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On the QED Limits in the Many-Server Systems

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Title:
On the QED Limits in the Many-Server Systems
Creator:
Motaei, Amir
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
MOMCILOVIC,PETAR
Committee Co-Chair:
PARDALOS,PANAGOTE M
Committee Members:
GARCIA,ALFREDO
BLIZNYUK,NIKOLAY A

Subjects

Subjects / Keywords:
heavy-traffic
machine-repair-model
multi-class-priority-model
qed-regime

Notes

General Note:
We study two classic queuing models operating in the Quality-and-Efficiency Driven asymptotic (QED) regime. First, a machine repair model under general operating/repair distributions is considered. Process-level convergence of the number of broken machines is established -- the limit is in terms of the corresponding tractable infinite-repairmen process, a stationary centered Gaussian process. In addition, a derived limit interchange establishes a connection between open and closed queuing systems in the QED regime. The second model is a multi-class many-server priority system with general arrival processes and general service times. The many-server heavy-traffic diffusion asymptotic is characterized in terms of the corresponding limiting infinite-server process.

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UFRGP
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All applicable rights reserved by the source institution and holding location.
Embargo Date:
12/31/2017

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ONTHEQEDLIMITSINTHEMANY-SERVERSYSTEMS By AMIRMOTAEI ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2016

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c 2016AmirMotaei

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Tomyparents

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ACKNOWLEDGMENTS IwouldliketoexpressmysinceregratitudetomyadvisorDr.PetarMomcilovicforthe continuoussupportofmyPh.D.studyandresearch,forhispatienceandimmenseknowledge.I havelearnedalotfromhimandhisguidancehelpedmeinallthetimeofresearchandwriting ofthisthesis.IcouldnothaveimaginedhavingabetteradvisorandmentorformyPh.D. study. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS...................................4 LISTOFFIGURES.....................................7 ABSTRACT.........................................9 CHAPTER 1INTRODUCTION...................................10 1.1Background...................................10 1.2Contributions...................................11 1.2.1Machine-repairmodel..........................11 1.2.2Multi-classpriorityqueueingmodel...................13 1.3Notations.....................................14 2LITERATUREREVIEW................................16 2.1QEDRegime...................................16 2.2MachineRepairModel..............................16 2.3Multi-ClassPriority...............................18 3MACHINEREPAIRMODEL.............................20 3.1Overview.....................................20 3.2Model......................................20 3.3Results......................................23 3.3.1Mainresults...............................23 3.3.2Limitinterchange.............................29 3.4ProofofTheorem3.1..............................32 3.4.1Outlineoftheproof...........................32 3.4.2Preliminaryresults............................37 3.4.3ProofofTheorem3.1..........................41 3.5Proofs......................................42 3.5.1ProofofLemma3.1...........................42 3.5.2ProofofProposition3.1.........................45 3.5.3ProofofLemma3.2...........................55 3.5.4ProofofLemma3.3...........................57 3.6NumericalExamples...............................68 3.6.1Performanceofnite-sizesystem....................68 3.6.2Residualservice/worktimes.......................71 4MULTI-CLASSPRIORITYMODEL..........................73 4.1Overeview....................................73 5

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4.2AssumptionsandModel.............................73 4.2.1Model...................................73 4.2.2QEDregime...............................74 4.3PreliminaryResults................................76 4.3.1Modiedsystem.............................76 4.3.2Innite-serversystem...........................78 4.4MainResults...................................81 4.5ProofofTheorem4.1..............................86 4.5.1Preliminaryresult.............................86 4.5.2Thediusionlimitforthemodiedsystem...............89 4.5.3Thediusionlimitfortheoriginalsystem................96 4.6Proofs......................................97 4.6.1ProofofLemma4.1...........................97 4.6.2ProofofLemma4.2...........................98 4.6.3ProofofLemma4.3...........................98 4.6.4ProofofLemma4.4...........................100 5CONCLUSION.....................................102 APPENDIX AADDITIONALMOTIVATIONFORTHEQEDREGIME...............104 BANCILLARYRESULTSANDPROOFS........................107 B.1AncillaryResultsforMachineRepairModel...................107 B.1.1Decomposition..............................107 B.1.2Tightness.................................122 B.1.3Dependencygraphforresultsonthemachinerepairmodel.......124 B.2AncillaryResultsforMulti-ClassPriorityModel.................124 CADDITIONALNUMERICALEXAMPLESFORMACHINEREPAIRMODEL....132 C.1PerformanceofaFinite-SizeSystem......................132 C.1.1Exponentialdistributions.........................132 C.1.2Uniformdistributions...........................140 C.2ResidualService/WorkTimes..........................148 REFERENCES........................................154 BIOGRAPHICALSKETCH.................................162 6

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LISTOFFIGURES Figure page 3-1AcommutativediagramillustratingtheinterchangeofPoissonandQEDlimits...30 3-2Anillustrationofthedecompositionfor X n i )]TJ/F38 11.9552 Tf 11.955 0 Td [(Z n i ..................35 3-3AdiagramillustratingrelationsbetweenquantitiesusedintheproofofLemma3.3.64 3-4Numericalexample M = M = 40 = 120 :Occupancyprocess...............69 3-5Numericalexample U = U = 40 = 120 :Occupancyprocess................70 3-6Numericalexample U = U = 20 = 60 :Residualservice/work...............72 C-1Numericalexample M = M = 18 = 60 :Occupancyprocess................133 C-2Numericalexample M = M = 20 = 60 :Occupancyprocess................134 C-3Numericalexample M = M = 22 = 60 :Occupancyprocess................135 C-4Numericalexample M = M = 37 = 120 :Occupancyprocess...............136 C-5Numericalexample M = M = 43 = 120 :Occupancyprocess...............137 C-6Numericalexample M = M = 194 = 600 :Occupancyprocess...............138 C-7Numericalexample M = M = 200 = 600 :Occupancyprocess...............139 C-8Numericalexample M = M = 206 = 600 :Occupancyprocess...............140 C-9Numericalexample U = U = 18 = 60 :Occupancyprocess.................141 C-10Numericalexample U = U = 20 = 60 :Occupancyprocess.................142 C-11Numericalexample U = U = 22 = 60 :Occupancyprocess.................143 C-12Numericalexample U = U = 37 = 120 :Occupancyprocess................144 C-13Numericalexample U = U = 43 = 120 :Occupancyprocess................145 C-14Numericalexample U = U = 194 = 600 :Occupancyprocess...............146 C-15Numericalexample U = U = 200 = 600 :Occupancyprocess...............147 C-16Numericalexample U = U = 206 = 600 :Occupancyprocess...............148 C-17Numericalexample U = U = 3 = 12 :Residualservice/work................149 C-18Numericalexample U = U = 4 = 12 :Residualservice/work................150 C-19Numericalexample U = U = 5 = 12 :Residualservice/work................150 7

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C-20Numericalexample U = U = 18 = 60 :Residualservice/work...............151 C-21Numericalexample U = U = 22 = 60 :Residualservice/work...............151 C-22Numericalexample U = U = 37 = 120 :Residualservice/work...............152 C-23Numericalexample U = U = 40 = 120 :Residualservice/work...............152 C-24Numericalexample U = U = 43 = 120 :Residualservice/work...............153 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ONTHEQEDLIMITSINTHEMANY-SERVERSYSTEMS By AmirMotaei December2016 Chair:PetarMomcilovic Major:Industrialandsystemsengineering WestudytwoclassicalqueueingmodelsoperatingintheQuality-and-EciencyDriven QEDasymptoticregime.First,amachine-repairmodelundergeneraloperating/repair distributionsisconsidered.Process-levelconvergenceofthenumberofbrokenmachinesis established{thelimitisintermsofthecorrespondingtractableinnite-repairmenprocess, astationarycenteredGaussianprocess.Inaddition,aderivedlimitinterchangeestablishesa connectionbetweenopenandclosedqueueingsystemsintheQEDregime.Thesecondmodel isamulti-classmany-serverprioritysystemwithgeneralarrivalprocessesandgeneralservice times.Themany-serverheavy-tracdiusionasymptoticischaracterizedintermsofthe correspondinglimitinginnite-serverprocess.Ourresultsgeneralizeresultsfromtheliterature ontherst-comerst-servedFCFSsystem. 9

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CHAPTER1 INTRODUCTION 1.1Background Inthisdissertation,westudytwolarge-scalequeueingmodels,namelythemachinerepair modelandmulti-classprioritymodel,intheQuality-and-Eciency-DrivenQEDregime. Informally,undersucharegime,theprobabilitiesthatajobndsanavailableserveranda serverndsajobawaitingservicearestrictlyin ,1 .IntheQEDregime,thecapacityofthe systemisroughlyequaltotheoeredload.StudiesoftheQEDregimearepartiallymotivated bythefactthatQED-basedapproximationsadequatelydescribenite-sizesystemsthat operateineciency-orquality-drivenregimes[27].WefurthermotivatetheQEDregimefor machine-repairmodelsinAppendixA.Underspecicassumptions,wearguethatQED-based approximationsofsteady-stateperformancemeasuresremainvalidforabroaderclassof machine-repairmodels,wherebehaviorofasinglemachinedependsonthesystemsize. Themachine-repairmodelisaclassicclosedqueueingmodel.Thesystemconsistsofa nitenumberofmachinesandrepairmen.Amachineoperatesforarandomamountoftime beforeitbreaksdown,andittakesarepairmanarandomamountoftimetoxamachine. Onceamachineisrepaired,itbecomesoperational,andthecyclerepeats.Themachine-repair modelisrelevantindescribingsystemswithanitepopulationofcustomersthatrepeatedly returnforservice.Thereexistsafundamentaldierenceinthebehaviorofopenandclosed systems.Inanopensystem,thearrivalrateisindependentofthenumberofcustomersin thesystem.Ontheotherhand,duetoafeedbackpresentinaclosedsystem,thearrivalrate decreasesasthenumberofcustomersinthesystemincreases.Ingeneral,ourmodelmightbe relevantforlarge-scalesystemswiththefollowingfeatures:ithesystemoperatorhasaxed constituencyfromwhichservicerequestsmaterialize,andiionceprocessed,customersreturn tothepoolofpotentialusersandseekadditionalservicelater. Thesecondmodelweconsiderisamulti-classpriorityqueueingsystem.Inparticular,we focusonanon-preemptive k -classpriorityqueueingsystemwiththerst-comerst-served 10

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FCFSservicedisciplinewithineachclass.Thesystemconsistsof k < 1 classesof customers,withapriorityorderingbetweenclasses.Uponanarrivalofacustomer,ifa serverisidle,thecustomerentersservice.Otherwise,customerentersawaitingreemof innitecapacity.Onceaserverbecomesidle,onecustomerfromhighestpriorityclass,awaiting service,entersservice. 1.2Contributions Inthissectionwefocusonourcontributionsinmachine-repairmodelSection1.2.1and multi-classpriorityqueueingmodelSection1.2.2. 1.2.1Machine-repairmodel InChapter3,westudyamachine-repairmodelwithgeneralrepairandworkingtimes intheQEDregime.OurmainresultTheorem3.1characterizesthelimitingcenteredand scalednumberofbrokenmachinesprocessintermsofanonlinearoperatorandanexplicitly characterizedstationaryGaussianprocess;thisGaussianprocesscorrespondstoalimiting centeredandscaledinnite-repairmenprocess,ananalogoftheinnite-serverprocess. Theorem3.1isanalogoustothemainresultin[77],wherethecaseofanopenQEDsystem istreated.Thatis,weestablisharelationinthelimitasthesizeofthesystemincreases betweenamachine-repairmodelandthecorrespondinginnite-repairmenmodelaprocess thatischaracterizedexplicitly.Duetotheclosednatureofoursystem,thisrelationismore intricatethanfortheopensystem.Inparticular,thecorrespondinginnite-servermodel providesalowerboundonthenumberofcustomersintheopensystemwithnitelymany servers,sincearrivalsinthetwoopenmodelsoccuratthesametime.Ontheotherhand,this isnotthecaseforclosedsystems:breakdowntimesinthetwoclosedmodelsdonotoccur atthesametime{anywaitforrepaircausesatimeshiftoffuturebreakdowntimesfora givenmachineinthemodelwithnitelymanyrepairmenseeFigure3-2inSection3.4.1 foranillustration.Hence,ournonlinearoperatoraccountsnotonlyforwaitingtimeslike intheopensystem,butfortimeshiftsinbreakdowntimesaswell.Thisoperatorisdened intermsoftherepairandworkingtimedistributions.Hence,wedemonstratetheimpactof 11

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thosedistributionsonmodelperformance.Specically,thenonlinearoperatorisdenedby aconditionalprobabilitythatanon-oprocessisintheonstate;onandoperiodsinthis processaredistributedaccordingtotherepairandworkingtimedistributions,respectively. Fromthetechnicalpointofview,weexpressthedierencebetweentheniteandinnite modelsasacountablesumofprocesseswithsomenontrivialdependencystructure.These processesareanalyzedbymeansofanextensionofamartingalemethodusedtostudyan opensystemin[77]originallydevelopedin[58].Inparticular,wedenetheseprocesses recursivelyinsuchawaythatnon-recursivepartsaresimilarweworkwithtime-varying arrival"processestoaprocessthatarisesintheanalysisofanopensystem.Inourcase, amartingaledecompositionisusednotonlytoprovetightness,butalsotoboundrelevant supremumnorms. Inaddition,weestablishaconnectionbetweenclosedandopensystemsintheQED regimeviaaPoissonlimit.Inparticular,weshowthatthetwolimitsareinterchangeable Section3.3.2.Ononehand,closednitepopulationqueueingsystemsmightseemmore relevantingeneralbecausetheworldisarguablynite[27].Ontheotherhand,theuseof openqueueingmodelsiswidespread,sinceclosedqueueingsystemsaretypicallyharderto analyzethantheiropencounterparts[96].Ourresultsprovideaguidelineonwhenemploying openmodelstoquantifyclosedqueueingsystemsmightbeappropriate.Thatis,weshed somelightonthefollowingquestion:giventhatthenumberofrepairmenislargethe limitingregimeweconsider,howlargethenumberofmachinesneedstoberelativetothe numberofrepairmenfortheopenapproximationtoberelevantonnitetimeintervals? {seeRemark3.5fordetails.Inthecontextofprocess-levelconvergence,thePoissonand heavy-traclimitsarenotinterchangeableinthegeneralcaseofsingle-serverqueuesdueto thedierenceinrelevanttimescalesforthetwolimits.Inparticular,forsingleserverqueues, thereexistsaseparationoftimescales:servicerepairtimesoccuronafastertimescalethan thetimescaleassociatedwithasinglesourcemachine.Ontheotherhand,inourmodel, 12

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repairandworkingtimesareassociatedwiththesametimescale.Foradetaileddiscussionof relevanttimescalessee[100,Section9.8]. 1.2.2Multi-classpriorityqueueingmodel InChapter4,westudyamulti-classpriorityqueueingmodelwithgeneralservicetimes andgeneralarrivalprocesses,intheQEDregime.Ourmainresult,Theorem4.1,describes thelimitingintheQEDregime,i.e.,diusionscale,many-serverlimitbehaviorofa k -class priorityG/GI/ n queueingsystemintermsofanonlinearconvolutionofthecorresponding limitinginnite-serversystem.Therelationbetweenthenumber-in-systemprocessandthe correspondinginnite-serverprocessismoreintricatethanundertheFCFSpolicy.However, weestablishunderacertainconditionthisrelationsimpliesasthoserelationundertheFCFS policy.Furthermore,ourresultsindicatesthatthisrelationcouldbestraightforwardinthecase ofthewell-knownShortest-Job-Firstdiscipline. Theproofofthemainresultisbasedonaxed-pointtechniquethatpotentiallycould beapplicableinanalysisofQEDsystemsundervariousservicepolicies.Onecanbreakdown thesourceofcomplexityintheanalysisofmulti-classpriorityqueueingmodeltotwofactors: 1randomnessofservicetimesandarrivalprocessesand2thestructuralcomplexitydueto thefactthatnumberofjobsfromeachclassinthesystem/queueaectsthenumberofjobs fromotherclassesinthesystem/queue.Intheproof,foreachnite-sizesystem,wecreate axedpointsequenceofsystems,convergenttothenite-sizesystemwhichallowsusto considereachfactor,separatelyviaalimitinterchange.Similartotherstpartofthethesis, ouranalysisofrandomnessfactorutilizesthemartingalemethoddevelopedin[58].Onthe otherhand,ineachsystemofourxedpointsequence,thenumberofjobsineachclassinthe system/queueisdeterminedintermsofthenumberofjobsfromotherclassesintheprevious system.Thistechniqueallowsustodealwiththestructuralcomplexity.Sinceonlyanalysis oftherandomnessfactorutilizesthefactthatnumberofclassesisnite,webelievethatour techniquefordealingwithstructuralcomplexitycanbeusedtoanalyzepolicieswherethe numberofclassesisunboundede.g.,theShortest-Job-Firstdiscipline. 13

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Fromapurelytheoreticalpointofview,wedevelopatechniquetoshowsomeweak limitsdonotexistinthe J 1 -metrictopologywhiletheymightexistinothermetrictypologies e.g. M 1 -metrictopology.ThefundamentalideainthistechniqueisbasedonLemmaB.6 whichprovidesarelativelysimplewaytondsomepropertiesofajumpatacertaintime t withoutknowingthecorrespondinglimit.Thislemmaallowsustocomparejumpsinthe prelimitsequencetojumpsinthelimitifthelimitexisted.Inparticular,weassumethatthe limitexistsandusethistechniquetocreateacontradiction,thatiswheneverjumpsinprelimit sequenceandjumpsinlimit,donotmatchwecanconcludethelimitdoesnotexist.This techniqueisusedtojustifyoneofourassumptions.Infact,allassumptionsusedinChapter4 arestandardassumptionsusedinthestudyofqueuesoperatingintheQEDregime,exceptone assumption. 1.3Notations Denoteby D [0, 1 thespaceofallreal-valuedfunctionson [0, 1 thatareright-continuous withleftlimitsr.c.l.l.,endowedwiththestandard J 1 topology.Let D 0 [0, 1 beasubsetof all x 2 D [0, 1 with x 0 .Let D [0, 1 bethesubsetoffunctionsin D 0 [0, 1 thatare nondecreasing. jj standsfortheEuclidianmetricon R ,andtheuniformmetricisdenedby thesupremumnormfor x 2 D [0, 1 and t 0 : k x k t =sup 0 s t j x s j Productmetricspaces D l [0, 1 d l aredenedby D [0, 1 D [0, 1 d d where d d referstothecorrespondingmaximummetric.Forasequenceofvariables f x i g weuse x i : j torepresent P j m = i x m .Thesymbols ^ and denotetheconvolution, minimumandmaximumoperators,respectively.Givenadistributionfunction H H n denotes the n -foldconvolutionof H withitself,i.e., H = H H ; H 1 .Dene x =1 )]TJ/F38 11.9552 Tf 12.665 0 Td [(x x + = x 0 and x )]TJ/F21 11.9552 Tf 11.427 -4.339 Td [(= )]TJ/F21 11.9552 Tf 9.298 0 Td [( x ^ 0 thatis, x = x + )]TJ/F38 11.9552 Tf 12.363 0 Td [(x )]TJ/F17 11.9552 Tf 7.084 -4.339 Td [(.Weuse x y toindicateforany t 0 x t y t .Let denoteweakconvergence{forstochasticprocessesin D l [0, 1 andforrandomvariablesin R l .Furthermore,let 1 fg and betheindicatorfunctionandthe 14

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standardnormaldistribution,respectively.Thecompositionmapisdenotedby ;namely,for x y 2 D [0, 1 D [0, 1 x y isdenedby x y t = x y t t 0 .Wedenethe identityfunctionby e := f e t := t t 0 g .For x 2 D [0, 1 and a b 2 [0, 1 ,weusethe followingnotation: x a b ]= x b )]TJ/F38 11.9552 Tf 11.955 0 Td [(x a x [ a b ]= 8 > > < > > : x b )]TJ/F38 11.9552 Tf 11.955 0 Td [(x a )]TJ/F21 11.9552 Tf 9.298 0 Td [(, b a > 0, x b b a =0, and x [ a b = 8 > > < > > : x b )]TJ/F21 11.9552 Tf 9.299 0 Td [( )]TJ/F38 11.9552 Tf 11.955 0 Td [(x a )]TJ/F21 11.9552 Tf 9.299 0 Td [(, b > a > 0, x b )]TJ/F21 11.9552 Tf 9.299 0 Td [(, b > a =0. Weuse d = toindicateequalityindistribution.Thesymbol f.d. isusedtodenote nite-dimensionalconvergence; P )778(! standsforconvergenceinprobability.Foranondecreasing function f on R ,wedenetheleftcontinuousinverseof f as f x =inf f s : f s x g ; theinmumofanemptysetissetto + 1 .Also,forsequence f x i i 0 g D [0, 1 and x y 2 D [0, 1 ,weused x i x ,as i !1 ,toindicateconvergenceundersupremumnormon compactintervals,i.e.,forany t 0 ,wehave k x i )]TJ/F38 11.9552 Tf 11.955 0 Td [(x k t 0 ,as i !1 15

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CHAPTER2 LITERATUREREVIEW Inthischapterweprovideabriefsurveyoftheliterature.InSection2.1wefocuson studiesintheQEDregime.Section2.2providesanoverviewofresultsonthemachine-repair andsimilarmodels.Finally,inSection2.3,wedescriberesultsrelatedtomulti-classpriority queues. 2.1QEDRegime OriginsoftheQEDregimecanbetracedbacktoErlang[31].Anearlymathematical analysisrelatedtotheregimecanbefoundin[50].TheregimewasformalizedbyHalnand Whittin[43]wheretheyanalyzedasystemwithexponentialservicetimes.Subsequent studiesfocusedonsystemswithphase-type[74],deterministic[52]seealso[84]and discrete[63]distributions.Thecaseofgeneralservicetimeswassolvedin[77]seealso[73]. Analternativetreatmentofthegeneralcasebasedonmeasure-valuedprocessescanbe foundin[54].Amodelthatcoversimpatientcustomersisanalyzedin[25,64].Steady-state distributionsofQEDsystemswasstudiedin[35,36].AsymptoticrenementsoftheQED regimecanbefoundin[51,105].Aspeedofconvergencetoastationarydistributionisa subjectof[34,90].SomeofresultsonthecontrolofvariousqueueingmodelsundertheQED regimeinclude[2,4{6,10,11,41,46,65,89]. 2.2MachineRepairModel ThepapersmentionedinSection2.1examinedopensystems.Severalstudiesexamined opensystemswithsomestate-dependentdynamics.Examplesinclude,service-ratedegradation studiedin[61],state-dependentprocessingtimesin[37]andstate-dependentrouting-policy in[1].Certainclosedsystemscanbeanalyzedbyconsideringopensystemswithstate-dependent arrivalrates[1]. Ithasbeenrecognizedthatretiringcustomershaveimpacttheoverallsystemperformance. Inparticular,inhealthcarecontext,stangandrecourseprovisioningproblemswherepatients canreturn[102,103]andcallcenterswherecustomersmaycallback[19,29,104].Asitwas 16

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demonstratedin[103],methodsthatdonottakeintoaccounttherepetitivenatureofservice suchasthePiecewiseStationaryApproximation[53]mayleadtopoorresultsintime-varying systems. AmachinerepairproblemwithsparesanadditionalfeatureintheEciency-Driven EDregimewasstudiedin[48].Closedmany-serversystemswithstate-dependentdriftsare exploredin[66].In[75],theauthorsconsideraclosedMarkovianmany-serverlosssystem withretrials.Thecaseofgeneralservicetimesiscoveredin[12],whereatransientregime inaclosedsystemisconsidered;inthismodel,eachcustomerreceivesservice"onlyonce andtheanalysisreducestoastudyofaninnite-serverprocess.Examplesofheavy-trac limittheoremsforclosedqueueingsystemswithaxednumberofserverscanbefound in[45,57,59]. Machinerepairmodelwidelyusedinsystemsbasedonmembership/subscription[28], healthcare,professionalandwarrantyservices,andcomputernetworks[27].Insuchsystems thenumberofusersisrelativelyconstantthegoalistondtheadequatestanglevel.A memorylessmachinerepairmodelintheQEDregimewasconsideredin[27].Theauthors analyzedbothtransientprocess-levelconvergenceaswellasconvergenceofsteady-state distributions. Inthehealthcarecontext,onecanthinkofpatientsrecoveringinahospitalbedand requiringassistancefrommedicalpersonnelmultipletimesthroughoutthedaythenumber ofoccupiedbedsisrelativelyconstantduringasingleshiftinsomemedicaldepartments. Outpatientfacilitiesprovidingdental,oncologyanddialysisservicescanmodeledunderthis frameworkaswell,albeitatadierenttimescale.Manyprimarycarephysiciansmanagea nitenumberofpatients[39]panelsize,andthusourmodelcanberelevantinthatsetting aswell.deVericourtandJenningsin[28]usedamachinerepairmodeltodetermineecient nursestangpolicies.Theystudytheperformanceofmedicalunitbasedonprobabilityof excessivedelaytheprobabilitythatdelayisbiggerthansomethreshold.Theydemonstrate thattheratioofthenumberofnursestothenumberofpatientsshouldbesomefunctionof 17

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numberofpatients,unlikesomealreadyexistinglegislationthatmandatesthisratiomustbea constant. 2.3Multi-ClassPriority Multi-classpriorityqueuesareclassicalqueuingmodelswithnumerousapplicants insystemswithneedtostreamthearrivalstodierentchannels[68].Inthecontextof telecommunicationsthesemodels,havebeenusedinthestudyofcellularmobilenetworks[40], ISDNnetworks[83],ATMnetworks[20],broadbandnetworks[69]andetc.Multi-classpriority queuesalsohavebeenusedinpolicepatrolallocation[18].Acasewheremultipledispatchis needed,hasbeenstudiedin[38]. Multi-classpriorityqueueswerealsohavebeenstudiedinthecontextofcontrolled queuingmodels.Typically,thegoalistondtheminimalinsomesensedelay/queue-length costsbyassigningprioritytoclassessee[67,91,92];abandonmentcostswereconsidered aswellsee[7,8].Suchstudiesledaformaljusticationoftheso-called c -ruleandits variants.Diusionanduidlimitsformany-serversMarkovianmulti-classqueuingnetworks wereestablishedin[62].In[9],amulti-classpriorityqueuingsystemwasstudiedwhenthe abandonmentisallowed;serviceandabandonmentdistributionsaregeneral.Theyestablished weakconvergenceresultsunderauidscalingandusedtheseresultstondtheasymptotically optimalpriorityassignmentforlinearabandonmentandqueue-lengthcosts. Somestudiesdevelopedapproximationsforsteady-stateperformanceofmulti-class priorityqueueingsystemswithnon-Poissonarrivalsandnon-exponentialservicetimes. Onecommonlyusedtechnique,developedin[82,95],aggregatesallclassesintoasingle classandonlythenitapproximatetheunderstudyquantitiese.g.,expectedsojourntime bytheparametric-decompositionapproximationmethodrstdevelopedbyReiserand Kobayashiin[78].Whilethistechniqueworksquitewellinmanycases,shortcomingsofthis techniquearewell-studied.Forexampleinthecaseofhighlyvariantarrivalprocess[15]orclass dependentservicetimes[98]thisapproximationisnotworkingproperlyseealso[32,33,87]. 18

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ThistechniquewasexpandedbyWhittin[99]inawaythatitdoesnotaggregateclassesinto oneclass. 19

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CHAPTER3 MACHINEREPAIRMODEL 3.1Overview Inthischapterwestudythemachinerepairmodel.Thechapterisorganizedasfollows. Inthenextsection,wedescribethemodel.OurmainresultscanbefoundinSection3.3. Sections3.4and3.5containtechnicalproofs.Finally,Section3.6providessomenumerical examples. 3.2Model Inthissection,wedescribethemodelandourassumptions.Weconsiderasequence ofmachinerepairmodelsindexedbythenumberofmachines n .Thenumberofrepairmen inthe n thmodelisgivenby k n .Atanyinstantoftime,amachineiseitherworking,or itisbroken.Amachineworksforarandomamountoftimedistributedaccordingtoa distribution F withmean 1 =< 1 F =0 .Oncebroken,amachinerequiresa repairthatlastsarandomamountoftimedistributedaccordingtoadistribution G withmean 1 =< 1 G =0 .Amachineentersarepairphaseassoonasarepairmanisavailable; repairmenservicemachinesaccordingtotheFCFSrule.Forconvenience,dene p = = + asthelong-runfractionoftimeamachinewouldspendbeingbrokenrepairedinthecaseit hadadedicatedrepairman.Workingandrepairtimesareindependentbothacrosstimeand machines/repairmen. Let X n t bethenumberofbrokenmachinesbeingrepairedorawaitingservicein the n thsystemattime t ; X n := f X n t t 0 g .Similarly,welet X n i t beanindicatorof the i thmachinenotworkinginthe n thsystemattime t ;then,wehave X n = n X i =1 X n i {1 where X n i := f X n i t t 0 g .Initially,attime t =0 n machinesarebrokeninthe n th system,i.e., X n = n .Inparticular, n ^ k n machinesarebeingrepairedthosewith indices i =1,..., n ^ k n n )]TJ/F38 11.9552 Tf 12.93 0 Td [(k + machinesareawaitingservicethosewithindices 20

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i = n ^ k n +1,..., n ,and n )]TJ/F24 11.9552 Tf 11.391 0 Td [( n machinesareworkingthosewithindices i = n +1,..., n Weassumethattheremainingworkingrepairtimesofmachinesworkingbeingrepairedat time t =0 areindependentanddistributedaccordingtotheresidualdistributionof F G Recallthattheresidualdistributions F and G aredenedby F x := Z x 0 F s d s and G x := Z x 0 G s d s {2 Now,consideranarbitrarymachine i .Forthismachine,wedenethe j thcycle j 2 to bethetimeperiodbetweenthe j thand j +1 stbreakdowns.Duringthe j thcycle,the i th machinerstawaitsforservicefor w i j 0 timeunits,thenreceivesservicefor s i j timeunits, andnallyremainsoperationalfor a i j timeunits. Fornotationalconvenience,weadopttheconventionthattherstcycleformachines workingattime t =0 n < i n consistsoftheirremainingworkingtimes a i ,1 is distributedaccordingto F w i ,1 0 s i ,1 0 ;formachinesbeingrepairedattime t =0 1 i n ^ k n ,therstcycleconsistsoftherestoftherepairtimeandthefollowing workingperiod s i ,1 isdistributedaccordingto G a i ,1 accordingto F w i ,1 0 ;therstcycle formachinesawaitingrepairattime t =0 n ^ k n < i n isthetimeperiodbetween t =0 andtheirrstbreakdownthatoccursafter t =0 s 1, i and a i ,1 aredistributedaccordingto G and F ,respectively.Finally,let c i j = s i j + a i j ,andnotethat f c i j g j 2 havethedistribution H = G F .Then, X n i canbewrittenasfollows: X n i t = 1 X j =0 1 f 0 t )]TJ/F39 7.9701 Tf 6.586 0 Td [(c i ,1: j )]TJ/F39 7.9701 Tf 6.587 0 Td [(w i ,1: j < w i j +1 + s i j +1 g ; {3 recallthat c i ,1: j := P j m =1 c i m and w i ,1: j := P j m =1 w i m WeconsiderasequenceofmachinerepairmodelsintheQEDregime.Insucharegime, thecapacityandtheoeredloadarerelatedviathesquare-rootrule: n := k n )]TJ/F38 11.9552 Tf 11.955 0 Td [(np p np p {4 21

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as n !1 ,forsome 2 1 ;recallthat p =1 )]TJ/F38 11.9552 Tf 12.147 0 Td [(p .Analogoustoopensystemsinthe QEDregime,thecapacityofthesystemisapproximatelyequaltotheoeredload.Thescaled totalnumberofbrokenmachinesisgivenby ^ X n := X n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n p np p {5 Remark 3.1 Typically,thescalingparameterusedforQEDregimeis p k n ratherthan p np p ; p n wasusedin[27].However, k n n and np p areofthesameorderofmagnitude,and np p ismoreconvenientintermsofstatingourresults.Inparticular,underthe p np p scaling, thestationarycomponentofthecorrespondinginnite-repairmenprocesshasunitvariance seeLemma3.1inSection3.3.1.Inaddition,suchascalingallowsonetoconsidermore generalsettings,where p or p vanishwhile n increases.Forexample,inthemodeldescribed inthissection,thedistributions F and G donotvarywiththesizeofthesystem n ;this isastandardassumption,e.g.,see[27].Consequently,theparameter p isxed.However,in someapplications,thevalueof p canbecloseto0,whiletheproduct np islargeforexample, p =0.001 and n =50000 .Forsuchcases,itisofinteresttoconsideramodelwhere p varies with n .WediscussoneaspectofsuchamodelinAppendix ?? WeassumethatthesystemisintheQEDregimeattime t =0 sincewefocuson diusionratherthanuidlimits.Inparticular,thenumberofbrokenmachinesattime t =0 satises,as n !1 ^ n := n )]TJ/F38 11.9552 Tf 11.956 0 Td [(k n p np p ^ {6 Notethatthislimitandtheassumptiononthedistributionofresidualrepairandworkingtimes at t =0 see3{2specifythestateofthesystemattime t =0 .Thesetworelativelysimple assumptionsensurethattheoursystemremainsintheQEDregimeonnitetimeintervals theweaklimitof ^ X n isnon-degenerate.Intheanalysis[77]oftheQEDGI/GI/ k queue,the openanalogueofourmodel,thesameassumption G onresidualservicerepairtimeswas used.Analternativewaytospecifythesystemat t =0 wouldbetoinitializeitinitssteady state.However,thelimitingQEDsteady-statedistributionforthemachinerepairmodelis 22

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notknown;moreover,eventhesteady-statedistributionfortheQEDGI/GI/ k queueisnot knownonlypartialresultsexists,e.g.,see[35]. 3.3Results 3.3.1Mainresults Aninnite-repairmenprocessiscentralinouranalysis.Wedene Z n i := f Z n i t t 0 g tobeanindicatorprocessofwhethermachine i isbrokeninthecasewhenadedicated repairmanisalwaysavailabletoit.Intermsofearlierintroducednotation,onehas,for t 0 Z n i t = 1 X j =0 1 f 0 t )]TJ/F39 7.9701 Tf 6.586 0 Td [(c i ,1: j < s i j +1 g {7 Wenotethatboth X n i and Z n i describethebehaviorofthesamemachine i amachine isdescribedbytwosequencesofworkingandrepairtimes.Indeed,theright-handside of3{7isobtainedfromtheright-handsideof4{36byeliminatingwaitingtimes f w i j g .In particular,amachine i n ^ k n < i n ,iswaitingrepairattime t =0 intheoriginalmodel X n i ,whileitrepairstartsat t =0 intheinnite-repairmenmodel Z n i Severalfunctionscanbeassociatedwiththeabovedenition3{7.Inparticular,let P 1 t t 0 ,betheconditionalprobabilityofamachinebeingbrokenattime t ,giventhata repairprocessofthemachinestartedattime t =0 .Then, P 1 := f P 1 t t 0 g isgivenby P 1 = R )]TJ/F38 11.9552 Tf 11.955 0 Td [(G R = G R {8 where R istherenewalfunctionassociatedwiththedistribution H ,i.e., R =1+ H + H H + e.g.,see[22,p.286].Similarly,let P 1 t and P 0 t betheconditionalprobabilitiesofa machinebeingbrokenattime t ,giventhatitisbeingrepairedat t =0 withthedistribution oftheremainingrepairtimeequalto G anditisworkingat t =0 withthedistributionof theremainingworkingtimeequalto F ,respectively.Consequently,if P 1 := f P 1 t t 0 g 23

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and P 0 := f P 0 t t 0 g ,onehas P 1 =1+ G F R )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G R =1 )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G F R {9 P 0 = F R )]TJ/F21 11.9552 Tf 13.174 2.657 Td [( F G R = F G R {10 Wenotethat,duetotheconstructionofstationaryon-oprocessesin[47], p P 1 + p P 0 = p holds.Moreover,ifthedistribution H isnonlattice,than P 1 t p ,as t !1 [3,p.173];the sameappliesto P 1 and P 0 [79,p.237].Finally,basedontheprecedingandtheindexingofthe machinesseeSection4.2.1,itfollowsthat E Z n i t = 8 > > > > > > < > > > > > > : P 1 t ,1 i n ^ k n P 1 t n ^ k n < i n P 0 t n < i n {11 Notethat E Z n i t dependsonthemachineindex i ,since i determinesthestateofamachine attime t =0 intheoriginalnite-repairmensystem:machines 1 i n ^ k n arebeing repaired,machines n ^ k n < i n areawaitingrepair,andmachines n < i n areworking. Thenextlemmadescribesthelimitingbehaviorofacenteredandscaledversionofthe innite-repairmenprocess Z n := f Z n t t 0 g := n X i =1 Z n i {12 Lemma3.1 Innite-repairmenprocess Suppose limsup t # 0 F t G t t < 1 {13 Then,as n !1 ^ Z n Z n )]TJ/F38 11.9552 Tf 11.955 0 Td [(np p np p ^ Z ^ Y + ^ + )]TJ/F21 11.9552 Tf 13.891 2.657 Td [(^ Y P 1 )]TJ/F21 11.9552 Tf 13.239 2.657 Td [( P 0 + ^ + P 1 )]TJ/F21 11.9552 Tf 13.239 2.657 Td [( P 1 {14 24

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where ^ and ^ Y := f ^ Y t t 0 g areindependent,and ^ Y isastationarycenteredGaussian processwiththecovariancefunction E ^ Y t ^ Y t + s = P 1 s )]TJ/F38 11.9552 Tf 11.955 0 Td [(p 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(p t s 0. {15 Proof. SeeSection3.5.1. Thestatementofthelemmaisconsistentwith ^ Z = ^ + ,since P 1 = P 1 =1 and P 0 =0 .Theterm ^ + )]TJ/F21 11.9552 Tf 14.087 2.656 Td [(^ Y P 1 )]TJ/F21 11.9552 Tf 13.435 2.657 Td [( P 0 stemsfromthedierencebetween ^ Z and ^ Y {seeRemark3.3fordetails.Theterm ^ + P 1 )]TJ/F21 11.9552 Tf 13.586 2.657 Td [( P 1 isdueto n )]TJ/F38 11.9552 Tf 12.302 0 Td [(k n machines awaitingrepairat t =0 {theirremainingrepairtimesat t =0 aredistributedaccordingto G ratherthan G forexponential G ,onehas P 1 = P 1 Remark 3.2 Thecondition3{13isneededforapplicationofHahn'stheorem[42],acentral limittheoremfor D [0,1] variablesstationaryon-oprocessesinourcontext.Inparticular, 3{13ensuresthataconditionofHahn'stheoremaresatised[88]seealso[26].Inthe caseofrenewalcountingprocessesratherthanalternatingrenewalprocesses,itwasshown in[97]thatananalogousconditionisnecessaryfortheconditionofHahn'stheoremtohold seealso[100,p.229].Suchaconditiondoesnotappearin[77],whereprocess-levelweak convergencewasconsideredforanopenQEDanalogueofourmodelG/GI/Nmodel{there, itwasassumedthatasequenceofexogenousarrivalprocessessatisesaFCLT.Interestingly, condition3{13fortheservicetimedistributionappearsinthestudy[35]ofthelimiting steady-state distributionoftheGI/GI/Nsystem. Remark 3.3Initialconditions Althoughtheinitialnumberofbrokenmachinescanbe describedbyasinglerandomvariable n ,thereexistsomebenetsindescribingtheinitial statewithtworandomvariables.Namely,suchachoiceallowsforamoreintuitivestatement ofthelemma.Suppose n = n + n ,where n isabinomialrandomvariablewithparameters n p .Thevariables n and n donothavetobeindependent;however,theyareindependent fromallothervariablesinthesystem.Forexample, n = k n )]TJ/F24 11.9552 Tf 12.306 0 Td [( n leadsto n = k n .Ifthe 25

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numberofbrokenmachinesattime t =0 satises,as n !1 ^ n ,^ n := n )]TJ/F38 11.9552 Tf 11.955 0 Td [(np p np p ,^ n := n p np p ^ =^ +^ )]TJ/F24 11.9552 Tf 11.955 0 Td [( ,^ ,^ where ^ isastandardnormalrandomvariable,then ^ Z n ^ Y +^ P 1 )]TJ/F21 11.9552 Tf 13.24 2.657 Td [( P 0 + ^ + P 1 )]TJ/F21 11.9552 Tf 13.24 2.657 Td [( P 1 as n !1 ,where ^ Y =^ inadditionto3{15.Here,theterm ^ P 1 )]TJ/F21 11.9552 Tf 13.961 2.657 Td [( P 0 explicitly capturesthedierenceinthenumberofbrokenmachinesattime t =0 andthesamenumber inthecorrespondingstationaryinnite-repairmenprocess.Ingeneral,givenadistributionof ^ ,thedistributionof ^ ,^ isnotunique.Forexample,both ^ ,^ =^ and ^ ,^ = ^ )]TJ/F21 11.9552 Tf 11.955 0 Td [(2^ yieldastandardnormal ^ Thesecondcomponentrequiredtostateourmainresultisthenonlinearconvolution operatordenedbelow.Theoperator G playsacrucialruleindescribingthebehaviorofopen QEDsystemswiththeservicedistribution G Denition3.1 Reed[77] Themapping P : D [0, 1 D [0, 1 issuchthat P x ,for each x 2 D [0, 1 ,istheuniquesolution y to y t = x t + Z t 0 y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d P s {16 Proposition3.1. Foreach x 2 D [0, 1 ,thereexistsaunique P 1 x .Thefunction P 1 : D [0, 1 D [0, 1 isLipschitzcontinuousinthetopologyofuniformconvergence overboundedintervals,anditismeasurablewithrespecttotheBorel -eldgeneratedbythe Skorokhod J 1 topology. Proof. SeeSection3.5.2. Thefollowingtheoremisthemainresultofthepaper.Itdescribesthelimitingnumberof brokenmachinesprocessintermsofthecorrespondinglimitinginnite-repairmanprocess ^ Z seeLemma3.1.Wenotethat,ingeneral, P 1 isnotamonotonicfunction.Thisreectsthe factthat ^ Z doesnotserveasalowerboundfor ^ X 26

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Theorem3.1. ConsiderasequenceofmachinerepairmodelsintheQEDregime.Ifthe sequenceofscaledandcenteredinnite-repairmenprocessesconvergesweakly ^ Z n ^ Z ,as n !1 ,then,as n !1 ^ X n ^ X P 1 ^ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( {17 Proof. SeeSection3.4. Remark 3.4Opensystem Let X k := f X k t t 0 g bethenumber-in-systemprocessin anopensystemwith k servers,Poissonarrivalsandthesameservicedistribution G .Here,we indexsystemswiththenumberofservers,sincethenumberofcustomersmachinesisnot xed.Thearrivalrate k inthe k -serversystemissuchthatthesequenceofsystemsisinthe QEDregime: k )]TJ/F24 11.9552 Tf 11.976 0 Td [( k = p k ,as k !1 .Theinitialnumberofcustomersinthesystem k satises k )]TJ/F38 11.9552 Tf 11.955 0 Td [(k = p k ,as k !1 .Themainresultin[77]implies,as k !1 X k )]TJ/F38 11.9552 Tf 11.955 0 Td [(k p k X G Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( {18 where Z = Y + + )]TJ/F21 11.9552 Tf 13.891 2.657 Td [( Y G + + G )]TJ/F21 11.9552 Tf 13.709 4.783 Td [( G {19 and Y := f Y t t 0 g isastationarycenteredGaussianprocessindependentof with E Y t Y t + s =1 )]TJ/F21 11.9552 Tf 13.829 2.657 Td [( G s t s 0 .Compare3{19to3{14.Process Y isastationary limitingcenteredandscaledM/G/ 1 processdenedbytheservicedistribution G .The resultin[77]isstatedintermsofatime-changedBrownianbridgeandanon-stationary limitingcenteredandscaledinnite-serverprocess;thosetwotermscorrespondtocustomers initiallyattime t =0 presentinthesystemandthosearrivingafter t =0 ,respectively. However,consideringallcustomerstogetherresultsin3{18and3{19. Wenotethatcomparison ^ X and X via3{17and3{18isnotstraightforward,in general.Thereasonisthatthedierencein ^ X and X isnotonlyduetothedierentnonlinear operators G and P 1 ,butdierentcorrelationstructuresof ^ Z and Z orratherdierent covariancefunctionsof ^ Y and Y aswell. 27

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Let W n := f W n t t 0 g bethevirtualwaitingtimeprocessinthesystemwith n machines.Thatis, W n t isthewaitingtimethatamachinewouldexperienceifitwereto breakattime t ,evenifsuchabreakdownisimpossibletooccur[27]i.e.,allmachinesare brokenattime t ; X n t = n .Itfollowsthat f X n t k n g and f W n t > 0 g areequivalent events.Ascaledversionof W n isdenedby ^ W n := f ^ W n t t 0 g = p np pW n .Acorollary characterizingthelimitingvirtualwaitingtimeprocessfollowsfromTheorem3.1and[72]. Corollary3.1. ConsiderasequenceofmachinerepairmodelsintheQEDregime.Ifthe sequenceofscaledandcenteredinnite-repairmenprocessesconvergesweakly ^ Z n ^ Z ,as n !1 ,then,as n !1 ^ W n 1 + ^ X + Next,wediscussanexamplewhereexplicitresultscanbeobtained. Example 3.1Memorylesssystem Assume G t = e )]TJ/F25 7.9701 Tf 6.586 0 Td [( t and F t = e )]TJ/F25 7.9701 Tf 6.586 0 Td [( t for t 0 .Inthis case, P 1 t = p + pe )]TJ/F22 7.9701 Tf 6.586 0 Td [( + t ,for t 0 e.g.,see[80,p.243],anditfollowsthat ^ X satises, for t 0 ^ X t = ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( + Z t 0 ^ X + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s e )]TJ/F22 7.9701 Tf 6.586 0 Td [( + s d s {20 where ^ Z t = ^ Y t + ^ + )]TJ/F21 11.9552 Tf 13.233 2.657 Td [(^ Y e )]TJ/F22 7.9701 Tf 6.586 0 Td [( + t and ^ Y isastationarycenteredGaussianprocess independentof ^ with E ^ Y t ^ Y t + s = e )]TJ/F22 7.9701 Tf 6.587 0 Td [( + s t s 0 .Alternatively, ^ Z ischaracterized bye.g.,see[17] d ^ Z t = )]TJ/F21 11.9552 Tf 9.298 0 Td [( + ^ Z t d t + p 2 + d B t ; {21 here, B := f B t t 0 g isastandardBrownianmotion.Relation3{20implies d ^ X t =d ^ Z t + ^ X + t d t )]TJ/F21 11.9552 Tf 11.955 0 Td [( + ^ X t )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z t + d t whichcombinedwith3{21yields d ^ X t = ^ X + t )]TJ/F21 11.9552 Tf 11.955 0 Td [( + ^ X t + d t + p 2 + d B t {22 Thatis, ^ X isadiusion,asearlierobtainedin[27,Theorem2]. 28

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BasedonRemark3.4,thelimitingnumberofcustomers X := f X t t 0 g inthe correspondingthesameinitialconditions: = ^ openQEDsystemsatises X = G Z )]TJ/F24 11.9552 Tf 12.004 0 Td [( where Z t = Y t + ^ + )]TJ/F21 11.9552 Tf 14.04 2.657 Td [( Y e )]TJ/F25 7.9701 Tf 6.586 0 Td [( t and Y := f Y t t 0 g isastationarycentered Gaussianprocessindependentof ^ with E Y t Y t + s = e )]TJ/F25 7.9701 Tf 6.586 0 Td [( s t s 0 .Deninga versionof X withatimespeedup ~ X := f ~ X t = X t = p t 0 g yieldsthat ~ X isequalin distributionto G = p ^ Z )]TJ/F24 11.9552 Tf 12.495 0 Td [( ,since E Y t = p Y t = p + s = p = E ^ Y t ^ Y t + s .Finally,we observe G = p ^ Z )]TJ/F24 11.9552 Tf 11.257 0 Td [( t P 1 ^ Z )]TJ/F24 11.9552 Tf 11.257 0 Td [( t ,whichimpliesthat ~ X t isstochasticallylargerthan ^ X t ,forany t > 0 .Weremarkthatthisisnottheonlywaytocomparetheclosedandopen systems.ComparisonoftheM/M/ k n / 1 / n machinerepairmodelwithmemorylessrepairand workingtimesandM/M/ k systemswiththesameservice/repairdistribution,andthesame stangparameter intheQEDregimewasconsideredin[27].Inparticular,theauthors establishedthatthestationaryprobabilityofdelayissmallerintheclosedsystemthaninthe opensystem. 3.3.2Limitinterchange Inthissubsection,wediscussarelationbetweenclosedandopensystemsintheQED regimeviaaPoissonlimitthelawofsmallnumbers.Undersuchalimit,thenumberof machinescustomersisincreasing,whiletheoeredloadandtheservicedistributionare keptconstant.WearguethatthePoissonandQEDlimitsareinterchangeable,asillustrated inFigure3-1.Tothisend,consideramachinerepairmodelwith k repairmen,therepair distribution G andtheworkingtimedistribution F m = F = m ,where G and F areasin theoriginalmodelSection4.2.1.Notethatinthisnewsystemtheaveragetimeamachine spendsworkingwithoutbreakingdownis 1 = m = m = ;dene p m := m m + = + m astheprobabilitycharacterizingasinglemachine.Let P m = R m )]TJ/F38 11.9552 Tf 12.687 0 Td [(G R m ,where R m is therenewalfunctionassociatedwiththedistribution G F m .Asequenceofnewmodelsis indexedbyapair k m .Thenumberofmachinesinthe k m modelisdenotedby n k m and 29

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Finitesystem k !1 QEDregime Closedsystem ^ X k m Theorem 3.1 P m 1 ^ Z 1 m )]TJ/F24 11.9552 Tf 11.955 0 Td [( m !1+ [21,93]Lemma3.2 + Opensystem ^ X k 1 [77] G ^ Z 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( Figure3-1.AcommutativediagramillustratingtheinterchangeofPoisson m !1 andQED k !1 limits. satises,forall m k )]TJ/F38 11.9552 Tf 11.955 0 Td [(n k m p m p n k m p m p m =: k {23 as k !1 .Dene X k m and Z k m tobetheprocessesofthenumberofbrokenmachinesin the k m systemandthecorrespondinginnite-repairmensystem;thescaledprocessesare denedas ^ X k m := X k m )]TJ/F38 11.9552 Tf 11.956 0 Td [(k p n k m p m p m and ^ Z k m := Z k m )]TJ/F38 11.9552 Tf 11.955 0 Td [(n k m p m p n k m p m p m Thenumberofbrokenmachinesattime t =0 inthe k m systemis k m .Residual repairtimesofmachinesbeingrepairedattime t =0 arei.i.d.withthedistribution G .The sequenceofrandomvariables f k m k m 2 Z + g satises,as k !1 k m )]TJ/F38 11.9552 Tf 11.955 0 Td [(k p n k m p m p m ^ 1 m and ^ 1 m ^ 1 ,as m !1 .Inaddition,onehas k m k 1 ,as m !1 ,and k 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(k p k ^ 1 as k !1 .Thisensuresthatthelimitinterchangeholdsforthesequenceofinitialconditions. Twolimitscanbeobtainedforatwo-dimensionalsequence f ^ X k m g seeFigure3-1. First,Theorem3.1impliesundertheassumption ^ Z k m ^ Z 1 m ,as k !1 ;seeLemma3.1 forasucientcondition,as k !1 ^ X k m P m 1 ^ Z 1 m )]TJ/F24 11.9552 Tf 11.955 0 Td [( 30

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where,accordingtoLemma3.1, ^ Z 1 m = ^ Y 1 m + ^ 1 m + )]TJ/F21 11.9552 Tf 13.891 2.657 Td [(^ Y 1 m P m 1 )]TJ/F21 11.9552 Tf 13.24 2.657 Td [( P m 0 + ^ 1 m + P m 1 )]TJ/F21 11.9552 Tf 13.239 2.657 Td [( P m 1 {24 and ^ Y 1 m := f ^ Y 1 m t t 0 g isastationarycenteredGaussianprocesswiththecovariance function P m 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(p m = p m .Second,dueto[21]and[93],as m !1 ^ X k m ^ X k 1 X k 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(k p k where X k 1 isthenumber-in-systemprocessforanopensystemwith k servers,theservice distribution G andPoissonarrivalswithrate k {dueto3{23, k solves k = k + k p k thenumberofcustomersinthesystemat t =0 is k 1 ;residualservicetimesof k 1 ^ k customersinservicearei.i.d.withthedistribution G .Wenotethat k )]TJ/F21 11.9552 Tf 14.916 2.656 Td [( G isthe covariancefunctionofthecorrespondingunscaledstationaryM/G/ 1 process. Now,consideringaQEDlimitforthesequenceofopensystemsyieldsseeRemark3.4, as k !1 ^ X k 1 G ^ Z 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( where ^ Z 1 = ^ Y 1 + ^ 1 + )]TJ/F21 11.9552 Tf 13.891 2.657 Td [(^ Y 1 G + ^ 1 + G )]TJ/F21 11.9552 Tf 13.709 4.782 Td [( G {25 and ^ Y 1 := f ^ Y 1 t t 0 g isastationarycenteredGaussianprocesswith E ^ Y 1 t ^ Y 1 t + s =1 )]TJ/F21 11.9552 Tf 13.807 2.656 Td [( G s t s 0 .Finally,thefollowinglemmajustiesthePoissonlimitforasequence ofclosedQEDsystems. Lemma3.2. Wehave P m 1 ^ Z 1 m )]TJ/F24 11.9552 Tf 11.955 0 Td [( G ^ Z 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( ,as m !1 Proof. SeeSection3.5.3. Remark 3.5 ThediagraminFigure3-1providesajusticationforapproximatingmachine repairmodelintheQEDregimewithitsopenanalogueinthesameheavy-tracregime. Performanceofamachinerepairmenmodelwithmanyrepairmeninanitetimeinterval [0, t ] approachesperformanceofthecorrespondingopenmodelM/G/ k n asthenumberof 31

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machinesincreaseswhiletheloadiskeptconstant.Informally,ourresultindicatethatsuch anapproximationmightbeappropriatewhenthefollowingquantityaresmall":i k P 1 )]TJ/F38 11.9552 Tf 12.02 0 Td [(G k t compare3{17and3{18;ii k P 1 )]TJ/F21 11.9552 Tf 11.573 2.657 Td [( P 0 )]TJ/F21 11.9552 Tf 10.288 0 Td [(1+ G k t and k P 1 )]TJ/F21 11.9552 Tf 11.573 2.657 Td [( P 1 )]TJ/F21 11.9552 Tf 12.043 2.657 Td [( G )]TJ/F38 11.9552 Tf 10.289 0 Td [(G k t compare3{14 and3{19;iii k P 1 )]TJ/F38 11.9552 Tf 12.391 0 Td [(p = p )]TJ/F21 11.9552 Tf 12.391 0 Td [(1+ G k t comparethecovariancefunctionsfor ^ Y and Y Theseconditionsaresatisedwhen = 1 t 1 and H t 1 3.4ProofofTheorem3.1 Intherstsubsectionweprovideanoutlineoftheproof.Westateandprovesome preliminaryresultsinthesecondsubsection.TheproofofTheorem3.1canbefoundinthelast subsection. 3.4.1Outlineoftheproof Wedescribetheproofintwoparts.Attheendofthesubsection,weprovidesome intuitionbehindourresult. PartI:Decomposition. Theorem3.1relatesthenumber-of-broken-machinesprocess X n andthecorrespondinginnite-repairmenprocess Z n ,inthelimitas n !1 .Inorder toquantify X n )]TJ/F38 11.9552 Tf 12.784 0 Td [(Z n ,weconsidersuchdierencesforindividualmachines,i.e., X n i )]TJ/F38 11.9552 Tf 12.784 0 Td [(Z n i Recallthat Z n i isanon-oprocesswithindependentonandoperiods.While X n i isan on-oprocessaswell,itsstructureismoreintricate.Inparticular,anywaitforrepairaftera breakdowncausesatimeshiftinallfuturetimeswhenthemachinestopsandstartsworking. WeillustratethisphenomenoninFigure3-2,whereweplot X n i Z n i andrelatedprocesses foramachine i n ^ k n < i n amachinethatisawaitingrepairat t =0 ;forother initialconditionsthegureissimilar.Asstatedearlier, Z n i doesnotprovidealowerbound for X n i ; X n i )]TJ/F38 11.9552 Tf 12.401 0 Td [(Z n i t takesvaluesin f)]TJ/F21 11.9552 Tf 15.276 0 Td [(1,0,1 g .Thedierencein X n i and Z n i isduetothe sequenceofmachine i waitingtimes f w i j g j 1 .Sincethestructureof X n i )]TJ/F38 11.9552 Tf 12.625 0 Td [(Z n i isinvolved, wedecomposethisprocessintoprocessesbasedonthecauseofdiscrepancy.Tothisend,we dene n i j = f n i j t t 0 g j 0 ,toaccountfordierencesin X n i and Z n i duetothe 32

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waitingtime w i j +1 seeFigure3-2,suchthat X n i )]TJ/F38 11.9552 Tf 11.955 0 Td [(Z n i = 1 X j =0 n i j Thethreemainfeaturesof n i j areilengthofall +1 and )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 periodsareequalto w i j +1 ; iibeginningof +1 and )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 periodsaredeterminedby f s i j + l g l 1 f a i j + l g l 1 andthestart" time i j := c i ,1: j + w i ,1: j ;and n i j does not dependon f w i l g l > j +1 .Next,wefurtherdecompose n l j itselfintermsofasdierenceofone-point"pointprocesses: n i j = X L 2f H G g 1 X l =0 )]TJ/F21 11.9552 Tf 9.298 0 Td [(1 1 f L = H g # n i j l L )]TJ/F21 11.9552 Tf 13.242 2.657 Td [(~ # n i j l L ; here, # n i j l L := f # n i j l L ; t t 0 g and ~ # n i j l L := f ~ # n i j l L ; t t 0 g .InFigure3-2we illustratethedecompositionof n i ,0 .Wenotethatone-pointpointprocessesarewell-studied objectsforexample,compensatorsforsuchprocessesarewellknown{see[49,p.98]. Thenextstepistocenterandscaleappropriatelyrelevantprocesses.Inparticular,we denesee3{29 M n l j L = 1 p n n X i =1 # n i j l L )]TJ/F35 11.9552 Tf 11.956 0 Td [(E [ # n i j l L j i j ] {26 where L 2f G H g ; ~ M n l j L isdenedanalogously{replace with ~ .Notethataconditional expectationisusedforcenteringintheprecedingequation,sincethesame i j isusedinthe denitionof f M l j L g l 0, L 2f G H g equivalently,thispreserves"thestructureof n i j ;recall that i j isthestart"timeof n i j .Capturingthisdependencyiscrucial.Then,onehas 1 p n n X i =1 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [( n i j )]TJ/F35 11.9552 Tf 11.955 0 Td [(E [ n i j j i j ,~ i j ] = X L 2f G H g 1 X l =0 )]TJ/F21 11.9552 Tf 9.298 0 Td [(1 1 f L = H g M n l j L )]TJ/F21 11.9552 Tf 14.565 2.656 Td [(~ M n l j L ; observethattheconditioningon i j ,~ i j intheprecedingequalityisequivalenttoconditioning onthestart"timeandthedurationof +1 and )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 periodsof n i j ,sincethatdurationis w i j +1 =~ i j )]TJ/F21 11.9552 Tf 12.352 0 Td [( i j seeFigure3-2. 33

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Finally,byusingtheabove, ^ X n see3{5canbewrittenintermsof ^ Z n see3{14by meansof P 1 seeDenition4.2and3{8: ^ X n = P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n + ^ n where n isdenedin3{4,and ^ n = 1 p p p X L 2f H G g 1 X l j =0 )]TJ/F21 11.9552 Tf 9.298 0 Td [(1 1 f L = H g M n l j L )]TJ/F21 11.9552 Tf 14.565 2.657 Td [(~ M n l j L {27 Wenotethat f M n l j L g l j L and f ~ M n l j L g l j L aretwosequenceswithdependentelements; moreover,thetwosequencesarenotmutuallyindependent. PartII:Convergence. Inviewofcontinuityof P 1 Proposition3.1, ^ Z n ^ Z Lemma3.1and n see3{4,therestoftheproofaddresses ^ n 0 Lemma3.6. Themainideaistoshowthat M n j l L )]TJ/F21 11.9552 Tf 14.153 2.656 Td [(~ M n j l L vanishesforxed i and j 1 Lemma3.3,as n !1 ,byarguing M n l j L ~ M n l j L M l j L M l j L where M l j L isanexplicitlycharacterizedGaussianprocesswitha.s.continuoussamplepaths. Thecase j =0 ishandledinLemma3.5{itisslightlydierentduetoinitialconditionsof machinesattime t =0 .Theremainingcaselarge l + j isaddressedinLemma3.4.There, insteadofconsideringjointconvergenceof M n l j L and ~ M n l j L ,weboundeachtermseparately basedonamartingaledecompositiondescribedbelow. Next,wediscusstheproofofLemma3.3indetail.Thestartingpointisthefactthat nite-dimensionalconvergenceandtightnessimplyweakconvergence,e.g.,see[14,p.139]. Thenite-dimensionalconvergenceisobtainedbyutilizingatechniquefrom[58]:stochastic integralsareapproximatedbynitesumsuidlimitsfromLemma3.10areutilizedthereas well;inturn,thoselimitsarebasedonuidlimitsinCorollary3.2andLemma3.9.Aldous' sucientcriterion[60,p.515]isthemaintoolforestablishingtightnessLemmaB.3and RemarkB.2inAppendixB.1.2.Onefeatureofourproofisthattightnessfor M n l j and ~ M n l j isestablishedrecursively,whichisduetoamartingaledecompositionseeLemmaB.1in 34

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Figure3-2.Anillustrationofthedecompositionfor X n i )]TJ/F38 11.9552 Tf 11.955 0 Td [(Z n i X n i t and Z n i t areindicators ofmachine i beingbrokenattime t inourmachinerepairmodelandthe correspondinginnite-repairmenmodel,respectively.Thisdierenceisrepresented asasumofprocesses f n i j g j 0 .Eachofthoseprocesses,inturn,isdecomposed intoone-pointpointprocesses f # i j l L g l 0 and f ~ # i j l L g l 0 L 2f G H g .Only thedecompositionfor i ,0 isshown. AppendixB.1.1;themartingaletermischaracterizedinLemmaB.2.Thebasicideabehind thispartofouranalysisistodeneanaloguesthemaintechnicaldicultyofarrivalprocesses intheopensystemsothatmachinerydevelopedin[58]isapplicable.Forthisreason,therst twotermsinthemartingaledecompositionof M n l j areofthesameformasthetermsthat 35

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appearintheanalysisoftheopensysteminourcasearrivals"aretime-varying.However, thedecompositionhasarecursivetermthatinvolves M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L ,where f L g = f G H gnf L g Thisisduetooursystembeingclosed{suchastructuredoesnotappearinanopensystem. Infact,theanalogueof ^ n foranopensystemcontainsasingledierenceratherthan innitelymanyasin3{27.Whileinanopensystemsarrivals"docorrespondtophysical arrivalsofcustomerstotheservicefacility,analoguesofarrivals"forourmodeldonot correspondtoarrivalsinthephysicalsystemeitherbreakdownsortimeswhenmachines startworking.Rather,arrivals"inouranalysisstemfromadecompositionofpointsthat deneone-pointpointprocesses # i j l L and ~ # i j l L .Sucharepresentationisneededto obtainorthogonalityofmartingalesthatcorrespondtoindividualmachinesseeRemarkB.1in AppendixB.1.1. Now,wedescribetheintuitionbehindourresult.Here,wedrawaparallelwithanopen system.There, X k t )]TJ/F21 11.9552 Tf 14.516 2.657 Td [( Z k t representsthenumberofcustomersthatarestillinthe nite-serversystem,butnotintheinnite-serverone,attime t Z k = f Z k t t 0 g isthe correspondingunscaledinnite-serverprocess.Allthosecustomersweredelayedupontheir arrival,i.e.,theywereawaitingserviceatsomepointintheinterval [0, t ] .Considercustomers awaitingserviceattime s 2 [0, t {thosecustomerarrivedtothesystemsattime s ,since waitingtimesvanishintheQEDregime.Hence,outofthose X s )]TJ/F38 11.9552 Tf 12.531 0 Td [(k + customers,only customerswithservicetimes t )]TJ/F38 11.9552 Tf 12.143 0 Td [(s areinthegroupof X k t )]TJ/F21 11.9552 Tf 13.548 2.657 Td [( Z k t customersthatare stillinthenite-serversystem,butnotintheinnite-serverone,attime t .Thisleadsto X k t )]TJ/F21 11.9552 Tf 13.36 2.657 Td [( Z k t Z t 0 X k t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(k + d G s ; here,wetookadvantageofthefactthat X s )]TJ/F38 11.9552 Tf 12.211 0 Td [(k + islargeproportionalto p k ,implying thatthefractionofjobswithservicetimes t )]TJ/F38 11.9552 Tf 12.741 0 Td [(s is d G t )]TJ/F38 11.9552 Tf 12.742 0 Td [(s .Theintuitionforthe closedsystemissimilar,yetoneneedstoaccountformorescenarios.Amachineawaiting repairattime s 2 [0, t cancauseadiscrepancyin X n t and Z n t inmultipleways.Namely, suchamachinecontributestothepositivepartof X n t )]TJ/F38 11.9552 Tf 12.323 0 Td [(Z n t ,ifitsnextrepairtime,say 36

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s i j ,is t )]TJ/F38 11.9552 Tf 12.682 0 Td [(s likeintheopensystem,butalsoif c i j : m + s i m +1 t )]TJ/F38 11.9552 Tf 12.683 0 Td [(s ,forsome m j .Thisreectsthefactthatamachinecancompletemultiplecyclesinthetimeinterval [ s t ] .Similarly,thissamemachinecontributestothenegativepartof X n t )]TJ/F38 11.9552 Tf 12.896 0 Td [(Z n t ,if c i j : m t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s ,forsome m j seeFigure3-2.Consequently,onehassee3{8 X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(Z n t 1 X m =0 Z t 0 X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + d G H m s )]TJ/F27 7.9701 Tf 16.745 14.944 Td [(1 X m =1 Z t 0 X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + d H m s = Z t 0 X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + d P 1 s whichaftercenteringandscalingyields ^ X n P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n 3.4.2Preliminaryresults Fornotationalsimplicity,let i j := c i ,1: j + w i ,1: j and ~ i j := c i ,1: j + w i ,1: j +1 ; {28 thesetwoquantitiescanbeinterpretedasthetimeofthe j thbreakdownandthetimewhen the j threpairstartsforthe i thmachine,respectively;for j =0 ,wehave i ,0 =0 and ~ i ,0 = w i ,1 recallthat w i ,1 > 0 onlyfor n )]TJ/F38 11.9552 Tf 12.559 0 Td [(k n + machines.Also,for L 2f G H g and l j 0 ,deneprocesses M n l j L := f M n l j L ; t t 0 g bysee3{26 M n l j L ; t := 1 p n n X i =1 1 f c i j +1: j + l + s i j + l +1 + a i j + l +1 1 f L = H g t )]TJ/F22 7.9701 Tf 6.944 0 Td [( i j g )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F21 11.9552 Tf 12.351 0 Td [( i j {29 Processes ~ M n l j L := f ~ M n l j L ; t t 0 g aredenedsimilarly{ i j 'sarereplacedwith ~ i j 's in3{29.Anasymptoticdescriptionoftheseprocessesisprovidedinthefollowinglemma. Lemma3.3. For L 2f G H g and j 1 ,wehave M n l j L ~ M n l j L M l j L M l j L as n !1 ,where M l j L := f M l j L ; t t 0 g isacenteredGaussianprocesswith a.s.continuouspathsandfor t s 0 E M l j L ; t M l j L ; t + s = Z t 0 L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u L H l t + s )]TJ/F38 11.9552 Tf 11.955 0 Td [(u d H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 u Proof. SeeSection3.5.4. 37

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Observethatthevarianceof M l j L ; t vanishesasindices l 0 and/or j 1 increase: E M l j L ; t 2 L H H l + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t ; indeed,thereexistsan m 1 suchthat H m t < 1 ,andhence E M l j L ; t 2 H l + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t H m t b l + j )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 m c 0, as l + j !1 .Thefactthatthevarianceof M l j L ; t vanishesmotivatesthefollowingresult. Lemma3.4. For L 2f G H g and > 0 ,wehave lim m !1 limsup n !1 P 2 6 6 4 X l + j > m j 6 =0 k M n l j L k t + k ~ M n l j L k t > 3 7 7 5 =0. Proof. SeeSectionB.1.1. Next,weconsider M n l j L and ~ M n l j L when j =0 .Inparticular,weexamineasum oftheirdierences,sincethetwoprocessesarenotcenteredonlyfor j =0 duetothe distributionof f c i ,1 g seeSection4.2.1;however,theirdierenceiscentered.Thecase j =0 isdierentfromthecase j 1 ,because i ,0 and ~ i ,0 arenotequalfor n )]TJ/F38 11.9552 Tf 12.427 0 Td [(k n + machines awaitingserviceattime t =0 onlyforallothermachines i ,0 =~ i ,0 =0 Lemma3.5. For L 2f G H g ,wehave,as n !1 X L 2f G H g 1 X l =0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 1 f L = H g M n l ,0 L )]TJ/F21 11.9552 Tf 14.565 2.657 Td [(~ M n l ,0 L 0. Proof. SeeSectionB.1.1. Remark 3.6 Notethat3{28and3{29yield k M n l j L k t + k M n l j L k t 2 p n n X i =1 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(1 f c i ,2: j + l t g + H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(c i ,2: j 38

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which,inturn,implies E 2 4 X L 2f G H g 1 X j l =0 k M n l j L )]TJ/F21 11.9552 Tf 14.564 2.656 Td [( M n l j L k t 3 5 4 p n H R t + R t +3 < 1 {30 Therefore,theinnitesuminthestatementofLemma3.5iswelldened. Thefollowinglemmaisthemainpreliminaryresult.Let ^ n := f ^ n t t 0 g ,where ^ n t := ^ X n t )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n t + n )]TJ/F29 11.9552 Tf 11.956 16.273 Td [(Z t 0 ^ X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s + d P 1 s {31 Lemma3.6. Wehave ^ n 0 ,as n !1 Proof. First,notethat n X i =1 1 X j =0 P 1 t )]TJ/F21 11.9552 Tf 12.351 0 Td [(~ i j )]TJ/F38 11.9552 Tf 11.955 0 Td [(P 1 t )]TJ/F21 11.9552 Tf 12.352 0 Td [( i j = n X i =1 1 X j =0 E n i j t j i j ,~ i j = n X i =1 1 X j =0 Z t 0 1 f i j t )]TJ/F39 7.9701 Tf 6.587 0 Td [(s < ~ i j g d P 1 s = Z t 0 n X i =1 1 X j =0 1 f i j t )]TJ/F39 7.9701 Tf 6.587 0 Td [(s < ~ i j g d P 1 s = Z t 0 X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(k + d P 1 s {32 wherethelastequalityfollowsfromthefactthatthedoublesumrepresentsthenumberof machinesawaitingserviceattime t )]TJ/F38 11.9552 Tf 12.31 0 Td [(s .Combining3{1,4{36,3{7and3{12renders seealsoFigure3-2 X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(Z n t = n X i =1 X n i t )]TJ/F38 11.9552 Tf 11.955 0 Td [(Z n i t = n X i =1 1 X j =0 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 f 0 t )]TJ/F39 7.9701 Tf 6.587 0 Td [(c i ,1: j )]TJ/F39 7.9701 Tf 6.586 0 Td [(s i j +1 < w i ,1: j +1 g )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 f 0 t )]TJ/F39 7.9701 Tf 6.587 0 Td [(c i ,1: j < w i ,1: j g = n X i =1 1 X j =0 j X l =0 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(1 f w i ,1: l t )]TJ/F39 7.9701 Tf 6.587 0 Td [(c i ,1: j )]TJ/F39 7.9701 Tf 6.587 0 Td [(s i j +1 < w i ,1: l +1 g )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 f w i ,1: l )]TJ/F22 5.9776 Tf 5.757 0 Td [(1 t )]TJ/F39 7.9701 Tf 6.587 0 Td [(c i ,1: j < w i ,1: l g = n X i =1 X n i )]TJ/F38 11.9552 Tf 11.956 0 Td [(Z n i t 39

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Next,achangeintheorderofsummation l and j and3{28resultin X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(Z n t = n X i =1 1 X j l =0 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 f w i ,1: j t )]TJ/F39 7.9701 Tf 6.586 0 Td [(c i ,1: l + j )]TJ/F39 7.9701 Tf 6.587 0 Td [(s i l + j +1 < w i ,1: j +1 g )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 f w i ,1: j t )]TJ/F39 7.9701 Tf 6.587 0 Td [(c i ,1: l + j +1 < w i ,1: j +1 g = n X i =1 1 X j =0 n i j t = n X i =1 1 X j l =0 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 f c i j +1: j + l + s i j + l +1 t )]TJ/F22 7.9701 Tf 6.944 0 Td [( i j < c i j +1: j + l +1 g )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 f c i j +1: j + l + s i j + l +1 t )]TJ/F22 7.9701 Tf 6.945 0 Td [(~ i j < c i j +1: j + l +1 g = 1 X i =1 1 X j l =0 # n i j l G ; t )]TJ/F21 11.9552 Tf 13.242 2.657 Td [( # n i j l H ; t )]TJ/F29 11.9552 Tf 11.955 13.271 Td [( ~ # n i j l G ; t )]TJ/F21 11.9552 Tf 13.242 2.657 Td [(~ # n i j l H ; t ; {33 seeFigure3-2foranillustration.This,togetherwith3{5,3{8,3{14and3{32,yields p p p ^ n = ^ n 1, m + ^ n 2, m + ^ n 3, m {34 where ^ n 1, m := X L 2f G H g X j + l m j 6 =0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 1 f L = H g M n l j L )]TJ/F21 11.9552 Tf 14.565 2.657 Td [(~ M n l j L ^ n 2, m := X L 2f G H g X j + l > m j 6 =0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 1 f L = H g M n l j L )]TJ/F21 11.9552 Tf 14.565 2.657 Td [(~ M n l j L ^ n 3, m := X L 2f G H g 1 X l =0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 1 f L = H g M n l ,0 L )]TJ/F21 11.9552 Tf 14.564 2.656 Td [(~ M n l ,0 L ; observethat ^ n 2, m and ^ n 3, m arewelldeneddueto3{30inRemark3.6. Notethat,foraxed m ,Lemma3.3andthecontinuityoftheadditionoperatorresultin, as n !1 ^ n 1, m 0. {35 Next,thenonnegativityofthesupremumnormandLemma3.4imply lim m !1 limsup n !1 P [ k ^ n 2, m k t > ]=0. {36 40

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Combining3{34,3{35,3{36andLemma3.5completestheproofofthelemma. 3.4.3ProofofTheorem3.1 Recallthedenition3{31of ^ n .Now,observethat ^ X n = ^ Z n + ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n = P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n + ^ n {37 andnotethat,as n !1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n + ^ n ^ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( dueto3{4,theassumptionofthetheorem,Lemma3.6, beingdeterministicsee Theorem11.4.5in[100,p.379]andthecontinuityoftheadditionoperator.Bythe Skorokhodrepresentationtheorem[85] D [0, 1 D isaseparablespace,thereexists analternativeprobabilityspacewith f ~ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n + ~ n g and ~ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( denedonit,suchthat ~ Z n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n + ~ n ~ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( {38 in D [0, 1 u almostsurely,as n !1 ;here, ~ Z n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n + ~ n d = ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n + ^ n and ~ Z )]TJ/F24 11.9552 Tf 11.956 0 Td [( d = ^ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( {39 Furthermore,dueto P 1 : D [0, 1 D D [0, 1 D beingmeasurableProposition3.1, itfollowsfrom3{37and3{39that,foreach n 1 ^ X n d = P 1 ~ Z n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n + ~ n {40 Ontheotherhand,thecontinuitypartofProposition3.1and3{38resultin P 1 ~ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n + ~ n P 1 ~ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( a.s. {41 in D [0, 1 u ,as n !1 .Thefactsthatconvergencein D [0, 1 u impliesconvergence in D [0, 1 d J 1 andthatalmostsureconvergenceimpliesweakconvergence,together 41

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with3{40,3{41andProposition3.1measurabilitypart,yield ^ X n P 1 ^ Z )]TJ/F24 11.9552 Tf 12.644 0 Td [( ,as n !1 .Thiscompletestheproof. 3.5Proofs ThissectioncontainsproofsofLemma3.1,Proposition3.1,Lemma3.2andLemma3.3. 3.5.1ProofofLemma3.1 Let n beabinomialrandomvariablewithparameters n p ,independentofallother variablesinthe n thmodel;set ^ n := n )]TJ/F38 11.9552 Tf 12.55 0 Td [(np = p np p .Weconstructastationaryprocess ^ Y n := f ^ Y n t t 0 g bymodifyingtheinnite-repairmenprocess Z n .Inparticular,let ^ Y n := 1 p np p n X i =1 Y n i )]TJ/F38 11.9552 Tf 11.955 0 Td [(p {42 where Y n i := 8 > > < > > : ~ Z n i n ^ k n ^ n < i n n Z n i otherwise with f ~ Z n i n ^ k n ^ n < i n g and f ~ Z n i n < i n n g beingi.i.d.processes correspondingtoabrokenandworkingmachineattime t =0 ,respectively.Namely,let ~ c i j =~ a i j +~ s i j j 1 ,where f ~ a i j j 2 g and f ~ s i j j 2 g bemutuallyindependenti.i.d. sequencesdenedbydistributions F and G ,respectively.Furthermore,let ~ s i ,1 i n ,be distributedaccordingto G ,and ~ s i ,1 0 for i > n .Similarly,let ~ a i ,1 bedistributedaccording to F and F for i n and i > n ,respectively.Then,wehave ~ Z n i t = 1 X j =0 1 f 0 t )]TJ/F22 7.9701 Tf 6.691 0 Td [(~ c i ,1: j < ~ s i j +1 g Basedontheaboveconstruction,itfollowsthat E ~ Z n i t = 8 > > < > > : P 1 t n ^ k n ^ n < i n P 0 t n < i n n {43 42

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Observethat,duetotheprecedingandthebinomialdistributionof n ,theprocess ^ Y n is stationarywith ^ Y n =^ n see3{42,and ^ Y n d = ~ Y n 1 p np p n X i =1 ~ Y i )]TJ/F38 11.9552 Tf 11.956 0 Td [(p where f ~ Y i g isasequenceofi.i.d.stationaryon-oprocessesdescribedbythedistributions G and F e.g.,see[47].Since ~ Y i isastationaryprocess,itfollowsthat E ~ Y i t )]TJ/F38 11.9552 Tf 11.956 0 Td [(p ~ Y i t + s )]TJ/F38 11.9552 Tf 11.955 0 Td [(p = E ~ Y i t ~ Y i t + s )]TJ/F38 11.9552 Tf 11.955 0 Td [(p 2 = p P 1 s )]TJ/F38 11.9552 Tf 11.956 0 Td [(p for t s 0 .Under3{13,conditionsoftheHahn'stheorem[42]seealso[100,p.226] hold[88],andthus ~ Y n ^ Y ,as n !1 ,where ^ Y isasinthestatementofthelemma; consequently, ^ Y n ^ Y ,as n !1 .Moreover,duetothecontinuousmappingtheoremand continuityofthemapping D [0, 1 u D 2 [0, 1 u 2 suchthat x 7! x x P 0 )]TJ/F21 11.9552 Tf 13.307 2.657 Td [( P 1 for x 2 D [0, 1 ,onehas ^ Y n ,^ n P 0 )]TJ/F21 11.9552 Tf 13.487 2.656 Td [( P 1 ^ Y ^ Y P 0 )]TJ/F21 11.9552 Tf 13.487 2.656 Td [( P 1 ,as n !1 ,which,in turn,implies[94] ^ Y n +^ n P 0 )]TJ/F21 11.9552 Tf 13.239 2.657 Td [( P 1 ^ Y + ^ Y P 0 )]TJ/F21 11.9552 Tf 13.24 2.657 Td [( P 1 {44 as n !1 ,since ^ Y isaGaussianprocess. Next,let E Z n i := f E Z n i t t 0 g and E ~ Z n i := f E ~ Z n i t t 0 g .Notethat,based on3{4,3{6,3{11and3{43,onecanconcludethat n n X i = n ^ k n ^ n +1 E Z n i )]TJ/F35 11.9552 Tf 11.955 0 Td [(E ~ Z n i )]TJ/F24 11.9552 Tf 11.955 0 Td [( n P 0 )]TJ/F21 11.9552 Tf 13.239 2.657 Td [( P 1 = n ^ k n P 1 + n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + P 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( n P 0 {45 and,therefore, 1 p np p n n X i = n ^ k n ^ n +1 E Z n i )]TJ/F35 11.9552 Tf 11.955 0 Td [(E ~ Z n i )]TJ/F21 11.9552 Tf 12.542 0 Td [(^ n P 0 )]TJ/F21 11.9552 Tf 12.879 2.657 Td [( P 1 )]TJ/F21 11.9552 Tf 12.602 2.657 Td [(^ )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [( P 1 + ^ + P 1 )]TJ/F21 11.9552 Tf 11.595 0 Td [( ^ + P 0 {46 as n !1 .Theindependenceof n and n ,Lemma3.7attheendofthissubsectionand 1 p np p n n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n ^ k n ^ n ^ ^ )]TJ/F24 11.9552 Tf 11.955 0 Td [( + ^ )]TJ/F20 11.9552 Tf 9.741 -4.936 Td [(_ ^ )]TJ/F24 11.9552 Tf 11.955 0 Td [( )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(, 43

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as n !1 ,yield,as n !1 1 p np p n n X i = n ^ k n ^ n +1 Z n i )]TJ/F35 11.9552 Tf 11.955 0 Td [(E Z n i )]TJ/F21 11.9552 Tf 13.361 2.656 Td [(~ Z n i + E ~ Z n i 0. {47 Finally,3{14and3{42imply ^ Z n = ^ Y n + ^ Z n )]TJ/F21 11.9552 Tf 13.891 2.657 Td [(^ Y n = ^ Y n + 1 p np p n n X i = n ^ k n ^ n +1 Z n i )]TJ/F21 11.9552 Tf 13.36 2.656 Td [(~ Z n i {48 Combining3{44,3{46,3{47,3{48,Theorems11.4.4and11.4.5in[100,pp.378-379] notethatinviewof3{45,theleft-handsideof3{46and ^ n areindependentandthe continuousmappingtheoremanditsextensionforsummation[94]yieldsthestatementofthe lemma. Lemma3.7. Let f U i g beani.i.d.sequenceofprocessesand u := f E U i t t 0 g .Suppose k U i )]TJ/F38 11.9552 Tf 11.955 0 Td [(u k t 1 and S n := n X i =1 U i )]TJ/F38 11.9552 Tf 11.955 0 Td [(u If f n g isasequenceofnonnegativerandomvariablesindependentof f U i g suchthat n as n !1 ,then,as n !1 k S b n n c = n k t P 0. Proof. Let 1 < beacontinuitypointofthedistributionof .For 0 < 1 < 2 < 3 and > 0 ,theunionboundand k S b n n c = n k t j n j resultin P [ k S b n n c = n k t > ] P [ 1 < n < 2 ]+ P sup 2 < x < 3 k S b xn c = b xn ck t > 3 + P [ n > 3 ]; thesecondtermontheright-handsidevanishes,as n !1 ,dueto[24].Letting n !1 ,and thenletting 2 1 and 3 !1 yieldsthestatementofthelemma. 44

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3.5.2ProofofProposition3.1 Theproofofthepropositionproceedsalongthelinesoftheproofofthecorresponding propositionfor G in[77].Themeasurabilitypartisarguedattheendoftheproof.Therest oftheproofcoversthreecases:degenerate G and F ;nondegenerate G ;anddegenerate G and nondegenerate F thesecasesimpactpropertiesof P 1 Degenerate G and F Assumethat G and F areconcentratedat c G and c F respectively.Then P 1 isperiodicwithperiod c = c G + c F : P 1 t = 1 X l =0 1 f lc + c G t < lc + c g Asolutionto3{16satisesthefollowingrecursion: y t = x t ,for 0 t < c G ,and y t = x t + b t + c F c c X i =1 y + t )]TJ/F21 11.9552 Tf 11.955 0 Td [( i )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 c )]TJ/F38 11.9552 Tf 11.955 0 Td [(c G )]TJ/F27 7.9701 Tf 13.89 17.007 Td [(b t c c X i =1 y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(ic {49 for t c G ;thisimpliesexistenceanduniqueness.Let y 0 :=0 andrecursivelydene y m +1 t := x t + Z t 0 y + m t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d P 1 s t 0, {50 for m 0 ;thatis, y m 7! y m +1 .Then,from3{49itfollowsthat k y m +1 )]TJ/F38 11.9552 Tf 12.627 0 Td [(y m k t =0 for all m d t = c G ^ c F e .Inaddition,3{49resultsin k P 1 x 1 )]TJ/F24 11.9552 Tf 12.496 0 Td [(' P 1 x 2 k t = k x 1 )]TJ/F38 11.9552 Tf 12.496 0 Td [(x 2 k t for 0 t < c G ,and k P 1 x 1 )]TJ/F24 11.9552 Tf 12.606 0 Td [(' P 1 x 2 k t k x 1 )]TJ/F38 11.9552 Tf 12.606 0 Td [(x 2 k t + k P 1 x 1 )]TJ/F24 11.9552 Tf 12.606 0 Td [(' P 1 x 2 k t )]TJ/F39 7.9701 Tf 6.586 0 Td [(c G for c G t < c see3{49.If t )]TJ/F38 11.9552 Tf 12.541 0 Td [(c G c G ,thisargumentcanbeappliedmultipletimesto obtain k P 1 x 1 )]TJ/F24 11.9552 Tf 12.176 0 Td [(' P 1 x 2 k t d c = c G ek x 1 )]TJ/F38 11.9552 Tf 12.175 0 Td [(x 2 k t ;thelastinequalityservesasabasisforour induction.Now,supposeforeverypositiveinteger l j forsomeinteger j onehas k P 1 x 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 2 k t 8 > > < > > : a l k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t l )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 c t < l )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 c + c G b l k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t l )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 c + c G t < lc {51 45

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Then,theinductivehypothesis3{51and3{49imply,for jc t < jc + c G k P 1 x 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [(' P 1 x 2 k t k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t + j X i =1 k P 1 x 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 2 k t )]TJ/F22 7.9701 Tf 6.586 0 Td [( i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 c )]TJ/F39 7.9701 Tf 6.587 0 Td [(c G + j X i =1 k P 1 x 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 2 k t )]TJ/F39 7.9701 Tf 6.587 0 Td [(ic k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t + j X i =1 a i k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t + j X i =1 b i k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t = 1+ a 1: j + b 1: j k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t a j +1 k x 1 )]TJ/F38 11.9552 Tf 11.956 0 Td [(x 2 k t ; also,for jc + c G t < j +1 c k P 1 x 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 2 k t k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t + j +1 X i =1 k P 1 x 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 2 k t )]TJ/F22 7.9701 Tf 6.586 0 Td [( i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 c )]TJ/F39 7.9701 Tf 6.587 0 Td [(c G + j X i =1 k P 1 x 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 2 k t )]TJ/F39 7.9701 Tf 6.586 0 Td [(ic k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t + j +1 X i =1 a i k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t + j X i =1 b i k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1+ a 1: j +1 + b 1: j k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t b j +1 k x 1 )]TJ/F38 11.9552 Tf 11.956 0 Td [(x 2 k t Therefore,theinductiveassumption3{51holdsforall j ,and P 1 isLipschitzcontinuous when G and F aredegenerate. Nondegenerate G Toshowuniqueness,weusethemethodofsuccessiveapproximations. Recall3{50;thatdenitionimplies y m +1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m t = Z t 0 )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(y + m t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(y + m )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d P 1 s {52 InLemma3.8seetheendofthissubsectionselecta > 0 suchthat t = isaninteger.Next, weuseinductiontoarguethat,foreachinteger 1 k t = k y m +1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k j j j m j m k x k t j =1,..., k m 1. {53 Thecase k =1 isstraightforward: k y 1 )]TJ/F38 11.9552 Tf 10.429 0 Td [(y 0 k = k x )]TJ/F21 11.9552 Tf 10.43 0 Td [(0 k k x k t ,3{52andLemma3.8result in k y m +1 )]TJ/F38 11.9552 Tf 10.923 0 Td [(y m k k y m )]TJ/F38 11.9552 Tf 10.922 0 Td [(y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ,which,inturn,renders k y m +1 )]TJ/F38 11.9552 Tf 10.923 0 Td [(y m k m k x k t m m k x k t 46

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for m 1 .Now,assumethat3{53holdsforsome k .Basedonthisinductivehypothesis, Lemma3.8and3{52,onehas k y m +1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k k +1 k X j =1 k y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k j + k y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k k +1 k X j =1 j j m )]TJ/F21 11.9552 Tf 11.956 0 Td [(1 j m k x k t + k y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k k +1 k k +1 m k m k x k t + k y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k k +1 {54 Notethat k y 1 )]TJ/F38 11.9552 Tf 12.042 0 Td [(y 0 k l = k x )]TJ/F21 11.9552 Tf 12.042 0 Td [(0 k l k x k t ,forall l =1,..., t = ,bydenition.This,combined with3{54,yields k y m +1 )]TJ/F38 11.9552 Tf 11.956 0 Td [(y m k k +1 k k +1 m k x k t m X j =0 j k k k +1 m k +1 +1 m k x k t k +1 k +1 m k +1 m k x k t Hence,theinductivehypothesisholds.Finally, f y m g isaCauchysequence,because 1 X m =1 k y m +1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k t 1 X m =1 k y m +1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k t = 1 X m =1 t = t = m t = m k x k t < 1 Inviewofthefactthat D [0, 1 equippedwiththesupremummetricisaBanachspace butnotunder J 1 [100,p.84],thereexistsalimitpoint y of f y m g e.g.,see[81,p.4]. Letting m !1 onbothsidesof3{50,oneconcludesthatthelimit y isasolution to3{16.Thiscompletestheproofofexistenceincaseofnondegeneratedistribution G Asfarasuniquenessisconcerned,supposethatboth u and v solve3{16,anddene := f t t 0 g ,where t := u t )]TJ/F38 11.9552 Tf 11.955 0 Td [(y t = Z t 0 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(u + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d P 1 s t 0. 47

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Then,Lemma3.8yields k k sup 0 t Z t 0 )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(u + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d P 1 s k k whichimplies t =0 for t 2 [0, ] .Similarly,onehas k k 2 k k + k k 2 = k k 2 and,hence, t =0 for t 2 [0,2 ] .Iteratingtheaboveargumentmultipletimescompletes theproofofuniqueness. Finally,weargueLipschitzcontinuity.NotethatbyLemma3.8onehas,for 0 t < k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k t k x 2 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 1 k t + k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k t whichimplies k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k t )]TJ/F24 11.9552 Tf 11.955 0 Td [( )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k x 2 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 1 k t .Similarly,for t < 2 ,onehas k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k t k x 2 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 1 k t + k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k t )]TJ/F25 7.9701 Tf 6.586 0 Td [( + k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k t )]TJ/F24 11.9552 Tf 11.956 0 Td [( )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k x 2 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 1 k t + k P 1 x 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(' P 1 x 1 k t which,inturn,implies k P 1 x 2 )]TJ/F24 11.9552 Tf 12.036 0 Td [(' P 1 x 1 k t )]TJ/F24 11.9552 Tf 12.036 0 Td [( )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 k x 2 )]TJ/F38 11.9552 Tf 12.036 0 Td [(x 1 k t ,for 0 t < 2 .Applying theaboveargumentmultipletimescompletesthispartoftheproof. Degenerate G andnondegenerate F Suppose G isconcentratedat c G .From Lemma3.8,itfollowsthat P 1 t = B t )]TJ/F21 11.9552 Tf 12.82 0 Td [(1 f t < c G g ,where B issuchthati B t =1 for t c G ,andiithereexist > 0 and < 1 suchthat,forevery t 2 [0, 1 sup t t 1 t 2 t + j B t 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(B t 2 j < {55 Therefore,3{16canberewritten: y t = 8 > > < > > : x t t < c G x t + y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(c G + R t c G y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d B s t c G {56 Fromtheprecedingequation,itisapparentthatthesolutionexistsandthatitisuniquefor t < c G .Now,supposethat,forsomeinteger i 1 y i istheuniquesolutionto3{56for t < ic G .Wearguethatthereexistsaunique y i +1 satisfying3{56for t < i +1 c G .Tothis 48

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end,let x i t := 8 > > < > > : x i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t t < ic G x t + y + i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(c G ic G t < i +1 c G where x 0 x .Observethat f x i t ,0 t < ic G g isuniquelydened,sincethereexistsa unique y i for t < ic G .Then,onecanrewritethesecondpartof3{56asfollows: y t = x i t + Z t c G y + t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s d B s t < i +1 c G {57 Thesameargumentusedtoshowexistenceanduniquenessofthesolutioninthecaseof nondegenerate G isapplicableinthecaseof3{57.Inparticular,3{55shouldbeused insteadofLemma3.8.Hence,3{57hastheuniquesolutionon t < i +1 c G {termthis solution y i +1 .Moreover,onehas y i +1 t = y i t for t < ic G ,since y i isuniqueon [0, ic G and y i +1 and y i solvethesameequationon [0, ic G .Iteratingtheprecedingreasoningmultiple timesyieldsthat3{16hasauniquesolutionwhen G isdegenerateand F isnondegenerate. TheproofofLipschitzcontinuityutilizes3{56.Itisstraightforwardthat k P 1 x 2 )]TJ/F24 11.9552 Tf -436.287 -23.908 Td [(' P 1 x 1 k t k x 2 )]TJ/F38 11.9552 Tf 12.161 0 Td [(x 1 k t ,for t < c G .Now,suppose k P 1 x 2 )]TJ/F24 11.9552 Tf 12.161 0 Td [(' P 1 x 1 k t e i k x 2 )]TJ/F38 11.9552 Tf 12.16 0 Td [(x 1 k t ,for i )]TJ/F21 11.9552 Tf 11.956 0 Td [(1 c G t < ic G ,i.e., e 1 =1 .Then,3{56implies,for ic G t < i +1 c G k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k t k x 2 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 1 k t + k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k ic G + d c G = e k P 1 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 x 1 k ic G d c G = e +1 e i +1 k x 2 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 1 k t e i +1 k x 2 )]TJ/F38 11.9552 Tf 11.956 0 Td [(x 1 k t Thiscompletesthispartoftheproof. Measurability. Let P : D [0, 1 D [0, 1 bedenedbythefollowing,for t 0 : P y t := Z t 0 y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d P s Theproofconsistsoftwoparts.Intherstpart,measurabilityof P 1 isarguednotethat P issimpler"than P .Thesecondpartdealswithmeasurabilityoftheoriginaloperator P 1 Fortherstpart,oneneedstoarguethat P 1 ismeasurablewithrespecttothe Borel -eld D generatedbytheSkorokhod J 1 topology.Giventhat D isequaltothe 49

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Kolmogorov -eldgeneratedbythenite-dimensionalcylindersets[100,p.385],itis sucienttocheckthatforeach m 1 and A 1 ,..., A m 2B R onehas f y 2 D [0, 1 : P 1 y t 1 ,..., P 1 y t m 2 A 1 ,..., A m g2D ,for 0 t 1 < t 2 < < t m .Moreover,it issucienttoverifythat P 1 t ismeasurable,forall t 0 ,since -eldsareclosedunder niteintersections.Tothisend,weconsideradecompositionof P 1 see3{8: P 1 = G R )]TJ/F38 11.9552 Tf 11.955 0 Td [(H R =C f G R g)]TJ/F21 11.9552 Tf 20.589 0 Td [(C f H R g +D f G R g)]TJ/F21 11.9552 Tf 20.589 0 Td [(D f H R g P C + P D {58 whereby C fg and D fg denotecontinuousanddiscreteparts,respectivelyboth G H and H R aremonotonic.Inwhatfollows,itisshownthatboth P C and P D aremeasurable functions,andthus P 1 = P C + P D ismeasurableaswell,sincethesumoftwomeasurable functions D [0, 1 D D [0, 1 D ismeasurable[30,p.119]. Asfarasmeasurabilityof P C isconcerned,itisenoughtoshowthat P C t isa continuous D [0, 1 d J 1 R jj function,foreach t 0 .Let f y m g beasequence,such that y m y underthe J 1 metric.Thisimpliesthat y m t y t forallbutacountable numberof t e.g.,see[14,p247].Inaddition,bythedenitionof P C above, R S d P C =0 for anycountableset S .Thus,sinceforeach t 0 ,thesequence fk y m k t m 1 g isbounded,it followsthat j P C y m t )]TJ/F24 11.9552 Tf 11.955 0 Td [( P C y t j = Z t 0 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(y + m t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.956 0 Td [(y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d P C s Z t 0 j y m t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(y t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s j d P C s 0, as m !1 ,i.e., P C ismeasurable. Let f d i g bethesetofdiscontinuitypointsof P D on [0, t ] .Thissetiscountable,sinceit isasubsetofcountableunionofsetsofdiscontinuitypointsof G H i and H i assuming theaxiomofchoice;thesetofdiscontinuitypointsof G H i or H i ,foreach i ,isa countableset,becausetheyaredistributionfunctions.Formeasurabilityof P D ,twocaseswill beconsidered.First,supposethatthecardinalityof f d i g is m < 1 .Inthatcase, P D isofthe 50

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form P D t = P m i =1 c i 1 f t d i g ,and,therefore, P D y t = m X i =1 c i y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(d i f t d i g Thatis, P D isanitesumofmeasurablefunctionstranslationbyaconstantandmultiplication byaconstantaswelltheircompositionaremeasurable,makingitmeasurableaswell.When thenumberofpointsin f d i g isinnite,then P D y t = 1 X i =1 c i y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(d i f t d i g Next,dene m P D y t := m X i =1 c i y + t )]TJ/F38 11.9552 Tf 11.956 0 Td [(d i f t d i g inordertoconsider,forevery y 2 D [0, 1 and t 0 k m P D y )]TJ/F24 11.9552 Tf 11.955 0 Td [( P D y k t =sup 0 s t 1 X i = m +1 c i y + s )]TJ/F38 11.9552 Tf 11.955 0 Td [(d i f s d i g k y k t 1 X i = m +1 j c i j 1 f s d i g 0, as m !1 ;thelimitisjustiedbysee3{58 1 X i =1 j c i j 1 f t d i g G R t + H R t < 1 Therefore, P D isapointwiselimitof f m P D g m in D [0, 1 u .Since m P D 'saremeasurable,it meansthat P D ismeasurableaswell[30,p.125]. Now,wearereadytostartthesecondpartofthemeasurabilityproof.Thebasicidea istoexploittheprecedingpartaswellasthepartoftheproofthatrelatestoexistence.Let : D [0, 1 D D [0, 1 D bedenedby,for t 0 y t := x t + P 1 y t ; 51

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ismeasurablesince P 1 ismeasurable.Themethodofsuccessiveapproximationsseethe existencepartoftheproofyields,foreach x 2 D [0, 1 P 1 x =lim m !1 m where m isthe m -foldcompositionof withitself,andthelimitistakenwithrespecttothe uniformmetric u .Notethat m ismeasurableforevery m ,becausethecompositionoftwo measurablefunctionsismeasurable[13,p.182].This,inturn,impliesthat P 1 measurable, sinceapointwiselimitofmeasurablefunctionsismeasurable.Thiscompletestheproof. Lemma3.8. Fornondegenerate G ,thereexist > 0 and < 1 suchthat sup j t 1 )]TJ/F39 7.9701 Tf 6.586 0 Td [(t 2 j < t 1 t 2 2 [0, T ] j P 1 t 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(P 1 t 2 j Proof. Equation3{8,togetherwithmonotonicityof R and G R ,implies j P 1 t 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(P 1 t 2 jj R t 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t 2 j_j G R t 1 )]TJ/F38 11.9552 Tf 11.956 0 Td [(G R t 2 j and sup j t 1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(t 2 j < t 1 t 2 2 [0, T ] j P 1 t 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(P 1 t 2 j sup t 2 [ T ] f R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t )]TJ/F24 11.9552 Tf 11.955 0 Td [( g_ sup t 2 [ T ] f G R t )]TJ/F38 11.9552 Tf 11.956 0 Td [(G R t )]TJ/F24 11.9552 Tf 11.955 0 Td [( g Therefore,inordertoprovethelemma,itissucienttoshowthatthereexist > 0 and < 1 suchthat sup t 2 [ T ] f R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t )]TJ/F24 11.9552 Tf 11.955 0 Td [( g and sup t 2 [ T ] f G R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(G R t )]TJ/F24 11.9552 Tf 11.956 0 Td [( g {59 Westartwiththerstinequalityin3{59.Thispartoftheproofhasseveralsteps.First, notethattherstequationin3{59holdsif sup t 2 T ] f R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t )]TJ/F21 11.9552 Tf 9.298 0 Td [( g < 1. {60 52

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Here,weremarkthatthereexistsa t ? thatachievessupremumin3{60: 0 := R t ? )]TJ/F38 11.9552 Tf -429.198 -23.908 Td [(R t ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( .Thecase 0 =0 istrivial.Otherwise 0 > 0 ,thereexistonlynitelymany t i 'ssuch that R t i )]TJ/F38 11.9552 Tf 12.143 0 Td [(R t i )]TJ/F21 11.9552 Tf 9.298 0 Td [( > 0 = 2 {infact,atmost d 2 R T = 0 e suchpointsexist.Theconclusion follows. Thesecondstepestablishesaboundonjumpsin R .Tothisend,onehas H i +1 t = Z [0, ] H i t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d H s + Z t ] H i t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d H s H + H i t )]TJ/F24 11.9552 Tf 11.955 0 Td [( for > 0 and i 1 .Letting 0 andrecalling H = F G =0 yields H i +1 t H i t )]TJ/F21 11.9552 Tf 9.299 0 Td [( .Now,thedenitionof R and R t < 1 imply R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t )]TJ/F21 11.9552 Tf 9.299 0 Td [(= H t + 1 X i =1 )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(H i +1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(H i t )]TJ/F21 11.9552 Tf 9.298 0 Td [( H t + m X i =1 )]TJ/F38 11.9552 Tf 5.479 -9.683 Td [(H i +1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(H i t )]TJ/F21 11.9552 Tf 9.298 0 Td [( {61 forany m 0 ;theinequalityfollowsfromthenon-positivityoftheelementsofthesum. Thethirdstepestablishespropertiesof H ,when H T =1 .Forthatpurpose,dene s ? :=inf f s : H s =1 g < 1 ,andnotethat H s ? )]TJ/F24 11.9552 Tf 12.801 0 Td [( < 1 ,forany > 0 .Dueto 1 H m ms ? H m s ? =1 and H m ms ? )]TJ/F24 11.9552 Tf 12.063 0 Td [( 1 )]TJ/F21 11.9552 Tf 13.626 2.657 Td [( H m s ? )]TJ/F24 11.9552 Tf 12.063 0 Td [(= m < 1 ,theconstant s ? satises ms ? =inf f s : H m s =1 g {62 Thenondegeneratenatureof H since G isnondegeneratebyassumptionensuresthat 0= H < H s ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( ,i.e.,thereexistsa > 0 suchthat H s ? )]TJ/F21 11.9552 Tf 9.299 0 Td [( )]TJ/F38 11.9552 Tf 12.856 0 Td [(H > 0 .Then, conditioningyields,forsome > 0 H m +1 m +1 s ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( )]TJ/F38 11.9552 Tf 11.955 0 Td [(H m +1 ms ? H s ? )]TJ/F21 11.9552 Tf 9.299 0 Td [( )]TJ/F38 11.9552 Tf 11.955 0 Td [(H )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(H m ms ? )]TJ/F38 11.9552 Tf 11.955 0 Td [(H m ms ? )]TJ/F24 11.9552 Tf 11.956 0 Td [( > 0, 53

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wherethelaststrictinequalityfollowsfrom3{62.Hence,since H m +1 isright-continuous, thereexistsa > 0 smallenoughsuchthat H m +1 m +1 s ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( > H m +1 ms ? + {63 Now,byusing [0, s ? ]=[0, [ [ s ? )]TJ/F24 11.9552 Tf 11.955 0 Td [( [ [ s ? )]TJ/F24 11.9552 Tf 11.955 0 Td [( s ? ] onehas H m +1 ms ? = Z s ? 0 H m ms ? )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d H s H + H m ms ? )]TJ/F24 11.9552 Tf 11.955 0 Td [( H s ? )]TJ/F24 11.9552 Tf 11.955 0 Td [( + H m m )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 s ? + H s ? )]TJ/F24 11.9552 Tf 11.955 0 Td [( which,afterletting # 0 ,renders H m +1 ms ? H m ms ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( H s ? )]TJ/F24 11.9552 Tf 11.955 0 Td [( + H m m )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 s ? + H s ? )]TJ/F24 11.9552 Tf 11.955 0 Td [( < H m ms ? )]TJ/F21 11.9552 Tf 9.298 0 Td [(, {64 wherethesecondinequalityisdueto3{63. Thefourthstepaddresses3{59.Thecase H T < 1 isstraightforward:3{61yields R t )]TJ/F38 11.9552 Tf 12.409 0 Td [(R t )]TJ/F21 11.9552 Tf 9.299 0 Td [( H t H T < 1 ,forall t 2 T ] ,andthedesiredresultfollows.On theotherhand,thecase H T =1 isslightlymorecomplicated.Inparticular,recallfrom therststepthatthereexistsa t ? 2 T ] thatachievesthesupremumin3{59.When t ? = s ? 2f 1,2,..., b T = s ? cg thatis, t ? = ms ? ,3{61,3{62and3{64yield R t ? )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( 1+ H m +1 ms ? )]TJ/F38 11.9552 Tf 11.956 0 Td [(H m ms ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( < 1. {65 However,when t ? = s ? 62f 1,2,..., b T = s ? cg ,itmustbethat t ? = s ? 2 m )]TJ/F21 11.9552 Tf 12.135 0 Td [(1, m forsome m Therefore,3{61and3{62resultin R t ? )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( H m t ? )]TJ/F21 11.9552 Tf 9.298 0 Td [( < 1. {66 Combining3{65and3{66impliesthattherstequationin3{59holds. 54

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Finally,wearguethatthesecondequationin3{59holdsaswell.Tothisend,let 1 > 0 besuchthat sup t 2 [ 1 T ] f G t )]TJ/F38 11.9552 Tf 11.955 0 Td [(G t )]TJ/F24 11.9552 Tf 11.955 0 Td [( 1 g = 1 < 1; sucha 1 existssince G isnondegeneratebytheassumptionofthelemma.Also,let 2 besuch that sup t 2 [ 2 T ] f R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t )]TJ/F24 11.9552 Tf 11.955 0 Td [( 2 g = 2 < 1; seetherstpartoftheproofofthislemma.Then,for 0 < 1 ^ 2 andall t 2 [ T ] ,one has G R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(G R t )]TJ/F24 11.9552 Tf 11.955 0 Td [( Z t )]TJ/F25 7.9701 Tf 6.587 0 Td [( 0 R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F38 11.9552 Tf 11.955 0 Td [(R t )]TJ/F24 11.9552 Tf 11.955 0 Td [( )]TJ/F38 11.9552 Tf 11.955 0 Td [(s dG s + Z t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( R t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s dG s 2 G t )]TJ/F24 11.9552 Tf 11.955 0 Td [( + )]TJ/F24 11.9552 Tf 5.479 -9.684 Td [( 1 ^ G t )]TJ/F24 11.9552 Tf 11.955 0 Td [( R 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( 1 + 1 R {67 wherethelastinequalityisobtainedbyfromconsideringwhichelementachievestheminimum in 1 ^ G t )]TJ/F24 11.9552 Tf 11.955 0 Td [( and R 1 > 2 .Since R =1 and R isright-continuous,itfollowsthat 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( 1 + 1 R 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F24 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( 2 < 1, {68 as # 0 .Combining3{67and3{68yieldstheexistenceofa ? > 0 suchthat sup t 2 [ ? T ] f G t )]TJ/F38 11.9552 Tf 11.955 0 Td [(G t )]TJ/F24 11.9552 Tf 11.955 0 Td [( ? g < 1, i.e.,3{59holds,asdoesthestatementofthelemma. 3.5.3ProofofLemma3.2 Intherststep,weargue k P m 1 )]TJ/F38 11.9552 Tf 12.738 0 Td [(G k t 0 ,as m !1 .Tothisend,3{8implies P m 1 = G )]TJ/F38 11.9552 Tf 11.955 0 Td [(P m 1 G F m ,andthus k P m 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(G k t = k P m 1 G F m k t G F m t F m t 55

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wherethesecondinequalityisdueto P m 1 t 2 [0,1] ,forall t ,andmonotonicityofdistribution functions.Thedesiredresultfollowsfrom F m t = F t = m 0 ,as m !1 recallthat F =0 .Here,wealsonotethat3{8and3{10render P m 0 = F m P m 1 ,whichresults in k P m 0 k t F m t t = m 0 ,as m !1 see3{2.Similarly,3{9,monotonicityof distributionfunctionsand R m G F m = R m )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 yield k P m 1 )]TJ/F21 11.9552 Tf 13.709 4.783 Td [( G k t = k G )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G F m R m k t k F m )]TJ/F21 11.9552 Tf 13.174 2.657 Td [( F m R m )]TJ/F21 11.9552 Tf 11.956 0 Td [(1 k t F m t + R m t )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 F m t +1 = F m t )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 0, as m !1 ,where R m t 1 = F m t wasusedtoobtainthelastinequality.Observethatthis implies,as m !1 G )]TJ/F21 11.9552 Tf 14.435 10.745 Td [( P m 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(p m 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(p m t 0; thatis,thecovariancefunctionsof ^ Y 1 and ^ Y 1 m converge. Thesecondstepoftheproofistoshow ^ Z 1 m ^ Z 1 ,as m !1 .FromtheGaussian natureof ^ Y 1 m and ^ Y 1 ,therststepoftheproofandindependenceofthefollowing components,itfollowsthat ^ Y 1 m P m 1 )]TJ/F21 11.9552 Tf 13.24 2.656 Td [( P m 0 P m 1 )]TJ/F21 11.9552 Tf 13.24 2.656 Td [( P m 1 ^ 1 m ^ Y 1 G G )]TJ/F21 11.9552 Tf 13.709 4.782 Td [( G ^ 1 as m !1 ,in D 3 [0, 1 R u 3 jj .Giventhata D 3 [0, 1 R u 3 jj D 3 [0, 1 u 3 function x 1 x 2 x 3 x 4 7! x 1 x 4 + )]TJ/F38 11.9552 Tf 12.699 0 Td [(x 1 x 2 x + 4 x 3 iscontinuous,thecontinuous mappingtheorem,3{24and3{25yield ^ Z 1 m ^ Z 1 ,as m !1 Now,wearereadyforthelaststepoftheproof.Resultsin[77]andthepreviousstepof thisproofimplythat,as m !1 G ^ Z 1 m )]TJ/F24 11.9552 Tf 11.955 0 Td [( G ^ Z 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( {69 56

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Inviewof3{69seealsoTheorem11.4.5in[100,p.379],inordertocompletetheproof,it issucienttoshowthat,as m !1 k P m 1 ^ Z 1 m )]TJ/F24 11.9552 Tf 11.956 0 Td [( )]TJ/F24 11.9552 Tf 11.955 0 Td [(' G ^ Z 1 m )]TJ/F24 11.9552 Tf 11.956 0 Td [( k t P 0. {70 Fornotationalsimplicity,let ~ y m = P m 1 ^ Z 1 m )]TJ/F24 11.9552 Tf 12.814 0 Td [( and y m = G ^ Z 1 m )]TJ/F24 11.9552 Tf 12.814 0 Td [( .Suppose rstthat G isanondegeneratedistribution,sothatthereexistsa > 0 andan < 1 asin Lemma3.8.Then,onehas k ~ y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k k P m 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(G k k ~ y m k + k ~ y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k which,afterletting m !1 andinvokingtherststepoftheproof,implies k ~ y m )]TJ/F38 11.9552 Tf 12.124 0 Td [(y m k P 0 Repeatingthisargumentmultipletimesyields3{70inthecaseofnondegenerate G .Onthe otherhand,if G isconcentratedon c G ,thenonehas k ~ y m )]TJ/F38 11.9552 Tf 10.394 0 Td [(y m k t =0 almostsurely,forall m for 0 t < c G ,since P m 1 t = G t forthose t .Next,for c G t < 2 c G ,itfollowsthat k ~ y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k t k P m 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(G k t k ~ y m k t + k ~ y m )]TJ/F38 11.9552 Tf 11.955 0 Td [(y m k t )]TJ/F39 7.9701 Tf 6.587 0 Td [(c G andtherefore k ~ y m )]TJ/F38 11.9552 Tf 12.033 0 Td [(y m k t P 0 ,as m !1 ,for 0 t < 2 c G .Iteratingthisproceduremultiple timesyields3{70inthecaseofdegenerate G .Thus,3{70holds. 3.5.4ProofofLemma3.3 Beforeweturnourattentiontotheproofofthelemma,westateafewresultsthatrelate touidlimitsofourmodel.Tothisend,let X n := X n = n Z n := Z n = n and n := f n t t 0 g ,where n t := X n )]TJ/F21 11.9552 Tf 13.36 2.656 Td [( Z n )]TJ/F29 11.9552 Tf 11.955 16.272 Td [(Z t 0 X n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n = n + d P 1 s {71 Ourrstlemmashowsthat n vanishesasymptotically. Lemma3.9. Wehave n 0 ,as n !1 57

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Proof. First,thesamestraightforwardalgebraasintheproofofLemma3.6see3{32 and3{33yields n = 1 p n X L 2f G H g 1 X j l =0 )]TJ/F21 11.9552 Tf 9.298 0 Td [(1 1 f L = H g M n l j L )]TJ/F21 11.9552 Tf 14.564 2.657 Td [(~ M n l j L Therefore,foraxed m ,theprecedingequality, ~ i j i j c i ,1: j see3{28and monotonicityresultin j n t j 2 n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + n + 1 p n X L 2f G H g X j + l m j > 0 j M n l j L ; t j + j ~ M n l j L ; t j + 4 n n X i =1 X j + l > m j > 0 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 f c i ,1: j + l t g + H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(c i ,1: j =:2 n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + n + n 1, m t + n 2, m t ; {72 set n 1, m := f n 1, m t t 0 g and n 2, m := f n 2, m t t 0 g .Intheprecedinginequality,we usedithefactthat i ,0 and ~ i ,0 dierfor n )]TJ/F38 11.9552 Tf 12.1 0 Td [(k n + indices n ^ k n < i n ;andiithe factthatacenteredon-oprocessisboundedinabsolutevalueby1. Second,recallthat f c i ,1 g areindependentbyconstruction,withthedistribution G F 1 i n ^ k n G F n ^ k n < i n or F i > n .Astronglawoflargenumbersfor D [0,1] -valuedrandomvariables[24]impliesalmostsurely,as n !1 k n 2, m k t 8 X j + l > m j > 0 H H j + l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t {73 Inaddition,wehave X j + l > m j > 0 H H j + l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t X i > m iH i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t = mH m R t + H m R t 0, {74 58

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as m !1 ;thelimitisduetothefactthat H m t decreasesexponentiallyin m since H m t H t m .Relations3{73and3{74resultin,forany > 0 lim m !1 lim n !1 P [ k n 2, m k t > ]=0. {75 Ontheotherhand,processes M n l j L ; t and ~ M n l j L ; t aretightseeLemmaB.3and RemarkB.2inAppendixB.1.2,andhence n 1, m 0 ,as n !1 .Theprecedinglimit, togetherwith3{6,3{72and3{75,yieldsthestatementofthelemma. Corollary3.2 Fluidlimit Wehave X n p ,as n !1 Proof. RecallDenition4.2.From3{71,itfollowsthat X n = P 1 Z n + n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n = n + k n = n Lemma3.9 n 0 ,[76] Z n p and3{4 k n = n p yieldseealsoTheorem11.4.5 in[100,p.379] Z n + n )]TJ/F38 11.9552 Tf 10.899 0 Td [(k n = n 0 ,as n !1 .Now,thesameargumentasinTheorem3.1 basedonpropertiesof P 1 statedinProposition3.1resultsin,as n !1 X n P 1 + p = p where P 1 =0 isduetothefactthat 0 isauniquesolutionto y t = R t 0 y + t )]TJ/F38 11.9552 Tf 12.02 0 Td [(s d P 1 s seetheproofofProposition3.1. Thefollowinglemmacharacterizesuidlimitsof A n j := f A n j t =# f i : i j t g t 0 g and ~ A n j := f ~ A n j t =# f i :~ i j t g t 0 g ,for j 1 .Recallthat H = F G and, consequently, H = p G F + p F = p G + p F G Lemma3.10. For j 1 ,wehave A n j = n H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 and ~ A n j = n H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,as n !1 Proof. First,notethat 0 A n j t )]TJ/F21 11.9552 Tf 13.163 2.657 Td [(~ A n j t X n t )]TJ/F38 11.9552 Tf 12.267 0 Td [(k n + ,for t 0 .Thus,Corollary3.2 implies,as n !1 A n j = n )]TJ/F21 11.9552 Tf 12.852 2.656 Td [(~ A n j = n 0. {76 59

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Therestoftheproofisbasedoninduction.Abaseisduetotheconstructionofourmodel see3{4and3{6: A n 1 = n H ,as n !1 .Consequently,3{76implies ~ A n 1 = n H ,as n !1 .Now,for > 0 ,let a j t = H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t + ^ 1 and a # j t = H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F24 11.9552 Tf 11.549 0 Td [( + Thenourinductiveassumptionyields,as n !1 P [ E ] 1, {77 where E := f a # j t ~ A n j t = n a j t 8 t 2 [0, T ] g .Next,notethat A n j +1 t = ~ A n j t X i =1 1 f ~ i j +~ c i j +1 t g = n X i =1 1 f ~ i j +~ c i j +1 t g where ~ i j :=inf f t 0: ~ A n j t i g and ~ c i j +1 isthedurationofthenextcycleofacustomer thatstartsits j +1 stcycleattime ~ i j .Theprecedingequalityandthedenitionof E result in E n X i =1 1 f a # j i = n +~ c i j +1 t g A n j +1 t n X i =1 1 f a j i = n +~ c i j +1 t g 8 t 2 [0, T ] ; {78 recallthat f x =inf f s : f s x g isaninverseofanondecreasing f Alawoflargenumberse.g.,see[24]and 1 n n X i =1 P h a j i = n +~ c i j +1 t i = 1 n n X i =1 H t )]TJ/F21 11.9552 Tf 11.955 0 Td [( a j i = n = 1 n Z t 0 H t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s d n X i =1 1 f na j s i g = Z t 0 H t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d b na j s c n Z t 0 H t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d a j s as n !1 ,yield 1 n n X i =1 1 f a j i = n +~ c i j +1 t g t 0 Z t 0 H t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d a j s t 0 {79 60

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as n !1 .Usingthesameapproachresultsin,as n !1 1 n n X i =1 1 f a # j i = n +~ c i j +1 t g t 0 Z t 0 H t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d a # j s t 0 {80 Finally,combining3{77,3{78,3{79and3{80with H H j t )]TJ/F24 11.9552 Tf 11.955 0 Td [( Z t 0 H t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d a # j s Z t 0 H t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d a j s H H j t + concludestheproofofthelemma. Finally,wearereadytopresenttheproofofLemma3.3. ProofofLemma3.3. TheproofofthelemmaisverysimilartotheproofofLemma5.3 in[58]seealsotheproofofProposition5.1in[77].Thebasicideaistoexpress M n l j L and ~ M n l j L intermsofempiricalprocessesthatconvergetoaKieferprocesstheproofalsoutilizes tightnessoftherelevantprocesses{seeLemmaB.3andRemarkB.2inAppendixB.1.2;it isbasedonamartingaledecomposition{see[71]forasurvey.Themaindierenceisthat theuidlimitsof A n j and ~ A n j arenotidentityfunctionsasin[58,77],but H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 see Lemma3.10inSection3.5.4.Adiagramthatoutlinesrelationsbetweenvariousobjectsused intheproofisshowninFigure3-3. First,weconsidermachinesintheincreasingorderiftheir j thbreakdowntimesthe j th breakdowntimeofmachine i occursat i j .Thatis,weintroduceanewlabelingofmachines basedon A n j .DuetotheFCFSpolicy,thelabelingbasedon ~ A n j isthesame.Tothisend, weintroducesomenotation.Let s m i m + j and a m i m + j betheserviceandworkingtimesinthe m + j thcycleofamachinethatwasthe i thamongallthemachinestostartits m thcycle alsoset c m i m + j := s m i m + j + a m i m + j ;forthiscustomerlet i j :=inf f t 0: A n j t i g bethe timeofits j thbreakdown.Now,for L 2f G H g ,let V n l j L := f V n l j L ; t x t 0, x 0 g and U n l j L := f U n l j L ; t x t 0, x 2 [0,1] g bedenedby V n l j L ; t x := 1 p n A n j t X i =1 1 f c j i j +1: j + l + s j i j + l +1 + a j i j + l +1 1 f L = H g x g )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l x 61

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and U n l j L ; t x := 1 p n b nt c X i =1 1 f L H l c j i j +1: j + l + s j i j + l +1 + a j i j + l +1 1 f L = H g x g )]TJ/F38 11.9552 Tf 11.955 0 Td [(x {81 respectively;here f c j i j +1: j + l + s j i j + l +1 + a j i j + l +1 1 f L = H g i > n g isasequenceofi.i.d.random variablesdistributedaccordingto L H l andindependentofallotherrandomvariablesin model.Then, V n l j L and U n l j L relatevia V n l j L ; t x = U n l j L ; A n j t = n L H l x {82 Moreover,theprocessesofinterest, M n l j L ,canbewrittenassee3{29 M n l j L = 1 p n A n j t X i =1 1 f c j i j +1: j + l + s j i j + l +1 + a j i j + l +1 1 f L = H g t )]TJ/F22 7.9701 Tf 7.108 0 Td [( i j g )]TJ/F38 11.9552 Tf 11.956 0 Td [(L H l t )]TJ/F21 11.9552 Tf 12.569 0 Td [( i j {83 anditadmitsthefollowingrepresentationsee[58]forthedenitionoftheintegral: M n l j L ; t = Z t 0 Z t 0 1 f s + x t g d V n l j L ; s x Showingdirectlythatprocesses M n l j L and M l j L areclose"forlarge n isdicult. Instead,weintroducerandomvariablesthatapproximate M n l j L ; t and M l j L ; t .Tothisend, considerasequence f r m q g suchthat 0= r m 0 < r m 1 < r m 2 < < r m m = t and max 1 q m j r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 j! 0, {84 as m !1 .Inturn,thissequenceisusedtodene M n l j m L ; t M n l j m L ; t and M l j m L ; t asfollows: M n l j m L ; t := m X q =1 V n l j L ; r m q t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F21 11.9552 Tf 13.822 2.656 Td [( V n l j L ; r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q {85 M n l j m L ; t := m X q =1 U n l j L ; H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 r m q L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(U n l j L ; H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q {86 62

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and M l j m L ; t := m X q =1 U l j L ; H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 r m q L H l t )]TJ/F38 11.9552 Tf 11.956 0 Td [(r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(U l j L ; H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q {87 where U l j := f U l j t x t 0, x 2 [0,1] g isaKieferprocesse.g.,see[55]or[23,Sect.4.2], atwo-parametercontinuouscenteredGaussianprocesson [0, 1 [0,1] withthecovariance function E U l j L ; s x U l j L ; t y = s ^ t x ^ y )]TJ/F38 11.9552 Tf 11.955 0 Td [(xy ; U l j x isaBrownianmotionforxed x ,and U l j t isaBrownianbridgeforxed t .The lastprocessweintroduceis M n l j L := f M n l j L ; t t 0 g : M n l j L ; t := Z t 0 Z t 0 1 f s + x t g d V n l j L ; s x {88 where V n l j L ; t x := U n l j L ; H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t L H l x .Here,wenotethatthecentered Gaussianprocessofinterestwitha.s.continuouspaths[58,Lemma2.1], M l j L ,canbe expressedintermsofaKieferprocessaswell[58]: M l j L ; t = Z t 0 Z t 0 1 f s + x t g d V l j L ; s x := l.i.m. m !1 M l j m L ; t {89 where V l j L ; t x := U l j L ; H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t L H l x andl.i.m.standsforthemean-square limit.Then,from3{87and3{89,itfollowsthat,as m !1 M l j m L ; t P M l j L ; t {90 Inaddition,3{85,3{86,Lemma3.10,Lemma3.1in[58] U n l j U l j aswellasthe continuityof U l j and H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 yield,for > 0 lim n !1 P h j M n l j m L ; t )]TJ/F38 11.9552 Tf 11.955 0 Td [(M n l j m L ; t j > i =0. {91 63

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M n l j L M n l j L M l j L l 3{95 l 3{96 l 3{90 M n l j m L ; t 3{91 M n l j m L ; t 3{97 M l j m L ; t Figure3-3.AdiagramillustratingrelationsbetweenquantitiesusedintheproofofLemma3.3. Second,wearguethat M n l j m L ; t convergesto M n l j L ; t inaparticularsense,as n !1 rst,and m !1 then.Tothisend,dene R n l j m t L := f R n l j m t L ; s s 0 g by R n l j m t L ; s := A n j s X i =1 m X q =1 1 f r m q )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 < i j r m q g 1 f t )]TJ/F39 7.9701 Tf 6.587 0 Td [(r m q < c j i j +1: j + l + s j i j + l +1 + a j i j + l +1 1 f L = H g t )]TJ/F22 7.9701 Tf 7.108 0 Td [( i j g )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F21 11.9552 Tf 12.57 0 Td [( i j + L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q Then,3{81,3{82,3{83and3{85imply R n l j m t L ; t = p n M n l j L ; t )]TJ/F21 11.9552 Tf 14.564 2.657 Td [( M n l j m L ; t {92 Twoadditionalobjectsthatrelateto R n l j m t L areofinterest:the -algebra F n s l j L = n i j ^ A n j s +1, j c j i ^ A n j s j +1: j + l + s j i ^ A n j s j + l +1 + a j i ^ A n j s j + l +1 1 f L = H g ,1 i n o _N andtheprocess h R n l j m t L i := fh R n l j m t L i s s 0 g denedby h R n l j m t L i s := A n j s X i =1 m X q =1 1 f r m q )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 < i j r m q g )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(L H l t )]TJ/F21 11.9552 Tf 12.569 0 Td [( i j )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F21 11.9552 Tf 12.57 0 Td [( i j + L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q A n j s X i =1 m X q =1 1 f r m q )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 < i j r m q g )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F38 11.9552 Tf 11.956 0 Td [(r m q m X q =1 )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q A n j r m q )]TJ/F21 11.9552 Tf 12.852 2.657 Td [( A n j r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 L H l t sup 1 q m n A n j r m q )]TJ/F21 11.9552 Tf 12.851 2.656 Td [( A n j r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 o {93 64

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Then,thesameargumentasintheproofofthethirdpartofLemma5.2in[58]yieldsthat R n l j m t isan f F n s l j L s 0 g -square-integrablemartingalewiththeprocess h R n l j m t L i aspredictablequadratic-variationprocess.Thisfact,togetherwith3{92,3{93andthe Lenglart-Rebolledoinequalitye.g.,see[60,p.66]or[101],yields,for > 0 P h j M n l j L ; t )]TJ/F21 11.9552 Tf 14.565 2.657 Td [( M n l j m L ; t j > i = P h j R n l j m t L ; t j = p n > i = 2 + P sup 1 q m n A n j r m q = n )]TJ/F21 11.9552 Tf 12.852 2.657 Td [( A n j r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = n o > {94 Now,Lemma3.10,thecontinuityof H H l and3{84resultin,for > 0 lim m !1 limsup n !1 P sup 1 q m n A n j r m q = n )]TJ/F21 11.9552 Tf 12.852 2.657 Td [( A n j r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = n o > =0. Theprecedingdoublelimitand3{94imply,for > 0 lim m !1 limsup n !1 P h j M n l j L ; t )]TJ/F21 11.9552 Tf 14.564 2.657 Td [( M n l j m L ; t j > i =0. {95 Third,asomewhatsimplerargumentcanbeusedtocompare M n l j L ; t and M n l j m L ; t Inparticular,3{86and3{88render p n )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(M n l j L ; t )]TJ/F38 11.9552 Tf 11.955 0 Td [(M n l j m L ; t = b n H H j )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 t c X i =1 m X q =1 1 f r m q )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 < i j r m q g 1 f t )]TJ/F39 7.9701 Tf 6.586 0 Td [(r m q < c j i j +1: j + l + s j i j + l +1 + a j i j + l +1 1 f L = H g t )]TJ/F25 7.9701 Tf 6.586 0 Td [( i j g )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F24 11.9552 Tf 11.955 0 Td [( i j + L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q where i j = H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 i = n .Duetotheindependenceofthesummandsinthepreceding expression,onehas n Var M n l j L ; t )]TJ/F38 11.9552 Tf 11.956 0 Td [(M n l j m L ; t sup 1 q m b n H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 r m q c)-222(b n H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 c andtherefore,duetothecontinuityof H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 and3{84,onehas,for > 0 lim m !1 limsup n !1 P j M n l j L ; t )]TJ/F38 11.9552 Tf 11.955 0 Td [(M n l j m L ; t j > =0. {96 65

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Fourth,consideraset f t i i =1,... h gf r m i i =1,..., m g .Sincetheonlycondition of f r m i g is3{84,theset f t i g canbearbitrary.Then,3{86,3{87,[58,Lemma3.1] U n l j U l j andthecontinuityof U l j yield,as n !1 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(M n l j m L ; t 1 ,..., M n l j m L ; t h M l j m L ; t 1 ,..., M l j m L ; t h {97 where r m m i = t i M n l j m L ; t i := m i X q =1 U n l j L ; H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 r m q L H l t i )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(U n l j L ; H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 L H l t i )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q and M l j m L ; t i := m i X q =1 U l j H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 r m q L H l t i )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F38 11.9552 Tf 11.956 0 Td [(U l j H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 L H l t i )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q Fifth,3{90,3{91,3{95,3{96,3{97and[14,Theorem3.2]imply,as n !1 M n l j L M n l j L f.d. M l j L M l j L {98 Furthermore,repeatingtheaboveargument,onecanshowthat,as n !1 ~ M n l j L M n l j L f.d. M l j L M l j L {99 Combining3{98and3{99renders,as n !1 M n l j L ~ M n l j L M n l j L f.d. M l j L M l j L M l j L Finally,theprecedinglimit,LemmaB.3intheappendixseealsoRemarkB.2,[101, Lemma3.2]itassumestheaxiomofchoiceand[14,p.139]resultin,as n !1 M n l j L ~ M n l j L M n l j L M l j L M l j L M l j L 66

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Sixth,wearguethat M l j L hasa.s.continuouspaths.Tobegin,weshowthat M l j L isstochasticallycontinuousi.e.continuityinthemeansquaresense.Thedenitions3{87 and3{89yield,for s < t thesamepartitionisused, E M l j L ; t )]TJ/F38 11.9552 Tf 11.955 0 Td [(M l j L ; s 2 =lim m !1 E M l j m L ; t )]TJ/F38 11.9552 Tf 11.955 0 Td [(M l j m L ; s 2 =lim m !1 m X q =1 \000 H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 r m q )]TJ/F21 11.9552 Tf 13.518 2.657 Td [( H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F38 11.9552 Tf 14.944 -9.684 Td [(L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F38 11.9552 Tf 11.956 0 Td [(L H l s )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1+ L H l s )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F38 11.9552 Tf 11.956 0 Td [(r m q = Z t 0 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l s )]TJ/F38 11.9552 Tf 11.956 0 Td [(u )]TJ/F21 11.9552 Tf 12.951 -9.684 Td [(1+ L H l s )]TJ/F38 11.9552 Tf 11.955 0 Td [(u )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u d H H j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 u {100 wherethelastequalityfollowsfromthecontinuityof H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,thefactthat L H l hasatmostacountablenumberofjumpsandthefactthat sup q m j r m q )]TJ/F38 11.9552 Tf 12.84 0 Td [(r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 j! 0 ,as m !1 .Now,byLemma4.9.6in[60,p.279],toconcludethat M l j L hasa.s.continuous path,itisenoughtoshowthatthefollowingholds,foranypartition f r m q g of [0, t ] suchthat sup q m j r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(r m q )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 j! 0 ,as m !1 lim !1 limsup m !1 P m X i =1 )]TJ/F38 11.9552 Tf 5.479 -9.683 Td [(M l j L ; r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(M l j L ; r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 > # =0. TheprecedingdoublelimitfollowsfromMarkov'sinequalityandthefollowing: m X i =1 E )]TJ/F38 11.9552 Tf 5.479 -9.683 Td [(M l j L ; r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(M l j L ; r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 m X i =1 Z t 0 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(L H l r m q )]TJ/F38 11.9552 Tf 11.955 0 Td [(u )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l r m q )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(u d H H j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 u 1, where3{100isusedintherstinequality. Thisconcludestheproof. 67

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3.6NumericalExamples Inthissection,weprovidesomenumericalexamplesformachinerepairmodel.InSection 3.6.1weevaluatetheaccuracyofthediusionapproximationforsystemsofnitesize.In Section3.6.2weevaluatetheassumptionmadeabouttheresidualservice/worktimesat t =0 3.6.1Performanceofnite-sizesystem Weconsidertwofamiliesofsystemswith n =120 machines.Inallsimulations p =1 = 3 =1 and =1 = 2 .Foreach n welet n =0 inthissection.Alsotheinitialnumber ofmachinesbrokenat t =0 hasadiscreteuniformdistributionbetween k n )-283(b p k n c and k n + b p k n c where k n isthenumberofrepairmen.Formoreexamples,includingpositiveand negative n anddierentvaluesfor n ,seeAppendixC.1.Ourconclusioninthischapteralso holdsforthosesimulationresultsrepresentedinAppendixC.1. Oneaspectofourevolutionismotivatedbyourmainresultformachine-repairmodel Theorem4.1,whichisaweakconvergenceresult.Here,theperformanceofourapproximation isevaluatedbyconsideringtheCDF'sof ^ X n ^ Z n )]TJ/F24 11.9552 Tf 12.086 0 Td [( n and P 1 ^ Z n )]TJ/F24 11.9552 Tf 12.087 0 Td [( n at t =5,20 and 50 Weuse ^ Z n )]TJ/F24 11.9552 Tf 12.322 0 Td [( n theinniteserversystemasabenchmark.Tothisend,weuseQ-Qplots. Thatis,weplottheinverseofempiricaldistributionfunctionsof ^ Z n )]TJ/F24 11.9552 Tf 12.317 0 Td [( n and P 1 ^ Z n )]TJ/F24 11.9552 Tf 12.317 0 Td [( n againsttheinverseofempiricaldistributionfunctionof ^ X n .Theseempiricaldistribution functionsareobtainedbasedon1000simulationruns.Thisaspectisillustratedinplotsb,d andffordierentvaluesof t inFigures3-4and3-5. ThesecondaspectofourevaluationismotivatedbytheproofofTheorem4.1.Informally, theproofutilizesthefactthaterrori.e. ^ X n )]TJ/F24 11.9552 Tf 13.106 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 13.107 0 Td [( n couldberepresentedas increments/decrementsofaprocesswhichisconvergenttoana.s.continuous-pathprocess, as n !1 overintervals,wherethelengthoftheseintervalsareconvergentto0,as n !1 Thelengthoftheseintervalsarerelatedtotheaveragewaitingtimepercustomerpernumber ofcycles.Thereforeinthispartweplottheaveragewaitingtimepercustomerpernumberof cyclesseeplotainFigures3-4and3-5,theheat-mapfor ^ X n )]TJ/F21 11.9552 Tf 13.771 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 12.366 0 Td [( n ploteinFigures 3-4and3-5and ^ X n )]TJ/F24 11.9552 Tf 12.414 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 12.414 0 Td [( n plotcinFigures3-4and3-5.Weillustrateasthe 68

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averagewaitingtimepercustomerpernumberofcyclesincreasesbytimethevarianceof errorincreases. a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 Figure3-4.Exponentialservice/work; n =120 k n =40 n =0 ;a.averagewaitingtimeper customerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e.Heat-map of ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.956 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 69

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 Figure3-5.Uniformservice/work; n =120 k n =40 n =0 ;a.averagewaitingtimeper customerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e.Heat-map of ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 Oursimulationresults,asweexpected,indicatesasaveragewaitingtimepercustomer pernumberofcyclesaquantityconvergentto0,as n !1 increasesbytimethevariance oferrorincreases.Basedonourproof,webelievethisquantityisthemainreasonoferror intheprelimitsequence.Oneotherinterestingaspectofoursimulationresults,isthat thedistributionofourapproximationisveryclosetotheactualdistributionintheprelimit 70

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sequence.Thatis,thedistributionsof ^ X n t and P 1 ^ Z n )]TJ/F24 11.9552 Tf 12.53 0 Td [( n t areclosedforaxed t eventhoughthequantity ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n t isnotnegligible. 3.6.2Residualservice/worktimes Here,weevaluateourassumptionthatresidualservice/worktimesat t =0 aredistributed accordingtoresidualdistributionfunctionsforservice/worknamely, G and F ,respectively. Tothisend,weconsideramachinerepairmodelwhereat t =0 allmachinesareworking andtheresidualworktimesaredistributedaccordingto F ratherthan F .Thenweplot theresidualserviceandworktimesatdierent t 'stoseewhentheyareroughlydistributed accordingtoresidualdistributionfunctionsforservice/work. Figure3.6.2illustratetheresidualservice/worktimesfor n =60 and n =0 ,where servicetimesandworktimesaredistributedaccordingtoauniform,1distributionanda uniform,2distribution,respectively.Inthisgure,wendtheresidualserviceandwork timesat t =0,1.5 and5for1000simulationsandusetheaverageCDF'sforeach t asthe CDFofresidualserviceandworktimesatthat t .ThenweusetheQ-Qplottocomparethis distributionstoresidualdistributionfunctions.Weinclude t =0 justtoseehowresidualtimes changeovertime.Formoreexamples,includingdierentvaluesfor n and n seeAppendix C.2. 71

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Figure3-6. n =60 k n =20 n =0 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. Oursimulationresults,showthateveninsystemswithrelativelysmallnumberof customers,theempiricaldistributionofresidualservice/worktimesinsmallperiodsoftime willbeveryclosetoresidualdistributionofservice/worktimes.Whichvalidatesourinitial assumptionthatat t =0 theresidualserviceandworktimesaredistributedaccordingto G and F ,respectively. 72

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CHAPTER4 MULTI-CLASSPRIORITYMODEL 4.1Overeview Inthenextsection,wedeneourmodelandprovideassumptionsusedthroughoutthe chapter.InSection4.3,weintroducetwocloselyrelatedmodelsandprovidesomebasic propertiesofthesesystems.OurmainresultTheorem4.1andsomerelateddiscussioncanbe foundinSection4.4.TheproofofthemainresultisprovidedinSection4.5. 4.2AssumptionsandModel 4.2.1Model Weconsiderasequenceofnon-preemptivepriorityqueuingsystemsintheQEDregime, indexedbythenumberofservers.Thereare k < 1 classesofcustomers.Aclassi customer haspriorityoveraclassj > i customer.Thatis,aclassj customerentersserviceattime t 0 onlyifnoclassi < j customersarepresentinthewaitingroom.Withineachclass, customersareservedonaFCFSbasis. Dene n betothetotalnumberofcustomersinthesystemat t =0 ,andlet n l be thenumberofclassl customersawaitingserviceat t =0 .Theserandomvariablessatisfy n )]TJ/F38 11.9552 Tf 12.125 0 Td [(n + = n 1: k a.s.Residualservicerequirements f s j ,1 j n ^ n g of n ^ n customers inserviceattime t =0 areindependent.Let f s l j l j j 1 g l =1,..., k ,beservice requirementsandthecorrespondingarrivaltimesclassl customersthatenterserviceafter t =0 .Withineachclass,customersareindexedinorderofarrivaltoservice.Weset l j =0 for j =1,..., n l ,since n l classj customersareawaitingserviceattime t =0 .Duetothe FCFSpolicywithineachclass,wehavethat f l j j 1 g isanon-decreasingsequenceforall l Fornotationalconvenience,let E n l := f E n l t t 0 g ,where E n l t :=# f l j t : j 1 g Dene A n l := f A n l t t 0 g tobearate n l arrivalprocessofclassl customerstothe system;then,wehave A n l t =# f l j 2 t ]: j 1 g .Notethat E n l = A n l + n l .Service requirementsofclassl customersthatenterserviceafter t =0 formani.i.d.sequence 73

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distributedaccordingto G l withmean 0 < 1 = l < 1 thatisindependentofallother variablesinthesystem;let G := f G t := k i =1 G i t t 0 g Let X n t bethenumberofcustomersinthesystemattime t ;set X n := f X n t t 0 g .Then, X n t admitsthefollowingrepresentation: X n t := k X l =1 E n l t X j =1 1 f l j + s l j + w l j > t g + n ^ n X j =1 1 f s j > t g {1 where w l j isthewaitingtimeofclassl customerwiththearrivaltime l j andservice requirement s l j .Thenumberofclass l customersawaitingserviceattime t 0 isgiven by K n l t := E n l t X j =1 1 f t )]TJ/F25 7.9701 Tf 6.586 0 Td [( i j < w i j g ; {2 let K n l := f K n l t t 0 g .Notethatthewaitingtimes f w l j g in4{1and4{2aresuch that K n 1: k t = X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(n + {3 4.2.2QEDregime Inthissubsection,weintroduceassumptionsonarrivalprocesses,residualservicetimesfor customersinserviceat t =0 ,andthenumberofcustomersinthesystemat t =0 .Those assumptionsensurethattheconsideredsystemisintheQEDregime. Supposethattheoeredloadandthecapacityarerelatedviathesquare-rootrule: n := p n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n 1: k 2 R n !1 {4 where n l := n l = n l .Itisassumedthatthearrivalrates f n l g increasewithoutabound,as n !1 : n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n 1 ,..., n k 1 ,..., k {5 where 1 ,..., k < 1 .Fornotationalconvenience,weintroducescaledarrivalrates: ^ n l := 1 p n n l 74

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Notethat4{4and4{5imply ^ n l !1 ,as n !1 ,foratleastoneindex l .Let k ? :=max n l :lim n !1 ^ n l = 1 o {6 andassume ^ n l ^ l ,for k ? < l k ,as n !1 .Observethat4{5andthedenitionof k ? imply l =0 for l > k ? .Inadditiontotheseassumptionsonrates,arrivalprocessessatisfya FCLT-likecondition,as n !1 : ^ A n 1 := A n 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( n 1 e p n ,..., ^ A n k := A n k )]TJ/F24 11.9552 Tf 11.955 0 Td [( n k e p n ^ A 1 ,..., ^ A k {7 where ^ A 1 ,..., ^ A k area.s.continuous-pathprocesses. Now,introducethefollowingdistribution: G n := k X l =1 n l n 1: k G l {8 Itisassumedthatrandomvariables f s j ,1 j n ^ n g areindependentandidentically distributedaccordingto G n ,theresidualdistributionof G n = f G n x x 0 g ;recallthat G n x := R x 0 G n s d s R 1 0 G n s d s Let G and G bethelimitsof G n and G n inthetopologyofuniformconvergence,respectively: k G )]TJ/F38 11.9552 Tf 11.955 0 Td [(G n k 1 0 and k G )]TJ/F21 11.9552 Tf 13.71 2.657 Td [( G n k 1 0, {9 as n !1 ;dueto4{5thesedistributionfunctionsarewell-dened. Finally,weassumethat n thenumberofcustomersinthesystemat t =0 and n l the numberofclassl customersawaitingserviceat t =0 satisfy,as n !1 ^ n := n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n p n ,^ n 1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 := n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 p n ,^ n k ? := n k ? p n ,...,^ n k := n k p n ^ ,0,^ k ? ,...,^ k {10 where ^ + =^ k ? : k a.s. Remark 4.1 Notethat ^ n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 ,as n !1 ,isassumed.Thisassumptionisconsistent withourmainresult,Theorem4.1.InExample4.2seeSection4.4,weshowthatthe 75

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assumptionisneededforascaledandcenteredversionof X n toconvergeweaklyinthe J 1 topology. 4.3PreliminaryResults Inthissection,weconsidertwosystemsthatarecloselyrelatedtoourmainmodel. Themodied"system,examinedinSection4.3.1,isobtainedfromtheoriginalmodelby eliminatingallcustomerswithzeroservicerequirements.Theinnite-serversystem,discussed inSection4.3.2,isderivedfromtheoriginalmodelbyeliminatinganycustomerwaiting. 4.3.1Modiedsystem Let X n t bethenumberofcustomerswithpositiveservicerequirementsintheoriginal modelattime t ; X n := f X n t t 0 g .Similarly,let K n l t l =1,..., k ,bethenumber ofclassl customersawaitingserviceintheoriginalmodelattime t withpositiveservice requirements; K n l := f K n l t t 0 g .Thatis,wehave X n t := k X l =1 E n l t X j =1 1 f l j + s l j + w l j > t g 1 f s l j > 0 g + n ^ n X j =1 1 f s j > t g {11 and K n l t := E n l t X j =1 1 f t )]TJ/F25 7.9701 Tf 6.586 0 Td [( l j < w l j g 1 f s l j > 0 g {12 Ingeneral,weusethedot"symboltodenotequantitiesthatrelatetothemodiedsystem. Sincecustomerswithzeroservicerequirementsdonotoccupyservers,themodiedsystemisa k -classprioritysystemwithFCFSdisciplinewithineachclass.Classl servicerequirementsare distributedaccordingto G l := G l )]TJ/F38 11.9552 Tf 11.955 0 Td [(G l f 0 g G l Asintheoriginalmodel,wedene A n l := f A n l t t 0 g and E n l := f E n l t t 0 g ,where A n l t :=# f l j 2 t ]: s l j > 0, j 1 g and E n l t :=# f l j t : s l j > 0, j 1 g .For notationalpurposes,wedene l j s l j w l j := l o j s l o j w l o j suchthat o j :=min f i :# f s l o > 0, o i g = j g for l =1,..., k and j 1 .Inotherwords, l j s l j w l j representthearrivaltime,the 76

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servicerequirement,andthewaitingtimeofthe j tharrivingtoserviceclassl customerinthe modiedsystem.Then,onecanrewrite4{11and4{12asfollows: X n t = k X l =1 E n l t X j =1 1 f l j + s l j + w l j > t g + n ^ n X j =1 1 f s j > t g {13 and K n l t = E n l t X j =1 1 f t )]TJ/F22 7.9701 Tf 6.612 -6.521 Td [(_ l j < w l j g {14 Notethat,for l =1,..., k ,thearrivalrateof A n l is n l := G l n l ,andthemeanof G l is )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 l = G l l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 .Thisresultsinrecall4{4 n := p n )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(1 )]TJ/F21 11.9552 Tf 12.608 -11.425 Td [(_ n 1: k = p n 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( n 1: k = n {15 Next,dene ^ A n l := f ^ A n l t t 0 g and ^ E n l := f ^ E n l t t 0 g by ^ A n l t := A n l t )]TJ/F21 11.9552 Tf 13.003 -9.099 Td [(_ n l e p n and ^ E n l t := E n l t )]TJ/F21 11.9552 Tf 13.003 -9.099 Td [(_ n l e p n ^ A n l t + E n l p n Thefollowinglemmacharacterizesthelimitingarrivalprocessforthemodiedsystem. Lemma4.1. Wehave,as n !1 ^ A n 1 ,..., ^ A n k G 1 ^ A 1 + B 1 1 e ,..., G k ? ^ A k ? + B k ? k ? e G k ? +1 ^ A k ? +1 ,..., G k ^ A k where f B l l =1,..., k ? g arezero-driftBrownianmotionsindependentofeachotherand ^ A 1 ,..., ^ A k ,suchthat E B 2 l t = G l G l t t 0 Proof. SeeSection4.6.1. Similarto4{8,weintroduce G n := P k l =1 n l = n 1: k G l .Thereexistaunique G such that k G )]TJ/F21 11.9552 Tf 13.952 -9.099 Td [(_ G n k 1 0 ,as n !1 .Then,itisstraightforwardtoverifythat G n and G satisfy thefollowing: G n = G n )]TJ/F38 11.9552 Tf 11.955 0 Td [(G n f 0 g G n and G = G )]TJ/F38 11.9552 Tf 11.955 0 Td [(G f 0 g G {16 77

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Thesetwoequalitiesimplythattheresidualdistributionsof G n i.e. G n and G i.e. G are G n and G ,respectively.Indeed,for t 0 ,wehave G t = R t 0 G s d s R 1 0 G s d s = R t 0 G s = G d s R 1 0 G s = G d s = R t 0 G s d s R 1 0 G s d s = G t {17 Finally,let n bethenumberofcustomersinthemodiedsystemat t =0 .Asinthe originalmodel,weuse n l todenotethenumberofclassl customersinthemodiedsystem awaitingserviceat t =0 .Wedenealsoscaledandcenteredversionsofthesevariables: ^ n := n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n p n and ^ n l := n l p n Thefollowinglemmadescribesthestateofthemodiedsystemattime t =0 Lemma4.2. Wehave,as n !1 ^ n ,^ n 1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,^ n k ? ,...,^ n k ^ )]TJ/F39 7.9701 Tf 18.741 14.944 Td [(k X l = k ? G l ^ l ,0, G k ? ^ k ? ,..., G k ^ k Proof. SeeSection4.6.2 Remark 4.2 Lemma4.1,Lemma4.2,thecontinuousmappingtheoremand n l = E n l imply, as n !1 ^ E n 1 ,..., ^ E n k ^ E 1 ,..., ^ E k where ^ E l 8 > > > > > > < > > > > > > : G l ^ A l + ^ B l l e l < k ? G l ^ A l + ^ B l l e + G l ^ l l = k ? G l ^ A l + G l ^ l l > k ? 4.3.2Innite-serversystem Here,weconsidertheinnite-serverprocess Z n := f Z n t t 0 g obtainedfrom4{1 byeliminatingallwaitingtimes: Z n t = k X l =1 E n l t X j =1 1 f l j + s l j t g + n ^ n X j =1 1 f s j t g {18 78

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Lemma4.3 Diusionlimitforinnite-serverprocess Wehave,as n !1 ^ Z n := Z n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n n 1: k p n ^ Z ^ G + V G )]TJ/F21 11.9552 Tf 12.963 2.657 Td [(^ )]TJ/F21 11.9552 Tf 8.839 -0.153 Td [( G + k X l = k ? ^ l G l + M + W where V ^ M and W aremutuallyindependent,and: V = f V t t 2 [0,1] g isaBrownianbridgewith E V s V t = s ^ t )]TJ/F38 11.9552 Tf 12.81 0 Td [(st ,for 0 s t 1 ; M = f M t t 0 g isacenteredGaussianprocesswitha.s.continuouspaths, M =0 and E M t M t + = k X l =1 l Z t 0 G l t + )]TJ/F38 11.9552 Tf 11.955 0 Td [(u G l t )]TJ/F38 11.9552 Tf 11.956 0 Td [(u d u 0; W = f W t t 0 g isana.s.continuous-pathprocesswith W t = k X l =1 Z t 0 G l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s ^ A l d s Proof. SeeSection4.6.3. Remark 4.3 Sincecustomerswithzeroservicerequirementsdonotoccupyanyservers, theinnite-serverprocessesthatcorrespondto X n and X n thatis, Z n and Z n areequal. Moreover,dueto n l = n l ,theirscaledandcenteredversionsareequalaswell: ^ Z n = ^ Z n := Z n )]TJ/F38 11.9552 Tf 11.956 0 Td [(n n 1: k p n Asaconsequence,wehave ^ Z n ^ Z ,as n !1 .WenotethatLemma4.3canbeappliedto ^ Z n directly.Inthatcase,oneobtainsaslightlydierentbutequivalentrepresentationof ^ Z Inparticular,onecanreplacethepair M W withapair M W ,where M t = M t )]TJ/F39 7.9701 Tf 14.657 14.944 Td [(k ? X l =1 Z t 0 G l t )]TJ/F38 11.9552 Tf 10.594 0 Td [(s B l l d s and W t = W t + k ? X l =1 Z t 0 G l t )]TJ/F38 11.9552 Tf 10.595 0 Td [(s B l l d s for t 0 ;here f B l l =1,..., k ? g arezero-driftmutuallyindependentBrownianmotions independentof M W ,suchthat E B 2 l t = G l G l t t 0 seeLemma4.1.The equivalenceofthetworepresentationsfor ^ Z isdueto M + W = M + W ,aswellas4{15, 79

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4{16,4{17andLemma4.2.Inparticular,onehas ^ l G l =^ l G l and ^ )]TJ/F21 11.9552 Tf 10.406 -5.147 Td [(= ^ + )]TJ/F21 11.9552 Tf 12.963 2.657 Td [(^ =^ k ? : k )]TJ/F21 11.9552 Tf 12.962 2.657 Td [(^ = k X l = k ? G l ^ l )]TJ/F21 11.9552 Tf 12.963 2.657 Td [(^ + k X l = k ? G l ^ l =^ k ? : k )]TJ/F21 11.9552 Tf 12.962 2.657 Td [(^ = ^ + )]TJ/F21 11.9552 Tf 12.963 2.657 Td [(^ = ^ )]TJ/F21 11.9552 Tf 7.085 -4.937 Td [(. Remark 4.4 Considerthecasewhenthearrivalprocesses A n l ,for l =1,..., k ,aresplit from A n inthei.i.d.fashion;thatis,acustomerbelongstoclassl withasplittingprobability p n l := n l = n 1: k .Then,4{7followsfromthefollowingweakerassumption: ^ A n A n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n 1: k e p n ^ A as n !1 ,where ^ A isana.s.continuouspathprocess.Inparticular,4{5implies p n l p l = l = 1: k ,as n !1 ,andwehave,as n !1 ^ A n 1 ,..., ^ A n k ^ A 1 ,..., ^ A k where ^ A l p l ^ A + B l 1: k e ,and B 1 ,..., B k isazero-drift k -dimensionalBrownianmotion with E B 2 i t = p i p i t t 0 ,and E B i t B j t = )]TJ/F38 11.9552 Tf 9.299 0 Td [(p i p j t i 6 = j t 0 .Furthermore,in thisspecialcase,anequivalentrepresentationof ^ Z exists.Namely,Lemma4.3holdswith processes M and W replacedwithprocesses N and Y ,respectively: N = f N t t 0 g isacenteredGaussianprocesswitha.s.continuouspaths, N =0 and E N t N t + = 1: k Z t 0 G t + )]TJ/F38 11.9552 Tf 11.955 0 Td [(u G t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u d u 0; Y = f Y t t 0 g isa.s.continuous-pathprocesswith Y t = Z t 0 G t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s ^ A d s 80

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Weconcludethissectionwitharelationbetween X n and Z n .Tothisend,straightforward algebraand4{14yield X n t = Z n t + k X l =1 E n l t X j =1 1 f 0 t )]TJ/F22 7.9701 Tf 6.612 -6.521 Td [(_ l j )]TJ/F22 7.9701 Tf 6.432 -6.521 Td [(_ s l j < w l j g = Z n t + n t + k X l =1 E n l t X j =1 G l t )]TJ/F21 11.9552 Tf 12.131 -9.1 Td [(_ l j )]TJ/F21 11.9552 Tf 13.773 -9.1 Td [(_ G l t )]TJ/F21 11.9552 Tf 12.131 -9.1 Td [(_ l j )]TJ/F21 11.9552 Tf 13.724 -9.1 Td [(_ w l j = Z n t + n t + k X l =1 Z t 0 E n l t )]TJ/F39 7.9701 Tf 6.587 0 Td [(s X j =1 1 f 0 t )]TJ/F39 7.9701 Tf 6.586 0 Td [(s )]TJ/F22 7.9701 Tf 6.611 -6.521 Td [(_ l j < w l j g G l d s = Z n t + n t + k X l =1 Z t 0 K n l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s {19 where n t := k X l =1 E n l t X j =1 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(1 f 0 t )]TJ/F22 7.9701 Tf 6.611 -6.521 Td [(_ l j )]TJ/F22 7.9701 Tf 6.432 -6.521 Td [(_ s l j < w l j g )]TJ/F21 11.9552 Tf 13.773 -9.1 Td [(_ G l t )]TJ/F21 11.9552 Tf 12.131 -9.1 Td [(_ l j + G l t )]TJ/F21 11.9552 Tf 12.131 -9.1 Td [(_ l j )]TJ/F21 11.9552 Tf 13.724 -9.1 Td [(_ w l j {20 4.4MainResults Inthissection,westateourmainresult,Theorem4.1,andprovideanoutlineofitsproof. Twoexamplesconcludethesection.Forthemodiedsystem,thefollowinglemmarelatesthe numberofcustomersawaitingservicefromeachclasswiththetotalnumberofcustomersa correspondingresultholdsfortheoriginalmodelaswell,sincetheproofdoesnotutilizethe factthatservicerequirementsarepositive. Lemma4.4. For l =1,..., k ,wehave K n 1: l t = sup 0 u t f X n )]TJ/F38 11.9552 Tf 11.956 0 Td [(n + [ u t ] )]TJ/F21 11.9552 Tf 13.325 -9.099 Td [(_ E n l +1: k [ u t ] g + Proof. SeeSection4.6.4. Theprecedinglemmamotivatesthefollowingdenition: 81

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Denition4.1. Denethemapping : D 2 [0, 1 D [0, 1 suchthat,for t 0 and x a 2 D [0, 1 x a t := sup u 2 [0, t ] f x + [ u t ] )]TJ/F38 11.9552 Tf 11.955 0 Td [(a [ u t ] g + Inordertostatethemainresult,wedenescaledandcenteredversionsofrelevant processes: ^ X n := f ^ X n t t 0 g ^ X n := f ^ X n t t 0 g ^ K n l := f ^ K n l t t 0 g and ^ K n l := f ^ K n l t t 0 g ,where ^ X n t := X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(n p n ^ X n t := X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(n p n ^ K n l t := K n l t p n and ^ K n l t := K n l t p n Informally,as n increases,onlyclass k ? ,..., k customersarepresentinthewaitingroomon thediusion p n scale.Thisfactisconsistentwiththeinitialassumption4{10,whichwe furtherdiscussinExample4.2attheendofthissection. Theorem4.1. Considerthesequenceof k -classpriorityqueuingmodelsintheQEDregime describedinSection4.2.Then,as n !1 ^ X n ^ K n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ^ K n k ? ,..., ^ K n k ^ X + k X l = k ? G l G l ^ K l ,0, ^ K k ? G k ? ,..., ^ K k G k {21 where ^ X ^ K k ? ,..., ^ K k istheuniquesolutionofthefollowingsystemofequations: 8 > > < > > : ^ X t = ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( + P k l = k ? R t 0 ^ K l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s t 0, ^ K k ? : l = ^ X ^ E l +1: k + ^ l +1: k e l = k ? ,..., k {22 Moreover,thefollowinglimit,as n !1 ,holdsforthesequenceofmodiedsystemsdescribed inSection4.3.1: ^ X n ^ K n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ^ K n k ? ,..., ^ K n k ^ X ,0, ^ K k ? ,..., ^ K k {23 Proof. Section4.5containsadetailedproof.Here,weprovideanoutline.Theproofconsists ofseveralsteps.IntherststepSection4.5.1,weshow n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 2 n 0 ,as n !1 ;thus, ^ Z n )]TJ/F24 11.9552 Tf 9.531 0 Td [( n + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 2 n ^ Z )]TJ/F24 11.9552 Tf 9.531 0 Td [( ,as n !1 see4{4,4{19andLemma4.3.Themainsecond stepSection4.5.2dealswiththediusionlimitforthemodiedsystem.Themodiedsystem 82

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isconsideredbecauseLemmaB.4inAppendixB.2guaranteesthat4{22hasaunique solution G := k l =1 G l =0 ,seeRemarkB.3.InthelaststepSection4.5.3,thelimit fortheoriginalsystemisderivedfromtheoneforthemodiedsystem.Thisisachievedby separatelyconsideringcustomerswithzeroandpositiveservicerequirements,andnotingthat customerswithzeroservicerequirementsdonotoccupyanyservers ^ X n )]TJ/F21 11.9552 Tf 10.406 -4.338 Td [(= ^ X n )]TJ/F17 11.9552 Tf 7.084 -4.338 Td [(. Inthemainstepoftheproof,weintroducesequences f ~ X n i i 0 g n 1 ,and f ~ X i i 0 g thatareusedtoapproximate ^ X n n 1 ,and ^ X ,respectively.Thesequences aredenedrecursively:i ~ X n 0 = ^ Z n and ~ X 0 = ~ Z here ^ Z n ^ Z d = ~ Z ,as n !1 ,due toLemma4.3;andii ~ X n i and ~ X i aredenedintermsof ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 and ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,respectively.The denitionsoftheseapproximationsaresuchthattheyallowustoshow:i ~ X n i ~ X i a.s.,as n !1 usinganinductiveargument;andii ~ X n i ~ X n d = ^ X n and ~ X i ~ X d = ^ X a.s.,as i !1 basedonLemmaB.4inAppendixB.2. Remark 4.5 Let ^ X ^ K 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ^ K k ? ,..., ^ K k bedenedastheright-handsideof4{21,i.e., ^ X ^ K 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ^ K k ? ,..., ^ K k := ^ X + k X l = k ? G l G l ^ K l ,0, ^ K k ? G k ? ,..., ^ K k G k {24 InAppendix ?? ,weshowthatthenewlydenedvectorsolvesacorrespondingversion of4{22fortheoriginalsystem: 8 > > < > > : ^ X t = ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( + P k l = k ? R t 0 ^ K l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s t 0, ^ K k ? : l = ^ X ^ E l +1: k + ^ l +1: k e l = k ? ,..., k {25 When k ? = k ,thestatementofTheorem4.1canbesimplied. Corollary4.1. Considerthesequenceof k -classpriorityqueuingmodelsdescribedinSection 4.2.If k ? = k ,then,as n !1 ^ X n ^ K n 1: k )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 G k ^ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( ,0, 83

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wherethemapping F : D [0, 1 D [0, 1 issuchthat F x ,foreach x 2 D [0, 1 ,isthe uniquesolution y to y t = x t + Z t 0 y + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s F d s Remark 4.6 IfthedescribedsystemweretooperateundertheFCFSpolicyinsteadofpriority scheduling,then[77]wouldyield ^ X n G ^ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( ,as n !1 ,where G isdenedin4{9. Proof. ItfollowsFromTheorem4.1that,as n !1 ^ X n ^ K n 1: k )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ^ X := ^ X + G k G k ^ X + ,0 where ^ X = G k ^ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( ;here,weusedthefactthat ^ X ,0= ^ X + : ^ X ,0 t =sup 0 u t ^ X + [ t u ]= ^ X + t Hence,for t 0 ,wehave ^ X t = ^ X t + G k G k ^ X + t = ^ Z t )]TJ/F24 11.9552 Tf 11.956 0 Td [( + Z t 0 ^ X + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G k d s + G k G k ^ X + t = ^ Z t )]TJ/F24 11.9552 Tf 11.956 0 Td [( + Z t 0 G k ^ X + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G k d s + G k ^ X + t = ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( + Z t 0 ^ X + t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s G k d s wherethethirdequalityisdueto ^ X + = 1+ G k G k ^ X + = 1 G k ^ X + Thecorollaryfollowsfromthefactthat G k iswell-dened,Lipschitzcontinuousinthe metrictopologyof D [0, 1 kk [77,Proposition3.1],andthereforemeasurablein B D [0, 1 d J 1 Weconcludethissectionwithtwoexamples. 84

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Example 4.1 Considerasequenceof k -classpriorityqueuingmodelswith k ? = k and G k [ s s ]=1 ,forsome s > 0 .Then,Corollary4.1yields ^ X t = ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( + ^ X + t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s = ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( + ^ X t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s = ^ Z t )]TJ/F24 11.9552 Tf 11.955 0 Td [( ^ Z t )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 + ^ Z t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s + ^ X + t )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 s = = b t = s c i =0 i X j =0 ^ Z t )]TJ/F38 11.9552 Tf 11.956 0 Td [(js )]TJ/F24 11.9552 Tf 11.955 0 Td [( ; {26 here,weusedthefactthat ^ X t = ^ Z t )]TJ/F24 11.9552 Tf 12.661 0 Td [( ,for t < s .When ^ k l =1 G l s =1 f ^ Z t )]TJ/F38 11.9552 Tf -433.798 -23.908 Td [(js j =0,..., b t = s cg areindependentrandomvariables.Underthiscondition,thestructure of4{26isthesameasforthelimitingprocessofaG/D/ n FCFSsystemintheQEDregime, withdeterministicservicerequirementsequalto s [52].Wetermthiscorrespondencethe reduced-supportequivalence .Weconjecturethatthisequivalenceappliestoasequenceof shortest-jobrstSJFqueuesintheQEDregime.Inparticular,weconjecturethatasequence ofappropriatelyinitializedSJFqueuesintheQEDregimesatises,as n !1 ^ X n := X n )]TJ/F38 11.9552 Tf 11.956 0 Td [(n p n 1 f s g ^ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( where s =sup f u : G u < 1 g ,and ^ Z isthecorrespondinglimitingscaledandcentered innite-serverprocesswith E ^ Z t ^ Z t + s =0 t 0 Example 4.2 Considerasequenceoftwo-classpriorityqueuingmodelswith k ? =2 ,where G 1 [1,1]= G 2 [2,2]=1 .ArrivalprocessesforthetwoclassesareindependentandPoisson. Suppose ^ n 1 ^ 1 ,as n !1 ,where P [ ^ 1 > 0 ] > 0 .Wearguebycontradiction that f ^ X n n 1 g doesnotconvergeweaklyunderthe J 1 metric.Tothisend,suppose ^ X n ^ X 2 D [0, 1 ,as n !1 ,anddene n :=inf f t > 0: K n 1 t =0 g .Since ^ X 2 D [0, 1 andclass-2jobsdonotenterserviceduringthetimeinterval [0, n ,weconclude n P )778(! 0 ,as n !1 otherwise,thesequence f ^ X n n 1 g isunbounded,since ^ n 2 n !1 as n !1 85

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Next, G 1 [1,1]= G 2 [2,2]=1 and4{20resultin n =0 .Consequently,4{19implies ^ X n t = ^ Z n t )]TJ/F24 11.9552 Tf 12.15 0 Td [( n t 2 [0,1 and ^ X n t = ^ Z n t )]TJ/F24 11.9552 Tf 12.15 0 Td [( n + ^ K n 1 t )]TJ/F21 11.9552 Tf 12.15 0 Td [(1 t 2 [1,2 .Thus,we have,as n !1 ^ X n )]TJ/F21 11.9552 Tf 10.836 0 Td [(1 = n ^ X n + n = ^ Z n )]TJ/F21 11.9552 Tf 10.835 0 Td [(1 = n )]TJ/F24 11.9552 Tf 10.836 0 Td [( n ^ Z n + n )]TJ/F24 11.9552 Tf 10.835 0 Td [( n ^ Z )]TJ/F21 11.9552 Tf 9.299 0 Td [( )]TJ/F24 11.9552 Tf 10.836 0 Td [( ^ Z )]TJ/F24 11.9552 Tf 10.835 0 Td [( where ^ Z [1,1]= )]TJ/F21 11.9552 Tf 9.53 0 Td [(^ 1 .Giventheprecedinglimit,LemmaB.6inAppendixB.2implies P [ j ^ X [1,1] j > ] > 0 ,forsome > 0 .However,sincethearrivalprocessesarePoissonand residualservicetimesat t =0 arecontinuousrandomvariables,atmostonearrival/departure occursatatime: P sup 0 < t < 2 j ^ X n [ t t ] j > 1 p n = P sup 0 < t < 2 j X n [ t t ] j > 1 =0. Therefore,jumpsin ^ X n and ^ X donotmatch{thisisacontradictionwiththeassumption ^ X n ^ X 2 D [0, 1 underthe J 1 metric,as n !1 .Toconclude, ^ 1 =0 a.s.isneededfor convergenceinthe J 1 -metrictopology. 4.5ProofofTheorem4.1 Intherstsubsection,weshow n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 2 n 0 ,as n !1 see4{19and4{20.The followingsubsectioncontainsaproofofadiusionlimitforthemodiedsystem.Finally,in Section4.5.3,weprovetheresultfororiginalsystem. 4.5.1Preliminaryresult Inthissubsection,weprovethefollowinglemma: Lemma4.5. Wehave n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 2 n 0 ,as n !1 TheproofforthislemmaisverysimilartotheproofofLemma6in[70]seealso[77]. Therefore,weomitsomedetails.For l =1,..., k ,dene E n l := f E n l t t 0 g ,where E n l t := E n l t X j =1 1 f l j + w l j t g 86

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and f l j := l j + w l j j 1 g arejumppointsof E n l .Then,onecanwrite n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 2 n = k X l =1 M n l )]TJ/F21 11.9552 Tf 14.564 2.656 Td [(~ M n l ; {27 here, M n l := f M n l t t 0 g and ~ M n l := f ~ M n l t t 0 g aredenedasfollows: M n l t := 1 p n E n l t X j =1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(1 f s l j t )]TJ/F22 7.9701 Tf 6.612 -6.521 Td [(_ l j g )]TJ/F21 11.9552 Tf 13.774 -9.099 Td [(_ G l t )]TJ/F21 11.9552 Tf 12.13 -9.099 Td [(_ l j and ~ M n l t := 1 p n E n l t X j =1 1 f s l j t )]TJ/F22 7.9701 Tf 6.944 0 Td [( l j g )]TJ/F21 11.9552 Tf 13.773 -9.1 Td [(_ G l t )]TJ/F21 11.9552 Tf 12.352 0 Td [( l j ByfollowingthestepsoftheproofofLemma6in[70]akeyobservationhereisthatFCFS schedulingisemployedwithineachclass,onecanshow M n l ~ M n l M l M l {28 as n !1 ,where M l := f M l t t 0 g isacenteredGaussianprocesswitha.s.continuous paths.Thislimitimplies M n l )]TJ/F21 11.9552 Tf 14.832 2.657 Td [(~ M n l 0 ,as n !1 ,andthestatementofthelemmafollows from4{27.Inordertoverify4{28,oneneedstochecktwoconditions:thesequences f n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 E n l t n 1 g and f n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 E n l t n 1 g areC-tight,andtheweaklimitofthe twosequencesisthesamea.s.continuous-pathprocess.Bothconditionsareimpliedbythe followingproposition: Proposition4.1. n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 E n l l e and n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 E n l l e ,as n !1 ,for l =1,..., k Proof. Theproofoftherstlimitfollowsfrom4{5,4{7,Lemma4.1andLemma4.2. Thus,wefocusonthesecondlimit.Theideaistoutilize 0 E n l )]TJ/F21 11.9552 Tf 14.223 2.657 Td [( E n l X n )]TJ/F38 11.9552 Tf 12.917 0 Td [(n + i.e.,itissucienttodemonstrate n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X n 1 ,as n !1 .Itfollowsfrom4{19and 87

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0 K n l X n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n + thatthefollowingholds: n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 X n t = n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Z n t + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n t + k X l =1 Z t 0 n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 K n l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Z n t + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n t + Z t 0 n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 X n t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 + G 1: k d s {29 Next,dene ~ : D 2 [0, 1 D [0, 1 by ~ z x t := z t + Z t 0 x + t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s G 1: k d s InviewofLemmaB.4inAppendixB.2consider x 7! ~ z x ,themapping : D [0, 1 D [0, 1 ,suchthat z istheuniquesolutionof x = ~ z x ,iswell-dened.Then,4{29 canberestatedas n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X n )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 y n 0 := ~ n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Z n )]TJ/F21 11.9552 Tf 11.955 0 Td [(1+ n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X n )]TJ/F21 11.9552 Tf 11.955 0 Td [(1. Deningasequence f y n i i 0 g by y n 0 and y n i := ~ n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Z n )]TJ/F21 11.9552 Tf 11.955 0 Td [(1+ n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 n y n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 i 1, yieldsthatthesequenceisnon-decreasingsince y n 0 y n 1 .Hence, n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 X n )]TJ/F21 11.9552 Tf 12.476 0 Td [(1 y n i forall i 0 .Moreover,LemmaB.4implies y n i n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Z n )]TJ/F21 11.9552 Tf 12.369 0 Td [(1+ n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 n ,as i !1 .Combining thesetwofactsresultsin n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X n )]TJ/F21 11.9552 Tf 11.956 0 Td [(1 n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Z n )]TJ/F21 11.9552 Tf 11.956 0 Td [(1+ n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 n {30 Next,Lemma4.3yields,as n !1 n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Z n = 1 p n ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n +1 1. {31 Ontheotherhand,from4{27oneobtains n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 n = k X j =1 1 p n M n j )]TJ/F21 11.9552 Tf 18.234 8.088 Td [(1 p n ~ M n j 88

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ByrepeatingthestepsofProposition4.2in[77],onecanshowthateachterminthepreceding sumconvergesweaklyto 0 .Theproofisbasedontwoconditionsthatholdinthiscase: FCFSpolicywithineachclass,and f n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 E n l t n 1 g and f n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 E n l t n 1 g being stochasticallyboundedoncompactintervals,whichfollowsfrom E n l E n l and n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 E n l l e as n !1 .Inconclusion,as n !1 n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n 0. {32 Now,LemmaB.4Lipschitzcontinuityof ,thecontinuousmappingtheorem,4{31and 4{32yield,as n !1 n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Z n )]TJ/F21 11.9552 Tf 11.955 0 Td [(1+ n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n =0= ~ ,0. {33 Finally, Z n X n ,4{30,4{31and4{33resultin n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X n 1 ,as n !1 .Thisconcludes theproofoftheproposition. 4.5.2Thediusionlimitforthemodiedsystem Inthissubsection,weprovethestatementofTheorem4.1thatrelatestothemodied system,namelythelimit4{23.Theproofconsistsofseveralsteps.Notethatthemetricused inthissectionistheuniformmetriconcompactintervalsi.e. d x y = k x )]TJ/F38 11.9552 Tf 12.399 0 Td [(y k t forsome t 0 ,whichisastrongermetricthanthe J 1 metriconcompactintervals. First,weintroducethefollowingdenitionfor m =1 or k ? .Theintroducedmapsare well-denedbyLemmaB.4andLemmaB.5inAppendixB.2. Denition4.2. Let a m +1 ,..., a k x z 2 D [0, 1 .Thendenemappings ~ m : D k )]TJ/F39 7.9701 Tf 6.586 0 Td [(m +2 [0, 1 D [0, 1 suchthatforany t 0 ~ m x z a m +1 ,..., a k t := z t + k X l = m Z t 0 f m l x a m +1 ,..., a k t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s where f m m : l x a m +1 ,..., a k = x a l +1: k l = m ,..., k 89

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Furthermore,themapping m : D k )]TJ/F39 7.9701 Tf 6.586 0 Td [(m +1 [0, 1 D [0, 1 issuchthat m z a m +1 ,..., a k is theuniquesolutionof ~ m x z a m +1 ..., a k = x Second,weintroduceseveralprocesses.BySkorokhod'srepresentationtheorem, Lemma4.3,Lemma4.5and4{4,thereexistasequence f ~ Z n ~ E n 2 ,..., ~ E n k n 1 g anda process ~ Z ~ E 2 ,..., ~ E k suchthat ~ Z n ~ E n 2 ,..., ~ E n k d = ^ Z n ^ E n 2 ,..., ^ E n k and ~ Z ~ E 2 ,..., ~ E k d = ^ Z ^ E 2 ,..., ^ E k ,aswellas ~ Z n ~ E n 2 ,..., ~ E n k ~ Z ~ E 2 ,..., ~ E k a.s. {34 as n !1 ,inthetopologyofuniformconvergence.Fornotationalsimplicity,wedene n l := ~ E n l + ^ n l e d = 1 p n E n l l =2,..., k and l := ~ E l + ^ l e l = k ? +1,..., k {35 Next,weintroduce ~ X n ~ K n 1 ,..., ~ K n k and ~ X ~ K 1 ,..., ~ K k asfollows: 8 > > < > > : ~ X n := 1 ~ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n + n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 2 n n 2 ,..., n k ~ K n 1: l := ~ X n n l +1: k l =1..., k and 8 > > > > > > < > > > > > > : ~ X := k ? ~ Z )]TJ/F24 11.9552 Tf 11.956 0 Td [( k ? +1 ,..., k ~ K k ? : l := ~ X l +1, k l = k ? ,..., k ~ K l :=0, l =1,..., k ? )]TJ/F21 11.9552 Tf 11.955 0 Td [(1. Then,thesedenitions,4{19,Lemma4.4andTheorem4.1imply ~ X n ~ K n 1 ,..., ~ K n k d = ^ X n ^ K n 1 ,..., ^ K n k and ~ X ~ K 1 ,..., ~ K k d = ^ X ^ K 1 ,..., ^ K k Finally,inordertorelate ~ X n ~ K n 1 ,..., ~ K n k and ~ X ~ K 1 ,..., ~ K k ,wecreateadditionalsequences indexedby i 0 : ~ X n i := 8 > > < > > : ~ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n i =0, ~ 1 ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n + n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 2 n n 2 ,..., n k i 1, {36 ~ K n i ,1: l := 8 > > < > > : 0, l =1..., k i =0, ~ X n i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 n l +1: k l =1..., k i 1, {37 90

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and ~ X i := 8 > > < > > : ~ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( i =0, ~ k ? ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( k ? +1 ,..., k i 1, ~ K i k ? : l := 8 > > < > > : 0, l = k ? ,..., k i =0, ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 l +1: k l = k ? ,..., k i 1, ~ K i l :=0, l =1,..., k ? )]TJ/F21 11.9552 Tf 11.956 0 Td [(1, i 0. Third,weuseinductiontoshow ~ X n i ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ~ X i ,0 a.s.,as n !1 ,for i 0 .The baseoftheinduction i =0 isprovidedby4{4,4{36,4{37andLemma4.3.Assume thatthelimitholdsforsomeindex i )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 .Then,for > 0 ,dene T n i t :=sup n u t :0 ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ n k ? : k [ u t ]+ ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t o ; {38 notethat T n i t iswell-dened,since ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = ~ X n i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 n k ? : k see4{37.Furthermore, since ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ n k ? : k hasaleftlimitatanypoint,wehave ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ n k ? : k [ T n i t ]+ ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t Notebythefactthat n =0 and G l =0 itfollows ~ X n i = ~ Z n )]TJ/F24 11.9552 Tf 12.112 0 Td [( n = n 1: k for i =0,1,... .Thisfactandtheprecedinginequalityimply,for > 0 k ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k t +sup 0 < s t ~ X n i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 + [ T n i s s ] )]TJ/F21 11.9552 Tf 17.9 0 Td [(inf 0 < s t n k ? : k [ T n i s s ]+ n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 +sup 0 s t ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ T n i s s ]+sup 0 s t ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ T n i s s ]+ n 1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 +2 k ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k t +sup 0 s 1 s 2 t s 2 )]TJ/F39 7.9701 Tf 6.587 0 Td [(s 1 n i ~ X + i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 [ s 1 s 2 ]+ n 1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 {39 Noteby4{10anddenitionof n l 'swehave n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 a.s.,as n !1 .Thesecond inequalityisduetonondecreasingsamplepathsof n k ? : k andthefactthat T n i s s for 91

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s 0 ,and 0 n i :=sup s t s )]TJ/F38 11.9552 Tf 11.955 0 Td [(T n i s sup s t 2 k ~ X n i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k s +2 k ~ E n k ? : k k s + ^ n k ? : k = 2 k ~ X n i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k t +2 k ~ E n k ? : k k t + ^ n k ? : k 0 a.s. {40 as n !1 ;here,theinequalityfollowsfrom4{35,4{38, ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t 0 ,and x [ s t ] 2 k x k t ;thelimitisduetotheinductiveassumption,4{6and4{34.Now, 4{39,4{40,theinductiveassumptionandLemma4.6attheendofthissubsection ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 is ana.s.lower-semi-continuous-pathprocessyield,as n !1 ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 a.s.{41 Next,thetriangleinequality,straightforwardalgebraandLemmaB.5inAppendixB.2yield,for t 0 k ~ X n i )]TJ/F21 11.9552 Tf 13.646 2.656 Td [(~ X i k t k ~ X n i )]TJ/F21 11.9552 Tf 13.256 2.656 Td [(~ k ? ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( k ? +1 ,..., k k t + k ~ k ? ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ Z )]TJ/F24 11.9552 Tf 11.956 0 Td [( k ? +1 ,..., k )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X i k t k ~ Z n )]TJ/F21 11.9552 Tf 13.36 2.657 Td [(~ Z k t +2 k ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k t +2 k X j = k ? +1 k ~ E n j : k )]TJ/F21 11.9552 Tf 13.261 2.657 Td [(~ E j : k k t + k ~ k ? ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ Z )]TJ/F24 11.9552 Tf 11.955 0 Td [( k ? +1 ,..., k )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X i k t Notethatcontinuityof x 7! ~ k ? x ~ Z )]TJ/F24 11.9552 Tf 12.247 0 Td [( k ? +1 ,..., k LemmaB.4inAppendixB.2,the continuousmappingtheorem,andtheinductiveassumption ~ X n i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 a.s.,as n !1 imply ~ k ? ~ X n i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ~ Z )]TJ/F24 11.9552 Tf 12.599 0 Td [( k ? +1 ,..., k ~ X i a.s.,as n !1 .Thislastlimit,Lemma4.3, 4{34and4{41resultin ~ X n i ~ X i a.s.,as n !1 .Thiscompletestheproofof ~ X n i ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ X i ,0 a.s. {42 as n !1 ,for i 0 92

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Fourth,thetriangleinequalityandLemmaB.4yieldthatthereexistsan i t whichdoes notdependon n ,suchthat,forall i i t k ~ X n )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X k t k ~ X n )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X n i k t + k ~ X n i )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X i k t + k ~ X i )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X k t c t k ~ Z n k t 4 k )]TJ/F21 11.9552 Tf 11.956 0 Td [(3 4 k )]TJ/F21 11.9552 Tf 11.956 0 Td [(2 i + k ~ X n i )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X i k t + C t k ~ Z k t 4 k )]TJ/F38 11.9552 Tf 11.955 0 Td [(k ? +1 4 k )]TJ/F38 11.9552 Tf 11.955 0 Td [(k ? +2 i forsome c t and C t thatdonotdependon n .Byletting n !1 rstandthen i !1 inthe precedinginequality,4{34and4{42resultin,as n !1 ~ X n ~ X a.s.{43 Similarly,thetriangleinequality,LemmaB.4andLemmaB.5yield,for > 0 k ~ K n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k t k ~ K n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F21 11.9552 Tf 13.734 2.656 Td [(~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k t + k ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k t 2 k ~ X n )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(~ X n i k t + k ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k t 2 c t k ~ Z n k t 4 k )]TJ/F21 11.9552 Tf 11.955 0 Td [(3 4 k )]TJ/F21 11.9552 Tf 11.956 0 Td [(2 i + k ~ K n i ,1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k t Theprecedinginequality,4{34and4{42resultin,as n !1 ~ K n 1: k )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 a.s.{44 Then,4{34,4{43,4{44,continuityof x a 7! x a LemmaB.5,thefactthat x 7! x f x iscontinuousif x 7! f x iscontinuous,andthecontinuousmappingtheorem resultin,as n !1 ~ X n n k ? +1 ,..., n k ~ K n 1: k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ K n k ? ,..., ~ K n k ~ X k ? +1 ,..., k ,0, ~ K k ? ,..., ~ K k a.s.{45 Finally,Skorokhod'srepresentationtheorem,4{43,4{44and4{45yield,as n !1 ^ X n ^ K n 1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ^ K n k ? ,..., ^ K n k ^ X ,0, ^ K k ? ,..., ^ K k Thiscompletestheproofof4{23. 93

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Lemma4.6. For i 0 ~ X i isana.s.lower-semi-continuous-pathLSCPprocess.Thatis, ~ X i 2 D [0, 1 and sup t > 0 ~ X i [ t t ] 0 a.s. Proof. Notebycontinuityofmapping x 7! ~ k ? x ~ Z ~ E k ? +1 + ^ k ? +1 e ,..., ~ E k + ^ k e Lemma B.4andthefactthat ~ X 0 ~ Z isana.s.LSCPprocessLemma4.3,itisenoughtoshowif ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 islower-semi-continuouspaththen ~ X i islower-semi-continuouspath. Theproofisbyinduction.AbaseisprovidedbyLemma4.3: ^ Z isLSCP,and ~ X 0 = ~ Z d = ^ Z .Suppose ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 isLSCP.Theinductivestepconsistsoftwoparts.First,wearguethat f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1 ,..., k isLSCP,for l = k ? ,..., k seeDenition4.2.Inthesecondpart,we considerpropertiesof ~ X i .Inparticular,for t > 0 ,onehas ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 l +1: k t = sup u t f ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ u t ] )]TJ/F21 11.9552 Tf 13.275 -9.1 Td [(_ l +1: k [ u t ] g + = sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ l +1: k [ u t g + ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] )]TJ/F21 11.9552 Tf 13.275 -9.1 Td [(_ l +1: k [ t t ] + ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] )]TJ/F21 11.9552 Tf 13.275 -9.1 Td [(_ l +1: k [ t t ] + = sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ l +1: k [ u t g + ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] + {46 wherethelastequalityisdueto l +1: k beingacontinuous-pathprocess l +1: k [ t t ]=0 and ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 beingLSCP ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] 0=0 .Then,for l = k ? ,itfollowsthat f k ? k ? ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k ? +1 ,..., k t = ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k ? +1: k t = sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.276 -9.099 Td [(_ k ? +1: k [ u t g + ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] + sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ k ? +1: k [ u t g + = ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1: k t )]TJ/F21 11.9552 Tf 9.298 0 Td [( = f k ? k ? ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1 ,..., k t )]TJ/F21 11.9552 Tf 9.299 0 Td [(, 94

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whereweusedtheinductiveassumption ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] 0 .Ontheotherhand,for l > k ? 4{46implies f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1 ,..., k t = ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 l +1: k t )]TJ/F24 11.9552 Tf 11.955 0 Td [( ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 l : k t = sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ l +1: k [ u t g + ~ X + i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 [ t t ] + )]TJ/F29 11.9552 Tf 11.956 16.857 Td [( sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ l : k [ u t g + ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] + sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ l +1: k [ u t g + )]TJ/F29 11.9552 Tf 11.955 16.857 Td [( sup u < t f ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ u t )]TJ/F21 11.9552 Tf 13.275 -9.099 Td [(_ l : k [ u t g + = ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 l +1: k t )]TJ/F21 11.9552 Tf 9.299 0 Td [( )]TJ/F24 11.9552 Tf 11.955 0 Td [( ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 l : k t )]TJ/F21 11.9552 Tf 9.299 0 Td [( = f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1 ,..., k t )]TJ/F21 11.9552 Tf 9.299 0 Td [(; theinequalityfollowsfrom ~ X + i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 [ t t ] < 0 0 l +1: k l : k ,and a + c + )]TJ/F21 11.9552 Tf 9.605 0 Td [( b + c + a + )]TJ/F38 11.9552 Tf 9.605 0 Td [(b + for a b and c 0 .Thiscompletestherstpartoftheinductivestep. Asfarasthesecondpartisconcerned,notethatLemma4.3implies ~ Z = ~ Z C + k X l = k ? ~ E l G l where ~ Z C isacontinuous-pathprocess.Then,for t 0 ,wehave ~ X i t = ~ Z t + k X l = k ? Z t 0 f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k ? +1 ,..., k t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s = ~ Z C t + ~ E k ? : k + k X l = k ? Z t 0 f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k ? +1 ,..., k t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s )]TJ/F21 11.9552 Tf 13.261 2.657 Td [(~ E l G l d s whichinturnyields,for t > 0 ~ X i [ t t ]= k X l = k ? Z [0, t f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1 ,..., k [ t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s ] G l d s + k X l = k ? f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k ? +1 ,..., k )]TJ/F21 11.9552 Tf 13.261 2.657 Td [(~ E l G l [ t t ] 0, wheretheinequalityisdueto f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1 ,..., k beingLSCPseetherstpartofthe inductivestep; f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k ? +1 ,..., k [ t )]TJ/F38 11.9552 Tf 12.093 0 Td [(s t )]TJ/F38 11.9552 Tf 12.093 0 Td [(s ] 0 ,and f k ? l ~ X i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k ? +1 ,..., k = 95

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~ E l since ~ X i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = ~ Z ;recall G l =0 ,for l = k ? ,..., k .Thiscompletestheproofof thelemma. 4.5.3Thediusionlimitfortheoriginalsystem Inthissubsection,weprove4{21{theproofisbasedontheresultforthemodied systemsee4{23andtheprevioussubsection.Thedenitionof K n l yields K n l t = E n l t X j = E n l t )]TJ/F39 7.9701 Tf 6.587 0 Td [(K n l t +1 1 f s l j > 0 g whichimplies ^ K n l = 1 G l ^ K n l )]TJ/F21 11.9552 Tf 13.414 2.656 Td [(^ B n l n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 E n l + ^ B n l )]TJ/F38 11.9552 Tf 5.48 -9.683 Td [(n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 E n l )]TJ/F38 11.9552 Tf 11.955 0 Td [(n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 K n l =: 1 G l ^ K n l )]TJ/F21 11.9552 Tf 23.799 8.087 Td [(1 G l n l {47 where ^ B n l 'sareintroducedintheproofofLemma4.1seeAppendix4.6.Thisequality, togetherwith ^ X n )]TJ/F20 11.9552 Tf 10.405 -10.594 Td [( ^ K n 1: k )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(^ X n = ^ X n )]TJ/F20 11.9552 Tf 10.405 -10.594 Td [( ^ K n 1: k )]TJ/F21 11.9552 Tf 13.647 2.657 Td [(^ X n whichfollowsfromthefactthatthenumberofcustomersinserviceinthemodiedsystem andtheoriginalsystemareequalatanytimecustomerswithzeroservicerequirementsdonot occupyservers,leadsto ^ X n = ^ X n + k X l =1 ^ K n l )]TJ/F21 11.9552 Tf 13.734 2.657 Td [(^ K n l = ^ X n + k X l =1 G l G l ^ K n l )]TJ/F39 7.9701 Tf 18.176 14.944 Td [(k X l =1 1 G l n l {48 Inviewof4{23,4{47and4{48,itissucienttoshow n 1 ,..., n k ,...,0, {49 as n !1 .However,dueto4{47,4{51,thefactthataBrownianmotionhas a.s.continuouspaths,and n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 E n l l e ,as n !1 ,see4{7and4{10,itsuces 96

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toshow n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 K n l 0 ,as n !1 ,toprove4{49.Tothisend,4{47yields,for t 0 k n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 K n l k t 1 G l k n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 K n l k t + 2 p n k ^ B n l n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 E n l k t Theprecedinginequalityand4{21resultin n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 K n l 0 ,as n !1 ,and,thus,4{49holds. Finally,4{21followsfrom4{23,4{47,4{48,4{49,thecontinuousmappingtheorem andcontinuityofsummationinuniformtopology. 4.6Proofs 4.6.1ProofofLemma4.1 For l =1,..., k ,onehas ^ A n l = ^ B n n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 E n l )]TJ/F21 11.9552 Tf 13.414 2.657 Td [(^ B n n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n l + G l ^ A n l {50 where ^ B n l := f ^ B n l t t 0 g isacenteredrandomwalk: ^ B n l t := 1 p n b nt c X j =1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(1 f s l j > 0 g )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G l Then,wehave,as n !1 ^ B n 1 ,..., ^ B n k ^ B 1 ,..., ^ B k {51 Inaddition,continuityof x 1 ,..., x k 7! G 1 x 1 ,... G k x k ,thecontinuousmapping theoremand4{7yield,as n !1 G 1 ^ A n 1 ,..., G k ^ A n k G 1 ^ A 1 ,..., G k ^ A k {52 Furthermore, E n l = A n l + n l ,4{7and4{10imply,as n !1 n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 E n 1 ,..., E n k 1 e ,..., k e and n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 n 1 ,..., n k ,...,0. 97

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Finally,thelemmafollowsfrom4{50,4{51,4{52,thelasttwolimitsand l =0 l > k ? ; bydenition, ^ B n l 'saremutuallyindependent,aswellasindependentof ^ A n 1 ,..., ^ A n k 4.6.2ProofofLemma4.2 Inviewof ^ n ,^ n 1: k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,^ n k ,...,^ n k = ^ n )]TJ/F39 7.9701 Tf 18.176 14.944 Td [(k X l =1 G l ^ n l k ? )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X l =1 G l ^ n l G k ? ^ n k ? ,..., G k ^ n k + k X l =1 G l ^ n l )]TJ/F21 11.9552 Tf 12.186 0 Td [(^ n l k ? )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 X l =1 ^ n l )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G l ^ n l ,^ n k ? )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G k ? ^ n k ? ,...,^ n k )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G k ^ n k continuityofmapping x x 1 ,..., x k 7! x )]TJ/F29 11.9552 Tf 12.092 8.966 Td [(P k l =1 G l x l G 1 x 1 ,..., G k x k and4{10, itisenoughtoshow,for l =1,..., k ^ n l )]TJ/F21 11.9552 Tf 13.71 2.657 Td [( G l ^ n l = 1 p n n l X j =1 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 f s l j > 0 g )]TJ/F21 11.9552 Tf 13.71 2.656 Td [( G l 0, as n !1 .Tothisend,onehas,for > 0 and b thatisacontinuitypointofthedistribution of ^ l P 2 4 1 p n n l X j =1 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(1 f s l j > 0 g )]TJ/F21 11.9552 Tf 13.709 2.656 Td [( G l > 3 5 P 2 4 sup c b 1 p n b p nc c X j =1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(1 f s l j > 0 g )]TJ/F21 11.9552 Tf 13.709 2.656 Td [( G l > 3 5 + P [ ^ n l > b ] P [ ^ l > b ] as n !1 ;thelimitfollowsfromaSLLNe.g.,see[ ? ]and4{10.Byselectingavalue of b ,theright-handsidecanbemadearbitrarilysmall.Hence,thelemmaholds. 4.6.3ProofofLemma4.3 Straightforwardalgebrayields,for t 0 ^ Z n t = ^ n G n t )]TJ/F29 11.9552 Tf 11.955 13.271 Td [( ^ n )]TJ/F21 11.9552 Tf 10.831 -5.811 Td [( G n t + k X l =1 ^ n l G l t + k X l =1 Z t 0 G l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s ^ A n l d s + 1 p n k X l =1 E n l t X j =1 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(1 f s l j + l j > t g )]TJ/F21 11.9552 Tf 13.71 2.656 Td [( G l t )]TJ/F24 11.9552 Tf 11.955 0 Td [( l j + 1 p n n ^ n X j =1 1 f s j > t g )]TJ/F21 11.9552 Tf 13.709 4.782 Td [( G n t {53 98

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Next,weconsiderelementsontheright-handsideof4{53.First,4{4,4{9and4{10 resultin,as n !1 ^ n G n )]TJ/F29 11.9552 Tf 11.955 13.27 Td [( ^ n )]TJ/F21 11.9552 Tf 10.831 -5.812 Td [( G n + k X l =1 ^ n l G l ^ G )]TJ/F21 11.9552 Tf 12.963 2.657 Td [(^ )]TJ/F21 11.9552 Tf 8.839 -0.154 Td [( G + k X l = k ? ^ l G l {54 Second,4{7implies,as n !1 k X l =1 Z t 0 G l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s ^ A n d s t 0 k X l =1 Z t 0 G l t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s ^ A d s t 0 {55 Third,let M n l := f M n l t t 0 g ,where M n l t := 1 p n E n l t X j =1 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 f s l j + l j > t g )]TJ/F21 11.9552 Tf 13.709 2.657 Td [( G l t )]TJ/F24 11.9552 Tf 11.955 0 Td [( l j ; furthermore,dene M n l := f M n l t t 0 g ,where M n l t := 1 p n b n l t c X j =1 1 f s l j + j = n l > t g )]TJ/F21 11.9552 Tf 13.71 2.656 Td [( G l t )]TJ/F38 11.9552 Tf 17.002 8.087 Td [(j n l Byfollowingthestepsoutlinedintheproofof[70,Lemma3],onecanshow M n l M n l M l M l ,as n !1 ,where M l := f M l t t 0 g isacenteredGaussianprocesswith a.s.continuouspaths, M l =0 and E M l t M l t + = l Z t 0 G l t + )]TJ/F38 11.9552 Tf 11.955 0 Td [(u G l t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u d u 0. Since f M n l l =1,... k g areindependentprocesses, f M l l =1,... k g areindependentaswell. Therefore,wehave,as n !1 M n 1: k M {56 Fourth,let ^ U n := f ^ U n t s t 0,0 s 1 g ,where ^ U n t s = 1 p n b nt c X j =1 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(1 f j s g )]TJ/F38 11.9552 Tf 11.955 0 Td [(s 99

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and f j j 1 g isasequenceofindependent [0,1] -uniformrandomvariables.Then,thefourth termin4{53canbeexpressedintermsof ^ U n : 1 p n n ^ n X j =1 1 f s j > t g )]TJ/F21 11.9552 Tf 13.709 4.782 Td [( G n t = ^ U n n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n ^ 1, G n t Now, ^ U n ^ U ,as n !1 ,where ^ U isaKieferprocess[58,Lemma3.1].Thelastlimit, togetherwithcontinuityof ^ U ,4{9and4{10,yields,as n !1 ^ U n n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n ^ 1, G n ^ U n G n V G V G {57 where V isasinthestatementofthelemma. Finally,thestatementofthelemmafollowsfrom4{54,4{55,4{56,and4{57.All limitingtermshavea.s.continuouspaths,exceptpossibly P k l = k ? ^ l G l .Independenceofthe limitingtermsisduetoindependenceof n M n 1: k ratherthan M n 1: k f A n l l =1,..., k g and ^ U n G n ratherthan ^ U n n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 n ^ 1, G n 4.6.4ProofofLemma4.4 Notethat K n 1: l t = E n 1: l t )]TJ/F39 7.9701 Tf 19.256 14.944 Td [(l X i =1 E n i t X j =1 1 f i j + w i j t g holdsfor l =1,... k .Utilizingtheprecedingequalitytwicefor l and k ,andcombiningitwith K n 1: k t = X n t )]TJ/F38 11.9552 Tf 11.955 0 Td [(n + resultsin K n 1: l [ u t ]= X n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n + [ u t ] )]TJ/F21 11.9552 Tf 13.325 -9.1 Td [(_ E n l +1: k [ u t ]+ k X i = l +1 E n i t X j =1 1 f u i j + w i j t g {58 Inaddition,priorityofclassi customersoverclassj j > i customersimplies f K n 1: l s > 0, 8 s 2 [ u t ] g 8 < : k X i = l +1 E n i t X j =1 1 f u i j + w i j t g =0 9 = ; ; {59 here,theeventontheleft-hand-sideistheeventthatthroughoutthetimeinterval [ u t ] at leastoneclassi 1 i l customerawaitsservice,andontheright-hand-sideistheevent thatnoclassj l +1 j k customerentersserviceintimeinterval [ u t ] 100

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Relations4{58and4{59areusedtoprovethestatementofthelemma.Tothisend, let u t :=inf f u 0: K n 1: l s > 0, 8 s 2 [ u t ] g .Then,twocasesareofinterest: u t = 1 :Itfollowsthat K n 1: l t =0 ,andhence K n 1: l [ u t ] 0 forany u t .This, togetherwith4{58andnon-negativityofthedoublesumin4{58,resultsin X n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n + [ u t ] )]TJ/F21 11.9552 Tf 13.325 -9.099 Td [(_ E n l +1: k [ u t ] 0, for u t .Therefore,thesupremuminthestatementoflemmaisnon-positive,andthe lemmaholdsinthiscase. 0 u t t :Itfollowsthat K n 1: l u t )]TJ/F21 11.9552 Tf 9.299 0 Td [(=0 alsointhecasewhen u t =0 ,duetoour notationand K n 1: l u > 0 for u 2 [ u t t ] .Then,4{58and4{59implythat,for u 2 [0, t ] X n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n + [ u t ] )]TJ/F21 11.9552 Tf 13.325 -9.099 Td [(_ E n l +1: k [ u t ] K n 1: l [ u t ] K n 1: l t = K n 1: l [ u t t ] = X n )]TJ/F38 11.9552 Tf 11.955 0 Td [(n + [ u t t ] )]TJ/F21 11.9552 Tf 13.325 -9.099 Td [(_ E n l +1: k [ u t t ]. Hence,thelemmaholdsinthiscaseaswell.Thiscompletestheproof. 101

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CHAPTER5 CONCLUSION Inthisdissertationwestudiedtwoclassicqueuingmodels,namelythemachinerepair modelandthemulti-classprioritymodel,intheQEDregime.OurmainresultsTheorem3.1 andTheorem4.1describethediusionlimitforeachmodelasanonlinearmappingdierent foreachmodeloftheircorrespondinginnite-servermodel.Thesetworesultshavevery similarrepresentationstoeachotherandthediusionlimitforopensystemsin[77].This suggeststhatthereexistaunifyingframeworkformany-serverlimits. Inaddition,inourstudyofmachinerepairmodel,weestablishaconnectionbetween closedandopensystemsintheQEDregimeviaaPoissonlimitSection3.3.2.This connectiontosomedegree,justiestheuseofopensystemstomodelreal-worldproblems. Giventhatqueuesintherealworldhaveanitepopulationofcustomers,itseemsnatturalto considerconditionsunderwhichopensystemsarereleventformodelingpurposes.Weprovide onepossibleanswerforthisquestionforsystemswithalargenumberofservers/repairmenand alargeratioofthenumberofcustomers/machinestothenumberofservers/repairmen. Fromthetechnicalpointofview,weintroducedtwotechniqueswhichcanpotentiallybe usedinaborderclassofproblems.Forthemachine-repairmodel,wefoundadecompositionto dealwiththecomplexitythatarisesduetothefactthatdelaysinclosedqueuingsystemsaect futurearrivals".Thistechniqueallowstousetoolsdevelopedforanalysisofopenqueueing systemsinthestudyofclosedqueueingsystems.Formulti-classpriorityqueuesweestablished axed-pointtechniquestodealwiththestructuralcomplexityduetothefactthatthenumber ofjobsfromeachclassinthesystem/queuedependsonthenumberjobsfromotherclassesin thesystem/queue.Webelievethistechniquecouldbeusefulintheanalysisofageneralservice decsplinesincetheassumptionthatthenumberofclassesisnite,wasusedinotherpartsof theproofonly. 102

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Finally,inSection3.6,weprovidedsomenumericalexamplesfornite-sizesystems.We illustratedthatrelativelysimpleresultsforQEDlimitaregoodapproximationsoftheprelimit sequence. 103

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APPENDIXA ADDITIONALMOTIVATIONFORTHEQEDREGIME Justinthisappendix,werestrictourattentiontoamodelwithexponentialrepairand workingtimes,butallow p tovarywith n .Ourfocusisonthelimitingstationarydistribution, and,thus,itisimmaterialwhether p variesduetochangesin F G orboth. Let X n 1 bearandomvariablewiththedistributionequaltothestationarydistribution of f X n t t 0 g .Thenextresultstatesthattheordersofmagnitudeofstochastic uctuationsofthestationarynumberofidlerepairman X n 1 )]TJ/F38 11.9552 Tf 11.251 0 Td [(k n )]TJ/F17 11.9552 Tf 7.085 -4.338 Td [(andwaitingmachines X n 1 )]TJ/F38 11.9552 Tf 12.043 0 Td [(k n + aredierent{ p np p and p np p = p p n ,respectively.Moreimportantly, comparingthepropositionbelowwithTheorem2in[27],indicatesthatthestationaryQED approximationfornite p remainsvalideveninthecasewhen p varieswith n PropositionA.1. ConsiderasequenceofmemorylessmachinerepairmodelsintheQED regime.Let bethesolutionto p n )]TJ/F24 11.9552 Tf 12.574 0 Td [( = p ^ k n ,and n = p p ,as n !1 .If np p !1 ,then lim n !1 P [ X n 1 k n ]= 1+ e 2 = 2 )]TJ/F24 11.9552 Tf 9.299 0 Td [( e 2 = 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ; A{1 moreover,for x + lim n !1 P X n 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( ^ k n p np p < x X n 1 k n = x )]TJ/F24 11.9552 Tf 11.955 0 Td [( )]TJ/F21 11.9552 Tf 7.085 -4.339 Td [( A{2 and,for x )]TJ/F21 11.9552 Tf 21.918 0 Td [( )]TJ/F45 11.9552 Tf 7.085 -4.339 Td [(, lim n !1 P X n 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( k n p np p p p n > x X n 1 k n = x 1 + + A{3 Remark A.1 Thereexistseveralcasesofinterest: If = =0 ,thentheratio = inA{1shouldbeinterpretedasthelimitof 1 = p p If p 0 and > 0 ,then !1 andA{1reducesto lim n !1 P [ X n 1 k ]= p 2 e 2 = 2 1+ p 2 e 2 = 2 104

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If p 0 and < 0 ,then andA{1reducesto P [ X n 1 k n ] 0 ,as n !1 If !1 ,thentheright-handsideofA{3reducesto e )]TJ/F39 7.9701 Tf 6.586 0 Td [(x x 0 Proof. Let n := n = p p i := P [ X n 1 = i ] andnotethate.g.see[56,Sect.3.10] i = 8 > > < > > : n i p = p i )]TJ/F39 5.9776 Tf 5.757 0 Td [(k n n k n k n i k n n i i p = p i )]TJ/F39 5.9776 Tf 5.757 0 Td [(k n k n i )]TJ/F39 5.9776 Tf 5.756 0 Td [(k n k n n k n k n i k n A{4 Next,weconsider b k n + x p np p c for x < 0 .RelationA{4,Stirling'sapproximation,3{4and sometediousalgebraresultin,for x < 0 b k n + x p np p c = e )]TJ/F39 5.9776 Tf 7.782 3.258 Td [(x 2 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [( n x k n + o A{5 as n !1 .Thisestimate,togetherwith ^ k n = k n )]TJ/F21 11.9552 Tf 12.631 0 Td [( n + p np p andtheassumption np p !1 ,yields P X n 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( ^ k n p np p < x X n 1 k n R x )]TJ/F25 7.9701 Tf 6.587 0 Td [( + e )]TJ/F39 5.9776 Tf 7.782 3.527 Td [(y 2 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [( y d y R 0 e )]TJ/F39 5.9776 Tf 7.782 3.527 Td [(y 2 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [( y d y as n !1 ,for x + ,andA{2follows.Similarly,A{4,Stirling'sapproximation,3{4, theTaylorexpansionof log+ x and k n = k n + n )]TJ/F20 11.9552 Tf 7.085 4.372 Td [(p n p imply b k n + x p n p = n c = e )]TJ/F22 5.9776 Tf 7.782 3.859 Td [( n + x 1 n )]TJ/F39 5.9776 Tf 18.956 3.258 Td [(x 2 2 n 2 b k n c + o A{6 as n !1 ,for x )]TJ/F21 11.9552 Tf 22.634 0 Td [( n n )]TJ/F17 11.9552 Tf 7.084 -4.338 Td [(.Consequently,passing n !1 resultinA{3.Finally, A{1canbeobtainedaswell.Tothisend,A{6renders k n = e )]TJ/F22 5.9776 Tf 7.782 3.859 Td [( n )]TJ/F22 5.9776 Tf 6.254 -2.508 Td [( 2 2 b k n c + o as n !1 ,which,combinedwithA{5,A{6and P [ X n 1 k n ]= 1+ P n i = k n +1 i P k n i =0 i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 105

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implies P [ X n 1 k n ] 0 @ 1+ e )]TJ/F22 5.9776 Tf 6.254 -2.509 Td [( 2 2 R 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [( )]TJ/F38 11.9552 Tf 8.745 1.882 Td [(e )]TJ/F39 5.9776 Tf 7.782 3.259 Td [(x 2 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [( + x d x R 0 e )]TJ/F39 5.9776 Tf 7.782 3.258 Td [(x 2 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [( x d x 1 A )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 as n !1 .Thisconcludestheproof. 106

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APPENDIXB ANCILLARYRESULTSANDPROOFS B.1AncillaryResultsforMachineRepairModel B.1.1Decomposition Inthissection,westateadecompositionfor M n l j L LemmaB.1;thereisananalogous decompositionfor ~ M n l j L {weomitdetails,andprovidesomepropertiesofelementsofthis decompositionLemmaB.2.TheproofsofLemma3.4andLemma3.5canbefoundatthe endofthissection.Inordertostateit,weintroducesomeadditionalnotation.Tothisend, let i l j L := i j + c j i j +1: j + l + s j i j + l +1 1 f L = H g B{1 and v i l j L :=1 f L = G g s j i j + l +1 +1 f L = H g a j i j + l +1 B{2 Randomvariables c j i j +1: j + l s j i j + l +1 and i j wereintroducedintheproofofLemma3.3.Recall that s m i m + j and a m i m + j aretheserviceandworkingtimesinthe m + j thcycleofamachine thatwasthe i thamongallthemachinestostartits m thcycle c m i m + j := s m i m + j + a m i m + j ;for thiscustomerlet i j :=inf f t 0: A n j t i g bethetimeofits j thbreakdowntheuidlimit of A n j isobtainedinLemma3.10.Notethat M n l j L see3{83canbewrittenasfollows: M n l j L ; t = 1 p n n X i =1 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(1 f v i l j L t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F21 11.9552 Tf 12.57 0 Td [( i j Furthermore,let E n l j L := f E n l j L ; t t 0 g E n l j L ; t := 1 p n n X i =1 E n i l j L ; t with E n i l j L := f E n i l j L ; t t 0 g and E n i l j L ; t :=1 f v i l j L t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g )]TJ/F29 11.9552 Tf 11.955 16.272 Td [(Z v i l j L ^ t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L + 0 d G L s G L s )]TJ/F21 11.9552 Tf 9.299 0 Td [( B{3 107

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where G L :=1 f L = G g G +1 f L = H g F isthedistributionfunctionof v i l j L .TheintegralinB{3isthestandardcompensatorfor theone-point"pointprocesstheindicatorinB{3[49,p.98];itiswelldenedforall distributionfunctions G L ,since G L s )]TJ/F21 11.9552 Tf 9.299 0 Td [( iszeroonlyonthenullsetof G L .Wealsodenea countingprocess J n l j L := f J n l j L ; t t 0 g with J n l j L ; t := A n j t X i =1 1 f i l j t g B{4 Thelastprocesswedenehereis B n l j L := f B n l j L ; t x t 0, x 2 [0,1] g : B n l j L ; t x := 1 p n b nt c X i =1 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(1 f G L v i l j L x g )]TJ/F38 11.9552 Tf 11.955 0 Td [(x where,for i n v i l j L = v i ? l j L i 7! i ? isabijectionand i ? l j L =inf f t 0: J n l j L ; t i g ;for i > n f v i l j L g isasequenceofi.i.d.randomvariablesdistributed accordingto G L ,independentofallotherrandomvariablesinmodel. Let L 2f G H g bedenedby f L g = f G H gnf L g .Westatethefollowingdecomposition withoutaproof,sinceitisbasedonstraightforwardalgebra.Bydenition M )]TJ/F22 7.9701 Tf 6.587 0 Td [(1, j L 0 LemmaB.1. Wehave M n l j L ; t = E n l j L ; t )]TJ/F29 11.9552 Tf 11.955 16.273 Td [(Z t 0 B n l j L ; J n l j L ; t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u = n G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( d G L u + Z t 0 M n l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g j L ; t )]TJ/F38 11.9552 Tf 11.956 0 Td [(u d G L u TheaboveresultinconjunctionwithLemmaB.2isanextensionofthemartingale decompositionin[77].Thersttwotermsofthedecompositionsaresimilartothosein[77], inthesensethattheirmartingalepropertiescanbeprovedwiththesametechnique,provided thatappropriateltrationscanbedened.Theapproachin[77]isitselfanextensionofa martingalemethodforempiricalprocesse.g.,see[49,p.99]forthecaseofdependent one-pointpointprocessesthedependency,inthatcases,isduetothenitenessofservice 108

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capacityofanopensystem,i.e.,waitingtimes.Themainideaemployedoriginallyused in[58]istonda -algebraforagiventimeintheltration,suchthattermscorresponding toindividualcustomersbecomeconditionallyindependentthecorrespondingmartingale termsareorthogonal.Inparticular,arrivalstothesystem,servicetimesoffuturearrivals,and residualservicetimesofcustomersinserviceareconditionallyindependentgiventhis -algebra. Inthecaseofamachinerepairmodel,dependenciesaremoreintricateduetotheclosed natureofthesystem:breakdowntimesarrivalsandrepairservicetimesaredependent. Thus,werstdecompose X n )]TJ/F38 11.9552 Tf 12.376 0 Td [(Z n seeSection3.4.1,andthenndappropriate -algebras withdesirablepropertiesforindividualpieces M n l j L and ~ M n l j L inthedecompositionof X n )]TJ/F38 11.9552 Tf 12.591 0 Td [(Z n .Itturnsout,the -algebrathatfacilitatestheanalysisof M n l j L resp. ~ M n l j L containsthe -algebrageneratedbytermsin M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L resp. M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L {thisisthe reasonforthelastterminLemmaB.1. Thenextresultcharacterizes E n l j L .Inordertostateit,wedenealtration E n l j L := f E n t l j L t 0 g associatedwiththefollowing -algebra: E n t l j L := n c j i j +1: j + l + s j i j + l +1 1 f L = H g i =1,2...., A n j t o n A n j s s t o 1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g s t i : i l j L t _N t 0; B{5 underthisdenition, E n l j L satisestheusualconditions[49,p.2]. Remark B.1 Notethereismorethanonewaytodene i l j L and v i l j L ,suchthat i l j L + v i l j L = i j + c j i j +1: j + l + s j i j + l +1 +1 f L = H g a j i j + l +1 whichisthepointthat denesoneofone-pointpointprocesses # m j l L 1 m n .Therearetworeasonsfor theparticularchoiceof i l j L and v i l j L seeB{1andB{2.First,suchachoiceleads toorthogonalmartingaleterms E n i l j L seeB{3thatcorrespondtodierentmachines dierent i .Second,thelastterminLemmaB.1canbeexpressedinarecursiveform.In viewofSection3.4.1, i l j L isananalogueofanarrivalinanopensystem.Thecomponents of E n t l j L inB{5describe i l j L and v i l j L .Inparticular,thersttwosigmaalgebras 109

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inB{5contain f i l j L t i =1,2,..., n g ,whilethethirdsigmaalgebracontains informationon 1 f i l j L + v i l j L t g LemmaB.2. Theprocess E n l j L isan E n l j L -squareintegrablemartingalewithpredictable quadratic-variationprocess h E n l j L i := fh E n l j L i t t 0 g ,where h E n l j L i t = 1 n n X i =1 Z v i l j L ^ t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L + 0 G L u G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 d G L u Proof. WefollowthethreestepsoftheproofofLemma3.5in[58].Intherststep,weargue thattheprocess E n i l j L 1 i n ,isan E n l j L -square-integrablemartingale.Thesecond stepinvolvesndingthepredictablequadratic-variationprocessof E n i l j L .Thethirdstepof theproofinvolvesshowingthatthemartingales E n i l j L and E n q l j L areorthogonal,for i 6 = q i.e.,that E n i l j L E n q l j L isan E n l j L -square-integrablemartingale. StepI. Notethat E n i l j L is E n l j L -adaptedand sup t 0 E E n i l j L ; t 2 1 Tocompletethisstepoftheproof,itissucienttoshowthat,for s < t 1 f i l j L > s g E h E n i l j L ; t j E n s l j L i =0 B{6 and 1 f i l j L s g E h E n i l j L ; t j E n s l j L i = E n i l j L ; s B{7 Observethat i l j L isan E n l j L -stoppingtime,and,thus,the -eld E n i l j L l j L is welldened[16,p.297]seeB{5.Then,onehas 1 f i l j L > s g E h E n i l j L ; t j E n s l j L i =1 f i l j L > s g E h E h E n i l j L ; t j E n i l j L l j L i j E n s l j L i B{8 Inviewof v i l j L beingapositiverandomvariable, v i l j L and E n i l j L l j L areindependent, whichimplies E h E n i l j L ; t j E n i l j L l j L i = E h E n i l j L ; t j i l j L i =0, 110

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wherethesecondequalityfollowsfromB{3.CombiningtheprecedingequalitywithB{8 resultsinB{6. Now,weconsidertheright-handsideofB{7: 1 f i l j L s g E h E n i l j L ; t j E n s l j L i =1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g E h E n i l j L ; t j E n s l j L i +1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g E h E n i l j L ; t j E n s l j L i Since 1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g and 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g areboth E n s l j L -measurable,fromB{3it followsthat 1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g E n i l j L ; t =1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g Z v i l j L ^ t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L + 0 d G L u G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( wherethelasttermis E n s l j L -measurable.Hence,theprecedingimplies 1 f i l j L s g E h E n i l j L ; t j E n s l j L i =1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g )]TJ/F21 11.9552 Tf 11.956 0 Td [(1 f v i l j L s )]TJ/F22 7.9701 Tf 6.785 0 Td [( i l j L g Z v i l j L 0 d G L u G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( +1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g E h E n i l j L ; t j E n s l j L i B{9 Next,weevaluatethelastterminB{9.Ontheevent f v i l j L > s )]TJ/F21 11.9552 Tf 12.342 0 Td [( i l j L 0 g v i l j L and f A n j u u s g areindependent.Toshowthis,notethatthereexistsaBorelfunction f suchthat A n j = f n s q m a q m q 1, m 1 .Now,ifwereplace v i l j L with 1 inthe function f ,weobtainanotherprocess A n j ,whichisindependentfrom v i l j L .Observethat, ontheevent f v i l j L > s )]TJ/F21 11.9552 Tf 12.792 0 Td [( i l j L 0 g A n j u coincides A n j u for u s .Recallthat v i l j L isindependentof c j q j +1: j + l + s q j + l +1 1 f L = H g for q =1,2,..., n .Thus,ontheevent f v i l j L > s )]TJ/F21 11.9552 Tf 11.638 0 Td [( i l j L 0 g E n i l j L ; t dependson E n s l j L onlythrough i l j L and v i l j L 111

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This,togetherwithB{5,implies E n s l j L f v i l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L 0 g f n s q m a q m q =1,2,..., n m 1 gnf v i l j L g _N f v i l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L 0 g Theprecedingrelationand[58,Lemma3.6]resultin 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g E h E n i l j L ; t j E n s l j L i =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g E h 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g E n i l j L ; t j i l j L i P [ v i l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L j i l j L ] = )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g Z s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 d G L s G L s )]TJ/F21 11.9552 Tf 9.298 0 Td [( CombiningtheprecedingequalitywithB{9yieldsB{7.Thiscompletestherststepofthe proof. StepII. Giventhatthesecondtermontheright-handsideofB{3is E n l j L -predictable, the E n l j L -predictablemeasureofjumpsoftheprocess f 1 f v i l j L t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g t 0 g isgiven by[58,p.259] e n l j [0, t ], A =1 f 1 2 A g Z t 0 1 f i l j L < u v i l j L + i l j L g d G L u )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L G L )]TJ/F21 11.9552 Tf 7.472 -9.85 Td [( u )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L )]TJ/F29 11.9552 Tf 11.955 9.684 Td [( 112

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Hence,thepredictablequadratic-variationprocessof E n i l j L satises[60,pp.192-193] h E n i l j L i t = Z t 0 Z R x 2 e n l j du dx )]TJ/F29 11.9552 Tf 16.404 11.357 Td [(X 0 < u t Z R x e n l j u dx 2 = Z t 0 1 f i l j L < u v i l j L + i l j L g d G L u )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L G L u )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L )]TJ/F21 11.9552 Tf 9.299 0 Td [( )]TJ/F29 11.9552 Tf 16.404 11.357 Td [(X 0 < u t 1 f i l j L < u v i l j L + i l j L g G L u )]TJ/F21 11.9552 Tf 12.179 0 Td [( i l j L G L u )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L )]TJ/F21 11.9552 Tf 9.298 0 Td [( 2 = Z v i l j L ^ t )]TJ/F22 7.9701 Tf 6.785 0 Td [( i l j L + 0 d G L u G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( )]TJ/F29 11.9552 Tf 54.912 11.357 Td [(X 0 < u v i l j L ^ t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L + G L u G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 = Z v i l j L ^ t )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L + 0 G L u G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( 2 d G L u ; here, G L u denotesthesizeofajumpof G L at u ,andthesumsareoverallthejumps. Thiscompletesthesecondstepoftheproof. StepIII. Itisenoughtoshowthat,for s < t 1 f i l j L q l j L > s g E h E n i l j L ; t E n q l j L ; t j E n s l j L i =0 B{10 and 1 f i l j L q l j L s g E h E n i l j L ; t E n q l j L t j E n s l j L i = E n i l j L ; s E n q l j L s ; B{11 notethat E h E n i l j L ; t E n q l j L ; t j E n s l j L i = E h E n i l j L ; t E n q l j L ; t j E n s l j L )]TJ/F39 7.9701 Tf 6.774 -1.793 Td [(i q i P )]TJ/F39 7.9701 Tf 6.775 -1.793 Td [(i q j E n s l j L + E h E n i l j L ; t E n q l j L ; t j E n s l j L )]TJETq1 0 0 1 437.548 223.648 cm[]0 d 0 J 0.478 w 0 0 m 6.775 0 l SQBT/F39 7.9701 Tf 444.323 222.433 Td [(i q i P )]TJETq1 0 0 1 476.634 223.648 cm[]0 d 0 J 0.478 w 0 0 m 6.775 0 l SQBT/F39 7.9701 Tf 483.409 222.433 Td [(i q j E n s l j L 113

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where )]TJ/F39 7.9701 Tf 6.774 -1.793 Td [(i q := f i l j L q l j L g and )]TJETq1 0 0 1 269.791 706.371 cm[]0 d 0 J 0.478 w 0 0 m 6.775 0 l SQBT/F39 7.9701 Tf 276.566 705.156 Td [(i q := f i l j L < q l j L g .TodemonstrateB{10, wewrite 1 f i l j L q l j L > s g E h E n i l j L ; t E n q l j L ; t j E n s l j L )]TJ/F39 7.9701 Tf 6.775 -1.793 Td [(i q i =1 f i l j L q l j L > s g E h E h E n i l j L ; t E n q l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.774 -1.794 Td [(i q i j E n s l j L )]TJ/F39 7.9701 Tf 6.775 -1.794 Td [(i q i B{12 where E h E n i l j L ; t E n q l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.774 -1.793 Td [(i q i =1 f v q l j L i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E h E n i l j L ; t E n q l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.775 -1.794 Td [(i q i +1 f v q l j L > i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h E n i l j L ; t E n q l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.774 -1.793 Td [(i q i B{13 Inviewofboth 1 f v q l j L i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g and 1 f v q l j L i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E n q l j L ; t =1 f v q l j L i l j L )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L g E n q l j L ; t ^ i l j L being E n i l j L l j L )]TJ/F39 7.9701 Tf 6.774 -1.793 Td [(i q -measurable,onehas 1 f v q l j L i l j L )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L g E h E n i l j L ; t E n q l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.774 -1.793 Td [(i q i = E h 1 f v q l j L i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E n i l j L ; t E n q l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.775 -1.793 Td [(i q i =1 f v q l j L i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E n q l j L ; t E h E n i l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.774 -1.794 Td [(i q i Since v i l j L isapositiverandomvariable,independentof E n i l j L l j L )]TJ/F39 7.9701 Tf 6.774 -1.794 Td [(i q ,onehas E h E n i l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.775 -1.794 Td [(i q i = E h E n i l j L ; t j i l j L i =0; thus,thersttermontheright-handsideofB{13isequalto 0 Next,weconsiderthesecondtermontheright-handsideofB{13.Thereexistsa randomvariable i l j L whichisaBorelfunctionof n andalltheservicetimesandworking timesexcept v q l j L and v i l j L ,suchthat f v q l j L > i l j L )]TJ/F21 11.9552 Tf 11.357 0 Td [( q l j L 0 g = f v q l j L > 114

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i l j L )]TJ/F21 11.9552 Tf 12.542 0 Td [( q l j L 0 g and i l j L = i l j L oneitherevent.Then,[58,Lemma3.6]and thefactthat v i l j L and v q l j L areindependentfrom i l j L and q l j L yield 1 f v q l j L > i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h E n i l j L ; t E n q l j L ; t j E n i l j L l j L )]TJ/F39 7.9701 Tf 6.775 -1.793 Td [(i q i =1 f v q l j L > i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h 1 f v q l j L > i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E n i l j L ; t E n q l j L ; t j i l j L q l j L i P [ v q l j L > i l j L )]TJ/F21 11.9552 Tf 12.178 0 Td [( q l j L 0 j i l j L q l j L ] where E n i l j L denotes E n i l j L with i l j L substitutedfor i l j L .Dueto v i l j L is independentof v q l j L i l j L and i l j L ,oneobtains E h 1 f v q l j L > i l j L )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L 0 g E n i l j L ; t E n q l j L ; t j i l j L q l j L i = E h 1 f v q l j L > i l j L )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E n q l j L ; t j i l j L q l j L i E h E n q l j L ; t j i l j L i wherethelastmultiplieris 0 bydenitionof E n q l j L andthefactthat v i l j L isindependent of i l j L .Inviewofthepreceding,theright-handsideofB{13iszero,aswellasthe right-handsideofB{12.AsimilarreasoningcanbeusedtoshowthatB{12remainstrue when )]TJ/F39 7.9701 Tf 6.775 -1.793 Td [(i q isreplacedwith )]TJETq1 0 0 1 201.984 376.951 cm[]0 d 0 J 0.478 w 0 0 m 6.775 0 l SQBT/F39 7.9701 Tf 208.758 375.735 Td [(i q .Consequently,B{10holds. FortheproofofB{11,weusethesameapproachasforproofofB{10;hence,some detailsareomitted.Onehas 1 f i l j L q l j L s g E h E n i l j L ; t E n q l j L t j E n s l j L i =1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g 1 f v q l j L s )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L g E h E n i l j L ; t E n q l j L t j E n s l j L i +1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h E n i l j L ; t E n q l j L t j E n s l j L i +1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E h E n i l j L ; t E n q l j L t j E n s l j L i +1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h E n i l j L ; t E n q l j L t j E n s l j L i B{14 115

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Thetermsintheprecedingequalitycanbeestimatedasfollows: 1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g 1 f v q l j L s )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L g E h E n i l j L ; t E n q l j L t j E n s l j L i = E h 1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g E n i l j L ; t f v q l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E n q l j L t j E n s l j L i =1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g 1 f v q l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E n i l j L ; s E n q l j L s B{15 1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h E n i l j L ; t E n q l j L t j E n s l j L i =1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L 0 g E n i l j L ; s E h E n q l j L t j E n s l j L i =1 f v i l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L 0 g E n i l j L ; s E n q l j L s B{16 and 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E h E n i l j L ; t E n q l j L t j E n s l j L i =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E n q l j L s E h E n i l j L ; t j E n s l j L i =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L g E n i l j L ; s E n q l j L s B{17 wherethemartingalepropertyof E n i l j L and E n q l j L wasusedinB{16andB{17. Now,weconsiderthelasttermontheright-hand-sideofB{14.Thereexistsrandom variables i l j L and q l j L whichareBorelfunctionof n andalltheservicetimesand workingtimesexcept v q l j L and v i l j L ,suchthat f v i l j L > s )]TJ/F21 11.9552 Tf 12.421 0 Td [( i l j L 0, v q l j L > s )]TJ/F21 11.9552 Tf 12.964 0 Td [( q l j L 0 g = f v i l j L > s )]TJ/F21 11.9552 Tf 12.964 0 Td [( i l j L 0, v q l j L > s )]TJ/F21 11.9552 Tf 12.964 0 Td [( q l j L 0 g ,and i l j L = i l j L and q l j L = q l j L oneitherevent.Hence,itfollowsthat 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h E n i l j L ; t E n q l j L t j E n s l j L i =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E E n i l j L ; t E n q l j L t j E n s l j L 116

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where v i l j L and v q l j L areindependentof q l j L and i l j L .Lemma3.6in[58]and E n s l j L f v i l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L 0 gf v q l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( q l j L 0 g f n s o m a o m o =1,2,..., n m 1 gnf v i l j L v q l j L g _N f v i l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L 0 gf v q l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( q l j L 0 g yield 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E h E n i l j L ; t E n q l j L t j E n s l j L i =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E n i l j L ; t E n q l j L t j i l j L q l j L P [ v i l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L 0, v q l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( q l j L 0 j i l j L q l j L ] =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E 1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g E n i l j L ; t j i l j L E 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E n q l j L t j q l j L P [ v i l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( i l j L 0 j i l j L ] P [ v q l j L > s )]TJ/F21 11.9552 Tf 12.178 0 Td [( q l j L 0 j q l j L ] =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g E E n i l j L ; t j E n s l j L 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.785 0 Td [( q l j L 0 g E E n q l j L ; t j E n s l j L =1 f v i l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( i l j L 0 g 1 f v q l j L > s )]TJ/F22 7.9701 Tf 6.786 0 Td [( q l j L 0 g E n i l j L ; s E n q l j L s B{18 wherethesecondequalityfollowsfromthemutualindependenceof v i l j L and v q l j L ,as wellastheirindependenceof q l j L and i l j L ;thelastequalityisduetothefactthat i l j L = i l j L and q l j L = q l j L on f v i l j L > s )]TJ/F21 11.9552 Tf 12.976 0 Td [( i l j L 0, v q l j L > s )]TJ/F21 11.9552 Tf 12.692 0 Td [( q l j L 0 g = f v i l j L > s )]TJ/F21 11.9552 Tf 12.692 0 Td [( i l j L 0, v q l j L > s )]TJ/F21 11.9552 Tf 12.692 0 Td [( q l j L 0 g ,andthe martingalepropertyof E n i l j L and E n q l j L SubstitutingB{15,B{16,B{17andB{18intoB{14,resultsinB{11.This completestheproof. WeconcludethissectionwiththeproofsofLemma3.4andLemma3.5. 117

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ProofofLemma3.4. Theproofisbasedonadecompositionof M n l j L ananalogous decompositionholdsfor ~ M n l j L .Inparticular,LemmaB.1yields E k M n l j L k t E k E n l j L k t + E sup s t Z s 0 B n l j L ; J n l j L ; s )]TJ/F38 11.9552 Tf 11.956 0 Td [(u = n G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( G u )]TJ/F21 11.9552 Tf 9.298 0 Td [( d G L u + E sup s t Z s 0 M n l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g j L ; s )]TJ/F38 11.9552 Tf 11.955 0 Td [(u d G L u B{19 Next,weboundthethreetermsontheright-handsideofB{19.First,Theorem1.9.5in[60], Jensen'sinequalityforconcavefunctionsandLemmaB.2resultin E k E n l j L k t 3 E h E n l j L i t 1 = 2 3 E h E n l j L i t 1 = 2 3 0 @ E 2 4 1 n X i : i l j L t Z v i l j L 0 d G L u G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( 3 5 1 A 1 = 2 3 )]TJ/F38 11.9552 Tf 5.479 -9.683 Td [(G f L = H g H H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t 1 = 2 3 )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g t 1 = 2 B{20 whereinthesecondtolastinequalityisbasedon E Z v i l j L 0 d G L u G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( =1; B{21 everywhereinthepaper G H )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 isinterpretedas 1 Second,byLemma3.1in[58]seealsop.252, f B n l j L ; t G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [(, t 0 g isalocally square-integrablemartingalewithrespecttoanappropriatelydenedltration;itspredictable quadratic-variationprocessisgivenby h B n l j L ; G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( i t = b nt c n G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [(. 118

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This,togetherwithTheorem1.9.5in[60],Jensen'sinequalityforconcavefunctionsandB{4, yields E sup s J n l j L ; t = n B n l j L ; s G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( 3 E h B n l j L ; G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( i J n l j L ; t = n 1 = 2 3 E h B n l j L ; G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( i J n l j L ; t = n 1 = 2 3 )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(G f L = H g H H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t 1 = 2 )]TJ/F21 11.9552 Tf 7.233 -7.027 Td [( G L x )]TJ/F21 11.9552 Tf 9.298 0 Td [( 1 = 2 3 )]TJ/F38 11.9552 Tf 5.479 -9.684 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g t 1 = 2 )]TJ/F21 11.9552 Tf 7.234 -7.027 Td [( G L x )]TJ/F21 11.9552 Tf 9.299 0 Td [( 1 = 2 TheprecedinginequalityallowsonetoboundthesecondterminB{19: E sup s t Z s 0 B n l j L ; J n l j L ; s )]TJ/F38 11.9552 Tf 11.955 0 Td [(u = n G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( G u )]TJ/F21 11.9552 Tf 9.299 0 Td [( d G L u Z t 0 E sup s J n l j L ; t = n B n l j L ; s G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( G u )]TJ/F21 11.9552 Tf 9.299 0 Td [( d G L u 3 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g t 1 = 2 Z t 0 d G L u G u )]TJ/F21 11.9552 Tf 9.298 0 Td [( 1 = 2 6 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g t 1 = 2 B{22 wherelastinequalityfollowsfromthefactthattheintegralisupperboundedby2.Abound forthethirdterminB{19isstraightforward: E sup s t Z s 0 M n l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g j L ; s )]TJ/F38 11.9552 Tf 11.955 0 Td [(u d G L u E sup s t Z s 0 k M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L k s )]TJ/F39 7.9701 Tf 6.587 0 Td [(u d G L u Z t 0 E k M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L k t )]TJ/F39 7.9701 Tf 6.587 0 Td [(u d G L u B{23 Next,weuseinductiontoprovethesameboundholdsfor ~ M n l j L E k M n l j L k t 9 l +1+1 f L = H g )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g t 1 = 2 B{24 Tothisend,for 2 l +1+1 f L = H g =1 i.e., l =0 and L = G ,theresultfollowsfromB{19, B{20,B{22and M n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1, j L 0 .Now,supposethatB{24holdsfor 2 l +1+1 f L = H g = q )]TJ/F21 11.9552 Tf 12.884 0 Td [(1 1 .Then,thisinductiveassumption,B{19,B{20,B{22andB{23imply 119

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thatB{24holdsfor 2 l +1+1 f L = H g = q : E k M n l j L k t 9 )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g t 1 = 2 + Z t 0 9 l )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 f L = G g +1+1 f L = H g )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(G f L = G g H l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u 1 = 2 d G L u 9 l +1+1 f L = H g )]TJ/F38 11.9552 Tf 5.48 -9.684 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g t 1 = 2 wherethelastinequalityfollowsfromJensen'sinequalityand G f L = G g H l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g G L u = G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g Finally,theunionbound,MarkovinequalityandB{24yiled P 2 6 6 4 X l + j > m j 6 =0 k M n l j L k t + k ~ M n l j L k t > 3 7 7 5 X l + j > m j 6 =0 P h k M n l j L k t > l j i + P h k ~ M n l j L k t > l j i X l + j > m j 6 =0 18 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 l j l +1+1 f L = H g )]TJ/F38 11.9552 Tf 5.479 -9.683 Td [(G f L = G g H l + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g t 1 = 2 B{25 wherepositive l j 'ssatisfy =2 X l + j > 0 j 6 =0 l j Since H l + j t decaysexponentiallyfastin l + j ,onecanselect f l j g insuchawaythatthe suminB{25isconvergentfor m =1 e.g., f l j g polynomiallydecreasingin l + j .The statementofthelemmafollows. ProofofLemma3.5. Straightforwardalgebraand3{28yield M n l ,0 L ; t )]TJ/F21 11.9552 Tf 14.564 2.657 Td [(~ M n l ,0 L ; t = N n l L ; t )]TJ/F21 11.9552 Tf 13.519 2.657 Td [(~ N n l L ; t B{26 120

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where N n l L := f N n l L ; t t 0 g ~ N n l L := f ~ N n l L ; t t 0 g N n l L ; t = 1 p n n X i = n ^ k n +1 1 f c i ,1: l + s i l +1 + a i l +1 1 f L = H g t g )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t and ~ N n l L ; t = 1 p n n X i = n ^ k n +1 1 f c i ,1: l + s i l +1 + a i l +1 1 f L = H g t )]TJ/F39 7.9701 Tf 6.586 0 Td [(w i ,0 g )]TJ/F38 11.9552 Tf 11.955 0 Td [(L H l t )]TJ/F38 11.9552 Tf 11.956 0 Td [(w i ,0 ApplyingLemma3.7resultin,as n !1 X L 2f G H g 1 X l =0 )]TJ/F21 11.9552 Tf 9.298 0 Td [(1 1 f L = H g N n l L 0. B{27 Next,weconsider ~ N n l L .BasedonananalogueofLemmaB.1, ~ N n l L canberepresented asasumofthreecomponents,wheretherstoneisasquare-integrablemartingalewithan appropriateltrationananalogueofLemmaB.2.Then,thesameargumentasintheproof ofLemma3.4yields E k ~ N n l L k t 9 E n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + n G f L = G g H l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g t ; B{28 inviewof 0 n )]TJ/F38 11.9552 Tf 12.304 0 Td [(k n + = n 1 ,theexpectedvalueofthisrandomvariableiswelldened. This,togetherwith3{6andthedominatedconvergencetheorem,resultsin,as n !1 1 n E n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + 0. B{29 Now,byMarkov'sinequalityandB{28,onehas limsup n !1 P 2 4 1 X l =0 X L 2f G H g )]TJ/F21 11.9552 Tf 9.298 0 Td [(1 1 f L = H g ~ N n l L t > 3 5 limsup n !1 1 X l =0 X L 2f G H g P h k ~ N n l L k t > l i limsup n !1 E n )]TJ/F38 11.9552 Tf 11.955 0 Td [(k n + n 1 X l =0 X L 2f G H g 9 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 l G f L = G g H l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g t 121

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where l arepositivenumberssuchthat 2 P 1 l =0 l = ;theycanbechoseninamannerthat thedoublesummationisnite.ThisfactandB{29imply,as n !1 X L 2f G H g 1 X l =0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 1 f L = H g ~ N n l L 0. EqualityB{26,thelastlimitandB{27yieldthestatementofthelemma. B.1.2Tightness Inthissubsection,westateatechnicalresult. LemmaB.3. Ifthesequence f M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L n 1 g resp. f ~ M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L n 1 g is C-tight,then f M n l j L n 1 g resp. f ~ M n l j L n 1 g istight. Remark B.2 Thesequences f M n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1, j H n 1 g and f ~ M n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1, j H n 1 g areC-tight, since M n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1, j H ~ M n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1, j H 0 .TheproofofLemma3.3isbasedonthefactthatthe sequence f M n l j L g n f ~ M n l j L g n istight.Underthiscondition,Lemma3.3statesthatthe limitof f M n l j L g n hasa.s.continuouspaths.Hence,thetightnessof f M n l j L g n impliesthe C-tightnessofthesamesequence. Proof. First,byAldous'sucientcriterion[60,p.515],wearguethat f E n l j L ; t n 1 g is tight.Inparticular,forall T > 0 and > 0 ,weshowthat lim !1 limsup n !1 P h k E n l j L ; t k T > i =0 B{30 and lim 0 limsup n !1 sup 2T n l j L P sup 0 t E n l j L ; + t )]TJ/F21 11.9552 Tf 13.261 2.657 Td [( E n l j L ; > =0, B{31 where T n l j L isisthesetofall E n l j L -stoppingtimesnogreaterthan T .TheproofofB{30 isstraightforward{B{20andMarkovinequalityyield P h k E n l j L ; t k T > i 3 = 122

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Next,observethat A n j + l t X i =1 1 f s j i j + l +1 1 f L = H g t g J n l j L ; t n X i =1 1 f c j i ,1: j + l + s j i j + l +1 1 f L = H g t g B{32 ApplyingtheargumentusedinLemma3.10resultsin,as n !1 8 < : 1 n A n j + l t X i =1 1 f s j i j + l +1 1 f L = H g t g t 0 9 = ; G f L = H g H H l + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ; B{33 aSLLNe.g.,see[24]canbeinvokedtoshow,as n !1 1 n n X i =1 1 f c j i ,1: j + l + s j i j + l +1 1 f L = H g t g t 0 G f L = H g H H l + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 B{34 Therefore,fromB{32,B{33,B{34andthefactthat G f L = H g H H l + j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 isa continuousfunction,itfollowsthat f J n l j L = n n 1 g isC-tight. Now,LemmaB.2andtheLenglart-Rebolledoinequalitye.g.,see[60,p.66]imply,for any > 0 andany 2T n l j L P sup 0 t E n l j L ; + t )]TJ/F21 11.9552 Tf 13.261 2.656 Td [( E n l j L ; > = 2 + P h h E n l j L i + )-222(h E n l j L i > i = 2 + P 2 4 1 n sup s T 0 < t )]TJ/F39 7.9701 Tf 6.587 0 Td [(s X i : s < i l j L t Z v i l j L 0 d G L u G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( > 3 5 B{35 Duetothefactthat # f i : s < i l j L t g = J n l j t )]TJ/F21 11.9552 Tf 12.313 2.657 Td [( J n l j s ,thefactthat f J n l j L = n n 1 g isC-tight,B{21andtheSLLN,onehas lim 0 limsup n !1 P 2 4 1 n sup s T 0 < t )]TJ/F39 7.9701 Tf 6.586 0 Td [(s X i : s < i l j L t Z v i l j L 0 d G L u G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( > 3 5 =0. B{36 Thus,B{31followsfromB{35andB{36. Second,combiningC-tightnessof f J n l j L = n n 1 g andtheargumentusedintheproof ofLemma3.4in[58]yieldsthatthesequence Z t 0 B n l j L ; J n l j L ; t )]TJ/F38 11.9552 Tf 11.955 0 Td [(u = n G L u )]TJ/F21 11.9552 Tf 9.299 0 Td [( G L u )]TJ/F21 11.9552 Tf 9.298 0 Td [( d G L u t 0 n 1 123

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isC-tight. Third,introduceamap f L : D [0, 1 D [0, 1 : f L x t = Z t 0 x t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s d G L s for t 0 and x 2 D [0, 1 .Then, f L iscontinuousinthetopologyofuniformconvergence. Moreover,if x 2 C [0, 1 ,then f L u 2 C [0, 1 .Thepreceding,theC-tightnessof f M n l )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f L = G g j L n 1 g andthefactthatconvergencetoacontinuousfunctionisequivalentin J 1 anduniformtopologiesimplythatthesequence f f L M n l )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 f L = G g j L n 1 g isC-tight. Theproofoflemmafollowsfromtheabove,LemmaB.1andCorollaryVI.3.33in[49]. B.1.3Dependencygraphforresultsonthemachinerepairmodel Lemma3.7 Lemma3.1 Lemma3.8 Proposition3.1 Lemma3.9 Corollary3.2 Lemma3.10 Lemma3.3 Lemma3.6 Theorem3.1 Corollary3.1 Lemma3.2 Lemma3.5 Lemma3.4 LemmaB.1/B.2 LemmaB.3 B.2AncillaryResultsforMulti-ClassPriorityModel LemmaB.4. Let f i : D [0, 1 D [0, 1 i =1,..., k ,bemeasurableandLipschitz continuouswiththeLipschitzconstant c i ,suchthat f i =0 .If z 2 D [0, 1 ,then: iThemapping g z : D [0, 1 D [0, 1 denedby g z x t := z t + k X l =1 Z t 0 f l x t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s is J 1 -measurableandLipschitzcontinuousin D [0, 1 kk Furthermore,if G := k l =1 G l =0 ,thenthefollowinghold: iiTheequation x = g z x hasauniquesolutionin D [0, 1 124

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iiiThemapping g : D [0, 1 D [0, 1 ,denedsuchthat g z istheuniquesolution of x = g z x ,is J 1 -measurableandLipschitzcontinuousin D [0, 1 kk ivDene x i +1 := g z x i i 0 ,where x 0 2 D [0, 1 isgiven.Then x i g z ,as i !1 ,in B D [0, 1 kk .Moreover,forany t 0 ,thereexistan i 0 > 0 afunctionof G and c > 0 afunctionof G and c 1: k suchthatfor i i 0 k x i )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z k t c c 1: k c 1: k +1 i k z )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 k t + k x 0 k t Remark B.3 Theassumption G =0 isessential.Withoutthisassumption,thereexist counterexamplesthatdemonstratethatthelemmadoesnothold. Proof.Parti. For t 0 ,onehas j g z x 1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z x 2 t j k X l =1 Z t 0 j f l x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(f l x 2 j t )]TJ/F38 11.9552 Tf 11.956 0 Td [(s G l d s k X l =1 Z t 0 c l k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t )]TJ/F39 7.9701 Tf 6.587 0 Td [(s G l d s k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t k X l =1 c l G l t B{37 andtherefore k g z x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z x 2 k t c 1: k k x 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2 k t B{38 Theproofofmeasurablityfollowsfrommeasurablityofmapping x 7! y where y t := R t 0 x + t )]TJ/F38 11.9552 Tf 12.439 0 Td [(s G l d s for l =1,..., k seeproofofProposition3.1in[77],thefactthat f l 's aremeasurablefunctionsandthefactthatcompositionandsumofmeasurablefunctionsare measurable. Partii. Wearguetheuniquenessrst.Suppose x y 2 D [0, T ] suchthat x = g z x and y = g z y .Since G =0 ,thereexistsa > 0 suchthat G < c 1: k +1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 .Then, B{37implies k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(y k = k g z x )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z y k k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(y k c 1: k c 1: k +1 125

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whichyields k x )]TJ/F38 11.9552 Tf 12.726 0 Td [(y k =0 .Next,suppose k x )]TJ/F38 11.9552 Tf 12.726 0 Td [(y k i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 =0 forsome i > 2 induction hypothesis.Then,for t 2 i )]TJ/F21 11.9552 Tf 11.956 0 Td [(1 i ] ,onehas j x t )]TJ/F38 11.9552 Tf 11.955 0 Td [(y t j = j g z x t )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z y t j k X l =1 Z t 0 k f l x )]TJ/F38 11.9552 Tf 11.955 0 Td [(f l y k t )]TJ/F39 7.9701 Tf 6.587 0 Td [(s G l d s k X l =1 Z 0 c l k x )]TJ/F38 11.9552 Tf 11.956 0 Td [(y k i G l d s k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(y k i c 1: k c 1: k +1 ; thesecondinequalityfollowsfromtheinductionhypothesis,whilethelastinequalityfollows fromdenitionof .Combiningthelastrelationshipand k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(y k i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 =0 resultsin k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(y k i k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(y k i c 1: k c 1: k +1 whichyields k x )]TJ/F38 11.9552 Tf 11.956 0 Td [(y k i =0 .Hence,theuniquenessfollows. Now,wedemonstratethatthereexistsasolution.Denethesequence f x i i 0 g : x i +1 = g z x i i 0 ,foragiven x 0 2 D [0, 1 .Wearguethat f x i i 0 g isaCauchy sequence.First,weprovebyinductionthat,for t 0 and i 1 k x i +1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x i k t c 1: k +1 c i 1: k k z )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 k t + k x 0 k t G i t B{39 Thebase i =0 oftheinductioncanbeveriedasfollows: j x 1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 t jj z t )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 t j + k X l =1 Z t 0 f l x 0 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G l d s k z )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 k t + c 1: k k x 0 k t c 1: k +1 k z )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 k t + k x 0 k t 126

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and,thus, k x 1 )]TJ/F38 11.9552 Tf 12.099 0 Td [(x 0 k t c 1: k +1 k z )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 k t + k x 0 k t .AssumingthatB{39holdsforsome i = j )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 yieldsthatB{39holdsfor i = j aswell: j x j +1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(x j t j k X l =1 Z t 0 k f l x j )]TJ/F38 11.9552 Tf 11.955 0 Td [(f l x j )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k t )]TJ/F39 7.9701 Tf 6.586 0 Td [(s G l d s c 1: k +1 c j 1: k k z )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 k t + k x 0 k t Z t 0 G j )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F38 11.9552 Tf 11.955 0 Td [(s G d s c 1: k +1 c j 1: k k z )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 0 k t + k x 0 k t G j t whichimpliesthatB{39holdsforall i 1 Second,weshowthatthereexist i 0 and d suchthat,forany i i 0 ,onehas 1 X j = i c j 1: k G j t < d c 1: k c 1: k +1 i B{40 Since G =0 ,thereexistsa > 0 ,suchthat G c 1: k +2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 .If F t := 8 > > < > > : c 1: k +2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,0 t < 1, t then G F and,for i d t = e = i 0 ,onehas c i 1: k G i t c i 1: k F i t = c i 1: k b t = c X j =0 i j 2 c 1: k +1 2 c 1: k +2 j 1 2 k 1: k +2 i )]TJ/F39 7.9701 Tf 6.586 0 Td [(j c i 1: k 2 i 1 2 c 1: k +2 i b t = c Therefore,onecanwrite,for i i 0 1 X j = i c j 1: k G j t 1 X j = i c j 1: k F j t c 1: k +2 b t = c 1 X j = i c 1: k c 1: k +1 j =2 b t = c c 1: k +1 b t = c +1 c 1: k c 1: k +1 i 127

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whichcompletestheproofofB{40. Next,B{39andB{40yieldthat f x i i 1 g isaCauchysequence.Consequently, since D [0, 1 isaBanachspaceunderthesupremumnorm, x i x ,as i !1 ,forsome x 2 D [0, 1 .Moreover,onehas k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z x k t k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(x i k t + k g z x )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z x i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 k t k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(x i k t + c 1: k k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(x i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k t wherethesecondinequalityfollowsfromLipschitzcontinuityof g z seeB{38.The right-handsidecanbemadearbitrarilysmallbyselectingalargeenough i {therefore, k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z x k t =0 and x = g z x .Thiscompletestheproofofthesecondpart. Partiii. Notethat g iswell-denedbythesecondpartofthelemma.For z 1 z 2 2 D [0, 1 ,denetwosequences f x l i i 0 g l =1,2 asfollows: x l i = g z l x l i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 i 1 ,where x l ,0 = z l .Recallfromtheproofoftheprecedingpartthat x l i g z l ,as i !1 .Thus,one has k g z 1 )]TJ/F38 11.9552 Tf 11.956 0 Td [(g z 2 k t = k lim i !1 x 1, i )]TJ/F21 11.9552 Tf 14.241 0 Td [(lim i !1 x 2, i k t = k lim i !1 x 1, i )]TJ/F38 11.9552 Tf 11.956 0 Td [(x 2, i k t =lim i !1 k x 1, i )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2, i k t B{41 wherethesecondequalityfollowsfromcontinuityofsubtractionintheuniformtopology,and thethirdequalityfollowsfromcontinuityofthesupremumnorm.Now,asintheproofofthe secondpart,thefollowingholds: k x 1, i )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2, i k t k z 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(z 2 k t + c 1: k k x 1, i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x 2, i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 k t G t k z 1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(z 2 k t i X j =0 c j 1: k G j t Lipschitzcontinuityof g followsfromB{40,B{41andtheprecedinginequality.The proofofmeasurablityfollowsfromthefactthatlimitofasequenceofmeasurablefunctionsis measurable. 128

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Partiv. Thestatementfollowsfrom x i g z ,as i !1 seePartiii,B{39, B{40and k x i )]TJ/F38 11.9552 Tf 11.956 0 Td [(g z k t k x i )]TJ/F38 11.9552 Tf 11.955 0 Td [(x i +1 k t + k x i +1 )]TJ/F38 11.9552 Tf 11.955 0 Td [(g z k t Thiscompletestheproofofthelemma. Recallthemapping : D 2 [0, 1 D [0, 1 introducedinDenition4.1. LemmaB.5. Let a b x y 2 D [0, 1 and t 0 .Then k x a )]TJ/F24 11.9552 Tf 11.956 0 Td [( y b k t 2 k x )]TJ/F38 11.9552 Tf 11.955 0 Td [(y k t +2 k a )]TJ/F38 11.9552 Tf 11.955 0 Td [(b k t Furthermore, isameasurablefunctionandif a 2 D [0, 1 ,then a =0 Proof. Let s t andsuppose sup u 2 [0, s ] f x + [ u s ] )]TJ/F38 11.9552 Tf 12.354 0 Td [(a [ u s ] g sup u 2 [0, s ] f y + [ u s ] )]TJ/F38 11.9552 Tf 12.354 0 Td [(b [ u s ] g withoutlossofgenerality.Then,wehave j y a s )]TJ/F24 11.9552 Tf 11.955 0 Td [( x a s j = y a s )]TJ/F24 11.9552 Tf 11.955 0 Td [( x a s sup u 2 [0, s ] f y + [ u s ] )]TJ/F38 11.9552 Tf 11.955 0 Td [(a [ u s ] g)]TJ/F21 11.9552 Tf 25.188 0 Td [(sup u 2 [0, s ] f x + [ u s ] )]TJ/F38 11.9552 Tf 11.955 0 Td [(a [ u s ] g sup u 2 [0, s ] f y + [ u s ] )]TJ/F38 11.9552 Tf 11.955 0 Td [(x + [ u s ] g 2 k y )]TJ/F38 11.9552 Tf 11.955 0 Td [(x k t Thesameideacanbeusedtoshow k y a )]TJ/F24 11.9552 Tf 12.043 0 Td [( y b k t 2 k a )]TJ/F38 11.9552 Tf 12.042 0 Td [(b k t .Therststatementof thelemmafollowsfromtheprecedingand k x a )]TJ/F24 11.9552 Tf 11.956 0 Td [( y b k t k x a )]TJ/F24 11.9552 Tf 11.955 0 Td [( y a k t + k y a )]TJ/F24 11.9552 Tf 11.955 0 Td [( y b k t Asfarasthesecondstatementofthelemmaisconcerned, a 2 D [0, 1 implies a [ u t ] 0 ,for u 2 [0, t ] .Hence,wehave,for t 0 a t =0 sup u 2 [0, t ] f)]TJ/F38 11.9552 Tf 15.277 0 Td [(a [ u t ] g =0. 129

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Theproofofmeasurablityfollowsfrommeasurablityofmapping z 7! w where w t := sup u t z t andthefactthatcompositionofmeasurablefunctionsismeasurable.This completestheproofofthelemma. LemmaB.6. Suppose: f x x n n 1 g D [0, 1 ,suchthat x n x ,as n !1 ,in D [0, 1 d J 1 ; f t n 0 t n 1 n 1 g R 2 ,suchthat t n 0 t n 1 P )778(! t t ,as n !1 ; x n t n 0 x n t n 1 # 0 # 1 ,as n !1 ,where # 0 # 1 isarandomvector; and arecontinuitypointsofthedistributionfunctionsof j x [ t t ] j and j # 0 )]TJ/F24 11.9552 Tf 12.812 0 Td [(# 1 j respectively. If >> 0 ,then P [ j x [ t t ] j > ] P [ j # 0 )]TJ/F24 11.9552 Tf 11.956 0 Td [(# 1 j > ] Proof. Thedenitionof J 1 metricimpliesexistenceofstrictlyincreasingcontinuousfunctions e n := f e n s s 0 g n 1 ,suchthat y n := x n e n x and e n e ,as n !1 ,in D [0, 1 kk .Let s n 0 s n 1 := e n t n 0 e n t n 1 anddene f t : D [0, 1 D [0, 1 by f t z s := 8 > > < > > : sup s u < t j z u )]TJ/F38 11.9552 Tf 11.955 0 Td [(z t )]TJ/F21 11.9552 Tf 9.298 0 Td [( j s < t sup t u s j z u )]TJ/F38 11.9552 Tf 11.955 0 Td [(z t j s t for s 0 and z 2 D [0, 1 .Itisstraightforwardtoverifythat f t isLipschitzcontinuousinthe topologyofuniformconvergence,and f t z [ t t ]= f t z t =0 .Then,wehave P [ j x n t n 0 )]TJ/F38 11.9552 Tf 11.955 0 Td [(x n t n 1 j > ] = P [ j y n s n 0 )]TJ/F38 11.9552 Tf 11.955 0 Td [(y n s n 1 j > ] = P [ j y n s n 0 )]TJ/F38 11.9552 Tf 11.955 0 Td [(y n s n 1 j > t 2 s n 0 ^ s n 1 s n 0 s n 1 ] ] + P [ j y n s n 0 )]TJ/F38 11.9552 Tf 11.955 0 Td [(y n s n 1 j > t 62 s n 0 ^ s n 1 s n 0 s n 1 ] ] P [ f t y n s n 0 + f t y n s n 1 + j y n [ t t ] j > t 2 s n 0 ^ s n 1 s n 0 s n 1 ] ] + P [ f t y n s n 0 + f t y n s n 1 > t 62 s n 0 ^ s n 1 s n 0 s n 1 ] ] P [ j y n [ t t ] j > ] + P [ f t y n s n 0 + f t y n s n 1 > )]TJ/F24 11.9552 Tf 11.955 0 Td [( ] + P [ f t y n s n 0 + f t y n s n 1 > ] B{42 130

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wheretherstinequalityfollowsfrom j y n s n 0 )]TJ/F38 11.9552 Tf 12.005 0 Td [(y n s n 1 j f t y n s n 0 + f t y n s n 1 + j y n [ t t ] j Thecontinuousmappingtheorem, f t z [ t t ]= f t z t =0 and t n 0 t n 1 P )778(! t t ,as n !1 yield f t y n s n 0 + f t y n s n 1 P )778(! 0, as n !1 .Thislimitimpliesthatthelasttwotermsontheright-handsideofB{42vanish, as n !1 .Thiscompletestheproof. 131

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APPENDIXC ADDITIONALNUMERICALEXAMPLESFORMACHINEREPAIRMODEL Inthischapterweprovidemorenumericalexamplesformachinerepairmodel.Similarto section3.6weprovideourexamplesforperformanceofourestimationintheprelimitsequence sectionC.1andsimulationoftheresidualservice/worktimessectionC.2. C.1PerformanceofaFinite-SizeSystem Forreasonsexplainedinsection3.6.1,inthissectionweprovidecompressionofaverage percustomerpercycledurationtotalwaitingtimesagainsterrorsofourapproximations heat-mapsandQ-QplotsforCDFofourapproximations,for n =60,120 and600, p =1 = 3 =1 and =1 = 2 and n > or = or < 0 C.1.1Exponentialdistributions Inthissectionweconsiderexponentialdistributionsfunctionforourservice/work time =1 and =1 = 2 132

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-1.Exponentialservice/work; n =60 k n =18 n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(0.55 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.06 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n o against ^ X n at t =5,20 and 50 133

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-2.Exponentialservice/work; n =60 k n =20 n =0 ;a.averagewaitingtimeper customerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e.Heat-map of ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 134

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-3.Exponentialservice/work; n =60 k n =22 n =0.55 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.06 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n o against ^ X n at t =5,20 and 50 135

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-4.Exponentialservice/work; n =120 k n =37 n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(0.58 ;a.averagewaiting timepercustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.059 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n o against ^ X n at t =5,20 and 50 136

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-5.Exponentialservice/work; n =120 k n =43 n =0.58 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.06 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n o against ^ X n at t =5,20 and 50 137

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-6.Exponentialservice/work; n =600 k n =194 n = )]TJ/F21 11.9552 Tf 9.299 0 Td [(0.52 ;a.averagewaiting timepercustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.06 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n o against ^ X n at t =5,20 and 50 138

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-7.Exponentialservice/work; n =600 k n =200 n =0 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.06 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n o against ^ X n at t =5,20 and 50 139

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-8.Exponentialservice/work; n =600 k n =206 n =0.52 ;a.averagewaiting timepercustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.06 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.654 0 Td [( n o against ^ X n at t =5,20 and 50 C.1.2Uniformdistributions Inthissectionweconsideruniformdistributionfunctionforourservice/worktimes =1 and =1 = 2 140

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-9.Uniformservice/work; n =60 k n =18 n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(0.55 ;a.averagewaitingtimeper customerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e.Heat-map of ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 141

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-10.Uniformservice/work; n =60 k n =20 n =0 ;a.averagewaitingtimeper customerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e.Heat-map of ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 142

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-11.Uniformservice/work; n =60 k n =22 n =0.55 ;a.averagewaitingtimeper customerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e.Heat-map of ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 143

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-12.Uniformservice/work; n =120 k n =37 n = )]TJ/F21 11.9552 Tf 9.299 0 Td [(0.58 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 144

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-13.Uniformservice/work; n =120 k n =43 n =0.58 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 145

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-14.Uniformservice/work; n =600 k n =194 n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(0.52 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 146

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-15.Uniformservice/work; n =600 k n =200 n =0 ;a.averagewaitingtimeper customerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e.Heat-map of ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 147

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a.Averagetotalwaitingtime c.Heatmapof ^ X n )]TJ/F25 7.9701 Tf 8.468 0 Td [(' P 1 ^ Z n )]TJ/F25 7.9701 Tf 8.468 0 Td [( n e.Heatmapof ^ X n )]TJ/F22 7.9701 Tf 9.426 1.771 Td [(^ Z n + n b.Q-Qplotsat t =5 d.Q-Qplotsat t =20 f.Q-Qplotsat t =50 FigureC-16.Uniformservice/work; n =600 k n =206 n =0.52 ;a.averagewaitingtime percustomerpernumberofcyclesc.Heat-mapof ^ X n )]TJ/F24 11.9552 Tf 11.955 0 Td [(' P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;e. Heat-mapof ^ X n )]TJ/F21 11.9552 Tf 13.361 2.657 Td [(^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n ;b,e,f:Q-Qplotsof ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n xand P 1 ^ Z n )]TJ/F24 11.9552 Tf 11.955 0 Td [( n oagainst ^ X n at t =5,20 and 50 C.2ResidualService/WorkTimes Thissectionprovidesmoreexamplesforsection3.6.2.Here,westudytheresidual service/worktimesviatimewhensystemsstartsfromanemptystate".Intheseexamples welet n =12,60 and 120 and n tobe0,positiveandnegative.Weassumeservicetimes andworktimesaredistributedaccordingtoauniform,1distributionandauniform,2 148

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distribution,respectively.Wendtheresidualserviceandworktimesat t =0,1.5 and5for 1000simulationsandusetheaverageCDF'sforeach t astheCDFofresidualserviceandwork timesatthat t .ThenweusetheQ-Qplottocomparethisdistributionstoresidualdistribution functions.Weinclude t =0 justtoseehowresidualtimeschangebytime. FigureC-17. n =12 k n =3 n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(0.61 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. 149

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FigureC-18. n =12 k n =4 n =0 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. FigureC-19. n =12 k n =5 n =0.61 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. 150

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FigureC-20. n =60 k n =18 n = )]TJ/F21 11.9552 Tf 9.299 0 Td [(0.55 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. FigureC-21. n =60 k n =22 n =0.55 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. 151

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FigureC-22. n =120 k n =37 n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(0.58 ;a-c:Q-Qplotsforresidualservicetimes against G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. FigureC-23. n =120 k n =40 n =0 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. 152

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FigureC-24. n =120 k n =43 n =0.58 ;a-c:Q-Qplotsforresidualservicetimesagainst G for t =0,1.5 and5;d-f:Q-Qplotsforresidualworktimesagainst F for t =0,1.5 and5. 153

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BIOGRAPHICALSKETCH AmirMotaeireceivedhisbachelor'sdegreeinIndustrialEngineeringfromSharifUniversity ofTechnologyin2008.HecontinuedhiseducationinIndustrialandSystemsEngineering programofSharifUniversityandgothismaster'sdegreein2011.Laterthatyear,hejoined theDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridaasaPhD studentunderthesupervisionofDr.PetarMomcilovic.HereceivedhisPh.D.inIndustrial andSystemsEngineeringfromUniversityofFloridainDecember2016.Hisresearchinterests includeheavytractheoryandQEDregimeinthequeuingtheory. 162