Fundamentals of Reliability Engineering E-Book

Indra Gunawan

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Performability Engineering Series
• Sprache: Englisch

Seitenzahl: 182

Veröffentlichungsjahr: 2014

Beschreibung

Provides fundamentals of reliability engineering and illustrates practical applications in the area of parallel/distributed systems (Multistage Interconnection Networks) The first part of the book (chapters 1-5) introduces the concept of reliability engineering, elements of probability theory, probability distributions, availability, and data analysis. The second part of the book (chapters 6-11) provides an overview of parallel/distributed computing, network design considerations, classification of multistage interconnection networks, network reliability evaluation methods, and reliability analysis of multistage interconnection networks including reliability prediction of distributed systems using Monte Carlo method. Fundamentals of Reliability Engineering meets the increasing demand for knowledge tools that practicing reliability professionals can use to optimize their reliability decisions. Reliability prediction is important as it determines the usability and efficiency of the network to provide services. Reliability evaluation methods discussed in this book can be applied to analyze the reliability of any other systems. As an example, reliability analysis of distributed systems that consist of layers of switching elements connected together in a predefined topology that provide the connectivity between the set of processors and the set of memory modules, are presented.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Preface

Chapter 1: Introduction to Reliability Engineering

1.1 The Logic of Certainty

1.2 Union (OR) operation

1.3 Intersection (AND) operation

1.4 Series systems

1.5 Parallel systems

1.6 General Series-Parallel System

1.7 Active Redundancy

1.8 Standby Redundancy

1.9 Fault Tree Analysis

1.10 Minimum Cut Sets and Path Sets

References

Chapter 2: Elements of Probability Theory

2.1 Basic Rules of Probability

2.2 Cumulative Distribution Function

2.3 Probability Mass Function

2.4 Probability Density Function

2.5 Moments

2.6 Percentiles

References

Chapter 3: Probability Distributions

3.1 Binomial

3.2 Poisson

3.3 Exponential

3.4 Weibull

3.5 Normal

3.6 Lognormal

3.7 Mean Time To Failure (MTTF)

References

Chapter 4: Availability

4.1 Definition

4.2 Summary

4.3 Availability of Systems with Repair

References

Chapter 5: Data Analysis

5.1 Theoretical Model and Evidence

5.2 Censored Samples

5.3 Bayesian Theorem

References

Chapter 6: Introduction to Network Systems

6.1 Parallel Computing and Networks

6.2 Network Design Considerations

6.3 Classification of Interconnection Networks

References

Chapter 7: Classification of Multistage Interconnection Networks

7.1 Background

7.2 Multistage Cube Network

7.3 Extra-Stage Cube Network

7.4 Shuffle-Exchange Network

7.5 Shuffle-Exchange Network with an Additional Stage

7.6 Gamma Network

7.7 Extra-Stage Gamma Network

7.8 Dynamic Redundancy Network

7.9 Improved Enhanced Augmented Data Manipulator Network

7.10 Improved Logical Neighborhood Network

7.11 Comparison

References

Chapter 8: Network Reliability Evaluation Methods

8.1 Overview of Network Reliability

8.2 Network Model

8.3 Network Operations

8.4 Approaches for Calculating Network Reliability

8.5 Summary

References

Chapter 9: Reliability Analysis of Multistage Interconnection Networks

9.1 Reliability Analysis of Shuffle-Exchange Network with Minimal Extra Stages

9.2 Terminal Reliability Improvement in Modified Shuffle-Exchange Network

9.3 Reliability Bounds for Large MINs

References

Chapter 10: Terminal Reliability Assessment of Gamma and Extra-Stage Gamma Networks

10.1 Introduction

10.2 Gamma Network

10.3 Terminal Reliability of Gamma Network

10.4 Extra-Stage Gamma Network

10.5 Comparison

10.6 Conclusions

References

Chapter 11: Reliability Prediction of Distributed Systems Using Monte Carlo Method

11.1 Introduction

11.2 Reliability Parameters

11.3 Monte Carlo Method

11.4 Confidence Interval for Monte Carlo Point Estimate

11.5 Numerical Results

11.6 Conclusion

References

Index

Fundamentals of Reliability Engineering

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106

Performability Engineering Series Series Editors; Krishna B. Misra ([email protected]) and John Andrews ([email protected])

Scope: A true performance of a product, or system, or service must be judged over the entire life cycle activities connected with design, manufacture, use and disposal in relation to the economics of maximization of dependability, and minimizing its impact on the environment. The concept of performability allows us to take a holistic assessment of performance and provides an aggregate attribute that reflects an entire engineering effort of a product, system, or service designer in achieving dependability and sustainability. Performance should not just be indicative of achieving quality, reliability, maintainability and safety for a product, system, or service, but achieving sustainability as well. The conventional perspective of dependability ignores the environmental impact considerations that accompany the development of products, systems, and services. However, any industrial activity in creating a product, system, or service is always associated with certain environmental impacts that follow at each phase of development. These considerations have become all the more necessary in the 21st century as the world resources continue to become scarce and the cost of materials and energy keep rising. It is not difficult to visualize that by employing the strategy of dematerialization, minimum energy and minimum waste, while maximizing the yield and developing economically viable and safe processes (clean production and clean technologies), we will create minimal adverse effect on the environment during production and disposal at the end of the life. This is basically the goal of performability engineering.

It may be observed that the above-mentioned performance attributes are interrelated and should not be considered in isolation for optimization of performance. Each book in the series should endeavor to include most, if not all, of the attributes of this web of interrelationship and have the objective to help create optimal and sustainable products, systems, and services.

Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Copyright © 2014 by Scrivener Publishing LLC. All rights reserved.

Co-published by John Wiley & Sons, Inc. Hoboken, New Jersey, and Scrivener Publishing LLC, Salem, Massachusetts. Published simultaneously in Canada.

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Cover design by Exeter Premedia Services Private Ltd., Chennai, India

Library of Congress Cataloging-in-Publication Data:

ISBN 978-1-118-54956-8

Preface

The purpose of this book is to provide readers with fundamentals of reliability engineering and demonstrate reliability approaches for evaluating system reliability with case studies in multistage interconnection networks.

The book can be used as an introductory book in reliability engineering for undergraduate/graduate students in Industrial/Electrical/Computer Engineering as well as engineers, researchers or managers. Practical applications are included to describe the importance of reliability measurement to achieve better systems.

In the first part of the book (chapters 1-5), it introduces the concept of reliability engineering, elements of probability theory, probability distributions, availability and data analysis.

The second part of the book (chapters 6-11) provides an overview of parallel/distributed computing, network design considerations, classification of multistage interconnection networks, network reliability evaluation methods, and reliability analysis of multistage interconnection networks including reliability prediction of distributed systems using Monte Carlo method.

It covers comprehensive reliability engineering methods and practical aspects in interconnection network systems. Students, engineers, researchers, managers will find this book as a valuable reference source.

The main key features of this book include:

Fundamental of reliability engineering.

Elements of probability and probability distributions.

Classification of network systems.

Reliability evaluation methods.

Reliability analysis of multistage interconnection network systems is illustrated as practical applications of reliability methods including reliability prediction of distributed systems using Monte Carlo method.

I would like to express my gratitude to Prof. K.B. Misra for his kind assistance in reviewing the book.

Finally, my heartfelt thanks go to my wife Donna, daughters Jessica and Cynthia for their continuous support and my parents Suwita and Effie Gunawan for their motivation and encouragement.

Chapter 1

Introduction to Reliability Engineering

Reliability is defined as the probability that a system (part or component) can perform its intended task under specified conditions and time interval. It is used normally as the quantitative measure of the performance of a designed part, component or system. Reliability is also a design parameter which can be improved by design modification, redesign, elimination of deficiencies, and addition of redundant components or units.

The first part of this book (chapters 1–5) describes fundamentals of reliability engineering and the second part (chapters 6–11) presents reliability methods and its applications in Multistage Interconnection Networks (MIN). Chapter 9–11 discusses in details reliability analysis of network systems. Reliability of MIN is an important parameter that can be used as a measure on how reliable the interconnected components in network systems.

1.1 The Logic of Certainty

Event is a statement that can be true or false. “It may rain today” is not an event. According to our current state of knowledge, we may say that an event is true, false, or possible (uncertain). Eventually, an event will be either true or false.

Sample Space is the set of all possible outcomes of an experiment [1-4]. Each elementary outcome is represented by a sample point. Examples: there are six possible outcomes/numbers {1, 2, 3, 4, 5, 6} from tossing a die; the failure time of a component is {0, ∞}. A collection of sample points is an event.

1.2 Union (OR) operation

(1.1)

(1.2)

(1.3)

Diagram Venn and fault tree for union (OR) operations are shown in Figure 1.1 below.

Figure 1.1 Diagram Venn and Fault Tree for Union (OR) Operation.

1.3 Intersection (AND) operation

(1.4)

(1.5)

(1.6)

Diagram Venn and fault tree for intersection (AND) operations are shown in Figure 1.2 below.

Figure 1.2 Diagram Venn and Fault Tree for Intersection (AND) Operation.

In A and B are mutually exclusive events (they are independent to each other) then

(1.7)

These two basic operations are implemented in real systems as below.

1.4 Series systems

Structure function of system failure and success in series systems can be defined as follows:

System failure:

(1.8)

System success:

(1.9)

Reliability block diagram and fault tree for series systems are shown in Figure 1.3 below.

Figure 1.3 Reliability Block Diagram and Fault Tree for Series Systems.

The system reliability Rs is the product of the individual element reliabilities:

(1.10)

If we assume that each of the elements has a constant failure rate, then the reliability of the ith element is given by the exponential relation:

(1.11)

Thus,

(1.12)

and failure rate of system of N elements in series

(1.13)

Then

If the individual Fi are small, i.e. Fi << 1,

(1.14)

1.5 Parallel systems

Structure function of system failure and success in parallel systems can be defined as follows:

System failure:

(1.15)

System success:

(1.16)

Reliability block diagram and fault tree for parallel systems are shown in Figure 1.4 below.

Figure 1.4 Reliability Block Diagram and Fault Tree for Parallel Systems.

The unreliability of parallel system is given by:

(1.17)

If the individual elements are identical:

(1.18)

This gives:

1.6 General Series-Parallel System

A general series-parallel system consists of n identical subsystems in parallel and each subsystem consists of m elements in series.

If Rji is the reliability of the ith elements in the jth subsystem, then the reliability of the jth subsystem is:

(1.19)

The corresponding unreliability of the jth subsystem is:

(1.20)

The overall system unreliability is:

(1.21)

1.7 Active Redundancy

A system is referred to as “k out of n” if the overall system will continue to function correctly when only k (k ≤ n) of the n elements/systems are working normally; the remaining (n − k) elements/systems ensure extra reliability.

In 2 out of 4 system, the overall system unreliability is:

(1.22)

The above result can also be obtained from the binomial expansion of (F + R)4:

(1.23)

If R is very close to 1 and F a lot smaller than 1, then:

(1.24)

1.8 Standby Redundancy

In this system, only one unit is operating at a time; the other units are shut down and are only brought into operation when the operating unit fails.

Assuming the switching system has perfect reliability, then the reliability of the standby system can be given by the cumulative Poisson distribution [5]:

(1.25)

The term exp(−λt) [λt] represents the increase in reliability due to adding one standby unit.

1.9 Fault Tree Analysis

A k out of n system means that at least k components should be working for the system to be operational. An example 2 out of 3 system is described in Figure 1.5 below:

Figure 1.5 Reliability Block Diagram for 2 out of 3 system.

Fault tree diagram is used to represent how the structure of the system works [6-7]. Fault tree diagram for this system is shown in Figure 1.6 above (symbol V represents OR operation and represents AND operation):

Figure 1.6 Fault Tree Diagram for 2 out of 3 system.

Structure function for system failure can be formulated as follows:

(1.26)

1.10 Minimum Cut Sets and Path Sets

Cut Set is any set of events (failures of components and human actions) that cause system failure [8-9]. Minimal cut set is a cut set that does not contain another cut set as a subset. On the other hand, Path Set represents any set of events that cause system success.

Figure 1.7 Simplified Fault Tree Diagram for 2 out of 3 system.

Minimal cut sets:

Another method to find the system failure function is by using the following formula:

(1.27)

Where:

Therefore:

This minimum cut set approach can be applied to find the system failure of the bridge system as shown in Figure 1.8 above:

Figure 1.8 Bridge System

There are four minimum cut sets for this system: {X1X2}, {X3X4}, {X2X3X5}, {X1X4X5}

Therefore the system failure function can be written as:

(1.28)

References

1. Walpole, R.E., Myers, R.H., and Myers, S.L., Probability and Statistics for Engineers and Scientists, Sixth edition, Prentice Hall, 1998.

2. Montgomery, D.C. and Runger, G.C., Applied Statistics and Probability for Engineers, Second edition, John Wiley & Sons, Inc., 1999.

3. Ross, S.M., Introduction to Probability Models, Sixth edition, Academic Press, 1999.

4. Montgomery, D.C., Runger, G.C., and Hubele, N.F., Engineering Statistics, John Wiley & Sons, Inc., 1998.

5. Bentley, J., Introduction to Reliability and Quality Engineering, Second edition, Pearson, 1999.

6. Rausand, M. and Hoyland, A., System Reliability Theory: Models, Statistical Methods, and Applications, John Wiley & Sons, Inc., 2004.

7. Kumamoto, H. and Henley, E.J., Probabilistic Risk Assessment and Management for Engineers and Scientists, Second edition, IEEE Press, 1996.

8. Misra, K.B., Reliability Prediction and Analysis: A Methodology Oriented Treatment, Elsevier, 1992.

9. Misra, K.B., New Trends in System Reliability Evaluation, Elsevier, 1993.

Chapter 2

Elements of Probability Theory

2.1 Basic Rules of Probability

The probability of an event A is a quantity that satisfies the following axioms [1-5]:

In general,

(2.1)

For the rare event approximation,

(2.2)

2.2 Cumulative Distribution Function

The cumulative distribution function (CDF) is F(x) Pr[X ≤ x]. This is valid for both discrete and continuous random variables.

Properties:

2.3 Probability Mass Function

For discrete random variable (DRV), probability mass function is

(2.3)

2.4 Probability Density Function

For continuous random variable (CRV), probability density function (PDF) is defined as:

(2.4)

(2.5)

(2.6)

The normalization condition gives

2.5 Moments

Expected (or mean, or average) value:

Variance (standard deviation σ):

2.6 Percentiles

For continuous random variable, the 100γ percentile is defined as that value of x for which

(2.11)

Example:

References

1. O’Connor, P.D.T. and Kleyner, A., Practical Reliability Engineering, Fifth Edition, John Wiley & Sons, Ltd., 2012.

2. Walpole, R.E., Myers, R.H., and Myers, S.L., Probability and Statistics for Engineers and Scientists, Sixth edition, Prentice Hall, 1998.

3. Montgomery, D.C. and Runger, G.C., Applied Statistics and Probability for Engineers, Second edition, John Wiley & Sons, Inc., 1999.

4. Ross, S.M., Introduction to Probability Models, Sixth edition, Academic Press, 1999.

5. Montgomery, D.C., Runger, G.C., and Hubele, N.F., Engineering Statistics, John Wiley & Sons, Inc., 1998.

Chapter 3

Probability Distributions

In this chapter, characteristics of discrete probability distributions such as binomial and poisson as well as common continuous probability distributions such as exponential, weibull, normal and lognormal are discussed [1-6].

These distributions are needed to formulate:

The probability that a component will start (fail) on demand.

The probability that a component will run for a period of time given a successful start.

The impact of repair on these probabilities.

The frequency of initiating events.

In general, it can be defined that:

3.1 Binomial

(3.1)

This is the probability mass function (PMF) of the Binomial Distribution.

It is the probability of

exactly

k failures in N demands.

The

binomial coefficient

is:

(3.2)

Mean number of failures:

(3.3)

3.2 Poisson

Used typically to model the occurrence of initiating events.

DRV: number of events in (0, t)

Rate is constant; the events are independent.

The probability of exactly k events in (0, t) is (PMF):

(3.7)

(3.8)

(3.9)

Example of the Poisson Distribution

Failure while running

3.3 Exponential

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

Example: 2-out-of-3 system

Each sensor has a MTTF equal to 2,000 hours. What is the unreliability of the system for a period of 720 hours?

Step 1: System Logic.

Step 2: Probabilistic Analysis.

For nominally identical components:

If