Table Of ContentSpringer Series in Reliability Engineering
Anatoly Lisnianski
Ilia Frenkel
Lev Khvatskin
Modern Dynamic
Reliability Analysis
for Multi-state
Systems
Stochastic Processes and
the L -Transform
z
Springer Series in Reliability Engineering
Series Editor
Hoang Pham, Department of Industrial and Systems Engineering, Rutgers
University, Piscataway, NJ, USA
More information about this series at http://www.springer.com/series/6917
Anatoly Lisnianski Ilia Frenkel
(cid:129) (cid:129)
Lev Khvatskin
Modern Dynamic Reliability
Analysis for Multi-state
Systems
Stochastic Processes and the L -Transform
Z
123
AnatolyLisnianski Ilia Frenkel
TheReliability Department CenterforReliabilityandRiskManagement
Israel Electric Corporation Ltd,Haifa, SCE-Shamoon Collegeof Engineering
Israel andCenterfor Reliability andRisk Beer Sheva,Israel
Management, SCE-Shamoon College
of Engineering
Beer Sheva,Israel
LevKhvatskin
CenterforReliabilityandRiskManagement
andIndustrialEngineeringandManagement
Department
SCE-Shamoon Collegeof Engineering
Beer Sheva,Israel
ISSN 1614-7839 ISSN 2196-999X (electronic)
SpringerSeries inReliability Engineering
ISBN978-3-030-52487-6 ISBN978-3-030-52488-3 (eBook)
https://doi.org/10.1007/978-3-030-52488-3
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To my son Roma and his wife Rima
my daughter Masha and her husband Igal
my beautiful granddaughters Yael, Liora,
Noa, Gaya and Maya
—Anatoly Lisnianski
To my grandchildren Idan, Sofia, Anastasia,
Thomas and Liam
—Ilia Frenkel
To my granddaughter Tomer
—Lev Khvatskin
Preface
This book is considered as a dynamic analysis of multi-state system (MSS) with
focus on reliability, availability and performability analysis of aging MSS, com-
ponentsimportancechanging(variety)duringthetime,initialconditionsimpacton
these issues, etc.
Using traditional methods for solving such problems for real-world multi-state
systemsleads toexplosion ofnumberofstatesthatshouldbeanalysed. Inorderto
avoidthisproblem,aspecialtypeoftransformforMarkovstochasticprocessthatis
called L -transform was suggested. By using this transform, the Universal
Z
Generating Function technique, which is widely used for steady-state MSS relia-
bility analysis, may be extended to dynamic reliability analysis and applied to
randomprocesses.RecentlytherearemanysuccessiveapplicationsofL -transform
Z
method to dynamic analysis of different real-world multi-state systems. L -trans-
Z
form was applied to determining age replacement policy in MSS, to provide reli-
ability, availability and performability analysis for different specific MSSs such as
power systems, refrigerating systems, air conditioning systems, different technical
systems in aviation and maritime engineering.
The aim of this book is to provide a comprehensive, up-to-date presentation of
anMSSdynamicanalysisbyusingL -transformandinverseL -transformbasedon
Z Z
currentachievementsin this field. For these purposes thebook is logicallydivided
into two parts. In the first part (Chaps. 1–3) theoretical issues are presented with
correspondingmethodologicalexamples.Thesecondpart(Chaps.4–8)presentsthe
method of implementation to solving practical problems.
The authors anticipate that the book will be attractive for researchers and
practical engineers in addressing issues related to reliability, availability and per-
formability analysis. In addition, it will be a helpful textbook for graduate courses
inappliedmathematics,industrial,electrical,mechanicalandcomputerengineering.
It should be noticed that it is impossible to describe all the achievements in the
field in a single book. Naturally some interesting results remained outside of the
book’sscope.Insuchcasestheauthorsprovidethereaderswiththecorresponding
references.
vii
viii Preface
The book is organized as follows.
Chapter 1 presents the generic MSS model as a set of models of stochastic pro-
cesses for every MSS’s element and the system structure function. There were
described indices for MSS reliability, availability and performability assessment.
In Chap. 2 in order to make the book self-contained, we have presented to the
reader a basic knowledge about Markov processes. It was considered a method-
ology of Markov models building for reliability, availability and performability
assessment.
Chapter 3 presents the approach for computation reliability, availability and per-
formabilityofMSSindynamicmodes.Forthispurpose,therewereintroducedand
mathematically defined L -transform and inverse L -transform for discrete-state
Z Z
continuous-time Markov process. The technique of their application for computa-
tion of MSS reliability, availability and performability indices was described in
detail. Corresponding methodological examples were presented.
Chapter4presentstheapplicationsofL -transformmethodtoashort-termanalysis
Z
of power systems. Here availability and performability assessment for coal-fired
power units as well as for combine cycle units were considered. By using inverse
L -transform for coal-fired power plant a short-term risk function and reliability
Z
evaluationandestimatingofmeantimetofailurewereperformed.Itisshownhow
these parameters may be used in power system dispatch for making important
operating decisions in real time.
In Chap. 5, the L -transform method is applied for the analysis of a redundant
Z
multi-stateairconditioningsystemforchemicallaboratorythatmustfunctionunder
different cases of hood functioning. Its availability is investigated when in the
system there are some aging elements.
In Chap. 6 a reserved cold-water supply system for a factory, specialized in pro-
ducingrawmaterialsfortheplasticsindustryisconsidered.Watercoolingsystems
often have aging components and so, one should consider in general a
non-homogeneous Markov model because some transition rates (intensities) are
being time-dependent. Such a model is complex—even in simple cases it has
hundredsofstates.Therefore,itisratherdifficulttobuildthemodelandtosolvethe
corresponding system of differential equations by using straightforward Markov
method.So,inordertomakeadynamicMSSreliabilityanalysisL -transformwill
Z
be used.
InChap.7themethodforanalysisofimpactofchangingdifferentfailuresorrepair
rates in different elements in aging MSS is presented. There were introduced sen-
sitivitymeasuresthatareusefulforinvestigationofanagingMSSandpresentedthe
methodfortheirevaluation,basedonusingL -transform.Themethodwasapplied
Z
to sensitivity analysis of real water cooling system. As sensitivity measures there
were proposed derivations of MSS life time with respect to restriction of avail-
ability, mean output performance and accumulated performance deficiency.
Preface ix
In Chap. 8 the method for computation of the Birnbaum B-availability importance
measure for components in an aging MSS under minimal repair was considered.
ThemethodisbasedontheL -transform.ItwasshownthatdynamicB-availability
Z
importance assessment for aging MSS is especially important, because the relative
importance of MSS’s components is changing over the time and strongly depends
on the system demand.
We would like to express our sincere gratitude and appreciation to our col-
leagues from The Israel Electric Corporation, Dr. G. Levitin, Dr. D. Elmakias,
Dr.H.BenHaimandDr.D.LaredoandourcolleaguesandstudentsfromShamoon
College of Engineering, Israel for providing a supportive and intellectually stim-
ulating environment. We also are very thankful to Prof. Yi Ding from Zhejiang
University, China for his collaboration and very interesting discussions.
We are especially thankful to our teachers and friends Prof. Ilya Gertsbakh and
Prof. Igor Ushakov that, unfortunately, recently have left us. Their fruitful ideas
have a great impact on our work. We hope that their ideas will be very helpful for
reliability researchers and engineers for a long time in the future.
Finally,itisindeedourpleasuretoworkwitheditorialstafffromSpringer,who
has assisted in the book publication.
Beer Sheva, Israel Anatoly Lisnianski
May 2020 Ilia Frenkel
Lev Khvatskin
Contents
1 Generic Model of Multi-state System. Reliability, Availability
and Performability in Dynamic Modes . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Generic Multi-state System Model. . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Reliability, Availability and Performability of Multi-state
System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Reliability Measures of Multi-state Systems . . . . . . . . . . . 9
1.2.2 Availability Measures of Multi-state Systems . . . . . . . . . . 10
1.2.3 Performability Measures of Multi-state Systems. . . . . . . . . 12
1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Stochastic Processes Methods for MSS Reliability, Availability
and Performability Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 General Concepts of Stochastic Processes Theory. . . . . . . . . . . . . 17
2.2 Markov Models: Discrete-Time Markov Chains . . . . . . . . . . . . . . 21
2.2.1 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Computation of n-Step Transition Probabilities
and State Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Markov Models: Continuous-Time Markov Chains. . . . . . . . . . . . 27
2.3.1 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Markov Models for Evaluating Reliability
of Multi-state Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 LZ-Transform and Inverse LZ-Transform of a Discrete-State
Continuous-Time Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 L -Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Z
3.1.1 L -Transform Definition. . . . . . . . . . . . . . . . . . . . . . . . . . 56
Z
3.1.2 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 57
xi
