Fundamentals of Reliability Engineering: Applications in Multistage Interconnection Networks

By Indra Gunawan

This book presents fundamentals of reliability engineering with its applications in evaluating reliability of multistage interconnection networks. In the first part of the book, it introduces the concept of reliability engineering, elements of probability theory, probability distributions, availability and data analysis.  The second part of the book provides an overview of parallel/distributed computing, network design considerations, and more.  The book covers a comprehensive reliability engineering methods and its practical aspects in the interconnection network systems. Students, engineers, researchers, managers will find this book as a valuable reference source.

• Language: English
• Publisher: Wiley
• Release date: Mar 10, 2014
• ISBN: 9781118914380

Book preview

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.

Indicator variables for events can be written in the following form. If an event i is true then Xi = 1 and if an event i is false then Xi = 0. Two basic operations, Union (OR) and Intersection (AND) are discussed.

1.2 Union (OR) operation

Suppose there are two events, A and B in the sample space. The equations below represent C as a union of the two events. XC = 1 means that an event C is true when either event A or B is true.

(1.1) equation

(1.2) equation

(1.3) equation

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

The equations below represent C as an intersection of A and B. XC = 1 means that an event C is true when both the events are true.

(1.4) equation

(1.5) equation

(1.6) equation

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) equation

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) equation

System success:

(1.9) equation

Where Xj = 1 or Yj = 1 represent when the component j is failed or working.

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) equation

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) equation

Thus,

(1.12) equation

and failure rate of system of N elements in series

(1.13) equation

Since Rs = 1 − Fs and Ri = 1 − Fi

Then

equation

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

(1.14) equation

1.5 Parallel systems

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

System failure:

(1.15) equation

System success:

(1.16) equation

Where Xj = 1 or Yj = 1 represent when the component j is failed or working.

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) equation

If the individual elements are identical:

(1.18) equation

This gives:

equation

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) equation

The corresponding unreliability of the jth subsystem is:

(1.20) equation

The overall system unreliability is:

(1.21) equation

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:

F = Prob. (A,B,C,D fail) + Prob. (A,B,C fail) + Prob. (B,C,D fail) + Prob. (A,C,D fail) + Prob. (A,B,D fail)

(1.22) equation

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

(1.23)

equation

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

(1.24) equation

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) equation

Thus for n = 1, R(t) = exp(−λt)

For n = 2, R(t) is increased to:

R(t) = exp(−λt) [1 + λt]

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

For n = 3, R(t) is further increase to:

equation

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:

equation

Expanding and using Xk = X we get

(1.26)

equation

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.

For 2 out of 3 system, we now can simplify the fault tree diagram into Figure 1.7 below: M1 = XAXB, M2 = XB XC,, M3 = XC XA

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

Minimal cut sets:

equation

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

(1.27)

equationequation

Where:

equation

Therefore:

equation

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)

equation

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]:

equation

For two events P(A∪B) = P(A) + P(B) − P(AB) and for two mutually exclusive events A and B P(A or B) = P(A) + P(B)

In general,

(2.1)

equation