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SPRINGER BRIEFS IN STATISTICS Mohamed Abdel-Hameed Lévy Processes and Their Applications in Reliability and Storage 123 SpringerBriefs in Statistics For furthervolumes: http://www.springer.com/series/8921 Mohamed Abdel-Hameed Lévy Processes and Their Applications in Reliability and Storage 123 MohamedAbdel-Hameed Department of Statistics College ofBusiness andEconomics UnitedArab Emirates University Al Ain UAE ISSN 2191-544X ISSN 2191-5458 (electronic) ISBN 978-3-642-40074-2 ISBN 978-3-642-40075-9 (eBook) DOI 10.1007/978-3-642-40075-9 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013954772 MathematicsSubjectClassification(2010):60K10,60K20 (cid:2)TheAuthor(s)2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To The memory of my parents Preface Overthelastfewdecades,Lévyprocesseshavebeenusedextensivelyinreliability, hydrology, and water resource engineering. Inreliabilityengineering,theyareusedtomodeldegradationofdevicesovertime. CertaintypesofLévyprocesseshavebeenfoundtoprovideagoodmodelforcreepof concrete, fatigue crack growth, corroded steel gates, and chloride ingress into concrete.Atthebeginningoftheworkdoneinreliability,engineersdescribedthe uncertaintiesaboutthefailuretimesusingthesurvivalfunction;knowingtheshape ofsuchafunctiontheycandetermineandstudythepropertiesofthefailurerateand, basedonthat,theycandeterminethebestpossiblemaintenancepolicies.Toestimate thesurvivalfunctionaccurately(fromastatisticalpointofview),onehastoobserve thefailuretimesofmanyitemsandthesefailurerandomvariablesareassumedtobe independent.Inpractice,itisnotalwayspossibletoobservemanyfailures,andeven ifsuchfailuretimesarepossibletoobtain,theyarenotindependentastheyallmight beaffectedbyanenvironment.Theotherapproachistoassessthefailureofadevice based on the characteristics of the process that caused its failure, normally a degradationprocess.Suchanapproachiscommoninassessingtheamountofcrack, theamountoferosionandcreep,andtheamountofcontamination. In hydrology and water resource engineering, they are used (among other things) to model the input of water in a reservoir over time. Brownian motion, compound Poisson processes, inverse Gaussian processes, and spectrally positive Lévyprocesseshavebeenusedtodescribesuchinput.Knowingtheinputprocess and its characteristics enables one to determine and properly improve the cost of running the dam over time. This monograph consists of three chapters, notations and terminology, and an appendix. In the appendix, we give some basic definitions and results. In Chap. 1 we discuss Lévy Processes and Their Characteristics. In Chap. 2, we discuss the applications of Lévy processes in describing Degradation Processes. In Chap. 3 we deal with the usage of Lévy processes to describe the input processes and controlling the cost of running reservoirs. Readersareadvisedtobeginwithatleastaquicklookattheappendix,andthe notations and terminology. They serve to review the prerequisite results and definitions. At the end of each chapter as well as in the appendix, relevant references are given. I did not attempt to compile comprehensive bibliographies, but rather give a list of those references that I used to write this book. vii Acknowledgments I want to thank my wife for her patience and understanding. I also thank the external reviewers for their helpful comments on an earlier version of this book. Many thanks to my editor for her insights and many suggestions. ix Contents 1 Lévy Processes and Their Characteristics. . . . . . . . . . . . . . . . . . . 1 1.1 Lévy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Lévy-Itô Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Strong Markov Property for Lévy Processes. . . . . . . . . . . . 3 1.4 Subordinators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Spectrally Positive Processes. . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Examples of Spectrally Positive Processes . . . . . . . . . . . . . . . . 12 1.7 The Compensation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.8 Non-homogeneous Lévy Processes . . . . . . . . . . . . . . . . . . . . . 16 1.9 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Degradation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Life Distributions of Devices Subject to Degradation. . . . . . . . . 30 2.4 Control-Limit Maintenance Policies for Continuously Monitored Degradable Systems. . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 One-Level Control-Limit Maintenance Policies for Non-Continuously Monitored Degradable Systems . . . . . . . . . . 41 2.6 Multi-Level Control-Limit Maintenance Policies for Non-Continuously Monitored Degradable Systems . . . . . . . . . . 59 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.8 Inference for the Parameters of the Degradation Process . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3 Storage Models: Control of Dams Using PM Policies. . . . . . . . . . . 77 k;s 3.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3 The PM Control Policies and Their Associated k;s Cost Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.4 Subordinator Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 xi xii Contents 3.5 Spectrally Positive and Spectrally Positive Reflected at its Infimum Inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Appendix: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Notation and Terminology We let R¼ð(cid:2)1;1Þ; R ¼ð0;1Þ; N ¼f1;2;...g; N ¼f0;1;...g;N ¼ þ þ þ N [f1g; R¼R[ f1g; R ¼R [f1g; and R ¼Rnf0g: We will use the þ þ þ 0 term ‘‘increasing’’ to mean ‘‘non-decreasing,’’ and the term ‘‘positive’’ to mean ‘‘non-negative.’’AllrandomvariablesaredefinedonaprobabilityspaceðX;F;PÞ. All processes used have the space of real numbers or subsets of it as their state space. For any process Y ¼fY; t(cid:3)0g, any Borel subset A of the state space and t any functional f; E ðfÞ denotes the expectation of f conditional on Y ¼ y 0 y; P ðAÞ denotes the corresponding probability measure and I is the indicator y A function of the set A. In the sequel we will write, indifferently, P or P and E or 0 0 E. For x; y 2 R, we define x _ y ¼ x max y and x^y¼x min y. For every t (cid:3)0, we define Y ¼ inf ðY ^0Þ;Y(cid:2) ¼ sup ðY _0Þ, For every t2R , the t s t s þ 0(cid:4)s(cid:4)t 0(cid:4)s(cid:4)t process obtained by reflecting Y at its infimum and supremum are defined as follows: Y (cid:2)Y and Y(cid:2) (cid:2)Y, respectively. For any space G, r(G) denotes the t t t t smallest sigma algebra of subsets of G. The following is a list of symbols that are used in this book. t Lévy measure / Laplace Exponent of a spectrally positive Lévy process p(t, x, y) Probability transition function W(a) a-scale function Z(a) Adjoint a-scale function Ua a-potential measure of a non-subordinator ua a-potential density non-subordinator F(cid:2) Survival probability IFR Increasing failure rate DFR Decreasing failure rate IFRA Increasing failure rate average DFRA Decreasing failure rate average TP Totally positive of order 2 2 Af Infinitesimal generator Qðx;A;tÞ Semi-Markov renewal kernel Rðx;A;tÞ Renewal function of a Markov renewal process xiii

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