Reliability Evaluation of Engineering Systems

In a modern society, professional engineers and technical managers are responsible for the planning, design, manufacture and operation of products and systems ranging from simple products to complex systems. The failure of these can often cause effects that range from inconvenience and irritation to a severe impact on society and on its environment. Users, customers and society in general expect that products and systems are reliable and safe. The question that arises is, ‘How reliable or how safe will the system be during its future operating life?’ This question can be answered, in part, by the use of quantitative reliability evaluation. In consequence a considerable awareness has developed in the application of such techniques to the design and operation of simple and complex systems together with an increasing amount of legal requirements, including product liability aspects and statutory directives. This book is primarily concerned with describing a wide range of reliability evaluation techniques and their application. However, it is first beneficial to discuss some of the issues and philosophy that relate to reliability in order to put these techniques into perspective and to identify the background from which the various techniques and measures have evolved, as well as to show why the techniques have been developed.

The word ‘probability’ is used frequently in a loose sense implying that a certain event has a good chance of occurring. In this sense it is a qualitative or subjective measure. It is important to realize that it has a strict technical meaning and is a scientific ‘measure of chance’, i.e., it defines quantitatively the likelihood of an event or events. Mathematically it is a numerical index that can vary between zero which defines an absolute impossibility to unity which defines an absolute certainty. This scale of probability is illustrated in Figure 2.1. A philosopher might argue that the two ends of the scale do not exist. An engineer might disagree from a pragmatic viewpoint. For example, the probability that a man will live for ever is zero, and the probability that one day he will die is unity.

Some of the general concepts and properties of distributions were introduced in Chapter 2. A number of standard distributions such as binomial, Poisson, normal, lognormal, exponential, gamma, Weibull, Rayleigh were also mentioned. Most of these distributions and their application in reliability evaluation are discussed in Chapter 6. However, it is useful to single out the binomial distribution at this stage.

The previous chapters have considered the application of basic probability techniques to combinational types of reliability assessment. In many types of problems these techniques may be all that is required to assess the adequacy of the system. However, in practice, a system is frequently represented as a network in which the system components are connected together either in series, parallel, meshed or a combination of these. This chapter considers series and parallel network representations (more complicated meshed networks are considered in the next chapter).

The techniques described in Chapter 4 are limited in their application to systems and networks that have a series and parallel type of structure. Many systems either do not have this simple type of structure or have complex operational logic. Additional modelling and evaluation techniques are necessary in order to determine the reliability of such systems. A typical system not having a series/parallel structure is the bridge-type network shown in Figure 5.1, a system that is used frequently to demonstrate techniques for complex systems and one that can occur often in many engineering applications.

The basic concepts associated with probability distributions were discussed in Chapter 2; terms such as probability mass function, probability density function, cumulative distribution function, expected or mean value, variance and standard deviation were described and illustrated. These concepts were extended in Chapter 3 to a specific distribution, the binomial distribution. The fundamental nature and application of the binomial distribution in reliability evaluation was also presented in Chapter 3.

A relatively standard procedure for evaluating the reliability of a system is to decompose it into its constituent components, estimate the reliability of each of these components and finally combine the component reliabilities using one or more numerical techniques to estimate the reliability of the complete system. The level to which the decomposition is taken must be such that the reliabilities of the resulting components are known within reasonable and acceptable precision. It may therefore not be necessary to decompose the system into individual real components but into a set of devices or subsystems, the reliabilities of which are known from experience.

Chapter 7 described and illustrated several analytical techniques for evaluating the reliability of systems. Although these techniques can be applied to both non-repairable and repairable systems, in the latter case they assume that the repair process is instantaneous or negligible compared with the operating time. This is an inherent restriction and additional techniques are required if this assumption is not valid. One very important technique that overcomes this problem and which has received considerable attention and use during the past few years is known as the Markov approach or Markov modelling. Several excellent texts [4, 25–27] are available on the subject of the application of Markov chains to reliability analysis.

Reliability problems are normally concerned with systems that are discrete in space, i.e., they can exist in one of a number of discrete and identifiable states, and continuous in time; i.e., they exist continuously in one of the system states until a transition occurs which takes them discretely to another state in which they then exist continuously until another transition occurs. The techniques described in this chapter pertain to systems that can be described as stationary Markov processes i.e., the conditional probability of failure or repair during any fixed interval of time is constant. This implies that the failure and repair characteristics of the components are associated with (negative) exponential distributions. In the case of a single component, or in systems composed of statistically independent components, the limiting or steady-state probabilities are not dependent on the state residence time distributions, only upon their mean values. This is discussed further in Chapter 12. It must be stressed however, that very considerable differences can exist in the values of the time-dependent state probabilities as these are very dependent on the distributional assumptions.

The Markov techniques described in the previous two chapters permit the probability of residing in each state of the system to be evaluated. The probability of being in the system up state, system down state and system derated state(s) can then be evaluated from these individual state probabilities knowing which of the individual system states contribute to the respective overall system states. These techniques can be used for either mission orientated or repairable systems, and can be extended to evaluate the MTTF of the system.

As discussed in Chapters 8 to 10, the Markov technique and the frequency and duration approach form sound and precise modelling and evaluation methods for reliability applications. They become less amenable, however, for hand calculations and even for digital computer solutions as the system becomes larger and more complex. In such cases, alternative methods are available which are based on the Markov approach and which use a set of appropriate but approximate equations. The essence of these approximate techniques is to derive a set of equations suitable for a series system in which all components must operate for system success and for a parallel system in which only one component need work for system success. These equations can then be used in conjunction with the network modelling techniques described in Chapters 4 and 5 to give rapid and sufficiently accurate results for a wide range of practical systems. In addition they are ideally suited for both hand calculations and digital computer implementation.

The derivations in many of the preceding chapters have been made assuming that the underlying distributions for the component state resident times are exponential. This is particularly the case in Chapter 8 onwards. It has been stressed in these preceding chapters that, although the assumption of exponential distributions may have been made, the results and equations are equally applicable to all distributions if only the limiting state or long term average values are being evaluated for systems containing statistically independent components. This is not true if the time dependent values are being evaluated. The effect of typical distributions on limiting state values is demonstrated in Section 12.6. If the underlying distribution is non-exponential, then the process becomes non-Markovian. The techniques proposed in Chapter 8 onwards are no longer applicable and the problem has to be treated differently. Before discussing these additional techniques, it is worth examining the assumption of exponential distributions a little more deeply.

As briefly stated in Chapter 1, the reliability indices of a system can be evaluated using one of two basic approaches, direct analytical techniques or stochastic simulation. Most chapters of this book are concerned with the analytical approach, but it is important to note that simulation is a very valuable method which is widely used in the solution of real engineering problems. This chapter is intended to provide an introduction to this approach, but for a more detailed and deeper discussion of the subject the reader should refer to specialist books such as References 53–55.

This book has been written primarily for the practising engineer, technical manager and engineering student who has little or no formal background in the area of statistics, probability theory and reliability evaluation techniques. It attempts to illustrate the theoretical concepts and practical formulae required to evaluate quantitatively the reliability of systems. It must be stressed that there is no single all-purpose technique that can be used to solve all reliability problems and great care must be given in selecting the most appropriate model and evaluation technique.