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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Quality and Reliability Engineering
- Sprache: Englisch
This complete resource on the theory and applications of reliability engineering, probabilistic models and risk analysis consolidates all the latest research, presenting the most up-to-date developments in this field.
With comprehensive coverage of the theoretical and practical issues of both classic and modern topics, it also provides a unique commemoration to the centennial of the birth of Boris Gnedenko, one of the most prominent reliability scientists of the twentieth century.
Key features include:
- expert treatment of probabilistic models and statistical inference from leading scientists, researchers and practitioners in their respective reliability fields
- detailed coverage of multi-state system reliability, maintenance models, statistical inference in reliability, systemability, physics of failures and reliability demonstration
- many examples and engineering case studies to illustrate the theoretical results and their practical applications in industry
Applied Reliability Engineering and Risk Analysis is one of the first works to treat the important areas of degradation analysis, multi-state system reliability, networks and large-scale systems in one comprehensive volume. It is an essential reference for engineers and scientists involved in reliability analysis, applied probability and statistics, reliability engineering and maintenance, logistics, and quality control. It is also a useful resource for graduate students specialising in reliability analysis and applied probability and statistics.
Dedicated to the Centennial of the birth of Boris Gnedenko, renowned Russian mathematician and reliability theorist
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Veröffentlichungsjahr: 2013
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Table of Contents
Series Page
Title Page
Copyright
Remembering Boris Gnedenko
References
List of Contributors
Preface
Part I Degradation Analysis, Multi-State and Continuous-State System Reliability
Part II Networks and Large-Scale Systems
Part III Maintenance Models
Part IV Statistical Inference in Reliability
Part V Systemability, Physics-of-Failure and Reliability Demonstration
Acknowledgements
Part One: Degradation Analysis, Multi-State and Continuous-State System Reliability
Chapter 1: Methods of Solutions of Inhomogeneous Continuous Time Markov Chains for Degradation Process Modeling
1.1 Introduction
1.2 Formalism of ICTMC
1.3 Numerical Solution Techniques
1.4 Examples
1.5 Comparisons of the Methods and Guidelines of Utilization
1.6 Conclusion
References
Chapter 2: Multistate Degradation and Condition Monitoring for Devices with Multiple Independent Failure Modes
2.1 Introduction
2.2 Multistate Degradation and Multiple Independent Failure Modes
2.3 Parameter Estimation
2.4 Important Reliability Measures of a Condition-Monitored Device
2.5 Numerical Example
2.6 Conclusion
Acknowledgements
References
Chapter 3: Time Series Regression with Exponential Errors for Accelerated Testing and Degradation Tracking
3.1 Introduction
3.2 Preliminaries: Statement of the Problem
3.3 Estimation and Prediction by Least Squares
3.4 Estimation and Prediction by MLE
3.5 The Bayesian Approach: The Predictive Distribution
Acknowledgements
References
Chapter 4: Inverse Lz-Transform for a Discrete-State Continuous-Time Markov Process and Its Application to Multi-State System Reliability Analysis
4.1 Introduction
4.2 Inverse -Transform: Definitions and Computational Procedure
4.3 Application of Inverse -Transform to MSS Reliability Analysis
4.4 Numerical Example
4.5 Conclusion
References
Chapter 5: On the Lz-Transform Application for Availability Assessment of an Aging Multi-State Water Cooling System for Medical Equipment
5.1 Introduction
5.2 Brief Description of the -Transform Method
5.3 Multi-state Model of the Water Cooling System for the MRI Equipment
5.4 Availability Calculation
5.5 Conclusion
Acknowledgments
References
Chapter 6: Combined Clustering and Lz-Transform Technique to Reduce the Computational Complexity of a Multi-State System Reliability Evaluation
6.1 Introduction
6.2 The Lz-Transform for Dynamic Reliability Evaluation for MSS
6.3 Clustering Composition Operator in the Lz-Transform
6.4 Computational Procedures
6.5 Numerical Example
6.6 Conclusion
References
Chapter 7: Sliding Window Systems with Gaps
7.1 Introduction
7.2 The Models
7.3 Reliability Evaluation Technique
7.4 Conclusion
References
Chapter 8: Development of Reliability Measures Motivated by Fuzzy Sets for Systems with Multi- or Infinite-States
8.1 Introduction
8.2 Models for Components and Systems Using Fuzzy Sets
8.3 Fuzzy Reliability for Systems with Continuous or Infinite States
8.4 Dynamic Fuzzy Reliability
8.5 System Fuzzy Reliability
8.6 Examples and Applications
8.7 Conclusion
References
Chapter 9: Imperatives for Performability Design in the Twenty-First Century
9.1 Introduction
9.2 Strategies for Sustainable Development
9.3 Reappraisal of the Performance of Products and Systems
9.4 Dependability and Environmental Risk are Interdependent
9.5 Performability: An Appropriate Measure of Performance
9.6 Towards Dependable and Sustainable Designs
9.7 Conclusion
References
Part Two: Networks and Large-Scale Systems
Chapter 10: Network Reliability Calculations Based on Structural Invariants
10.1 First Invariant: D-Spectrum, Signature
10.2 Second Invariant: Importance Spectrum. Birnbaum Importance Measure (BIM)
10.3 Example: Reliability of a Road Network
10.4 Third Invariant: Border States
10.5 Monte Carlo to Approximate the Invariants
10.6 Conclusion
References
Chapter 11: Performance and Availability Evaluation of IMS-Based Core Networks
11.1 Introduction
11.2 IMS-Based Core Network Description
11.3 Analytic Models for Independent Software Recovery
11.4 Analytic Models for Recovery with Dependencies
11.5 Redundancy Optimization
11.6 Numerical Results
11.7 Conclusion
References
Chapter 12: Reliability and Probability of First Occurred Failure for Discrete-Time Semi-Markov Systems
12.1 Introduction
12.2 Discrete-Time Semi-Markov Model
12.3 Reliability and Probability of First Occurred Failure
12.4 Nonparametric Estimation of Reliability Measures
12.5 Numerical Application
12.6 Conclusion
References
Chapter 13: Single-Source Epidemic Process in a System of Two Interconnected Networks
13.1 Introduction
13.2 Failure Process and the Distribution of the Number of Failed Nodes
13.3 Network Failure Probabilities
13.4 Example
13.5 Conclusion
Appendix D: Spectrum (Signature)
References
Part Three: Maintenance Models
Chapter 14: Comparisons of Periodic and Random Replacement Policies
14.1 Introduction
14.2 Four Policies
14.3 Comparisons of Optimal Policies
14.4 Numerical Examples 1
14.5 Comparisons of Policies with Different Replacement Costs
14.6 Numerical Examples 2
14.7 Conclusion
Acknowledgements
References
Chapter 15: Random Evolution of Degradation and Occurrences of Words in Random Sequences of Letters
15.1 Introduction
15.2 Waiting Times to Words' Occurrences
15.3 Some Reliability-Maintenance Models
15.4 Waiting Times to Occurrences of Words and Stochastic Comparisons for Degradation
15.5 Conclusions
Acknowledgements
References
Chapter 16: Occupancy Times for Markov and Semi-Markov Models in Systems Reliability
16.1 Introduction
16.2 Markov Models for Systems Reliability
16.3 Semi-Markov Models
16.4 Time Interval Omission
16.5 Numerical Examples
16.6 Conclusion
Acknowledgements
References
Chapter 17: A Practice of Imperfect Maintenance Model Selection for Diesel Engines
17.1 Introduction
17.2 Review of Imperfect Maintenance Model Selection Method
17.3 Application to Preventive Maintenance Scheduling of Diesel Engines
17.4 Conclusion
Acknowledgement
References
Chapter 18: Reliability of Warm Standby Systems with Imperfect Fault Coverage
18.1 Introduction
18.2 Literature Review
18.3 The BDD-Based Approach
18.4 Conclusion
Acknowledgments
References
Part Four: Statistical Inference in Reliability
Chapter 19: On the Validity of the Weibull-Gnedenko Model
19.1 Introduction
19.2 Integrated Likelihood Ratio Test
19.3 Tests based on the Difference of Non-Parametric and Parametric Estimators of the Cumulative Distribution Function
19.4 Tests based on Spacings
19.5 Chi-Squared Tests
19.6 Correlation Test
19.7 Power Comparison
19.8 Conclusion
References
Chapter 20: Statistical Inference for Heavy-Tailed Distributions in Reliability Systems
20.1 Introduction
20.2 Heavy-Tailed Distributions
20.3 Examples of Heavy-Tailed Distributions
20.4 Divergence Measures
20.5 Hypothesis Testing
20.6 Simulations
20.7 Conclusion
References
Chapter 21: Robust Inference based on Divergences in Reliability Systems
21.1 Introduction
21.2 The Power Divergence (PD) Family
21.3 Density Power Divergence (DPD) and Parametric Inference
21.4 A Generalized Form: The -Divergence
21.5 Applications
21.6 Conclusion
References
Chapter 22: COM-Poisson Cure Rate Models and Associated Likelihood-based Inference with Exponential and Weibull Lifetimes
22.1 Introduction
22.2 Role of Cure Rate Models in Reliability
22.3 The COM-Poisson Cure Rate Model
22.4 Data and the Likelihood
22.5 EM Algorithm
22.6 Standard Errors and Asymptotic Confidence Intervals
22.7 Exponential Lifetime Distribution
22.8 Weibull Lifetime Distribution
22.9 Analysis of Cutaneous Melanoma Data
22.10 Conclusion
22.A1 Appendix A1: E-Step and M-Step Formulas for Exponential Lifetimes
22.A2 Appendix A2: E-Step and M-Step Formulas for Weibull Lifetimes
22.B1 Appendix B1: Observed Information Matrix for Exponential Lifetimes
22.B2 Appendix B2: Observed Information Matrix for Weibull Lifetimes
References
Chapter 23: Exponential Expansions for Perturbed Discrete Time Renewal Equations
23.1 Introduction
23.2 Asymptotic Results
23.3 Proofs
23.4 Discrete Time Regenerative Processes
23.5 Queuing and Risk Applications
References
Chapter 24: On Generalized Extreme Shock Models under Renewal Shock Processes
24.1 Introduction
24.2 Generalized Extreme Shock Models
24.3 Specific Models
24.4 Conclusion
Acknowledgements
References
Part Five: Systemability, Physics-of-Failure and Reliability Demonstration
Chapter 25: Systemability Theory and its Applications
25.1 Introduction
25.2 Systemability Measures
25.3 Systemability Analysis of -out-of- Systems
25.4 Systemability Function Approximation
25.5 Systemability with Loglog Distribution
25.6 Sensitivity Analysis
25.7 Applications: Red Light Camera Systems
25.8 Conclusion
References
Chapter 26: Physics-of-Failure based Reliability Engineering
26.1 Introduction
26.2 Physics-of-Failure-based Reliability Assessment
26.3 Uses of Physics-of-Failure
26.4 Conclusion
References
Chapter 27: Accelerated Testing: Effect of Variance in Field Environmental Conditions on the Demonstrated Reliability
27.1 Introduction
27.2 Accelerated Testing and Field Stress Variation
27.3 Case Study: Reliability Demonstration Using Temperature Cycling Test
27.4 Conclusion
References
Index
Applied Reliability Engineering and Risk Analysis:
Probabilistic Models and Statistical Inference
Ilia B. Frenkel, Alex Karagrigoriou, Anatoly Lisnianski and Andre Kleyner
Electronic Component Reliability:
Fundamentals, Modelling, Evaluation and Assurance
Finn Jensen
Measurement and Calibration Requirements
For Quality Assurance to ASO 9000
Alan S. Morris
Integrated Circuit Failure Analysis:
A Guide to Preparation Techniques
Friedrich Beck
Test Engineering
Patrick D. T. O'Connor
Six Sigma: Advanced Tools for Black Belts and Master Black Belts*
Loon Ching Tang, Thong Ngee Goh, Hong See Yam and Timothy Yoap
Secure Computer and Network Systems: Modeling, Analysis and Design*
Nong Ye
Failure Analysis:
A Practical Guide for Manufacturers of Electronic Components and Systems
Marius Bâzu and Titu Bjenescu
Reliability Technology:
This edition first published 2014
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Library of Congress Cataloging-in-Publication Data
Applied reliability engineering and risk analysis : probabilistic models and statistical inference / Ilia B. Frenkel, Alex Karagrigoriou, Anatoly Lisnianski, Andre Kleyner. – First edition.
1 online resource.
Includes bibliographical references and index.
Description based on print version record and CIP data provided by publisher; resource not viewed.
ISBN 978-1-118-70189-8 (ePub) — ISBN 978-1-118-70193-5 (MobiPocket)
ISBN 978-1-118-70194-2 — ISBN 978-1-118-53942-2 (hardback) 1. Reliability (Engineering)
2. Risk assessment — Mathematical models. I. Frenkel, Ilia, Ph.D., editor of compilation.
II. Gnedenko, B. V. (Boris Vladimirovich), 1912-1995.
TA169
620′.00452 — dc23
2013025347
A catalogue record for this book is available from the British Library.
ISBN: 978-1-118-53942-2
Remembering Boris Gnedenko
Andre Kleyner1 and Ekaterina Gnedenko2
1Editor of Wiley Series in Quality and Reliability Engineering,
2Granddaughter of Boris Gnedenko, Faculty, Department of Economics, Tufts University
Boris Gnedenko was one of the most prominent mathematicians of the twentieth century. He contributed greatly to the area of probability theory, and his name is permanently linked to pioneering and developing mathematical methods in reliability engineering. Gnedenko is best known for his contributions to the study of probability theory, such as the extreme value theorem (the Fisher–Tippett–Gnedenko theorem). He first became famous for his work on the definitive treatment of limit theorems for sums of independent random variables. He was later known as a leader of Russian work in applied probability and as the author and coauthor of outstanding textbooks on probability, mathematical methods in reliability and queuing theory.
Boris Gnedenko was born on January 1, 1912, in Simbirsk (later Ulianovsk), a Russian city on the Volga River. He was admitted to the University of Saratov at the young age of 15 by special permission from the Minister of Culture and Education of the Soviet Union.
After graduation from the university, Gnedenko took a teaching job at the Textile Institute in Ivanovo, the city east of Moscow, which for many years was the center of the Soviet textile industry. While lecturing at the university, Gnedenko simultaneously was involved in the solution of some practical problems for the textile industry. This is when he wrote his first works, concerning queuing theory, and became fond of the theory of probability. That triggered his later works on the applications of statistics to reliability and quality control in manufacturing.
In 1934, Gnedenko decided to resume his university studies at the graduate level. He was awarded a scholarship which allowed him to undertake research at the Institute of Mathematics at Moscow State University. He became a graduate student under the direction of Alexander Khinchin and Andrei Kolmogorov. The latter became one of the most famous mathematicians of the twentieth century (the Kolmogorov-Smirnov test in statistics, the Kolmogorov-Arnold-Moser theorem in Dynamics, the Kolmogorov Complexity, and other landmark achievements). As a graduate student, Gnedenko became interested in limiting theorems for the sums of independent random variables. In 1937, he defended his dissertation on “Some Results in the Theory of Infinitely Divisible Distributions,” and, soon after the defense, he was appointed a researcher at the Institute of Mathematics at Moscow State University.
Years later, Kolmogorov would say:
Boris Gnedenko is recognized by an international mathematics community as one of the most prominent mathematicians who is currently working in the area of probability theory. He combines an exceptional skill and proficiency in classical mathematical methods with deep understanding of a wide range of modern probability problems and a perpetual interest to their practical applications.
In 1937, during the infamous Stalin's Purges, Gnedenko was falsely accused of “anti-Soviet” activity and thrown in jail. The NKVD (the Soviet secret police at that time) were trying to coerce him to testify against Kolmogorov, who was not yet arrested but was under investigation by the NKVD for running a conspiracy “against the Soviet people,” a very common bogus charge at the time. However, Gnedenko survived brutal treatment at the hands of the NKVD and refused to support the false accusations against his mentor. He was released six months later, though his health suffered.
He returned to Moscow University as an assistant professor in the Department of the Theory of Probability in 1938 and as a research secretary (an academic title in Russia) at the Institute of Mathematics. During this period at Moscow State University, he solved two important problems. The first problem involved the construction of asymptotic distributions of the maximum term of the variation series and defining the nature of limit distributions and the conditions for convergence (Gnedenko 1941b). The second problem involved the construction of the theory of corrections to the Geiger–Muller counter readings used in many fields of physics and technology (Gnedenko 1941a). This paper is a landmark in what later became “the theory of reliability”.
In 1939, Gnedenko married Natalia Konstantinovna and subsequently they had two sons.
During World War II, Gnedenko continued his research work at Moscow State University, although for two years, together with all the university colleagues, he had to temporarily relocate to Turkmenistan and later to the Ural mountains because of Moscow's proximity to the front lines. Some of his work during the war was of national defense nature, including quality and process control at military plants. During that time Gnedenko continued working on a variety of mathematical problems, including the limit theorem for the sums of independent variables, and discovering the classification of the possible types of limit behavior for the maximum in an increasing sequence of independent random variables. The Weibull distribution, one of the most popular time-to-failure distributions in reliability engineering, is occasionally referred to as ‘Weibull-Gnedenko’ (see, e.g. Pecht 1995, or Chapter 19 in this book). Gnedenko developed this model at about the same time as Waloddi Weibull in Sweden, however, due to the relative isolation of the Soviet Union at that time, this was not common knowledge. Only in 1943, two years after Gnedenko had published his results in Russian, due to the warming relationship between the USA and the Soviet Union during World War II, did Gnedenko receive an invitation to publish his hallmark paper on extreme value limit theorem in the American journal, Annals of Mathematics. His research into limit theorems continued and in 1949 this resulted in a monograph with Kolmogorov, entitled “Limit Distributions for Sums of Independent Random Variables”. This monograph was awarded the Chebyshev Prize in 1951 and was translated into many languages. It was later published in English by Addison-Wesley publishing (Gnedenko and Kolmogorov 1954) and underwent a second edition in 1968.
Boris Gnedenko and his wife Natalia, 1978
During World War II, the western part of the Soviet Union was devastated by the German occupation, therefore after the war in 1945, Gnedenko was sent to Lviv, the largest city in Western Ukraine, to help rebuild Lviv University and undertake the restoration of the overall Ukrainian system of higher education. He accepted this challenging job with great energy and enthusiasm.
Building on the works of Kolmogorov and Smirnov establishing the limit distributions for the maximum deviation of an empirical distribution function from the theoretical, Gnedenko developed effective methods to obtain the exact distributions in the case of finite samples in these and other related problems. This work received worldwide recognition, because it served as the basis for compiling tables which were very valuable in applied statistics at that time.
In 1948, Gnedenko was elected a full member of the Ukrainian Academy of Science and in 1950 he was transferred to Kiev, the capital of the Ukraine, to become Head of the recently created Department of the Theory of Probability at the Institute of Mathematics of the Ukrainian Academy of Sciences and also Head of the Physics, Mathematics and Chemistry Section of the Ukrainian Academy of Sciences. At the same time he served as the Chair of the Department of Probability Theory and Algebra at Kiev State University. Later he became Director of the Kiev Institute of Mathematics.
His work in Kiev followed several directions. Besides mathematics and statistics, his contribution was instrumental in developing computer programming and setting up a computing laboratory and encouraging his younger colleagues to study programming. His earlier efforts at the Ukrainian Academy of Science helped to create, in 1951, one of the first fully operational electronic computers in continental Europe.
In 1958, Gnedenko was a plenary speaker at the International Congress of Mathematicians in Edinburgh with a talk entitled, “Limit Theorems of Probability Theory”. One of Gnedenko's most famous books is called Theory of Probability, which first appeared in 1950. Written in a clear and concise manner, the book was very successful in providing an introduction to probability and statistics. It has undergone six Russian editions and has been translated into English (Gnedenko 1998), German, Polish and Arabic. Earlier, in 1946, Gnedenko also co-authored, with Khinchin, the book, Elementary Introduction to the Theory of Probability, which also has been published many times in the USSR and abroad.
In 1960, Boris Gnedenko returned to Moscow State University and later, in 1966 became Head of the Department of Probability holding this post until his death in 1995. He took over from Andrei Kolmogorov, who became Head of the Interdepartmental Laboratory of Probability and Statistics in Moscow.
During the sixties, Gnedenko's interests turned to mathematical problems with industrial application, namely, the queuing theory and mathematical methods in reliability. In 1961, with several of his students and colleagues, he organized and chaired the Moscow Reliability Engineering Seminars. This was a very successful undertaking with around 800 participants: academics, engineers and mathematicians. Many attendees traveled from other cities, and besides academic activities, it resulted in a number of practical consultations helping engineers in various industries. This seminar was also a big promoter of reliability engineering, which at that time was in its infancy.
As Professor Vere-Jones, then a British graduate student at Moscow State University, remembers: “It was into this seminar that I strayed in 1961. I was much impressed, not so much by the academic level, which varied from excellent to indifferent, as by the strong impression I received that this was an environment in which everyone's contribution was valued” (Vere-Jones 1997). This “owed a great deal to Boris Vladimirovich Gnedenko's own personality and convictions, and the influence he had on his colleagues”. Later, at the end of 1965, Vere-Jones managed to arrange for Gnedenko a two-month trip “down under”. Gnedenko was invited by the Australian National University in Canberra, which had a special exchange agreement with Moscow State University. Unfortunately, Gnedenko was not able to take his family with him on this exciting trip-the Soviet government, afraid of losing Gnedenko to the foreign capitalistic country, did not allow his wife and sons to accompany him. According to Vere-Jones, Gnedenko lectured on two themes: one was reliability theory (estimation and testing of the life-time distribution); and the second was mathematical education in the Soviet Union, which generated the greatest interest. Throughout his visit, Gnedenko displayed great interest in everything: the people, the birds and animals, the scenery, the universities and schools, shops, etc. He was particularly fascinated by the koala in Australia and by the kiwi and its huge egg in New Zealand. “His interest in matters ‘down under’ continued well past this visit. I believe he became a president or patron of the USSR side of the New Zealand-USSR friendship society”.
Most people who met Gnedenko remember their personal interactions very fondly and pay tribute to his disposition and personal qualities. Vere-Jones says:
I consider myself extraordinarily fortunate in having happened to drift into his seminar in October 1961. Not only was it a chance to step inside the legendary world of Russian probability theory, it was a chance to come to know a rare human being, to see him at home and with his family, and to work briefly alongside him.
(Vere-Jones 1997)
In 1991, Gnedenko visited the USA. Professor Igor Ushakov, his longtime colleague, who was his host during this trip, remembers:
I was blessed in life to have an opportunity to work closely with Boris Gnedenko. I accompanied him on various academic business trips and spent many evenings at his house and in the company of his family. It would not be an exaggeration to say that I've never met another person with more zest for life and more willingness to share his kindness and help others in need.
(Ushakov 2011)
During that trip Gnedenko, accompanied by his son Dmitry, visited the University of North Carolina, where he lectured and had several research meetings. Then he went to Washington, DC, to give a lecture at the Department of Operations Research of George Washington University. Upon learning about Gnedenko's arrival, the university photographers took a lot of photos and the university newspaper published an article about his visit. While in Washington, DC, Gnedenko met quite a few local mathematicians. Many invited him to their homes and he was always the center of attention. Since this was an international crowd, his knowledge of German and French (in addition to English) came in very handy.
During that visit he was also interviewed by Professor Nozer Singpurwalla at George Washington University. Answering questions, Gnedenko recalled many important events in his life, his work with Kolmogorov and other prominent Russian mathematicians, as well as other significant milestones (Singpurwalla and Smith 1992). During his professional life he held a number of high administrative positions, both at a university and the Academy of Science levels, however, his heart was clearly in academic work and research. Gnedenko said: “I prefer scientific work, lecturing and writing. I enjoy working with students. I have had over a hundred doctoral students, of whom 30 are professors in my country or in other countries”. Seven of his students became members of the Academy of Science, the highest academic distinction in Russia and the former Soviet Republics.
His teaching activities extended beyond academia, Boris Gnedenko had also contributed greatly to popularizing math and science. Besides the book on history of mathematics mentioned earlier, he also wrote for primary and secondary school. Gnedenko said: “This year I have also written a short book for school children on mathematics and life”, and later, “The second book I plan is for school children—a trip into a mathematical country”.
Boris Gnedenko and his colleague and friend, Igor Ushakov, at Gnedenko's home, 1970
Besides studying and doing research in mathematics, Gnedenko also took a keen interest in the history of this discipline, which he considered very important to a future development of mathematics. According to O'Connor and Robertson (2000), Gnedenko's interest in the history of mathematics extended well beyond his text aimed at secondary school pupils. He published much on this topic, including the important Outline of the History of Mathematics in Russia which was not published until 1946, although he wrote it before the start of World War II. It is a fascinating book which looks at the history of mathematics in Russia in its cultural background.
Later in 1993, Gnedenko visited the USA again, now by invitation from the MCI Corporation, at that time a telecommunication giant, where Professor Ushakov was a consultant at the time. Gnedenko was 81, but despite his health problems he put together a rigorous plan to visit all the technical and academic centers he was invited to. The first visit was to the MCI Headquarters in Dallas, where he lectured to a large audience about statistical problems in the telecommunications industry. Introducing him to the audience, Chris Hardy, the MCI Chief Scientist, said: “I did not have any difficulties inviting Prof. Gnedenko, I just said to our CEO that for us hosting Professor Gnedenko would be like for Los Alamos Labs hosting Norbert Wiener.”
Next stop was at Harvard University, hosted by Eugene Litvak, professor at the School of Public Health. For his lecture topic Gnedenko chose, “Probability and Statistics from Middle Ages to Our Days”. Gnedenko always had a sixth sense and a feel for his audience; therefore, because this time he was not speaking to expert mathematicians, he chose one of his favorite subjects: the history of mathematics.
During his lifetime Gnedenko produced a remarkable number of published works. One of the most complete lists of his publications can be found in Gnedenko D.B. (2011). Interestingly enough, all his life, even after the introduction of word processors, Gnedenko still used a typewriter. When asked during one interview how many drafts it took for a paper or book, Gnedenko answered, “one draft only”. It was difficult to believe one draft would have no errors or need for improvement. Gnedenko replied, “it is necessary to think first and only then to write. At this stage I am almost finished” (Singpurwalla and Smith 1992).
Many of the facts and events of Gnedenko's life that are presented here are also recounted in greater detail by the great man himself in his memoirs (Gnedenko 2012). More information about Boris Gnedenko can be found on a dedicated website, Gnedenko Forum, an informal association of specialists in reliability (Gnedenko Forum 2013). The Forum, created by Gnedenko's followers, I. Ushakov and A. Bochkov, is designed to commemorate his legacy and also to promote contacts between members of the global reliability community. It contains the latest professional news in the areas of probability theory, statistics, reliability engineering, risk analysis, mathematical methods in reliability, safety, security and other related fields. Many contributors of this book are members of the Gnedenko Forum and were inspired or influenced in some way by the lifelong work of this great mathematician. This book commemorates the centennial of his birth and pays tribute to his immense contribution to the probability theory and reliability mathematics and also celebrates his legacy.
References
Gnedenko, B.V. (1941a) To the theory of GM-counters, Experimental and Theoretical Physics 11: 101–106. (In Russian).
Gnedenko, B.V. (1941b) Limit theorems for maximum term of variational series. Moscow, DAN; Sc. Sci. of the USSR 32 (1): 231–234. (In Russian).
Gnedenko, B.V. (1943) Sur la distribution limite du terme maximume d'une série aléatoire, Annals of Mathematics 44 (3): 423–453.
Gnedenko, B.V. (1998) Theory of Probability, 6th edition. Boca Raton, FL: CRC Press.
Gnedenko, B.V. (2005) Essays on History of Mathematics in Russia, 2nd ed. Moscow: KomKniga. (In Russian).
Gnedenko, B.V. (2012) My Life in Mathematics and Mathematics in My Life: Memoirs. Moscow: URSS. (In Russian).
Gnedenko, B.V. and A.Ya. Khinchin (1962) Elementary Introduction to the Theory of Probability. New York: Dover Publications, pp. 1–130.
Gnedenko, B. and Kolmogorov, A. (1954) Limit Distributions for Sums of Independent Random Variables. Cambridge, MA: Addison-Wesley.
Gnedenko, B.V. and Kolmogorov, A. (1968) Limit Distributions for Sums of Independent Random Variables, 2nd ed. Cambridge, MA: Addison-Wesley.
Gnedenko, B.V. and Ushakov, I. (1995) Probabilistic Reliability Engineering. New York: John Wiley & Sons.
Gnedenko, D. B. (2011) Gnedenko's bibliography. Reliability: Theory and Applications 6 (4). (December). Accessible at: http://gnedenko-forum.org/Journal/2011_4.html. (In Russian).
Gnedenko Forum (2013) http://gnedenko-forum.org/
O'Connor, J. and Robertson, E. (2000) Boris Vladimirovich Gnedenko. Available at: http://www-history.mcs.st-and.ac.uk/Biographies/Gnedenko.html
Pecht, M. (ed.) (1995) Product Reliability Maintainability Supportability Handbook. Boca Raton, FL: CRC Press.
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Vere-Jones, D. (1997) Boris Vladimirovich Gnedenko, 1912–1995: A personal tribute. Australian Journal of Statistics 39 (2): 121–128.
List of Contributors
Preface
This book is a collective work by a number of leading scientists, analysts, mathematicians, and engineers who have been working on the front end of reliability science and engineering. This work is dedicated to the memory of Boris Gnedenko and commemorates the 100-year anniversary of his birth. Boris Gnedenko was a remarkable mathematician and one of the pioneers of reliability science, thus most of the material presented in this book in some way have been influenced by his work. More about Gnedenko and his life can be found in the introductory chapter “Remembering Boris Gnedenko”.
All chapters in this book are written by the leading researchers and practitioners in their respective fields of expertise and present quite a number of innovative methods, approaches and solutions not covered before in the literature. The amount of research and study in these areas, which represent conventional, contemporary, and recently emerged fields of reliability science have been growing rapidly due to ample opportunities to apply their results in various areas of academic research and in industry.
Despite the large number of contributing authors, this manuscript presents a continuous story of modern reliability engineering, shows the recent advances in mathematical methods in reliability and also celebrates the legacy of Boris Gnedenko. This book has been divided into five logically contiguous parts:
Part I Degradation Analysis, Multi-State and Continuous-State System Reliability
Multi-state and continuous-state system reliability, along with degradation analysis, have undergone vast growth and development in recent years and now is one of the most intensively developing areas in reliability science.
Chapter 1 presents a stochastic process, which models a degraded system behavior, as inhomogeneous continuous time Markov chain. It compares four possible approaches to the problem solution: Monte-Carlo simulation, uniformization, state-space enrichment and numerical differential equations solvers.
Chapter 2 studies multiple independent failure modes for a device under condition monitoring and suggests how to assess the key reliability measures for such a device. In addition, a method for estimation of characteristic parameters associated with the multistate degradation structure of each failure mode is presented.
Chapter 3 covers applications of time series regression with exponentially distributed errors to acceleration test models, degradation monitoring, and financial risk tracking. The chapter also explores inference and prediction modeling under both Bayesian and frequentist methods.
Chapter 4 introduces the inverse Lz-transform for a discrete-state continuous-time Markov process. It shows that the application of inverse Lz-transform to multi-state system reliability analysis significantly expands the range of problems that can be solved by using Ushakov's universal generating operator and corresponding universal generating function (UGF).
Chapter 5 presents a case study of Lz-transform applied to a real world problem: the availability assessment of an aging cooling system of MRI equipment under stochastic demand. The method presented is used to determine the system structure and to achieve the required availability level for complex multi-state aging systems. The proposed solution offers a significant decrease in computational burden compared with conventional Markov methods and also can be used as a tool in engineering decision-making.
Chapter 6 combines clustering and the Lz-transform technique to reduce the computational complexity of multi-state system reliability analysis. The proposed method reduces the computational dimension for a complex system and consequently decreases computational complexity.
Chapter 7 presents three new types of multi-state systems that generalize a linear multi-state sliding window system in the case of allowed gaps (groups of consecutive elements with insufficient cumulative performance). These models can be applied in manufacturing technology, radar and military systems, system integration testing and multi-target tracking.
Chapter 8 presents a new way to model reliability of a component or system using the theory and methods for fuzzy sets. A comprehensive literature survey for traditional fuzzy reliability is also given. This chapter discusses the use of a substitute variable to fuzzify the states of a component or a system. The approach and the methods presented constitute a generalization of traditional binary and multi-state reliability and can also be applied to model the continuous state behavior for systems that degrade over time. The applications of fuzzy reliability models and their advantages in decision-making processes are demonstrated in several case studies.
Chapter 9 explores the relationship between multi-state system reliability and performability. The fast depleting resources of the world with an exponentially increasing population prompted the development of sustainable products, systems and services that should also be dependable. This chapter discusses the necessity of introducing a performance index based on an overall consideration of the problem suitable to be used as a criterion for developing future products, systems and services.
Part II Networks and Large-Scale Systems
Part II covers various aspects of reliability analysis of complex and large systems (both continuous and discrete), including networks.
Chapter 10 discusses monotone binary systems with binary components and its structural invariants as multidimensional parameters. The three invariants are: (1) cumulative D-spectrum, which numerically matches the cumulative signature; (2) the importance spectrum; and (3) the set of border states. The authors present a simple formula to compute the Birnbaum Importance Measure using the second structural invariant, which can be used to approximate the system reliability function.
Chapter 11 presents six hierarchical analytic models to evaluate the performance and availability of mobile networks, especially useful for the next generation of networks based on the Third Generation Partnership Project IP Multimedia Subsystem.
Chapter 12 presents an empirical method to estimate the probability of the first failure, the rate of occurrence of failures, and the steady-state availability for a discrete-time semi-Markov system. A numerical application is given to illustrate the stability of this process.
Chapter 13 presents a system of two finite networks with a certain topology of the network nodes failures. Failure of a randomly chosen node of the first network causes a failure of another randomly chosen node of the second network, which in turn, causes a failure of a randomly chosen node of the first network, and so on. As a result, a random number of nodes will fail in each network. Using the distribution of the failed nodes in each network and the D-spectra (cumulative signature) technique, it generates the failure probability models for both networks.
Part III Maintenance Models
Part III covers repairable systems along with various maintenance models and their applications, placing emphasis on both mathematical rigor and practical applications.
Chapter 14 presents a comparative analysis of four preventive replacement policies with minimal repairs: (1) the unit is replaced at scheduled regular times; (2) the unit is replaced at a planned time or at a random operating time, whichever occurs first; (3) the unit is replaced at a planned time or at a random operating time, whichever occurs last; (4) the unit is replaced upon completion of some operating times over a planned time. Optimal replacement policies of each model are summarized and compared.
Chapter 15 uncovers the similarities between waiting times to occurrences of words in a random sampling of letters from an alphabet and random degradation of devices or systems. It utilizes this “words sequences” approach to model systems with aging, degradation, or reliability-maintenance in discrete time.
Chapter 16 covers distributions of occupancy times in subsets of Markov and semi-Markov processes in terms of Laplace-Stieltjes transforms. It discusses the applications of the well-studied ion-channel theory to system reliability analysis and systems that suffer terminal failure.
Chapter 17 discusses the imperfect maintenance model selection method with a novel GoF test method and a tailored Bayesian model selection method. An example of scheduling the preventive maintenance plan for diesel engines is presented.
Chapter 18 studies the reliability of a warm standby system with consideration of imperfect fault coverage based on binary decision diagrams (BDD). It shows applications of a BDD-based approach to warm standby systems with any type of time-to-failure distribution.
Part IV Statistical Inference in Reliability
The statistical inference Part consists of six chapters which cover important issues associated with goodness of fit tests, divergence measures, cure rate models, perturbed discrete time renewal equations, and extreme shock models.
Chapter 19 offers a comprehensive survey of statistical methods on validation of the two-parameter Weibull-Gnedenko distribution and its connection with the extreme value theory.
Chapter 20 deals with heavy-tailed distributions which are of great interest in reliability systems where the interest lies in the occurrence of rather exceptional and rare events associated with tails of the distributions.
Chapter 21 explores two important families of divergence measures, namely, the power divergence and the density power divergence, which have proven to be useful tools in robust inference analysis.
Chapter 22 investigates cure rate survival models where the competing cause scenario is considered and the exponential and Weibull distributions for the time-to-event are assumed.
Chapter 23 presents asymptotic expansions for solutions of perturbed discrete time renewal equations.
Chapter 24 examines a generalization of the classical extreme shock model where the probability of failure depends not only on the history of the shock process but also on the operational history of a system as well.
Part V Systemability, Physics-of-Failure and Reliability Demonstration
Part V is comprised of several miscellaneous topics to cover other important areas of modern reliability science, such as physics-of-failure (PoF), accelerated testing and operating environments.
Chapter 25 defines the systemability of complex systems and shows how it is affected by the uncertainty of operating environments. A simple systemability function approximation using the Taylor series expansion is presented. Numerical examples are given to illustrate and compare the results of reliability and systemability functions for systems with loglog distribution.
Chapter 26 presents a review of the physics-of-failure (PoF) reliability modeling approach via a step-by-step discussion on how this methodology is applied. The key advantage of the PoF approach is that the root causes of individual failure mechanisms are studied and corrected to achieve the desired lifetimes. Various applications of PoF are also discussed.
Chapter 27 considers the effect of variance in field environmental conditions on the demonstrated reliability. In many cases the acceleration factor for a reliability demonstration test is calculated based on a high percentile field stress level, corresponding to severe user conditions. This chapter presents a mathematical approach to calculating the “true” field reliability, which in those cases will be higher than that demonstrated by the test.
We hope that this book will appeal to a wide range of reliability practitioners including engineers, scientists, researchers, statisticians and postgraduate students.
Ilia B. Frenkel Alex Karagrigoriou Anatoly Lisnianski Andre Kleyner
Acknowledgements
We would like to thank all the contributing authors for sharing their technical expertise, their engineering foresight and their hard work on this book.
We would also like to thank the following individuals for their invaluable help in reviewing the book chapters and their timely feedback and constructive suggestions:
Professor Vilijandas Bagdonaviius (Vilnius University, Lithuania), Professor Maxim Finkelstein (University of the Free State, South Africa), Professor Ilya B. Gertsbakh (Ben-Gurion University, Israel), Professor Gregory Levitin (The Israel Electric Corporation Ltd., Israel), Professor Ying Ni (Malardalen University, Sweden), Professor Dmitrii Silvestrov (Stockholm University, Sweden), Dr Lina Teper (Rafael, Israel), Professor Zhigang (Will) Tian (Concordia University, Canada), Professor Marian Grendar (Slovak Academy of Sciences, Slovakia) and Professor Ilia Vonta (National Technical University of Athens, Greece).
We would like also to thank the SCE–Shamoon College of Engineering (Israel), and its president, Professor Jehuda Haddad, and the SCE Industrial Engineering and Management Department and its dean, Professor Zohar Laslo, for their help and continuous support at all stages of the work.
It was indeed our pleasure to work with Laura Bell, Assistant Editor at John Wiley & Sons.
Part One
Degradation Analysis, Multi-State and Continuous-State System Reliability
Chapter 1
Methods of Solutions of Inhomogeneous Continuous Time Markov Chains for Degradation Process Modeling
Yan-Fu Li,1 Enrico Zio1,2 and Yan-Hui Lin1
1European Foundation for New Energy-Electricité de France, Ecole Centrale, France
2Politecnico di Milano, Italy
1.1 Introduction
Degradation process modeling has been an active field of research in reliability engineering for some time (Barata et al. 2002; Black et al. 2005; Hosseini et al. 2000; Li and Pham 2005; Chryssaphinou et al. 2011). Multi-state modeling (MSM) (Lisnianski and Levitin 2003; Ding et al. 2009; Lisnianski et al. 2010) is often applied in degradation process modeling, because it offers the possibility of describing the degradation state through a number of consecutive levels from perfect working to complete failure. To model the dynamics of such multi-state degradation process, Markov models have often been used (Black et al. 2005; Kim and Makis 2009; Chryssaphinou et al. 2011). In doing this, it is typically assumed that the rates of transition among the degradation states are constant, which implies that the degradation process is memoryless. The resulting stochastic process is called a homogeneous continuous time Markov chain (HCTMC). In many realistic situations, e.g. cracking of nuclear component (Unwin et al. 2011), battery aging (Cloth et al. 2007), cancer patients' life quality (Liu and Kapur 2008), etc., with varying external factors influencing the degradation processes, the transition rates can no longer be considered as time-independent. Under these circumstances, the inhomogeneous CTMC (ICTMC) is more suited to modeling the degradation process.
One of the drawbacks of ICTMC is that its closed-form solution is difficult, if not impossible, to obtain. Therefore, a number of numerical solution techniques have been proposed. Four representative methods are: numerical solver of differential equations (Telek et al. 2004) (e.g. the Runge–Kutta method), uniformization (van Moorsel and Wolter 1998; Arns et al. 2010), Monte Carlo simulation (Lewis and Tu 1986), and state-space enrichment (Unwin et al. 2011). To the best of our knowledge, very few studies have been carried out to analyze the performances of these numerical approaches with the special application background of degradation process modeling.
This work aims to carry out a comparative study of the above-mentioned methods, from the point of view of accuracy and efficiency. The rest of the chapter is organized as follows. Section 1.2 introduces the formalism of the ICTMC model. Section 1.3 briefly introduces the numerical techniques for its solution. Section 1.4 presents empirical comparisons with reference to two degradation case studies. Section 1.5 analyzes the comparison results and provides guidelines for the use of the methods. Section 1.6 draws the conclusions of the study.
1.2 Formalism of ICTMC
Let be a Markov process on a finite state space . The primary quantity of interest in many applications of CTMC is the state probability vector at any time instant , . By the definition of probability, we have .
In the case of HCTMC, is typically found by solving the following system of differential equations:
1.1
where is the state index ranging from 0 to , is the rate which characterizes the stochastic transition from state to state . The transition rate is defined as:
1.2
In the case of ICTMC, the transition rate is dependent on time : . Due to the time dependency, it is in general difficult to obtain the closed-form solutions to the ICTMC differential equations:
1.3
To obtain the state probability vector , Equation (1.3) has to be solved by numerical methods (Arns et al. 2010).
1.3 Numerical Solution Techniques
1.3.1 The Runge–Kutta Method
Runge–Kutta methods (Butcher 2008) are an important family of iterative approximation methods used to solve the differential equations of ICTMC. Let denote the transition matrix of ICTMC, Equation (1.3) can be rewritten as:
1.4
The main idea of the Runge–Kutta methods is to compute by adding to the product of the weighted sum of derivatives at different locations within the time interval . Mathematically, can be expressed as follows:
1.5
1.6
where is the first-order derivative of at , , and and are the coefficients which are usually arranged in a Butcher tableau (Butcher 2003):
The Runge–Kutta method is consistent if . It is also noted that the Runge–Kutta method is explicit if the Butcher tableau is lower triangular, while if the Butcher tableau is not necessarily lower triangular, then the Runge–Kutta method is implicit, which is more general than the explicit case (Butcher 2003).
The coefficients in Butcher tableau are chosen to match as many as possible of the terms in the Taylor series:
1.7
in order to minimize the approximation error. The vector quantity can be expressed by and its derivatives, for example, and . On the other hand, can also be expressed by and its derivatives using the Taylor series: . The coefficients in the Butcher tableau can be obtained by setting the right-hand side of Equation (1.5) equal to the Taylor series of in Equation (1.7). For example, a general form of an explicit two-stage Runge–Kutta method is:
1.8
where the coefficients are in a lower triangular Butcher tableau. By Taylor expansion, we obtain and . Therefore, , and .
1.3.2 Uniformization
Uniformization for HCTMC has been known to be the most efficient approach to obtain the state probabilities of HCTMC since it was proposed by Jensen (Jensen 1953). In a later study, van Dijk (van Dijk 1992) was the first to formulate uniformization for ICTMC. Subsequently, van Moorsel and Wolter (van Moorsel and Wolter 1998) have proposed three numerical algorithms to realize the uniformization formulation of ICTMC. More recently, Arns et al. (2010) have proposed two novel variations of uniformization for ICTMC which are shown to outperform standard differential equation solvers if the transition rates change slowly.
The idea of uniformization is to represent the behaviors of CTMC by a Poisson process and an embedded discrete time Markov chain (DTMC) (Jensen 1953). Let denote the transition matrix of HCTMC and , then is a stochastic matrix of the DTMC. Furthermore, let
1.9
denote the probability of a Poisson process with rate to have events in the time interval . According to Peano-Baker series, the state probability vector can be rewritten as (Antsaklis and Michel 1997):
1.10
In the case of ICTMC, the transition matrix is dependent on time. Suppose that there are different time-dependent events/transitions in and the rate of the -th event/transition is described by the function . At the state level of CTMC, the -th event leads to the transitions described by the matrix . Consequently, defines the overall transition rate from state to state at time caused by event m. Moreover, let . By an appropriate scaling of , we can achieve . Therefore, the matrix can be decomposed into time-dependent transition rates multiplying their corresponding time-independent DTMC transition probability matrices:
1.11
To compute the state probabilities , the total time horizon T has to be discretized into equal time intervals of size , during which the transition rate matrix is constant and the uniformization formulation of HCTMC can be utilized. After the discretization, starting from , the state probabilities at any time instant can be computed by the following recursive formula:
1.12
where is the average transition rate of event m within the interval , is the summation of all the averaged transition rates in , and . To implement this method, two parameters have to be determined: the truncation point for the summation which can be computed from the Poisson probabilities, and the size of time interval , which can be obtained by minimizing a local error estimate. More detailed descriptions of the theory and implementation of the uniformization of ICTMC can be found in (Arns et al. 2010).
1.3.3 Monte Carlo Simulation
In the Monte Carlo (MC) simulation approach, Equation (1.3) is rewritten as (Lewis and Tu 1986):
1.13
where and . The quantity is regarded as the conditional probability that, given the transition out of state at time , the transition arrival state will be . To rewrite Equation (1.13) into integral form, an integrating factor is used. Multiplying both sides of Equation (1.13) by the integrating factor, we obtain:
1.14
Taking the integral of both sides, we obtain:
1.15
In the MC simulation of the Markov process, the probability distribution function is not sampled directly. Instead, the process holding time at one state is sampled and then the transition from state to another state is determined. This procedure is repeated until the accumulated holding time reaches the predefined time horizon. The resulting time sequence consists of the holding times at different states.
Therefore, the holding time is of interest in the MC simulation. To sample the holding time, the probability density (or total frequency) of departing state , , can be obtained by multiplying to both sides of Equation (1.15):
1.16
where
1.17
is defined as the conditional probability density function that the process will depart state at time , given that the process is at state at time . Equation (1.16) indicates that the probability density function consists of the sum of contributions from the random walks with transitions passing through all the states (including state ) from time . From Equation (1.16), the MC simulation procedure mentioned above can be derived.
The cumulative probability distribution function of the holding time is obtained by integrating (1.17):
1.18
Given the current time at state , the holding time can be sampled through direct inversion sampling, acceptance-rejection sampling, and other sampling techniques (Zio and Zoia 2009). Following the departure, the sampling of the arrived state can be achieved by choosing a uniformly distributed random number and selecting the state which satisfies the following condition:
1.19
1.3.4 State-Space Enrichment
The state-space enrichment method has recently been applied to model the degradation process of a repairable nuclear component, where the transition rates are dependent on the state holding times (rather than the system time) (Unwin et al. 2011). In this method, the ICTMC is embedded into another stochastic process , which is a discrete time Markov chain (DTMC) with an enriched state space. This enriched state space can be described by a tuple where is the finite set of original states and is the discretized holding time at each state. Let T denote the total system time and denote the interval size of discretization. Then is the total number of intervals. Let denote all the possible numbers of time intervals that the process could reside in at each state. can be rewritten as .
The transition probability of is defined as:
1.20
which is the probability of transition to state from the current state . In relation to the embedded ICTMC , has the following non-zero transition probabilities:
1.21
which is the probability of the one-step transition from state to a new state , and
1.22
which is the probability of the process remaining at state . Given the definitions of the individual transition probabilities, the transition probability matrix of can be written as:
1.23
Given that the process starts from state 0 when , the initial state probability vector of is defined as follows:
1.24
By the recursive property of CTMC, we can obtain the state probability vector after steps as follows:
1.25
Give , we can obtain the state probability of as:
1.26
where is the th element of the vector .
1.4 Examples
To compare the techniques for ICTMC numerical solution, two examples of degradation processes are considered. Each numerical solution technique has been run 20 times on each example. All the experiments were carried out in MATLAB on a PC with an Inter core i5 CPU at 2.67 GHz and a RAM of 4 GB.
1.4.1 Example of Computing System Degradation
The first example is taken from (van Moorsel and Wolter 1998). It concerns a non-repairable computing system with two processors. Both of them have a time-dependent failure rate , where is the system time. The failure of each processor can lead to a system crash or a safe shutdown of the other processor. The probability of a safe shutdown is 0.6 for both processors. The transition diagram is shown in Figure 1.1.
Figure 1.1 Degradation process of a two-processor computing system
The corresponding transition rate matrix is:
The numerical solution techniques are applied to this problem. Because the state-space enrichment method was designed to handle the changing transition rate dependent on holding time, it is not applicable to this example. The parameter setting of each technique is presented as follows: in the Runge–Kutta method, the time interval size is 0.1 time unit; in the uniformization method, the truncation point is 100 and the time interval size is 0.1 time unit; for the MC simulation, the number of repetitions is 5000. Table 1.1 summarizes the probabilities of the safe and unsafe failure states at the time steps from 1 to 5, in one experiment run. It is shown that the results obtained by the uniformization and Runge–Kutta methods are very close, that is, the mean absolute error (MAE) between them is 1.1035E-6, whereas the results of MC simulation are relatively distant from those of the Runge–Kutta method, with the MAE between them equal to 0.0013.
Table 1.1 State probabilities obtained by the different solution techniques in one experiment run
To investigate the impacts of different variation speeds of transition rates on the techniques' accuracies and efficiencies, we have considered four additional examples in which the transition rates are 2, 4, 8, and 16 times those of the original case, respectively. Table 1.2 summarizes the results, in terms of MAE with reference to the Runge–Kutta method and of average computation time. The results confirm the finding of Table 1.1, that uniformization is the closest to the Runge–Kutta method, this latter being the most efficient followed by uniformization and MC simulation. It is also seen that the accuracy of uniformization might deteriorate if the variation speed of transition rate increases, whereas that of MC simulation does not exhibit any significant tendency. As to the average computation time, the Runge–Kutta method and MC simulation show small-sized steady increases when the variation speeds of the transition rates grow, while uniformization does not have a clear trend in this respect.
Table 1.2 Results of accuracy and computation efficiency
1.4.2 Example of Nuclear Component Degradation
The second case study refers to the cracking process in an Alloy 82/182 dissimilar metal weld in a primary coolant system of a nuclear power plant (Unwin et al
