Table Of ContentContributions to
Hardware and
Software Reliability
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Contributions to
Hardware and
Software Reliability
P K Kapur & R B Garg
University of Delhi, India
S Kumar
National University of Science & Technology,
Zimbabwe
World Scientific
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CONTRIBUTIONS TO HARDWARE AND SOFTWARE RELIABILITY
Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd.
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About the Authors
Dr. P. K. Kapur is a Professor in the Department of Operational Research, University of
Delhi, Delhi 110 007. He obtained his Ph.D. in Operational Research in 1976 from University of
Delhi. He has published extensively in the areas of Reliability theory, Optimisation, Queueing
theory and Software reliability. His work is cited in books and research journals. He has guided
number of Ph.D. and M.Phil theses in these areas. He has also guided Ph.D./M.Tech theses in
Computer Science. He is widely travelled in India and abroad. He visited IMSOR, Denmark
during 1976-78, GDR on a cultural exchange programme for three months in 1990 and Royal
Military College of Canada, Kingston for six months in 1992. He is a reviewer of several national
and international journals. He has been the vice president of ORSI, (Delhi chapter) in 1991-
92 and president in 1992-94 and is currently the vice president of the national body. He has
successfully organised national seminars and regularly contributed in International conferences.
He is currently president of International society of Reliability Engineers (ISRE), Delhi Section.
He has recently edited a volume - Operations Research : Theory and Practice, published by
Spaniel Publishers, New Delhi. He is also an author of forthcoming book "General Stochastic
Modelling in Reliability with Computer Science Applications" to be published by Wiley Eastern
Limited, New Delhi.
Dr. R. B. Garg is working as Systems Incharge in the Delhi University Computer Centre,
University of Delhi, Delhi 110 007. He obtained his Ph.D. in Computer Science in 1990 from
University of Delhi. He has published extensively in the areas of Software reliability, Optimi
sation and Queueing theory. His work is cited in books and research journals. He is currently
treasurer of International society of Reliability Engineers (ISRE), Delhi Section.
Dr S. Kumar is a Professor and Chair of Applied Mathematics, National University of Science
and Technology, Bulawayo, Zimbabwe. He obtained MSc and PhD degrees from Vikram and
Delhi universities, respectively.
Before joining the present appointment he was in the Department of Mathematics, Royal Mel
bourne Institute of Technology, Australia. His visiting appointments ranging for a duration of a
few weeks to six months include the University of Southern California, Case Western Researve
University, University of Louvain, Indian Institute of Technology and Czechslovak Management
Center.
He has edited one book, "Recent Developments in Mathematical Programming" published by
Gordon and Breach Scientific Publishers, 1991 and published articles in refereed journals and
presented papers in many international conferences.
He has served as Vice President (Administration), National Council of the Australian Society
for Operations Research (ASOR), President of the ASOR Melbourne Chapter, President of
the Asia-Pacific Operations Research Societies within IFORS for the period 1995-97, and the
foundation President of the Zimbabwean Association of Applied Mathematics for the period
1996 and 1997.
He is a life Fellow of the Institute of Applied Mathematics and its Applications, UK.
He is a Foreign Editor of the International Journal of Management and Systems, and Associate
Editor of the Asia-Pacific Journal of the Operational Research Societies and the International
Journal of Operations and Quantitative Management. Recently he has become the founder
editor of a new electronic journal entitled "Zimbabwe Journal of Science and Technology."
v
ACKNOWLEDGEMENT
The book contains mostly the research work of the authors done over a decade or so, yet the
contributions of several other researchers are gratefully acknowledged. The researchers namely,
Prof. S. Osaki, Prof. Yamada, Prof. Nakagawa, Prof. Park and Prof. Min Xie whose research
contributions in the area of reliability theory have always been a source of great inspiration and
prompted us to do more and more in this area. The authors are particularly gTateful to Prof.
Min Xie for suggesting us to undertake this venture. Ultimately, we accepted his suggestion
and the challenge and it is gratifying that we have been able to come up with this book which
gives an ample idea of how an interdiscipliniary approach to solving problems can work. The
authors in particular, Dr. Kapur acknowledge with thanks the help rendered by Dr. S. H. Sim
while the author was visiting Canada.
Professors Kapur and Kumar are particularly thankful to their numerous M. Phil / Ph.D
students in Operational Research, Computer Science and Mathematics. Professor Kapur is
thankful to Dr. N. L. Butani, Dr. V. K. Bhalla, Dr. Sanjay Agarwala and Dr. Said Younes
whose research work done jointly with him has also immensly contributed to the growth of the
book.
The authors gratefully acknowledge their contributions. The authors also acknowledge with
thanks the help rendered by Dr. V. K. Sehgal, Mr. A. K. Jha and Mr. A. K. Bardhan during
different stages of writing the book.
Professor Kumar would like to thank Mr. G. Xia, Dr. P. Zeephongsekul, Ms. R. Kalyan, Dr.
H. Arora, Mr. R. Bappoo and Ms. G. Raviganesh. He is also thankful to Dr N. Singh for
introducing the field of Reliability to him. He is thankful to Mr P.J.P. Ewer and Dr. Prince
Ndlouve for their help with Latex and to Miss Ska Dube and Miss Jackie Moyo for their secre
tarial help.
Dr. Kapur expresses his deep sense of love and respect to his father and late mother who always
have been a pillar of strength. He is also grateful to his sisters who always reposed great faith
in him and did not loose hope for all his failings. Lastly ha owes a lot to his late friend Dev
whose fond memories he always cherishes.
The authors are grateful to their wives and children for putting up with great difficulties during
their long absense from home.
VI
PREFACE
A boom in computer technology has led to increased dependance on computers in almost every
human activity . As a result, reliability theory has assumed greater importance. Hardware
reliability though, has reached high level of precision, yet stochastic modelling in this area,
continues to grow probably aiming at filling the gaps wherever they exist. However, a new area
of research is fast emerging and is of utmost importance in order that a software is able to meet
the objectives for which it has been designed and developed. Such an area of immense present
research activity in the foreseable future is called software reliability. It aims at ensuring that
the software will be able to run bug-free in the user friendly environment.
This book aims at revisiting some of the chosen concepts in hardware reliability and highlighting
them with a view to their having new applications in computing systems. Concepts like replace
ment and ordering policies under minimal repairs, interval reliability computations, inspection
policies etc. have wide applications in computing systems. The book further, emphasises at
finding the transient solutions in renewal and reliability theory. Results are discussed both
in continuous and discrete time, a very distinct feature of the book. No presently available
book covers these aspects. The book further lays emphasis on reviewing and developing soft
ware reliability growth models based on Non-Homogeneous Poisson Process. A new approach
is underlined called the flexible modelling approach. A new direction in modelling software
reliability growth phenomenon is also presented. Finally, the problem of release time has also
been discussed at length.
The book is divided into six chapters.
Chapter one describes some of the basic concepts in renewal theory. Besides, an extensive
review of software reliability modelling is also given, though the book is only restricted to soft
ware reliability growth models under non-homogeneous Poisson process.
Chapter two deals with the concepts of minimal repairs which has wide applications in com
puters, electronics etc. and discusses replacement and ordering policies related with it. The
policies discussed in this chapter are limited to those which are relevant to the book. However,
others have also been reviewed.
Chapter three deals with some related problems with applications in computing systems. Prob
lems discussed are (1) Interval Reliability Computation (a performance related reliability mea
sure), (2) Inspection Policies and (3) A General Replacement Policy.
Chapter four first reviews several known software reliability growth models based on Non-
Homogeneous Poisson Process both in continuous and discrete times and then finally a software
reliability growth model is presented which is flexible enough to account for any variability in
the software reliability growth curve. Model is validated on several software test data sets ob
tained from different sources. Predictive validity of the model is also discussed at length. A new
direction in modelling is also provided which is commensurate with the software development
process.
Chapter five discusses at length the important problem of release time of the software in terms
of cost and reliability criteria, though other criteria also exist in terms of the software reaching
a given level of failure intensity or reliability ignoring the cost involved.
Finally chapter six deals with finding the transient solutions in continuous and discrete times
for mean and variance of the renewal process. Though this problem has been attracting the
vii
Vlll PREFACE
attention of several researchers for the past decade or so, yet no simple and easy to use solutions
are available. In fact no available book on reliability theory has ever discussed the numerical
solutions of these problems. Besides, these distinct features of this chapter, another very im-
protant problem in reliability theory has also been discussed which has been abegging a solution
for a long time. The problems under considerations are cold/warm standby multi-unit systems,
parallel systems, K-out-of-N: G systems etc. They have been discussed in discrete time and
explicit closed form analytic solutions are presented and their numerical solutions are discussed
at length. Parallel results in continuous time are also obtained. An application of this is also
shown in software reliability. It would not be out of place to remark here that these problems
have their origin in the literature as early as the origin of renewal and reliablity theories, yet
solutions were not available in closed form which were easy to compute.
Lastly, we may add that the book would be of immense use to researchers, teachers, scientists
and practitioners particularly working in the areas of Computer Science, Operations Research,
Statistics, Industrial Engineering and Business Administration with its multi disciplinary ap
proach.
MORE ABOUT THE BOOK
With ever increasing dependence on computers in almost every walk of life, two areas of re
search have simultaneously developed e.g. Hardware and Software Reliability. Hardware reli-
ability,though has reached a high level of precision and covers approximately 30% of the cost.
Yet the efforts continue to grow with special emphasis on computational aspects of the existing
stochastic models. With greater computing facilities now available, there is an ever increasing
demand to make elegant theoretical results computationally available. Therefore, keeping this
in mind, we have discussed only those aspects from hardware reliability which have relevance to
computing systems and those where numerical computations was a problem. We may mention
some of the topics discussed in this book: namely
1. Optimal Replacement and Ordering policies with Minimal Repairs,
2. Interval Reliability Computations,
3. Inspection Policies,
4. A General Replacement Policy,
5. Numerical Computations of Exact Transient solutions for problem in Renewal and Reli
ability Theory.
In order to keep the book self contained, a brief review of the concepts in renewal theory is
given. These concepts are used in later chapters either for application in computing systems or
developing algorithms to compute results numerically Besides, it may also be mentioned that
nearly 70% of the cost goes into solftware development and hence software reliability, assumes
special importance. Therefore, in section II of the Introduction, an extensive review of the
literature on software reliability is given. This discipline requires strong partnership between
computer scientists, statisticians and operational researchers. This is broadly highlighted in
this book.
Let us now briefly explain why only the above mentioned problems have been chosen in hard
ware reliability. If we look at the literature, two methods of analysis have been used e.g.
difference - differential equations technique and regenerative point technique of Markov Re
newal Process (MRP). In both cases, invariably only the steady state solutions are available
despite having developed elegant theoretical results. Renewal type equations which have been
used had no takers because users have no idea, how to use them. As the pressure grew especially
from the practitioners, with excellent computing facilities available, researchers were forced to
look at the computational aspects of the well established problems in Reliability Theory and
other related fields which further helped in finding numerical solutions to the current state
of the art in the subject. We are of the considered opinion that this is an era of providing
good numerical solutions especially when greater computing facilities are available and several
user friendly softwares are available. Keeping this in view, these problems were chosen and
nice elegant numerical results have been provided which also have application in other fields.
If we look at the algorithm for Interval Reliability computation, one may immediately notice
that same with little alteration can be similarly used for finding numerical solutions to several
problems in Reliability Theory e.g. one/two unit Cold Standby System, Parallel Redundant
Systems under more general condition of non- exponentiality. It also has applications in Inven-
tory/Queuening Theory. As there is continuing interest in maintenance policies, topics, such as
optimal replacement and ordering policies under minimal repairs have been chosen. Moreover,
"A General Replacement Policy" is discussed at length with a view to acquaint the readers with
its applications in computing systems. Similar is the case for section on Inspection Policies.
The Book not only reviews extensively the Software Reliability Growth Models (SRGM's) but
also provides direction in developing models which are flexible and can be used in a wide variety
IX
X MORE ABOUT THE BOOK
of testing environments. Primarily, these SRGM's help in finding a relationship between the
errors detected and the CPU time and thus help in predicting the reliability of the Software.
Besides, there is a separate chapter on Release Time problems. As it is of utmost importance to
decide when to release the Software in the market after it has been developed and tested, as the
testing of the Software can not go on endlessly Several alternative formulations of the Release
Time problem have been discussed along with variants like allocation of testing effort resources
to different modules of the software or the testing effort control problems. This chapter should
be extremely useful to researchers in Operational Research. It throws open plethora of new
problems and challenges to them.
