Reliability Analysis and Plans for Successive Testing: Start-up Demonstration Tests and Applications

By Narayanaswamy Balakrishnan, Markos Koutras and Fotios Milienos

Reliability Analysis and Plans for Successive Testing: Start-up Demonstration Tests and Applications discusses all past and recent developments on start-up demonstration tests in the context of current numerical and illustrative examples to clarify available methods for distribution theorists and applied mathematicians dealing with control problems. Throughout the book, the authors focus on the panorama of open problems and issues of further interest. As contemporary manufacturers face tremendous commercial pressures to assemble works of high reliability, defined as ‘the probability of the product performing its role under the stated conditions and over a specified period of time’, this book helps address testing issues.

• Unites the tools and methodologies of applied statistics and stochastic modeling to aid the determination of device reliability for better performing consumer goods
• Clearly articulates how successive testing methods can be used in practice
• Comments not only on distribution sequences closed, but also on open problems and issues of further interest for researchers

• Language: English
• Publisher: Academic Press
• Release date: Jan 15, 2021
• ISBN: 9780128043622

Narayanaswamy Balakrishnan

Narayanaswamy Balakrishnan is a distinguished university professor in the Department of Mathematics and Statistics at McMaster University Hamilton, Ontario, Canada. He is an internationally recognized expert on statistical distribution theory, and a book-powerhouse with over 24 authored books, four authored handbooks, and 30 edited books under his name. He is currently the Editor-in-Chief of Communications in Statistics published by Taylor & Francis. He was also the Editor-in-Chief for the revised version of Encyclopedia of Statistical Sciences published by John Wiley & Sons. He is a Fellow of the American Statistical Association and a Fellow of the Institute of Mathematical Statistics. In 2016, he was awarded an Honorary Doctorate from The National and Kapodistrian University of Athens, Athens, Greece. In 2021, he was elected as a Fellow of the Royal Society of Canada.

Book preview

2020

Chapter 1: Introduction

Abstract

In this chapter, we discuss the early history of start-up demonstration tests, the need for start-up demonstration tests, and the scope of the book. We mention that at the end of each chapter, as a final section, we give a detailed literature survey concerning the subject matter of that chapter and also present a chronological account of all pertinent developments. We have used throughout the book plethora of figures, tables, and numerical examples to explain the ideas in a simple and succinct way to enhance the understanding and experience for the readers. Needless to say, while going through the material presented in the book, one may identify some potential problems and issues that are worthy of further attention and work from researchers. In this regard, the extensive bibliography that we have provided at the end of this book, we hope, will be of great assistance.

Keywords

Start-up demonstration tests; discrete distributions; waiting time distributions

1.1 Early history

Hahn and Gage (1983), by assuming the experimental results to be independent and identically distributed as Bernoulli outcomes, were the first to discuss start-up demonstration testing with the requirement of the occurrence of a prespecified number of consecutive successful start-ups. To facilitate the computation of associated probabilities and reliability characteristics as a function of the probability of success p of the underlying trials, they derived the probability mass function of the corresponding waiting time in a recursive form. Even though this work has been mentioned and has received great attention in many of the publications that ensued on start-up demonstration tests during the last four decades, it is important to mention the work of Feder (1974), nearly a decade earlier than the paper of Hahn and Gage (1983). Feder (1974) employed the Markov chain method to discuss the distribution of a (prespecified) number of defective fasteners adjacent to one another resulting in a failure in a conveyor belt in an assembly process. As a matter of fact, Hahn and Gage (1983) gave due reference to the earlier work of Feder, and mentioned that their approach presented a simpler formula for the calculation of associated probabilities.

Interestingly enough, as this book clearly demonstrates, it is the Markov chain method that has become an effective and essential tool in studying various tests plans that have been considered in the context of start-up demonstration testing, and also in successfully exploring several generalizations of these plans such as those based on multistate trials. It is for this reason that the Markov chain embedding technique has been adopted throughout this book, and has also been amply illustrated with many numerical examples.

1.2 Need for start-up demonstration tests

It is important to highlight that reliability plans for start-up demonstration tests are what are essential in evaluating the reliability properties and characteristics of many equipments and tools such as lawn mowers, crank shafts, backup generators, car batteries, mechanical saws, back-up battery systems, fasteners in manufacturing processes, etc. These are all equipments wherein the success or failure of operation of the equipment is based on the outcome of a binary trial, resulting in a success if the equipment were to start successfully or a failure if the equipment failed to start. Therefore all different models and methods of analysis of data obtained from such demonstration tests that are discussed in this book will be useful not only in evaluating the reliability of such equipments, but also in devising suitable strategies for their acceptance in terms of quality and performance in the context of acceptance sampling and statistical quality control methods. Of course, each model and method will have its own advantages and disadvantages, and so we have made a sincere attempt in this book to discuss their merits and demerits; in addition, wherever possible, we have also given guidance about the implementation of test plans and as to how to make a suitable choice of the associated design parameters.

1.3 Scope of the book

This book contains nine chapters in total, including this Introduction chapter. Below, we present a synopsis of what is discussed within each chapter.

We start in Chapter 2 with some preliminaries on basic techniques and theory that a reader will need for following and understanding the developments in rest of the chapters. We specifically describe some basic principles on combinatorial methods and use of generating functions. We then explain in detail the finite Markov chain embedding technique, which is what is used primarily in all subsequent chapters. Finally, some elements of reliability theory, used to derive results on the start-up demonstration tests, are introduced, focusing mainly on binary systems and systems with multiple failure modes.

The main discussion of the book starts with a detailed exposition of various binary start-up demonstration tests in Chapter 3. Throughout this chapter, as shorthand and convenient notation, we use C to denote consecutive and T to denote total (meaning, frequency) for occurrences of concerned events, S to denote success as outcomes, and F to denote failures as outcomes. Thus, for example, the basic model of Hahn and Gage (1983), described earlier in Section 3.2.1, regarding the requirement of consecutive successes, will naturally be denoted by CS model. Similarly, one of the early models discussed by Balakrishnan and Chan (2000), in the context of acceptance plans of start-up equipments with acceptance based on the requirement of consecutive successes and rejection based on a total (or frequency) number of failures, will be denoted conveniently by CSTF model. Ten different such plans based on consecutive or total number of successes and failures are described and discussed in detail in this chapter. Then the models involving multiple runs and frequencies, scan-type statistics, and extended runs are also discussed. The computation of the distribution of the stopping time and relevant reliability characteristics are all developed at length and several illustrative examples are presented throughout.

The results and methods for binary start-up demonstration tests discussed in detail in Chapter 3 are then generalized in Chapter 4 to multistate start-up demonstration tests. Specifically, the focus is first made on tests possessing two types of successes and one type of failure, and analogously on tests possessing two types of failure and one type of success. Many plans that have been studied in the literature in this setting are reviewed in this chapter and illustrated with numerical examples. Finally, the chapter concludes with a general discussion on some models in which there are multiple types (more than two) of successes and failures.

An unified approach for studying the general theory of start-up demonstration tests is described next in Chapter 5. The binary and multistate cases are then taken up to exemplify this unified approach. The chapter finally concludes with a discussion on some bounds and approximations that could be used fruitfully in the assessment of reliability of the concerned equipments, and also as to how they compare with some known results in the reliability theory literature.

Statistical inferential results, based on the likelihood method and method of moments, were first offered by Viveros and Balakrishnan (1993) for the basic CS model considered by Hahn and Gage (1983). In Chapter 6, such inferential methods are outlined for different binary and multistate models. In this context, a mixture model is also considered in the context of start-up demonstration testing and inferential procedures for this mixture model, including the identifiability issues of model parameters, are all discussed.

In Chapter 7, some extensions of start-up demonstration test procedures that have been proposed in the literature are described. These include tests with corrective actions that are taken upon observing a failure (or a certain number of failures), a two-stage start-up demonstration procedure in which an early rejection of the equipment (based on observing a certain number of failures) is permitted, a parallel start-up demonstration test plan based on two sequences of outcomes, and a weighted start-up demonstration test plan. Finally, the chapter concludes with a discussion on the design of optimal start-up testing