Table Of ContentMATHEMATICAL
MODELS FOR
SYSTEMS RELIABILITY
© 2008 by Taylor & Francis Group, LLC
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MATHEMATICAL
MODELS FOR
SYSTEMS RELIABILITY
Benjamin Epstein
Ishay Weissman
© 2008 by Taylor & Francis Group, LLC
C0822_FM.indd 3 3/31/08 12:03:36 PM
Chapman & Hall/CRC
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Library of Congress Cataloging-in-Publication Data
Epstein, Benjamin, 1918-
Mathematical models for systems reliability / Benjamin Epstein and Ishay
Weissman.
p. cm.
“A CRC title.”
Includes bibliographical references and index.
ISBN 978-1-4200-8082-7 (alk. paper)
1. Reliability (Engineering)--Mathematics. 2. System failures
(Engineering)--Mathematical models. I. Weissman, Ishay, 1940- II. Title.
TA169.E67 2008
620’.00452015118--dc22 2008010874
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Dedicated to
my wife Edna
and my children Shoham, Tsachy and Rom
IshayWeissman
© 2008 by Taylor & Francis Group, LLC
Preface
ThisbookhasevolvedfromthelecturesofProfessorBenjamin(Ben)Epstein
(1918–2004)attheTechnion—IsraelInstituteofTechnology.Throughouthis
tenure at the Technion, from 1968 until his retirement in 1986, he designed
andtaughttwocoursesonReliabilityTheory.One,whichheconsideredtobe
fundamental in reliability considerations, was Mathematical Models for Sys-
temsReliability.ThesecondcoursewasStatisticalMethodsinReliability.As
thesetitlesindicate,althoughtherewassomeoverlapping,thefirstcoursecon-
centrated on the mathematical probabilistic models while the second course
concentratedonstatisticaldataanalysisandinferenceappliedtosystemsreli-
ability.
Epsteinwasoneofthepioneersindevelopingthetheoryofreliability.Hewas
the first to advocate the use of the exponential distribution in life-testing and
developed the relevant statistical methodology. Later on, when Sobel Milton
joined Epstein’s Math Department in Wayne State, they published their joint
workinasequenceofpapers.HereiswhatBarlowandProschansayintheir
nowclassicalbook[3]:
In1951EpsteinandSobelbeganworkinthefieldoflife-testingwhichwas
toresultinalongstreamofimportantandextremelyinfluentialpapers.This
workmarkedthebeginningofthewidespreadassumptionoftheexponential
distributioninlife-testingresearch.
Epstein’scontributionswereofficiallyrecognizedin1974,whentheAmerican
SocietyforQualitydecoratedhimwiththeprestigiousShewhartMedal.
Epstein’s lecture notes for Mathematical Models for Systems Reliability have
neverbeenpublished.However,in1969theyweretyped,duplicatedandsold
toTechnionstudentsbytheTechnionStudentAssociation.Soonenough,they
were out of print, but luckily, five copies remained in the library, so students
couldstilluse(orcopy)them.AfterEpstein’sretirementandoverthelasttwo
decades,Itaughtthecourse,usingEpstein’snotes.Duringtheyears,Iadded
some more topics, examples and problems, gave alternative proofs to some
results,butthegeneralframeworkremainedEpstein’s.Inviewofthefactthat
the Statistical Methods in Reliability course was no longer offered, I added a
© 2008 by Taylor & Francis Group, LLC
brief introduction to Statistical Estimation Theory, so that the students could
relatetoestimationaspectsinreliabilityproblems(mainlyinChapters1–3and
6).
Itismyconviction,thatthematerialpresentedinthisbookprovidesarigorous
treatmentoftherequiredprobabilitybackgroundforunderstandingreliability
theory.Therearemanycontemporarytextsavailableinthemarket,whichem-
phasizeotheraspectsofreliability,asstatisticalmethods,life-testing,engineer-
ing, reliability of electronic devices, mechanical devices, software reliability,
etc.TheinterestedreaderisadvisedtoGoogletheproperkeywordstofindthe
relevantliterature.
Thebookcanserveasatextforaone-semestercourse.Itisassumedthatthe
readersofthebookhavetakencoursesinCalculus,LinearAlgebraandProba-
bilityTheory.KnowledgeofStatisticalEstimation,DifferentialEquationsand
LaplaceTransformMethodsareadvantageous,thoughnotnecessary,sincethe
basicfactsneededareincludedinthebook.
ThePoissonprocessanditsassociatedprobabilitylawsareimportantinrelia-
bilityconsiderationsandsoitisonlynaturalthatChapter1isdevotedtothis
topic.InChapter2,anumberofstochasticmodelsareconsideredasaframe-
work for discussing life length distributions. The fundamental concept of the
hazard or force of mortality function is also introduced in this chapter. For-
mal rules for computing the reliability of non-repairable systems possessing
commonly occurring structures are given in Chapter 3. In Chapter 4 we dis-
cussthestochasticbehaviorovertimeofone-unitrepairablesystemsandsuch
measures of system effectiveness as point-availability, interval and long-run
availabilityandintervalreliabilityareintroduced.TheconsiderationsofChap-
ter 4 are extended to two-unit repairable systems in Chapter 5. In Chapter 6
weintroducethegeneralcontinuous-timeMarkovchains,purebirthanddeath
processesandapplytheresultston-unitrepairablesystems.Weintroducethe
transitionsandratesdiagramsandpresentseveralmethodsforcomputingthe
transition probabilities matrix, including the use of computer software. First
passage time problems are considered in Chapter 7 in the context of systems
reliability. In Chapters 8 and 9 we show how techniques involving the use of
embeddedMarkovchains,semi-Markovprocesses,renewalprocesses,points
ofregenerationandintegralequationscanbeappliedtoavarietyofreliability
problems,includingpreventivemaintenance.
IamextremelygratefultoMalkaEpsteinforherconstantencouragementand
unflaggingfaithinthisproject.IwouldalsoliketothanktheTechnionatlarge,
and the Faculty of Industrial Engineering and Management in particular, for
havingprovidedsuchasupportiveandintellectuallystimulatingenvironment
over the last three decades. Lillian Bluestein did a a superb job in typing the
book,remainingconstantlycheerfulthroughout.
© 2008 by Taylor & Francis Group, LLC
Finally, and most importantly, I want to thank my wife and children for their
continualloveandsupport,withoutwhichIdoubtthatIwouldhavefoundthe
strengthtocompletethebook.
IshayWeissman
Haifa,Israel
April2008
© 2008 by Taylor & Francis Group, LLC
Contents
1 Preliminaries 1
1.1 ThePoissonprocessanddistribution 1
1.2 WaitingtimedistributionsforaPoissonprocess 6
1.3 Statisticalestimationtheory 8
1.3.1 Basicingredients 8
1.3.2 Methodsofestimation 9
1.3.3 Consistency 11
1.3.4 Sufficiency 12
1.3.5 Rao-Blackwellimprovedestimator 13
1.3.6 Completestatistic 14
1.3.7 Confidenceintervals 14
1.3.8 Orderstatistics 16
1.4 GeneratingaPoissonprocess 18
1.5 NonhomogeneousPoissonprocess 19
1.6 Threeimportantdiscretedistributions 22
1.7 Problemsandcomments 24
2 Statisticallifelengthdistributions 39
2.1 Stochasticlifelengthmodels 39
2.1.1 Constantriskparameters 39
2.1.2 Time-dependentriskparameters 41
2.1.3 Generalizations 42
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2.2 Modelsbasedonthehazardrate 45
2.2.1 IFRandDFR 48
2.3 Generalremarksonlargesystems 50
2.4 Problemsandcomments 53
3 Reliabilityofvariousarrangementsofunits 63
3.1 Seriesandparallelarrangements 63
3.1.1 Seriessystems 63
3.1.2 Parallelsystems 64
3.1.3 Thekoutofnsystem 66
3.2 Series-parallelandparallel-seriessystems 67
3.3 Variousarrangementsofswitches 70
3.3.1 Seriesarrangement 71
3.3.2 Parallelarrangement 72
3.3.3 Series-parallelarrangement 72
3.3.4 Parallel-seriesarrangement 72
3.3.5 Simplifications 73
3.3.6 Example 74
3.4 Standbyredundancy 76
3.5 Problemsandcomments 77
4 Reliabilityofaone-unitrepairablesystem 91
4.1 Exponentialtimestofailureandrepair 91
4.2 Generalizations 97
4.3 Problemsandcomments 98
5 Reliabilityofatwo-unitrepairablesystem 101
5.1 Steady-stateanalysis 101
5.2 Time-dependentanalysisviaLaplacetransform 105
5.2.1 Laplacetransformmethod 105
5.2.2 Anumericalexample 111
5.3 OnModel2(c) 113
5.4 ProblemsandComments 114
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6 Continuous-timeMarkovchains 117
6.1 Thegeneralcase 117
6.1.1 Definitionandnotation 117
6.1.2 Thetransitionprobabilities 119
6.1.3 ComputationofthematrixP(t) 120
6.1.4 Anumericalexample(continued) 122
6.1.5 Multiplicityofroots 126
6.1.6 Steady-stateanalysis 127
6.2 Reliabilityofthree-unitrepairablesystems 128
6.2.1 Steady-stateanalysis 128
6.3 Steady-stateresultsforthen-unitrepairablesystem 130
6.3.1 Example1—Case3(e) 131
6.3.2 Example2 131
6.3.3 Example3 131
6.3.4 Example4 132
6.4 Purebirthanddeathprocesses 133
6.4.1 Example1 133
6.4.2 Example2 133
6.4.3 Example3 134
6.4.4 Example4 134
6.5 Somestatisticalconsiderations 135
6.5.1 Estimatingtherates 136
6.5.2 Estimationinaparametricstructure 137
6.6 Problemsandcomments 138
7 Firstpassagetimeforsystemsreliability 143
7.1 Two-unitrepairablesystems 143
7.1.1 Case2(a)ofSection5.1 143
7.1.2 Case2(b)ofSection5.1 148
7.2 Repairablesystemswiththree(ormore)units 150
© 2008 by Taylor & Francis Group, LLC
