to a specified continuous distribution $G$ and propose and study a selection procedure which is different from the non-parametric procedure of Rizvi and Sobel (1967). We assume (a) $F_{\lbrack i\rbrack} (x) \geqq F_{\lbrack k\rbrack}(x), i = 1, 2, \cdots, k$ and all $x$. (b) $\mathbf{\exists}$ a continuous distribution $G \ni F_{\lbrack i\rbrack} \underset{\sim}{\prec} G, \mathbf{\forall}i = 1, 2, \cdots, k$, where $\underset{\sim}{\prec}$ denotes a partial ordering relation on the space of distributions. A relation $\underset{\sim}{\prec}$ on the space of distributions is a partial ordering if \begin{equation*}\begin{split}F \underset{\sim}{\prec} F\quad \mathbf{\forall} \text{distributions} F \\ F \underset{\sim}{\prec} G,\quad G \underset{\sim}{\prec} H \text{implies} F \underset{\sim}{\prec} H.\\ \end{split}\end{equation*} Note that $F \underset{\sim}{\prec} G$ and $G \underset{\sim}{\prec} F$ do not necessarily imply $F \equiv G$. Various special cases in addition to stochastic ordering are: (i) $F \prec_\ast G \operatorname{iff} F(0) = G(0) = 0$ and $G^{-1}F(x)/x$ is nondecreasing in $x \geqq 0$ on the support of $F$. (ii) $F \prec_c G \operatorname{iff} G^{-1}F(x)$ is convex on the support of $F$. (iii) $F \prec_r G \operatorname{iff} F(0) = G(0) = \frac{1}{2}$ and $G^{-1}F(x)/x$ is increasing (decreasing) for $x$ positive (negative) on the support of $F$. (iv) $F \prec_s G \operatorname{iff} F(0) = G(0) = \frac{1}{2}$ and $G^{-1}F$ is concave-convex about the origin, on the support of $F$; i.e., $\{x\mid 0 < F(x) < 1\}$. If $G(x) = 1 - e^{-x}$ for $x \geqq 0$, then (i) defines the class of IFRA distributions studied by Birnbaum, Esary and Marshall (1966) while (ii) defines the class of IFR distributions studied by Barlow, Marshall and Proschan (1963). For any distribution $G, F \prec_\ast G \operatorname{iff} F(x)$ crosses $G(\theta x)$ at most once and from below if at all as a function of $x$ for all $\theta > 0$. If $G(x) = 1 - \exp (-x^\lambda)$ for $x \geqq 0$ and $\lambda > 0$, then $F \prec_\ast G$ implies that $F$ is "sharper" than the family of Weibull distributions with shape parameter $\lambda$. Implications of orderings defined by (iii) were studied by Lawrence (1966). Van Zwet (1964) studies orderings defined by both (ii) and (iv). Clearly $\prec_c$ ordering implies $\prec_\ast$ ordering and $\prec_s$ ordering implies $\prec_r$ ordering. If $\mathbf{X}_i = (X_{i1}, X_{i2}, \cdots, X_{in})$ is the observed sample from the $i$th population, then we restrict ourselves to the class of statistics $T_i = T(\mathbf{X}_i)$ that preserve both ordering relations (a) and (b), i.e., $(a') P_{F_\lbrack i\rbrack}\{T(\mathbf{X}) \leqq x\} \geqq P_{F_\lbrack k\rbrack}\{T(\mathbf{X}) \leqq x\}$ for all $x$ and $i = 1, 2, \cdots, k$. $(b') F_{T(X_i)} \precsim G_{T(\mathbf{Y})}, i = 1, 2, \cdots, k$, where $F_{T(\mathbf{X}_i)}$ represents the cdf of $T(\mathbf{X}_i)$ under $F_{\lbrack i\rbrack}$ and $G_{T(\mathbf{Y})}$ is the cdf of $T(\mathbf{Y})$ under $G, \mathbf{Y} = (Y_1, Y_2, \cdots, Y_n)$ being a random sample from $G$. In Section 2 of this paper, we propose and study procedures $R (R')$ for selecting the population with the largest (smallest) $\alpha$-quantile for distributions which are $\prec_\ast$ ordered with respect to a specified distribution $G$. The infimum of the probability of a correct selection is obtained in Theorem 2.1 and asymptotic evaluation is given in Theorem 2.2. Section 3 deals with quantile selection procedures for the class of IFRA distributions. In Section 4, we study the efficiency of procedure $R$ with respect to a procedure studied by Rizvi and Sobel (1967) under scale type slippage configurations. Asymptotic relative efficiency of $R$ with respect to a selection procedure for the gamma populations proposed by Gupta (1963) is also investigated. Section 5 deals with selection procedures for the median for distributions that are $\prec_r$ ordered with respect to a specified $G$. In Section 6 we propose a selection procedure with respect to the means for distributions that are $\prec_c$ ordered with respect to $G(x) = 1 - e^{-x}$. Application to the selection of gamma populations is also given in Section 6.