Discrete Stochastic Models and Applications for Reliability Engineering and Statistical Quality Control

183 Pages·2022·12.448 MB·English

by  Serkan Eryilmaz

Discrete Stochastic
Models and
Applications for
Reliability Engineering
and Statistical
Quality Control
Discrete stochastic models are tools that allow us to understand, control, and optimize
engineering systems and processes. This book provides real-life examples and illustrations
of models in reliability engineering and statistical quality control and establishes a connec-
tion between the theoretical framework and their engineering applications.
This book describes discrete stochastic models along with real-life examples and explores
not only well-known models, but also comparatively lesser-known ones. It includes defi-
nitions, concepts, and methods with a clear understanding of their use in reliability en-
gineering and statistical quality control fields. Also covered are the recent advances and
established connections between the theoretical framework of discrete stochastic models
and their engineering applications.
This book is an ideal reference for researchers in academia and graduate students working
in the fields of operations research, reliability engineering, quality control, and probability
and statistics

Discrete Stochastic
Models and
Applications for
Reliability Engineering
and Statistical
Quality Control
Serkan Eryilmaz
First edition published 2023
by CRC Press
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and by CRC Press
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CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Serkan Eryilmaz
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ISBN: 978-0-367-74983-5 (hbk)
ISBN: 978-0-367-74984-2 (pbk)
ISBN: 978-1-003-16063-2 (ebk)
DOI: 10.1201/9781003160632
Typeset in Times New Roman
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Dedication
TomybelovedparentsNurhanandNecatiEryilmaz,
andmylovelydaughterBuseCeylinEryilmaz

Contents
Preface.......................................................................................................................ix
AuthorBiography.....................................................................................................xi
Chapter1 AnOverviewofDiscreteStochasticModeling...............................1
1.1 Discreterandomvariable........................................................1
1.2 Conditioning...........................................................................6
1.3 Discretization..........................................................................8
1.4 Combinatorialmethods...........................................................9
1.5 Recursiveequations..............................................................11
1.6 Simulation.............................................................................12
Chapter2 DiscreteProbabilityDistributions.................................................17
2.1 Binomialmodel....................................................................17
2.2 Waitingtimemodels.............................................................22
2.3 Poissonmodel.......................................................................28
2.4 Compoundmodels................................................................31
2.5 Modelsviadiscretizationofcontinuousdistributions..........33
2.6 Bivariatebinomialandgeometricmodels............................34
2.7 Discretereliabilityindices....................................................37
2.8 Discreterenewalprocess......................................................39
Chapter3 MarkovChains..............................................................................43
3.1 Markovbinomialmodel........................................................51
3.2 SimulationofaMarkovchain..............................................53
3.3 TestingtheorderofMarkovdependence.............................57
3.4 Markovchainwithachangepoint........................................59
Chapter4 Phase-typeDistributions................................................................63
4.1 Definitionandcharacteristics...............................................63
4.2 Closureandotherproperties.................................................71
4.3 Bivariatephase-typedistributions.........................................81
Chapter5 Matrix-geometricDistributions.....................................................85
5.1 Definitionandcharacteristics...............................................85
5.2 Properties..............................................................................87
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viii Contents
5.3 Two-unitcoldstandbyrepairablesystemvia
matrix-geometricdistributions..............................................93
Chapter6 SomeParticularModels/ClassesofDiscreteDistributions.........101
6.1 Orderstatistics....................................................................101
6.2 Systemsignature.................................................................104
6.3 Powerseriesclassofdistributions......................................108
6.4 Panjer’sfamilyofdistributions...........................................109
Chapter7 OtherDiscreteStochasticModelsInvolvingDependency..........113
7.1 Exchangeabledependence..................................................113
7.2 Previous-sumdependentmodel..........................................118
7.3 Thedistributionofthenumberofsuccesses.......................119
7.4 m-dependentsequenceofrandomvariables.......................120
Chapter8 Applications.................................................................................123
8.1 Systemreliabilitymodels...................................................123
8.2 Shockmodels......................................................................145
8.3 Agereplacementpolicies....................................................151
8.4 Qualitycontrol....................................................................154
8.5 Start-updemonstrationtests...............................................160
Index......................................................................................................................169
Preface
It has become increasingly apparent that many problems involving uncertainty in
various fields of science provide important examples to discrete stochastic models.
Therearemanyreal-lifeengineeringsystemsandprocesseswhichmaybedescribed
bydiscretestochasticvariables. Thenumberoffailed windturbinesduringaspec-
ified period of time in a wind farm, the total number of collected subsamples until
anout-of-controlsignalinaspecificproductionprocess,thenumberoffailedcom-
ponentsinatechnicalsystematthetimewhenthesystemfails,andthenumberof
arrivalstoacertainservicestationcanbeallmodeledbyadiscretestochasticvari-
able.Further,thelifetimesofmanyreal-lifesystems/componentsaredefinedasthe
number of cycles until their failure. For example, the lifetime of car tires are mea-
suredintermsofthenumberofkilometersorthenumberofdays;thelifetimeofa
switchismeasuredbythenumberoftimesitisswitchedon/off.
Asitiswellknown,themostimportantprobabilistictoolforunderstandingsuch
astochasticvariableisitsprobabilitydistribution.Thus,thedevelopmentofsuitable
discretestochasticmodelsisofspecialimportanceforproperanalysisofthevariable
underconcern,andhence,thecorrespondingreal-lifephenomenonorsystem.
Thepurposeofthisbookistopresentthewell-knownandlesser-knowndiscrete
stochastic models with an emphasis to their applications in reliability engineering
and quality control. It is not intended to present the complete theory of the topics
mentioned. Rather, a lucid description of the models is presented with simple real-
lifeexamples.Intheproofsandexplanations,clarityandsimplicityarefavored.Ex-
tensivesubchaptersarepresentedonspecificresearchfieldsofreliabilityengineering
andstatisticalquality/processcontrol.
Thisbookhelpsthereaderdeveloptheunderstandingandintuitionnecessaryto
applydiscretestochasticmodelsinreliabilityengineeringandoperationsresearch.It
isorganizedintoeightchapters.Chapter1providesanoverviewanddiscussionofthe
discretestochasticmodeling.InChapter2,well-knownandlesser-known(butwith
manyreal-lifeapplications)discreteprobabilitydistributionsarereviewed.Particular
discretestochasticmodelsaredescribedthroughChapters3–7withsimpleandclear
examples.Chapter8containsextensiveapplicationsconcernedwithimportantfields
including system reliability models, shock models, age replacement policies, and
statisticalqualitycontrol.
Iwishtoexpressmygratitudetoallwhohavesupportedmeinonewayoranother
throughoutmyacademicjourney,especiallymyPh.D.supervisorProfessorIsmihan
Bairamov (Bayramoglu) who has greatly contributed to my learning of probability
theoryandstochasticmodeling.IextendmyappreciationtomyformerPh.D.student
Dr. Cihangir Kan and Dr. Murat Ozkut for their great assistance in formatting the
book. My sincere thanks goes to my dear friends and colleagues, Professor Yılser
DevrimandProfessorFatihTank,whohavealwaysbeenthereforme.Iwouldalso
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