Table Of ContentProbability Concepts
in Eng·ineering·
Planning· and Desig·n
VOLU1l1E II DECISI01V, RISK,
AND RELIABILITY
ALFREDO H-S. ANG
Professor of CiL·il Engineering
Unh·ersity of Illinois at Urbana-Champaign
WILSON H. TANG
Professpr of Civil Engineering
University oflllinois at Urbana-Champaign.
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JOHN WILEY & SONS f
New York • Chichester Brisbane Toronto Singapore
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Dedicated to Myrtle and Bernadette
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Copyright l!:l 1984 by John Wiley & Sons. Inc.
All righis reserved. Published simultaneousi)· in Canada.
Reproduction or translation of any pan "of
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Librtzry of onxrtss llttzloxinx in Publictztion DIIUI:
!Revised for volume 2)
Ans. Alfredo Hua-Sin~;. 193(}-
Probability concepts in engineering planninl! and
design.
Includes bibliographical references and in~exes.
Contentl: , .. I. Basic principles-v. ::!. Decision.
risl.:. and reliability.
I. Engineering-Statistical methods. !!. Probabilities.
I. Tang. Wilson H.
TA340.AS 620 .. 00422"01519::! 75-589.:::
Printed in the Republic of Sinppore
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10 9 8 7 6 4 3 2
preface. _
Like Volume I, this volume emphasizes the applications of probability and statistics
in engineering. It is built on the basic principles contained in Volume I but
additional more advanced tools are developed. including statistical decision
analysis. Markov and queueing models. the statistics of extreme values, Monte
Carlo simulation. and system reliability. Again, the necessary concepts are
introduced and illustrated within the context of engineering problems. Indeed, the
developrr.ent of the logical concepts and the illustration of established principles
in engineering applications are· the main objectives of the volume. The tools
provided should form the quantitative bases for risk evaluation, control, and
management, and intelligent decision making under uncertainty.
In some instances. problems in several diverse areas are used to illustrate the
same or similar principles. This was purposely done to demonstrate the universal
application of the pertinent concepts and also to provide readers with wider
choices of illustrative appii~ations.
Much of the mathematica( material is, of course, available in the literature and
some of it is in text form: for example, Gumbel (1958) on extreme value statistics;
Rubinstein ( 1981) on Monte Carlo simulation; Raiffa (1970) and Schlaifer ( 1969)
on statistical decision analysis: and Parzen (1962) and Saaty (1961) on Markov
and queueing processes. However, the concepts that are especially useful for
engineering planning and design are emphasized: moreover, these are presented
purposely in the context of engineering significance and in terms that are more
easily comprehensible to engineers (supplemented with numerous illustrative
examples). Much of the material in Chapters 6 and 7 on basic and systems relia
bility is. for the most part, only currently available in scientific and technical
journals-in this regard. only typical references to the available literature are cited. .
as there is no intention to be exhaustive.
The material is suitable for two advanced undergraduate or graduate lrvel
courses. One would cover engineering decision and risk analysis, based on Chapters
2. 3, and 5 of the present volume supplemented by Chapter 8 of Volume I; the
other would focus on system reliability and design and may be developed· using
the material contained in Chapters 4, 6. and 7. Both courses would require a pre
requisite .background in introductory applied probability and statistics, such as the
material in Volume I.
During the preparation a:nd development of the material for this volume, the
authors are indebted in many ways to colleagues and students-to former students
who had· to endure imperfect and incomplete versions of the material; to our
colleague Professor Y. K. Wen for numerous discussions and suggestions over
the years; and to Professors A. der Kiureghian, M. Shinozuka, E. Vanmarcke,
and J. T. P. Yao for constructive suggestions and incisive reviews of the manuscript.
A number of former graduate students also contributed to the development ·or
some of the material including numerical calculations of the examples; in par
ticular. the contributions and assis.tance of R. M. Bennett, C. T. Chu, H. Pearce,
v
vi PREFACE
J. Pires, I. Sidi, M. Yamamoto, and A. Zerva are greatly appreciated. Finally, our
thanks to Claudia Cook for her patience and expert typing of several versions of
the manuscript material, and to R. Winburn for his professional artwork of the
figures.
A. H-S. Ang AND W. H. Tang
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contents
l. Introduction
1.1 Aims and Scope of Volume II
1.2 Essence and Emphasis I
1.2.1 Decision Analysis I
1.2.2 Markov, Queueing, and Availability Models 2
1.2.3 Statistical Theory of Extremes 3
1.2.4 Monte Carlo Simulation 3
1.2.5 Reliability and Reliability-Based Design 3
1.2.6 Systems Reliability 4
2. Decision AJIO:lysis 5
.......
2.1 Introduction 5
2.1.1 Simple Risk-Decision Problems 5
2.1.2 Characteristics of a General Decision Problem 9
2.2 The Decision Model 10
2.2.1 Decision Tree 10
2.2.2 Decision Criteria 13
2.2.3 Decision Based on Exis~ing Information-Prior Analysis 16
2.2.4 Decision with Additio'nal Information-Terminal Analysis 25
2,.2.5 Preposterior Analysis 28
2~.6 Value of Information 30
2.2.7 Sensitivity of Decision to Error in Probability Estimation 32
2.2.8 More Examples 33
2.3 Applications in Sampling and Estimation 47
2.3.1 Bayes Point Estimator 47
2.3.2 Optimal Sample Siz~ 50
2.4 Elementary Concepts of Utility Theory 56
2.4.1 Axioms of Utility Theory 56
2.4.2 Utility Function 57
2.4.3 Determination of Utility Values 60
2.4.4 Utility Function of Monetary Value 63
2.4.5 Utility Function of Other Variables 66.
2.4.6 Maximum Expected Utility 67
2.4.7 Common Types of Utility Functions 70
2.4.8 _ ~nsitivity of Expected Utility to Form of Utility Function 73
viii CONTENTS
2.5 Assessment of Probability Values 74
2.5.1 Bases of Probability Estimation 74
2.5.2 Subjective Probability 75
'2.5.3 ~ubjective Distribution of a Random Variable 76
2.6 Decisions with Multiple Objectives 77
2.6.1 Weighted Objective Decision Analysis 78
2.6.2 Multiattribute Utility Approach 84
Problems for Chapter 2 96
3. . Markov, Queueing, ~nd Availability Models 112
3.1 The Markm· Chain 112
3.1.1 Introduction 112
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3.1.2 The Basic Model 112
3.1.3 State Probabilities 113
3.1.4 Steady State Probabilities 118
3.1.5 First Passage Probabilities li2
3.1.6 Recurrent, Tran~ient, and Absorbing States 125
3.1.7 Random Time Between Stages 136
...
... . 3.1.8 Continuous Parameter Markov Chain 138
3.2 Queueing Models 140
3.2.1 Introduction 140
3.2.2 Poisson Process Arrivals and Depanures 140
3.2.3 Poisson Arrivals and Arbitrarily Distributed Service Time 149
3.2.4 Distribution of Waiting Time 154
3.3 Availability Problems 156
3.3.1 Introduction 156
3.3.2 Continuous Inspection-Repair Process 156
3.3.3 Inspection and Repairs at Regular Intervals 166
3.3.4. Deteriorating Systems 176
Problems for Chapter 3 . 178
4. Statistics of Extremes 186
4.1 Introduction 186
4.1.1 Engineering Significance of Extreme Values 186
4.1.2 Objective and Coverage 187
4.2 Probability Distribution of Extremes 187
4.2.1 Exact Distributions 187'
COiHENTS _;~
4.2..2 Asymptotic Distributions 194
4.2.3 The Symmetry Principle .:!04
4.3 The Three Asymptotic Forms 206
4.3.1 Gumbel's Classification 206
4.3.2 The Type I Asymptotic Form 207
4.3.3 The Type II Asymptotic Form 216
4.3.4 The Type III Asymptotic Form 222
4.3.5 Convergence Criteria 228
4.4 Extremal Probability Papers 235
4.4.1 The Gumbel Extremal Probability Paper 235
4.4.2 The Logarithmic Extremal Probability Paper 238
4.5 Exceedance Probability 246
4.5.1 Distribution-Free Probability 246
4.5.2 Significance of Probability Distribution 249
4.6 Estimation of Extremal Parameters 255
4.6.1 Remarks on Extreme Value. .E stimation 255
4.6.2 The Method of Moments 256
4.6.3 The Method of Order Statistics 256
4.6 •. £ Parameters of Other Asymptotic Distributions 261
4.6.5 Graphical Method 266
Problems for Chapter 4 268
5. Mont~ Carlo Simulation 274
\ 5.1 Introduction 274
5.1.1 An Example of Monte Carlo Simulation 275
5.2 Generation of Random Numbers 278
5.2.1 Random Numbers with Standard Uniform Distribution · 279
5.2.2 Continuous Random Variables 281
5.2.3 Discrete Random Variables 288
5.2.4 Generation of Jointly Distributed Random Numbers 290
5.2.5 Error Associated with Sample Size 291
5.3 Variance Reduction Techniques 292
5.3.1 Antithetic Variates 292
5.3.2 Correlated Sampling 296
5.3.3 Control Variates 298
5.4 Application Examples '301
Problems for Chapter 5 326
X CONTENTS
6. Reliability and Reliability-Based Design 333
6.1 Reliability of Engineered Systems 333
6.2 Analysis and Assessment of Reliability 334
6.2.1 Basic Problem 334
6.2.2 Second-Moment Formulation 340
6.2.3 Linear Perforniance Functions 347
6.2.4 Nonlinear Performance "Functions 359
6.3 Modeling and Analysis:ofUncertainty 383
6.3.1 Sources and Types of Uncertainty 384
6.3.2 Quantification ofUncertainty Measures 387
6.3.3 Analysis of Uncertainty · 390
6.3.4 An Alternative Representation of Uncenaimy 406
6.4 hobability-Based Design· Criteria 412
6.4.1 On Design and Design Criteria. 412
6.4.2 Second-Mc;>ment Criteria 414
Prob_lems for Chapter 6 434
·< .
7. Systems Reliability 448
7.1 Introduction 448
7.2 Multiple Failure Modes 448
7.2.1 Probability Bounds 450
7.3 Redundant and Nonredundant Systems 471
7.3.1 Series Systems 473
7.3.2 Parallel Systems · 474
7.3.3 Combined Series-Parallel Systems 478
7.4 Fault Tree, Event Tree Analyses 485
7.4.1 The Fault Tree Diagram 486
7.4.2 Probability Evaluation 495
7.4.3 Event Tree Analysis 498
7.5 Approximate Methods 504
7.5.1 The PNET Method 505
Problems for Chapter 1 517
CONTElVTS xi"
Appendix A-Tables 531
Table A."I Values of the Gamma Function 532
Table A.2 CDF of the Standard Extremal VariateS 533
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Table A.3 The Three Types of Asymptotic Extremal Distributions 537
Table A.4. Weights a and b for Leiblein's Order Statistics Estimator 538
1 1
Table A.S Relations Among Parameters of the T)i:,e III Asymptotic
Distribution 539
Appendix B-TransforTTUJtion of Nonir.orrnal Variates
to Independent Normal Variates 540
B.l The Rosenb_latt Transformation 540
B.2 Determination· of Safety Index 544
References 549
Inde;c 559
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