Table Of ContentSpringer Series in Reliability Engineering
Ji Hwan Cha
Maxim Finkelstein
Point Processes
for Reliability
Analysis
Shocks and Repairable Systems
Springer Series in Reliability Engineering
Series editor
Hoang Pham, Piscataway, USA
More information about this series at http://www.springer.com/series/6917
Ji Hwan Cha Maxim Finkelstein
(cid:129)
Point Processes
for Reliability Analysis
Shocks and Repairable Systems
123
JiHwan Cha Maxim Finkelstein
Department ofStatistics Department ofMathematical Statistics
EwhaWomansUniversity University of the FreeState
Seoul Bloemfontein
Korea (Republicof) SouthAfrica
and
ITMO University
Saint Petersburg
Russia
ISSN 1614-7839 ISSN 2196-999X (electronic)
SpringerSeries inReliability Engineering
ISBN978-3-319-73539-9 ISBN978-3-319-73540-5 (eBook)
https://doi.org/10.1007/978-3-319-73540-5
LibraryofCongressControlNumber:2017962972
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To My Family and Friends
—Ji Hwan Cha
To My Family
—Maxim Finkelstein
Preface
This book is about point processes and their applications in reliability and risk
analysis. In the first part, we provide a general introduction to the ‘basic’ point
processes and discuss their properties, and then consider the corresponding appli-
cationslater.However,intheapplicationspart,wedonotaimatthebroadcoverage
of the field but rather focus on the models and applications that have been inves-
tigatedbytheauthorsmostlyinthelast10yearsandwhichweconsiderimportant
fromtheoreticalandpracticalpointsofview.Thosearevariousshockmodelswhen
apoint processdescribesanimpact ofanenvironmentonanoperatingsystem and
modelsoffailure/repairwhentheconsecutiveinstantsoffailure/repairaremodelled
by a point process.
Wedealwith variousmodels basedonPoissonandrenewalprocessesandtheir
generalizations. For instance, a renewal process that describes a sequence of
instantaneousperfect repairsof asystem isgeneralized tothecase when the repair
isimperfect thus forming apoint process ofimperfect (general) repairs. Reliability
characteristics of repairable systems under a general repair can be of a paramount
importancewhenplanningoperationofasystemforlongperiodsoftime.Minimal
repair, when a system is repaired to a statistical state it had prior to failure, is a
specific case of a general repair. It is well known that the process of instantaneous
minimalrepairsischaracterizedbythenonhomogeneousPoissonprocess(NHPP).
Poisson processes are also very popular in numerous reliability engineering
applications. The NHPP possesses the independent increments property, which is,
in fact, restrictive, especially in applications, as stochastic phenomena in many
real-worldproblemsdonotmeetthisassumption.Therefore,inourbook,werelax
thisassumptionanddescribeindetailthegeneralizedPolyaprocess(GPP)recently
introduced and studied by the authors. This process has dependent increments and
the probability of an event occurrence in an infinitesimal unit interval of time
alreadydependsonthenumberofeventsoccurredpreviously.Thisprocess,onone
hand,ismuch morerealistic inmanypracticalsituationsand,ontheother hand,is
mathematicallytractable.Asignificantpartofourbookisdevotedtothetheoryand
applications of the univariate and multivariate GPP.
vii
viii Preface
Allrealobjectsareoperatinginachanging(stochastic)environment.Oneofthe
waystomodelanimpactofthisenvironmentisviatheexternalshocksoccurringin
accordance with some stochastic point processes. We understand the term ‘shock’
in a very broad sense as some instantaneous and potentially harmful event (e.g.
electrical impulses of large magnitude, demands for energy in biological objects,
insurance claims in finance, earthquakes, etc.). Shock models are widely used in
practical and theoretical reliability and in the other disciplines as well. Numerous
shock models have been studied in the literature during the last decades. Shock
models in reliability mostly deal with survival characteristics of engineering sys-
tems subject to external point events. Therefore, under different assumptions and
settings, we consider a number of survival models for systems operatingunder the
shock processes (renewal, NHPP, GPP).
We hope that our book will be useful to reliability specialists and graduate
students in reliability or applied probability. It also contains numerous stochastic
models that can be of interest to applied mathematicians and statisticians.
Ji Hwan Cha’s work was supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2-
B2014211). Ji Hwan Cha acknowledges the support of the Ewha Womans
University (Seoul, Republic of Korea). Ji Hwan Cha also greatly appreciates the
help, assistanceand sacrifice ofCocoDaram(Seungsin Lee).
MaximFinkelsteinacknowledgesthesupportoftheUniversityoftheFreeState
(Bloemfontein,SouthAfrica),theNationalResearchFoundation(SouthAfrica)and
the ITMO University (St. Petersburg, Russia).
Wearealsogratefultoourcolleagues,co-workersandtheformerstudentsofJi
Hwan Cha (Dr. Hyunju Lee, Dr. Juyoung Kim). Their support and discussions
contributed a lot to this project. Finally, we are indebted to Dhanaganapathy
Madhavan, Anthony Doyle and the Springer stafffor their editorial work.
Seoul, Korea (Republic of) Ji Hwan Cha
Bloemfontein, South Africa/Saint Petersburg, Russia Maxim Finkelstein
November 2017
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Aim and Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Brief Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Preliminaries: Reliability and Point Processes. . . . . . . . . . . . . . . . . 9
2.1 Distribution Function and Failure Rate. . . . . . . . . . . . . . . . . . . 9
2.2 Mean Remaining Lifetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Monotonicity of the Failure Rate and the MRL Function . . . . . 17
2.4 Stochastic Ordering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Point Processes and Their Characterization. . . . . . . . . . . . . . . . 24
2.6 Perfect and Minimal Repair. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Shocks and Extreme Shock Model. . . . . . . . . . . . . . . . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Renewal Processes and Applications. . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Definition and Main Properties . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Limiting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Alternating Renewal and Renewal Reward Processes . . . . . . . . 47
3.4 Applications of Renewal Theory . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.1 Renewal Reward Process with Continuous Output . . . . 51
3.4.2 Alternating Renewal Process with Gradual Repair . . . . 53
3.4.3 Classical Optimal Replacement Problem and Its
Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.4 Using Renewal Equations in Shocks Modeling . . . . . . 64
3.4.5 Renewal in Populations of Technical Items and
Organisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
ix
x Contents
4 Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Homogeneous Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Nonhomogeneous Poisson Process. . . . . . . . . . . . . . . . . . . . . . 76
4.3 Minimal Repair and Applications to Optimal Maintenance . . . . 86
4.4 Simple Poisson Shock Models. . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4.1 ‘Classical’ Extreme Shock Model . . . . . . . . . . . . . . . . 93
4.4.2 Direct Impact on the Failure Rate . . . . . . . . . . . . . . . . 95
4.5 General Compound Poisson Process
and a Shot Noise Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6 Mixed Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Advanced Poisson Shock Models . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1 The Terminating Shock Process with Independent Wear
Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.2 Exponentially Distributed Boundary . . . . . . . . . . . . . . 114
5.1.3 Deterministic Boundary . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 History-Dependent Termination Probability . . . . . . . . . . . . . . . 120
5.3 Shot Noise Process for the Failure Rate . . . . . . . . . . . . . . . . . . 127
5.3.1 Shot Noise Process Without Critical Shocks . . . . . . . . 127
5.3.2 Shot Noise Process with Critical Shocks and
Deterioration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 Extreme Shock Model with Delayed Termination. . . . . . . . . . . 139
5.5 Cumulative Shock Model with Initiated Wear Processes . . . . . . 142
5.6 ‘Curable’ Shock Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.7 Stress-Strength Model with Delay and Cure . . . . . . . . . . . . . . . 151
5.8 Survival of Systems with Protection Subject
to Two Types of External Attacks . . . . . . . . . . . . . . . . . . . . . . 153
5.9 Information-Based Thinning of Shock Processes. . . . . . . . . . . . 160
5.9.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.9.2 Formal Description of the Information-Dependent
Thinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.9.3 Stress-Strength Type Classification Model . . . . . . . . . . 164
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6 Poisson Shock Model with Applications to Preventive
Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1 Conditional Characteristics and Interpretation . . . . . . . . . . . . . . 169
6.1.1 Conditional Characteristics . . . . . . . . . . . . . . . . . . . . . 169
6.1.2 A Specific Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1.3 The Dynamics of Conditional Distributions . . . . . . . . . 177
