Reliability Modelling and Analysis in Discrete Time

By Unnikrishnan Nair, P.G. Sankaran and N. Balakrishnan

Reliability Modelling and Analysis in Discrete Time provides an overview of the probabilistic and statistical aspects connected with discrete reliability systems. This engaging book discusses their distributional properties and dependence structures before exploring various orderings associated between different reliability structures. Though clear explanations, multiple examples, and exhaustive coverage of the basic and advanced topics of research in this area, the work gives the reader a thorough understanding of the theory and concepts associated with discrete models and reliability structures. A comprehensive bibliography assists readers who are interested in further research and understanding.

Requiring only an introductory understanding of statistics, this book offers valuable insight and coverage for students and researchers in Probability and Statistics, Electrical Engineering, and Reliability/Quality Engineering. The book also includes a comprehensive bibliography to assist readers seeking to delve deeper.

• Includes a valuable introduction to Reliability Theory before covering advanced topics of research and real world applications
• Features an emphasis on the mathematical theory of reliability modeling
• Provides many illustrative examples to foster reader understanding

• Language: English
• Publisher: Academic Press
• Release date: May 15, 2018
• ISBN: 9780128020067

Unnikrishnan Nair

N. Unnikrishnan Nair obtained his Ph.D. from the University of Kerala, India and was conferred the degree of Doctor of Human Letters (honoris causa) by the Juniata College, USA. He was Professor and Chair, Department of Statistics, Dean, Faculty of Science and the Vice-Chancellor of the Cochin University of Science and Technology in India. He is a Fellow and past President of the Indian Society for Probability and Statistics, as well as an elected member of the International Statistical Institute. Dr. Nair has published 150 peer reviewed research papers and is an author, editor, or contributor of several books for publishers including Birkhauser, Boston and Education Book Distributors (and others in foreign languages).

Book preview

work.

Chapter 1

Reliability Theory

Abstract

In this chapter, we provide a brief account of the background materials that are required for the discussions in the ensuing chapters. The meaning, objectives and importance of reliability theory, as a discipline, is discussed. We then point out why reliability modelling and analysis in discrete time is separately viewed in this work. Life distributions play an important role in reliability modelling. Accordingly, various properties of life distributions that are of interest in reliability modelling are briefly reviewed. We also discuss the role of mixture distributions and weighted distributions. Among the weighted distributions, special attention is paid to equilibrium distributions and their higher orders. Some basic topics like convolutions and shock models are reviewed along with some essential results in this connection. This is followed by describing some geometric concepts like convexity, concavity, and star-shapedness used in the derivation of certain properties of reliability concepts. In order to model complex devices or systems with several components, multivariate distributions become necessary and for this reason some basic tools and techniques in multivariate analysis are presented. Multivariate equilibrium distributions of different forms are defined in the discrete case. An important aspect to be considered in modelling and analyzing multivariate data is the dependence relation that exists between the components. So, we review several dependence measures like correlation, Kendall's tau, Spearman's rho and Blomquist's β and also dependence concepts that indicate positive or negative association like tail monotonicity, stochastic monotonicity, and total positivity. Moreover, the multivariate extensions of these notions are also presented. Following this, discrete versions of time-dependent measures of Clayton, Anderson, Louis, Holm and Harvald, Bairamov, Kotz and Kozubowski, and Nair and Sankaran are discussed. This chapter finally concludes with the definitions of Schur-convexity, concavity and constancy that form an essential tool in Bayesian reliability analysis.

Keywords

Reliability modelling; Weighted and mixture distributions; Geometric concepts; Multivariate distributions; Dependence concepts; Equilibrium model

1.1 Reliability Theory

In ordinary usage, an object is meant to be reliable if it works upto expectation in times of need. However, for the development of a proper theory out of this intuitive meaning, one needs to have a more precise definition that would be capable of utilizing our scientific knowledge to characterize, quantify and measure reliability. A commonly used definition of reliability is that it is the probability that an equipment or device will perform the functions intended of it under conditions specified for its operation for a given period of time. When the device does not carry out its intended function, we may say that it has failed. The notion of failure can be more elaborately described, of course. Failure can be any incident or condition that causes an industrial plant, manufactured product, process, material or service to degrade or become unsuitable for performing satisfactorily, safely, reliably and cost-effectively (Witherell, 1994). An important objective of reliability theory is to understand the pattern in which failures occur for different mechanisms under different operating conditions. The uncertainty in the times at which failures occur demands a probabilistic framework for dealing with reliability of a device or system.

Rapid strides in technology during the last few decades have given rise to new products, devices and services that aim to make human life more comfortable. These include a wide range of products, from consumer goods for daily use to sophisticated devices for inter-planetary missions. From the producer to the customer, everyone is interested in ensuring that the product undergoes failure-free operation at least for a specified period of time. Reliability theory is a body of concepts, practices and methodology that attempts to meet this objective. Blischke and Murthy (2000) have identified five main areas through which the tasks in reliability can be accomplished, and these are reliability modelling, reliability analysis, reliability engineering, reliability science and reliability management. Of these, reliability modelling deals with the identification of mechanism that generates observations on the failure times in a specific study. Various concepts that describe the failure mechanism and their properties can shed some light into the patterns in the observations thus enabling the determination of the law that governs the lifetimes of the device. Many results in probability theory, statistics and stochastic processes become the main inputs in this connection. A suitable mathematical model, along with the real data, can then be used to draw inferences about the reliability. This is the main topic in reliability analysis. Reliability engineering aims at producing more reliable products through testing, designing and construction of products. Some key properties of materials that may cause failure and the effects of the manufacturing process are studied in reliability science. Emphasis in reliability management is on the consequences of unreliable products such as cost, loss of goodwill and repair, and to ensure quality of the product through management of design, operation and manufacture. Of these different areas, this book focuses only on reliability modelling and analysis. The relevant mathematical theory is described in detail with numerous examples in Chapters 2 through 8.

In the discussions held so far, emphasis has been made on the reliability of products and the role of reliability theory in assessing it. The time interval of failure free operation of a device is often termed as the lifetime of the device. However, lifetime can be viewed within a more general framework. If the duration spent by an element or object in a given state is taken to be the lifetime, then the theory of reliability can readily find applications in life tables, nuptiality studies, period of service before leaving a particular job in manpower planning, and so on. Reliability concepts have also found many useful applications in the construction of income distributions, analysis of human settlements, under-reported incomes, and bibliometry. A survey of some important applications is presented in Chapter 9. Thus, a discussion of the basic concepts and methodology in reliability theory assume importance in a plethora of areas of scientific activity.

In most of the discussions on reliability theory, the lifetime is treated as continuous. The modelling and analysis aspects use distributions and processes that commonly represent a non-negative continuous random variable. Comparatively, much less literature is available when the lifetime is discrete. However, there are some compelling reasons to consider failure times as discrete random variables taking on non-negative integer values. When a piece of equipment operates in cycles and the observation is the number of cycles completed before failure, the lifetime is clearly discrete. So also is the case when the device is monitored only in completed units of time, like how many failures have occurred at the completion of one hour, two hours, and so on. The lack of accuracy of measuring devices may also generate discrete lives. There are occasions to prefer counts over clock time even when the latter is available. In weapons reliability, the number of rounds fired is more important than the age at failure. The same is the case with lifetimes of car tires wherein the number of kilometres run before it becomes out of use is preferred to the number of days before failure. These types of problems provide a strong impetus for studying reliability in discrete time. The concepts in continuous and discrete times are the same, but the definitions and interpretations may differ between the two. To derive reliability properties analogous to the continuous case, occasionally continuous distributions are discretized. But, it is not necessary in such cases that the distributional properties are the same nor the discretization process will always result in meaningful discrete models. Consequently, there are conceptual and mathematical problems in developing discrete reliability theory. These points will be deliberated at appropriate places in the ensuing chapters.

The build-up of reliability theory for discrete lifetimes requires inputs from other disciplines, predominantly tools from mathematics, statistics and probability theory. These are described briefly in the remaining sections of this chapter.

1.2 Discrete Life Distributions

Let X , where b can be finite or infinite, representing the lifetime of a device. Reliability, being defined in terms of probability, involves the basic function of interest in describing the probabilities of X, which is the distribution function.

. Of equal importance is the survival function defined by

representing the probability that the device has a lifetime of at least xis also called the reliability function. Note that

The probability mass function of X is

is the probability mass function of some discrete random variable X. In the sequel, we will often make use of the following relationships:

of X is

, we get the rth moment

(1.1)

which is said to exist when the series in (1.1) it is absolutely convergent. The mean of the distribution, conceived as a measure of location or central point of the distribution, is the first moment

(1.2)

To define other measures, we need the central moments defined by

For measuring the dispersion, or spread of the values from the mean, the variance is defined as

(1.3)

Similarly, for capturing the shape characteristics of the distribution, measures of skewness and kurtosis are defined as

(1.4)

These measures are generally used though there are some other measures introduced in the literature for this purpose. For discrete models, sometimes the descending factorial moments defined by

(1.5)

where

is the descending factorial expression, may be more convenient to derive and use. Another similar type of moments are the ascending factorial moments expressed as

(1.6)

. The characteristic function is defined, for all real t, as

(1.7)

If the moment of order r ,

or

through the inversion formula (Johnson et al., 1992)

The probability generating function is defined as

(1.8)

We can obtain the distribution of X using

(1.9)

and also the moments from

Similarly, the factorial moment generating function is defined as

(1.10)

From (1.10), the rth descending factorial moment is calculated from the formula

(1.11)

Given the factorial moments, the survival function can be expressed as

1.3 Mixture Distributions

A mixture . Then,

(1.12)

is called a finite mixture if n is finite or an infinite mixture if n is infinite. The simplest form of mixture is a two-component mixture of the form

(1.13)

are called the components of the mixture and α is said to be the mixing constant.

, the mixture is formed as

(1.14)

One can easily recognize that , the resultant mixture has its distribution function as

(1.15)

.

In either case,

where the expectation is taken with respect to the distribution of Θ. . As a second example, an important quality characteristic of a product depends on the amount θ with θ satisfies the equation

Some of the standard distributions of importance in reliability analysis can also arise as mixtures. For example, let X be geometric with

where p has a beta distribution with probability density function,

denoting the complete beta function. Then, from (1.15), we get

(1.16)

. Eq. (1.16) is the so-called Waring distribution, whose properties are studied in the subsequent chapters.

As a second example, let X be binomial with

with parameters a and k for p, we get

(1.17)

Eq. (1.17) is the so-called negative hyper geometric distribution, the distribution with

(1.18)

is of special interest in the sequel.

The reliability functions of mixture models will be discussed in Chapter 2 and the role of mixtures in generating models with non-monotonic hazard rates will be discussed in Chapter 5.

1.4 Weighted Distributions

The concept of weighted distributions was introduced first by Fisher (1934) and was later popularized by Rao (1965); see also Patil (2002). When observations are recorded, they will not have the assumed model unless every observation has the same probability of being included in the set of observations. When there is a probability of recording an observation x of a discrete random variable X has its mass function as

(1.19)

where C is an arbitrary non-negative weight function which may exceed unity, (1.19) is called the weighed distribution of X . Patil et al. (1986) provided a structural perspective of discrete weighed model and have then presented several examples to show how such models arise in various real-life situations. They have also provided some general properties of weighted distributions and weight functions that are of specific interest. Gupta and Kirmani (1990) have pointed out several instances from survival analysis, wherein weighted distributions are of natural interest, while Pakes et al. (2003) have established many characterization results using weighted distributions, and Kokonendji et al. (2009) have developed semiparametric estimation method for count data through weighted distributions. In this and the rest of this work, we designate X as the original distribution. The distribution in (1.19) can also be presented as

(1.20)

, or by the survival function

(1.21)

Apart from the general forms in (1.20) or (1.21), there are two special cases that are of interest in reliability modelling. The first one is the length-biased model defined as

(1.22)

, which arises when the sampling mechanism selects units with probability proportional to the lengths of lifetimes of the units. , are the following (Patil and Rao, 1971 and Patil et al., 1986):

;

is monotone increasing or decreasing in xby the variance to mean ratio of X ;

denote the characteristic functions of X , respectively, then

(Sen and Khattree, 1996);

(Patil and Rao, 1978);

in is the length-biased version of order n of Xis form-invariant if it has the same form of distribution of X. Let X which is form-invariant under length-bias of order nwhich is also form invariant under length-bias of order n. Then, the mixture distribution in (1.15) is form-invariant under length-bias of order n.

One particular family of weighted distributions that has received considerable attention in the literature is the weighted Poisson distributions; see, for example, , is not covered by the above result.

To further examine the properties of the weighted Poisson distribution, Kokonendji et al. (2008) started with its general form of the probability mass function given by

is the natural parameter of the Poisson (λ, and established the mean-variance relationship

is log-convex or log-concave, respectively.

A useful and interesting example of the Weighted Poisson distribution is the COM-Poisson family, originally introduced by Conway and Maxwell (1962) with probability mass function

, respectively. As demonstrated by ) distributions as special cases, and is therefore quite useful for fitting count data which may be overdispersed or underdispersed. For example, it has been used in developing flexible cure rate models in survival analysis; see Rodrigues et al. (2009), and Balakrishnan and Pal (2013, 2016).

A further extension has been provided by Minkova and Balakrishnan (2013) in terms of compound weighted Poisson distribution 's are i.i.d. discrete random variables. They have then discussed the overdispersion and underdispersion of these distributions as well as properties of several special cases of these compound weighted Poisson distributions.

has equilibrium distribution corresponding to X if it has its probability mass function of the form

(1.23)

Applications of (1.23) are found in many diverse topics that include characterizations of probability distributions, criteria for ageing, maintenance policies, income analysis, and risk theory. There are different interpretations for (1.23). As a weighted distribution, (1.23) has a slightly different meaning than in the continuous case, where the weight function is the reciprocal of the hazard rate of X. However, the discrete counterpart has weight function

is the discrete hazard rate defined in Section specified by

(1.24)

is the probability mass function of X . The marginal distributions of (1.24) are seen to be

and

Notice that

the discrete uniform distribution. In other words, if the sampling is done proportional to the lifelength x and given that sample points fall in an interval of length xis given