Reliability Engineering

Alessandro Birolini

Reliability Engineering

Theory and Practice

Seventh Edition

With 190 Figures, 60 Tables, 140 Examples,and 70 Problems for Homework

123

Prof. Dr. Alessandro Birolini*

Centro Storico—BargelloI-50122 FirenzeTuscany, Italy

birolini@emeritus.ethz.chwww.ethz.ch/people/whoiswho,www.birolini.ch

*Ingénieur et penseur, Ph.D., Professor Emeritus of Reliability Eng.at the Swiss Federal Institute of Technology (ETH), Zurich

ISBN 978-3-642-39534-5 ISBN 978-3-642-39535-2 (eBook)DOI 10.1007/978-3-642-39535-2Springer Heidelberg New York Dordrecht London

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"La chance vient à l'esprit qui est prêt à la recevoir." 1)

Louis Pasteur

"Quand on aperçoit combien la somme de nos

"ignorances dépasse celle de nos connaissances,

"on se sent peu porté à conclure trop vite." 2)

Louis De Broglie

"One has to learn to consider causes rather than

"symptoms of undesirable events and avoid hypo-

"critical attitudes." Alessandro Birolini

1) "Opportunity comes to the intellect which is ready to receive it."

2) "When one recognizes how much the sum of our ignorance2) "exceeds that of our knowledge, one is less ready to draw rapid2) "conclusions."

Preface to the 7 th Edition

The large interest granted to the 6th edition (over 2000 on-line requests per year) incited me for a 7thand last edition of this book (11 editions with the 4 German editions 1985 - 97).

The book shows how to build in, evaluate, and demonstrate reliability, maintainability, andavailability of components, equipment, and systems. It presents the state-of-the-art of reliabilityengineering, both in theory and practice, and is based on the author's more than 30 years experiencein this field, half in industry (part of which in setting up the Swiss Test. Lab. for VLSI, 1979 - 83 inNeuchâtel) and half as Professor of Reliability Engineering at the Swiss Federal Institute of Technology(ETH), Zurich. Considering that performance, dependability, cost, and time to market are key factorsfor today's products and services, but also that failure of complex systems can have major safetyconsequences, reliability engineering becomes a necessary support in developing and producingcomplex equipment and systems.

The structure of the book has been conserved through all editions, with main Chapters 1 to 8 andAppendices A1 to A11 (A10 & A11 since the 5th Edition 2007). Chapters 2, 4, and 6 deal carefullywith analytical investigations, Chapter 5 with design guidelines, Chapters 3 and 7 with tests,and Chapter 8 with activities during production. Appendix A1 defines and comment on the termscommonly used in reliability engineering. Appendices A2 - A5 have been added to support managers inanswering the question of how to specify and achieve high reliability (RAMS) targets for complexequipment and systems. Appendices A6 - A8 are a compendium of probability theory, stochasticprocesses, and mathematical statistics, as necessary for Chapters 2, 4, 6, and 7, consistent from amathematical point of view but still with reliability engineering applications in mind (demonstration ofestablished theorems is referred, and for all other propositions or equations, sufficient details forcomplete demonstration are given). Appendix A9 includes statistical tables, Laplace transforms, andprobability charts. Appendix A10 resumes basic technological component's properties, and AppendixA11 gives a set of 70 problems for homework.

This structure makes the book self contained as a text book for postgraduate students or courses inindustry (Fig. 1.9 on p. 24), allows a rapid access to practical results (as a desktop reference), and offersto theoretically oriented readers all mathematical tools to continue research in this field.

The book covers many aspects of reliability engineering using a common language, and has beenimproved step by step. Methods & tools are given in a way that they can be tailored to cover differentreliability requirement levels, and be used for safety analysis too. A large number of tables (60),figures (190), and examples (210 of which 70 as problems for homework), as well as comprehensivereference list and index, amply support the text. This last edition reviews, refines, and extends allprevious editions. New in particular includes:

• A strategy to mitigate incomplete coverage (p. 255), yielding new models (Table 6.12 c & d, p. 256).

• A comprehensive introduction to human reliability with a set of design guidelines to avoid humanerrors (pp. 158-159) and new models combining human errors probability and time to accomplish atask, based on semi-Markov processes (pp. 294-298).

• An improvement of the design guidelines for maintainability (pp. 154-158).

• An improvement of reliability allocation using Lagrange multiplier to consider cost aspects (p. 67).

• A comparison of four repair strategies (Table 4.4, p. 141).

• A comparison of basic models for imperfect switching (Table 6.11, p. 248).

• A refinement of approximate expressions, of concepts related to regenerative processes, and of theuse and limitations of stochastic processes in modeling reliability problems (e.g. Table 6.1, p. 171).

• New is also that relevant statements and rules have been written cursive and centered on the text.

Furthermore,

• Particular importance has been given to the selection of design guidelines and rules, the devel-opment of approximate expressions for large series-parallel systems, the careful simplification ofexact results to allow in-depth trade off studies, and the investigation of systems with complexstructure (preventive maintenance, imperfect switching, incomplete coverage, elements with morethan one failure mode, fault tolerant reconfigurable systems, common cause failures).

VII

VIII

• The central role of software quality assurance for complex equipment and systems is highlighted.

• The use of interarrival times starting by x = 0 at each occurrence of the event considered, instead ofthe variable t, giving a sense to MTBF and allowing the introduction of a failure rate λ( )x and amean time to failure MTTF also for repairable systems, is carefully discussed (pp. 5-6, 41, 175,316, 341, 378, 380) and consequently applied. Similar is for the basic difference between failurerate, (probability) density, and renewal density or intensity of a point process (pp. 7, 378, 426, 466,524). In this context, the assumption as-good-as-new after repair is critically discussed wherevernecessary, and the historical distinction between nonrepairable and repairable items is scaled down(removed for reliability function, failure rate, MTTF, and MTBF); national and internationalstandards should better consider this fact and avoid definitions intrinsically valid only for constant(time independent) failure rates.

• Also valid is the introduction since the 1st edition of indices Si for reliability figures at system level (e. g. ,MTTFSi) where S stands for system and i is the state entered at t = 0 (system referring to thehighest integration level of the item considered, and t = 0 being the beginning of observations, x = 0for interarrival times). This is mandatory for judicious investigations at the system level.

• In agreement with the practical applications, MTBF is reserved for MTBF = 1 / λ.

• Important prerequisites for accelerated tests are carefully discussed (pp. 329-334), in particular totransfer an acceleration factor A from the MTTF ( ).MTTF MTTFA1 2= to the (random) failure-free time τ ( ).τ τ1 2= A .

• Asymptotic & steady-state is used for stationary, by assuming irreducible embedded chains; repairfor restoration, by neglecting administrative, logistical, technical delays; mean for expected value.For reliability applications, pairwise independence assures, in general, totally (mutually, statisti-cally, stochastically) independence, independent is thus used for totally independent.

The book has growth from about 400 to 600 pages, with main improvements in the 4th to 7th Editions.

• 4th Edition: Complete review and general refinements.• 5th Edition: Introduction to phased-mission systems, common cause failures, Petri nets, dynamic

FTA, nonhomogeneous Poisson processes, and trend tests; problems for homework.• 6th Edition: Proof of Eqs. (6.88) & (6.94), introduction to network reliability, event trees & binary

decision diagrams, extensions of maintenance strategies and incomplete coverage,refinements for large complex systems and approximate expressions.

The launching of the 6th Edition of this book coincided with my 70th anniversary, this wascelebrated with a special Session at the 12th Int. Conf. on Quality and Dependability CCF2010 held inSinaia (RO), 22-24 September 2010. My response to the last question at the interview [1.0] given toProf. Dr. Ioan C. Bacivarov, Chairman of the International Scientific Committee of CCF2010, can helpto explain the acceptance of this book:

" Besides more than 15 years experience in the industry, and a predisposition to be a self-taughtman, my attitude to life was surely an important key for the success of my book. This is bestexpressed in the three sentences given on the first page of this book. These sentences, insistingon generosity, modesty and responsibility apply quite general to a wide class of situations andpeople, from engineers to politicians, and it is to hope that the third sentence, in particular, willbe considered by a growing number of humans, now, in front of the ecological problems we arefaced and in front of the necessity to create a federal world wide confederation of democraticstates in which freedom is primarily respect for the other ."

The comments of many friends and the agreeable cooperation with Springer-Verlag are gratefullyacknowledged. Looking back to all editions (1st German 1985), thanks are due, in particular, to K.P.LaSala for reviewing the 4th & 6th Editions [1.17], I.C. Bacivarov for reviewing the 6th Edition [1.0],book reviewers of the German editions, P. Franken and I. Kovalenko for commenting Appendices A6 -A8, A. Bobbio F. Bonzanigo, M. Held for supporting numerical evaluations, J. Thalhammer forsupporting the edition of all figures, and L. Lambert for reading final manuscripts.

Zurich and Florence, September 13, 2013 Alessandro Birolini

Contents

1 Basic Concepts, Quality & Reliability (RAMS) Assurance of Complex Equip. & Systems . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Failure Rate, MTTF, MTBF . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Maintenance, Maintainability . . . . . . . . . . . . . . . . . . . . . 81.2.5 Logistic Support . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.6 Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.7 Safety, Risk, and Risk Acceptance . . . . . . . . . . . . . . . . . . . 91.2.8 Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.9 Cost and System Effectiveness. . . . . . . . . . . . . . . . . . . . 111.2.10 Product Liability . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.11 Historical Development . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Basic Tasks & Rules for Quality & Rel. (RAMS) Assurance of Complex Eq. & Systems . 17

1.3.1 Quality and Reliability (RAMS) Assurance Tasks . . . . . . . . . . . . . 171.3.2 Basic Quality and Reliability (RAMS) Assurance Rules . . . . . . . . . . . 191.3.3 Elements of a Quality Assurance System. . . . . . . . . . . . . . . . . . 211.3.4 Motivation and Training . . . . . . . . . . . . . . . . . . . . . . . 24

2 Reliability Analysis During the Design Phase (Nonrepairable Elements up to System Failure) . . 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Predicted Reliability of Equipment and Systems with Simple Structure . . . . . . . 28

2.2.1 Required Function . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.2 Reliability Block Diagram . . . . . . . . . . . . . . . . . . . . . . 282.2.3 Operating Conditions at Component Level, Stress Factors . . . . . . . . . 332.2.4 Failure Rate of Electronic Components . . . . . . . . . . . . . . . . . 352.2.5 Reliability of One-Item Structures . . . . . . . . . . . . . . . . . . . 392.2.6 Reliability of Series-Parallel Structures . . . . . . . . . . . . . . . . . 41

2.2.6.1 Systems without Redundancy . . . . . . . . . . . . . . . . . 412.2.6.2 Concept of Redundancy . . . . . . . . . . . . . . . . . . . 422.2.6.3 Parallel Models . . . . . . . . . . . . . . . . . . . . . . 432.2.6.4 Series - Parallel Structures . . . . . . . . . . . . . . . . . . 452.2.6.5 Majority Redundancy . . . . . . . . . . . . . . . . . . . . 49

2.2.7 Part Count Method . . . . . . . . . . . . . . . . . . . . . . . . . 512.3 Reliability of Systems with Complex Structure . . . . . . . . . . . . . . . . . 52

2.3.1 Key Item Method . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.1.1 Bridge Structure . . . . . . . . . . . . . . . . . . . . . . 532.3.1.2 Rel. Block Diagram in which Elements Appear More than Once . . . 54

2.3.2 Successful Path Method . . . . . . . . . . . . . . . . . . . . . . . 552.3.3 State Space Method . . . . . . . . . . . . . . . . . . . . . . . . 562.3.4 Boolean Function Method . . . . . . . . . . . . . . . . . . . . . . 572.3.5 Parallel Models with Constant Failure Rates and Load Sharing . . . . . . . 612.3.6 Elements with more than one Failure Mechanism or one Failure Mode . . . . 642.3.7 Basic Considerations on Fault Tolerant Structures . . . . . . . . . . . . 66

2.4 Reliability Allocation and Optimization . . . . . . . . . . . . . . . . . . . 67

IX

X Contents

2.5 Mechanical Reliability, Drift Failures . . . . . . . . . . . . . . . . . . . . 68

2.6 Failure Modes Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.7 Reliability Aspects in Design Reviews . . . . . . . . . . . . . . . . . . . . 77

3 Qualification Tests for Components and Assemblies . . . . . . . . . . . . . . . . 81

3.1 Basic Selection Criteria for Electronic Components . . . . . . . . . . . . . . . 81

3.1.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.1.2 Performance Parameters . . . . . . . . . . . . . . . . . . . . . . 843.1.3 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.1.4 Manufacturing Quality . . . . . . . . . . . . . . . . . . . . . . . 863.1.5 Long-Term Behavior of Performance Parameters . . . . . . . . . . . . . 863.1.6 Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.2 Qualification Tests for Complex Electronic Components . . . . . . . . . . . . . 87

3.2.1 Electrical Test of Complex ICs . . . . . . . . . . . . . . . . . . . . 883.2.2 Characterization of Complex ICs . . . . . . . . . . . . . . . . . . . 903.2.3 Environmental and Special Tests of Complex ICs . . . . . . . . . . . . . 923.2.4 Reliability Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.3 Failure Modes, Mechanisms, and Analysis of Electronic Components . . . . . . . 101

3.3.1 Failure Modes of Electronic Components . . . . . . . . . . . . . . . . 1013.3.2 Failure Mechanisms of Electronic Components . . . . . . . . . . . . . 1023.3.3 Failure Analysis of Electronic Components . . . . . . . . . . . . . . . 1023.3.4 Present VLSI Production-Related Reliability Problems . . . . . . . . . . 106

3.4 Qualification Tests for Electronic Assemblies . . . . . . . . . . . . . . . . . 107

4 Maintainability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.1 Maintenance, Maintainability . . . . . . . . . . . . . . . . . . . . . . . 112

4.2 Maintenance Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2.1 Fault Detection (Recognition) and Localization. . . . . . . . . . . . . . 1164.2.2 Equipment and Systems Partitioning . . . . . . . . . . . . . . . . . . 1184.2.3 User Documentation . . . . . . . . . . . . . . . . . . . . . . . . 1184.2.4 Training of Operation and Maintenance Personnel . . . . . . . . . . . . 1194.2.5 User Logistic Support . . . . . . . . . . . . . . . . . . . . . . . 119

4.3 Maintainability Aspects in Design Reviews . . . . . . . . . . . . . . . . . . 121

4.4 Predicted Maintainability . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4.1 Calculation of MTTRS . . . . . . . . . . . . . . . . . . . . . . . 1214.4.2 Calculation of MTTPMS . . . . . . . . . . . . . . . . . . . . . . 125

4.5 Basic Models for Spare Parts Provisioning . . . . . . . . . . . . . . . . . . 125

4.5.1 Centralized Logistic Support, Nonrepairable Spare Parts . . . . . . . . . . 1254.5.2 Decentralized Logistic Support, Nonrepairable Spare Parts . . . . . . . . . 1294.5.3 Repairable Spare Parts . . . . . . . . . . . . . . . . . . . . . . . 130

4.6 Maintenance Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.6.1 Complete renewal at each maintenance action . . . . . . . . . . . . . . 1344.6.2 Block replacement with minimal repair at failure . . . . . . . . . . . . . 1384.6.3 Further considerations on maintenance strategies . . . . . . . . . . . . 139

4.7 Basic Cost Considerations . . . . . . . . . . . . . . . . . . . . . . . . 142

5 Design Guidelines for Reliability, Maintainability, and Software Quality . . . . . . . 144

5.1 Design Guidelines for Reliability . . . . . . . . . . . . . . . . . . . . . . 144

5.1.1 Derating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Contents XI

5.1.2 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.1.3 Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.1.4 Electromagnetic Compatibility, ESD Protection . . . . . . . . . . . . . 1485.1.5 Components and Assemblies . . . . . . . . . . . . . . . . . . . . . 150

5.1.5.1 Component Selection . . . . . . . . . . . . . . . . . . . . 1505.1.5.2 Component Use . . . . . . . . . . . . . . . . . . . . . . 1505.1.5.3 PCB and Assembly Design . . . . . . . . . . . . . . . . . . 1515.1.5.4 PCB and Assembly Manufacturing . . . . . . . . . . . . . . . 1525.1.5.5 Storage and Transportation . . . . . . . . . . . . . . . . . . 153

5.1.6 Particular Guidelines for IC Design and Manufacturing . . . . . . . . . . 1535.2 Design Guidelines for Maintainability . . . . . . . . . . . . . . . . . . . . 154

5.2.1 General Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 1545.2.2 Testability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2.3 Connections, Accessibility, Exchangeability . . . . . . . . . . . . . . . 1575.2.4 Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.2.5 Human, Ergonomic, and Safety Aspects . . . . . . . . . . . . . . . . 158

5.3 Design Guidelines for Software Quality . . . . . . . . . . . . . . . . . . . 159

5.3.1 Guidelines for Software Defect Prevention . . . . . . . . . . . . . . . 1625.3.2 Configuration Management . . . . . . . . . . . . . . . . . . . . . 1655.3.3 Guidelines for Software Testing . . . . . . . . . . . . . . . . . . . 1665.3.4 Software Quality Growth Models . . . . . . . . . . . . . . . . . . . 166

6 Reliability and Availability of Repairable Systems . . . . . . . . . . . . . . . . 169

6.1 Introduction, General Assumptions, Conclusions . . . . . . . . . . . . . . . 169

6.2 One-Item Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.2.1 One-Item Structure New at Time t = 0 . . . . . . . . . . . . . . . . . 1766.2.1.1 Reliability Function . . . . . . . . . . . . . . . . . . . . . 1766.2.1.2 Point Availability . . . . . . . . . . . . . . . . . . . . . 1776.2.1.3 Average Availability . . . . . . . . . . . . . . . . . . . . 1786.2.1.4 Interval Reliability . . . . . . . . . . . . . . . . . . . . . 1796.2.1.5 Special Kinds of Availability . . . . . . . . . . . . . . . . . 180

6.2.2 One-Item Structure New at Time t = 0 and with Constant Failure Rate λ . . . 1836.2.3 One-Item Structure with Arbitrary Conditions at t = 0 . . . . . . . . . . 1846.2.4 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . 1856.2.5 Steady-State Behavior . . . . . . . . . . . . . . . . . . . . . . . . 187

6.3 Systems without Redundancy . . . . . . . . . . . . . . . . . . . . . . . . 189

6.3.1 Series Structure with Constant Failure and Repair Rates . . . . . . . . . . 1896.3.2 Series Structure with Constant Failure and Arbitrary Repair Rates . . . . . . 1926.3.3 Series Structure with Arbitrary Failure and Repair Rates . . . . . . . . . . 193

6.4 1-out-of-2 Redundancy (Warm, one Repair Crew) . . . . . . . . . . . . . . . . 196

6.4.1 1-out-of-2 Redundancy with Constant Failure and Repair Rates . . . . . . . 1966.4.2 1-out-of-2 Redundancy with Constant Failure and Arbitrary Rep. Rates . . . . 2046.4.3 1-out-of-2 Red. with Const. Failure Rate in Reserve State & Arbitr. Rep. Rates . 207

6.5 k-out-of-n Redundancy (Warm, Identical Elements, one Repair Crew) . . . . . . . . 213

6.5.1 k-out-of-n Redundancy with Constant Failure and Repair Rates . . . . . . . 2146.5.2 k-out-of-n Redundancy with Constant Failure and Arbitrary Repair Rates . . . 218

6.6 Simple Series - Parallel Structures (one Repair Crew) . . . . . . . . . . . . . . 220

6.7 Approximate Expressions for Large Series - Parallel Structures . . . . . . . . . . 226

6.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.7.2 Application to a Practical Example . . . . . . . . . . . . . . . . . . 230

XII Contents

6.8 Systems with Complex Structure (one Repair Crew) . . . . . . . . . . . . . . . 238

6.8.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . 2386.8.2 Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . 2406.8.3 Imperfect Switching. . . . . . . . . . . . . . . . . . . . . . . . 2436.8.4 Incomplete Coverage . . . . . . . . . . . . . . . . . . . . . . . . 2496.8.5 Elements with more than two States or one Failure Mode . . . . . . . . . 2576.8.6 Fault Tolerant Reconfigurable Systems . . . . . . . . . . . . . . . . 259

6.8.6.1 Ideal Case . . . . . . . . . . . . . . . . . . . . . . . . 2596.8.6.2 Time Censored Reconfiguration (Phased-Mission Systems) . . . . . . 2596.8.6.3 Failure Censored Reconfiguration . . . . . . . . . . . . . . 2666.8.6.4 Reward and Frequency / Duration Aspects . . . . . . . . . . . 270

6.8.7 Systems with Common Cause Failures . . . . . . . . . . . . . . . . 2716.8.8 Basic Considerations on Network-Reliability . . . . . . . . . . . . 2756.8.9 General Procedure for Modeling Complex Systems . . . . . . . . . . . 277

6.9 Alternative Investigation Methods . . . . . . . . . . . . . . . . . . . . . 280

6.9.1 Systems with Totally Independent Elements . . . . . . . . . . . . . . 2806.9.2 Static and Dynamic Fault Trees . . . . . . . . . . . . . . . . . . . 2806.9.3 Binary Decision Diagrams . . . . . . . . . . . . . . . . . . . . . . . 2836.9.4 Event Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2866.9.5 Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2876.9.6 Numerical Reliability and Availability Computation . . . . . . . . . 289

6.9.6.1 Numerical Computation of System's Reliability and Availability . . . 2896.9.6.2 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . 290

6.9.7 Approximate expressions for Large, Complex Systems: Basic Considerations. . 2936.10 Human Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

7 Statistical Quality Control and Reliability Tests . . . . . . . . . . . . . . . . . 299

7.1 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . 299

7.1.1 Estimation of a Defective Probability p . . . . . . . . . . . . . . . . 3007.1.2 Simple Two-sided Sampling Plans for Demonstration of a Def. Probability p . . 302

7.1.2.1 Simple Two-sided Sampling Plan . . . . . . . . . . . . . . . 3037.1.2.2 Sequential Test . . . . . . . . . . . . . . . . . . . . . . 305

7.1.3 One-sided Sampling Plans for the Demonstration of a Def. Probability p . . . 3067.2 Statistical Reliability Tests . . . . . . . . . . . . . . . . . . . . . . . . . 309

7.2.1 Reliability and Availability Estimation & Demon. for a given fixed Mission . . 3097.2.2 Availability Estimation & Demonstration for Continuous Operation (steady-state) 311

7.2.2.1 Availability Estimation (Erlangian Failure-Free and/or Repair Times) . . . 3117.2.2.2 Availability Demonstration (Erlangian Failure-Free and/or Repair Times) 3137.2.2.3 Further Availability Evaluation Methods for Continuous Operation . . 314

7.2.3 Estimation and Demonstration of a Const. Failure Rate λ (or of MTBF =1 / )λ . . 3167.2.3.1 Estimation of a Constant Failure Rate λ . . . . . . . . . . . . 3187.2.3.2 Simple Two-sided Test for the Demonstration of λ . . . . . . . . 3207.2.3.3 Simple One-sided Test for the Demonstration of λ . . . . . . . . 324

7.3 Statistical Maintainability Tests . . . . . . . . . . . . . . . . . . . . . . . 325

7.3.1 Estimation of an MTTR . . . . . . . . . . . . . . . . . . . . . . . 3257.3.2 Demonstration of an MTTR . . . . . . . . . . . . . . . . . . . . . 327

7.4 Accelerated Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

7.5 Goodness-of-fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

7.5.1 Kolmogorov-Smirnov Test . . . . . . . . . . . . . . . . . . . . . 3347.5.2 Chi-square Test . . . . . . . . . . . . . . . . . . . . . . . . . . 338

Contents XIII

7.6 Statistical Analysis of General Reliability Data . . . . . . . . . . . . . . . . . 3417.6.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . 3417.6.2 Tests for Nonhomogeneous Poisson Processes . . . . . . . . . . . . . . 3437.6.3 Trend Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

7.6.3.1 Tests of a HPP versus a NHPP with increasing intensity . . . . . . 3457.6.3.2 Tests of a HPP versus a NHPP with decreasing intensity . . . . . . 348

7.6.3.3 Heuristic Tests to distinguish between HPP and Monotonic Trend . . . 3497.7 Reliability Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

8 Quality & Reliability (RAMS) Assurance During Production Phase (Basic Considerations) . 357

8.1 Basic Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

8.2 Testing and Screening of Electronic Components . . . . . . . . . . . . . . . 358

8.2.1 Testing of Electronic Components . . . . . . . . . . . . . . . . . . 3588.2.2 Screening of Electronic Components . . . . . . . . . . . . . . . . . 359

8.3 Testing and Screening of Electronic Assemblies . . . . . . . . . . . . . . . . 362

8.4 Test and Screening Strategies, Economic Aspects . . . . . . . . . . . . . . . 364

8.4.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . 3648.4.2 Quality Cost Optimization at Incoming Inspection Level . . . . . . . . . . 3678.4.3 Procedure to handle first deliveries . . . . . . . . . . . . . . . . . . 372

Appendices (A1 - A11)

A1 Terms and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

A2 Quality and Reliability (RAMS) Standards . . . . . . . . . . . . . . . . . . . 387

A2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

A2.2 General Requirements in the Industrial Field . . . . . . . . . . . . . . . . 388

A2.3 Requirements in the Aerospace, Railway, Defense, and Nuclear Fields . . . . . . 390

A3 Definition and Realization of Quality and Reliability (RAMS) Requirements . . . . . 391

A3.1 Definition of Quality and Reliability (RAMS) Requirements . . . . . . . . . . . 391

A3.2 Realization of Quality & Reliability (RAMS) Requirements for Complex Eq. & Syst. . 393

A3.3 Elements of a Quality and Reliability (RAMS) Assurance Program . . . . . . . . 398A3.3.1 Project Organization, Planning, and Scheduling . . . . . . . . . . . 398A3.3.2 Quality and Reliability (RAMS) Requirements. . . . . . . . . . . . . 399A3.3.3 Reliability, Maintainability, and Safety Analysis . . . . . . . . . . . 399A3.3.4 Selection and Qualification of Components, Materials, Manuf. Processes . . 400A3.3.5 Softwaer Quality Assurance . . . . . . . . . . . . . . . . . . . 400A3.3.6 Configuration Management . . . . . . . . . . . . . . . . . . . 401A3.3.7 Quality Tests . . . . . . . . . . . . . . . . . . . . . . . . . 402A3.3.8 Quality Data Reporting System . . . . . . . . . . . . . . . . . . 404

A4 Checklists for Design Reviews . . . . . . . . . . . . . . . . . . . . . . . . 405

A4.1 System Design Review . . . . . . . . . . . . . . . . . . . . . . . . . 405A4.2 Preliminary Design Reviews . . . . . . . . . . . . . . . . . . . . . . . 406A4.3 Critical Design Review (System Level) . . . . . . . . . . . . . . . . . . . 409

A5 Requirements for Quality Data Reporting Systems . . . . . . . . . . . . . . . . 410

A6 Basic Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 413

A6.1 Field of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

A6.2 Concept of Probability . . . . . . . . . . . . . . . . . . . . . . . . . 415

XIV Contents

A6.3 Conditional Probability, Independence . . . . . . . . . . . . . . . . . . . 418

A6.4 Fundamental Rules of Probability Theory . . . . . . . . . . . . . . . . . . 419A6.4.1 Addition Theorem for Mutually Exclusive Events . . . . . . . . . . . 419A6.4.2 Multiplication Theorem for Two Independent Events . . . . . . . . . 420A6.4.3 Multiplication Theorem for Arbitrary Events . . . . . . . . . . . . . 421A6.4.4 Addition Theorem for Arbitrary Events . . . . . . . . . . . . . . . 421A6.4.5 Theorem of Total Probability . . . . . . . . . . . . . . . . . . . 422

A6.5 Random Variables, Distribution Functions . . . . . . . . . . . . . . . . . 423

A6.6 Numerical Parameters of Random Variables . . . . . . . . . . . . . . . . 429A6.6.1 Expected Value (Mean) . . . . . . . . . . . . . . . . . . . . . 429A6.6.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 432A6.6.3 Modal Value, Quantile, Median . . . . . . . . . . . . . . . . . . 434

A6.7 Multidimensional Random Variables, Conditional Distributions . . . . . . . . . 434

A6.8 Numerical Parameters of Random Vectors . . . . . . . . . . . . . . . . . 436A6.8.1 Covariance Matrix, Correlation Coefficient . . . . . . . . . . . . . 437A6.8.2 Further Properties of Expected Value and Variance . . . . . . . . . . 438

A6.9 Distribution of the Sum of Indep. Positive Random Variables and of τmin , τmax . 438

A6.10 Distribution Functions used in Reliability Analysis . . . . . . . . . . . . . 441A6.10.1 Exponential Distribution . . . . . . . . . . . . . . . . . . . 441A6.10.2 Weibull Distribution . . . . . . . . . . . . . . . . . . . . 442A6.10.3 Gamma Distribution, Erlangian Distribution, and χ2 -Distribution . . 444A6.10.4 Normal Distribution . . . . . . . . . . . . . . . . . . . . 446A6.10.5 Lognormal Distribution . . . . . . . . . . . . . . . . . . . 447A6.10.6 Uniform Distribution . . . . . . . . . . . . . . . . . . . . 449A6.10.7 Binomial Distribution . . . . . . . . . . . . . . . . . . . . 449A6.10.8 Poisson Distribution . . . . . . . . . . . . . . . . . . . . 451A6.10.9 Geometric Distribution . . . . . . . . . . . . . . . . . . . 453A6.10.10 Hypergeometric Distribution . . . . . . . . . . . . . . . . . 454

A6.11 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 454A6.11.1 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . 455A6.11.2 Central Limit Theorem . . . . . . . . . . . . . . . . . . . 456

A7 Basic Stochastic-Processes Theory . . . . . . . . . . . . . . . . . . . . . . 460

A7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

A7.2 Renewal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 463A7.2.1 Renewal Function, Renewal Density . . . . . . . . . . . . . . . . 465A7.2.2 Recurrence Times . . . . . . . . . . . . . . . . . . . . . . . 468A7.2.3 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . 469A7.2.4 Stationary Renewal Processes . . . . . . . . . . . . . . . . . . . 471A7.2.5 Homogeneous Poisson Processes (HPP) . . . . . . . . . . . . . . . 472

A7.3 Alternating Renewal Processes . . . . . . . . . . . . . . . . . . . . . . 474A7.4 Regenerative Processes with a Finite Number of States. . . . . . . . . . . . . 478A7.5 Markov Processes with a Finite Number of States . . . . . . . . . . . . . . . 480

A7.5.1 Markov Chains with a Finite Number of States . . . . . . . . . . . . 480A7.5.2 Markov Processes with a Finite Number of States . . . . . . . . . . . 482A7.5.3 State Probabilities and Stay Times in a Given Class of States. . . . . . . 491

A7.5.3.1 Method of Differential Equations . . . . . . . . . . . . . 491A7.5.3.2 Method of Integral Equations . . . . . . . . . . . . . . . 495A7.5.3.3 Stationary State and Asymptotic Behavior . . . . . . . . . 496

A7.5.4 Frequency / Duration and Reward Aspects . . . . . . . . . . . . . 498A7.5.4.1 Frequency / Duration . . . . . . . . . . . . . . . . . . 498A7.5.4.2 Reward . . . . . . . . . . . . . . . . . . . . . . . . 500

Contents XV

A7.5.5 Birth and Death Process . . . . . . . . . . . . . . . . . . . . . 501

A7.6 Semi-Markov Processes with a Finite Number of States . . . . . . . . . . . . 505

A7.7 Semi-regenerative Processes with a Finite Number of States. . . . . . . . . . . 510

A7.8 Nonregenerative Stochastic Processes with a Countable Number of States . . . . . 515A7.8.1 General Considerations . . . . . . . . . . . . . . . . . . . . . 515A7.8.2 Nonhomogeneous Poisson Processes (NHPP) . . . . . . . . . . . . 516A7.8.3 Superimposed Renewal Processes . . . . . . . . . . . . . . . . . 520A7.8.4 Cumulative Processes . . . . . . . . . . . . . . . . . . . . . . 521A7.8.5 General Point Processes . . . . . . . . . . . . . . . . . . . . . 523

A8 Basic Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . 525

A8.1 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 525A8.1.1 Empirical Distribution Function . . . . . . . . . . . . . . . . . . 526A8.1.2 Empirical Moments and Quantiles . . . . . . . . . . . . . . . . . 528A8.1.3 Further Applications of the Empirical Distribution Function . . . . . . . 529

A8.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 533A8.2.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . 533A8.2.2 Interval Estimation . . . . . . . . . . . . . . . . . . . . . . . 538

A8.2.2.1 Estimation of an Unknown Probability p . . . . . . . . . . 538A8.2.2.2 Estimation of Param. λ for Exp. Distrib.: Fixed T, instant. repl. . 542A8.2.2.3 Estimation of Param. λ for Exp. Distrib.: Fixed n, no repl. . . . 543A8.2.2.4 Availability Estimation (Erlangian Failure-Free and/or Repair Times) 545

A8.3 Testing Statistical Hypotheses . . . . . . . . . . . . . . . . . . . . . . 547A8.3.1 Testing an Unknown Probability p . . . . . . . . . . . . . . . . . 548

A8.3.1.1 Simple Two-sided Sampling Plan . . . . . . . . . . . . . 549A8.3.1.2 Sequential Test . . . . . . . . . . . . . . . . . . . . 550A8.3.1.3 Simple One-sided Sampling Plan . . . . . . . . . . . . . 551A8.3.1.4 Availability Demonstr. (Erlangian Failure-Free and/or Rep. Times) . . 553

A8.3.2 Goodness-of-fit Tests for Completely Specified F ( )0 t . . . . . . . . . 555A8.3.3 Goodness-of-fit Tests for F ( )0 t with Unknown Parameters . . . . . . . 558

A9 Tables and Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

A9.1 Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . 561

A9.2 χ2- Distribution (Chi-Square Distribution) . . . . . . . . . . . . . . . . . 562

A9.3 t - Distribution (Student distribution) . . . . . . . . . . . . . . . . . . . . 563

A9.4 F - Distribution (Fisher distribution) . . . . . . . . . . . . . . . . . . . 564

A9.5 Table for the Kolmogorov-Smirnov Test . . . . . . . . . . . . . . . . . . 565

A9.6 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

A9.7 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

A9.8 Probability Charts (Probability Plot Papers) . . . . . . . . . . . . . . . . . 569A9.8.1 Lognormal Probability Chart . . . . . . . . . . . . . . . . . . . 569A9.8.2 Weibull Probability Chart . . . . . . . . . . . . . . . . . . . . 570A9.8.3 Normal Probability Chart . . . . . . . . . . . . . . . . . . . . 571

A10 Basic Technological Component's Properties . . . . . . . . . . . . . . . . . . 572

A11 Problems for Homework . . . . . . . . . . . . . . . . . . . . . . . . . . 576

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605