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Volume 14 - 2018

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Detailed review of: 

The Universal Generating Function in Reliability Analysis and Optimization





Springer Verlag, London




Gregory Levitin




The Universal Generating Function in Reliability Analysis and Optimization


Year of Publication














Krishna B. Misra




Review published in IJPE, Vol. 3, No. 1, January, 2007, p. 198.

Table of Contents
  • Preface  (4 pages)
  • General Notation and Acronyms  (2 pages)
  1. Basic Tools and Techniques  (28 pages)
  2. Universal Generating Function in Reliability Analysis of Binary Systems (39 pages)
  3. Introduction to Multi-state Systems (33 pages)
  4. Universal Generating Function in Analysis of Series-Parallel  Multi-state Systems (93 pages)
  5. Universal Generating Function in Optimization of Series-Parallel  Multi-state Systems (73 pages)
  6. Universal Generating Function in Analysis and Optimization of  Special Types of Multi-state Systems (103 pages)
  7. Universal Generating Function in Analysis and Optimization of  Consecutively Connected  Systems and Networks (45 pages)
  8. Universal Generating Function in Analysis and Optimization of Fault-tolerant Software (21 pages
  • References  (9 pages)
  • Index  (2 pages)

This book is different in many ways. Firstly, it is rather a research monograph than a text book, which

The Universal Generating Function in Reliability Analysis and Optimization

lists a total of 203 references and is mainly based on author's 44 research papers and a book published by him between 1997 and 2005. The research papers published between 1997 and 2002 were mostly in coauthorship of Dr. Anatoly Lisnianski and therefore an earlier book, Multi-state System Reliability Assessment, Optimization and Applications, published by World Scientific in the author has Dr. Lisnianski as a coauthor. The present book is an enlarged and extended version of his 2003 book and the author has tried to cover as much ground as was possible in the reliability and performance analysis and optimization of multi-state systems. This is a most comprehensive book ever written on the subject and can also be called as a tribute to Prof. Igor Ushakov, who introduced and indicated the usefulness of Universal Generating Function in the reliability analysis of complex systems. Dr. Levitin carried forward the work and has demonstrated beyond doubt the versatility of Universal Generating Function (UGF) in system reliability analysis and the utility of Genetic Algorithms in optimizing system reliability, particularly in case of systems having multi-state components or components having several modes of failure. 

This is the first book, which presents UGF (except the earlier book by the same author, where UGF was introduced in a chapter) comprehensively. Therefore, it is not without reason that the titles of his book as well as all the 8 chapters of his book except the first chapter on Basic Tools and Techniques and the third chapter on Introduction to Multi-state Systems carry UGF on all of them. Therefore, the objective of the book is to familiarize and popularize the use of UGF in system analysis and Genetic Algorithms in optimization, particularly in case of multi-state systems. Of course there can be better approaches available for the analysis and optimization of binary state systems but for MSS, probably the authors approach may be more acceptable.

Secondly, this book is different in approach as it not only allows usual binary state system analysis but also extends it to systems with components that may have several discrete states depending on the performance level and failure modes. Analysis of such systems is usually tedious and time consuming. Levitin demonstrates in this book that the analysis can be carried out with ease and effectively with the help of UGF. Not only that, the author has been quite successful in demonstrating that a variety of problems with various underlying assumptions can be carried out without much difficulty. For example, the author considers controllable multi-state systems, systems with dependent elements, common cause failures and also discusses various importance analyses in case of series-parallel multi-state systems (MSS) in chapter 4. The author goes further to extend his discrete performance levels approach to continuum–state systems by discretizing the continuum performance distribution. This of course makes one wonder whether fuzzy set approach and other possibilistic approaches would be more appropriate for such systems unless one does not want to leave the probabilistic frame of analysis. 

Other chapters progressively provide the analysis of MSS with varying degree of assumptions and complexities. Any optimization of system performance is done using Genetic Algorithm (GA) which the author considers universal in handling such optimization problems whether they concern structure optimization with or without common cause failures or reliability optimization including optimization of reliability growth testing or even minimization of total maintenance cost in case of imperfect maintenance. Subsequently, the author goes on to deal with the analysis and optimization of several special MSS systems such as bridge system, system with two failure modes , weighted voting system and sliding window system. With all these systems the analysis is done using UGF and the optimization problem is solved using GA. No effort is made at any point to quantitatively assess the efficacy of the approaches used. The reliability and steady state availability are computed using the UGF assuming that all reparable components are repaired independently without specifying any restrictions on the repair facilities available. The last two chapters concern consecutively connected MSS systems, multi-state networks and the optimization is about optimal element allocation with or without common cause failures The last chapter is on  fault tolerant software with various performance models and the optimization determines the optimal version sequencing in programs and optimal structure of fault tolerant systems.

The book is useful to researchers, practitioners and serious minded teachers and students who would like to know and compare new techniques with what the author has demonstrated successfully is possible with UGF and GA, particularly in respect of flexibility and ease of handling large systems, the author's approach offers, in the reliability analysis and optimization of MSS

Krishna B. Misra


Review published in the International Journal of Performability Engineering, Vol. 3, No. 1, January 2007, p.198. 

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