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Real-Time Scheduling of Ensemble Systems with Limited Resources

Permanent Link: http://ufdc.ufl.edu/UFE0043541/00001

Material Information

Title: Real-Time Scheduling of Ensemble Systems with Limited Resources
Physical Description: 1 online resource (224 p.)
Language: english
Creator: Rattanatamrong, Prapaporn
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: adaptive -- autonomic -- brain -- control -- cyberphysical -- dynamic -- embedded -- ensemble -- experts -- fuzzy -- interface -- limited -- machine -- management -- mixture -- optimization -- real -- resource -- responsibilities -- scheduler -- scheduling -- system -- time -- uncertainty
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Inspired by the strategy of divide and conquer, ensemble systems replace a single complex computational model by multiple simple models (called 'experts') that can, individually or in some combination, generate solutions for a larger range of input cases than their single original model. Real system requirements including size/weight constraints, power limitations or cost factors of ensemble systems often lead to limited availability of computational resources required to support concurrent execution of all experts. This dissertation addresses the problem of scheduling the execution of experts in ensemble systems in the way that the overall system performance is minimally affected by limited resources. To achieve this goal, the dissertation proposes a generalized scheduling architecture, called Elastic Ensemble Scheduling (EES) manager, which consists of a Task Utilization Adaptor (TUA) and an adaptive Real-Time Scheduler (RTS). Any given ensemble system has a gating component that is capable of determining the contribution (called 'responsibility') of each expert to the final solution and these responsibilities can vary as the system operates from one cycle to another. By formulating the resource utilization adaptation of experts corresponding to their responsibilities as an optimization problem, the TUA uses efficient optimization techniques to dynamically determine the time-varying resource utilization required by each expert to ensure that critical experts, achieve their best performance while guaranteeing minimum execution time needed by other experts. Based on the determined resource demands, the RTS creates a schedule for expert execution that allows each task to achieve resource utilization as close as possible to its demand without any deadline miss or violation of ensemble system's policies. First, the dissertation presents an implementation of the EES manager for ensemble systems in which resources are dedicated and each expert has an accurate and uniform worst-case execution time (WCET). Two proposed linear-time algorithms for the TUA, called TT-TC* and TT-Top, can efficiently adapt task utilization demands as responsibilities change by avoiding expensive reoptimization in the critical path of expert execution by up to 90% through the use of the optimal-solution sensitivity analysis technique. the RTS implements EPOC, a novel real-time multiprocessor algorithm that creates a schedule by assigning priorities to experts within fixed-length time durations (called epochs) based on their adapted demands and system policies. Experts in a test ensemble system with limited resources scheduled with the EES manager are shown to produce system outputs closely similar (= 8% error) to those of the system with sufficient resources. Next, another real-time scheduling algorithm, called EAGLE-T, is proposed as an improved RTS implementation for more generalized ensemble systems with non-uniform experts' WCETs. EAGLE-T is the first T-L plane-based scheduling algorithm with a mode-transition protocol. It's mode-transition protocol allows tasks to adapt their resource utilizations progressively rather than in a step-wise manner (i.e., the adaptation is delayed until the exact value from the new allocation can be achieved), typically used in other existing mode-transition schemes. Analytical and empirical results show that EAGLE-T is optimal, efficient and its progressive adaptation outperforms conventional step-wise adaptation (averaged utilization drift and delay during transition are reduced by 68.75% and 32.16%). EAGLE-T can enable tasks to achieve utilization within 56% to 90% of their demands, compared to 11%-81% when the step-wise scheme is used. Finally, an approach using co-design of fuzzy feedback control and real-time scheduling is derived and applied to the EES manager (called FuzzyEES). In order to support the deployment of resource-constrained ensemble systems in environments with uncertain resource availability and imprecise expert execution time, an additional fuzzy-logic inference component intelligently determine the total utilization allocation for the TUA's adaptation in order to achieve the resource utilization of the system as close as possibly to the actual resource capacity available for the system's operation. Simpler greedy-based optimization algorithm enables a more efficient implementation for the TUA, while also taking into consideration the performance measurement of each expert (i.e., how far the expert is from its equilibrium state). The EAGLE-T algorithm is extended to support the anticipation of the actual execution times and slack-time reclamation when WCETs are uncertain. From performance evaluation, the fuzzyEES manager can tolerate execution-time imprecision and occasional fluctuation of resource capacity and provide significant improvement (about 45% or more on average) over the open-loop approach. The results reported in this dissertation provide a basis upon which further research could be pursued to enable full deployment of the EES manager in real ensemble systems, such as a resource manager in a collaborative research platform in Brain-Machine Interfaces or switching-control systems. In future work, switching costs (when migrating experts between processing cores) and other sources of uncertainty (e.g., responsibility prediction error and inaccuracy of system timekeeping) should be taken into consideration when adapting resource-utilization demands, allocating resources and creating schedules.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Prapaporn Rattanatamrong.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Fortes, Jose A.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043541:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043541/00001

Material Information

Title: Real-Time Scheduling of Ensemble Systems with Limited Resources
Physical Description: 1 online resource (224 p.)
Language: english
Creator: Rattanatamrong, Prapaporn
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: adaptive -- autonomic -- brain -- control -- cyberphysical -- dynamic -- embedded -- ensemble -- experts -- fuzzy -- interface -- limited -- machine -- management -- mixture -- optimization -- real -- resource -- responsibilities -- scheduler -- scheduling -- system -- time -- uncertainty
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Inspired by the strategy of divide and conquer, ensemble systems replace a single complex computational model by multiple simple models (called 'experts') that can, individually or in some combination, generate solutions for a larger range of input cases than their single original model. Real system requirements including size/weight constraints, power limitations or cost factors of ensemble systems often lead to limited availability of computational resources required to support concurrent execution of all experts. This dissertation addresses the problem of scheduling the execution of experts in ensemble systems in the way that the overall system performance is minimally affected by limited resources. To achieve this goal, the dissertation proposes a generalized scheduling architecture, called Elastic Ensemble Scheduling (EES) manager, which consists of a Task Utilization Adaptor (TUA) and an adaptive Real-Time Scheduler (RTS). Any given ensemble system has a gating component that is capable of determining the contribution (called 'responsibility') of each expert to the final solution and these responsibilities can vary as the system operates from one cycle to another. By formulating the resource utilization adaptation of experts corresponding to their responsibilities as an optimization problem, the TUA uses efficient optimization techniques to dynamically determine the time-varying resource utilization required by each expert to ensure that critical experts, achieve their best performance while guaranteeing minimum execution time needed by other experts. Based on the determined resource demands, the RTS creates a schedule for expert execution that allows each task to achieve resource utilization as close as possible to its demand without any deadline miss or violation of ensemble system's policies. First, the dissertation presents an implementation of the EES manager for ensemble systems in which resources are dedicated and each expert has an accurate and uniform worst-case execution time (WCET). Two proposed linear-time algorithms for the TUA, called TT-TC* and TT-Top, can efficiently adapt task utilization demands as responsibilities change by avoiding expensive reoptimization in the critical path of expert execution by up to 90% through the use of the optimal-solution sensitivity analysis technique. the RTS implements EPOC, a novel real-time multiprocessor algorithm that creates a schedule by assigning priorities to experts within fixed-length time durations (called epochs) based on their adapted demands and system policies. Experts in a test ensemble system with limited resources scheduled with the EES manager are shown to produce system outputs closely similar (= 8% error) to those of the system with sufficient resources. Next, another real-time scheduling algorithm, called EAGLE-T, is proposed as an improved RTS implementation for more generalized ensemble systems with non-uniform experts' WCETs. EAGLE-T is the first T-L plane-based scheduling algorithm with a mode-transition protocol. It's mode-transition protocol allows tasks to adapt their resource utilizations progressively rather than in a step-wise manner (i.e., the adaptation is delayed until the exact value from the new allocation can be achieved), typically used in other existing mode-transition schemes. Analytical and empirical results show that EAGLE-T is optimal, efficient and its progressive adaptation outperforms conventional step-wise adaptation (averaged utilization drift and delay during transition are reduced by 68.75% and 32.16%). EAGLE-T can enable tasks to achieve utilization within 56% to 90% of their demands, compared to 11%-81% when the step-wise scheme is used. Finally, an approach using co-design of fuzzy feedback control and real-time scheduling is derived and applied to the EES manager (called FuzzyEES). In order to support the deployment of resource-constrained ensemble systems in environments with uncertain resource availability and imprecise expert execution time, an additional fuzzy-logic inference component intelligently determine the total utilization allocation for the TUA's adaptation in order to achieve the resource utilization of the system as close as possibly to the actual resource capacity available for the system's operation. Simpler greedy-based optimization algorithm enables a more efficient implementation for the TUA, while also taking into consideration the performance measurement of each expert (i.e., how far the expert is from its equilibrium state). The EAGLE-T algorithm is extended to support the anticipation of the actual execution times and slack-time reclamation when WCETs are uncertain. From performance evaluation, the fuzzyEES manager can tolerate execution-time imprecision and occasional fluctuation of resource capacity and provide significant improvement (about 45% or more on average) over the open-loop approach. The results reported in this dissertation provide a basis upon which further research could be pursued to enable full deployment of the EES manager in real ensemble systems, such as a resource manager in a collaborative research platform in Brain-Machine Interfaces or switching-control systems. In future work, switching costs (when migrating experts between processing cores) and other sources of uncertainty (e.g., responsibility prediction error and inaccuracy of system timekeeping) should be taken into consideration when adapting resource-utilization demands, allocating resources and creating schedules.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Prapaporn Rattanatamrong.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Fortes, Jose A.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043541:00001


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REAL-TIMESCHEDULINGOFENSEMBLESYSTEMSWITHLIMITEDRESOURCESByPRAPAPORNRATTANATAMRONGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011PrapapornRattanatamrong 2

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Tomyfamilyandothers,whoalwayshavegivenmetheirsupport 3

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ACKNOWLEDGMENTS Somanycontributed,directlyorindirectly,totherealizationofthisdissertationandmylifeduringthePh.D.programingeneralthatitwouldbehardtonamethemall,butmygratitudegoestoeachoneofthemalthoughsomeoftheirnamesmightbeomittedhere.Firstandforemost,Iwouldliketothankmyadvisor,Dr.JoseA.B.Fortes,forhisextremepatience,wiseadviceandguidanceinmyresearchthroughoutmyPh.D.years.Ihavegrownfromagraduatestudenttoaresearcherbecauseofhistremendoussupport.Iwouldalsoliketothanktherestofmycommittee:Dr.JoseC.Principe,Dr.J.ColeSmith,Dr.RenatoFigueiredoandDr.JustinC.Sanchez.Iamsoblessedtohavesuchaqualiedcommittee,andIdeeplyappreciateallofthehelpandfeedbacktheyhaveprovidedmeovertheyears.IappreciateallofmanycolleaguesandstaffsattheAdvancedComputingandInformationSystems(ACIS)LaboratorywhohavemadewhereIusuallyspendmosttimeofthedaysuchapleasantandproductiveplace.Specialthanksaregiventomylabmateswhohavealsobecamemybestfriends,SelviKadirvel,Dr.GirishVenkatasubramanian,Dr.DavidWolinsky,YuchuTong,Dr.XinFu,KeyuChen,Dr.JingXu,Dr.MingZhaoandDr.VineetChadha.Theirfriendshipsandsupportsmeansomuchtome.SincerethanksgotoDr.AndreaMatsunagaandDr.MauricioMatsunagawhohavetaughtmesomuchaboutresearchandworkingdisciplines,sincemydayoneinthelab.Ialsowanttothankmycollaboratorsinthepast,Dr.JackDiGiovanniandDr.BabakMahmoudi,forsharingtheirknowledgeinBrain-MachineInterfaceswithme.Myacknowledgmentsgotothosewhosponsoredmystudyandresearchwork:theRoyalThaiGovernment,theNationalScienceFoundation(underGrantsNo.CNS-0540304,CNS-0821622andIIP-0758596),theDefenseAdvancedResearchProjectsAgencyDefenseSciencesOfce(undertheauspicesofDr.GeoffreyLingthroughtheSpace 4

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andNavalWarfareSystemsCenter,PacicContractNo.N6600110-C-2008)andtheBellSouthFoundation.TheothergroupofpeopleIwouldliketothankisallofmyThaibestfriendsinGainesvillethatkindlycareaboutmethatmademesometimesforgetthatIhavebeenawayfrommyhome(e.g.,Dr.DonruethaiLaphasradakulandherfamily,Dr.SiripornKobnithikulwong,KittipatandSutheenatKampa,Dr.SiripornKamontum,Dr.RisaPatarasuk,WuttichaiLeelavorawongandtheSchroderfamily).IalsowouldliketothankJintanaDamrongwajasatandPanidaSenajit,mybestfriendsinThailandwhohavebeenprovidingtheirgreatsupportsallalong.Mostimportantofall,Iwouldliketoexpressmydeepgratitudetomyboyfriend,PanoatChuchaisri,andmybelovedfamily(includingmyaunt,Dr.WilaiSilapa-acha)forbeinganunstintingsourceofsupportandencouragement.TheyalwaysknowhowtomakemesmileandlaughregardlessofhowdifcultofthesituationIamin.ItwouldbeimpossibletogettowhereIamtodaywithoutthestrengthfromallofthem. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 LISTOFIMPORTANTABBREVIATIONS ........................ 14 ABSTRACT ......................................... 16 CHAPTER 1INTRODUCTION ................................... 18 1.1EnsembleSystemsandMotivatingApplications ............... 19 1.1.1Brain-MachineInterfaces ....................... 21 1.1.2Switching-ControlSystems ...................... 23 1.1.3VirtualWorldsandthe3DInternet .................. 25 1.2Real-TimeSystems .............................. 27 1.3SchedulingofStaticReal-TimeSystems ................... 29 1.3.1UniprocessorScheduling ....................... 29 1.3.2MultiprocessorScheduling ....................... 30 1.3.3OverloadHandling ........................... 33 1.4SchedulingofAdaptiveReal-TimeSystems ................. 34 1.4.1UniprocessorScheduling ....................... 34 1.4.2MultiprocessorScheduling ....................... 36 1.5Real-TimeSchedulingUnderUncertainty .................. 36 1.6ThesisStatement ................................ 37 1.7Contributions .................................. 38 1.8Organization .................................. 39 2RELATEDWORK .................................. 46 2.1ElasticTaskModel ............................... 46 2.2Rate-BasedExecutionModel ......................... 49 2.3FeedbackControlReal-TimeSchedulingFramework ............ 51 2.4TaskReweightingSchemes .......................... 53 2.5Mode-ChangeProtocolsforSchedulingMulti-ModeReal-TimeSystems 56 2.6DeterminingAdaptiveTaskParameters ................... 58 3ELASTICENSEMBLESCHEDULING(EES)MANAGER ............ 61 3.1EnsembleSchedulingModel ......................... 61 3.2ProposedEnsembleSchedulingArchitecture ................ 63 3.2.1TaskUtilizationAdaptor(TUA) ..................... 63 6

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3.2.2Real-TimeTaskScheduler(RTS) ................... 64 3.3EnsembleSchedulingProcedure ....................... 65 3.4Summary .................................... 66 4ADAPTIVEENSEMBLESCHEDULINGWITHLIMITEDRESOURCES ..... 69 4.1AdaptingResourceUtilizationDemandsofTasks .............. 69 4.1.1ProblemFormulation .......................... 70 4.1.2WeightsofEnsembleTasks ...................... 74 4.1.2.1PolicycombinationI:kwinners/unrankednon-winners .. 75 4.1.2.2PolicycombinationII:kwinners/rankednon-winners ... 76 4.1.2.3PolicycombinationIII:relevantexperts/rankednon-winners 77 4.1.3SensitivityAnalysisTest ........................ 78 4.1.4TaskThrottling(TT)Heuristics ..................... 81 4.2Ensemble-Policy/Objective-Conscious(EPOC)SchedulingAlgorithm ... 82 4.2.1SchedulingTasksinEachEnsembleCycle ............. 83 4.2.2FromLocalEpochSolutionstoaGlobalSolution .......... 85 4.3TestEnsembleSystem ............................. 88 4.3.1DescriptionofTestEnsembleSystem ................ 89 4.3.2Implementation ............................. 90 4.4PerformanceEvaluation ............................ 91 4.4.1EnsembleSchedulingTimeliness ................... 92 4.4.1.1TCsavingfromsensitivityanalysisofcandidatesolutions 92 4.4.1.2Critical-TCoverheadsoftheTT-basedalgorithms .... 93 4.4.2CorrectnessbasedonEnsembleExecutionPolicy ......... 94 4.4.3CorrectnessbasedonEnsembleLearningPolicy .......... 94 4.5Discussion ................................... 97 4.6Summary .................................... 98 5ENHANCEDADAPTIVEREAL-TIMETASKSCHEDULER ............ 111 5.1ProblemFormulation .............................. 112 5.2AnomaliesinMultiprocessorScheduling ................... 113 5.2.1Dhall'sEffects .............................. 114 5.2.2CumulativeOfoadingFactorEffects ................. 115 5.2.3SequentialTaskModelEffects ..................... 116 5.3ModelsforOptimalMultiprocessorScheduling ............... 116 5.3.1FluidSchedule ............................. 117 5.3.2T-LPlane-basedSchedulingModel .................. 117 5.3.3DP-FAIRSchedulingModel ...................... 119 5.4EfcientApproximationofGradualLoadExecution(EAGLE)SchedulingAlgorithm .................................... 120 5.4.1OverviewofEAGLE'sSchedulingFlow ................ 121 5.4.2AssignmentofTaskParametersforLocalExecution ........ 121 5.4.3SchedulingLocalTaskExecution ................... 123 5.4.4MappingTaskstoProcessors ..................... 124 7

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5.4.5ExampleofEAGLEScheduling .................... 124 5.4.6OptimalityoftheEAGLESchedulingAlgorithm ........... 126 5.5PerformanceEvaluationofEAGLE ...................... 127 5.5.1Schedulability .............................. 128 5.5.2PreemptionandMigrationOverheads ................ 129 5.6SchedulabilityAnalysisofEAGLEunderModeChanges .......... 129 5.7EAGLEwithMode-TransitionProtocol(EAGLE-T) ............. 132 5.7.1SchedulingExample .......................... 134 5.7.2TransitionDelayandUtilizationDrift ................. 135 5.7.3EAGLE-TversusEPOC ........................ 137 5.8PerformanceEvaluationofEAGLE-T ..................... 138 5.8.1Drift ................................... 139 5.8.2Delay .................................. 141 5.9Summary .................................... 141 6CONTROL-SCHEDULINGAPPROACHFORENSEMBLESYSTEMSUNDERUNCERTAINTY .................................... 160 6.1ProblemFormulationofRobustEnsembleSchedulingunderUncertainty 161 6.2ArchitectureoftheFuzzyEESManager ................... 162 6.3TheModiedImplementationoftheTUA ................... 164 6.4ExtendedEAGLE-TAlgorithm ......................... 165 6.4.1FromGeneralAdaptiveReal-TimeSystemstoLimited-ResourceEnsembleSystemswithUncertainty ................. 166 6.4.2SlackReclamation ........................... 168 6.4.3ComputationTimeAnticipationandMode-ChangeRequests ... 169 6.5TheFuzzy-LogicController .......................... 171 6.5.1BasicsofFuzzy-LogicInference .................... 171 6.5.2TheDesignoftheFuzzy-LogicController(FZ) ............ 172 6.6ACase-StudyEnsembleSystem ....................... 174 6.7PerformanceEvaluation ............................ 176 6.7.1UncertaintyinTaskWCETsandBenetsoftheExtendedEAGLE-T 177 6.7.2UncertaintyinAvailableResourceCapacity ............. 181 6.7.3UncertaintyinBothResourceCapacityandTaskWCETs ..... 183 6.8Summary .................................... 183 7CONCLUSIONSANDFUTUREWORK ...................... 207 7.1SummaryofContributions ........................... 207 7.2FutureWork ................................... 210 REFERENCES ....................................... 211 BIOGRAPHICALSKETCH ................................ 224 8

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LISTOFTABLES Table page 1-1Descriptionsoffeaturesforcomparingthisresearchandrelatedwork. ..... 44 1-2AcomparisonbetweenthepriorapproachesandtheElasticEnsembleScheduling(EES)manager. ................................... 45 3-1Listofnotationsusedinensembleschedulingmodel. .............. 68 4-1DenitionofvariablesusedinSection 4.2 ..................... 108 4-2Critical-TCoverheads(i.e.,percentagesofcycleswithcritical-TCexecutions)intheTT-TC,TT-TC*andTT-Topalgorithms. ................... 109 4-3Statisticsoftheabsolutepercentageerror()ofoutputsofthetestsystemwithlimitedresourcesrelativetooutputsusingunlimitedresourceswhenschedulingwiththeTT-TC,TT-TC*andTT-Topalgorithms. .................. 110 5-1ThetaskparametersoftheEAGLEschedule. ................... 158 6-1Thelistofimportantnotations. ........................... 201 6-2TherulebaseoftheFZ. ............................... 202 6-3PerformancecomparisonbetweenEESandFuzzyEESwhenUsvariesbetween8and18. ....................................... 203 6-4PerformancecomparisonbetweenEESandFuzzyEESwhenUsvariesbetween8and18andtheactualtaskexecutiontimevariesbetween5to10. ...... 206 9

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LISTOFFIGURES Figure page 1-1Structureofanensemblesystemwithlimitedresources ............. 41 1-2Aclosed-loopBrain-MachineInterfaces(BMIs) .................. 41 1-3Anexampleoftheavionicsofanaircraft ...................... 42 1-4Anexamplesettingofacomplexdatacenter ................... 42 1-5Anillustrationoftasktimingparameters ...................... 43 3-1ThearchitectureandworkowoftheEESmanager ............... 67 4-1Theformulationoftheensembleschedulingproblem ............... 100 4-2CombinedowchartofthreeTask-Throttling-basedalgorithms ......... 100 4-3AnexampleofEPOCschedulingwhenallocationepochscoincidewithschedulingepochs. .................................. 101 4-4AnexampleofEPOCschedulingwhenallocationepochspartiallyoverlapwithschedulingepochs. ............................... 101 4-5Diagramsofhowdifferentexpertsindifferentcontextsmapcommandsintonextpositionsoftheagent. ............................. 101 4-6Testensemblesystemforcontrollingthemovementoftheagent. ........ 102 4-7Contextualvalues(y-axis)of500cycles(x-axis)forcontexttypes1through5(toptobottom) .................................... 103 4-8Responsibilitiesofexperts(y-axis)forcontextualsignalsinFigure 4-7 ..... 104 4-9Taskcompression(TC)savingsduetotheuseofsensitivityanalysisheuristic 105 4-10Comparisonofx-axisoutputstodemonstratetheimportanceoflearningpolicyinensemblescheduling ............................... 106 4-11ResponsibleregionsofexpertsfortheexperimentwhoseresultsareshowninFigure 4-12 .................................... 107 4-12Absolutepercentageerrorofthelimited-resourcetestensemblesystem .... 107 5-1Dhall'seffectsillustration ............................... 143 5-2AddressingDhall'seffectsusingzero-laxitypromotion .............. 143 5-3Anexamplescenariowithcumulativeofoadingfactoreffects .......... 144 5-4Resolvingcumulativeofoadingfactoreffectsbyanticipatingslacktime .... 144 10

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5-5Sequentialtaskmodeleffects ............................ 145 5-6Fluidscheduleversusapracticalschedule .................... 145 5-7Illustrationof(a)T-Lplanesacrosstimeand(b)taskexecutionwithinaT-Lplane ......................................... 146 5-8FlowdiagramandmaindatastructuresoftheEAGLEalgorithm ......... 146 5-9Anillustrationofparametersrepresentingataskstate. .............. 147 5-10Pseudo-codeoftheLD-PandLS-PphasesofEAGLE. .............. 148 5-11AnexampleofEAGLEschedule .......................... 149 5-12AschedulabilitycomparisonofveschedulingalgorithmsinschedulingrandomlygeneratedtasksetswhosetotalutilizationequalsU .......... 150 5-13Averagesandstandarddeviationsofthepercentagesofoverheads(i.e.,preemptions,migrationsandinvocations)ofeachschedulingalgorithm .. 151 5-14Pseudo-codeofEAGLE'smode-transitionprocedure. .............. 152 5-15AnexampleofanEAGLE-TscheduleforschedulingadynamictasksetT=f1:[0.2,1],2:[0.2,1],3:[0.2,1],4:[0.2,1],5:[0.2,1]gon3processors. .. 153 5-16Theutilizationdriftoftask4fromtheexampleinFigure 5-15 .......... 153 5-17Anexampleof(a)EPOCand(b)EAGLE-TschedulesforthestatictasksetinFigure 5-11 ...................................... 154 5-18Anexampleof(a)EPOCand(b)EAGLE-TschedulesforthedynamictasksetinFigure 5-15 .................................. 155 5-19TheaveragesandstandarddeviationsofmaximalutilizationdriftanditspercentagepertimeunitoftheEAGLE-SandEAGLE-Tschedules ....... 156 5-20Theaveragesandstandarddeviationsofmode-transitiondelayintheEAGLE-SandEAGLE-Tschedules .............................. 157 5-21Theaveragesandstandarddeviationsofcyclesinmodetransition(representedas%)intheEAGLE-SandEAGLE-Tschedules ................. 157 6-1AfeedbackschedulingarchitectureoftheFuzzyEESmanager. ......... 185 6-2AnexampleillustratingtwokindsofWCETsinensemblesystemswithlimitedresources. ....................................... 185 6-3ThecasewhenWCETsareuncertain. ....................... 186 6-4Astructureofafuzzy-logicsystem. ......................... 186 11

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6-5Themembershipfunctionsoftheinput/outputlinguisticvariablesofuDi,uRateand. ......................................... 187 6-6IllustrationofhowcrispscalarvaluesofuDianduRatecanbefuzziedtofuzzyvalues. ..................................... 187 6-7Anexampleofaruleinference. ........................... 188 6-8Defuzzicationusingthecentroidcalculationmethod. .............. 188 6-9TheoverallstructureoftheFZ. ........................... 188 6-10TheoutputsurfaceoftheFZ. ............................ 189 6-11Anexampleofatwo-jointroboticarm. ....................... 189 6-12Expertresponsibilitiesofthecase-studyensemblesystem ............ 190 6-13Input-spacecoverageofexpertsinthecase-studyensemblesystem ...... 191 6-14Performanceofthecase-studyensembleasafunctionofnumberofavailableprocessors. ...................................... 192 6-15PerformanceofEAGLE-TandXEAGLE-TwhenWCETsarecertain ...... 193 6-16PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareoverestimated .. 193 6-17ThepercentageofimprovementinthenumberofexecutedinstancespertimeunitwhenWCETsareoverestimated. ....................... 194 6-18ThepercentageofimprovementinthetotalnumberofcompletedjobswhenWCETsareoverestimated. ............................. 194 6-19ThepercentageofimprovementinthemeanoutputerrorwhenWCETsareoverestimated. .................................... 195 6-20ThepercentageofimprovementinjUa)-222(UsjwhenWCETsareoverestimated. 195 6-21PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareunderestimated 196 6-22ThepercentageofimprovementinthenumberofexecutedinstancespertimeunitwhenWCETsareunderestimated. ...................... 196 6-23ThepercentageofimprovementinthetotalnumberofcompletedjobswhenWCETsareunderestimated. ............................ 197 6-24ThepercentageofimprovementinthemeanoutputerrorwhenWCETsareunderestimated. ................................... 197 6-25ThepercentageofimprovementinjUa)-222(UsjwhenWCETsareunderestimated. 198 12

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6-26PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareoverestimatedtemporarily ...................................... 198 6-27PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareoverestimatedtemporarily ...................................... 199 6-28Thepercentagesofimprovementinmeanoutputerrorundervariousdegreesofresource-capacityuncertainty. .......................... 199 6-29Thepercentagesofimprovementinmeanoutputerrorundervariousdegreesofresource-capacityandWCETuncertainty. ................... 200 13

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LISTOFIMPORTANTABBREVIATIONS ASEDZL AnticipatingSlackEDZLBMI Brain-MachineInterfaceCriticalTC TCexecutioninthecriticalpathofexpertexecutionDP-FAIR Deadline-Partitioning-FairnessschedulingEAGLE Efcient-Approximation-of-Gradual-Load-ExecutionEAGLE-T EAGLEwithatransitionalmode-changeprocedureEAGLE-S EAGLEwithastep-wisemode-changeprocedureEDF EarliestDeadlineFirstEDZL EarliestDeadlineuntilZeroLaxityEES ElasticEnsembleSchedulingEPOC Ensemble-Policy/Objective-ConsciousschedulingalgorithmFuzzyEES theFuzzy-FeedbackEESmanagerFZ Fuzzy-logicinferencesystemKKT Karush-Kuhn-TuckerconditionsLD-P LocalparameterDecisionPhaseofT-Lplane-basedschedulingLLF LeastLaxityFirstLLREF LeastLocalRemainingExecutiontimeFirstLS-P LocalSchedulingPhaseofT-Lplane-basedschedulingLUB LeastUpperboundoftotalUtilizationachievablebyaschedulingalgorithmMCR Mode-ChangeRequestMoE Mixture-of-Experts 14

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NABLR NotionalApproximationforBalancingLoadResiduePfair ProportionatefairschedulingRTS Real-timeTaskSchedulerTC TaskCompressionT-Lplane Time-and-Local-execution-timeplaneTT TaskThrottlingTT-TC aTTalgorithmthatrequirescritical-TCexecutionsineverycycleTT-TC* aTTalgorithmthatrequirescritical-TCexecutionsinsomecycleTT-Top aTTalgorithmthatusesthetop-responsibilityheuristicTUA TaskUtilizationAdaptorWCET Worst-CaseExecutionTimeXEAGLE-T theextendedEAGLE-Talgorithm 15

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyREAL-TIMESCHEDULINGOFENSEMBLESYSTEMSWITHLIMITEDRESOURCESByPrapapornRattanatamrongDecember2011Chair:JoseA.B.FortesMajor:ElectricalandComputerEngineering Inspiredbythestrategyofdivideandconquer,ensemblesystemsutilizemultiplesimplecomputationalmodels(called`experts')thatcan,individuallyorinsomecombination,generatesolutionsforalargerrangeofinputcasesthantheirsingleoriginalmodel.Realsystemrequirementsofensemblesystems(e.g.,size,weight,powerandcostconstraints)oftenleadtolimitedavailabilityofcomputationalresourcesrequiredtosupportconcurrentexecutionofallexperts.Thisdissertationproposesageneralizedarchitecture,calledElasticEnsembleScheduling(EES)manager,toaddresstheproblemofschedulingexpertsinensemblesystemsinthewaythattheoverallsystemperformanceisminimallyaffectedbylimitedresources. TheEESmanagerconsistsofaTaskUtilizationAdaptor(TUA),anadaptiveReal-TimeScheduler(RTS)andaFuzzyfeedbackcontroller(FZ).TheTUAusesoptimizationtechniquestodeterminethetime-varyingresourceutilizationrequiredbyeachexperttoensurethatcriticalexpertsachievetheirbestperformancewhileguaranteeingminimumexecutiontimeneededbyotherexperts.TheRTScreatesascheduleofexpertexecutionthatallowseachtasktoachieveresourceutilizationascloseaspossibletoitsdemandwithoutanyviolationoftimeconstraintorensemblesystem'spolicies.Inordertocopewithuncertaintyinthesystemanddeploymentenvironment,theFZdeterminesthetotalutilizationallocationfortheTUAsothatthesystemfullyutilizestheavailableresourcecapacity.Thedissertationrstconsiders 16

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whenresourcesarededicatedandeachexperthasanaccurateworst-caseexecutiontime(WCET)andpresentstheimplementationsoftheEESmanagerforsystemswithuniformandnon-uniformWCETs.Then,theEESmanagerisextendedtosupportschedulingunderuncertainresourceavailabilityandimpreciseWCETs. Fromperformanceevaluation,expertsinaresource-constrainedcase-studyensemblesystemscheduledwiththeEESmanagerareshowntoproducesystemoutputscloselysimilar(8%error)tothoseofthesystemwithsufcientresources,althoughthelimited-resourcesystemhasupto40%lessresources.Thesimulationresultsalsoshowthatexecution-timeimprecisionandoccasionaluctuationofresourcecapacitycanbetoleratedanddemonstratetheEESmanager'sefciencywithreasonablysmalloverheadsinoptimization,preemptionandmigration. 17

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CHAPTER1INTRODUCTION Ensemblesystemshavebeendevelopedandappliedtomanydomainsofapplicationsforthepastfewdecades,includingmolecularbiology[ 35 64 147 ],nancial/business[ 34 63 73 ],signalprocessing[ 55 79 133 ],imageprocessing[ 71 119 ]andmedicine[ 60 65 117 144 ].Ingeneral,theobjectiveoftheensembleapproachistoreplaceasinglecomplexcomputationalmodelthatmightbeapplicabletoonlyasubsetofthepossibleinputsbymultiplesimplerandmoretailoredmodels(eachcorrespondingtoanexpert)thatcan,individuallyorinsomecombination,generatesolutionsforalargerrangeofinputcases.Inmanyapplications,includingBrain-MachineInterfaces,whichoriginallymotivatethiswork,ensemblesystemscouldconsistofalargenumberofexpertstoachievecomplexgoalsorbedeployedasembeddedsystemsinremotearea.Thissubsequentlyraisesanimportantissueinresourcemanagementsinceconcurrentexecutionsofallexpertsineverycycleareimpracticalduetoresourcelimitations.Thisdissertationaddresseshowexecutionofallrelevantexpertsinreal-timeensemblesystemscanbeprovidedwhileminimizingtheimpactoflimitedresourcesonthesystemperformance.Priortotheresearchinthisdissertation,nomechanismhadbeenproposedforhandlingdynamicreal-timeschedulinginthecontextofensemblesystems. Thischapterprovidesabriefoverviewofensemblesystemsandtheirapplicationsthatmotivatetheresearchinthisdissertation,namelyBrain-MachineInterfaces,switching-controlsystemsandvirtualworlds.Bythecharacteristicsoftheseexampleapplications,resourcelimitationscouldexistandhinderthesystemstoachievetheirtargetperformancegoals.Then,theclassicalmodelofreal-timesystemsandrelatedpriorworkonreal-timeschedulingarepresented.Next,thethesisofthisdissertationisstatedtoemphasizethenecessityofensembleschedulingunderresourcelimitations. 18

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Thechapterisconcludedbyasummaryofdissertation'scontributionsandanoutlinefortheremainderofthedissertation. 1.1EnsembleSystemsandMotivatingApplications AsillustratedinFigure 1-1 ,insteadofseekingasolutionthatworksforallinstancesofaproblem,anensemblesystemusesvaryingcombinationsofNindividualcomputationalmodels(calledexperts)tosolvetheproblemforsubsetsofthoseinstances.Ensemblescombiningmultiplemodelshavebeentheoreticallyandempiricallyprovedtoensuresignicantlybetterperformancethantheirsingleoriginalmodels[ 43 67 ].Ensemblesystemsgenerallyoperateincycles(referredlateras`ensemblecycles')inwhichtheirexpertsprocessasysteminputandproducetheiroutputsyi,i=1,..,Nwithineachcycle.Theseexpertscanbeminorvariantsofthesamebasicmodelordiversemodelsthatarenotfromthesamefamily[ 128 ].Bagging[ 28 ],boosting[ 131 ],AdaBoost[ 57 ]andmixtureofexperts[ 77 ]aregroupsoftechniquesproposedforcreationofensemblemodels. ByassigningaresponsibilityvalueRitoeachexpert,agatingcomponentdetermineswhichandhowexpertoutputsy1,..,yNneedtobeaggregatedtogeneratetheoutputforanygivenprobleminstance.Thenaloutputcanbegeneratedbyeitherfusingtheoutputsofsomeexpertstogetherorselectingtheoutputofasingleexpert.Withoutlossofgenerality,thisdissertationparticularlyfocusesonahybridofbothmethodsinwhichkexpertsareselected(theseexpertsarecalled`winners')bythegatingcomponentandtheiroutputsareweightedlycombinedtogethertocreateanaloutput.Manyapproachescanbeusedfordesigningandimplementinggatingcomponentsofensemblesystemsincludingdistanceofaninputfromexperts'responsiblesubspaces,geneticalgorithmsandreinforcementlearning[ 128 ].Thegatingcanbedoneatthebeginningofeachcyclebasedonapartitionofthesystem'sinputspaceand/ortherelativeperformancesoftheexperts.Inaninput-gatingensemble,themostresponsibleexpertsaredeterminedbythedistancebetweenthepositionofthe 19

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currentinputintheinputspaceandtheinputregionsforwhichexpertsareresponsible;onlyasubsetofexpertsneedstobeexecuted.Inaperformance-gatingensemble,eithertheexperts'performancesneedtobepredictable,ortimeandresourcesmustbeavailableforallexpertstoexecuteandtheirperformancestobeknown.Thefocusofthisworkisonthecaseswhenexpertresponsibilitiesareknownafterinputsarerealizedbutpriortoexpertexecution;responsibilitiescanbestaticallydetermined(staticinput-gating)ordynamicallypredicted(dynamicinput-gatingorperformance-gating). Eachensemblesystemimplementsexecutionandlearningpoliciesduringitsoperation.Theexecutionpoliciesdeterminethegatingcomponent'scriteriaforselectingwhichexpertsshouldhavetheiroutputscombinedtogenerateasystemoutput.Examplesofexecutionpoliciesincludethoseusedtocomputeweightedaveragesofallorasubsetoftheexpertoutputs,suchasthek-winnerpolicieswhereoutputsfromtheexpertswithtop-kresponsibilitiesarecombinedtoproducethesystem'soutput.Inmanycases,includingthesystemsofinteresttobepresentednext,conceptdriftsmayoccurandexpertsneedtohandlethedriftsbyrunninganadaptationprocedurethatallowsthemtolearnandmaintaintheirperformancesovertime.Thus,theoperationofsuchensemblesystemsalsoentailslearningpolicies,whichdeterminewhenandwhichexpertsshouldbeadapted.Thelearningcanbebasedonsupervisedlearningusingtheexpert'soutputerrorsinrelationtothedesiredoutput,orreinforcementlearning,whichneedsrewardinformationfrompreviousexecution;hencetheeffectiveadaptationisonlypossiblewhentheexpert'sexecutionistimelywithrespecttotheeventsitlearnsfrom,regardlessofwhetheritcontributestothenaloutputofthesystematthattime. Ingeneral,resourcelimitationsmaybeinherentinseveraltypesofensemblesystems,namely: Ensemblesystemswithaverylargenumberofexpertstoexecuteinparallelthatcaneasilyexceedthecapacityofavailableresources. 20

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Ensemblesystemswhereeachexpertisacomputationallychallengingtask.Resourceconstraintslimittheexecutionofsuchsystemswithevenasmallnumberofexperts(inthescaleoftens). EmbeddedEnsemblesystems. Sincecomputingresources,e.g.,processorsormemory,continuouslybecomeaffordable,sufcientresourcesmaybeclaimedforthersttwotypesofensemblesystems.However,itisanticipatedthatmanagementcostsofsufcientlylargeresourcescouldstillbeprohibitiveand,hence,therewillbelimitationsofresourcecapacityforthesesystems.Forembeddedensemblesystems,itisundeniablethatresourcelimitationsoftencomefromrestrictionsonsystemsize,powerandcost.Brain-MachineInterfacesystems[ 42 75 157 ]wouldfallintherstandlastcategories,forexample,whenimplementedasamobileassistivedevicetohelpaparaplegicachievesophisticatedcontrolofroboticprosthetics.Examplesofthesecondcategoryaretheensembleforecastingsystemsreportedin[ 51 122 156 ].Whenresourcesareinsufcienttoexecuteallexpertsineverycycle,ensembleschedulingisneededtoexecuteonlytherightsubsetofexperts.Next,threetimelyapplicationsofensemblesystemswithlimitedresourcesaredescribedtoillustratetheneedandchallengesofensemblescheduling. 1.1.1Brain-MachineInterfaces TheworkinthisdissertationisprimarilymotivatedbyanapplicationofensemblesystemstoBrain-MachineInterfaces(BMIs)forneuralprostheticcontrol[ 55 83 130 ].Thistypeofsystemstranslatesneuralactivitiesintomotor-controlcommandsforprostheticlimbs,roboticarmsorotherelectromechanicaldevices.Thetranslationrequiresusingabrain-inspiredmotor-controlarchitecture[ 157 ],whichadvocatestheuseofmultiplemodelscapableofcollectivelyderivingmotorcommandsfromneuraldata,sensorialinputsandresponsibilitiesofindividualmodels.Intuitively,eachcomputationalmodelorexpertspecializesincontrollingtheroboticarmforaparticulartypeorpartofdesiredmovementtrajectories.Theresponsibilityofanexpert(i.e.,thedegreetowhich 21

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anexpertissuitedforaspecictypeofmovement)changesdynamically,dependingonthepatternsoftheneuralactivitiesofthesubjectataparticularmoment.Inaddition,expertsmustbetrainedthroughouttheoperationofthesystem,sotheycanadaptandlearnbasedonpastexperiences(i.e.,implementingexecutionandlearningpoliciesasmentionedintheprevioussection). ThewholeBMIcycle(showninFigure 1-2 ),fromsignalrecordinganddatatranslationtoroboticcontrol,mustbecompletedperiodically,witheachofthesuccessivecycleslastingaxedamountoftime,atypicalvaluebeing60ms[ 47 ].Thistightcycletimemustaccommodatetheexecutionofexpertsandtrainingproceduresinaccordancewiththeexecutionandlearningpolicies.Therefore,onlyveryfewmillisecondsareavailabletomakedecisionsofwhentoscheduleeverysingleexpertanditslearningprocedure.Sincemanytrainingproceduresneedtoknowthevaluesoftheexpertoutputs,itishereonassumed,unlessotherwisestated,thattheexecutionofanexpertalsoimpliestheexecutionofitslearningprocedureandviceversa.Inotherwords,inagivencycle,someexpertswillexecuteandlearnbecausetheexecutionpolicyrequiresthattheybeexecuted,andotherexpertswillexecuteandlearnbecausethelearningpolicyrequiresthattheyundergoadditionaltraining.Thechallengeofashortcycle-timeisexacerbatedwhenthenumberofexpertsislarge,thusrequiringcomputationalresourcesthatexceedthoseavailableonBMIsystems.Thecaseofalargenumberofexpertsisanimportantlikelyscenario,sincetheuseofasubstantialnumberofmodelscangreatlyimprovetheoverallsystem'sperformance,andisanexpectednecessityoffutureBMIsystemstargetingcomplexhuman-likemotortasks.Sincenotallexpertscanexecuteineverycycle,theproblemtobesolvedconsistsofquicklyimplementingtheexecutionandlearningpolicies,i.e.,dynamicallyschedulingthetimelyexecutionoftheexpertswhoseoutputsareneededbasedondynamicallychangingexpertresponsibilities,andasmanyadditionalexpertsasoftenaspossiblesothattheycancontinuetolearn. 22

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Inpreviouswork[ 48 124 125 ],auniquegridmiddlewareinfrastructurecalledCyberWorkstation(CW)hasbeendevelopedbytheauthorofthisthesisandotherresearchersoftheAdvancedComputingandInformationSystems(ACIS)laboratoryattheUniversityofFlorida.IthasenabledthedeploymentandmanagementofknownparallelimplementationsofBMIalgorithmsoncampuscomputingresourcesforclosed-loopBMIexperiments.Theproposedensemble-schedulingsolutionpresentedinthisdissertationwillnotonlybebenecialforthecaseofembeddedBMIs,buttheyalsolaythegroundworkforscalablemiddlewaretechniquesthatcaneventuallysupportincreasinglyelaborateBMI-researchtestbedsinwhichsubjectscancarryoutmorecomplextasks.TheseadvancedBMItestbedsareessentialforthedevelopmentofthecomputationalcomponentsofBMIsandtheoptimizationoftheirresourceutilizationonfuturecomputerswithmultiplemulti-coreprocessors,whichcorrespondstotheanticipatedhardwarethatwillcollectivelyprovidethenecessaryresourcesforexecutionofasubsetofexpertsinaffordableandeffectivepatient-dedicatedassistivedevices. 1.1.2Switching-ControlSystems Asmodernsystemsbecomeincreasinglycomplex,thewaytomanagethemshiftsfromasinglesystemadministratortoanensemble-basedautonomicsystemmanagementsoftware[ 76 82 146 ]comprisingmultiplesensorsandactuatormodules.Insuchsystems,thedynamicsofthesystemsarecontrolledbyswitchingfromasubsetofactuatorstoanotherinasystematicwaysothat,ineachinstantoftime,onlyonesetofactuatorsisactive.Thissubjectisstudiedinliteratureas`switchingcontrol'[ 7 ].Switching-controlsystemscanbeviewedashybridsystemsthatcouplebetweencontinuousanddiscreteevents;thecontinuousdynamicsofthesystemsaresupervisedbyswitchingpoliciesgeneratedbydiscrete-eventsupervisors.Eachmodule(i.e.,expert)implementstheinternallogicofaself-managementdiagnostic/prognosticcontrollerthatcanhaveuniqueresourcedemandandexecutiontimedependingonhowcloseitistoitsequilibriumpoint.Inaddition,thesesystemsrepresentageneralizedclassof 23

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ensemblesystemsinwhichallexpertsarenotrequiredtoproduceresultswithineachcycle,theirresourcedemandscouldbetotallydifferentfromoneanother,andchangingacrosscycles. Oneexampleofswitching-controlsystemsisareal-timesystemcontrollingtheavionicsofanaircraft,asshowninFigure 1-3 .Thesystemhastodealwithdifferenttypesofactivitiesduringtake-off,cruisingandlandingstages.Oneorseveralmodulesmaymonitoreachsubsystemoftheaircraft[ 127 ].Theresourcedemandofeachmodulevariesbythemodeinwhichthesystemisoperating.Inaddition,thesystemmustrespondinatimelymannertoavoidcollisionswithobstaclesorothervehicles.Boskovicetal.[ 26 ]addresstheproblemofcontrolrecongurationinthepresenceofwingdamageusingonlineestimationofdamage-relatedaircraftparameterstodecideswitchingamongmultiplecontrollersforaparticularmodeofoperation.Astructuralhealth-monitoringsystemofanaircraft[ 56 ]canbeconstructedusingmultiplesensorswhichcontributetheirobservations,measurementsordecisionstoacentralfusionnodeinordertoperformsituationassessment.Insuchsystems,thechoiceofcomplementarysensorsrequiresproperresourcemanagementinorderforthesesensorstooptimizetheircollectiveperformancerelativetotheapplication'scriteria. Figure 1-4 showsanensembleapplicationforalarge-scalecomputingfacility.Theensemblesystementailsaconsiderablenumberofreal-timefault-analysismodulestoprovidehigh-resolutionandaccuratemonitoringofthedatacenter'sstate.Foranintricatesystem,thecostsofrunningthousandsofthesemodulescontinuouslyandinreal-timecouldeasilyexceedtheavailableresourcesandbenetsinitiallyexpectedfromusingtheautonomicmanagement.Inpractice,allofthemmightnotneedtobeexecutedatthesametimewiththesamequalityofserviceinordertoeffectivelymanagetheoverallsystem.Forexample,whenanirregularityisdetectedinaclusternode,onlythesubsetofmodulesassociatedtothesubsystemsrelatedtotheoperationofthatparticularnodemaybeneeded.Allothermodulesjustneedtobeexecutedaspossible, 24

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incasetheyarelaterfoundcontributingtothecauseoftheabnormalbehavioroccurredinthenode.Coordinatingtheensembleofcontrollersthatprocessinformationfromtheubiquitoussensingandknowledgediscoverylayers,andadjustlower-levelcontrollersinaccordancewithmanagementpoliciesdenedbyadministrators,endusers,andhigher-levelcontrollersisnecessarytoensurethatallcomponentsworktogetherinharmony,andcanwhollyoptimizeadatacentertothedesiredmixofoperationalcostandenvironmentalimpact[ 89 154 160 166 ]. 1.1.3VirtualWorldsandthe3DInternet Thedebutofmassivelymulti-playeronlinegames(MMOGs)inthelate90shaveopenedupthepossibilityforthousandsofplayerstointeracttogetherinapersistentworld[ 13 150 ].Overtheyears,thepopularityofMMOGscontinuestoincrease,andtheyhavebecomemorethanjustonlinegamesbutvirtualworldsinwhichuserscansocializeforsomereal-lifepurposes.Withthemassadoptionof3Dimageryandvirtualworldsonwebsites,the3DInternetisemergingfasterthanourexpectations[ 5 ].Intheverynearfuture,virtualworldscouldbeapartofeverydayrealityjustascellphonesandemailsaretoday. Anissuearisingwiththewideadoptionofvirtualworldtechnologyisitsscalability,i.e.,theabilitytosupporthundredsofthousandsofplayersandmillionsofobjectsinacontinuousanduniqueworld.Inatypicalvirtualworld,eachclientviewsa3Dgraphicalrepresentationoftheworldandcontrolsaplayeranavatarwhichcanperformactions.Thebasicbuildingblocksofsuchactionsincludemovingtheavatar,pickingupobjectsandcommunicatingwithotherplayers.Thescalabilityproblemsthatvirtualworldsfacearisefromthehandlingofmassiveamountsofobjectsandconnectedplayers,presentingthemwithareal-timeconsistentviewoftheworld,goodreal-timeperformanceandanenjoyableexperience. Relatedworkintheeldpartiallyovercomesthischallengeofscalebyreducingtheunnecessaryworkloadinvirtualworlds.Interest-managementtechniques[ 4 72 25

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112 163 ]areusedtodetermineandsendonlyrelevantstatechangestoeachplayer;theypotentiallyreducetheamountofinformationexchangebetweenplayersandtheresourcesitconsumes.Tominimizetheheavyworkloadassociatedwithrenderingtasks,Level-Of-Detail(LOD)schemes[ 8 9 96 107 ]eliminategeometricprimitivesingraphicrenderingthataretoosmalltomakevisibleimpactstonalimages.Despitehavingallthesetechniques,therearestillsomehighcostsinhostingvirtualworlds.SecondLife[ 2 70 129 ]requires20,000processorcoresformaintaining20,000regionsregardlessofwhethertheregionsareutilizedorempty.ForOpenSim[ 1 54 ],althoughthecostsofhostinganemptyregionarenearlyzero,continuousprocessinginphysicsandscriptingevenifthereisnoneedordesiretodosoposeshighexpenses.Theimplementationofaresource-managementschemethatcanarrangecomputingresourcesbasedontheloadofthesystematanyparticularmoment,i.e.,allocatingmoreresourcestobusyregionsoftheworldthantoinactiveones,inordertoprovideaseamlessworldtomassiveusersremainsanopenresearchproblem[ 87 ]. Usingagatingcomponentthatimplementstechniquessimilartothoseusedinloadreduction,expertsortasksinphysicsenginesofthevirtualworldscanbeprioritizedatanymoment.Theauthorsof[ 12 ]investigatetheeffectofvaryingtheupdaterateofacomputer-generatedsimulationwithvariousfrequenciesonthesenseofpresencewithinstereoscopicvirtualenvironments.Theirresultsindicatethatwhilethesubjectivereportofpresencewithinthevirtualenvironmentwithlowupdateratesissignicantlylessincomparisontohigherupdaterates,thereisacertainrangeofupdatefrequencieswithinwhichthereportedlevelofpresencewasperceivedsimilarlybyusers.Preliminaryworkin[ 145 ]investigatestheuseoftheelastictaskmodelandscheduling[ 32 ]toprovideLODeffectforthegamecharactersaccordingtotheworkloadofthegameonMicrosoftXbox360.Thissuggeststhatcomputationalresourcescouldbesavedusingaslowerupdateratewhilemaintainingagivenlevelofpresencetousers;thegoalistodevisean 26

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onlinepriority-basedschedulerthatcanselectmostrelevanttasksatanymomentforexecutionwhileschedulingtheremainderswhennecessaryandpossible. 1.2Real-TimeSystems Sinceensemblesystemsinreal-timeapplicationsareofparticularinteresttotheresearchinthisdissertation,thissectionprovidesabackgroundonreal-timesystemsandpriorworkinschedulingthem.Thesuccessofreal-timeapplicationsdependsnotonlyontheavailabilityoftheirresults,butalsoonthetimewhentheseresultsareproduced.Theconsequencesofmissingdeadlinesofhardreal-timeapplicationscanbecatastrophic,whereassuchconsequencesforsoftreal-timeapplicationsarerelativelylessdamaging.Thegoalofreal-timeschedulingistoensurethattasksareallocatedresourcesandtimeintervalsinsuchawaythattheirtimelinessrequirementscanbemet[ 141 ].Areal-timesystemisoftenmodeledasasetofNconcurrenttasks(1toN),whichcanbeminimallycharacterizedbyaworst-caseexecutiontime(WCET)Ciandatimeconstraintinwhichtheexecutionmustbecompleted.Eachtaskconsistsofaninnitesequenceofinstances(called`jobs').Theterms`taskinstances'and`jobs'areusedinterchangeably.Thejthjobofataskireleasesattimerijanditsabsolutedeadlineisdij=rij+DiwhereDiisarelativedeadlineofthetask.Thetasksetissynchronous,i.e.,alltasksarrivesimultaneouslyatstartuptime(ri1=0).Ifthereisaxedperiodbetweentwoconsecutivejobreleasesofthesametask,thetaskisconsideredperiodic.Alternatively,ifthetaskreleasetimeisirregularatsomeunboundedinterval,thetaskiscalledaperiodic.Ifsuchtheintervalhasaknownboundary,thetaskissporadic.Forperiodicandsporadictasks,taskperiods(denotedasPi)representtheirxedperiodsandboundedintervals,respectively.TherelativedeadlineDithatcanbelessthan,equaltoorgreaterthanitsperiod.Ui(=Ci=Pi)iscalledthetask'sresource-utilizationdemand,referredinshortastaskdemand.Figure 1-5 illustratesthemeaningofthesetimingparameterswhenDi=Pi. 27

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Anyreal-timetaskwhosealltimingparameters(i.e.,Ci,Pi,andUi)havestaticvaluesisconsideredastaticreal-timetask.Staticreal-timetasksaretypicalinclassicalreal-timesystems.Ontheotherhand,ifareal-timetaskcanbecharacterizedbytimingparametersthathavedynamicvalues,thetaskisclassiedasanadaptivereal-timetask.Adaptivereal-timetaskscanbecategorizedfurthertobemultiple-version,skippable,andtransitional.Multiple-versionreal-timetaskshavemorethanoneversionofimplementation,eachwithitsownuniquesetoftimingparameters.Atmostoneversioncanbeexecutedatanytime.Skippabletaskshaveonlyoneversionofcodeandtheycanfunctionnormallyalthoughsomeoftheirinstancesareskipped(i.e.,notexecuted).Transitionaltasksalsohavesingleversionofcodeandsequentialexecutionoftheirinstancesarecriticalforthecorrectnessofthetasks,howevertheexecutionratesoftheirinstancescanvarywithinpredenedrangesdependingontheiroperatingconditions.Astaticreal-timesystemconsistsofonlystaticreal-timetasks,whileanadaptivereal-timesystemhasoneormoreadaptivereal-timetasks.Theadaptivereal-timesystemsareinthefocusofthisdissertation.Ifthereisasingleprocessorfortheexecutionofalltasks,thesystemiscalledauniprocessorsystem.Otherwise,thesystemisamultiprocessorsystemwithM(>1)processors.Whenthecomputationaldemandofareal-timetasksetexceedstheavailableresourcecapacity,thesystemissaidtobeoverloaded,whichcancauseoneormorereal-timetaskstomisstheirdeadlines.Theoverloadconditioncanoccurduetomanyreasons,suchasadditionofnewtasks,changesofoperatingmodes,taskexecutionoverrunandfailureofsystemcomponents.Ifanoverloadisnotproperlyhandled,itmaydisruptthesystem'sfunctionality.Giventhevastamountofliteratureavailableinschedulingreal-timesystems,thenextsectionsonlysummarizetheexistingapproachesrelevanttothescopeofthisdissertation. 28

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1.3SchedulingofStaticReal-TimeSystems 1.3.1UniprocessorScheduling Mostuniprocessorschedulingalgorithmsusuallydetermineadistinctprioritytoeachactivetaskinstanceandtheinstancewiththehighestpriorityamongtheseactiveinstancesischosenforexecutionateachtimeinstant[ 95 ].IfU=Pi=1,...,NUi1,thetasksetisconsideredschedulablewithsomeuniprocessoralgorithm.Uniprocessoralgorithmscanbeclassiedintostatic-anddynamic-priorityalgorithms.TherelativejobprioritiesofeverypairoftasksAandBwhereA6=Bdeterminedbystatic-priorityalgorithmsdonotchange,i.e.,anyjobofAalwayshasahigherprioritythananyjobofB.RateMonotonic(RM)schedulingalgorithm[ 100 ]isanexampleofstatic-priorityalgorithmsinwhichthehighestpriorityisassignedtoataskwiththeshortestperiod.ItisknownthatifUN(21=N)]TJ /F6 11.955 Tf 12.74 0 Td[(1),thetasksetisschedulablebyRM.AsN!1,theleastupperboundoftotalutilization(LUB)achievablebyRMisapproximately0.69.Manymethods,suchassub-taskanddual-prioritymethods,havebeenproposedtoimprovetheRM'sLUBwiththetradeoffonthecomplexitycostandsystemoverload[ 80 ].DeadlineMonotonic(DM)schedulingalgorithmusestheinverse-deadlinepriorityassignment,i.e.,jobprioritiesareinverselyproportionaltotheirrelativedeadlines.DMschedulingissimilartoRMschedulingwhenatask'srelativedeadlineequaltoitsperiod(i.e.,Di=Pi). Fordynamic-priorityalgorithms,thereisnorestrictionuponwhichprioritiesareassignedtoeachindividualjob.ForanypairoftasksAandB,itispossiblethatajobofAhasahigherprioritythanajobofBattimet1butanotherjobofAhasalowerprioritythanaB'sjobattimet2.TheEarliest-Deadline-First(EDF)schedulingalgorithmisanexampleofanoptimalschedulingalgorithmforschedulingreal-timetasksbyassigningdynamicpriorities.EDFassignsthehighestprioritytothetaskthathastheearliestabsolutedeadline,andschedulesitrst.ItsLUBisproventobe1,i.e.,anytasksetwithU1canbesuccessfullyscheduledusingEDFonasingleprocessor. 29

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TheLeast-Laxity-First(LLF)algorithm[ 111 ]isanotherexampleofdynamic-priorityschedulers.Tasks'prioritiesaredeterminedbasedontheiridletimes,calledtasklaxities.Atask'slaxityisdenedasthetimetoitsjob'sdeadlineminusesitsremainingcomputationtime.Taskswhoseidletimesareshorterhavehigherpriorities.LLF'sLUBissimilartoEDF's.BothEDFandLLFareoptimalalgorithmsforuniprocessorscheduling. 1.3.2MultiprocessorScheduling Asmulti-coreplatformsbecomemoreandmorecommonintoday'scomputingenvironments,researchonreal-timemultiprocessorschedulingreceivesincreasingattentionduetotheneedtodetermineanassignmentandexecutionorderofthesetasksonmultiprocessorsinawaythatallowsmosteffectiveuseoftheavailableprocessingcapacity(i.e.,accommodatingasmanytasksaspossible)whilemeetingtheirrealtimerequirements.Whilemanyoptimalschedulersexistforuniprocessorsystems(e.g.,EDFandLLF),theiroptimalityisknowntobreakdownonmultiprocessorsystems.TheEDFalgorithmcannotguaranteetosuccessfullyscheduleasetofperiodictasksonMidenticalprocessorsifthetotalutilizationexceedsM(1)]TJ /F3 11.955 Tf 10.9 0 Td[(Umax)+Umax,whereUmaxisthemaximumutilizationofeveryindividualtask[ 62 ],whichislessthanorequalto1.Forexample,ifUmax=0.8,EDFisguaranteedtobeabletoschedulealltaskstomeettheirdeadlineson16processorsiftheoverallutilizationofalltasksinthesystemdoesnotexceed4.Thisleavesthreequartersoftheavailableresourcesunderutilized. Manyalternativealgorithmshavebeenproposedtoimprovetheworst-caseresourceutilizationonmultiprocessorscheduling.Theycanbeclassiedintopartitioningandglobalapproaches.Inpartitioningscheduling,alltheinstancesofataskareexecutedonthesameprocessor.Tasksareassignedtoprocessorsinadvanceusingsometask-assignmentalgorithms.Oneachprocessor,differentorsimilaruniprocessorschedulingalgorithmscanbeusedtoscheduleasubsetoftasksassignedtoit.Examplesofpartitioningmultiprocessorschedulingalgorithmscanbefoundin[ 44 ]. 30

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Partitioningapproacheshavetwodrawbacks:(1)ndinganoptimalassignmentoftaskstoprocessorsisNP-hard,sonon-optimalheuristicsaregenerallyused,and(2)thereexistmanytasksetsthatareschedulableifandonlyiftasksarenotpartitioned(i.e.,tasksarerestrictedtoexecutesolelyontheassignedprocessor).Carpenteretal.[ 36 ]showthattheoptimalityofmultiprocessorschedulingisknowntobeachievedbyonlysomealgorithmsthatallowfulljobmigrationandunrestricteddynamicjobpriorities,whicharedenitelyimpossibleinpartitioningapproaches.Thesepartitioningapproachesarenonethelessattractivebecauseonceanallocationoftaskstoprocessorshasbeenachieved,real-timeschedulingtechniquesandanalysesforuniprocessorsystemscanbedirectlyapplied. Incontrast,inaglobalschemeataskcanmigratefromoneprocessortoanotherduringitsexecution.Proportional-fairorPfair[ 14 ],LLREF(standsforLargestLocalRemainingExecutiontimeFirst)[ 39 ]areoptimalglobalmultiprocessorschedulingalgorithmsthatcanachievethefullusageofprocessortimewhileguaranteeingthatalltasksmeetdeadlines.Pfairschedulingdividesthetimeintoequallengthquanta.Ateachtimequantaoflengthl,theschedulerallocatestaskstoprocessors,suchthattheaccumulatedprocessortimeallocatedtoeachtaskwithademandUiwillbeeitherblUicordlUie;theconstraintdenes`proportionalfairness'.LLREFdividesthetimeintosectionsseparatedbytaskreleases.Theremaininglocal-executiontimeforataskatthestartofsectionistheamountofexecutiontimethatthetaskwouldbeallocatedduringthatsectioninauidschedule,i.e.,lUiwherelisthelengthofthesectionandUiisthetask'sresourcedemand.Atask'sremaininglocal-executiontimedecrementsasthetaskexecutesduringthesection.Atthestartofeachsection,Mtasksareselectedtoexecuteonthebasisofthelargest-remaining-local-execution-timerstwithaprioritypromotiontoanytaskwithzerolaxity.Unfortunately,theoptimalityofthesealgorithmscomeswithhighruntimeoverheads(i.e.,preemptions,migrationsandschedulerinvocations)whichmakethemunattractiveforrealimplementation.Forexample,inthe 31

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caseofPfair,theuseoftimequantaanduid-likeschedulingraisestheneedforveryne-grainedtimersandincreasesthenumberofschedulerinvocationssignicantly[ 81 ].Manyextensionsonthesealgorithmswereproposedtoimprovetheirefciency.Severalschedulingmodelswereproposedforunderstandingoptimalmultiprocessorschedulers,includingTime-and-Local-execution-time(orT-L)plane-based[ 40 ]andDeadline-Partitioning(orDP-FAIR)[ 97 ]scheduling.Bypartitioningtaskexecutionintoseriesofsmallperiodsdeterminedbyanytwotaskarrivalinstants,signicantschedulingoverheads(i.e.,schedulerinvocations,preemptionsandmigrations)canbereduced[ 97 110 ]. Alternatively,severalsuboptimalbutefcientglobalalgorithms,suchasEarliestDeadlineFirstuntilZeroLaxity(EDZL)[ 94 ]andAnticipatingSlackEDZL(ASEDZL)[ 53 ],andNotionalApproximationforBalancingLoadResidues(NABLR)[ 123 ]wereproposedduetotheirlowcomplexitiesandoverheads.TopreventDhall'seffect[ 46 ],whenataskreacheszerolaxity(i.e.,ifthetaskdoesnotexecuteatthattime,itwillmissthedeadline)EDZLassignstoitthehighestpriority.ASEDZLextendsEDZLbyconsideringeachintervalbetweentwoconsecutivejobarrivalsofanytaskandplanningtaskexecutiontimeswithintheintervalbasedonearliestdeadlinepriority.Byanticipatingpossibleslacktimeduringintervals,ASEDZLcanhandleadditionalmultiprocessor-schedulinganomaliesandimprovetheschedulabilityofEDZLbyabout20%[ 123 ].NABLR,oneofthepriorworkofthisdissertation'sauthor,usesaheuristicthatconsidersboththeremainingexecutiontimesandlaxitiesoftaskswhenplanningtaskexecutionbetweenanyconsecutivejobreleaseinstantssothattaskshavemoderatelybalancedloadresidues(i.e.,lessdeadlinemisses).TheevaluationconrmsthatNABLRgreatlyoutperformsEDZLandASEDZLbyupto53%insuccessfullyschedulingalargenumberofrandomlygeneratedtasksets,particularlywhenthetotalutilizationofeachtasksetequalsavailableresourcecapacity(i.e.,U=M).NABLRincursonlysmalladditionaloverheadsbeyondthoseofotheralgorithms.Insomecases, 32

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thepercentagesofpreemptionsandmigrationsofNABLRturnouttobesmallerthanthoseofEDZL[ 123 ].Highschedulabilityisnecessaryforefcientresourceutilizationconcernedinthisdissertationsinceitimpliesthatmoretaskscanbeaccommodatedtomeetdeadlinesusingthesameamountofresources. 1.3.3OverloadHandling Admissioncontrolpoliciesarewidely-usedfordealingwithoverloadconditionsforstaticreal-timetasks.Byrejectingorcancelingsometasks,thetotalsystemloadcanbekeptunderavailableresourcecapacityandahardreal-timeguaranteecanbeprovidedtothoseacceptedintothesystem.Thesimpleadmissioncontrolapproachesuseschedulabilitytestsofschedulingalgorithmsbasedontasks'worst-caseexecutiontimesuponeachtaskarrival[ 6 22 135 140 ];anewtaskisacceptedintothesystemonlyifitdoesnotcauseanoverload.Theseapproaches,however,donottakeimportanceoftasksintoaccountandalwaysrejectnewlyarrivedtasksnomatterhowimportanttheyare.Dover[ 85 ],theheuristicusedin[ 141 ]andRobustEarliestDeadline(RED)[ 31 ]areexamplesofbest-effortadmissioncontrolalgorithmsthatincorporatetaskvaluesintheirrejectionpolicies.Doverpartitionstasksintotwodisjointsets:privilegedandwaitingtasks.Whenanytaskreachesitslateststarttime,i.e.,itsruntimelaxityiszero,thetaskisdiscardedifitsvalueisnotgreaterthanthetotalvalueofalltasksintheprivilegedset.Otherwise,thetaskisexecutedandallprivilegedtasksbecomewaitingtasks.Theheuristicusedin[ 141 ]usesadynamicguaranteepolicytopredictanoverloadconditionandarejectionpolicybasedontheimportancevaluesoftaskstoavoidthepredictedoverloadsituation.Duringtheoverloadcondition,REDdetermineswhetherataskisacceptedintothesystemusingacombinationofitsresiduallaxity,whichistheintervalbetweenthetask'sestimatednishingtimeanditsabsolutedeadline,anditsdeadlinetolerance.Tasksthatarerejectedarestoredtemporarilyinarejectqueueandcanbereacceptedbythealgorithmifarunningtaskcompletesitsexecutionearlierthanits 33

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worst-caseexecutiontime.ThereclaimingmechanismofREDincreasesschedulabilitybytakingadvantageofthesystem'ssparetime. 1.4SchedulingofAdaptiveReal-TimeSystems Priortechniquesinschedulingofadaptivereal-timetasksismuchrelatedtoagroupofoverload-handlingtechniques,calledperformance-degradationmethods.Insteadofrejectingtasksintheexistenceofanoverloadcondition,thesemethodsadjustfractionsofresourcesallocatedtotasksinordertoaccommodateallofthemwithavailableresources.Hence,real-timetasksareassumedtobeadaptiveinthecontextoftheperformance-degradationmethods. 1.4.1UniprocessorScheduling Therstgroupofperformance-degradationmethodsusesserviceadaptationtocopewithoverloads.Imprecisecomputation[ 101 ]assumesthateverytaskconsistsofmandatoryandoptionalparts.Themandatorypartofeachtaskmustbecompletedinordertoguaranteeaminimumlevelofperformance,whiletheoptionalpartcanbeleftincompleteifthesystemlacksofresourceswiththepriceofthequalityofthetask'sresult.TheIncreasingRewardwithIncreasingService(IRIS)[ 45 ]issimilartotheimprecisecomputation,butinsteadofminimizingerror,itsgoalistomaximizerewardthatisdenedasafunctionoftasks'accruedexecutiontimes.Theseservice-adaptationschemesrequiremultiple-versiontasksinwhicheachversionhasitsownexecutioncostandprovidesadifferentbenettothesystem.Thesecondtypeofperformance-degradationmethods,calledjobskipping,abortssometaskinstancesbasedonsomecriteria,butguaranteesthataminimumnumberofper-taskinstancesareexecuted.ExamplesoftheseapproachesareSkip-overalgorithms[ 84 ]and(m,k)rmguarantee[ 66 ].Generally,eachtaskisassociatedwithaspecicparameterthatcontrolsaminimumnumberofskipsallowedpertask.Usingtheseapproachesdoesnotmeanthatthesystemsreacttooverloadconditionsimmediately,butonlyaftertheoverloadshaveoccurredpastcertainthresholds. 34

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Insomeapplicationsinwhichtaskscanincreaseordecreasetheirexecutionratesindifferentoperatingconditions(i.e.,transitionaltasks),period-adaptationapproachesarethemostcommonperformance-degradationcountermeasureforoverloadconditions.Ingeneral,period-adaptationapproachescanbethoughtasconsistingoftwomainstages:adaptingtaskexecutionratesaccordingtoaperformancegoalandenablingtheadaptationintheactualtaskschedule.Manyalgorithmshavebeenproposedtodeterminepropertaskperiodsthatcouldmaximizethesystemperformanceduringoverloadsinuniprocessorsystems[ 19 27 30 88 90 114 134 ].In[ 30 ]periodictasksaremodeledasspringswithgivenelasticcoefcientsandminimumlengths.Duringoverloadconditions,taskexecutionratescanbechangedbasedontheirspringelasticcoefcients.Setoetal.[ 132 ]introduceaperformanceindexassociatedtoeachtaskthatisamonotonicallydecreasingandconvexfunctionofthetaskrate.UsingacombinationofsearchandLagrangianmultipliers,taskexecutionratescanbedeterminedtomaximizetheoverallcontrolperformance.Severalvariationsofrate-modulationtechniquesofperiodicsoftreal-timetasksinautonomousrobotcontrolsystemsarepresentedin[ 18 ]andtheyareextendedforgeneralizedsoftreal-timetasksin[ 19 ].Theproportional-shareresource-allocationalgorithmproposedby[ 142 ]assignsashareoftheprocessortoeachtaskinthewaythattasksareguaranteedtomakeprogressatawell-dened,uniformrate.TheExtensionsoftheEDFalgorithmareproposedtosupportadaptivereal-timetasks,suchastheRate-BasedEarliest-Deadlinerst(RBED)scheduler[ 27 ]andtheratetransitionmechanismin[ 32 ].ByutilizingtheRBEDschedulerandtheproportional-shareschedulingconcept,theResourceAllocation/Dispatching(RAD)integratedschedulingmodel[ 27 ]providesabilitiestodealwithtaskarrivals,departures,anddynamicadjustmentoftaskparametersatruntimeaswellasdynamicslackmanagement.Draco[ 108 ]employsthetaskmodelinRADforitsunderlyingresourcemanagementofcontrollersandisbuiltontopoftheRBEDscheduler. 35

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1.4.2MultiprocessorScheduling Whileapplyinguniprocessortechniquesinexecution-rateadaptationtomultiprocessorcasesmightbemanageable,i.e.,justchangingthedesiredtotalutilizationtobewithintheLUBoftheschedulerthatisusedinthesystem,enactingthechangesinmultiprocessorschedulesismorechallengingandcomplicatedthanjustmodifyingsometechniquesfromtherichsetofresultsintheuniprocessorcase[ 126 ].Asmallnumberofworkhasbeendonetohandleadaptivetasksetswhosetimingparametersmaychangeovertimeinthecaseofmultiprocessorsystems.SrinivasanandAnderson[ 137 ]havegivensufcientleave-and-joinconditionsfortasksinarunningPfair-scheduledsystemwithoutcausinganymisseddeadlines.Ane-grainLeave/Joinreweightingmechanism[ 139 ]extendsthepriorworkin[ 15 ]and[ 142 ]bypresentingconditionsunderwhich(1)tasksmayjoinorleaveatjobboundariesinmultiprocessorsystemsunderanydeadline-basedPfairalgorithmwithoutcausingmisseddeadlinesand(2)aconstantdriftisguaranteed.Reweightingrulesforpartitionedandglobalmultiprocessorschedulingalgorithmsatarbitrarytimesarepresentedin[ 25 ].In[ 116 ],synchronousandasynchronousmode-changeprotocolsareprovidedforhandlingtaskswithvariousresourcerequirementschangingincorrespondencetothesystem'smodeofoperation,inreal-timeidentical-multiprocessorsystems.Differentglobalschedulersareassumedtobeavailableforschedulingtasksindistinctmodesin[ 116 ]ratherthanasingleintegratedscheduler.Thesepriorapproachesforapplyingadaptationintaskschedulesonlyconsidertransitionaltasksthatcanadaptinastep-wisemanner,i.e.,thereallocationofresourcestotasksisdoneonlyifitcanfullyachievetasks'targetresourceutilizations. 1.5Real-TimeSchedulingUnderUncertainty Recentresearchinschedulinghasfocusedinsoftreal-timeapplicationsexecutinginopenandunpredictableenvironments,likecertainclassesofembeddedsystems.Theuncertaintyofworkloadsandthereal-timeconstraintinherentinsuchsystems 36

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disguisetheschedulingprocessesevenfurther.Theproblemofuncertaintaskexecutiontimes,hasbeenaddressedpreviouslyinuniprocessorandmultiprocessorsystems.Real-timesystems'uncertaintyparametersanddisturbancewhentheworkloadchangesdramaticallyforcontroluniprocessorsystemsinnondeterministicenvironmentsarehandledin[ 49 ].UncertainexecutiontimeoftasksonaprocessoristreatedusingaLinearMatrixInequalities(LMIs)controlschemein[ 136 ].Formultiprocessoravionicsystems,atwo-loopapproachin[ 98 ]utilizesaninnerfeedbackcontrolloopforhandlingmildresource/workloadvariations,whiletheoutercontrolloopprovidessubtaskallocationandmigrationalgorithms,tocopewithdrasticvariations.EUCON[ 106 ]andDEUCON[ 152 ]algorithmsuseamodelpredictivecontrolapproachtomaintaindesiredCPUutilizationofeachprocessorthroughratemodulation.Othertechniquesthatapplyfeedback-controlapproachestoaddressuncertaintyinreal-timeschedulinginclude[ 3 33 41 69 102 103 113 118 151 158 159 ]. 1.6ThesisStatement UsingthefeatureslistedinTable 1-1 ,thecomparisonsoftherelatedexistingapproachesandtheproposedapproachinthedissertation,calledtheElasticEnsembleScheduling(EES)manager,aresummarizedinTable 1-2 .Whilethereisaconsiderableamountofpriorworklistedintheprevioussection,theworkpresentedinthedissertationistherstdealinginparticularwithensemblesystemswithlimitedresourcesandpossessesuniquecharacteristicsasdiscussedbelow. Whileitisgenerallyacceptableforsoftreal-timetaskstobecomputedatvaryingrates,fewornoneoftheexpertsinensemblesystemscanbeignoredforanarbitraryamountoftimewithoutaseriousimpairmentoftheoverallsystem'scapabilities.Thus,theperformance-degradationmethodusedintheensemble-schedulingalgorithmsmustensurethattheoverallsystemloadiskeptmanageablewhilethesystem'sexecutionandlearningpoliciesarenotviolated.Asparametersinensemblesystems(e.g.,expertresponsibilities,resourceavailability)neededformakingschedulingdecisionsmay 37

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changeasfrequentlyasineveryensemblecycle,eithertheexistingapproachesmustbeenhancedornewschedulingmethodsmustbedevisedinordertoperformefcientlyforensemblesystemswithconsiderablylargenumberofexpertsandverytighttimeconstraints(whichcouldbeassmallasmilliseconds).Furthermore,existingschemesthatadjusttheongoingtaskschedulebyassumingonlystep-wiseadaptationarenotacceptablesincetasksinensemblesystemshavetooperatecontinuouslyinclosed-loopcycles;theycanpotentiallycauseintolerabledelaytocriticaltaskswhoseexecutionsareessentialforthesystemperformance.Anewadaptiveschedulingschememustbedevisedtoprovidearesponsiveandprogressiveresourceadaptationtotasksintheongoingscheduleswithboundedperformancelossandsmallschedulingoverheads.Whilemuchpreviousworkusespartitioningschedulingapproaches,thisdissertationadoptstheglobalschedulingapproachinordertomaximizetheachievableresourceutilization.Lastly,thisworkcopeswithvariouskindsofuncertaintyinensemblesystemsinclusively. 1.7Contributions Inthisdissertation,aschedulingarchitecture,calledtheEESmanager,isproposedtoenableon-lineadjustmentofcomponents'resourcedemandsinresponsetodynamicoperatingconditionsofsuchsystemsandensurethatthesystems'componentscaneffectivelyutilizetheamountofresourcesaccordingtothedeterminedallocation.Byincorporatingmathematicaloptimizationtechniques,thearchitecturecandetermine,atruntime,whenaresourcereallocationisneededandwhatanappropriateamountofresourceallocationshouldbe.Thechangeinresourceallocationisthensafelyenactedtocreateascheduleconformingtotimingconstraintsandpoliciesoftheensemblesystems.Tosupportageneralizedensemble-taskmodel,existingworkinreal-timemultiprocessorschedulingisextendedtoaccommodateadaptivetaskswhoseexecutioncharacteristicschangeovertimeandtheirexecutionratescanbeanyvaluewithinpredenedranges.Furthermore,bothuncertaintyinresourcecapacityand 38

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executiontimesaretakenintoaccountwhenmakingschedulingdecisionsinordertoprovidearobustscheduleoftaskexecution.Theproposedadaptiveschedulingschemeisevaluatedusingacase-studyapplicationofensemblesystems,motivatedbyresearchonBrain-MachineInterfacesattheUniversityofFlorida.Insummary,themaincontributionsofthedissertationarelistedasfollows. 1. AgeneralizedarchitectureoftheEESmanagerforschedulingexpertsofensemblesystemswithlimitedresourcestosatisfytheirreal-timeandquality-of-servicerequirements(Chapter 3 ). 2. Efcientapproachesforcomputingandapproximatingoptimalvaluesoftaskresourcedemandswhenresourcesarelimited,butdedicated(TT-TC*andTT-TopapproachesinChapter 4 ). 3. Anoveladaptiveschedulingalgorithmthatperformsresourceutilizationadaptationinschedulesofensembletasksonmultiprocessorsystems(EPOCinChapter 4 ). 4. Anoveladaptivemultiprocessorschedulingalgorithmforresourceutilizationadaptationofgeneralizedensembletasks,i.e.,WCETsoftaskscanbedistinctandvaryacrosscycles(EAGLE-TinChapter 5 ). 5. Ancontrol-schedulingapproachforschedulingexpertsofensemblesystemsinuncertainenvironments(FuzzyEESinChapter 6 ). 1.8Organization Thisremainderofthedissertationisorganizedasfollows.Chapter 2 providesadetailedsurveyofrelatedworkinadaptivereal-timescheduling.Chapter 3 presentsthearchitectureandworkowoftheEESmanager.ThecasewhenresourcesarelimitedbutdedicatedisconsideredinChapter 4 .Heuristics(TT-TC*andTT-Top)forhandlingresource-demandadaptationandamultiprocessorschedulingalgorithm(EPOC)forcreatingafeasiblescheduleoftasksaccordingtotheadaptedresourcedemandsarepresented.Chapter 5 providesadesignandimplementationofanoptimalmultiprocessorschedulingalgorithm(EAGLE-T)withasystematicprocedurehandlingsafemodechanges.ThisalgorithmimprovestheschedulabilityandefciencyofschedulesoftheEESmanagerforgeneralizedensemblesystemswithnon-uniformexperts'WCETs.InChapter 6 ,resource-capacityandtask-WCETuncertaintiesin 39

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ensemblesystemsandtheirimpactsonresourceutilizationadaptationarestudied.Anewprocedureforresource-demandadaptationandschedulecreationisderivedbasedonfeedback-controlandschedulingco-design.Chapter 7 providesconclusionsofresearchpresentedinthedissertationandadiscussionofpossiblefuturework. 40

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Figure1-1. Structureofanensemblesystemwithlimitedresources.Eachexpertprocessesasysteminputandproducesoutputyi.ThegatingcomponentassignsresponsibilitiesRitoexpertsandusesthemtocombineexpertoutputsintoasystemoutput.Duetoinsufcientresourcestorunallexpertsineachcycle,ensembleschedulingisneededtoexecuteonlytherightsubsetofexperts. Figure1-2. Aclosed-loopBrain-MachineInterfaces(BMIs).Therearethreemainfunctionalphases:dataacquisition,dataprocessingandprostheticcontrol.In-vivobrainsignalsaresensedfromalivesubjectandprocessedbyanonlinedigitalsignalprocessingschemetodetectneuralactivitiesduringdataacquisition.Then,theneuralactivitiesaretranslatedtoanappropriatemotor-controlcommand.Theresultofdataprocessingissenttocontroltheprostheticdevice. 41

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Figure1-3. Anexampleoftheavionicsofanaircraft.Multiplesensorscanbeusedtocollectusefulinformationfromdifferentpartsoftheaircraft.Thechoiceofcomplementarysensorsmustcontributetheirmeasurementstoacentralnodetoperformsituationanalysis. Figure1-4. Anexamplesettingofacomplexdatacenter.Theensembleofcontrollersmustbecoordinatedtoensurethatallcomponentsworktogetherinharmony,andcanwhollyoptimizetheoperationofthedatacenter. 42

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Figure1-5. Anillustrationoftasktimingparameters.Eachtaskconsistsofaninnitesequenceofinstances.Thejthinstancesofataskireleasesattimerijanditsabsolutedeadlineisdij=rij+Pi. 43

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Table1-1. Descriptionsoffeaturesforcomparingthisresearchandrelatedwork. FeaturesDescriptions Taskactivation(Act)P=Periodic S=Sporadic Systemtype(Type)S=Static A=Adaptive Environment(Env)U=Uniprocessor M=Multiprocessor Taskadaptability(Adapt)M=Multiple-version S=Skippable ST=Step-wisetransitional PT=Progressivelytransitional Adaptationrequirement(Req)R1=Keepthesystemloadwithinavailablecapacity R2=Maximizethesystemperformance R3=Satisfyotherspecicconstraintsofapplications Adaptationsolutionapproach(Sol)O=Optimal H=Heuristic C=Control ScopeD=Determineadaptivetaskparameters T=Handletransitionofparametervalues Uncertainty(Unc)X=Impreciseexecutiontime R=Fluctuatingresourcecapacity W=Time-varyingapplicationpriority 44

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Table1-2. AcomparisonbetweenthepriorapproachesandtheElasticEnsembleScheduling(EES)manager. TechniquesActTypeEnvAdaptReqSolScopeUnc Classicaluniprocessorscheduling:RM,EDF[ 100 ],LLF[ 111 ]P,SSU----Classicalmultiprocessorscheduling:partitioningscheduling(e.g.,[ 44 ])andglobalscheduling(e.g.,[ 14 40 ])P,SSU,M----Admissioncontrolschemes:schedulabilitytest(e.g.,[ 6 22 ]),RED[ 31 ],Dover[ 85 ]P,SSU,M-R1--Serviceadaptationap-proaches:Imprecisecomputation[ 101 ],IRIS[ 45 ]PAUMR1,R2HD,TSkip-overalgorithm[ 84 ]PAUSR1,R2HD,TElasticscheduling[ 32 38 ]PAUSTR1,R2OD,TRate-basedexecution:RBED[ 27 ],Draco[ 108 ]PAUSTR1,R2HD,TX,R Taskreweighting[ 15 25 139 ]P,SAU,MSTR1,R2-TMode-changeprotocols(e.g.,[ 116 162 ])P,SAU,MSTR1,R2-TPeriodadaptation(e.g.,[ 19 ])PAUSTR1,R2,R3O,HD,TFeedback-controlscheduling:EUCON[ 106 ],DEUCON[ 152 ],others(e.g.,[ 98 ])P,SAU,MST,PTR1,R2O,CD,TX,R EESmanagerP,SAU,MST,PTR1,R2,R3O,H,CD,TX,R,W 45

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CHAPTER2RELATEDWORK Thischapterreviewspriorapproachesinadaptivereal-timeschedulingthatarecloselyrelatedtotheworkinthisdissertation.AsmentionedinChapter 1 ,real-timeschedulingofadaptivesystemsusingperformance-degradationtechniquesgenerallyneedstwocomponents:therstdeterminesthevaluesoftherunningtasks'parameters(e.g.,periodsandutilizationshares)andthesecondactuallyenactsthechangesofsuchparameters.Fortheconvenienceofexplanationandcomparison,priorworkisclassiedintothreegroups.Therstgroupcontainsapproachesthatprovideintegratedsolutionsthathavebothoftheabovetwocomponents,includingelastictaskmodel[ 32 38 ],rate-basedexecutionmodel[ 27 59 78 108 ]andfeedbackcontrolreal-timeschedulingframework[ 104 106 153 ].Theothertwofocusonaddressingissuesoftherstcomponent[ 18 19 99 134 ],orthesecondcomponentsuchastaskreweightingmechanisms[ 25 ]andmode-changeprotocols[ 116 162 ].ThecomparisonsbetweenthesepriorapproachesandtheresearchinthisdissertationaresummarizedinTable 1-2 andtheyarediscussedalongwiththedetaileddescriptionofeachapproachintherestofthischapter. 2.1ElasticTaskModel Elastictaskmodel[ 30 32 ]considerseachtasktobeexibleasaspringwithagivenrigiditycoefcientandlengthconstraints.Theactuallengthoftheithspringxicanbecompressedorextendedwithintherange[ximin,ximax],whereximinandximaxaretheminimumandmaximumlengthofthespring.ThelengthofaspringisanalogouslyequivalenttotheutilizationdemandUioftheithtaskthatmustbeadjustedwithintherange[Uimin,Uimax]whereUiminandUimaxaretheminimumandmaximumvaluesofthetask'sutilizationdemand.Buttazzoetal.proposedtheTask-Compression(TC)algorithmtocompressthetotalutilizationoftasksdowntoadesiredvalueUdasaresultoftemporarysystemoverloads.TheelasticitycoefcientEi,similartotheinverseofthe 46

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spring'srigiditycoefcient,determinesagiventask'sutilizationdecreaseasafractionofthetotalutilizationdecrease(forallthetasks)neededtobringdowntheoverallutilizationofthesystem. TheTCalgorithmdividesalltasksintotwosubsets:TvandTf.ThesubsetTvincludesvariabletaskswhoseutilizationscanstillbecompressed.ThesubsetTfcontainstaskswhoseutilizationscannotbecompressedanyfurther,i.e.,thetaskswithxedutilizations(Ei=0orUimin=Uimax)andthetasksthatreachedtheirminimumutilizations(Ui=Uimin).ThealgorithmcalculatesthecompressionneededforiinthesubsetTvinproportiontoitselasticitycoefcientEi,dividedbythesummation(denotedEv)ofelasticitycoefcientsofalltasksinTv.Morespecically,tasksareadjustedaccordingtothefollowingequation: 8i2Tv,Ui=Uimax)]TJ /F6 11.955 Tf 11.96 0 Td[((Uvmax)]TJ /F3 11.955 Tf 11.95 0 Td[(Ud+Uf)Ei Ev(2) whereUvmax=Pi2TvUimax,Uf=Pi2TfUiminandEv=Pi2TvEi.Theexpertswiththesameelasticitycoefcientarecompressedequallyregardlessoftheircurrentutilizationdemands.IfthecompressioncanbedonewithoutcausinganytasktohaveutilizationdemandlowerthanUimin,thealgorithmperformsthecompressionandterminates.However,ifthecompressionresultsinsometaskshavingutilizationdemandslessthantheirminimums,thenthealgorithmonlycompressestheutilizationdemandsofthosetasksdowntotheirminimums,movesthemfromthevariablegrouptothexedgroupandrecalculatesthecompressionagain.TheTCalgorithm'stimecomplexityisO(N2)whereNisthenumberofalltasks. Whenarequestoftask-periodchangeisissued,aQuality-of-Service(QoS)managercalculatesthenewtaskperiodsaccordingtotheelasticmodel.TheQoSmanagerincreasestheperiodsofthedecompressedtasksimmediately,butdecreasestheperiodsofthecompressedtasksonlyattheirnextreleasetimestokeepthesystemschedulable.Thesameauthorsproposedin[ 29 ]smoothrateadaptationusing 47

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impedancecontrolbyextendingtheelastictaskmodeltohaveeachspringcoupledwithadampingdevice,whichpreventsabruptperiodchanges.Whenanactivetaskwantstochangeitsperiod(eitherlowerorhigherthanthecurrentone),alineartransitionlawenablessmootherperiodvariationontheothertasks.Whenanewtaskactivationisneeded,anexponentiallawisusedtovaryothertasks'periodsmoregracefully. ItwaslaterprovenbyHuetal.thattheTCalgorithmyieldsanoptimalsolutiontoaquadraticprogrammingproblemofminimizingthetotalperturbationoftaskutilizationsinthesystem[ 74 ].TheelastictaskmodelisusedtoscheduletasksetsaccordingtotheEDF-schedulingpolicywhentaskdeadlinesequaltaskperiods.Howevertheworkdoesnotdescribesystematicallyhowthetask-periodtransitionisperformedtopreventtransientoverloadspossiblyoccurredduringtransition.Recentworkbythesamegroupofauthors[ 38 ]extendstheTCalgorithmusinganiterativeapproachtosolvetheperiod-selectionproblemforreal-timetaskswithdeadlinessmallerthanrespectiveperiods. Theelastictaskmodelapproacheshavebeenappliedtoprovidequalityofservicesformanyapplications,includinggaming[ 145 ],ethernet-basedsurveillancesystems[ 120 ]andmultimediasystems[ 164 ].Inmostpreviousworkrelatedtoelasticscheduling,theelasticitycoefcientsarestaticallysetatsystemdesignorcongurationtimeandaregenerallyrelatedtotheutilizationrequirementsofthetasks.Thetaskelasticitiesrepresentthereciprocalofthetaskimportancevalues.However,theTT-TC*andTT-Topalgorithmsproposedinthisdissertationarebasedonanelastic-schedulingapproachwheretaskelasticitiesarenotstaticinnatureand,instead,theycanchangeineverycycle.Theelasticitiesarefunctionsoftheresponsibilitiesoftheexpertsthatcanbeassignedinaccordancewiththeensemble'sexecutionandlearningpolicies.Schedulingdecisionsforagivencyclecannotbedonepriortothebeginningofthecyclewhentaskresponsibilitiesareupdatedforeverycycle.Anotherdifferenceisinthenatureofthesystemoverloadingwhich,inthecaseofensemblesystemswithlimitedresources,is 48

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permanentratherthantemporary.Hence,itisveryimportanttodeviseamoreefcientapproachthatcanavoidexcessiveoverheadsofrepetitivelyexecutingTCineachcycle. 2.2Rate-BasedExecutionModel TheRate-BasedExecution(RBE)taskmodelwasproposedbyJeffayandGoddard[ 78 ]topresentevent-driventasksinwhichonlytheexpectedarrivalratesoftheirjobsareknown,suchastasksindistributedmultimediaanddigitalprocessingapplications.IntheRBEmodel,processesspecifytheirresourceutilizationdemandsintermsofthedesirednumberofexecutiontimeunitsineveryintervalofaspeciclength.NecessaryandsufcientconditionsfordeterminingthefeasibilityofanRBEtasksetonsingleprocessorareprovidedanditisshownthattheschedulabilityanalysisholdsfortheEDFschedulingforbothpreemptiveandnon-preemptiveexecutionenvironments.In[ 102 ],theRBEmodelisextendedtosupportresourcesharingbetweenreal-timeandnon-real-timetasks.AnEDFwithDeadline-CeilingInheritance(EDF-DCI)algorithmwasdevelopedtoassigntaskdeadlinesusingadynamicdeadlineceiling,whichisdenedastheminimumrelativedeadlineofallmembersinadynamictasksetassociatedtoeachresource.ThegoalofEDF-DCIistosatisfymutual-exclusionconstraintsonsharedresourceswhilepreventingpossibledeadlockamongtasks.TheVariable-RateExecution(VRE)taskmodel[ 59 ]providesaprimaryextensiontotheRBEmodel.VREprovidesschedulabilityconditionstodeterminewhenWCETsandperiodsoftasksmaychange,andwhentasksinadynamictasksetareallowedtoenterandleavethesystem. Independentlyfromtheabovework,asimilartaskmodelisalsosupportedbyaRate-BasedEarliestDeadline(RBED)scheduler[ 27 ].Thegoaloftheworkistoprovideanintegratedsystemthatcanseamlesslysupportexecutionofhardreal-time,softreal-time,andnon-real-timetasks.ThesystemconsistsofaresourceallocatorandanEDF-basedscheduler.Theresourceallocatorassignsthetargetresourceutilizationandperiodforeachtaskinthesystem,basedonitsspecicprocessingrequirements.The 49

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EDF-basedschedulerdispatchesprocessesintheEDForderbutinterruptsthemviaaprogrammableone-shottimerwhentheirworst-caseexecutiontimeareconsumed.Hardreal-timeprocessesareguaranteedtoreceivetheirdesiredexecutionrates,whilesoftreal-timeprocessesreceiveexecutionrates,equaltheirdesiredratesorless,dependingontheavailabilityofresources.Properratesforsoftreal-timetasksaredeterminedbyaformofproportionalsharescheduling.Theminimumresourceutilizationisreservedtoguaranteenostarvationofbest-effortprocesses.Draco[ 108 ]employsthetaskmodelin[ 27 ]foritsunderlyingresourcemanagementofcontrollersandisbuiltontopoftheRBEDscheduler.Itallocatesaminimallysufcientamountofresourcestoeachcontroltaskandthendynamicallyreallocatesslack,orexcessresources.TheproblemaddressedbyDracoistodecideforeachcontroltaskhowtherateshouldbeincreasedorshortenedsuchthattheoverallcontrolperformanceisimproved,consideringthedynamicsofthecontrolledsystems.Twoadaptivefeedback-basedslack-managementtechniques,calledproportionalanddiscrete,areproposed.Theproportionalpolicydistributesslacksamongcontrollersinproportiontotheirfairshares.Thediscretepolicytunestaskratestovaluesinasetofpredeneddiscreteratesincorrespondencetothesystem'sstate.Draco'sperformanceisevaluatedincomparisonwithabaselinepolicythatallocateslimitedresourcestotasksaccordingtostaticinformation. AlthoughtheRBE-basedapproachesmentionedaboveallowperiodictaskstodynamicallychangeutilizationsandperiods,theydonotconsiderthecaseofmultiprocessorsystemsinwhichtheoptimalityofEDFdoesnotapply.Inaddition,atask'sexecutionratemaybekeptunchangeduntilalltaskspendingjobsarecompleted,andhencetheirdeadlinesarenotmodiedaftertheirreleases.Asaresult,acceptancesofnewtaskscouldbedelayeduntilsomerunningtasksterminateandenoughresourcecapacityisreleased.Insomecasesincludingensemblesystems,however,thenewtasksmightneedtohavesmallerresponsetimesthansomerunningtasks,anditisnecessarytopromptlylowertheexecutionratesofthelow-priorityrunningtasksto 50

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achievesmalldegradationinthesystemperformance.Inthesecases,low-prioritytasks'pendingjobsshouldbeadaptedsuchthattheirminimumdemandsaresatisedandtheirresourcesharesarereleasedimmediately. 2.3FeedbackControlReal-TimeSchedulingFramework Feedbackcontrolreal-timescheduling(FCS)framework[ 104 ]usescontrol-theory-basedmethodologytoprovidetransientandsteady-stateperformanceguaranteestoreal-timesystems.TheFCSarchitecturefeaturesafeedback-controlloopconsistingofamonitor,acontrollerandaQoSactuator.Themonitormeasurescontrolledvariables,suchasdeadline-missratioandutilization,andfeedstheirvaluestothecontroller,whichcomparesperformancereferencesofthecontrolledvariablesandcomputesthecorrectvaluesofmanipulatedvariables,likethetotalestimatedutilizationoftheprocessor.Then,theQoSactuatorenforcesthenewtotalestimatedutilizationbyadjustingtheQoSlevelsoftasks.EDFandextendeddeadlinemonotonicalgorithmsareusedasschedulingpolicies.ThreeFCSalgorithmswereproposed,namelyFeedbackUtilizationControl(FC-U),FeedbackMissRatioControl(FC-M)andIntegratedUtilization/MissRatioControl(FC-UM).FC-Uperiodicallysamplestaskutilizationvaluesandcomputesachangeinthetotalestimatedutilization.Atthesaturationpoint,FC-Ucannotdetecthowseverelythesystemisoverloadedifthetotalutilizationisat100%.Itcanguaranteezerodeadlinemissesifautilizationboundofthesystemisknownaprioriandnotpessimisticallydetermined.UnlikeFC-U,FC-Mcontrolsthemissratiodirectlyfromamiss-ratiocontrolloopand,hence,doesnotdependontheknowledgeabouttheutilizationbound.WhileFC-Mcanachievealowdeadline-missratioandhighCPUutilizationevenifthesystem'sutilizationboundisunknownortimevarying,ithasthesamerestrictiononthemissratioasFC-Uduetosaturation.Ifthemissratioisatzero,FC-Mcannotdetecthowseverelythesystemisunderutilized.Unfortunately,thisimpliesthatthereferencevalueofthemissratiomustbesettoapositivevalue,whichmeansthatthezeromissratiocannotbeguaranteedinasteadystate.Combiningthe 51

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advantagesfrombothFC-UandFC-M,boththemissratioandutilizationaremonitoredbyFC-UM.Twocontrollerstakecareofcomputingcontrolsignalsforthemissratioandutilizationindividually.SincetheutilizationcontrolofFC-UMdominatesinthesteadystate,zerodeadlinemissescanbeguaranteedifthetotalrequestedutilizationiswithintheutilizationthreshold,otherwisethemissratioiskeptclosetoitsreferencevalue. TheproposedFCSalgorithmshavebeenimplementedasamiddlewareservice[ 105 ].However,sincetheirdesignsarebasedonsingle-input-single-output(SISO)linear-controltechniques,theycannotprovideend-to-endutilizationcontrolinenvironmentswherecouplingamongmultipleprocessorsexists.FCSwassubsequentlyextendedtohandleDistributedReal-timeEmbedded(DRE)systems[ 106 ].Theworkassumestheend-to-endtaskmodelinwhichataskiscomprisedofsubtasksexecutingondifferentprocessorsandtheirexecutionsinvolveprecedenceconstraints.TheEnd-to-endUtilizationCONtrol(EUCON)algorithmisproposedtomaintaindesiredCPUutilizationoneachprocessorthroughonlineadaptationusingaModel-Predictive-Control(MPC)approach.Rateconstraintsandutilizationsetpointsaresuppliedtoacontrollerthatconnectstothepairoftheutilizationmonitorandtheratemodulatoroneachprocessor.Basedonthemonitoredvaluesofutilization,thecontrollercalculatesnewtaskratesandsendsthemtotheratemodulatoroneachprocessorforenforcingthechanges.EDFschedulingisusedforschedulingtaskspartitionedtorunonanyindividualprocessor.Insteadofusingacentralizedcontroller,DEUCON[ 152 ]proposestheuseofdecentralizedcontrolalgorithmstoimprovethescalabilityandreliabilityofadaptiveutilizationcontrolinDREsystems.Controllabilityandfeasibilityareproventobedependentonend-to-endtaskallocations,andalgorithmsareproposedtoensurecontrollabilityandfeasibilityofDREsystemswithdynamicworkloadvariations[ 153 ].Twoonlineadaptive-optimal-controltechniquesarepresentedin[ 161 ]:therstoneusingrecursive-least-squaresmodelidenticationwithalinearquadraticoptimalcontrollerandtheotherbasedonadaptive-criticdesign.Whentheestimationerror 52

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oftaskexecutiontimeislarge,thesetechniquescanachievebetterperformanceinutilizationcontrolthantheMPC-basedcontrollerssincethesystemmodelisestimatedonlineratherthanusingaxedmodelasthosein[ 106 ]and[ 152 ]. InthegeneralformofMPC,aquadraticoptimizationproblemissolvedforeveryinputsample,usingtechniquesbasedonsequentialunconstrainedminimization.Giventhatthesamplingtimeconsideredinensemblesystemscouldpotentiallybeveryne-grained,theneedtosolvetheoptimizationproblemwithinthesamplingtimeistooexpensiveorsimplyinfeasiblealthoughafastleast-squaresolverisused.Thepreviouslypresentedapproachesmainlyadjusttaskutilizationreactivetotheerroroftaskexecutionestimationanductuatingworkload,butdoesnotaccountforchangingresourcedemandsraisedbyapplications.Inaddition,thepartitioning-schedulingapproachisoftenusedinthesepriorapproaches.Untilthetimethatthisdissertationwasprepared(i.e.,August2011),onlyfewapproachesincorporateoptimalglobal-schedulingreal-timemultiprocessorschedulerstocontrollers[ 58 ]andnoneofthemconsidersaT-Lplane-basedschedulerasusedinthisdissertation.TheT-Lplane-basedschedulingmodel[ 40 ]providesguidelinesfordevisingamultiprocessorschedulingalgorithmthatisnotonlyoptimalbutalsomuchmoreefcientthanpureuid-basedschedulingalgorithms,suchasPfairandLLREFalgorithms. 2.4TaskReweightingSchemes Ataskweightisdenedasaprocessorshareofeachtask(similartotheresource-utilizationdemand).Changesoftaskweightsareenactedthroughaprocesscalled`reweighting'.SrinivasanandAnderson[ 139 ]derivesufcientconditionsunderwhichtasksmaydynamicallyjoinandleaveasystemscheduledunderPfairschedulingwithoutcausinganydeadlinemiss.Ataskmayjoinsuchasystemifitsinclusiondoesnotcauseanoverload.Ifaleavingtaskhasoverutilizedresources(incomparisontoitsuid-scheduleutilization),thenitsdeparturemustbedelayed.Theseconditions,calledleave/joinrules,canbeappliedtoprovidecoarse-grainedtaskreweighting,i.e.,ataskleaveswithits 53

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oldweightandreenterswithitsnewweightwhenitissafetodoso.Thedelayfortheleavingtasks,however,issusceptibletoanon-constantbound. Fine-grainedreweightingonmultiprocessorsthatensureconstantdriftispossiblebyusingreweightingrulesforthePD2Pfairalgorithm,aspresentedin[ 23 ].Ataskiis`ow-changeable'atareweightingtimetcfromweightutovifitsactivejobisscheduledbeforetc.Otherwise,itis`omission-changeable'attimetcfromutov.Omission-changeabletaskscanbeviewedashavingleftthesystemsincethecompletiontimeofitslastjob,soitcanrejointhesystemimmediatelyattc,whileow-changabletasksmightneedtodelayitsweightenactmentuntillaterusingsimilarrulestothecoarse-graintaskreweighting.Byusingthesenewrules,reweightingcanbeaccomplishedfaster.Bothreweightingtechniquesbasedonfairschedulingcanachievehighaccuracyinenactingweightchangesattheexpenseofpotentiallyfrequenttaskpreemptionsandmigrationsamongprocessors. Blockandhiscolleagues[ 24 ]thenconsideredtheuseofbothpartitioning-andglobal-schedulingalgorithmstoschedulehighly-adaptiveworkloadsonmultiprocessors.ThePartitioned-AdaptiveScheduling(PAS)algorithmanditsnon-preemptivecounterpart(NP-PAS)areproposedascandidatesforpartitioningschemes.PASisderivedfromtheearliest-eligible-virtual-deadline-rstschedulingalgorithm[ 142 ].Whilebothpartitioning-basedalgorithmsentailloweroverheadsthanfairapproaches,theyprovidepoorer(butsometimesacceptable)accuracy.Forglobalscheduling,twoalgorithmsbasedonthewell-knownglobalEDFalgorithmareproposed,namelychangeableEDF(CNG-EDF)andnon-preemptiveCNG-EDF(NP-CNG-EDF)[ 25 ].Thesesuboptimalglobal-schedulingalgorithmsarefavorablewhenpreemption/migrationcostsarehighandagoodmixofaverage-caseperformanceandfairnessyieldssomebenetsforthesystemperformance.Therearetradeoffsthatmustbemadebetweenaccuracyandefciencybecauseofthecharacteristicsofthebaseschedulingalgorithmsusedintheseapproaches. 54

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Task-reweightingapproachespresentedinthissectionaretheclosesttotheapproachproposedforenactingtaskallocationchangesinthisdissertation.However,allofthesepreviousapproachesassumethattasksmustadaptinastep-wisemanner;thereallocationofresourcestotasksisdoneonlyifitcanfullyachievetasks'targetresourceutilizations.Moreover,theresourcereallocationcannotbeappliedtoalready-startedtaskinstances,soanexistinginstanceofataskmusteithercomplete(topreserveconsistencyofthesystem)orterminateearlybeforetheprotocolcanreallocateresourcesandallowareleaseofanewinstance.Thisispossiblyduetotheusageofdifferenttaskversionsfordifferentmodes.Inensemblesystems,taskscanchangetheirresourcedemandsovertimeandtheiractiveinstancescancontinuouslyutilizeprogressivelyadaptedresources.Thisoccursinmanyscenariossuchastime-shareschedulinginoperatingsystemsorreal-timearticialintelligencesystems.Forexample,inalarge-scaleairportsurveillancesystem[ 71 ],multipletasksperformpatternrecognitionofsuspiciouspersonsorobjectsintheeldofviewofmultiplesurveillancecameras.Whenananomalyissuspectedordetected,tasksforasubsetofthecamerasmayneedtoperformfurthercomputation(e.g.,fortimelysceneanalysis)requiringadditionalresourceswithoutadisruption.AnotherexampleistheapplicationoftheMixture-of-ExpertsapproachtoBrain-MachineInterfaces[ 50 55 ].Acollectionofsignal-processingmodelsareexecutedconcurrentlytodecodeasequenceofcontinuousneuralsignals.Agatingcomponentexecutesadecision-makingalgorithmtodeterminetherelevancyofeachmodelinprovidinganappropriateprostheticcommandforanyparticularsample.Forcomplexhumanmotortasks,e.g.,liftingorgraspingobjects,severalhundredsofsuchmodelsmayberequiredandthecombinationofthemforspecicsituationsisdeterminedonlinebythegatingcomponent.Inadditiontotheirexecutions,thesekindsofmodelsalsorequiretheexecutionoflearningproceduresthatenablethemtoadaptovertime.Insuchsystems,promptandgradualadaptationtowardsthedesiredresourceutilizationisveryimportanttomaintainacceptablesystem 55

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performancewhenresourcesarelimitedsincethesystemsmustoperateunremittinglyfromonemodetoanother. 2.5Mode-ChangeProtocolsforSchedulingMulti-ModeReal-TimeSystems Tasksystemsthatrequirechangesofprocessorsharesovertimecanbegeneralizedasmulti-modesystemsinwhichboththeexecutiontimesandperiodsoftasksmaychangesimultaneouslyovertime.Inmulti-modereal-timesystems,taskscanbeclassiedintoold-modeandnew-modetasks.Old-modetasksaretasksthathavestartedpriortoaModeChangeRequest(MCR).Thesetaskscanbecompletedoraborteddependingontheapplicationrequirements.Forexample,taskswhosecompletionsarenecessaryformaintainingdataconsistencyorcorrectnessoffutureexecutionsmustnotbeabortedinordertokeepthesysteminasafecondition.Ontheotherhand,new-modetasksrepresentfunctionalitydesiredbythesystemafterthemodechange.Thesenew-modetaskscanbewhollynewtasks(i.e.,tasksthatarenotactivepriortothemodechange),changednewtasks(i.e.,previouslyactivetaskswhosebehaviorschangeafterthemodechange)orunchangednewtasks(i.e.,previouslyactivetasksthatoperateexactlythesamebeforeandafterthemodechange).Amode-changeprotocolisconsideredtobesynchronousifitdoesnotscheduleold-modeandnew-modetaskssimultaneously,otherwiseitisasynchronous.Periodicityisanotherpropertyofmode-changeprotocols,whichindicateswhethertheprotocolcanhandlemode-independenttasks(i.e.,unchangednewtasks). Acomprehensivesurveyofmode-changeprotocolsforsingle-processorsystemswithxed-priorityschedulingisprovidedin[ 126 ].Synchronousprotocolsdelaythereleasesofnew-modetasksuntilacertaininstantaftertheMCRoccurred(e.g.,noCPUloadoraxedtimeoffset).Theidletimeprotocol[ 148 ]suspendstheold-modetasksattherstidletime-instantoccurringduringthetransitionandthen,releasesnew-modetasks.Themaximum-periodoffsetprotocol[ 10 ]isasynchronousprotocolwithperiodicity,whichdelaystherstactivationofallthenew-modetasksforatime 56

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thatisequaltotheperiodoftheleast-frequenttaskinbothmodes.TheMinimumSingleOffsetprotocol[ 126 ]completesthelastactivationofalltheold-modetasksandthen,releasesthenew-modeones.Bothversionswithandwithoutperiodicitywereproposed.Thesesynchronousprotocolsaregenerallysimplebutnotveryprompttomodetransition.Alternatively,asynchronousapproachesofferfastertransition,butrequiremorecomplexschedulabilityanalysis.Previouslyproposedanalysismethodsforadaptivemulti-modesystemswithrate-monotonicordeadline-monotonicschedulinginclude[ 121 143 ]. NelisandGoossensrecentlyproposedasynchronousmode-transitionprotocolwithoutperiodicityforidenticalmultiprocessors,calledMS-MSO,whichextendstheMinimalSingleOffsetprotocol[ 115 ].TheworkconsidersanM-processorsystemwithasetofoperatingmodeswhoseasetoftaskfunctionalitiesispredened.Ineachmode,thesystemusesitsownschedulerthatcanguaranteetomeetalldeadlinesoftasksassociatedwiththemodewhenexecutedonMprocessors.Theglobal-schedulingbasedDeadline-MonotonicandEDFschedulersareusedasexamplesoflegitimateschedulingalgorithms.Thegoaloftheproposedprotocolwhenthesystemtransitionsfrommodemitomjistocompletetheremainingjobsofold-modetasksandtoenableeverynew-modetasksbelongingtothenewmodemj,whilemeetingeveryjobdeadlineandenablementdeadline(i.e.,therelativedeadlineontheenablingtimeofanew-modetaskduringtransition).Anasynchronousprotocolwithoutperiodicityforthesamecase,calledAM-MSO,ispresentedin[ 116 ].Wheneveraprocessorcompletesaremainingjobofanyold-modetasksandtherearenootherremainingjobswaitingforexecution,AM-MSOimmediatelyenablessomenew-modetasksratherthanwaitingforallremainingjobstonish.Bothprotocolswereextendedtouniformmultiprocessorplatformsin[ 162 ]. Theparticularinterestoftheworkinthedissertationisinrealisticadaptivesystemswithanunboundednumberofoperatingmodes,non-deterministicMCRtimesand 57

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dynamictaskutilizationdemands(i.e.,theirvaluesareonlyknownatthetimewhenanMCRoccurs).Thereisahighpossibilitythatperiodicityisnecessaryformodetransitionofensemblesystemssincewinnerscouldremainaswinnersforaconsiderablylongperiodwhileothernon-winnerscouldtaketurnsinadaptingtheirknowledge,especiallyinanysystemwithalargenumberofexpertsorslowlychangingdynamics.Periodicityisnotyethandledintheexistingmode-changeprotocolsformultiprocessorsystems.Inaddition,theprotocolsuseseparateschedulersforeachmodeinsteadofjustasingleintegratedscheduler.Themechanismsthatallowsswitchingbetweentheseschedulersduringtransitionandtheoverheadindoingsoarenotpreviouslydiscussed,soitishardtodeterminehowpracticaltheapproachis.Similartotask-reweightingtechniquespresentedintheprevioussection,thesemode-changeprotocolssupportonlytasksthatarestep-wisetransitional. 2.6DeterminingAdaptiveTaskParameters Setoetal.[ 132 ]allowtaskratestovarywithingivenranges.Associatedwitheachtaskisamonotonicallydecreasingperformanceindexthatisaconvexfunctionofthetaskrate.Byformulatinganoptimizationproblemasanonlinearprogrammingproblem,taskratesarecalculatedusingacombinationofsearchandLagrangianmultipliers.Sincetheapproachisinthecontextofcontrolsystemtogetherwithtaskscheduling,itcannotdealwithothergeneraltasksforwhichaperformanceindexcannotbederived,suchasdata-processingtasks.Moreover,itscomputationalcomplexityanditsinteractionwithdigitalcontrolsynthesismaketheapproachonlysuitableforoff-line,staticschedulabilityanalysis[ 19 ].ShinandMeissner[ 134 ]provideanextensionbymakingitsuitableforonlineoperationinmultiprocessorsystems.Loadadaptationisperformedbytaskreallocationtootheruniformprocessorsandperiodextensions.However,thisapproachremainsrestrictedtodigitalcontrolsystemsorsimilarapplications. 58

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Beccarietal.[ 18 ]provideanalternativeapproachforsoftreal-timetaskadaptationbasedonrateadjustmentonauniprocessor.Theyregardautonomousrobotcontrolarchitecturesassoftreal-timesystemsandpresentaschedulingtechniqueusingratemodulationforadaptationofsoftreal-timetaskstoavailableresourcecapacity.Byrateadjustment,computationalresourcescanbedynamicallyredistributedtobettersuitthecurrentrequirementsofapplications.Severaladaptationpoliciesareconsideredbycastingconstraintsonraterangesalongwithadditionalconstraintsreectingapplicationrequirementsinalinearprogrammingformulation.Thisworkwaslaterextendedtosupportrateadaptationforgeneralsoftreal-timetaskscharacterizedbyarangeofadmissiblerates[ 19 ].Severaladaptationpolicies,emphasizingdifferentcriteriarelevanttoapplications,aredescribed.Thepaperalsopresentsarate-transitionscheme,whichcansafelyperformrateadaptationofrate-increasingtasksattheirnextreleaseinstantsfollowingthetimetheadaptationrequestwasmade.Still,theoverheadoftheadaptationthroughoptimizationcanberatherhighand,hence,theuseofhysteresisthresholdstokeeparateadaptationasaninfrequenteventissuggestedbytheauthors. RecentworkbyLimaetal.[ 99 ]considersdynamicrecongurationofapplicationsstructuredasasetofmultiple-versiontasks.Eachtaskhasasetofversions;eachofwhichhasanexecutioncostandprovidesacertainbenettothesystem.Thegoaloftheworkistoselecttheappropriatetaskversionsthatmaximizetheglobalbenetforthesystem.Acomplexoptimizationproblematruntimesubjecttothesystemschedulabilityconditionsissolvedusingdynamicprogrammingtechniques.Theoptimalsolutionhasapseudo-polynomialruntimecomplexity,whichisthenimprovedbyafasterapproximationalgorithmtunablebythesystemdesigners.Similartotheworkinmode-changeprotocols,itisassumedthatthenumberofoperatingmodesisniteandtheexecutionrequirementsofthesemodesareknownaheadoftime.Generally,expertsinensemblesystemsareimplementedwithonlyoneversionofsourcecode,whichcouldexhibitdifferentexecutionbehaviordependingonwhichmodeitisoperating 59

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andthenumberofmodescouldbeinnite.Anobviousexamplesofsuchexpertsareneural-networkmodelsneededinBMIs.Hence,techniquesthatrequiremultipleversionsoftaskcodearenotquiteapplicable. 60

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CHAPTER3ELASTICENSEMBLESCHEDULING(EES)MANAGER Whenresourcesareinsufcienttorunallexpertsofanensemblesystem,therightsetofexpertsmustexecuteineachcycleinordertominimizelossoftheoverallsystemperformance.ThischapterpresentstheproposedElasticEnsembleScheduling(EES)manager,whichcanbeusedtodeterminewhichandwhenexpertsaretobeexecutedwhenresourcesarelimited.First,Section 3.1 introducestheformalmodelofensembleschedulingtodeneimportantterminologyusedthroughouttherestofthisdissertationandhighlightsimportantchallengesinensembleschedulingwithlimitedresources.Next,Section 3.2 presentsthearchitectureoftheEESmanagerproposedtoaddressthesechallenges.ThefunctionalitiesandrequirementsoftwomaincomponentsoftheEESmanager,aTaskUtilizationAdaptor(TUA)andanadaptiveReal-TimeScheduler(RTS),arepresentedindetail.TheschedulingprocedureusingtheEESmanagerisdescribedinSection 3.3 .TheconclusionsofthischapterareprovidedinSection 3.4 3.1EnsembleSchedulingModel Thisdissertationconsidersareal-timeensemblesystemwithNadaptivetasks,1,...,N.Fromthispointon,theterms`task'and`expert'areusedinterchangeably,unlessotherwisestated.Areal-timeensembletaskiischaracterizedbyapair:[Uimin,Uimax]whereUiminandUimaxdenotethetask'sminimumandmaximumresourceutilizationdemands.Thenotationi:[Uimin,Uimax]isusedtodenotethetaskinthesystem'sconguration.Anyresource-utilizationdemandislessthanorequaltooneandassumedtohavearationalvalue(i.e.,representableasaratiooftwointegers).Atanyparticularcyclet,Ui(t)representsi'sdynamicresource-utilizationdemand.Dependingonavailableresources,eachtaskiisallocatedacertainamountoftimeCi(t)foritsexecutionwithinacertainperiodoftimePi(t).Atthesystem'sstartuptime,Ci(t)andPi(t)havethesamevaluesasthetask'sworst-caseexecutiontime(WCET)andtaskperiod,respectively.ThefractionCi(t)=Pi(t)representsthetask'sresource-utilization 61

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allocationVi(t).Forarestrictedensemble-systemmodel,sinceeachexpertmustbeabletocompleteitsexecutionwithinanensemblecycle,alltimingparametersoftasksaregenerallysettobeonecycleandtheWCETofeachtaskisassumedtobeonecycle.Inthegeneralizedsystemmodel,WCETsoftaskscanbedifferentandsmallerthanonecycle.Thelengthoftheensemblecycleisusuallydenedbytheinput-samplingintervalofthesystem.Forbrevity,theterms`demand'and`allocation'areusedwhenreferringtoresource-utilizationdemandandresourceutilizationallocatedtotasksbythesystem. Eachtaskiconsistsofaninnitesequenceofinstances,eachinstancebeingcalledajob.Throughoutthedissertation,jobsandtaskinstancesareusedexchangeably.Thetask'srelativedeadlineDi(t)equalsitsperiodPi(t).Theactivationpatternofi'sjobsisdictatedbyVi(t);thereleasetimeofthejthjobofi(ori,j)isdenotedasri,jandthejob'sabsolutedeadlinedi,j(t)equalsri,j+Pi(t).Tasksareassumedtobeindependentofeachotheranddonotshareanyresourceexceptprocessortime.Eachtaskcanexecuteonasingleprocessoratanygiventime.Inaddition,tasksareconsideredtobeprogressivelyadaptabletoanyallocationwithintheallowablerange,i.e.,theycanfunctionproperlyiftheirVi(t)areassignedanyrationalvaluewithin[Uimin,Uimax]. TheconsideredensemblesystemhasMunit-capacityandhomogeneousprocessors(M
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Duetothedynamicnatureofensemblesystems(i.e.,changingresponsibilities)andtheirresourcelimitations,ensembleschedulingmustaddressthefollowinghigh-levelresearchgoals: G1.GiventheupdatedexpertresponsibilitiesRi(t)fromthegatingcomponent,determinehowmuchresourceutilizationisdemandedbyeachexpert(i.e.,Ui(t))inordertominimizetheimpactoflimitedresourcesonthesystemperformance. G2.Giveneachtask'sresourcedemandUi(t),dynamicallyallocateresourcestotasks,i.e.,assigningvaluesofVi(t),anddecidewhentasksshouldbedispatchedforexecutionsothattheresultingscheduleisgloballyfeasible(i.e.,donotviolatethesystem'sexecutionandlearningpolicies)andatthesametime,enablestaskstoutilizetheamountofresourcesascloseaspossibletotheirdemandsinG1. Duetothecharacteristicsofthesystemsofinterest,thisdissertationfocusesonreal-timesystemswiththefollowingassumptions:(1)tasksareperiodicorsporadic,(2)tasksareindependentintheirexecutionsornoprecedencerelationamongtasks,(3)taskpreemptionandmigrationisallowedandhaveboundedcosts,(4)eachtaskissequential,(5)taskssharenootherresourcesexceptprocessortimeand(6)anygiventaskcanhaveonlyoneactiveinstancewithineachperiod(i.e.,ifadeadlinemissoccurs,eithertheinstanceisabortedoranewinstanceisnotreleasedinthenextperiod). 3.2ProposedEnsembleSchedulingArchitecture TheschedulingarchitectureforensemblesystemswithlimitedresourcesisshowninFigure 3-1 .ThearchitectureincludesanensemblesystemandanElasticEnsembleScheduling(EES)manager.TheEESmanagerhastwomaincomponents:aTaskUtilizationAdaptor(TUA)andaReal-timeTaskScheduler(RTS).ThechallengesG1andG2intheprevioussectioncanbeaddressedbytheTUAandRTS,respectively. 3.2.1TaskUtilizationAdaptor(TUA) Basedonchangingsystemstate,theTUAutilizesresponsibilitiesassignedbythegatingcomponenttodynamicallyadaptresource-utilizationdemandsofensembletasks.Aspreviouslymentioned,theobjectiveoftheTUAistominimizetheimpactoflimitedresourcesonthesystemperformancewhilesatisfyingtheensemblesystem's 63

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resource-capacityconstraints,executionpoliciesandlearningpolicies.Tobemorespecic,theTUAsolvesanoptimizationproblemofthefollowingform: ProblemP:minimizeUiPerformancelosssubjecttoResource-capacityconstraintsExecution-policyconstraintsLearning-policyconstraints Theresource-capacityconstraintsensurethattheoverallresourcedemandofthesystemdoesnotexceedtheavailableresources.Inthescopeofthisdissertation,resourcesrefertoavailableprocessortimefromMprocessors.Theconstraintsrelatedtoexecutionandlearningpoliciesoftheensemblesystemarenecessaryforthefunctionalityandqualityofthesystem.Inthiscontext,optimizationyieldsbest-effortperformanceamongthosesolutionsthatfullysatisfythespeciedconstraints.Giventheneedtopotentiallychangetheexperts'demandsineverycycleandthelimitedtimeavailablefortheircomputations,theTUAmustimplementanefcientalgorithmthatcanassignexactlyorapproximatelyoptimaltasks'resourceutilizationdemandswithminimaloverheads. AnoptimizationschemefortheTUAisproposedinChapter 4 .Inthatparticularsetting,allinformationneededformakingdecisions,exceptexpertresponsibilities,areassumedtobeknownprecisely.Anexampleofsuchsettingcanbefoundinensemblesystemswithlimited,butdedicatedresources.Chapter 6 providesanotheroptimizationschemeforabroadercaseinwhichotheruncertaintiesmayexistinensemblesystems. 3.2.2Real-TimeTaskScheduler(RTS) TheRTSreallocatesresourcestotasksinaccordancewithtotheirresourcedemandsandcreatesaglobally-feasibleschedule.Ascheduleisconsideredgloballyfeasiblewhenbothexecutionandlearningpoliciesoftheensemblesystemarenotviolatedacrosstime.Thedifferencebetweenresourceutilizationachievedbyeachtask 64

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iinthescheduleandthetask'schangingresourcedemandshouldalsobeboundedasclosetozeroaspossible(i.e.,theactualresourceutilizationofthetaskatanytimetshouldbeequalorrelativelyclosetoUi(t)). Inunlimited-resourceensemblesystems,tasksprocessanewinputinstanceineachcycleregardlessofwhethertheiroutputscontributetothenalsystemoutput.Whenresourcesarelimited,asubsetoftaskscannotexecuteinsomecycles.Dependingonapplications,thesequenceofdatainstancesseenbyanindividualtaskmightneedtobepreservedintheorderinwhichtheseinstancesareoriginallycreated.Forexample,thedatainstancesmustbeprocessedintheiroriginalsequenceiftheexpertsimplementtime-seriesdataprocessingalgorithms.Ontheotherhand,onlythelatestdatainstancesareworthbeingprocessedforsomeclassicationtasks.Asaresult,theRTSmustbecapableofassociatingappropriateinputdatainstancestotaskinstanceswhendispatchingthemforexecution. Chapter 4 presentsasimpleyetefcientalgorithmthatcanbeimplementedastheRTScomponent.Thealgorithmisabletoallocateresourcesandcreateaglobally-feasibleschedulefortasks.Chapter 5 proposesanenhancedimplementationoftheRTSwhichhandlesmoregeneraltasksetstheirWCETscanbegreaterthanoneandarenotnecessarilythesame.InChapter 6 ,theRTSimplementationismodiedtosupportthecaseswhentaskWCETsareuncertain. 3.3EnsembleSchedulingProcedure Figure 3-1 showssteps(identiedasnumberedcyclesinFigure 3-1 )intheproposedensembleschedulingscheme.Atthebeginningofeachensemblecycle,anewinputinstancearrivesinthesystem.Asshowninstep1,thisinputinstanceissenttothegatingcomponentandtheEESmanager.TheRTSappendsthenewlyarrivedinputintoitsdatarecord.Forinput-gatingensemblesystems,theTUAaddstheinputdatatotheinputhistoryandmayusethishistorytopredictnext-cycleresponsibilities.Instep2,thegatingcomponentuseseithertheinputvaluesorexpertperformance 65

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evaluationbasedonthelasttaskoutputstodetermineexpertresponsibilitiesRiandpickswinnersofthecycleaccordingtothesystem'sexecutionpolicy.TheresultingresponsibilitiesandwinnerstatusoftasksaresenttotheoutputaggregatorandtheEESmanager(instep3). Next,theTUAassignstotasksresource-demandvaluesUi(t)inaccordancewiththeirresponsibilitiesandwinningstatusreceivedfromthegatingcomponent,asshowninstep4.Basedontheresourcedemands,theRTSassignstasks'resourceutilizationallocationVi(t)(step5)andcreatesascheduleoftaskexecutiononMprocessors(step6).Attheendofeachcycle,thewinnerexperts'outputsareavailableandaggregatedintoanalsystemoutputaccordingtoanexecutionpolicy.Thisisshownasstep7. Byusingthesystemfeedback,e.g.,desiredoutputs,taskperformancecanbeevaluated(step8).Forperformance-gatingensemblesystems,theresultofthetaskperformanceevaluationmustbeloggedbythegatingcomponentandcanbeusedbytheTUAtopredictthenext-cycleresponsibilities,asshowninstep9.Thenext-cycleresponsibilitiescanbesuppliedtotheTUAforthepurposeinimprovingoptimization'sefciency(i.e.,avoidunnecessaryadaptationofresourcedemands). 3.4Summary Thischapterprovidesanoverviewoftheensembleschedulingmodelandtheproposedschemeforschedulingtasksinensemblesystemswithlimitedresources,i.e.,theEESmanager.TheEESmanagerusesaTaskUtilizationAdaptor(TUA)toadjusttaskresourcedemandsaccordingtothedynamicresponsibilitiesandsubsequentlycreatesafeasibleexecutionscheduleusingaReal-timeTaskScheduler(RTS).WiththemodulararchitectureoftheEESmanager,thechallengesoftheTUAandRTScanbesolvedindependentlyandtheirimplementationscanbetunedtoanyconsideredapplications.TheTUA'sandRTS'simplementationsfordedicatedenvironmentsareprovidedinChapter 4 ,andtheirextensionstomoregeneralizedensemblesystemsareprovidedChapters 5 and 6 66

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Figure3-1. ThearchitectureandworkowoftheEESmanager.TheTaskUtilizationAdaptor(TUA)assignsresource-utilizationdemandsofexpertsaccordingtotheexperts'responsibilities,determinedbyagatingcomponentofanensemblesystem,inordertominimizetheimpactoflimitedresourcesonthequalityofsystemoutputs.TheReal-timeTaskScheduler(RTS)allocatesresourcestoexpertsbasedontheirassignedresource-utilizationdemandsandselectsonlyasubsetofexpertstoexecuteineachcycle.Circlednumberscorrespondtoschedulingactionsdiscussedinthetext. 67

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Table3-1. Listofnotationsusedinensembleschedulingmodel. VariablesDescriptions iTheithtaskoftheensemblesystem UiminTheminimumresourceutilizationdemandofi UimaxThemaximumresourceutilizationdemandofi Ui(t)Thedynamicresourceutilizationdemandofi Pi(t)Thedynamicperiodoftaskexecution(Pi(0)=taskperiod) Ci(t)ThedynamicamountoftimethatimustexecuteinPi(t)(Ci(0)equalstheworst-caseexecutiontimeofi) Vi(t)ThedynamicresourceutilizationallocationassignedtoibytheRTS Di(t)Thedynamicrelativedeadlineofi,whichequalsitsperiod ri,jThereleasetimeofthejthjobofi di,j(t)Thedynamicabsolutedeadlineofthejthjobofi(di,j(t)=ri,j+Pi(t)) NTotalnumberoftasksorexperts MTotalnumberofavailableunit-capacityprocessors Ri(t)Thedynamicresponsibilityofideterminedbythegatingcomponent 68

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CHAPTER4ADAPTIVEENSEMBLESCHEDULINGWITHLIMITEDRESOURCES Thischapterconsidersadaptiveschedulingofensemblesystemswithlimitedresourceswhenallinformationnecessaryformakingschedulingdecisions,exceptexpertresponsibilities,ispreciselyknownbeforehand.ThreenovelalgorithmsareproposedforimplementingtheEESmanagerinthisscenario.Thersttwolinear-timetask-throttling(TT)algorithms,calledTT-TC*andTT-Top,caneachbeusedtoimplementaTUA.Section 4.1 formulatestheoptimizationproblemofadaptingtaskresourcedemandsinensemblesystemsandshowshowexpertresponsibilitiescanbemappedtothetaskweightsusedintheformulation.Bothalgorithmsutilizeatestbasedonoptimal-solutionsensitivityanalysisinordertoeliminateunnecessarypolynomial-timeoptimizationinthecriticalpathofexpertexecution.Section 4.2 introducesthelastalgorithm,namedtheEnsemblePolicy/ObjectiveConscious(EPOC)algorithm,whichcanefcientlyallocateresourcesandscheduleexpertsonmultipleprocessorsbasedonthesolutionsobtainedfromeitherTT-TC*orTT-Topalgorithms.Section 4.3 describesatestensemblesysteminmotorcontrolusedtoevaluatetheproposedapproachdiscussedinSection 4.4 .ConclusionsaresummarizedinSection 4.6 4.1AdaptingResourceUtilizationDemandsofTasks Inthissection,thefocusisontheresource-demandassignmentforexpertsinensemblesystemswhenresourcesarenotsufcienttoallocatetoallexpertstheirmaximallydesiredutilization.Thegoalistochooseresourcedemandsthatcanbeservicedbytheavailableresourcesandleadtotheexecutionofasubsetofexpertsaccordingtothesystem'sexecutionandlearningpolicies.Notethattheterm'utilization'inthissectionreferstoresource-utilizationneededbyatask(insteadofitsactualresourceutilization). 69

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4.1.1ProblemFormulation TheTUAcomponentaimstominimizetheimpactoflimitedresourcesonthesystemperformancewhilesatisfyingtheensemblesystem'sresourcecapacityconstraints,executionpoliciesandlearningpolicies,intheformoftheproblemPasstatedinthepreviouschapter.Inthecasewhenthereareonlycapacity(i.e.,resource)constraints,weareinterestedinprovidingschedulesforreal-timetaskswhileminimizingthecumulative(weighted)changesoftheutilizationvaluesandkeepingchangeswithintheirallowedranges.Thisconstraintoptimizationproblemcanbeexpressedasthefollowingquadraticprogrammingproblemrstdescribedin[ 38 ]: ProblemP(w): minimizeUiNXi=1wi(Uimax)]TJ /F3 11.955 Tf 11.96 0 Td[(Ui)2subjecttoNXi=1UiUdUiminUiUimax,8i=1,...,N whereanon-negativeweightofiatcyclet,orwi(t),isdenedasafunctionoftheexpert'sresponsibilityRi(t),w=[w1,...,wN],andUdisthedesiredtotalutilization,i.e.,thetotalresourcesavailabletodeploythesystem.Forbrevity,thetimeindextmaybedroppedwhenitisclearthattheparameters'currentvaluesarereferred.Thefollowingtheorem(rstprovidedin[ 38 ])showsthattheoptimalsolutionofP(w)canbeprovidedbytheTask-Compression(TC)algorithm,proposedin[ 32 ],whenthetaskelasticitiesarethereciprocaloftheobjectivefunction'scoefcientswi. Theorem4.1. TheTCalgorithmyieldsa(optimal)solutiontoP(w)whenPNi=1Ui=Ud. Proof. TheKarush-Kuhn-Tucker(KKT)conditions[ 17 ]areknowntobesufcientfortheoptimalityofasolutionofaconvexoptimizationproblem,suchasP(w).Toprovetheabovestatement,weneedtoshowthattaskutilizationdemandsUoptiobtainedbythe 70

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taskcompressionalgorithmsatisfytheKKTconditionsofthisproblem,whicharestatedasfollows: NXi=1Uopti)]TJ /F3 11.955 Tf 11.95 0 Td[(Ud=0(4) )]TJ /F4 11.955 Tf 11.96 0 Td[(i+i)]TJ /F6 11.955 Tf 11.95 0 Td[(2wi(Uimax)]TJ /F3 11.955 Tf 11.95 0 Td[(Uopti)=0(4) i0,i0(4) (Uimin)]TJ /F3 11.955 Tf 11.96 0 Td[(Uopti)i=0(4) (Uopti)]TJ /F3 11.955 Tf 11.96 0 Td[(Uimax)i=0(4)fori=1,...,N where,iandiforalliareunknownLagrangemultipliers. Theresourceutilizationdemandofataskcanfallintooneofthreecases:Uopti=Uimin,Uopti=Uimax,andUimin
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=20BBB@Xj:Ujmin
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ibetrainedatleastonceeverytrainicycles,i.e.,itsutilizationmustbenolessthanUitrain=1=traini.Consideralsothetypicalexecutionpolicyrequiringanexperttobeexecutedineverycycleforwhichitisawinner,i.e.,itsutilizationmustequalUiwinner=Cisinceitsperiodmustbeonetimeunit.Addingthesetwoconstraintsyieldsthefollowingoptimizationproblem: ProblemP0(w): minimizeUiNXi=1wi(Uimax)]TJ /F3 11.955 Tf 11.96 0 Td[(Ui)2subjecttoNXi=1UiUdUiminUiUimax,8i=1,...,NUiUitrain,8i=1,...,NUiUiwinner,8i2winners TheaboveproblemP0(w)reducestoP(w)byreplacingUimininthesecondconstraintofP(w)bymax(Uitrain,Uimin)fornon-winnersandmax(Uiwinner,Uimin)forwinners.SimilarmathematicalprogrammingproblemscanbeformulatedfordifferentpoliciesasdiscussedinSection 4.1.2 .ForalloftheseproblemsthefollowingconsiderationsregardingP(w)apply. Solvingtheproblemofschedulingensembletasksinsuccessivecyclescanbeviewedassolvingasuccessionofquadraticprogrammingproblems,i.e.,P(w(t))toP(w(t+n)).TheseproblemsaresimilartoP(w),differingonlyintheweightsoftheirobjectivefunctions(Figure 4-1 ).UsingtheexplicitmappingsbetweenRi(t)towi(t)proposedinthefollowingsection,eachoftheseproblemscanbesolvedinO(N2)time,whichisexcessiveifneededatthebeginningofeverycycle.ThechallengeisthentoneveruseTCatthebeginningofacycle(oruseitonasfewcyclesaspossible)andtoinsteaduseafasterapproach.ThisdissertationproposestheTT-TC*andTT-Topheuristicswhichreduceoreliminatetheneedtodelaytaskexecutiondueto 73

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theexecutionofTCinthecriticalpath(i.e.,atthebeginningofeverycycle).TheseheuristicsarediscussedinSection 4.1.4 .Insummary,tosolvetheproblemP(w)ofallocatingtaskutilizationsoflimitedresourcesineverycyclet,weneedtodeviseamappingofresponsibilitiestoweights(Section 4.1.2 )andafastheuristicforitssolution(Section 4.1.3 ).Thisapproach(i.e.,mapping+solvingP(w(t)))iscalledtask-throttling(TT)and,dependingonhowP(w(t))issolved,ityieldsheuristicsTT-TC,TT-TC*andTT-Top(discussedinSection 4.1.4 ). 4.1.2WeightsofEnsembleTasks AnaveapproachtoderivetaskweightswfromresponsibilitiesRwouldbetosimplyletw=aRforsomeconstanta.Itturnsoutthatsuchmappingsleadtoincorrectschedules.Forexample,ifoneofthewinnershasalowresponsibilityvalue,itwouldgetalowweight.Thiswouldimplyalowexecutionratefortheexpertanditsfailuretoruninthecycleforwhichitisawinner,henceviolatingtheensemble'sexecutionpolicy.Intuitively,eachweightusedinP(w)shouldreectthecontributionofonespecicexpertinenablingtheensemblesystemtoprovideaccuratesolutions.Thequalityofeachexpert'scontributiontoasolutionisafunctionofboth(1)theresponsibilityoftheexpertand(2)howwelltheexpertshavelearnedtoadapttochanges.Inotherwords,justlikeconstraintsreectpoliciesforexecutionandlearning,mappingapproachespresentedinthissectionalsodeterminethenatureofthebest-effortsolutionthatoptimizestheobjectivefunctionofP(w)bydeterminingweightsbasedonensembleexecutionandlearningpolicies.Althoughthedesirablemappingsmayalsoexhibitdependenciesontheensemblesystem,theproposedapproachescanbeusedasabasisuponwhichapplication-specicconstraintscanbeadded. Withoutlossofgenerality,ensembleexecutionpoliciescanbecategorizedintothefollowingtwogroups:k-winnerandrelevant-expertpolicies.Withthek-winnerpolicy,somecriteriaareusedtoalwaysselectonlykoutofNexpertstogenerateanaloutputofthesystem.Forexample,theexpertswiththekhighest-valuedresponsibilitiescould 74

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bechosen.Thesepoliciescanbefurtherclassiedintotwodistinctlearningpoliciesaccordingtowhethertheimportanceoflearningbythe(N)]TJ /F3 11.955 Tf 12.76 0 Td[(k)non-winnerexpertsshouldbeweightedequallyorinproportiontotheexperts'responsibilities.Incontrast,therelevant-expertpolicycombinesoutputsfromallexperts.Inthispolicy,althoughoutputsfromallexpertsareneededtogeneratetheaccuratesystemoutput,thesystemoutputmaybelargelydeterminedbyonlyasubsetoftheexpertswhoseresponsibilitiesexceedacycle-dependentthreshold(eitherindividuallyorcollectively).Insuchcases,thenumberofexpertsthatmustbeexecutedineachcyclevariesacrosscycles.Thispolicycanthusbeviewedasthek-winnerpolicywherekchangesineverycycle. Foranyofthepoliciesunderconsideration,theviabilityofanensemblesystemwithlimitedresourcesrestsontheassumptionthatthereareatleastenoughresourcesforexecutionofkoutofNexpertsandfortheminimalresourceutilizationneededforlearningbytheother(N)]TJ /F3 11.955 Tf 13.02 0 Td[(k)experts.Werefertothisamountofresourcesasreasonableresources.Accordingtoitsdenition,reasonableresourcesmustbeenoughtoexecutek+d(N)]TJ /F3 11.955 Tf 11.96 0 Td[(k)max(Uimin)eexperts.Givenreasonableresources,theschedulerhastomakesurethattherightkexpertscandelivertheirresultswithinthestrictreal-timedeadline.Therefore,themaximumutilizationrequiredmustbeallocatedtothewinnerexperts,andtheremainingresourceutilizationcanbedistributedamongnon-winnersbasedontheensemble'slearningpolicies.Thetwovariantsofthek-winnerpolicyareconsideredrst,followedbytherelevant-expertpolicy(thecombinationsofexecutionandlearningpoliciesareidentiedbytheirexecutionpolicyfollowedby/followedbytheirlearningpolicy). 4.1.2.1PolicycombinationI:kwinners/unrankednon-winners Theweightsofthekwinnersaresettoinnity(i.e.,averylargenumber)sincetheseexpertsmustbeexecutedtoensureproperoperationofthesystem.ThischoiceleadstotheutilizationofawinnerbeingUimax.Forthenon-winners,sincetheyare 75

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unranked,differentnotionsoffair-sharingofresourcesarepossible;twoareconsideredhere,namelyequalweightsandequalutilization. EQUALWEIGHTS.Thisrequirementismetbysettingtheweightsofnon-winnersastheinverseofthetotalnumberofnon-winners(i.e.,wi=1=((N)]TJ /F3 11.955 Tf 13.21 0 Td[(k))).Thisassignsequalimportancetothetrainingofeachexpertwithoutexplicitlyconstrainingitsutilization.Thus,equaldeductionismadefromthemaximumutilization(i.e.,Uimax)requiredforeachnon-winner'sexecutioninacycle;theresultingUiforeachnon-winner,however,mayormaynotbeequaltheutilizationsassignedtoothers. EQUALUTILIZATION.Intuitivelythisrequiresthateachexpertbetrainedwiththesamefrequency.ItcanbemetbyaddingtoP(w)theconstraintUi=Ujforalli,jsuchthatexpertsiandjarenon-winners.Inthiscasetheweightsassociatedwithnon-winnersareirrelevantandcanthusbesettozero,leadingtothesolutionwherethebestvalueofUiismax((Ud)]TJ /F3 11.955 Tf 11.95 0 Td[(kUimax)=(N)]TJ /F3 11.955 Tf 11.96 0 Td[(k),max(Uimin)). Thereisaclosed-formsolutionfortheequal-utilizationcase,sincewecanderivethevaluesofUiforbothwinnersandnon-winners.However,inpractice,itmaybedesirabletorequirethateachexpertbetrainedwithadifferentminimumfrequency,asspeciedbythethirdconstraintinP0(w).Byusingtheseconstraintsand,inaddition,choosingequalweightsasspeciedabove,itbecomespossibletoobtainbest-effortsolutionsthatmeetminimumlearningrequirements.Thustheequal-weightsformulationispreferabletotheadditionofequal-utilizationconstraints.Itcanbeextendedfurthertoyieldbetterresourceutilizationandmoreexibility,e.g.,favoringtrainingofthoseexpertsforwhichtheminimumtrainingfrequencyyieldspoorperformance.Thiscanbedonebyusingnon-equalweightsthatarerelatedtoexpertperformance,asdescribedforPolicycombinationII. 4.1.2.2PolicycombinationII:kwinners/rankednon-winners AsforcombinationI,theweightsofwinnersaresettoinnity.Thenon-winnersgetafractionoftheresourceutilizationbasedonaperformanceindicatorsuchastheirresponsibilities(othersarepossibleasdiscussedbelow).Thustheirweightsaresetaswi=Ri=(Pj2non)]TJ /F5 7.97 Tf 6.59 0 Td[(winnersRj).Thisfavorsexpertsthatcouldbecomewinnersinfuturecyclesifadditionallearningismadepossible.Itisalsoconceivablethat,insome 76

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ensemblesystems,itmaybebettertoassignmoreutilizationtotheleastresponsibleexpertssothattheycancatchupintheirlearning.Fromaschedulingstandpoint,thiswouldrequirethemappingtouse(1)]TJ /F3 11.955 Tf 12.94 0 Td[(Ri)insteadofRiinboththenumeratoranddenominatoroftheabovenon-winner'sweightmapping.Insteadofusingweightsbasedonperformanceindicators(i.e.,Ri)toimplicitlydetermineutilization,onecouldintroduceexplicitconstraints,e.g.,Ui=Ri=Uj=Rjforalli,jsuchthatexpertsiandjarenon-winners.However,justasfortheequal-utilizationcase,thiseliminatesexibilityandviablebest-effortsolutions.Thereforethisapproachisalsonotconsideredfurtherinthisdissertation. Performanceindicatorsotherthanresponsibilitiescouldalsobeusedsimilarly.Thefollowingareexamples: errori:thegoodnessofexpertiisquantiedbythedifferencebetweentheexpertoutputandthecorrectoutput;byusingitasaperformanceindicator,expertaccuracydetermineswhentrainingmustoccur. 1=Uimax:thisindicatorfavorsthetrainingofthefastest(i.e.,withshortestexecutiontime)experts;intuitively,thegoalistoobtaingoodfastsystemresponses;unlikeresponsibilitiesanderrors,thisindicatortypicallydoesnotvaryfromcycletocycle. 4.1.2.3PolicycombinationIII:relevantexperts/rankednon-winners Withthisexecutionpolicy,eachandeveryexpertcon-tributestothesystem'snaloutput.Missingtheexecutionofanygivenexpertcouldresultinaninaccuratenaloutputifthatexpertisrelevantenough.However,ifatmostkexpertscouldberelevantenough,thentheweightsforthispolicycanbeassignedinthesamemannerasforPolicycombinationII.Inotherwords,suchsystemswouldrstrequiretheidenticationofkasanupperboundofthenumberofrelevantexpertsandthenusethemappingsproposedforPolicycombinationII. Insummary,whileseveralexecutionandlearningpoliciescanbeconsidered,(1)theresourceutilizationallocationproblemcanbereducedtoP(w)onceweightsarechosenaccordingtopoliciesandconstraintsarecastintheformoftheconstraints 77

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ofP(w),and(2)PolicycombinationIisaparticularcaseofcombinationIIwhenallnon-winnershaveequalweights,andthemappingsforthelattercanbeusedforcombinationIII.Thus,hereontheoptimizationproblemandpolicycombinationofinterestareassumedtobeP(w)andPolicyII,respectively. Aftertheexperts'weightsareassignedusingtheaboveapproach,TCcanbeusedtoproperlyadjusttheutilizationsUiofexperttaskssuchthattheentiretasksetisschedulablewiththeavailableresources.TheassignedutilizationvaluescanthenbeusedbytheEnsemblePolicy/ObjectiveConscious(EPOC)scheduler,presentedinSection 4.2 ,toselectexpertsforexecutionineachcycle. 4.1.3SensitivityAnalysisTest WhileTCtakesO(N2)timetosolveP(w),itisstilltooexpensivetorunatthebeginningofeverycycle,possiblycausingscheduledexpertstonotcompletetheirexecutionintime.ItisdesirabletoavoidrecomputationofTCinthecriticalpathofexpertexecution(denotedascriticalTC)asmuchaspossible.ThissectionprovidesasensitivityanalysisproceduretodeterminewhethertheoptimalityofaknownsolutionofP(w)isdisturbedbythechangesofweightsintheoptimizationfunction.Iftheoptimalityisnotaffectedthentheknownsolutioncanbereusedforanewcyclewithoutneedforre-optimization. Theorem4.2. TheKKTconditions(( 4 )to( 4 ))arenecessaryforoptimalityofasolutionoftheproblemP(w). Proof. Foranoptimizationprobleminthefollowingformulation:minimizef(x)subjecttogi(x)0hj(x)=0 Thestrongdualityoftheaboveoptimizationproblemholdswhentheproblemsatisesanadditionalconstraintqualication,calledSlater'scondition[ 52 ].LetFbeasetof 78

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feasiblesolutionsoftheaboveproblem.AvectorexFiscalledaSlaterpointofF,orsatisesSlater'scondition,when: gi(x)0,foralliwheregi(x)isalinearfunction gi(x)0,foralliwheregi(x)isanon-linearfunction hj(x)=0,forallj Toprovetheabovestatement,weneedtoshowthatthesetoffeasiblesolutionsoftheproblemsatisesSlater'scondition,whichimpliesthattheprimalproblemisstrictlyfeasible.SinceallconstraintsintheproblemP(w)arelinearfunctions,everypointexFisaSlaterpointwhenF6=.ItfollowsthattheKKTconditionsarenecessaryandsufcientforauniqueglobaloptimalsolutionofP(w). TheKKTconditions(( 4 )to( 4 ))areusedtoderivetheprocedureoftheTTalgorithmtoanalyzethesensitivityofagivenoptimalsolutionwhenweightschange.FromTheorem 4.1 ,theoptimaltaskutilizationdemands(Uopti)obtainedbyTCmustsatisfyalloftheseKKTconditions.SincetheseKKTconditionsaresufcientandnecessaryfortheoptimalityofasolutionofaconvexoptimizationproblem(asstatedinTheorem 4.2 ),thesensitivityanalysisproceduredescribedbelowisexact,i.e.,ityieldsneitherfalsenegativesnorfalsepositives.WhentheobjectivefunctioncoefcientsofP(w)changefromwtow0,theKKTconditionin( 4 )becomes )]TJ /F4 11.955 Tf 11.95 0 Td[(i+i)]TJ /F6 11.955 Tf 11.96 0 Td[(2w0i(Uimax)]TJ /F3 11.955 Tf 11.96 0 Td[(Uopti)=0(4) IftheknownoptimalsolutionUoptialsosatisesthismodiedKKTcondition,thenitisstilltheoptimalsolutionforP(w0).Foreachexpert,threecasescanoccur: 79

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Case1:Uopti=Uimax Wehavei=0,=)]TJ /F4 11.955 Tf 9.3 0 Td[(iand,from( 4 ),wehave)]TJ /F6 11.955 Tf 9.3 0 Td[(2w0i(0)=0.Asaresult,forthetaskwiththemaximumutilizationdemand,changingitsweightwillnotaffecttheoptimalityofthecurrentsolution.Itisthennotnecessarytochecktheoptimalityconditionofthistask. Case2:Uimin0)UoptIsOptimalForP0w=false;endend 80

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Accordingtotheaboveanalysisprocedure,wheneverthereisanupdateofthetasks'responsibilities,theoptimalityconditionforthosetaskswithutilizationdemandslessthanUimaxneedstobeveried.Iftheweightswiofallthosetaskssatisfytheconditionjw0i)]TJ /F3 11.955 Tf 11.96 0 Td[(wij0,wedonotneedanotherTCexecutiontoadjustthetasks'periods(inpractice,theprocedurechecksforjw0i)]TJ /F3 11.955 Tf 11.95 0 Td[(wijwheredependsonthenumericresolutionusedtorepresentweights).Thisstatementisalwaystruebecausethemappingofresponsibilitiestoweights(Section 4.1.2 )guaranteesthattheremustbekexpertswithinnityweightsineverycycle.ThesensitivityanalysisprocedureisveryinexpensiveasittakesO(N)time.Fork-winnerpolicies,itcanbefurthersimpliedbecausetheassignmentofwi=1tothewinnersimpliesthemaximumallocationofutilization(Uimax)bytheTCalgorithm.Thereforeawinner'sutilizationneverneedstobecheckediftheexpertremainsawinner.Forthek-winnerpolicywherethenon-winnersaregivenequalweights,itsufcestocheckwhetherthesetofwinnerexpertsremainsthesame.TheidenticationofwinnersandthemappingofresponsibilitiesintoweightsalsotakeO(N)time. 4.1.4TaskThrottling(TT)Heuristics Figure 4-2 presentsaowchartoftheTT-basedalgorithmsincludingTT-TC,TT-TC*andTT-Top.TheTT-TCalgorithmisthebaselinealgorithmintroducedforperformancecomparisonpurposes.Inanygivencyclet,theTT-TCalgorithmalwaysprovidestotheoptimalresourceutilizationallocationcomputedbyTC.TT-TC*andTT-Topusethepreviouslydescribedsensitivityanalysisproceduretotestwhethereitheroneoftwopre-computedoptimalsolutionsisthesolutionoftheoptimizationproblemtoavoidre-optimizationwhenUoptIsOptimalForP0wremainstrue.Thesetwocandidatesolutionsarethedemandsusedinthepreviouscycle(denotedasUoptprev(t))andthedemandspre-computedbytheTCalgorithmduringthepreviouscycleusingpredictedresponsibilitiesforthefollowingcycle(denotedasUoptpred(t)). 81

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InthecasewhennoneofthesesolutionscanbeusedasanoptimalsolutionforP(w),TT-TC*andTT-Toptakedifferentsteps.TT-TC*usesTCtocomputeanewoptimalsolutionandsuppliesittotheEPOCscheduler(describedinSection 4.2 ).Ontheotherhand,TT-TopquicklydeterminesanapproximatedU(t)usingTopResp(R(t))andallowstheschedulertousethispotentiallysuboptimalsolutiontoimmediatelyscheduleexperts.TheTopRespfunctiontakesO(N)timetocompute;itinitiallyassignsUi=Uimaxforallwinners,Ui=Uiminforallnon-winnersandthenreadjustsUi=Ui+min(Uimax)]TJ /F3 11.955 Tf 12.46 0 Td[(Ui,M)]TJ /F3 11.955 Tf 12.46 0 Td[(M0)whereM0=PNi=1Uiforallnon-winnersiinadescending-responsibilityorderuntilMprocessorsarefullyutilizedornoexpertisreadyforexecution.TheperformanceevaluationinSection 4.4 conrmsthattheTopRespassignmentproducesagoodapproximationoftheoptimalsolutionobtainedfromTCwhenareoptimizationisneeded. Lemma4.2.1. Foranygivencyclet,anyoftheTT-TC,TT-TC*andTT-TopalgorithmsyieldsafeasiblesolutionofP(w). Proof. TT-TCandTT-TC*useTCandsensitivityanalysistoyieldsolutionswhich,accordingtoTheorem 4.1 ,mustbefeasibleandoptimal.WhenTT-topdoesnotuseTCorsensitivityanalysis,itusestheTopRespassignmentwhichisdesignedtosatisfytheconstraintsofP(w).Thus,TT-TopalsoyieldsfeasiblesolutionsofP(w). 4.2Ensemble-Policy/Objective-Conscious(EPOC)SchedulingAlgorithm ThissectionpresentsanewschedulerthatselectsensembletasksforexecutiononmultipleprocessorsbasedontheresourceutilizationallocationassignedbytheTUAusingtheTT-basedalgorithmsexplainedintheprevioussection.NotationusedinthissectionissummarizedinTable 4-1 .TheEPOCalgorithmutilizesnotionsofallocationandschedulingepochsinallocatingresourcesandschedulingtasks.Anallocationepoch(abbreviatedasa epoch)isasetofconsecutiveensemblecyclesinwhichthetaskdemandsUiassignedbytheTUAremainunchanged(i.e.,anepochduring 82

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whichtheresourceallocationdoesnotchange).Aschedulingepoch(abbreviatedass epoch)isdenedasasetofsuccessivecyclesstartingatsomecyclea2Z+andendingatcyclea+whereisthelowestcommondenominatorofalltheresourceutilizationallocationsVi(=Ui)atcycleawhentheyareexpressedasaratiooftwointegers,i.e.,(a)=LCDi=1,..,N(Ui(a)).Thetimeindexmaybedroppedandonlyisusedwhenitisclearwhenthestartingcycleis.AlowerboundofisdNW=MNWewhichistheminimumnumberofcyclesneededforachievingtheminimumutilizationrequiredbyeachexpert.Schedulingepochsdonotoverlapwitheachothers,i.e.,anewschedulingepochcanonlystartwhentheexistingoneends.Anallocationepochcancoincideand/orpartiallyoverlapwithoneormoreschedulingepochs.Therstschedulingepochstartsatthesametimeastherstallocationepoch.Dependingonthedynamicsofthesystem,thelengthsofallocationepochsmaydifferfromoneanother. 4.2.1SchedulingTasksinEachEnsembleCycle TheEPOCschedulermaintainsNrst-in-rst-outqueues;eachqueueisassociatedtoonedistincttaski.Ateverycycle,anewdatainstancearrivesandiscopiedintoeverytaskqueue.WhenajobofiisdispatchedforexecutionbyEPOC,thedatainstancetobeprocessedbythejobistheoneattheheadofi'squeue.TheEPOCscheduleralsomaintainsthefollowinginformationforeachtaskineachcycle: Taskidenticationnumber(idi); Winningstatus(!i)aagindicatingwhetherthetaskisawinner(!i=1)ornot(!i=0); Numberofjobs(i)requiredtoexecuteineachschedulingepochtomeettheminimumutilization(Uimin)constraintanintegerwhosevalueisdUimine; Numberofjobs('i)thatcanpotentiallyexecuteinthecurrentschedulingepochinaccordancewiththeutilizationdemandUiofthecurrentallocationepochanintegerwhosevalueisdUiewhereUiistheutilizationdemandassignedinthecurrentallocationepoch. Whilethetaskidenticationnumberisstatic,theotherthreevaluesaredynamicandcanchangeovertime. 83

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Atthebeginningofanygivencycle,theoldestdatainstanceinanygiventaskqueueisacandidateforprocessing.Ifanon-winnerbecomesawinner,theremainingdatainstancesinitsqueuearediscardedbeforeaddingthenewinstance.Thereleaseofajobisallowedonlyatthebeginningofthecycletoguaranteenopreemptionaftertheexpertsareselectedtoexecuteinthatcycle.Thisisrequiredbecauseexpertoutputsmustbeavailableduringthecycleinwhichtheirexecutionsarestarted.Therateofjobreleasesforagiventaskisfi=1=Pi=Ui=Ci. Todeterminetaskswhosejobsmustbereleasedatthebeginningofeachcyclet,EPOCdoesthefollowingsteps: STEP1.Update!iaccordingtoexperts'responsibilities STEP2. if(t=0)or(t=a+)//thebeginningofans epocha t, LCDi=1,...,N(Ui(a))i dUimine;8i=1,...,NelseifUichange//thebeginningofana epoch'i 'i+dUieend STEP3.chooseMtaskswithpositive'itoexecute(i.e.,releasejobsfromqueues)byapplyingthefollowingrules: WINNERRULE.if!i>!j,choosei;elseif!ij,choosei;elseifi'j,choosei;elseif'i<'j,choosej;elseif'i='j,seethenextrule. IDRULE.ifidi>idj,choosei;elsechoosej. TheaboverulesinSTEP3canbemoresuccinctlystatedasfollows:choosetheMtaskswiththehighestlexicographicalvaluesof(!i,i,'i,idi).Attheendofthecycle, 84

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theschedulerdecrementsbyonebothiand'iforeverytaskselectedinthelaststepunlesstheyarealreadyzeros. 4.2.2FromLocalEpochSolutionstoaGlobalSolution ThereremainsaquestionastowhetherthesolutionsofthemultipleproblemsdepictedinFigure 4-1 yieldaglobalsolutiontotheproblemofschedulingexpertssothatconstraintsderivedfromexecutionandlearningpoliciesaresatisednotonlyineachcyclebutalsoacrosscycles.Sincetheresourceutilizationallocationandthescheduleforanygivenepochguaranteestimelyexecutionofeachwinnerexpertinthatepoch,thetimingandexecutionpolicyrequirementsareclearlymetforallepochs.However,thelearningpolicyrequirementscouldconceivablybeviolated,e.g.,whenanallocationepochistooshortforanexpertwithalargeperiodtoeverexecute.Insuchcase,dependingonhowtheschedulerhandlestaskselectioninsuccessiveepochs,thelearningrequirementsmayormaynotbemet.ToshowthattheanswertothisquestionisdenitelyyeswhenusingtheproposedimplementationoftheEESmanager,thefollowingtheoremsshowthattheEPOCschedulesachieveglobalfeasibilitywhenreasonableresourcesareavailableandsimpleexamplesillustratetheoperationofEPOCinsupportofthereasoningusedintheproofs. Theorem4.3. GivenresourcesMMrandafeasibleutilizationallocationU=[Ui;i=1,...,N],EPOCgeneratesschedulesthatalwaysmeetboththeconstraints(1)UiUiminforalliand(2)Pi=1,...,NUiUdattheboundariesofschedulingepochs. Proof. LetdeneN,andK,NWasthesetsofallexperts,winnersandnon-winners,respectively.Eachschedulingepochconsistsofcycles.Considertheworst-casescenariowhenthereislittleornoslackavailable:theavailableresourcesaretight(i.e.,M=Mr=k+minMNW)andtheschedulingepochisminimal(i.e.,=).ConsideranindividualschedulingepochthatsatisesbothconstraintsinP(w)atitsbeginningcycleandletnbethenumberofallocationepochsoverlappingwiththisscheduling 85

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epoch.Whenn=1,theiand'iofalltasksareinitializedonceintherstcycleoftheschedulingepoch.The`winnerrule'guaranteesthatallwinnersgetexecutedineachcycleusingMkresources.Inanygivenschedule,thenumberofnon-winnerexecutionsmustbeatleastPi2NWi=Pi2NW(Uimin)=Pi2NWUimin,whichdoesnotexceedthenumberofslotsminMNWavailableintheschedulingepoch.TheorderofexecutionforthesejobsdependsonthevaluesofiaccordingtotheUminsatisfactionrule.Thisruleensuresthatallibecomezerosbeforetheendoftheepoch,whichimpliestheutilizationUiminbyeachnon-winnerexperti,i.e.,(1)issatised.(2)isalsosatisedbecausePi2NUi=Pi2KUi+Pi2NWUik+(minMNW=)Ud.Whenn=s>1(i.e.,theschedulingepochcontainssallocationepochs),'iofeachtaskisincreased(butnoti)whenchangingfromoneallocationepochtoanother.SinceiindicatestheminimumnumberofjobsneededtobeexecutedinordertosatisfytheUiminconstraint,achangeofutilizationallocationduringtheschedulingepochdoesnotcauseanyviolationoftheUiminrequirement.Asforn=1,thetotalutilizationcannotexceedUd. Asshownbytheaboveproof,thestatementinTheorem 4.3 istrueregardlessofthenumberoftheallocationepochsoverlappingwiththeschedulingepoch.Example 4.1 illustratesthescenariowhenallocationepochscoincidewithaschedulingepoch(n=1).Example 4.2 considersallocationepochspartiallyoverlappingwithschedulingepochscausingn=2intherstschedulingepochandn=1inthesecondschedulingepoch,respectively. Example4.1. LetUd=M=2,N=5,k=1andUimin=0.25fori=1,...,5.Inthe1stallocationepoch,4isthewinner,U1=U2=U3=U5=0.25,andU4=1.Inthe2ndallocationepoch,1isthewinner,U1=1,andU2=U3=U4=U5=0.25.Wehave==4. TheEPOCschedulercreatesthescheduleshowninFigure 4-3 .Ins epoch1,1to5execute1,1,1,4and1outof4cycles,respectively.Ins epoch2,1to5execute4,1,1, 86

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1and1cycles,respectively.Hence,withineachschedulingepoch,allexpertsachieveresourceutilizationUiUimin. Example4.2. AssumeExample 4.1 'scongurationoftasksinbothallocationepochs.Instead,the2ndallocationepochstartsafteronecycleofthe1stallocationepoch(i.e.,cycle2). FromtheschedulegeneratedbytheEPOCschedulerinFigure 4-4 ,1to5execute3,1,1,2and1timesoutof4cyclesins epoch1,respectively.Durings epoch2,1to5executethesamenumberofcyclesasins epoch2inExample 4.1 .BothconstraintsinTheorem 4.3 arethensatised. Corollary4.3.1. Theorem 4.3 holdstruefortheTT-TC,TT-TC*andTT-TopalgorithmsdescribedinSection 4.1.4 Proof. FromLemma 4.2.1 ,theutilizationUassignedbyanyoftheabovealgorithmsisalwaysafeasiblesolution;thusCorollary 4.3.1 isclearlytrue. WhilesatisfyingtheUiminconstraintofeachtaskisimportant,anotheressentialquestioniswhetherexpertsexecutedaccordingtoEPOCschedulescanachievetheutilizations(Ui)assignedbytheTUA.Theorem 4.4 andassociatedcorollariesanswerthisquestion. Theorem4.4. GivenresourcesMMr,theschedulegeneratedbytheEPOCscheduleralwaysachievestheUiassignedbytheTUAwhentheallocationepochcoincideswithoneormoreschedulingepochswithtotallengthb,whereb2Z+. Proof. IfeachUiisexpressedasui=vwhereui,v2Z+,wehave=v.Whenanallocationepochcoincideswithschedulingepochs,thelengthoftheallocationepochequalsbv.Thetotalutilizationofiinthisallocationepochisbdv(ui=v)e=bv=Ui. 87

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Corollary4.4.1. GivenresourcesMMr,asinTheorem 4.4 ,andanallocationepochwithb+dcycleswhereb,d2Z+andd<,theutilizationachievedbytheEPOCschedulerforiisnolessthan(bUi)=(b+1)whereUiisthevaluedeterminedbytheTTalgorithm. Proof. Anallocationepochwithb+dcyclesfullyoverlapswithbschedulingepochsandpartiallyoverlapswithoneortwoschedulingepochs.Considertheworstcaseofataskiwhen=,Ui=1=andd=)]TJ /F6 11.955 Tf 12.72 0 Td[(1.TheamountofresourceutilizationachievedbythistaskisUib=(b+d)=(1=)[b=(b+)]TJ /F6 11.955 Tf 12.03 0 Td[(1)]=b=[(b+1))]TJ /F6 11.955 Tf 12.03 0 Td[(1]b=[(b+1)]bUi=(b+1) Corollary4.4.2. Theutilizationsoftasksexecutedinanysufcientlylargeallocationepocharenearlyoptimal. Proof. FromTheorem 4.4 andCorollary 4.4.1 ,sincetheutilizationofataskisatleastApproxUi=[b=(b+1)]Ui.Forbsufcientlylarge,b=(b+1)1andApproxUiUi. Corollary4.4.3. Theorem 4.4 ,Corollary 4.4.1 and 4.4.2 holdtruefortheTT-TC,TT-TC*andTT-Topalgorithms. 4.3TestEnsembleSystem Theperformanceoftheproposedschedulingapproachnecessarilydependsontheapplicationofaspecicensemblesystem.However,itisusefultoconsideraparticulartestapplicationthatincludeskeycharacteristicsanddynamicsofrealensemblesystemstogaininsightsonboththeaccuracyandtimelinessoftheproposedschedulingmechanismandissuesthatrequirefurtherresearch.Inthissection,thetestsystemusedfortheperformanceevaluationisdescribedinSection 4.3.1 anditsimplementationareexplainedinSection 4.3.2 88

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4.3.1DescriptionofTestEnsembleSystem AsimpliedMixture-of-Experts(MoE)applicationforcontrollingthemovementofanagent(e.g.,aroboticarm)in2Dspaceisdevelopedbasedonthearchitectureproposedin[ 157 ].Eachexpertiisaforwardmodeloftheagentmotorcontrol.Ittakestheagentposition(xt,yt)attimetandamotorcontrolcommandSttocreateanagentposition(x0t+1,y0t+1)iattimet+1.ThegatingcomponentofthisMoEsystemusesthevalueofthecontextsignal,Ct,todeterminetheresponsibilitiesofforwardmodelsattimet.Theexpertresponsibilitiesarethenusedinaccordancewithanexecutionpolicytocombineoutputsfromforwardmodelsintothenalpredictednextposition(x0t+1,y0t+1). Thereareeightpossiblecommandsfortheagent'smovementinoneoffouraxes(N,S,E,W)oroneoffourquadrants(NE,SE,SW,NW).Thebehavioralcontext,suchastheordinalpositionorproximityoftheagenttoitsreward[ 93 ],affectstheangleanddisplacementintheaxisorquadrantassociatedwiththecommand.Applyingthesamecommandatsometimettothesamepositionoftheagentindifferentcontextsleadstodifferentpositionsattimet+1.Figure 4-5 showsanexampleofpossiblemappingsofcommandsintonextpositionsfordifferentforwardmodelsindifferentcontexts.Ineachcommandmapping,ablackdotrepresentsthecurrentpositionoftheagentandgreydotsaretheestimatednextpositionsoftheagentaftereightcommandsareapplied. Inthissystem,anexpertperformssoftlearning-learninginproportiontoitsresponsibilitytoproducemoreaccurateoutputs.Itadjustsitselfusingtheerrorofitsoutputcomparedtotheactualnextposition(xt+1,yt+1)oftheagent.AcompletediagramofcomponentsofthistestensemblesystemisshowninFigure 4-6 .Unlessstatedotherwise,therearetenforwardmodels(N=10)andveprocessors(M=5).Theexecutionandlearningpoliciesofthisensemblesystemarekwinnerswhenkequals3andrankednon-winnersasdiscussedinthepolicycombinationIIinSection 4.1.2 89

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4.3.2Implementation Eachforwardmodel(i.e.,expert)isimplementedasasingle-layerneuralnetworkusingtheleastmeansquare(LMS)methodforgradient-descentlearning[ 68 ].Allexpertshavesimilarstructurebutdifferentnetworkweights.Duringcyclesinthebatch-trainingphase,thegatingcomponentappliestheK-Meanclusteringalgorithm[ 68 ]tosamplesofthecontextualsignalinordertoclusterthemintoclusters.Thegatingcomponentthenmapsexpertstoeachcluster.Eachexpertistrainedwithvaluesofthe5-tuple(xt,yt,St,x0t+1,y0t+1),representingcommandmappingsfromitsassociatedcluster,togetaninitialassignmentofitsneuralnetworkweights.Duringepochsinthetestingphase,thegatingcomponentassignstheresponsibilitiestoexpertsaccordingtotheEuclidiandistancesbetweenatestsampleofthecontextualsignalandthecentroidsoftheexperts'clusters.Thesmallerthedistance,thehighertheexpert'sresponsibilitywillbe.Inthiscasethegatingcomponentdenesasoftboundaryofthecontextualsignalandassignseachsubspacetotherelevantexpert.Withoutlossofgenerality,thestartingpositionoftheagentisat(0,0).Thecommandsignalisauniformlydistributedrandomsignal.TheactualpositionsignalisgeneratedusingasimilarMoEsystemwhoseexpertshaveidealnetworkweights.Thefollowingvetypesofcontextualsignals(inFigure 4-7 )areusedtocapturebroadrangeofdynamicbehaviorsofrealisticensemblesystems(inFigure 4-8 ). CT1.thesignalisverypredictableandrarelychangesfromoneclustertotheother.Asaresult,therankingofexpertsaccordingtotheirresponsibilitiesremainsthesameformanysuccessivecycles.Thistypeofsignalisrepresentedasaridgedsquarewaveinourevaluation. CT2.thesignalisstillverypredictableandchangesgradually,butthechangesfromoneclustertotheotheroccurmorefrequently,potentiallycausingmoresuddenswitchingbetweenwinnerandnon-winnerexperts.TheunivariatetimeseriesdatasetDfromtheSantaFeTimeSeriesCompetition[ 155 ]andasine-wave-likesignalwereaddedtogethertogeneratethissignaltype. 90

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CT3.thecontextsignalstillchangesgradually,butitisnotaseasytopredictasCT1andCT2.Thesignalinthiscategoryissimulatedasaboundedrandomwalksignal. CT4.thecontextualsignalhasrelativelyhigh-dimensionalandnonlineardynamics.Therankingofexpertresponsibilitieschangesarbitrarilymostofthetime.ThissignalissampledfromthepreviouslymentionedSantaFeCompetitiondatasetD. CT5.thesignalistotallyrandomandtheresponsibilitiesofexpertschangearbitrarily.AwhiteGaussiansignalisusedasanexample.Predictionisthehardestsincenotmuchcanbeinferredfrompastresponsibilities. 4.4PerformanceEvaluation ThetestsystemdescribedintheprevioussectionisusedtoevaluatetheperformanceoftheproposedEESmanager.Performancemetricsofinterestare(1)thetimelinessofschedulingdecision,(2)thecorrectnessofensembleschedulingaccordingtothegivenexecutionpolicy,and(3)thequalityofthelimited-resourcesystem'soutputscomparedtothatoftheunlimited-resourcesystem.Specically,theperformanceevaluationanswersthefollowingquestions: WhatistheEESmanageroverhead(intermsofthenumberofcriticalTCs)whenusingTT-TC,TT-TC*,andTT-Top?(Section 4.4.1 ) CantheEESmanagerproduceaschedulethatexecutesthemostresponsibleexpertsaccordingtothegivenexecutionpolicyineachcycle?(Section 4.4.2 ) Howaccurateareoutputsofthelimited-resourcesystemcomparetothatoftheunlimited-resourcesystem?(Section 4.4.3 ) Atestensemblesystemwithtenexpertsisconsidered.AlltaskshavethesamesetoftimingparametersCi=1,Uimin=0.2,andUimax=1onM=5processors.Thetimeunitisacycle.Asmentionedpreviously,vecontextualsignalsareusedtorepresentensemblesystemsrangingfromawell-behavedcase,i.e.,whenresponsibilitiesvarygradually,toahypotheticalworstpossiblecase,i.e.,whenresponsibilitieschangecompletelyrandomly.TheTUAusesanadaptiveFIRlterwithgradient-descentlearningtopredictthenextcontextfrom3-tapdelayofpastcontextualvaluesandusesafunctionsimilartothegatingcomponenttogeneratenextresponsibilitiesfromthepredicted 91

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contextualvalue.ForeachcombinationofcontextualsignaltypesandtheTT-basedalgorithms(i.e.,TT-TC,TT-TC*andTT-Top),ftydistinct500-cycleexperimentsareperformedandmetricsofinterestsforeachquestionarerecorded. 4.4.1EnsembleSchedulingTimeliness Incomparisontothebaselinealgorithm(i.e.,TT-TC),theoverheadsoftheEESmanagerwhenusingTT-TC*andTT-Toparestudied.TheconsideredoverheadcanbeinferredfromthenumberofTCexecutionsinthecriticalpathoftheexpertexecution.Withtheworst-casetimecomplexityofTCbeingO(N2),thecritical-TCexecutioncanpotentiallycausesignicantdelaywhentherearemanyexperts,tightcycletimeanduncertainexecutiontimeofexperts.Asaresult,systemoutputsmaymissdeadlines,signicantlydegradingtheperformanceoftheensemblesystem. 4.4.1.1TCsavingfromsensitivityanalysisofcandidatesolutions AsdescribedinSection 4.1.4 ,bothTT-TC*andTT-TopusethesensitivityanalysisproceduretocheckifareoptimizationbyTCisnecessary.Inthisrststudy,thenumbersofcyclesinthenumberofcyclesinwhichanexecutionoftheTCalgorithmatthebeginningofthecycle(i.e.,acriticalTCexecution)canbeavoidedarerecordedforthefollowingthreecases:whenapplyingthesensitivityanalysistochecktheoptimalityof(1)onlythesolutionfromthepriorcycle(Prev),(2)onlythetentativesolutionfrompredictedresponsibilities(Pred),(3)bothprevious-cycleandpredictedsolutions(Both).Theavoidedcyclesarecalculatedasapercentagefromall500cyclesineachexperiment.TheresultingaveragesandstandarddeviationsoftheTC-savingpercentagesfromallcasesareshowninFigure 4-9 ForCT1andCT2,whereresponsibilitiesofexpertschangerathergraduallyandaremathematicallysimpletopredict,itismorelikelythatoneoftwopre-computedoptimalschedules(i.e.,prior-cycleorpredictedschedules)canbeusedtoavoidacriticalTCexecution.FromFigure 4-9 ,about75-85%ofTCexecutionscanbeskippedwhenusingbothsolutions.ForCT3,thedynamicsofthetestensemblesystemare 92

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notrepetitiveand,hence,moredifculttopredict,butstillchangerathergradually,so45.87%savingisobserved.Ontheotherhand,CT4andCT5exhibithigherchancesinwinnerandnon-winnerswitching,soTCsavingsusingbothsolutionsfromthesetwocasesarenotassignicantasthosefromthersttwocontexttypes(i.e.,36.61%and10.30%,respectively).Overall,combiningPrevandPredcancertainlyincreaseTCsavingbecausethecyclessavedbyeachcandidatesolutioncomplimentthosesavedbytheother,especiallyforthemostarbitrarilychangingcontextasCT5.TheadditionalbenetsarelessnoticeableinCT3becausethecyclesinwhichTCcanbeavoidedfromPrevandPredarehighlyoverlapped.Fortherestoftheperformanceevaluation,bothprior-cycleandpredictedsolutionsareusedinTT-TC*andTT-Top. 4.4.1.2Critical-TCoverheadsoftheTT-basedalgorithms Table 4-2 shows,fordifferentalgorithmsandcontexttypes,thepercentagesofcycleswithcriticalTCs.AhighpercentageforanalgorithmAinacontexttypemathcalBindicatesthatAhashighoverheadandisnotsuitableforschedulingatime-sensitiveensemblesystemwithitsdynamicbehaviorcapturedbythecontexttypeB.WhenTT-TCisused,TCrunsatthebeginningofeverycycleforallcontexttypes.ForTT-TC*,theremainingpercentageofcycleswithcritical-TCexecutionsgreatlydependsonthedynamicnature(orthecontexttype)oftheensemblesystem.IncorrespondencewiththeresultsshowninFigure 4-9 ,Table 4-2 showsthatcriticalTCsforCT1toCT5are15.40,24.48,54.13,63.39and89.70percent,respectively.SinceTT-Topalwaysusesthetop-responsibilityheuristicwhenevernoneofthetwopre-computedsolutionsremainsoptimal,itsnumberofcritical-TCexecutioniszeroasshownintheright-mostcolumnofTable 4-2 InsystemswheretheneedtorunTCinaparticularcycleimplieseithertheviolationofadeadlineoranerroneousresult,thecritical-TCoverheadhasadirectimpactonsystemquality.Insomecasestheensemblesystemapplicationcantolerateuptox%ofcycleswithdeadlinemissesordegradedperformance.Thevalueofx,thekindof 93

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policiesandthetypeofcontextualsignalswilldeterminewhethertheheuristicproposedinthispaperwillenablethedeploymentoftheensemblesystemapplicationwithlimitedresources.Table 4-2 suggeststhatTT-TC*canbeusedinsystemswherexisassmallas15.4,whileTT-TCandTT-Topcansupportsystemsthatcanalways(x=100)andnever(x=0)toleratedeadlinemissesordegradedperformance,respectively.TheTT-TCalgorithmmaybeusefulfornonreal-timeensemblesystems.TheTT-TC*algorithmcanpotentiallybeusedforsoftreal-timeensemblesystemswhoseexpertresponsibilitiesarenothighlydynamicandhard-to-predict,asshowninthelastcontextcaseofTable 4-2 .Ontheotherhand,theTT-Topalgorithmismoresuitablethantheformertwoalgorithmsinschedulinghardreal-timeensemblesystemssinceitnevercausesoverheadthatcouldleadtodeadlinemisses. 4.4.2CorrectnessbasedonEnsembleExecutionPolicy Acycleisconsideredsatisfyingtheexecutionpolicyifthethreeexpertswiththelargestresponsibilities(i.e.,thek-winnerpolicywhenk=3)areexecuted.GiventhemappingdiscussedinSection 4.1.2 ,reasonableresources,andtheEPOCschedulingalgorithm,winnersareguaranteedbydesigntoexecuteineverycycle.Thisisthecaseinalloftheexperiments'results,notonlyconrmingthetheorybehindthedesignoftheEESmanagerbutalsovalidatingitsimplementationintheexperimentalsetup. 4.4.3CorrectnessbasedonEnsembleLearningPolicy Inspiteofacorrectimplementationoftheexecutionpolicy,systemoutputscanstillbeincorrectduetothecombinedeffectof(1)possibledeadlineviolationscausedbycritical-TCexecutionsand(2)inadequatepriorlearningbythewinnerexperts.Figure 4-10 showstheplotsoftheagent'sx-axispositionwhenresourcesareunlimitedversuswhenresourcesarelimited,and(a)schedulingtasksusingtheEESmanager,(b)schedulingonlywinnersand(c)schedulingwinnersandrandomly-chosennon-winners.Thesolidanddottedlinesrepresentthetrackingoftheagent'spositionsintheunlimited-andlimited-resourcecases,respectively.Theoutputsfrom(a)and(b)aresimilartothe 94

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unlimited-resourceoutput.However,whenanexpertthathashardlyeverbeenawinnerinpriorcyclesbecomesawinner,thedifferenceoflimited-andunlimited-resourceoutputsincase(b)canbeupto3timesofthedifferenceincase(a);thedegradationhappensbecausethewinnerhasnotlearnedenoughtoproduceagoodresult.Forthecase(c),althoughnon-winnerexpertsgotarandomchancetoexecuteandadapt,oneexpertcouldgetscheduledmuchmoreoftenthanothers.ThisbehaviordiffersfromwhentheEESmanagerisusedinwhichexpertsgetfairchancesinlearning;expertswhicharerarelyselectedforlearningcannotperformwellwhentheybecomewinners.Moreover,itispossiblethatanexpertlearnstoomuchknowledgethatisirrelevanttoitsassociatedcontextcluster;thismaycompromisetheexpert'sabilitytocreateaccurateoutputswhenthesystemoperateswithinitscontext. Inordertoverifythataccuracydegradationisminimal(whencomparedtotheensemblesystemwithunlimitedresources),Table 4-3 showstheaveragesandstandarddeviationsofthedistanceerrorsofthesystemoutputsinthetestcasesfromSection 4.4.1 .Anabsolutepercentageerror()isdenedasfollows: =j[d(pl,po))]TJ /F3 11.955 Tf 11.96 0 Td[(d(pu,po)]=d(pu,po)j100(4) wherepoisthepointoforiginandpuandplarethepredictednextagentpositionsfromtheunlimited-resourceandlimited-resourcesystems,respectively.Thevalued(p1,p2)=p (x1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2)2+(y1)]TJ /F3 11.955 Tf 11.96 0 Td[(y2)2istheEuclidiandistancebetweenpointsp1:(x1,y1)andp2:(x2,y2). Table 4-3 providesstatisticsoffrom50500-cycleexperimentsofeachalgorithm/contextcombination.Whenawinnermissesadeadlinebecauseofcritical-TCexecution,twopossiblechoicesareconsideredtocreateasystemoutputforaparticularcycle:oneusestheprevious-cycleoutputandtheotherusesthedelayedcurrent-cycleoutput.The`Previousoutput'columnsinthetablerepresenttheworst-casepossibleinwhichadeadlinemissalwaysoccurswhenacritical-TCexecutionisneededandthusthe 95

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previous-cycleoutputisused.The`Delayedoutput'columnsrepresentbest-caseresultswhenthelimited-resourcesystemcanaffordtowaitfordelayedoutputswhichhavethesameaccuracyaswhennodeadlinemissoccursduetoacritical-TCexecution. ForthebaselinealgorithmTT-TC,thevalueofcanbeashighas99percentintheworstcasesincecritical-TCexecutiontakesplaceineverycycle,andaslowaswithin0.5percentinthebestcase.TT-TC*hassignicantimprovementoverTT-TCfortheworstcaseineverycontexttypes(rangesfrom0.82%to39.18%)duetotheuseofpreviousandpredictedUiassignments.LikeTT-TC,TT-TC*producesalmostasaccurateoutputsasoutputsoftheunlimited-resourcesystemforthebestcase.ForTT-Top,critical-TCexecutionisneverneeded,soitalwaysachievesthebest-caseaccuracy,i.e.,systemoutputsarenearlyasaccurateastheunlimited-resourceoutputs(0.02%to0.45%)andthereisnoadditionaldelay.Notethatmostoftheseerrorsoccurinthecycleswhenexpertsarelearning1;theoutputerroriszeroaftertheweightsoftheexpertsconvergesincetherightexpertsalwaysgetscheduledaccordingtotheexecutionandlearningpolicies. TheaccuracyofTT-Topsuggeststhatwecouldpossiblyconsideraschedulingapproachthat,forbothexecutionandlearningpurposes,alwayspicksexpertswithtopresponsibilities(insteadofonlydoingsowhenacriticalTCexecutionisotherwiseneeded).However,experimentalevidenceshowsthat,forsomeensemblelearningpolicies,choosingonlytop-responsibilityexpertsforexecutiondoesnotgivethemostaccuratesystemoutputs.AnexampleexperimentconsidersasimplecasewiththreeexpertsandtwoprocessorsincontextCT1.Thecentroidsofexperts'responsibleregionsareshowninFigure 4-11 .Thetasks'timingparametersareCi=1,Uimax=1forallexperts,U2min=1,andUimin=0.2for1and3.Theensemble'sexecutionpolicy 1Intheseexperiments,noiseisperiodicallyinjectedintotheweightsoftheidealexpertstocapturetheneedfordynamiclearning 96

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iswinner-take-all(i.e.,thek-winnerpolicywhenk=1).Itisclearthatthesystemisalternatingmostlybetween1and2.Theresults(Figure 4-12 )showthattheaveragevalueforthecasewhenthetop-responsibilityheuristicisalwaysused(i.e.,TopResp)is0.24%withstandarddeviation=0.34%,whiletheaverageofTT-Topis0.03%withstandarddeviation=0.14%.Inthiscase,TT-Topprovidesabout8timesbetteraccuracythanTopResp.Thisperformancedeteriorationisgreaterforensemblesystemswithlargernumberofexperts.Systemswhoseperformanceisverysensitivetotheiroutputaccuracy(e.g.,BMIs)couldeasilyfailifthepuretop-responsibilityschedulingisused. 4.5Discussion WhiletheimplementationoftheEESmanagerpresentedinthischaptercanprovidegreatperformanceasshownintheperformanceevaluation,pointsforimprovementcanbestatedasfollows.TheTT-basedalgorithmsoptimizetheassignmentoftask-resource-utilizationdemandswiththeassumptionthatallparametersexcepttheresponsibilitiesarepreciselyknownwithoutanyuncertainty.Real-worldensemblesystemsaresubjecttovarioussourcesofuncertainty,whichcanimpactschedulingdecisions.Someexamplesofthesecomplexitiesincludeuncertaintask-executiontimes,imprecisesystems'timekeeping,thecostofmovinganexperttaskfromoneprocessortoanotherandnon-deterministicresultsoftaskexecutions(eithersuccessorfailure).Moreover,thedesiredgoaloftheoptimizationproblemforensembleschedulingisgenerallynotlimitedtominimizationofdifferencebetweendesiredandactualexecutionrates,butalsocombinedwithvariousotherobjectives.Examplesoftheseobjectivescouldbeminimizingdeadlinemisses,schedulingoverheadandswitchingcosts,maximizingcompletedtaskinstances,andoverallsystemutilityandfairness.ThecapabilityoftheEPOCschedulerinachievingtheactualresourceutilizationoftasksduringanallocationepoch,asassignedbytheTUA,highlydependsontheselectionofthelengthofschedulingepochs(i.e.,).FromCorollary 4.4.2 ,thenumberofschedulingepochs(i.e.,b)overlappingwithanallocationepochmustbesufcientlylarge(i.e., 97

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b9forguaranteeingthattaskscanachievenolessthan90%oftheassignedUi).Inpractice,canbesetaccordingtothenumericresolutionusedtorepresenttheutilizationvalues.If=10,thisimpliesthatanallocationepochshouldbeatleast90cyclesfortaskstoachievesufcientlyhighresourceutilization.Inhighlydynamicensemblesystemsinwhichcontextchangeshappenfrequentlyandexpertsrequirecontinuousandextensiveonlinelearning,anevenlargerandbmayberequired.Thismakesitraretoachieveclose-to-optimalutilizationssincethemultiplicationofandbisrelativelyhigh,whiletheallocationepochisquiteshort.Insuchcases,theperformanceprovidedbyEPOCmightnotbeacceptableduetothesuboptimalityofnon-winners'resourceutilization.Inaddition,whenthetaskWCETsaregreaterthanone,thenumberofpreemptionsandmigrationsintheEPOCschedulecanbehighbecauseoftheround-robin-likeschedulingwhenseveraltaskshaveaboutthesameUiorUimin. 4.6Summary ThischapterpresentsanimplementationoftheEESmanagertoaddresstheproblemofschedulingreal-timeensemblesystemswhoseresourcesareinsufcientfortheexecutionofallexpertsineverysinglecycle.Theobjectiveoftheproposedschemeistoexecuteexpertsdeemedmostimportantforoutputgenerationaccordingtoanensembleexecutionpolicyineverycyclewhileallowingotherexpertstoexecutelessfrequentlytoperformnecessaryadaptationcorrespondingtoanensemblelearningpolicy.Thisrequiresdeterminingoptimalresource-utilizationdemandsbysolvingasuccessionofoptimizationproblemsandcreatingafeasibleschedulebasedonthesedemandsineverycycle.Eachoftheseproblemsminimizesperformancelosssubjecttoresource-capacityconstraintsandsystem-specicconstraintsthatcaptureexecutionandlearningpoliciesofensemblesystems.Ourfocusisonabroadclassofensemblesystemswiththefollowingcharacteristics:relativeexpertperformancecanberankeddirectlyfromresponsibilities,k-winnerpoliciesareusedforsystemexecution,andminimumlearningfrequenciesmustbeguaranteedtoensurethatsystemoutputshave 98

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acceptablequality.However,extensionstootherclassesofensemblesystemsarealsodiscussed. TheamountofresourceutilizationdemandedbyensembletasksareadaptedbyanyoftwoO(N)algorithms,namelyTT-TC*andTT-Top.TheseresourcedemandsarethenusedbyanovelEPOCschedulertogeneratecorrespondingfeasibleschedules.ExperimentalevaluationconrmsthattheEESmanagerimplementingTT-basedapproachesandEPOCcanbeusedtodeployensemblesystemswithlimitedresources(butinareasonableamount)andachievenegligibledegradationinperformance(i.e.,averageerrorsneverexceed0.5%forTT-Top)relativetousingunlimitedresources.Inaddition,theproposedTUAimplementationcangreatlyreduceoverheadcausedbyreoptimization(upto85%forTT-TC*and100%forTT-Top)throughtheuseofaresponsibilitypredictorandsensitivityanalysis.Giventheuseofaresponsibilitypredictor,ensemblesystemsfordifferentapplicationswillperformdifferently.Thevariationsinperformancearecapturedbyincludingvecontexttypesdifferinginthedegreetowhichresponsibilitiescanbepredicted,rangingfromsystemswithpredictable-and-progressivetoarbitrarilychangingdynamics.Theevaluationresultsshowthatevenforthemostchallengingcase,TT-TC*andTT-Topprovidesignicantimprovementoverthebaselineapproach(TT-TC)whichalwaysschedulestherightexpertsbutmissesapercentageofdeadlines.Differentensemblesystemsmayalsodifferinhowlearningtakesplace(e.g.,inwhatandwhendatainstancesarerequired).However,althoughmodicationsmayberequiredinwhatconstitutesalearningtask,theydonotinvalidatetheproposedapproach. 99

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Figure4-1. Theformulationoftheensembleschedulingproblem.Theadaptationoftaskresourcedemandscanbeviewedasasequenceofmathematicalprogrammingproblemswithdifferentvaluesofwiforcyclesstartingatt,t+1,...,t+n. Figure4-2. CombinedowchartofthreeTask-Throttling-basedalgorithms.TT-TC(usedasabaselineapproachforcomparisonagainsttheproposedapproaches),TT-TC*(oneproposedapproach)andTT-Top(anotherproposedapproach).Ineverycyclet,givencurrentvaluesofresponsibilitiesR(t),eachofthesealgorithmscomputesautilizationallocation(superscript`opt'indicateswhensolutionisoptimal)usingadifferentmethodandusestheEPOCschedulerEPOC(U(t))toderiveaschedule.TC(P(w(t))andTopResp(R(t))standforsolutionsofP(w(t))usingTCandthetop-responsibilityheuristic,respectively.Subscriptspredandprevidentifypredictedvaluesandvaluesfromapreviouscycle,respectively.InthecaseofTT-TC*andTT-Top,previousorpredictedvaluesofUoptmaybeoptimalforcyclet,inwhichcasenewsolutionstoP(w(t))neednotbefound.ThecomputationofthosevaluestakesplaceoutsidethecriticalpathforcomputingthescheduleEPOC(U(t)). 100

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Figure4-3. AnexampleofEPOCschedulingwhenallocationepochscoincidewithschedulingepochs. Figure4-4. AnexampleofEPOCschedulingwhenallocationepochspartiallyoverlapwithschedulingepochs. Figure4-5. Diagramsofhowdifferentexpertsindifferentcontextsmapcommandsintonextpositionsoftheagent. 101

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Figure4-6. Testensemblesystemforcontrollingthemovementoftheagent. 102

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Figure4-7. Contextualvalues(y-axis)of500cycles(x-axis)forcontexttypes1through5(toptobottom).Thecirclesshowexpertcentroidsclusteredalongy-axisvalues. 103

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Figure4-8. Responsibilitiesofexperts(y-axis)forcontextualsignalsinFigure 4-7 .Only4outof10expertsandonlytherst200outof500cycles(x-axis)areshowntoavoidclutteringtheplots. 104

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Figure4-9. Taskcompression(TC)savingsduetotheuseofsensitivityanalysisheuristic.Whenresponsibilitieschange,thesensitivityanalysischeckstheoptimalityoftheprevious-cyclescheduleonly(Prev),withthepredictedscheduleonly(Pred)andwithbothpre-computedschedules(Both). 105

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Figure4-10. Comparisonofx-axisoutputstodemonstratetheimportanceoflearningpolicyinensemblescheduling.Trackingofnalsystemoutputsareshownforresource-limitedcases(dottedlines),when(a)schedulingbytheEESmanager,(b)schedulingonlywinnersand(c)schedulingwinnersandarbitrarily-chosennon-winners,versustheunlimited-resourcecase(solidlines). 106

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Figure4-11. ResponsibleregionsofexpertsfortheexperimentwhoseresultsareshowninFigure 4-12 Figure4-12. Absolutepercentageerrorofthelimited-resourcetestensemblesystem.ExpertsareselectedforexecutionusingtheTT-Topalgorithmversususingonlytop-responsibilitycriteria. 107

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Table4-1. DenitionofvariablesusedinSection 4.2 VariablesDescriptionsValuesandConstraints NNo.ofexpertsintheensemblesystem kNo.ofwinnersintheexecutionpolicy NWNo.ofnon-winnersNW=N)]TJ /F26 10.909 Tf 10.91 0 Td[(k MkTotalresourceutilizationneededbywinnersMk=k minMNWMinimumutilizationneededbynon-winnersminMNW=dNWmax(Uimin)e MNWTotalresourcecapacityavailablefornon-winnersMNWminMNW MrReasonableresourceMr=Mk+minMNW MNo.ofavailableresources(theunitofresourcesis1processor)Ud=min(schedulerLUB,M) schedulerLUBLowestachievableutilizationboundofthealgorithmimple-mentedbytheRTSschedulerLUBM Lowerboundofthelengthofaschedulingepoch=dNW=MNWe=dNW=(M)]TJ /F26 10.909 Tf 10.91 0 Td[(Mk)e (a)Lengthofaschedulingepochstartingatcyclea(a)=LCDi=1,...,N(Vi(a)) 108

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Table4-2. Critical-TCoverheads(i.e.,percentagesofcycleswithcritical-TCexecutions)intheTT-TC,TT-TC*andTT-Topalgorithms. Critical-TCoverheads TT-TCTT-TC*TT-Top AverageStandarddeviationAverageStandarddeviationAverageStandarddeviation CT1100.000.0015.400.000.000.00 CT2100.000.0024.480.290.000.00 CT3100.000.0054.1323.390.000.00 CT4100.000.0063.396.530.000.00 CT5100.000.0089.701.800.000.00 109

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Table4-3. Statisticsoftheabsolutepercentageerror()ofoutputsofthetestsystemwithlimitedresourcesrelativetooutputsusingunlimitedresourceswhenschedulingwiththeTT-TC,TT-TC*andTT-Topalgorithms. TT-TCTT-TC*TT-Top PreviousoutputDelayedoutputPreviousoutputDelayedoutput Avg.MedianAvg.MedianAvg.MedianAvg.MedianAvg.Median (Stdev.)(MAD)(Stdev.)(MAD)(Stdev.)(MAD)(Stdev.)(MAD)(Stdev.)(MAD) CT196.85(6.24)98.80(2.40)0.29(0.78)0.07(0.29)17.08(23.35)7.48(14.52)0.29(0.78)0.07(0.29)0.29(0.80)0.07(0.26) CT296.25(6.24)98.31(2.31)0.45(1.25)0.13(0.42)3.00(7.73)0.82(2.73)0.46(1.25)0.13(0.42)0.45(1.24)0.10(0.37) CT397.52(5.89)98.99(1.79)0.19(0.67)0.02(0.18)21.61(26.46)4.77(18.31)0.19(0.67)0.02(0.18)0.21(0.70)0.02(0.20) CT495.91(7.10)98.04(2.95)0.33(1.31)0.04(0.40)15.20(24.34)5.90(13.31)0.33(1.32)0.04(0.41)0.35(1.34)0.03(0.31) CT596.51(5.85)98.54(2.32)0.48(1.54)0.13(0.48)39.18(23.75)35.36(17.64)0.50(1.67)0.13(0.48)0.40(1.02)0.11(0.38) 110

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CHAPTER5ENHANCEDADAPTIVEREAL-TIMETASKSCHEDULER ThischapterconsidersadesignandimplementationofageneralizedandimprovedschedulingalgorithmoftheEESmanager'sRTSforgeneralizedensemblesystemsandforaddressingtheissuesoftheEPOCscheduler,i.e.,itssuboptimalityconditionandschedulingoverheads,mentionedinChapter 4 .ToachievethedesiredRTSimplementation,theEfcient-Approximation-of-Gradual-Load-Execution(EAGLE)SchedulingAlgorithmanditsextensionfortransitionaltasksareproposedinthischaptertotackletwochallengingsubproblems:(1)howstatictasksetscanbeefcientlyandoptimallyscheduledonmultiprocessorsand(2)whenconsideringadaptivetasksets,howonlineparameterchangescanbeperformedresponsivelyandsafely. Section 5.1 formulatestheproblemofschedulingstaticandmulti-modetasksetsonmultiprocessors.Knownanomaliesinreal-timemultiprocessorschedulingthatcannotbehandledbyexistingsub-optimal-but-efcientmultiprocessorschedulingalgorithmsarestudiedinSection 5.2 .Existingmodelsforoptimalreal-timeschedulingonmultiprocessorsarereviewedinSection 5.3 .UsingtheinsightslearnedfromSections 5.2 and 5.3 ,areal-timemultiprocessorschedulingalgorithm,calledEAGLE,isproposedtoaddresstherstsubproblem.Section 5.4 presentsthedesignandimplementationoftheEAGLEalgorithm.Itsoptimalityforstatictasksetsisvalidatedusingtherulesfromtwoknownmodelsofoptimalmultiprocessorscheduling.Section 5.5 presentstheeffectivenessevaluationofEAGLEagainstotherexistingmultiprocessoralgorithmsthroughsimulationstudies.Byconsideringtheimpactofmodechanges(i.e.,changesinperiodsanddemandedresourceutilization)onschedulabilityofEAGLEinSection 5.6 ,asetofconditionscanbederivedtoadjustresource-to-taskallocationswithoutcausingtheresultingscheduletobecomeinfeasible.Anovelmode-transitionprocedurederivedfromtheseconditionsextendstheEAGLEalgorithmtoanewscheduler(calledEAGLE-T)thatcanaddressthesecondsubproblem,i.e.,supportinganew 111

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multi-modetaskmodel.EAGLE-TisexplainedinSection 5.7 .Themulti-modetaskmodelacceptsprogressiveresourceadaptationinsteadofthestep-wiseapproachesdescribedinrelatedwork.Theupperboundofthedelayinmodetransitionandthedifferencebetweenresourceutilizationachievedbytheproposedalgorithmandtheidealschedulerareprovided.PerformanceevaluationinSection 5.8 showsthatprogressiveresourceallocationadaptationoutperformsthestep-wisecounterpartofEAGLE(calledEAGLE-S).Finally,conclusionsoftheworkpresentedinthischapterareprovidedinSection 5.9 5.1ProblemFormulation Thissectionpresentsageneralizedmodelofanadaptivereal-timesystemwithNtasks(1toN)andMprocessors(1toM).Specicationofeachreal-timetaskcanbeexplainedsimilarlytotheoneprovidedinChapter 3 .Alongwiththeexplanationofthesystemmodel,adescriptionofhowthisgeneralizationcanbemappedtothecaseofensemblesystemsisalsoprovided.Then,theproblemofinterestisstatedattheendofthesection. Inadaptivereal-timesystems,thevaluesofresourceutilizationrequiredbytasksmayvaryfromonemodetoanotherbuttheymustliewithinthespeciedinterval[Uimin,Uimax].TherequiredutilizationoftaskUiinamodemisrepresentedbyUmi.PmiisthesmallestintegerthatUmiPmi=Cmiisalsoaninteger.Inanyparticularmodem,iisinactiveifUmi=0andactiveotherwise.Thetotaldesiredutilizationrequiredbyalltasksineachmodecannotexceedtheavailableresourcecapacityofthesystem(i.e.,PNi=1UmiM).Otherwise,itisassumedthatthereexistsanadditionalcomponentcapableofadjustingtasks'utilizationdemandsproportionallytomeetthisconstraint.Atthesystem'sstarttimeto,Vi(to)issetequaltothevalueofUoiwhere`o'referstotheoriginalmodeofthesystem.Duetochangesinthesystem'soperatingconditions,amode-changerequest(MCR)mayoccuratanytimeandtriggeratransitionfromonemodeintoanothermodem.AmodetransitionstartsattheinstantthattheMCR 112

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occurs,ortMCR,andendsatsometimetsuchthatVi(t)=UmiforalliforthersttimeaftertMCR.ApropermodetransitionfromamodextothemodemresultsinchangingallocatedresourceutilizationofalltasksVi(t)fromUxitoUmiwithoutviolatingtasks'utilizationrangerequirementsandcausinganytasktomissitsdeadline.IftheMCRoccursduringapreviouslyinitiatedmodetransition,thepreviousMCRisabortedandtreatedasifithadnotoccurred. Theparticularinterestofthisworkisinensemblesystems,whichisrealisticadaptivesystemswithanunboundednumberofoperatingmodes,non-deterministicMCRtimesanddynamictaskutilizationvaluesUmiineachmode(i.e.,theirvaluesareonlyknownatthetimewhenanMCRoccurs).Insuchsystems,Umiaredeterminedatruntimebasedontheexpertresponsibilitiesassignedbyagatingcomponent.Hence,thetotalutilizationdemandisalwayswithintheavailableresourcecapacity.Insuchadaptivereal-timesystems,itisessentialtoderiveanefcientproceduretoensurepropertransitionacrossmodechangesinwhichalldeadlinesmustbemetnotonlyintheindividualoperatingmodes,butalsoduringthetransitionsbetweenmodes.Furthermore,thetransitionmustyieldboundeddelayanddegradationofsystemperformance(i.e.,attainedutilizationoftasksinthesystem). Totacklethisproblem,studiesaboutanomaliesinmultiprocessorschedulingandmodelsforoptimalmultiprocessorschedulingarepresentedinSections 5.2 and 5.3 .Usingtheacquiredknowledge,abasealgorithmEAGLEisdevisedforstatictasksets(i.e.,typicalsingle-modereal-timesystems).Thenthealgorithmisextendedformulti-modesystemsusingaproposedmode-transitionprotocolthroughschedulabilityanalysis. 5.2AnomaliesinMultiprocessorScheduling EversincethediscoveryofsuboptimalityoftheEarliestDeadlineFirst(EDF)schedulerinthemultiprocessorcase,manyalternativealgorithmshavebeenproposedtoimprovetheworst-caseresourceutilization.Theycanbeclassiedintopartitioning 113

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andglobalapproaches.Inpartitioningscheduling,alltheinstancesofataskareexecutedonthesameprocessor.Incontrast,inaglobalschemeataskcanmigratefromoneprocessortoanotherduringitsexecution.Pfair[ 14 ]andLLREF[ 39 ]areoptimalglobalalgorithmsthatcanachievethefulluseofprocessortimewhileguaranteeingthatalltasksmeetdeadlines.However,theiroptimalitycomeswithhighruntimeoverheads(i.e.,preemptions,migrationsandschedulerinvocations)whichmakethemunattractiveforrealimplementation.Alternatively,severalsuboptimalbutefcientglobalalgorithms,suchasEarliestDeadlineFirstuntilZeroLaxity(EDZL)[ 94 ]andAnticipatingSlackEDZL(ASEDZL)[ 53 ],wereproposedduetotheirlowcomplexityandoverhead.Inordertodeviseanewefcientschedulingalgorithmwithimprovingschedulabilitytowardoptimality,itisnecessarytounderstandwhythesealgorithmsfailtoachievetheiroptimality. 5.2.1Dhall'sEffects Dhalletal.[ 46 ]statethatthetotalutilizationoftasksetsunschedulablebytherate-monotonic,deadline-monotonicandearliest-deadline-rstschedulingalgorithmscanbearbitrarilycloseto1regardlessofthenumberofavailableprocessorsusedtoschedulethem.TheyconsiderM+1periodictasksthatshouldbeschedulableonMprocessors.TherstMtasksi(where1iM)havePi=1,Ci=2andthelasttaskM+1hasPM+1=1+,CM+1=1,where!0. Thetotalutilizationofthistasksetgetscloserto1asgetscloserto0.Figure 5-1 showstheglobalEDFscheduleofthiskindoflow-utilizationtasksetswithM=2.Atthetime1+,3missesitsdeadline,sothetasksetisunschedulable.Thisproblemiscausedbytheexistenceofatleastonetaskwithrelativelyhighutilizationandlowprioritywhencomparedtoothertasks[ 44 ]. Manyexistingalgorithmsincludestaticprioritypromotionforhigh-utilizationtasksintasksetstocircumventthisanomalyandimproveschedulability[ 20 62 ].EDZL,oneofthesealgorithms,promotescriticaltasks(i.e.,tasksthatcannottolerateanyblocking)to 114

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havethehighestpriority[ 94 ]andusesEDFpriorityforothertasks.Ataskisconsideredcriticalwhenitsruntimelaxity(i.e.,theremainingtimetoitsdeadlineminusitsremainingcomputationtime)reacheszero.Figure 5-2 showsasuccessfulscheduleoftheabovetaskseton2processorswhenscheduledwithEDZL.EDZLisknowntobeatleastaseffectiveasEDFwhilethenumberofschedulerinvocationsisnotfarbeyondEDF.ManytheoreticalandsimulationresultsdemonstratethatEDZLcanimprovethetotalutilizationboundofEDFonMprocessorsfromM(1)]TJ /F3 11.955 Tf 12.19 0 Td[(Umax)+Umax,whereUmax1isthemaximumutilizationofeveryindividualtask,to0.6321M. 5.2.2CumulativeOfoadingFactorEffects Letdeneanofoadingfactorofataskiastheratiooftasklaxitydividedbytaskperiod,i.e.,apercentageofprocessortimethatcouldbeusedtoexecutetasksotherthaniwithoutaffectingthetimelinessofi.ForanytasksetthathasU=M,ithasbeenshownin[ 53 ]thatifawaitingtaskwhasutilizationequaltothecumulativevalueofofoadingfactorsofMexecutingtasksxandPw>Pxforallx,thenatleastoneprocessorwillbecomeidlethiscausestaskstomisstheirdeadlinesintheirsubsequentperiods.OneexampleinwhichT=f1:=[2,3],2:=[2,3],3:=[4,6]gandM=2isillustratedinFigure 5-3 .Fromthisexample,wehavethesummationoftheofoadingfactorsof1and2equaltoU3(i.e.,1=3+1=3=4=6).Asaresult,oneprocessorisidleatt=3,whichtriggersadeadlinemissby2initssecondperiod.TheglobalEDFandEDZLcannotsuccessfullyschedulethetaskset,asshowninFigure 5-3 .ThereexistsatimeinstantinwhichmorethanMtasksbecomecriticaltasks.ASEDZL[ 53 ]aimstowardsoptimalitybyextendingEDZLtosolvethisissue.ASEDZLplanssomeexecutiontimeforMearliest-deadlinetasksbetweenthereleasesoftwoconsecutivetaskinstances.ThetotalslacktimeofthechosenMEDFtasksisanticipatedandASEDZLpromotesprioritiesofsomeremainingtasksbasedontheirdeadlinesbyallowingthemtofullyutilizetheremainingslacktime.Duringeachinterval,tasksareselectedinthefollowingorder:zerolaxitytasksfollowedbyearliest-deadline 115

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taskswithnon-zeroplannedexecutiontime.ASEDZL'sscheduleisillustratedinFigure 5-4 5.2.3SequentialTaskModelEffects AsstatedinChapter 3 ,anygiventaskisrequiredtoexecuteatanygiventimeinatmostasingleprocessoralthoughseveralprocessorsmaybefreeatthattime.Unfortunately,thissequentialtaskmodelcomplicatesmultiprocessorschedulingandisamajorcausewhysomeexistingalgorithmsareunabletoschedulesometasksetssuccessfully,includingtheanomaliesinSection 5.2.2 and 5.2.3 .Figure 5-5 illustratesoneadditionalsituationwhenEDF,EDZLandASEDZLallfailtoscheduleatasksetT=f1:=[12,12],2:=[2,3],3:=[3,6],4:=[10,12]gon3processors.Sincethetotalutilizationofthetasksetis3,allprocessorsshouldbekeptbusyatalltime;howeveritturnsoutthatduringt=5to6,oneprocessorisleftidle.Although4hasremainingcomputationtime(=7)atthattime,itcannotbeexecutedontwoprocessorssimultaneously.Asaresult,therearemorethanMcriticaltasksinthesubsequentperiod.Inthisexample,the4thjobof2missesitsdeadline. Amongthethreealgorithms,ASEDZListheclosesttotheproposedapproachtobepresentednext,anditisclaimedtobeanoptimalalgorithm.However,thisexampleshowsitsinfeasibilityforsometasksetsandidentiesthemisstepinitspreviouslyproposedoptimalityproofinSection 5.4.2 .SinceEDZLandASEDZL(andotherEDF-basedalgorithms)relymainlyontaskdeadlineinformation,theyareunabletohandlewellmanyanomalousscenariosduetothesequentialtaskmodel.Itisnecessarytotakeintoaccountinformationaboutboththedeadlinesandthetaskloadtomitigateinfeasibilitycausedbythesequentialtaskmodeleffect. 5.3ModelsforOptimalMultiprocessorScheduling Fromstudyingsequentialtaskmodeleffects,itisobservablethatadeadlinemisslikelyoccursifremainingcomputationoftasksaregreatlyimbalanced.Usingthisinsight,anextensionoftheASEDZLalgorithmcalledNABLRwaspreviouslyproposed 116

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in[ 123 ].NABLRshowsgreatimprovementintermsofschedulabilityfromthoseofEDZLandASEDZL.ThissectionsummarizesimportantconceptsfromastudyoftheFluidschedulingmodelandtworecently-proposedmodels,T-Lplane-based[ 40 ]andDP-FAIRscheduling[ 97 ],thatgivefurtherunderstandinginoptimalmultiprocessorscheduling.Withtheknowledgeacquiredfromthestudyoftheseschedulingmodels,abasealgorithm,calledEfcientApproximationofGradualLoadExecution(orEAGLE),presentedinSection 5.4 canbedevelopedtoaddresstherstsubproblem(i.e.,howtodeviseanefcientandoptimalschedulerforschedulingstatictasksetsonmultiprocessors). 5.3.1FluidSchedule Intheuidschedulingmodel,eachtaskexecutesataconstantratesimilartoitsutilizationdemands[ 138 ]asshownintheFluidschedulingratecurve(i.e.,)]TJ /F3 11.955 Tf 9.29 0 Td[(Ui)inFigure 5-6 .Theuidscheduleallowstaskstoproportionallymakeprogresstowardstheirdeadlines.Schedulingdecisionscanonlybemadeaftereachcertainperiodoftime(i.e.,thegranularityoftimeimplementedbythesystem).When!0,itmayappearasifataskconstantlyexecutesusingafractionofaprocessorequivalenttoitsutilizationdemand.Asaresult,giventhattasksdonotexecutemorethantheirrequestedutilizationdemandsandtheirtotalutilizationdemandiswithintheavailableresourcecapacity,alljobscancompletetheircomputationbydeadlines.Inpracticalsystems,however,0andtheuidschedulemustbeapproximated. 5.3.2T-LPlane-basedSchedulingModel T-Lplane-basedschedulingmodelviewsschedulingformultiprocessorsasschedulingonrepeatedlyoccurringTimeandLocal-execution-time(abbreviatedasT-L)planes.FeasiblyschedulingonasingleT-LplaneresultsinanoptimalscheduleforallT-Lplanesacrosstime.TheT-Lplaneprovidesavisualmodelfortaskexecutionbehavioronmultiprocessors.Figure 5-7 (a)showsanexampleofNtasks.Betweenanytwoconsecutiveschedulingevents,sayst1andt2,aT-Lplanecanbeestablishedand 117

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representedasarightisoscelestriangle.TheT-Lplanesforalltasksbetweent1andt2areofthesamesize.Whenoverlappingthesetrianglesoveroneanother,itcanbeviewedasonetrianglewithtasks'uidschedulelinesinsideasshowninthebottomtriangles,TLkandTLk+1wherekincrementsovertime.Ascanbeseenfromthegure,thesizesofT-Lplanesforadifferentkarenotnecessarilyequal. Figure 5-7 (b)detailsasingleT-Lplane.Eachblackcirclerepresentsatasktoken,whichdenotesthecurrentstatusofatask.Thex-axisoftheT-Lplaneindicatesprogressionintime,whilethey-axisrepresentstheremainingcomputationtimeoftasksthatmustbecompletedbytheendoftheT-Lplane(calledlocalexecutiontime).WhenataskisexecutedduringtheT-Lplane,itstokenmovesverticallytowardthex-axis.Theeventthatatokenhitsthex-axisiscalledabottom-hittingeventorEvent-B.Thismeansthetaskcompleteditslocalexecutiontime.Theverticaldistancefromthezero-laxitydiagonallinetoeachtokendenotesthelocallaxityofeachtasklli(t).LocallaxityofataskisdenedastheendoftheT-Lplaneminusestheremaininglocalexecutiontimeofthetask.Whenataskisnotselectedforexecution,itstokenmoveshorizontallytowardthezero-laxitydiagonalline.Whenthetokenhitstheline,i.e.,lli=0,aceiling-hittingevent(orEvent-C)occurs.Anactivetokenchangestoinactivewhenitliesonthebottomoftheplane.Ifalltokensbecomeinactivebythetimetk+1,theyarelocallyfeasible.WhenalltokensarelocallyfeasibleforeachT-Lplane,alltaskscanbescheduledsuccessfullythroughoutallT-Lplanesacrosstime. T-Lplane-basedschedulingconsistsoftwophases:localparameterdecision(LD-P)andlocalscheduling(LS-P)phases.LD-PestablishesaT-LplanebetweentwoconsecutivejobdeadlinesanddeterminesappropriatelocalexecutiontimeoftasksduringtheestablishedT-Lplane.LS-Plocallyschedulestasksaccordingtotheassignedexecutiontimesothatalltokensarelocallyfeasible.InordertodesignaschedulingalgorithmfortheLS-Pphase,guidelineforlocally-feasiblescheduling(GLS)were 118

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proposedtoprovidesufcientconditionsforlocalfeasibility[ 40 ].Theseconditionscanbestatedasfollows. 1. Ateveryschedulingevent,uptoMactivetokensshouldbeselected. 2. AtEvent-C,thetokeninvokingtheeventshouldbeselectedimmediately. 3. AtEvent-B,thetokeninvokingtheeventshouldnotbeselectedifthereareatleastMactivetokens. 5.3.3DP-FAIRSchedulingModel DP-FAIRschedulingmodelprovidesaunifytheoryforalgorithmsusingthedeadline-partitioningtechnique.Deadlinepartitioning(DP)sub-dividestimeintosliceswherealltasksareallocatedalocalworkloadtobedonewithinthetimesliceandtheyhavethesamedeadline.DPreliesonallocatinglocalworkloadforalltasksforeachtimesliceandschedulingwithinatimeslice.AnalgorithmusingDPapproachisconsideredoptimalorDP-CORRECTwhenthefollowingconditionsaresatised: thetimesliceschedulermustexecutealljobs'localworkloadsbytheendofthetimeslice. jobsareallocatedworkloadsforeachslicesuchthatitispossibletocompletetheworkwithinthesliceandcompletionofalltheseworkloadsleadtonodeadlinemiss. Aminimallyrestrictivesetofschedulingrules,calledDP-FAIR,isproposedtoguaranteethatanalgorithmisDP-CORRECT.DP-FAIRschedulingconditionscanbestatedasfollows. 1. Alwaysrunajobwithzerolocallaxity. 2. Neverrunajobwithnoremaininglocalwork. 3. Neverallowaprocessortoidleiftherearetaskswaitingtoexecute. DP-FAIRscheduling'sconceptiscomparabletoT-Lplane-basedscheduling.AtimesliceinDP-FAIRisequivalenttoaT-Lplane.Besides,theDP-FAIRconditionscanbedirectlymappedintotheGLS'srequirementsstatedintheprevioussection.LaterinSection 5.4.6 ,theoptimalityoftheEAGLEalgorithmisprovedusingbothmodels. 119

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5.4EfcientApproximationofGradualLoadExecution(EAGLE)SchedulingAlgorithm Figure 5-8 showstheowdiagramofthebaseschedulingalgorithm,calledEfcientApproximationofGradualLoadExecutionorEAGLE.Itutilizesthreedatastructuresforschedulingtasks:TaskInfo,ReadyQueueandProcList.TaskInfoisamatrixstoringstateinformationofalltasks(i.e.,eachrowrepresentsthestateofeachtask)ateachschedulingpoint.Aschedulingpointreferstoaneventwhenaschedulerisinvoked.Thiscorrespondstowheneithersomejobarrives,nishesitscomputation,orbecomescritical(i.e.,itslaxitybecomeszero).Thetimeinstantofthesthschedulingpointisdenotedasts.Followingtheterminologyin[ 40 ],eachschedulingintervaliscalledaT-Lplane(denotedasTL).AT-Lplaneisdenedasaninterval[bk,bk+1)wherek0andb0=0.Foreachk>0,bk=mint>bk)]TJ /F31 5.978 Tf 5.75 0 Td[(1fdi,j(t)ji2T,j>0g.ExamplesofT-LplanesofatasksetareshowninFigure 5-9 (a). Thestateofataskiatthesthschedulingpointincludesthreevectors:i(ts),0i(ts)and#i(ts).Forbrevity,siswrittenwhenreferringtots,e.g.,i(s)referstoi(ts).Thevectori(s)consistsoftheremainingtaskcomputationtimeci(s)andremainingtimetothetaskabsolutedeadlinedi(s).Thevector0i(s)hassimilardenitionofparametersasi(s)butonlyconsideredforthedurationofaT-Lplane:lci(s)istheremaininglocalexecutiontimeofiwithintheT-Lplaneandldi(s)istheremainingtimeuntiltheendoftheT-Lplane.Thevector#i(s)containsaagXi(s)indicatingwhetherthetaskiscurrentlyexecuting(=1)ornot(=0)andtheindexoftheprocessortowhichthetaskwasassigned(}i(s)2[0,M]).Thehyper-period(denotedasHPT)ofthetasksetequalsLCMi=1,...,N(Pi).ThemeaningsoftheseparametersareillustratedinFigure 5-9 (b).ReadyQueueisasortedlistofactivetaskinstancesindecreasingorderoftheirutilizationdemands.Taskswithlowertaskindiceswiniftiesarise.Taskinstancesthatwerereleasedbuthavenotyetcompletedtheircomputationarecalled'active'.EachtaskinstanceisaddedintoReadyQueueatitsreleasetimeandremovedfromthequeue 120

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whenitscomputationnishes.Thelastdatastructure,calledProcList,keepstrackofwhichtaskiscurrentlyrunningoneachprocessor.Sinceeachtaskcanhaveonlyonejobactiveatanytime,ajobnumbersubscriptinalloftheseparametersmaybeomitted.Thetimeindexmayalsobeleftoutifitisclearfromthecontext. 5.4.1OverviewofEAGLE'sSchedulingFlow Whenthesystemstartsrunning,i.e.,t=0,ProcListentriesareallinitializedtozerosmeaningthatallprocessorsareoriginallyidle.Therstsetoftaskinstancesarrive(ri0=0;8i)andtheyareaddedintoReadyQueue.Everyjobarrivalassignsci=Ci,di=ri0+Pi.Taskruntimelaxityli(s)=di(s))]TJ /F3 11.955 Tf 12.33 0 Td[(ci(s).TheinitialvaluesofallXiand}iarezerosmeaningthatnotaskisinitiallyexecutingandassignedtoanyprocessor.Ateachschedulingpoint,theschedulerisinvokedviathesystem'sinterruptforselectingappropriatetaskstoexecute.Iftheschedulingpointisduetoajobarrival(i.e.,thebeginningofanewT-Lplane)thenaroutinefordetermininglocaltaskexecution(calledLD-PanddiscussedinSection 5.4.2 )assignsvaluesofldiandlcitoensurethatalltasksprogresstowardtheircompletionratherfairly.Then,Mreadytaskswithnon-zeroremaininglocalcomputationtime(i.e.,lci>0)ortasksthathavezerolocalruntimelaxity(i.e.,lli=ldi)]TJ /F3 11.955 Tf 12.36 0 Td[(lci=0)areselectedforexecution(usingtheLS-ProutinediscussedinSection 5.4.3 ).EAGLEassignstaskstoprocessorssothattheassignmentincursasfewtaskmigrationsaspossible(discussedinSection 5.4.4 ).Thepseudo-codesforbothLD-PandLS-PareprovidedinFigure 5-10 .Thechosentasksexecuteuntilthefollowingschedulingpoint.Aftertaskexecution,alldatastructuresareupdated.Ifaninstanceofihasfullyexecuted,XiandProcList(}i)areresettozerosandtheinstanceisremovedfromtheReadyQueue.Thesamestepsrepeatateveryschedulingpoint. 5.4.2AssignmentofTaskParametersforLocalExecution LD-Pdeterminesparametersforlocaltaskexecution(i.e.,executionduringeachT-Lplane)ateachschedulingpointthatcoincideswithT-Lplaneboundaries.TheroutineusesasimilarmechanismtotheoneusedinBoundaryFair(BF)scheduling[ 165 ]. 121

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RatherthanguaranteeingfairnessineverytimeunitlikePfair,BFenforcesfairnessonlyattaskperiodboundaries(i.e.,T-Lplaneboundaries).ItisshownthatBFisfairenoughanditcanachieveoptimality,whilereducingupto75%ofschedulingpoints,whichresultsinmuchlessoverallschedulingoverheads(i.e.,preemptionsandmigrations)thanthoseofotherproportionalfairalgorithms[ 165 ].TheessenceoftheLD-Pissummarizedasfollows. Atanyperiodboundarybk(k=0,1,...),theLD-ProutineestablishesaT-Lplaneanddeterminestasks'localparametersfortimesintheinterval[bk,bk+1).TheamountofavailableexecutiontimeunitsfortheT-Lplaneequalstheproductofthesizeoftheplane(i.e.,bk+1)]TJ /F3 11.955 Tf 12.64 0 Td[(bk)bythenumberofprocessors(i.e.,M).Thelagofiattimet,orlagi(t),isdenedasthedifferencebetweentheexecutiontimeallocatedtothetaskaccordingtouidschedulingandtheactualtimeallocatedtothetask[ 14 ],i.e.,Vi(t)(t)]TJ /F3 11.955 Tf 12.2 0 Td[(ri,j))]TJ /F3 11.955 Tf 12.2 0 Td[(ei,j(t)whereri,jandei,jarethereleasetimeandtheactualexecutiontimeuntiltimetofthelatestjobofi.Zerolagindicatesthatthetaskgetspreciseresourceutilization,whilenegativeandpositivelagsmeanthatthetaskover-utilizesandunder-utilizesprocessortime,respectively.Sincestatictasksareconsideredandatanytimeonejobexistsforagiventask,theindextmaybeomittedfromanyparameters,e.g.,ViisusedinsteadofVi(t),andthesubscriptjofanyjob-relatedparameters(suchasri,j,ei,janddi,j)maybedroppedwhenitisclearfromthecontextthattheirlatestvaluesarereferred. Forbrevity,lagkiisusedtodenotelagi(bk),whichcanbeconsideredastheremainingworkfromthepreviousT-Lplane,i.e.,[bk)]TJ /F7 7.97 Tf 6.58 0 Td[(1,bk).Toensurefairnessatthenextboundary(i.e.,lagk+1i1),theLD-Proutinerstallocatesmandatoryexecutionunitsforeachtask(line3inFigure 5-10 ).Thetasks'mandatoryworks(denotedasmwk+1i)areintegerunitsofexecutiontimethatmustbeaccomplishedbyeachtaskbytimebk+1andtheircorrespondingdecimalpartsarecalledpendingworks(denotedaspwk+1i).Taskswhosependingworksarepositivenumbersandmwk+1i<(bk+1)]TJ /F3 11.955 Tf 12.08 0 Td[(bk)are 122

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eligibletoreceiveanadditionaltimeunitifthereareremainingunusedtimeunits(i.e.,RU>0). Eachtaskhasitscharacteristicstringattimebk:(i,k)=k+1(i)k+2(i)...k+n(i)whereeachcharacterk(i)2f`+',`0',`-'gisdenedas[bk+1Vi)-170(bbkVic)]TJ /F6 11.955 Tf 18.7 0 Td[((bk+1)]TJ /F3 11.955 Tf 11.34 0 Td[(bk)]andnisthesmallestintegerthatisgreaterthanorequalto1andk+n(i)6=`+'.Basedonthetaskprioritiesdeterminedfromthesecharacteristicstrings,thetopRUeligibletasks(denotedasOTinline7ofFigure 5-10 )areselectedtoreceiveanoptionalunit(i.e.,owk+1i=1).Therelativeprioritiesofanytwotaskscanbedeterminedbycomparingtheircharacteristicstringsstartingfromtherstcharacter.Thevaluesofcharactersareinthefollowingorder:`+'>`0'>`-'.Thetaskwhosecharacterhasthehighestvaluehasthehighestpriority.Ifbothcharactersare`+',theirnextcharactersarecompared.Ifbothcharactersare`0',thetaskwithasmalleridentiergetshigherpriority.Ifbothcharactersare`-',thetaskwithhigherurgencyfactor(UFk+1i=[1)]TJ /F6 11.955 Tf 12.46 0 Td[((bk+sVi)-264(bbk+nVic)]=Vi)hashigherpriority.Iftheirurgencyfactorsareequal,thesmaller-identiertaskwinsthetiebreaker.lagk+1icanbeimmediatelycalculatedfrompwk+1i)]TJ /F3 11.955 Tf 12.36 0 Td[(owk+1i.Theoverallworst-casecomplexityoftheLD-ProutineisO(N)[ 165 ]whenimplementingthecomparisonofcharacteristicstringsbetweentwotasksusingtheconstantprioritycomparisonfunctiontechniquesuggestedin[ 16 ]. 5.4.3SchedulingLocalTaskExecution LS-PusesthelocalparametersassignedbyLD-PtoscheduletasksduringeachT-Lplane.InsteadofusingMcNaughton'salgorithm[ 109 ]asinBFscheduling,theLS-Proutineusesadifferentselectionmechanisminordertoimproveschedulingefciencybyreducingunnecessarypreemptionsandmigrations.ThevalidationofimprovementisprovidedinSection 5.4.6 Eachtaskhastwotime-varyinglocalparameters:lci(t)containstheremainingexecutiontimeunitsofiintheestablishedT-Lplane,andldi(t)representstheremainingtimeunitsuntiltheendoftheplane.Thevalueoflci(bk)equalsthetotal 123

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numberoftimeunitsthaticanexecuteduring[bk,bk+1),i.e.,thesumofmwk+1iandowk+1i.ldi(bk)ofalltasksareequaltothesizeoftheplane.Tasklocallaxitylli(t)=ldi(t))]TJ /F3 11.955 Tf 13.39 0 Td[(lci(t).LS-Putilizesbothlocalparameters,lci(t)andldi(t),forschedulingtasksduringeachT-Lplane.TheseparametersareshowninFigure 5-9 (b)forclarication.Ateachschedulingpoint,Mtaskswithpositivelci(t)areselectedforexecution.Thetasksforwhichlli(t)iszerogetthehighestpreference.LS-Phasworst-casetimecomplexityO(N). 5.4.4MappingTaskstoProcessors Conceptually,theschedulerattemptstokeeptasksintheprocessorstowhichtheywerelastassignedif}iisoneofthecurrentlyidleprocessors,otherwisethetaskswillbeassignedtotherstprocessorthatisnotyetreassignedtoanytask.Criticaltasks(i.e.,havingzerolaxity)havethehighestpreference.Otherlower-preferencetasksmightneedtomigratetootherprocessorsiftheirprocessorswerealreadyreassignedtoexecuteothertasks.Theworst-casetimecomplexityoftask-to-processormappingroutinesisO(N). 5.4.5ExampleofEAGLEScheduling TodemonstratehowEAGLEschedulingworks,thissectionshowsanexampleofanEAGLEscheduleforatasksetT=f1:[5,5],2:[4,5],3:[7,10],4:[3,10],5:[1,5]on3processors.TheresourceutilizationdemandsU1toU5are1,0.8,0.7,0.3and0.2,respectively.Theresourceutilizationsallocatedtotasksequaltotheirdemands,i.e.,Vi=Ui.ThevaluesofvariablescalculatedbyLD-PareshowninTable 5-1 .TheGanttchartsinFigure 5-11 showthescheduleoftaskexecutionovertime(horizontalaxis)fromthetaskperspective(thetopchart)andtheprocessorperspective(thebottomchart)whendispatchingtasksforlocalexecutionusing(a)McNaughton'sruleasinBFand(b)theLS-Proutine. The1stT-Lplanestartsatb0=0andendsatb1=min(5,5,10,10,5)=5.DuringthisT-Lplane,themandatoryworkunitsmw1iof1to5are5,4,3,1and1,respectively. 124

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Thus,RU=(5*3)-(5+4+3+1+1)=1,soonlyonetaskcanbeselectedtoreceiveanoptionalunit.3and4arethetwopossiblecandidatessincetheirpw1iaregreaterthan0andthevaluesoftheirmw1iaresmallerthanthelengthoftheplane.Therstcharactersinthecharacteristicstringsofbothtasksare`-',soUF13andUF14mustbedeterminedtodecidetaskpriorities.SinceUF130arechosenforexecutionduringb0t
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McNaughton'srulecomefromtheminimizationofschedulelength[ 40 ],whichisnottheconcerninLS-Paslongasalltaskscancompletetheirexecutionbydeadlines.ThenumberofschedulerinvocationsinFigure 5-11 (a)isalsoslightlysmallerthanthatofFigure 5-11 (b).Evaluationresultsin[ 40 ]provideathoroughcomparisonbetweenMcNaughton'sruleandaschedulingapproachfollowingtheproposedGLS.Inthenextsection,theLS-PisshowntosatisfytheGLS'srequirements. 5.4.6OptimalityoftheEAGLESchedulingAlgorithm Theterm'optimality'referstotheabilityofaschedulertocreateascheduleoftasksthatcanfullyutilize100%ofeachresourceinasystem.EAGLE'soptimalityinschedulingstatictasksetsonMprocessorscanbeshownusingeitherT-Lplane-based[ 40 ]orDP-FAIR[ 97 ]schedulingmodels.ThesetwoschedulingmodelsprovidesetsofsufcientconditionsforoptimalityofmultiprocessorschedulingalgorithmsasdescribedinSection 5.3 .ThissectionprovidestwotheoremsshowingthatEAGLEsatisesGLSinT-Lplane-basedmodelandrulesofDP-FAIRmodel.Consequently,EAGLEisanoptimalmultiprocessorscheduler. Lemma5.0.1. EAGLE'sLS-PmeetstherequirementsofGLS. Proof. SinceLS-PselectsasmanyasMactivetasks,givesthehighestprioritytozero-local-laxitytasksandexecutethemimmediately,anddoesnotallowtaskswithzerolocalremainingexecutiontimetoexecute,LS-PsatisesGLS. Theorem5.1. Foranystatictaskset,theEAGLEalgorithmdeterminesafeasiblescheduleifPi=1,...,NViM. Proof. SinceEAGLEadoptsapartoftheBFalgorithmasLD-PanditsLS-PmeetstherequirementsofGLS(asstatedinLemma 5.0.1 ),Lemma1in[ 165 ]andTheorems14and15in[ 40 ]implythatthestatementistrue. 126

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Theorem5.2. EAGLEisDP-FAIRand,hence,optimal. Proof. Rules1and2fromDP-FAIRschedulingconditionsaresatisedbyLS-P.FromLemmas3and4in[ 165 ],itcanbeimpliedthatrule3isalsosatised.Hence,EAGLEisDP-FAIRandanoptimalschedulingalgorithm. 5.5PerformanceEvaluationofEAGLE Inthissection,EAGLEiscomparedagainsttheEDZL,ASEDZL,NABLRandLLREFschedulers.Theessenceofeachalgorithmissummarizedbelow: EDZListheEDFschedulingwithzerolaxitypolicy.TaskswithzeroruntimelaxityaregiventhehighestpriorityandtheremainingtasksarerankedbyEDFpriority.ItisknowntoneverperformworsethanEDFscheduling[ 44 ]. ASEDZLisanextensionofEDZL.Betweentwoconsecutivejobarrivals,Mearliestdeadlinetasksarechosentoexecute.Ifthereisanyslackprocessortime,othertasksareselectedtoexecuteusingEDFpriority.Theslacktimeisanticipatedbeforeitoccurs.TheintervalbetweentwoconsecutivejobarrivalsisdenedsimilarlytoaT-Lplane. NABLRisourearlyextensionofASEDZL.Itusesaheuristicthatconsidersboththeremainingexecutiontimeandruntimelaxityoftaskswhenplanningtaskexecutionbetweenanyconsecutivejobreleaseinstantssothattaskshavemoderatelybalancedremainingloads(i.e.,lessdeadlinemisses). LLREFisaknownoptimalschedulingalgorithm.Italsoconsidersexecutionplanningbetweentwoconsecutivejobarrivals.Insteadofplanningtoexecuteonlyasubsetofjobswithintheinterval,LLREFassignslocalexecutiontimetoeachjobproportionaltotheirutilizationdemands(i.e.,lci=Ui(bk+1)]TJ /F3 11.955 Tf 12.47 0 Td[(bk)).Ateachschedulingpoint,taskswiththehighestlcivaluesareselectedtoexecute. TheSimulationTOolforReal-timeMultiprocessorscheduling(STORM)[ 149 ]developedbyresearchersofReal-TimeSytemsgroupfromtheIRCCyNLaboratoryisusedforsimulatingschedulesofpseudo-randomgeneratedtasksetscreatedbyallalgorithms.Eachtaskcanbecreatedbyrstassigningapseudo-randomperiodPiaccordingtotheperiodgenerationalgorithmin[ 61 ],selectingtaskutilizationUifromauniformdistributionbetween0and1,andtaskworst-caseexecutiontimecan 127

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becalculatedfrommax(1,round(UiPi)).Atasksetisgeneratedusingasimilarthefollowingprocedure[ 21 ]. STEP1.ExtractasetofM+1tasks. STEP2.Checkifatasksetisfeasible(i.e.,U=Pi=1,...,NUiM). STEP3.Ifthetasksetisfeasible,addittothecollection.Thenextractanadditionaltaskandaddittothetasksettocreateanewtasksetandgobacktostep2. STEP4.Ifthetasksetisinfeasible,discardthetasksetandgobacktostep1. Foreachsimulation,thefollowingmetricswererecorded:numberofdeadlinemisses,preemptions,migrationsandschedulerinvocations. 5.5.1Schedulability First,theschedulabilityofveschedulingalgorithmsareevaluatedfortwodistinctcollectionsof10,000randomlygeneratedtasksetson4and8processors.Foranyperiodicandsynchronoustaskset,ifaschedulingalgorithmdoesnotcauseanydeadlinemisswithinthehyper-periodofthetasksetthenthetasksetisschedulablebythatalgorithm[ 44 ].TheschedulabilityofallalgorithmsareshowninFigures 5-12 (a)and 5-12 (b).Fromthehistograms,everytasksetwhosetotalutilizationislessthan0.9Mcanbesuccessfullyscheduledbyallalgorithms.Forhigh-utilizationtasksets(i.e.,0.9MUM),suboptimalalgorithmsperformdifferently.ThedifferencesarelargerasUapproachesM.WhenU=MandM=4,theschedulabilityofEDZL,ASEDZLandNABLRare67.11,80.34and95.96percent,respectively.ForM=8,thesevaluesare39.27,57.22and92.99percentforEDZL,ASEDZLandNABLR.NABLRhashigherschedulabilitythanEDZLby28.85and53.71percent.WhencomparedtoASEDZL,NABLR'sschedulabilityisalso28.85%and35.76%higherwhenM=4andM=8.Anothersetofsimulationsisperformedwithother7,500randomlygeneratedhigh-utilizationtasksetsinwhichtasksaredividedintothreegroupsof2,500tasksets.EverytasksetinthethreegroupshasU=3,U=4andU=8,respectively.Onaverage,NABLR'sschedulabilityisabout50%higherthanEDZLandabout37%higher 128

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thanASEDZL.Byfairlydistributingexecutiontimetotasksbasedontheirutilization,theschedulabilityofschedulingalgorithmscanbesignicantlyimprovedascanbeseenfromtheresultsobtainedfromNABLR,whichissignicantlybetterthantheothertwoefcientalgorithms,andfromtwooptimalschedulers,LLREFandEAGLE. 5.5.2PreemptionandMigrationOverheads Figure 5-13 presentspercentagestatisticsofpreemptions,migrationsandschedulerinvocationsfortherstsetofsimulationsinSection 5.5.1 .LLREFincursconsiderablylargeroverheadsthanallotheralgorithms.Whencomparedtoothersuboptimalalgorithms,NABLRandEAGLEareconsideredcomparativelyefcient(ifnotlessthanthoseofEDZLandASEDZL,theiradditionaloverheadsarewithin0.12%and0.3%,respectively).AnadditionaltimerneededforzerolaxityisalsolesscostlythanthatneededforLLREF.Inaddition,thetimermanagementisonlyrequiredforzerolaxitywhichislesscostlythanthatneededforLLREF.Possibleimplementationofazero-laxitytimerneededforbothalgorithmsisdiscussedin[ 11 ]. 5.6SchedulabilityAnalysisofEAGLEunderModeChanges Amodechangeresultsineithertheincreaseordecreaseofoneormoretasks'utilizationsand/orperiods.ThissectionconsiderstheimpactofchangesineitherperiodorutilizationofataskonEAGLE'sschedulabilitywhiletheotherparameterremainsunchanged.Sincethescheduleralwaysworkswiththecurrentvaluesofalltime-varyingtaskparameters,thetimeindexmaybeomittedforbrevity.First,modechangesarerestrictedtoonlyoccuratjobdeadlines. Theorem5.3. Giventhattheperiodofataskicanchangeonlyatitsjobdeadlines,theEAGLEalgorithmdeterminesafeasiblescheduleifPi=1,...,NViM. Proof. Assumethatthetaskmissesitslthjobdeadline.Atthedeadlineofataskinstancey,l,thetotaltimeusedbytaskyisVy(dy,1)]TJ /F3 11.955 Tf 13.07 0 Td[(ry,1)+Vy(dy,2)]TJ /F3 11.955 Tf 13.07 0 Td[(dy,1)+...+Vy(dy,l)]TJ /F3 11.955 Tf 12.06 0 Td[(dy,l)]TJ /F7 7.97 Tf 6.59 0 Td[(1)=Vy(dy,l)]TJ /F3 11.955 Tf 12.07 0 Td[(ry,1).Thetotalamountofexecutiontimeusedby 129

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yandallothertasksisA=Vy(dy,l)]TJ /F3 11.955 Tf 12.31 0 Td[(ry,1)+Pz6=yVz(dz,latest)]TJ /F3 11.955 Tf 12.31 0 Td[(rz,1)wheredz,latestdenotesthedeadlineofthelatestjobofzcompletedbeforedy,l.Sincery,1andrz,1mustbenon-negative,wehaveVy(dy,l)+Pz6=yVz(dz,latest)X.Fromthedenition,dz,latestdy,l,itfollowsthatVydy,l+Pz6=yVzdy,l=dy,lPi=1,...,NViX.Sinceadeadlinemissoccursatdy,l,itfollowsthatdy,lPi=1,...,NVi>dy,lMwhichcontradictsPi=1,...,NViM. Corollary5.3.1. GivenafeasibleEAGLEschedule,ataskthatneedsprocessorutilizationbyuptoM)]TJ /F10 11.955 Tf 12.48 8.96 Td[(Pi=1,...,NVimayenterthesystematanytimewithoutcausinganytasktomissadeadlineintheresultingEAGLEschedule. Proof. SincetherstjobsoftasksconsideredinTheorem 5.3 canstartatanyarbitrarytime,theproofinTheorem 5.3 alsoappliesfortheabovestatement. Theorem 5.3 ,Corollary 5.3.1 andthefactthatanytaskmayleavesafelyatjobdeadlinesconrmthatatanyjobdeadlines,ataskmayincrease/decreaseitsperiodand/orutilizationaslongasthetotalutilizationdemandoftheresultingscheduledoesnotexceedtheavailableresourcecapacity,andtheremainingscheduleisstillfeasible.Next,therestrictionthatmodechangescanonlyoccuratjobdeadlinesisrelaxedandconditionsunderwhichamodechangecanoccursafelyatarbitrarytimesarepresented. Theorem5.4. GivenafeasibleEAGLEschedule,ataskimayincreaseitsutilizationtoV0=Vi+VatanytimetifVM)]TJ /F10 11.955 Tf 12.1 8.97 Td[(Pi=1,...,NViandtheresultingEAGLEscheduleremainsfeasible. Proof. Ascheduleresultingfromanincreaseinutilizationofthealready-releasedjobofiisequivalenttoascheduleinwhichanothertask0iwithVentersthesystemattimetanditsabsolutedeadlineisthesameasi'sjob.ThetotalexecutiontimerequiredbythetwojobsisVi(di)]TJ /F3 11.955 Tf 12.35 0 Td[(ri)+V(di)]TJ /F3 11.955 Tf 12.35 0 Td[(t),whichequalsVi(t)]TJ /F3 11.955 Tf 12.35 0 Td[(ri)+V0(di)]TJ /F3 11.955 Tf 12.36 0 Td[(t). 130

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AslongasVM)]TJ /F10 11.955 Tf 12.5 8.97 Td[(Pi=1,...,NVi,Corollary 5.3.1 provesthattheresultingscheduleremainsfeasible. Theorem5.5. GivenafeasibleEAGLEschedule,atanytimeataskimayincreaseitsperiodandtheresultingEAGLEscheduleremainsfeasible. Proof. Ascheduleresultingfromaperiodenlargementofataskwithanalready-releasedjob(i.e.,diischangedtod0i=di+d)canbeconsideredasascheduleinwhichthejobisfollowedbyanotherjobwithVi;thelatterjobisreleasedatdiandhasitsdeadlineatd0i.ThetotalexecutiontimeVi(di)]TJ /F3 11.955 Tf 11.89 0 Td[(ri)+Vi(d0i)]TJ /F3 11.955 Tf 11.89 0 Td[(di)isequivalenttoVi(d0i)]TJ /F3 11.955 Tf 11.89 0 Td[(ri).UsingTheorem 5.3 ,thistheoremfollows. Theorem5.6. GivenafeasibleEAGLEschedule,ataskmayleaveatanytimeaslongasitslagisnon-negativeandthereclaimedutilizationcanbeusedbyothertaskswithoutcausinganytasktomissadeadlineintheresultingEAGLEschedule. Proof. Letthetaskyleavethesystemwhenitsjthjobhasnon-negativelagattimetandthenewjobwiththesameutilizationdemandentersatthattime(thesameidentierisusedforthenewtask).Assumethatthenewtaskmissesitsnthjobdeadline.Attimet,theexecutiontimeusedbyy,jisVy(t)]TJ /F3 11.955 Tf 11.87 0 Td[(dy,j)ifitslagiszeroandVy(t)]TJ /F3 11.955 Tf 11.87 0 Td[(dy,j))]TJ /F6 11.955 Tf 11.88 0 Td[(xifitslagispositive.Atdy,l,thetotalexecutiontimeoftaskyisVy(dy,l)]TJ /F3 11.955 Tf 13 0 Td[(ry,1)orVy(dy,l)]TJ /F3 11.955 Tf 11.95 0 Td[(ry,1))]TJ /F6 11.955 Tf 11.96 0 Td[(x.TheremainderoftheprooffollowsthatofTheorem 5.3 Theorem5.7. GivenafeasibleEAGLEschedule,ataskimaydecreaseitsutilizationtoV0=Vi)]TJ /F6 11.955 Tf 10.53 0 Td[(VwhereVViatanytimetsuchthatei(t)Vi(t)]TJ /F3 11.955 Tf 10.53 0 Td[(ri)anditsreclaimedutilizationVcanbeusedbyothertaskswithoutcausingtheresultingEAGLEscheduletobecomeinfeasible. Proof. Ascheduleresultingfromthetaskdecreasesitsutilizationwhilethereisanalready-releasedjobissimilartoascheduleinwhichtherearetwotasks:ywithVyand 131

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zwithVz=VwhereVy+Vz=Vi.Thejobsofyandzarereleasedatthesametimeriandhavethesamedeadlinesatdi.Attimetsuchthatei(t)Vi(t)]TJ /F3 11.955 Tf 12.01 0 Td[(ri),zcanbeconsideredasleavingthesystemwhileitslagiszero(ifVei=(t)]TJ /F3 11.955 Tf 12.11 0 Td[(ri)),andwhileitslagispositive(otherwise).AccordingtoTheorem 5.6 ,theprooffollows. Theorem5.8. GivenafeasibleEAGLEscheduler,ataskmaydecreaseitsperiodtoatmostei=ViwithoutcausinganytasktomissadeadlineintheresultingEAGLEschedule. Proof. AscheduleafterthetaskdecreasesitsperiodfromPitoP0i=Pi)]TJ /F6 11.955 Tf 13.08 0 Td[(PissimilartoascheduleinwhichthetaskleavesthesystembeforeitsjobdeadlineandanewtaskwithareducedperiodP0ienters.FromTheorem 5.6 ,thetaskcanleavethesystemsafelywhenitslagisnon-negative.AslongasP0iei=Vi,thetaskdenitelyhasnon-negativelagwhenitleavesand,hence,Theorem 5.8 istrue. ByconsideringschedulabilityoftheEAGLEschedulingalgorithmunderdifferentscenariosoftaskparameterchanges(i.e.,periodandutilization),anovelmode-transitionprocedurecanbederivedandintegratedintotheEAGLEalgorithm.Theoremsinthissectionprovideabasistoenablemodetransitionsthatmayinvolvebothutilizationandperiodchangesofoneormoretaskswhilemaintainingthefeasibilityoftheresultingschedule.ThesubsequentalgorithmiscalledEAGLE-T.Ratherthantakingthestep-wiseapproach,EAGLE-Tenablesprogressiveutilizationadaptationoftaskswhileguaranteeingnodeadlinemisshappensintheresultingschedule. 5.7EAGLEwithMode-TransitionProtocol(EAGLE-T) AnMCRtomodemprovidesanewsetofdesiredutilizationsUmiforalltasks.SinceVicontrolstheactivationpatternsoftaskiasdescribedinSection 5.1 ,changingVitoUmicanresultinchangesofbothtaskutilizationandperiod.Forexample,ifVi(t)]TJ /F6 11.955 Tf 12.73 0 Td[(1)=3=10andUmi=1=2,Pi(t)]TJ /F6 11.955 Tf 12.73 0 Td[(1)isequalto10whiletheperiodforUmiis2.Thisrequirescarefulapplicationandcombinationofthetheoreticalresultsshownintheprevioussection.Thepseudo-codeofthemode-transitionroutineisshownin 132

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Figure 5-14 .Withoutlossofgenerality,MCRsarerstassumedtohappenonlyatperiodboundaries,andthenminormodicationstothealgorithmneededwhenthisassumptionisremovedareexplained.TheroutineisinvokedwhenanMCRoccurs(tMCR),andateveryperiodboundarybkduringthemodetransition(i.e.,thereexistsatleastonetaskwhoseViisnotthesameasUmi).TheMode TransitionroutineprovidesasafeandprogressiveutilizationadjustmentfromtheexistingschedulingutilizationVi(t)]TJ /F6 11.955 Tf 12.58 0 Td[(1)toVi(t)whileensuringthatVi(t)2[Umi,Vi(t)]TJ /F6 11.955 Tf 12.67 0 Td[(1)]fordecreasing-utilizationtasksandVi(t)2[Vi(t)]TJ /F6 11.955 Tf 11.96 0 Td[(1),Umi]forincreasing-utilizationtasks. FortaskswhoseUmiVi(t)]TJ /F6 11.955 Tf 12.45 0 Td[(1))areconsideredinthedecreasingorderoftheirUmi.Theroutineappliestoeachofthesetasksoneofthefollowingrulesbasedonlagki: INCREASEWITHZEROLAG(IZ).thealready-startedjobofthetaskishalted,Vi(t)=min(Umi,Vi(t)]TJ /F6 11.955 Tf 12.21 0 Td[(1)+Q)andanewjobcanbereleasedimmediately.PiisthesmallestintegersuchthatVi(t)Piisalsoaninteger. INCREASEWITHPOSITIVELAG(IP).thecurrentjobishalted.Vi(t)=min(Umi,Vi(t)]TJ /F6 11.955 Tf 12.81 0 Td[(1)+Q)andanewjobcanbereleasedimmediately.PiisthesmallestintegersuchthatVi(t)Piisalsoaninteger.Atthenewjob'sdeadline,resetlagki=0. 133

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INCREASEWITHNEGATIVELAG(IN).V0i=min(Umi,Vi(t)]TJ /F6 11.955 Tf 13.08 0 Td[(1)+Q)andCi=min(bV0iPic,dVi(t)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(t)]TJ /F3 11.955 Tf 11.96 0 Td[(ri)+V0i(di)]TJ /F3 11.955 Tf 11.95 0 Td[(t)e).Vi(t)isCi=(di)]TJ /F3 11.955 Tf 12.14 0 Td[(ri).Piisthesame,andupdatelagki=Vi(t)(t)]TJ /F3 11.955 Tf 11.96 0 Td[(ri))]TJ /F3 11.955 Tf 11.95 0 Td[(ei. Inthecasethatataskinstanceishaltedbeforeitsdeadlineaccordingtooneoftheaboverules,theexecutioncostofthetaskCicanbeconsideredasadjustedtoei.Thus,thereisnodeadlinemiss. WhenMCRsareallowedtohappenatanarbitrarytimet(notnecessarilyattheperiodboundaries),itcanbeconsideredasifthealready-startedT-Lplane[bk,bk+1)isshortenedto[bk,t)andanewT-Lplanemustbeestablishedattimet.lagk+1imustbeupdatedtoreectthechangeoftheendingtimeoftheT-Lplane.Atthispoint,theabsolutevaluesofalltasklagsarenotnecessarilywithinone,butthefairnessstillholdsatotherregularT-Lplaneboundaries(notcausedbyMCRs).Becauseofthischange,additionalvalidationmustbeperformedinLD-Ptoensurethatmwk+1i(bk+1)]TJ /F3 11.955 Tf 12.09 0 Td[(bk).AstherestrictionofhavingMCRsonlyatboundarytimesislifted,thenumberofschedulerinvocationscanbelessenedbyassigningPi=minP,foranytaskiwhoseVi(t)=1,wherePmin=min(Pj)andj6=i.Themode-transitionprocedurehasO(NlogN)becauseofthesortingoftasksinanascendingorderoftheirtargetutilizations.ThesortingcanbeskippedtoreducethecomplexitydowntoO(N)withatradeoffinhighutilizationdriftofimportanttasks. 5.7.1SchedulingExample Figure 5-15 showsanexampleofanEAGLE-Tscheduleforasetof5adaptivetaskswiththesamerangeofallowableutilization[0.2,1].Thevectorsnexttotheblackcirclesdenotesthevaluesofthedesiredtaskutilizationinthemodesinwhichthesystemisoperating.Thetop-righttableshowstheactualchangesoftaskutilizations.Thebottomtabledisplaysallscheduler'sparametersateachT-Lboundary.Thegrayshadedperiodsdenotethecyclesduringmodetransitions.Thesymbol'x'indicatesthatavalueneedsnotbecalculatedbecauseitisnotnecessaryformakingaschedulingdecisionatthetime. 134

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DuringthersttwoT-Lplanes(t=0tot=10),thelocalparametersaredeterminedasdescribedearlierinSection 5.4.5 .Att=8,anMCRsuddenlyoccurswithUm=f0.5,1,0.4,0.3,0.8g.The2ndT-Lplaneisthenshortenedto[5,8)insteadof[5,10)andthelagsoftasks1to5areupdatedto0,-3/5,-14/10,14/10and3/5,respectively.Thedecreasing-utilizationtasks(i.e.,1and3)areconsideredrst.Since1haszerolag,theDZruleisappliedandV1ischangedto0.5.ApplyingDNto3andV3stillremainsthesameuntilthejobdeadline.Atthispoint,Q=0.5.ApplyingINto2,wehaveV2=1andQ=0.3.Then,V5=0.5afterapplyingIPto5.SinceVi6=Umifori=3and5,themodetransitioncontinues.Atthenextperiodboundary(t=10),bothV3andV5arechangedto0.4and0.8sincethetaskshavezerolag.Atthispointthemodetransitionendsandthesystemoperatesinthetargetmode.ThreemoreMCRsoccurattime16,29and38.AtableshowninFigure 5-15 providesvaluesofallschedulingparameterstoillustratethealgorithmwhentheseMCRsoccur. Theorem5.9. GivenafeasibleEAGLE-Tschedule,themode-transitionroutineinFigure 5-14 doesnotcauseanyviolationofanytasks'minimumutilizationrequirementsintheresultingschedule. Proof. ApplyingtheDZ,DP,DN,IZorINalwaysresultinascheduleinwhichalltaskinstancesreceiveutilizationwithintherange[Uimin,Uimax].ItisthereforesufcienttoverifytheabovestatementonlyforthecasewhentheIPruleisapplied.Sincelagkiiscarriedalongforthenewjob,bytheoriginaldeadline,thetaskwouldhaveexecutedforatleastasmanytimeunitsaswiththeoriginalutilizationbeforeapplyingtherule.Hence,Theorem 5.9 isalsotruefortheIPrule. 5.7.2TransitionDelayandUtilizationDrift AmodetransitionstartswhenanMCRoccursandendswhenalltaskshavetheirVi(t)equaltothedesiredutilizationsofthetargetmode.Hence,thedurationofmodetransitionsdependsonhowinstantaneouslyeachtaskcanadapttothenewmode's 135

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utilization.Thefollowingtheoremprovidestheupperboundofthemodetransition'sduration. Theorem5.10. EachmodetransitionintheEAGLE-TschedulelastsnomorethanmaxP=max(Pi)(i=1,...,N). Proof. ForallrulesintroducedinSection 5.7 ,thelatesttimethatataskcanadapttoanewmodeisbyitsdeadline.Hence,themodetransitionisboundedbythelargestpossiblesizeofthetaskperiods,i.e.,maxP. Notonlythesystemmusttransitionintothetargetmodewithoutdeadlineviolations,butalsothetransitionmustbecarriedoutinawaythattheoverallsystemperformanceisminimallyaffected.Intuitively,thesystemperformancehasadirectrelationshiptotheamountofprocessorutilizationachievedbytasksinthesystem.Itisthereforeimportanttocharacterizethedurationofmodetransitionsandtheutilizationdriftduringthetransitions. LetanominalscheduleISdenoteascheduleinwhicheachtaskcanswitchtoatargetmodeinstantaneously(i.e.,ViisalwaysequaltoUmi).Utilizationdriftofataskiattimet,orD(i,t),isdenedastheabsolutevalueofthedifferencebetweenViintheISscheduleandViintheEAGLE-Tscheduleattimet.Asanexample,theutilizationdriftsbetweentheISscheduleandtheEAGLE-Tscheduleof4,orD(4,t),fromtheexampleshowninFigure 5-15 ateachtimeunitareshowninFigure 5-16 Theorem5.11. Theupperboundofthetotalutilizationdriftpermodetransitionforanytaskiis(Uimax)]TJ /F3 11.955 Tf 11.95 0 Td[(Uimin)maxP. Proof. ThemaximaldriftineachtimeunitduringthemodetransitionisUimax)]TJ /F3 11.955 Tf 12.57 0 Td[(Uimin.FromTheorem 5.10 ,thetotalutilizationerrorforeachchangeisthenboundedat(Uimax)]TJ /F3 11.955 Tf 11.96 0 Td[(Uimin)maxPasstatedabove. 136

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5.7.3EAGLE-TversusEPOC TherearesubtledifferencesbetweenEAGLE-TandEPOCwhichcomplicatesadirectcomparisonofutilizationdriftandtransitiondelaybetweenschedulescreatedbybothalgorithms.TheEPOCscheduler(presentedinChapter 4 )isspecicallydevelopedfortasksinensemblesystemsinwhichalltaskssharethesameworst-caseexecutiontime(WCET).Timeispartitionedintoslicesofxedsize,calledschedulingepochs.EPOCensuresthattheresourceutilizationsgainedbytasksduringeachschedulingepochareexactlythesameamountastasks'utilizationdemandsinthecaseofstatictasksetsordynamictasksetsinwhichallocationchangescoincidewithschedulingepochs.Whenanallocationepochpartiallyoverlapswithschedulingepochs,EPOCusesabest-effortapproachtoscheduletaskssothattheirutilizationsduringeachschedulingepochareclosetotheirresourcedemands.Ontheotherhand,EAGLE-Tisdesignedforgenericreal-timeadaptivetasksandmakesnoassumptionregardingWCETsoftasks.EAGLE-T'sgoalistocreateaschedulethatallowsresourcereallocationaccordingtotaskdemandsandguaranteethatnodeadlinemissoccursacrosstime.Duetothesedifferences,thissectionprovidesonlyexamplestogiveanintuitiononthecomparativeperformanceofthetwoalgorithms. Whenconsideringastatictaskset,liketheoneusedinFigure 5-11 ,tasks'resourceutilizationsachievedbytheEPOCschedule(inFigure 5-17 (a))canbedeterminedateachschedulingepochboundary,whilethoseachievedbytheEAGLE-Tschedule(inFigure 5-17 (b))aredecidedattaskdeadlines.Bothschedulescanscheduletaskstoreceiveresourceutilizationsasdictatedbytheirdemands.FortaskswhoseWCETsaregreaterthanonetimeunit,likeinthisexample,EPOCcanschedulethesetasksifthenexttaskinstancecanbereleasedimmediatelyafterthecurrentonecompletesitscomputationoristerminatedvoluntarilybecauseofutilizationchanges.ThegraystarsintheGanttchartofFigure 5-17 showthecompletionofeachtaskinstanceintheEPOCschedule.Thenumberofmigrationsoccurredinbothschedulesareequal 137

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(=1).EAGLE-Thasnopreemptions,whileEPOChas6preemptionsduringtheshownschedulingepoch.SchedulingeventsoftheEPOCscheduleoccurateverytimeunit,whiletheEAGLE-Tschedulehasonly6schedulerinvocations.Figure 5-18 (a)and(b)providetheEPOCandEAGLE-Tschedules,respectively,forthedynamictasksetinFigure 5-15 .TheEPOCschedulehas43schedulerinvocations,28preemptionsand15migrations,whiletheEAGLE-Tschedulehas35schedulerinvocations,19preemptionsand14migrations.EAGLE-TismoreefcientthanEPOCsinceitcanreducetheschedulingoverheadsinEAGLE-Tby18.60,32.14and6.66percent,respectively. 5.8PerformanceEvaluationofEAGLE-T Allofthepreviousworkinmultiprocessorsystemsassumesthattasksmustadaptinastep-wisemanner;thereallocationofresourcestotasksisdoneonlyifitcanfullyachievetasks'targetresourceutilization.Moreover,theresourcereallocationcannotbeappliedtoalready-startedtaskinstances,soanexistinginstanceofataskmusteithercomplete(topreserveconsistencyofthesystem)orterminateearlybeforetheprotocolcanreallocateresourcesandallowareleaseofanewinstance.Thisispossiblyduetotheusageofdifferenttaskversionsfordifferentmodes.Theworkinthisdissertationfocusesonadistincttaskmodelnotconsideredinpreviousworkinwhichtasksmaychangetheirresourcedemandovertimeandtheiractiveinstancescancontinuouslyutilizeprogressivelyadaptedresources.Promptandgradualadaptationtowardthedesiredresourceutilizationisveryimportanttomaintainacceptablesystemperformancesincethesystemsmustbeoperatingunremittinglyfromonemodetoanother. InthissectionwestudytheperformanceofEAGLE-Tinschedulingadaptivetasksets.Sincetheconsideredtaskmodelisdifferentfromthetaskmodelsconsideredinthepreviousworkasdescribedabove,abaselinealgorithmrepresentingastep-wiseapproachisintroducedforcomparisonpurposes.Thisbaselinealgorithmwithstep-wiseadaptation,calledEAGLE-S,usesthesetofrulesintroducedinSection 5.7 toadjusttheactualallocatedutilizationsVi(t)]TJ /F6 11.955 Tf 12.58 0 Td[(1)toVi(t)onlywhenVi(t)canbesafelyadjusted 138

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toUmiwhereUmipresentsthetask'sdesiredutilizationissuedbythelatestMCR.Otherwise,Vi(t)remainsthesameasVi(t)]TJ /F6 11.955 Tf 11.95 0 Td[(1). TheperformanceoftheEAGLE-Talgorithmisevaluatedwhen(A)varyingtheprobabilityofmodechange,orProbMCR,ineachcycle(from0.1to1)and(B)varyingthenumberoftasksinthesystem(from4to12).Withoutlossofgenerality,Uimin=0.2andUimax=1foralli=1,...,N.InsettingA,N=5andM=3.ForsettingB,ProbMCR=0.5,andM=3.Foreachsetting,100pseudo-randomtasksetsarecreatedusingthefollowingprocedure: CreateasetofMCRtimes.UsingProbMCR,itcanbedeterminedwhetheranMCRwilloccurineachcycle. ForeachMCR,generateasetoftasks'desiredresourceutilizationsusingsimilarprocedurestothosein[ 21 61 ].Denominatorsofutilizationarepseudo-randomlyselectedbetween1to60cycles.Resourcesarefullyutilizedateverymode(i.e.,Pi=1,...,NUmi=M). Atrialisdenedasa1000-cycleEAGLE-*scheduleofacreatedtaskset.Fromeachtrial,twoperformancemetricsaremeasured:tasks'utilizationdriftsandmode-transitiondelay.TheresultsdiscussedintherestofthissectiongiveinsightonhowfrequencyofMCRsandsizeofthetasksetscanimpacttheperformanceofEAGLE-SandEAGLE-T.Theseresultsalsoconrmthattherecanbemajordifferenceinperformancebetweenthestep-wiseandprogressiveapproachalthoughtheirtransitionrulesarethesame.Dependingonthereal-timeapplications,thestep-wisemodetransitionmightnotbeacceptable. 5.8.1Drift TheutilizationdriftiscalculatedpercycleusingtheapproachdiscussedinSection 5.7.2 .ThepercentageofutilizationdriftistheratiooftheutilizationdriftandtheutilizationachievedbytheISschedule.Figure 5-19 showstheaveragesofmaximalvalueandpercentageofutilizationdriftfrom100trialsofeachconguration.Maximalutilizationdriftofagiventasksetattimetisdenedasmaxi(D(i,t))where 139

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i=1,...,N.Similarly,maximalpercentageofutilizationdriftisthemaximumvalueofthetasks'percentagesateachtimeunit.Forbrevity,maxDandmaxPdenotethemaximaldriftandthemaximalpercentageofdrift,respectively.Overall,theresourceutilizationachievedbytasksscheduledusingEAGLE-Tarewithin56%to90%ofthedesiredutilization(comparedto11%-81%whenEAGLE-Sisused). FromFigure 5-19 (a),theEAGLE-S'smaxDisabout2.4to3.2timeslargerthanthatoftheEAGLE-TregardlessofthefrequencyofMCRs.Fromourrecord,per-cycledriftsintheEAGLE-TtrialsarealwaysnolargerthantheyareinthecorrespondingEAGLE-Strials.ThisisbecauseEAGLE-Sneedstodelaytheadjustmentifthetarget-modeutilizationisnotachieved.Insomeapplications(likeBrain-MachineInterfacesandothersmentionedinChapter2)theutilizationdriftcausedbythisdelaycanbeunacceptablesincenon-criticaltasksmightcontinueutilizeresourcesthatshouldbeallocatedtoimprovetheresponsetimeofothersignicanttasks.Astheprobabilityofmodechangeincreases,bothalgorithmshaveincreasingmaxDandmaxP.ThisisananticipatedbehaviorsincemodechangesmayhappenonecycleafteranotherandthesystemcannotswitchtoanytargetmodebeforeanotherMCRoccurs.TheincreasesofmaxDandmaxPwhenusingEAGLE-TarehoweverlessevidentthanwhenusingEAGLE-S. Figure 5-19 (b)presentstheaveragesandstandarddeviationsofmaxDandmaxPasthenumberoftasksgrows.Atataskperiodboundary,taskutilizationcanbeimmediatelyadjustedtothenewmode'sutilization,asshowninSection 5.6 .Whenthetasksetsaresmall(N=4and6),itisarelativelyhighpossibilitythatMCRsoccurattimeotherthanperiodboundariesandthuspossiblycausesomedelaytotheadjustmentsofothertasks.Tobespecic,increasing-utilizationtasksrelyonfreedutilizationbydecreasing-utilizationtasks.Whenalargernumberoftasksexistsinthesystem,itislikelythatMCRscoincidewithoneormoretaskdeadlines;theutilizationcanbeadjustedpromptlyandprecisely(i.e.,Vi=Umi)forthosetasks.Inaddition,the 140

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simulationindicatesthatfortasksthatarenotattheirdeadlineswhenanMCRoccurs,theclosertothereleasetimeofitsactiveinstance,thesooneritsVicanbeadjustedtothenew-modeutilization.Asaresult,theaveragesmaxDandmaxPdecreaseforthecasewhenN>6. 5.8.2Delay Theaveragesandstandarddeviationsofthemode-transitiondelay,i.e.,thenumberofcyclesafteranMCRoccursuntilVi=Umiforalli,areshowninFigure 5-20 .AsMCRshappenmorefrequently,theaveragedelaydecreasesasshowninFigure 5-20 (a)sinceeachMCRcanbereplacedbythenextone.Incontrary,thenumberofcyclesthatthesystemspendsinmodetransitionincreases(inFigure 5-21 (a)). WhenthenumberoftasksgrowsasshowninFigure 5-20 (b)andFigure 5-21 (b),themode-transitiondelayalsoincreasesinitially(thecasewhenN=4to8).HoweverforN>8,utilizationadjustmentcanbecompletedmoretimelybecauseofthereasonsmentionedinSection 5.8.1 .Asanticipated,EAGLE-SperformsslightlyworsethanEAGLE-TintermsoftransitiondelayinbothsettingsAandBsinceitneedstodefertheutilizationadjustmentuntilthenewvalueofVicanmatchthetarget-modeutilization. 5.9Summary Thischapterpresentsanovelreal-timemultiprocessorschedulingthatsupportsmodechangesofadaptivereal-timesystems.Fromstudyingsequentialtaskmodeleffects,welearnthatadeadlinemisslikelyoccursiftaskloadresiduesaregreatlyimbalanced.Usingthisinsighttogetherwithfurtherstudyofmultiprocessorschedulingmodels,anoptimalschedulingalgorithm,calledEfcientApproximationofGradualLoadExecutionorEAGLE,isderivedasabasealgorithm.TheLD-ProutineoftheEAGLEalgorithmcarefullyplanslocalexecutionduringanyintervalbetweenanytwoconsecutivedeadlinesusingtheBoundaryFairschedulingscheme.TasksarescheduledbyEAGLE'sLS-Ptomakesurethattheplannedlocalexecutionsoftasksarecompletedbytheendoftheinterval.TheevaluationconrmsthattheEAGLEalgorithm 141

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isoptimalandgreatlyimproveefciencyovertheotheroptimalalgorithmsby13.35%to39.24%. SchedulabilityoftheEAGLEalgorithmisstudiedunderdifferentscenariosoftaskparameterchanges(i.e.,periodandutilization).Byconsideringanewadaptivetaskmodelinwhicheachtaskisprogressivelyadaptableratherthanstep-wiseadaptable,amode-transitionprocedureisderivedandintegratedintotheEAGLEalgorithm,resultinginanewalgorithmcalledEAGLE-T.Fromevaluationresults,EAGLE-Tcanprovidedeadline-miss-freemodetransitionwithmuchlessaverageutilizationdrift(over50%inmostcases)thanEAGLE-S.TheseevaluationresultsconrmthatEAGLE-TcanbeusedtoscheduleadaptivetasksetsefcientlyandprovideamoregenericimplementationoftheRTStosupportschedulinginabroaderclassofensemblesystems. 142

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Figure5-1. Dhall'seffectsillustration.TheEDFscheduleofT=1:[2,1],2:[2,1],3:[1,1+]on2processorswhereisasmallnumbercloseto0.AlthoughthetotalutilizationofTisapproximatelycloseto1,3missesadeadlineattime1+. Figure5-2. AddressingDhall'seffectsusingzero-laxitypromotion.TheEDZLschedulercansuccessfullyschedulethesametasksetusedinFigure 5-1 .Itusesthesameearliest-deadline-rstpolicyasEDF,butpromotestaskstothehighestprioritywhentheirlaxitiesreachzero.Bypromoting3att=,adeadlinemisscanbeavoided. 143

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Figure5-3. Anexamplescenariowithcumulativeofoadingfactoreffects.WhenEDZLschedulesT=1:[2,3],2:[2,3],3:[4,6]on2processors,theutilizationdemandofthewaitingtask3equalsthesumofallofoadingfactorsofrunningtasks(i.e.,1and2).Since3alsohasthelargestperiodamongthreetasks,itcausesoneprocessortobeidleduringt=2tot=3andadeadlinemissoccursatt=6when2cannotnishitscomputationintime. Figure5-4. Resolvingcumulativeofoadingfactoreffectsbyanticipatingslacktime.ASEDZLusestheearliest-deadline-rstprioritytoselecttasksforexecutionandanticipatesidletimeleftbetweentwoconsecutivejobarrivals.Itassignstheidletimetotheunselectedtaskswiththeearliestdeadlinetokeepprocessorsoccupiedandavoiddeadlinemisses. 144

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Figure5-5. Sequentialtaskmodeleffects.ThisexampleillustratesoneofthescenariothatcannotbesuccessfullyscheduledbyASEDZL.WhenT=1:[12,12],2:[2,3],3:[3,6],4:[10,12]isscheduledon3processors,oneprocessorisleftidleduringt=5tot=6.Although1and4hasenoughremainingcomputationtime(=7)atthattime,theyarenotallowedtoexecuteontwoprocessorssimultaneously.Asaresult,therearemorethan3criticaltasksinthesubsequentperiodandthe4thjobof2missesitsdeadline. Figure5-6. Fluidscheduleversusapracticalschedule.ThisgureisadaptedfromFigure1in[ 40 ]andFigure3in[ 97 ]). 145

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Figure5-7. Illustrationof(a)T-Lplanesacrosstimeand(b)taskexecutionwithinaT-Lplane.ThisgureisadaptedfromFigure2andFigure3in[ 40 ]. Figure5-8. FlowdiagramandmaindatastructuresoftheEAGLEalgorithm.EAGLEutilizesinformationinReadyQueue(RQ),TaskInfo(TI)andProcList(PL).RQcontainsreleasedandactivejobsindescendingorderoftheirresourcedemands.TIkeepstrackofeverytask'sstatevariables.PLstoresindicesoftasksrunningoneachprocessor.EAGLEplanstaskexecutionineachT-LplaneusingaheuristicfromtheBoundaryFairalgorithm.Then,tasksareselectedbasedontheplannedlocalparametervalueswithprioritypromotionforzero-laxitytasks. 146

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Figure5-9. Anillustrationofparametersrepresentingataskstate. 147

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Figure5-10. Pseudo-codeoftheLD-PandLS-PphasesofEAGLE. 148

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Figure5-11. AnexampleofEAGLEschedule.ThetasksetT=f1:[5,5],2:[4,5],3:[7,10],4:[3,10],5:[1,5]gisscheduledon3processors.Tasksaredispatchedusing(a)McNaughton'sruleand(b)theLS-Proutine. 149

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Figure5-12. AschedulabilitycomparisonofveschedulingalgorithmsinschedulingrandomlygeneratedtasksetswhosetotalutilizationequalsU.(a)whenM=4and1UM,(b)whenM=8and1UM,and(c)whenM=UandU=3,4and8.TheresultsfromLLREFandEAGLEisdenotedbytheoptimalline(=100%). 150

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Figure5-13. Averagesandstandarddeviationsofthepercentagesofoverheads(i.e.,preemptions,migrationsandinvocations)ofeachschedulingalgorithm.ThesimulationsettingsarethesameasthoseinFigures 5-12 (a)and 5-12 (b).(a)whenM=4and(b)whenM=8. 151

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Figure5-14. Pseudo-codeofEAGLE'smode-transitionprocedure. 152

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Figure5-15. AnexampleofanEAGLE-TscheduleforschedulingadynamictasksetT=f1:[0.2,1],2:[0.2,1],3:[0.2,1],4:[0.2,1],5:[0.2,1]gon3processors. Figure5-16. Theutilizationdriftoftask4fromtheexampleinFigure 5-15 153

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Figure5-17. Anexampleof(a)EPOCand(b)EAGLE-TschedulesforthestatictasksetinFigure 5-11 154

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Figure5-18. Anexampleof(a)EPOCand(b)EAGLE-TschedulesforthedynamictasksetinFigure 5-15 155

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Figure5-19. TheaveragesandstandarddeviationsofmaximalutilizationdriftanditspercentagepertimeunitoftheEAGLE-SandEAGLE-Tschedules.Eachexperimentcreatesandschedules1500pseudo-randomlygeneratedtasksetswithvarying(a)probabilitiesofMCRand(b)numbersoftasksinthesystem. 156

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Figure5-20. Theaveragesandstandarddeviationsofmode-transitiondelayintheEAGLE-SandEAGLE-Tschedules.Thegeneratedtasksetshavevarying(a)probabilitiesofMCRand(b)numbersoftasksinthesystem. Figure5-21. Theaveragesandstandarddeviationsofcyclesinmodetransition(representedas%)intheEAGLE-SandEAGLE-Tschedules.Thegeneratedtasksetshavevarying(a)probabilitiesofMCRand(b)numbersoftasksinthesystem. 157

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Table5-1. ThetaskparametersoftheEAGLEschedule.ThetasksetT=1:[5,5],2:[4,5],3:[7,10],4:[3,10],5:[1,5]isscheduledon3processors. k 0(t=0)1(t=5)2(t=10) lagi 1000 2000 30)]TJ /F7 7.97 Tf 6.59 0 Td[(5 100 405 100 5000mwi+owi 1x5+05+0 2x4+04+0 3x3+13+0 4x1+02+0 5x1+01+0pwi 1x00 2x00 3x5 100 4x5 100 5x00 158

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Table 5-1 .Continued. k 0(t=0)1(t=5)2(t=10) i 1xxx 2xxx 3x-x 4x-x 5xxxUFi 1xxx 2xxx 3x5 7x 4x5 3x 5xxx 159

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CHAPTER6CONTROL-SCHEDULINGAPPROACHFORENSEMBLESYSTEMSUNDERUNCERTAINTY Thischapterconsidersvarioussourcesofuncertaintyinadditiontothechangesinexpertresponsibilitieswhenmakingschedulingdecisionsforensemblesystems.Innon-dedicatedenvironmentswhereensemblesystemsshareresourceswithotherapplications,theamountofresourcesavailablefortheensemblesystems'operationcoulductuateconsiderablyovertime.Previouslypublishedworkusuallyassumeswell-denedtasks'characteristics,i.e.,thetasks'worst-caseexecutiontimes(WCETs)canbeestimatedaccurately.Duetomanyreasons,theexecutiontimesofrealisticandcomplextaskscanbeuncertainandimprecise,anddeterminingWCETswithoutbeingoverlypessimisticisanon-trivialproblem.Tohandlethesekindsofuncertainty,thisdissertationproposestheuseoffuzzy-logicandfeedback-controltechniquestointelligentlydeterminehowtheEESmanagershouldadjustresourceallocationtotaskswithrespectto(1)changingresourceavailabilityand(2)imprecisetaskexecutiontimes. Section 6.1 providesaformulationofthefeedback-schedulingproblemofensemblesystemsunderuncertainty.Section 6.2 presentsanarchitectureofanefcientfeedback-schedulingco-designmanager,calledFuzzyEES,forensembleschedulingunderresourcelimitationsanduncertainty.TheFuzzyEESmanageristhecombinationofaFuzzy-logicinferencesystem(FZ),aTaskUtilizationAdaptor(TUA)andaReal-timeTaskScheduler(RTS)workingharmoniouslyinaclosedloopthatincludestheensemblesystemtobecontrolled.InSection 6.3 ,animplementationoftheTUAusedinthischapterisexplained;itisrathersimilartotheTT-TopalgorithmdiscussedinChapter 4 withaslightmodicationtoincludeafastandsimplegreedyheuristicforadjustingresourceallocationwhenachangeduetoeitheruncertainsystemworkloadorvaryingtaskexecutionbehavioroccurswithinanensemblecycle.TheEAGLE-Talgorithm(presentedinChapter 5 )isalsoextendedconsiderablytobeusedasanimplementationoftheRTS.TheextendedEAGLE-Tsupportingensembletaskswithuncertainexecution 160

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timesisexplainedinSection 6.4 .Section 6.5 providesabackgroundonfuzzy-logicinferenceanddescribesthedesignoftheFZindetail.Acase-studyensemblesystemwithanapplicationinsimpliedroboticcontrolisintroducedinSection 6.6 .Thiscase-studysystemisusedforFuzzyEES'sperformanceevaluation.Theperformanceofthecase-studyensemblesystemundertheschedulingcontrolofFuzzyEESiscarefullystudiedwheneachtypeofuncertainty,i.e.,resourcecapacityandexperts'executiontime,existsexclusivelyandinclusively.Thesimulationresultsofschedulingthelimited-resourcesystemwithFuzzyEESarepresentedanddiscussedincomparisonwiththelimited-resourcessystemscheduledwiththeopen-loopEESmanager.Finally,theconclusionsareprovidedinSection 6.8 6.1ProblemFormulationofRobustEnsembleSchedulingunderUncertainty Beforeformulatingtherobustensemblescheduling,notationsareintroducedinTable 6-1 .ConsideranensemblesystemconsistingofNreal-timetasksonMprocessorswhereM0and"c>0,theensemblesystemmayunderutilize(i.e.,Us>Ua)oroverutilize(i.e.,Us
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minimizejUs(t))-221(Ua(t)j (6)subjecttoNXi=1Ui(t)Us (6)UiminUiUimax;8i=1,...,N (6) Theoptimizationgoaloftheaboveproblemavoidsunderutilizingoroverutilizingthesystemresourcesbymakingthetotalutilizationoftheresourcesasclosetoitsallowableamountaspossible.Atthesametime,theutilizationconstraintsensurethatnotaskreceivesfewerresourcesthanitminimallyrequiresandtheexcessiveoverloadisprevented. 6.2ArchitectureoftheFuzzyEESManager Figure 6-1 showsthearchitectureoftheproposedrobust-schedulingmanager,calledFuzzyEES,featuringafuzzyfeedback-controlloopthatdynamicallyadjuststhetotaltask-resourcedemandusedbytheTUAinordertofullyutilizetheavailablesystemresourcesandachieveasgoodperformanceaspossiblewhenuncertaintyexists.FuzzyEEShasanadditionalcomponentbesidestheTUAandRTSoftheoriginalEESmanager:aFuzzy-logiccontroller(FZ).ThisFuzzyEESmanagermaybelocatedonaseparateprocessororshareaprocessorotherapplications.Ithowevermustbescheduledasthehighestprioritytasktoeffectivelycontrolresourceutilizationofthesystemunderoverloadconditionspossiblycausedbytheconsidereduncertainty.Thedeployenvironmentisequippedwithautilizationmonitor,aresource-distributionpolicyandaratemodulator.Theutilizationmonitorsensestheamountofresourcesactuallybeconsumedbyapplicationsresidingintheenvironment.Usinginformationearnedfromtheutilizationmonitor,theadministratoroftheenvironmentcandecidehowmuchresourcesshouldbedistributedforthedeploymentoftheensemblesystemaccording 162

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totheresource-distributionpolicy.Theratemodulatoradjuststherateofexecutionforindividualtasks. TheFuzzyEESmanagerisinitializedbythenominalnumberofprocessorsavailablefortheensemblesystem'sdeployment(denotedasM),thedegreeofresourceuncertainty(denotedas"m),thenominaltaskWCETs(Ciforalli=1,...,N),thedegreeofWCETuncertainty(i.e.,"c),andtheutilizationconstraintsofeachtask(i.e.,UiminandUimaxforalli=1,...,N).Thefollowingfeedback-controlloopisinvokedattheendofeachsamplingperiodt: 1. Theamountofprocessorsavailableforthedeploymentoftheensemblesystem,denotedasUs(t)isupdatedbasedonanenvironment'spolicy. 2. Theutilizationmonitorsendsthemeasuredutilizationvaluesfromallprocessorsusedfortheensemblesystem'sdeploymentinthelastsamplingperiodtotheFZ.Thesumoftheseutilizationvaluesrepresentstheactualresourceutilizationofthesystem,denotedasUa(t). 3. TheFZcomputesUdfromtheinformationabouthowfaroffthecurrentresourceutilizationofthesystemisfromtheamountofresourceavailable(i.e.,Us(t))-66(Ua(t))andhowUachangesfromtheprevioussamplingperiod(i.e.,Ua(t))-269(Ua(t)]TJ /F6 11.955 Tf 12.52 0 Td[(1))(Note:inthecasewherenouncertaintyexists,UdisintuitivelysettoUs(t)). 4. TheTUAdeterminesasetofUi(t)values(i.e.,avectorU(t))thatsatisestheutilizationconstraints 6 and 6 mentionedintheprevioussection. 5. IfU(t)changesfromU(t)]TJ /F6 11.955 Tf 12.29 0 Td[(1),theratemodulatorsendsarequesttochangethetasks'resourceallocationsfromU(t)]TJ /F6 11.955 Tf 11.98 0 Td[(1)toU(t)(calledamode-changerequestinSection 5.1 )totheRTS. 6. TheRTSperformsamodetransitionandschedulestasksforexecutingandproducingoutputs. Next,extensionsoftheTUAandtheRTSbeyondthosethathavebeenpresentedinthepreviouschaptersarealsodescribed.Theseextensionsaremandatoryforfullyachievingthebenetofthefuzzy-logiccontrollerinFuzzyEES.Tobespecic,sincethesamplingperiodoftheFuzzyEESispreferablyinsmallerthananensemblecycle,itismoreefcienttouseasimpleandfastheuristicintheTUAthatcanaccountforadaptingtask-resourcedemandsduetotheoccurreduncertainty.Moreover,theRTSneedstobe 163

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awareofuncertainWCETsandcanallowreclamationofunusedexecutiontimewhentasksconsumelesstimethanexpected.Lastly,thedetaileddescriptionoftheFZarelaterprovidedinSection 6.5.2 6.3TheModiedImplementationoftheTUA InChapter 4 ,theTT-Topalgorithmisshowntobeanefcientalgorithmforadjustingresource-utilizationallocationtotaskswithineachensemblecycle.Insummary,TT-TopusestheO(N)sensitivity-analysisproceduretoexamineifeithertheprior-cycleallocationorthepredictedallocationcalculatedoffthecriticalpathofexpertexecutionduringthepriorcyclecanbereusedforthechangedresponsibilities.Ifoneofthesepredeterminedallocationscanbereused,anexpensiveexecutionoftheTask-Compressionalgorithmisavoided.Otherwise,TT-Topapproximatesanewallocationbyusingasimpleheuristicthatassignsthemaximumamountofresourcesrequiredbyexpertsinthedescendingorderoftheirresponsibilities.Infact,thesimpleheuristicthatTT-Topusesisthegreedysolutionofthefollowingconstrainedoptimizationproblem.maximizeNXi=1wi(t)Ui(t)subjecttoNXi=1Ui(t)UdUiminUi(t)Uimax;8i=1,...,N ThegeneralformoftheaboveproblemisknownasFractionalKnapsackproblemanditcanbesolvedinalineartimewithrespecttothenumberoftasks.InChapter 4 ,wi(t)=ri(t).Withoutlossingenerality,wi(t)canbesubstitutedbyanotherfunctionand,inthischapter,wi(t)isassignedtober2i(t)Ei(t)whereEi(t)indicatestheerrormeasurementofeachtaskattimet.Themotivationforthisfunctioncomesfromexperimentalevidencethatitcanbettercapturethebehaviorofthesystemwhentheresourcemanagementcannotrelysolelyontheexpertresponsibilities.Anexampleof 164

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suchthecaseisanensemblesystemwhoseexpertresponsibilitiesaredeterminedfrominputspacepartitioningandconceptdriftsoccurovertime.Asimplealgorithmshownbelowcanbeusedtogetanoptimalsolutiontotheaboveproblem. TheTUA'sGreedyHeuristicsSorttasksinthedescendingorderofr2iEii=1;remainingU=Ud)]TJ /F10 11.955 Tf 11.96 8.97 Td[(PNi=1Uiminwhile(iN)Ui=min(Uimax,extraU+Uimin)remainingU=remainingU)]TJ /F6 11.955 Tf 11.96 0 Td[((Ui)]TJ /F3 11.955 Tf 11.96 0 Td[(Uimin)end Becauseofthesorting,thecomplexityofthealgorithmisO(NlogN),howeveritcanbeimprovedfurthertoO(N)timeusingtheweightedmedianalgorithmsimilartotheonepresentedin[ 86 ]. 6.4ExtendedEAGLE-TAlgorithm Inthepreviouschapter,theEAGLE-Talgorithmispresentedinthecontextofgeneraladaptivereal-timesystemswherethereisnouncertaintyinvolved.WhenapplyingEAGLE-TtoscheduletasksinensemblesystemswithuncertaintyinresourcecapacityandtaskWCETs,someconceptualconnectionandmodicationinmechanismsareneeded.First,Section 6.4.1 identiesthedifferencebetweenWCETsofgeneralreal-timetasksandWCETsoflimited-resourceensembletaskswhenthereisnoWCETuncertainty.TheunderstandingaboutthedifferencecanbeusedtoderivetheconditionstodetermineifataskinstancewillbeconsideredascompletedwhenscheduledbyEAGLE-T,anddetermineproperactionswhentasksnishtheircomputationssoonerorlaterthantheirnominalexecutiontimes.Afterthat,Sections 6.4.2 and 6.4.3 considerwhenWCETuncertaintyexistsandpresenttheextensionsof 165

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EAGLE-Tsothattheschedulercanallowtaskstoutilizeresourcesproperlywhenthesystem'sresourcesareeitherunderutilizedoroverutilizedduetotheWCETuncertainty. 6.4.1FromGeneralAdaptiveReal-TimeSystemstoLimited-ResourceEnsembleSystemswithUncertainty Mostexistingreal-timeschedulersdescribedintheliteraturewerederivedupontheassumptionthattaskWCETsareaccuratelyspecied,i.e.,theactualcomputationtimesoftaskscanneverexceedtheirWCETs.WhileEAGLE-TpresentedinChapter 5 isnoexception,itoffersmoreexibilitythanmanyotherschedulerssinceitrelaxestheconstraintthattheexecutionpatternofeachtaskmustremainstaticthroughoutitslifetime.EAGLE-Tallowstaskstohavevaryingbutprecisely-knownWCETsovertime.Duringatask'sintervalD1,ifatask'sWCETisCi(D1),thetaskcanalwayscompleteitscomputationifitexecutesfornolessthanCi(D1)withinthatinterval.DuringanothertimeintervalD2,theWCETofthetaskchangestoCi(D2)(whichcanbeindicatedinthemostrecentmode-changerequestasdescribedinthepreviouschapter)andexecutingthetaskforthatmuchtimeguaranteesthatataskinstanceiscompleted.Inotherwords,thedescribedsituationcanbethoughtofasascenariowherethetaskactuallyhasonepessimistically-determinedWCETCi(),whichislongerthanCi(D1)andCi(D2).Astimeprogresses,moreinformationmightbecomeavailableandatighterWCETduringaspecicintervalofthetask'slifetime(i.e.,Ci(D1)andCi(D2))canbeknown(eitherbypredictionalgorithmsorsuggestionsofapplicationexperts). Whenconsideringtasksinthecontextofensemblesystemswithlimitedresources,therearetwokindsofWCETs:therstkindofWCETsthatactuallycomefromthetasks'specication,andtheotherWCETsthatareassignedbytheTUAwhentheTUAadjuststhetotalresourceutilizationdemandtomatchwithresourceavailability.LetconsiderthescenarioinFigure 6-2 asanexample.Similarnotationstothoseintroducedinthepreviouschapterareused(i.e.,i:[Ci,Pi]).Theguredemonstratestheschedulesof3expertson3processors(theunlimited-resourcesystem)and3 166

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expertson2processors(thelimited-resourcesystem).Inthisexample,eachensemblecycleconsistsof10timeunits.Thescaleshowninthegureis2timeunitspertickonalltimelines.EachexperthasthesameWCET,whichequals10timeunits.Fortheunlimited-resourcesystem,alltasksalwaysget10timeunitsineachcycle.Incontrast,thelimited-resourcesystemcanonlyexecute20timeunitsaltogetherwithineachcycle,sotheTUAhastoassigntotaskstheirnewspecicationsforschedulingpurposes.Thesetasks'schedulingspecicationsarewrittenineachtimeunitabovethetasks'timelines.FromFigure 6-2 ,1getsall10unitssinceitisthewinner,while2and3receives5timeunitseach(giventhattheunrankednon-winnerpolicyintroducedinChapter 4 isused).TherstkindofWCETsforallthetasksarethesame,i.e.,10timeunits(writtenasCiinthetasks'specicationsontheleftoftimelines).ThesecondkindofWCETsfor1,2and3are10,5and5respectively.Asexpertresponsibilitieschange,say3becomesthewinnerandtheothersarenon-winners,thevaluesofthesecond-typeWCETsforthreetaskschangeto5,5and10,respectively.Hence,whenusingEAGLE-Tforschedulingensembletasks,EAGLE-Talwaysworkswiththesecond-typeWCETs.Whilethisscenariomightappeardifferentfromthescenarioexplainedinthepreviousparagraph,acorrespondencecanbemadebyconsideringtherst-typeWCETsasCi()andthesecond-typeWCETsasCi(D1)andCi(D2),respectively.Fromthispointon,therst-typeWCETsarethemainfocuswhenmentioninguncertaintyanditisdenotedasCi.Generally,Ciisthesameasthelengthofanensemblecycle.Thesecond-typetime-varyingWCETsaredenotedwithC0i(t)toavoidconfusion. ThethickbarsonthetimelinesinFigure 6-2 indicatethecompletionoftaskinstances.Inthegeneral-taskcase(whichissimilartothecaseoftheunlimited-resourcesystem),ataskinstancecompleteswheneachtaskreceivesC0i(t),whosevaluesarealwaysxedtoCi.However,taskinstancesof2and3inthelimited-resourcesystemareconsideredcompletedoncetheyacquireexactlytheirnominalexecutiontimes 167

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Ci,i.e.,10timeunits.Whenschedulinggeneralreal-timetasks,EAGLE-Tistypicallyinvokedbyasysteminterruptwhenanyofthreeeventsoccurs:ajobarrives,ajobcompletesandajob'slaxityreacheszero.WhenEAGLE-Tschedulesensembletasks,alltheseeventsareassociatedwithUi(t)assignedbytheTUA.LettheensemblecyclehavethelengthP,theTUAassignstheperiodsofalltasksasPi(t)=Pforallt.Eachtask'sutilizationcanberoundeddownifnecessarytogetanintegerWCETci(t)=Pi(t)Ui(t).Asaresultofthisassignment,ajob-arrivalinterruptisalwaysissuedatthebeginningofanensemblecycle.Ajob-completioninterruptinvokestheschedulingroutinewhenataskinstancenisheditscomputation.Thezero-laxityinterruptisalsoraisedwhenataskneedstoexecuteimmediatelytoaccumulateenoughexecutiontimeasindicatedbyci(t).Atthebeginningofthecycle,unnishedtaskinstancesaretypicallyabortedandthenextinstanceisreleasedimmediatelyafterthat.Dependingonthetypeofensemblesystems,insteadofabortingtheseunnishedinstances,someofthemshouldcontinuetheirexistingcomputationsduringthefollowingensemblecycle.Forexample,ifthereislearninginvolved,non-winnerscankeepexecutingtheirtrainingroutinesinthefollowingensemblecyclesincetheydonotrequiretocontributetothenaloutputattheendofthatcycle. 6.4.2SlackReclamation Next,letusconsiderthecasewhenWCETsareuncertain,i.e.,ratherthanusingthenominalexecutiontimeCieachtaskhasatime-dependentWCETthatmayshiftfromCi.Thetime-dependentanduncertainexecutiontimeofiisdenotedaseCi(t)[Ci)]TJ /F4 11.955 Tf -458.7 -23.9 Td[("c,Ci+"c].Figure 6-3 showsanexampleofsuchacasewhenWCETsoftaskschangeforthe3rd,4thand5thcycles.Inreality,whensometasksnishtheircomputationssoonerthanexpected,theremainingtimesareleftidle(denotedasgreyboxesinthegure)andcannotbeutilizedtoexecuteothertasksthatmighttakelongertimetocompletetheircomputations.Thus,itisnecessaryforEAGLE-Ttobeawareoftheseslacktimes,sothatitcanfullyutilizeresourcesthatbecomeavailableovertime,which 168

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cansimultaneouslyimprovetheensemblesystem'sperformance.ThemodicationsmadetoEAGLE-Tsothatitappliestoensembletaskswithuncertainexecutiontimesarediscussedinthisandthefollowingsubsections. TheEAGLE-Tscheduleronlydispatchestaskswhoselocalexecutiontime(denotedascifromthepreviouschapter)isnonzero.SincetheselocalexecutiontimesareassignedonlyatthebeginningoftheT-Lplane(denedinSection 5.3.2 ),whenataskinstancecompletesitscomputationusinglessexecutiontimethanthatspeciedbyci,someprocessorsareleftidle.Theseidletimeunitsarecalledslacktimes.EAGLE-Tismodiedsothatanadditionalcheckisperformedwhenthejob-completioninterruptisraisedandlocalexecutiontimeisresettozeroifthejobnishesearlierinordertokeeptrackofanypossibleslacktimepriortotheendofeachT-Lplane. Ontheotherhand,whenanytaskinstancerequireslongertimetonishbutithasfullyconsumeditslocalexecutiontimequota,anadditionalone-shottimerinterrupthandlerisusedtopreempttheinstanceandkeepitinthereadyqueuewithotherjobswhoselocalexecutiontimesarezero(ratherthanjustabortingit).Thepriorityofthesuspendedjobishoweversettobehigherthanothertasksthathavenotoriginallyreceivedanylocalexecutiontime.TheschedulingruleintheLS-Pphase(discussedinSection 5.4.3 )isalsomodiedtoallowexecutionofreadytaskswhoselocalexecutiontimeiszerowhenitdetectsthatsomeprocessorsarenotoccupiedatanytimewithintheT-Lplane,whichonlyhappenswhensomeexistingjobsconsumelesstimeunitsthanexpected. 6.4.3ComputationTimeAnticipationandMode-ChangeRequests Slackreclamationcanincreaseresourceutilizationofthesysteminthecasewhenthereisanytask(withthecurrentallocationbytheTUA)thatdemandsmoreresourcesandcanclaimtheuseofexistingslacktimes.Otherwise,theslacktimesareleftunused.ThiscouldbeanindicationthattheTUAmightneedtoincreasetheoverallresourceallocationfortheensemblesystem(i.e.,Ud)tokeepllinguptheseslacktimes.In 169

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general,whenamode-changerequest(MCR)isissuedbytheTUA(typicallyatthebeginningofeachensemblecycleoratthetimewhenUdchanges),EAGLE-Tusesthetasks'expectedutilizationdeterminedfromC0i(t)toevaluatetheschedulabilityduringitsmode-transitionprocedure.Thisbehaviorcanbetooconservativeiftasks'actualcomputationtimesshiftawayfromtheirnominalvaluesforalongperiodoftime;thesystemcanremainunderutilizedoroverutilizediftheactualcomputationtimesaresmallerorhigherthantheirnominalcomputationtimes,respectively. InordertoimproveEAGLE-T,theinformationabouttheactualutilizationconsumedbytheensemblesystemcanbeusedbytheschedulertohaveanupdatedknowledgeoftheactualbehavioroftasks.Thenumbersoftimesthattasksuseexactlytheirnominalexecutiontimesarerecordedover10consecutiveensemblecyclesandtheyareusedtocalculatetheprobabilitiesthatthenominalexecutiontimeofeachtaskinthefollowingensemblecyclewillbemis-estimated.Iftheprobabilitiesarehigherthanacertainthreshold()]TJ /F5 7.97 Tf 6.78 -1.79 Td[(oor)]TJ /F5 7.97 Tf 6.78 -1.79 Td[(uforwhenthetask'scomputationtimeexceedsoriswithinthenominalexecutiontime,respectively),thelatestknowncomputationtimesareusedwhenthemode-transitionprocedureestimatestheamountofresourcescurrentlyoccupiedbyexistingtasks.Adjusting)]TJ /F5 7.97 Tf 6.78 -1.79 Td[(otoasmallvaluemakestheschedulertobemoreconservativesinceitbelievesthattaskWCETsareunderestimated.Incontrast,small)]TJ /F5 7.97 Tf 6.78 -1.79 Td[(udrivestheschedulertoberatheroptimisticandgeneroustogiveawaymoreresourcesifthemode-changerequestrequireshigherresourceutilizationforsometasks.Thistechniqueismotivatedbythenatureofthemanyensembleapplicationsthatgenerallydonotchangetooabruptlybutrathersmoothlyovertime.Usingprobabilitiesmakestheschedulerawareofthechangingtasks'executionbehaviorsandassistsonhowitshouldefcientlyallocateutilizationsforupcomingmode-changerequests.Whenthesystemisunderutilized,modetransitioncanbedonefastersinceresourcesmightbereclaimedsoonerfromsometaskswhonishtheirexecutionsusingsmalleramount 170

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oftime.Increasingoverallworkloadshouldbedelayediftheexistingtasksalreadyusemorethananticipated. 6.5TheFuzzy-LogicController 6.5.1BasicsofFuzzy-LogicInference Fuzzy-logicinferenceisawell-knowntechniqueusedinresearchinmanyapplicationareas.Thethoroughexplanationaboutitsmethodologycanbefoundin[ 91 92 ].Thissectionprovidesonlyabriefsummaryofhowfuzzy-logicinferenceworks.Ratherthansolvingacontrolproblembasedonmathematicalsystemmodels,fuzzy-logicinferenceallowstheuseofdescriptivelanguageandasimplerule-basedinference.Themechanismoffuzzy-logicinferenceissimilartousinghuman'sintuitioninsolvingdaily-lifeproblems.Foreachvariableinvolvedinthefuzzy-logicsystem,meaningfullinguisticvariablesandtheirmembershipfunctionsaredened.Anonlinearmappingbetweenthefuzzy-logicsystem'sinputsandoutputscanbeimplementedbyutilizingfourcomponents:fuzzier,rulebased,inferenceengineanddefuzzier.Aschematicrepresentationofafuzzy-logicsystemispresentedinFigure 6-4 .Acrisp(ornumeric)inputentersthesystem,andthefuzzierconvertsthecrispinputvaluesintofuzzyvaluesdenedbythelinguisticvariablesandtheirmembershipfunctions.Theinferenceengineusesthefuzziedinputsandtherulesstoredintherulebasetoproduceafuzzyoutput.Fuzzyrulesaredenedassimpleif-thenrules,i.e.,IFXANDYTHENZwhereX,YandZareinaformofaproposition.Thesepropositionsmustbemeaningfulsuchthateitheratrueorfalsevaluecanbedeterminedbyapplyingthreefundamentaloperationsintraditionalpropositionallogic,i.e.,conjunction(AND),disjunction(OR)andimplication.TheIFpartcontainspropositionsthatincludeinputvariables,whiletheTHENpartinvolvesoutputvariables.Theseoutputsearnedfromtheappliedrulesneedtobeaggregatedandconvertedbacktoacrispvalueusingthedefuzzier. 171

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6.5.2TheDesignoftheFuzzy-LogicController(FZ) TherststepinthedesignoftheFZistoidentifytheinputandoutputvariables.AsstatedearlierinSection 6.2 ,theinputsoftheFZareuDianduRate.Thesevariablescanbecalculatedusingthefollowingformulas. uDi=(Ua(t))-222(Us(t))=uDiRange(6) uRate=(Ua(t))-221(Ua(t)]TJ /F6 11.955 Tf 11.96 0 Td[(1))=uRateRange(6) whereuDiRange=max("c,"m)anduRateRange=2uDiRange.ThecrispvaluesofuDicanbefrom-7to7,whileuRatevariesfrom-10to10.TheFZhasoneoutputvariable:(0.21.8),whichcanbeusedasascalingfactorforcalculatingUd(t)=Ud(t)]TJ /F6 11.955 Tf 11.96 0 Td[(1). Besidesthecrisp-valuespecication,fuzzysetsmustalsobeidentiedforeachofthesevariables.Afuzzysetisasetoflinguisticvariableswithoutacrisp,clearlydenedboundary.Itcancontainelementswithonlyapartialdegreeofmembershipdenedbymembershipfunctions.Amembershipfunctionrepresentsacurvethatdeneshoweachpointinthevariablespaceismappedtoadegreeofmembership()between0and1.ThelinguisticvariablesforuDiindicateifthesystemresourcesareExtremelyOverutilized(EO),VeryOverutilized(VO),Overutilized(O),Exactlyutilized(X),Underutilized(U),VeryUnderutilized(VU)andExtremelyUnderutilized(EU).TheirfuzzymembershipfunctionsareshowninthetopgraphofFigure 6-5 .Torepresentthechangingdirectionofresourceusage,FastDecrease(FD),Decrease(D),Same(S),Increase(I)andFastIncrease(FI)areusedasuRate'slinguisticvariables(themiddlegraphinFigure 6-5 ).For,themembershipfunctionsforExtremelySmall(ES),VerySmall(VS),Small(S),RatherSmall(RS),Medium(M),RatherBig(RB),Big(B),VeryBig(VB),ExtremelyBig(EB)areshowninthebottomgraphinFigure 6-5 .Table 6-2 describesallpossiblerulesusedintherulebaseoftheFZ.Eachofthemcanbe 172

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expressedasIFuDiisuDirANDuRateisuRatecTHENisr,c,forallr=1,...,7andc=1,...,5. Themembershipfunctionsandtherulesintherulebaseareconstructedbasedontheauthor'sknowledgeandexperienceinsimulationstudiesofensemblesystemsandreal-timescheduling.Whenthesystemresourcesareoverutilized,uDiispositivesincetheactualutilizationexceedstheavailableresourcecapacity.Ontheotherhand,whentheactualresourceutilizationislessthantheavailableamountofresources,uDibecomesnegativeandtheensemblesystemisconsideredunderutilizingthesystemresources.Theresourceutilizationdecreasesifthecurrentresourceutilizationislessthanthatintheprevioussamplingperiod.Otherwise,theresourceutilizationincreases.Intuitively,whenresourcesareoverutilized,thesystemloadshouldbereducedsothatthesystemcanbebroughtdowntoastablecondition.Hence,mustbeasmallnumbersothatUd(t)alsobecomessmallerthanUd(t)]TJ /F6 11.955 Tf 12.88 0 Td[(1).Asbecomesbigger,Ud(t)isenlargedandthesystemcanutilizemoreresourcesmadeavailabletoit.Thesefuzzymembershipfunctionsandruleswerene-tunedviaextensivesimulationrunswiththecase-studyensemblesystem(presentedlaterinSection 6.6 ).WhiletheperformanceresultsshownlaterinSection 6.7 arespecictothestudiedensemblesystem,thedesignschemeoftheFZisgeneralandalsoapplicabletootherensemblesystems. AsmentionedbrieyinSection 6.5.1 ,thescalarvaluesofinputvariablesmustrstbetransformedintofuzzyvaluesbytheFZ'sfuzzierusingthedenedmembershipfunctions.AsshowninFigure 6-6 ,anx-coordinateoneachmembershipfunction'scurverepresentsacrispvalueofaninputvariable,whileay-coordinategivesafuzzydegreeofmembershipofthevariable.Theantecedentofafuzzyrule(i.e.,theIFpart)mayrequireapplyingoneormorelogicaloperations,suchasANDandOR.Forevaluatingtheantecedentinvolvingfuzzy-logicvaluesratherthanbinary-logicvalues,ANDandORoperationscanbeachievedbytheminandmaxfunctions,respectively.Forexample,thestatementuDiisAANDuRateisBisevaluatedbymin((uDi==A), 173

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(uRate==B)).Then,thefuzzyoutputisdeterminedbyapplyingthefuzzy-logicvalueoftheantecedenttotheconsequent(i.e.,THENpart)usingimplication.ThisprocessisdonebytheinferenceengineoftheFZthroughtruncation(Figure 6-7 ).Analogoustothebinary-logiccase,iftheantecedentistruetosomedegreeofmembership,thentheconsequentisalsotruetothatsamedegree.TwoormorerulescouldbeappliedtoproducetheoutputoftheFZ.Eachruleinferenceresultsinanoutputfuzzysetandallthesefuzzysetsarethenaggregatedintoasingleoutputfuzzyset.Finallytheresultingsetisdefuzzied,orresolvedtoasinglenumber(asshowninFigure 6-8 ).TheoverallstructureoftheFZisshowninFigure 6-9 andasurfaceplotofitsoutputsisprovidedinFigure 6-10 6.6ACase-StudyEnsembleSystem Thissectiondescribesanensemblesystemusedforevaluationpurposeoftheworkinthischapter.Thecase-studyensemblesystemisasyntheticensemblesystemwithanapplicationinasimpliedroboticcontrol.Aroboticarmiscomposedoftwoseriallinkswhichareafxedtoeachotherbyrevolutejointsfromthebaseframethroughtheend-effector,asshowninFigure 6-11 .Tocontrolthemotionofthisroboticarm,twotypesofkinematicrelationshipsforwardandinversekinematicsareneeded.Forwardkinematicscalculatethepositionandorientationofthearm'send-effectorwhenitsjointanglesareknown.Ontheotherhand,inversekinematicsrepresentthereverseoftheforward-kinematicprocess.Givenadesiredeffector'spositionandorientation,inversekinematicsdeterminetheanglesofallthejointstoachievethedesiredposition.Theensemblesystemconsistsof48experts,eachexecutingalinearneuralnetworkforperformingplanar-basedrobotinversekinematics.Theneuralnetworkhasasinglelayerwith2inputsand1output.Foradesiredpositionina2-dimensionalinputspace,theensemblesystemneedstodeterminethetwoanglesinvolved:1betweentherstlinkandthebase,and2betweenthetworoboticlinks. 174

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Thetrainingdataiscreatedasfollows.LetthelengthoftherstlinkbeL1andthatofthesecondlinkbeL2.Itisassumedthattherstjointcanrotatebetween0and90degrees,whilethesecondjointhasmoreexibilityandthepossiblevaluesofitsanglearebetween0and180degrees.Foreverypossiblecombinationof1and2,the(x,y)coordinatesarededucedusingthefollowingforwardkinematicsformulas. x=L1cos(1)+L2cos(1+2)(6) y=L1sin(1)+L2sin(1+2)(6) TheexpertsandgatingnetworkintheensemblesystemaretrainedofineusingtheExpectation-Maximization(EM)algorithm[ 37 ].Expertsaredividedintotwogroups:onecomputing1andtheothercomputing2asafunctionofarobot's(x,y)coordinate.Eachexperthasanunknownprobability,calledaresponsibilityri,ofitprovidingtherightvalueofoneofthejointangles.EMusesaniterativeandstatisticalapproachfordeterminingnetworkweightsoftheexpertsandgatingcomponentthatgivesthemaximumlikelihoodsbyperformingtwostepsineachiteration:E-StepandM-Step.Startingfromsomeinitialweights,E-Stepattemptstoguesstherobotjointanglesandcomputeaprobabilitydistributionoverallpossiblevaluesofriusingthecurrentweights.Theseprobabilitiescreateasetofweightedtrainingsamplesthatamaximum-likelihoodestimationprocedureinM-Stepcanusetore-estimatethenewmodelweights.ByalternatingbetweentheseE-andM-Stepsinseveraliterations,thenetworkweightsconverge.Figure 6-12 showsthe3-dimensionalsurfaceplotsofexpertresponsibilitiesontheinputspace.Toreducecluttering,eachsubplotshowsresponsibilitiesof8expertsatatime.Theinput-spaceregionscoveredbyeachexpertareprovidedinFigure 6-13 .Thecentroidsofexperts'regionsaredenotedbysmallwhitecirclesinthegure.Foranytargetposition(x,y),outputsfromthreeexpertswithtopresponsibilitiesfromeachgroup(of24experts)areaveragedtocreateanalvalueofeachjointangle. 175

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Inenvironmentswithsufcientresources,all48expertsexecuteconcurrentlyinindividualprocessors.Whenresourcesarelimited,Figure 6-14 showstheperformanceofthecase-studyensemblesystemscheduledwiththeEESmanagerondifferentnumberofprocessorsintheidealcasewherenouncertaintyinvolves.Anerrorforeach(x,y)inputiscalculatedfromthefollowingformula. error=q (1)]TJ /F4 11.955 Tf 11.96 0 Td[(ref1)2+(2)]TJ /F4 11.955 Tf 11.96 0 Td[(ref2)2(6) whereref1andref2arethecorrectjointanglesoftheroboticarmthatyieldthedesiredend-effector'sposition. FromFigure 6-14 ,whenthereare10processorsormore,theEESmanagercanschedulethelimited-resourcesystemtoperformrelativelywellwhencomparedtotheunlimited-resourcesystem(theperformancefor48processors).Hence,intheperformanceevaluation,Missettoeither10or13processors.Becauseoftheunit-capacityassumption,thereisnoperformancegainwhenavailableresourcecapacityislessthanone.Inaddition,itisassumedthattheensemblecycleforthisensemblesystemis10timeunits.Tasknominal-executiontimesCiarelessthanorequalto10timeunits.TheUiminandUimaxforalltasksare0andCi=10respectively.ThedegreeofuncertaintyinresourcecapacityandtaskWCETs(i.e.,"mand"c)are5and5. 6.7PerformanceEvaluation Inanidealenvironment(wherethereisnouncertainty),theEESmanagercanscheduletasksofanensemblesystemsothattheactualresourceutilizationoftheensemblesystemisgenerallythesameastheavailableresourcecapacity.WheneithertheresourcecapacityisuctuatedorthetaskWCETsareimprecise,theactualresourceutilizationcanbelessormorethantheavailableresourcecapacitysincetheEESmanagerstillperceivesthatthesystem'sresourcecapacityandWCETswillbeasexpected.Sofar,thischapterhasprovideddescriptionsabouttheexistingcomponents'extensionsandtheadditionalcomponent'sdesignforimplementingthe 176

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FuzzyEESmanagertohandlecaseswhentheaforementioneduncertaintyexists.Hence,theobjectiveofthissectionistodemonstratethattheFuzzyEESmanagercanscheduletasksinensemblesystemstotolerateuncertaintyinbothresourcecapacityandtaskWCETsanditsperformanceexceedthoseoftheopen-loopEESmanager.First,XEAGLE-TisspecicallycomparedagainstEAGLE-TwhenonlytaskWCETsareuncertain.WhileEAGLE-TalwaysassumethatUdmustbeexactlythesameasUs,XEAGLE-TrelaxessuchanassumptionandallowsthesystemtoadaptseamlesslytoUdupdatedbytheTUA.Asaresult,XEAGLE-Tcanachievebetterperformancebycloselytrackingtheactualresourcecapacity.Then,thewholeFuzzyEESmanager(i.e.,theclosed-loopcontrolwiththemodiedTUA,XEAGLE-TandFZ)isstudiedincomparisonwiththeEESmanager(i.e.,theopenloopwiththeTUAandEAGLE-T)whenonlyresourcecapacityisuncertainandwhenthereisuncertaintyinbothresourcecapacityandtaskWCETs. 6.7.1UncertaintyinTaskWCETsandBenetsoftheExtendedEAGLE-T TheEAGLE-TwiththeextensionsexplainedinSections 6.4.2 and 6.4.3 isnamedtheextendedEAGLE-T(XEAGLE-T).ThissectionpresentsaperformancecomparisonstudybetweenEAGLE-TandXEAGLE-TtovalidatethebenetsoftheextensionswhentaskWCETsareuncertain.Thecase-studyensemblesystem(introducedinSection 6.6 )isscheduledusingbothalgorithms.Theresourcecapacityisconsideredstaticinthisstudy.Eachensemblecyclehas10timeunits.Fourmainscenariosareexamined:accurateWCETs,overestimatedWCETs,underestimatedWCETsandtemporarilymis-estimatedWCETs.Toillustratewhathappensineachscenario,twographswillbeshownineachgure:oneforEAGLE-TandtheotherforXEAGLE-T.Ineachgraph,Udisshowninadashedline,Usdenotesthenumberofavailableprocessors(showninasolidline),Uarepresentstheactualresourceutilizationsofthesystem(shownasasolidlinewithcrossmarks)andthenumberoftasksexecutedwithineachtimeunit(ornExecuted)isshowninadottedline.Therearefourperformancemeasurementsof 177

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interestinthisstudy.Themeannumberofexecutedinstancespertimeunitisequivalenttoanaverageofthenumberofprocessorsoccupiedineachtimeunit.Thetotalnumberofcompletedinstancesreferstothenumberoftaskinstancesthatreceivecumulativeexecutiontimesnotlessthantheiractualcomputationtimes.Thesystem'sperformanceisindicatedbythemeanerroraccordingtoEq. 6 .ThemeanjUa)-308(Usjprovidesavalidationofhowwelltheschedulerperformsinachievingthesystem'sresourceutilizationthatisascloseaspossibletoUs.Inallscenarios,)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(oand)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(uaresetto0.1,whichmeansthatthepreviousexecutiontimesoftasksarebelievedtorepeat. Scenario1:WCETsareaccurate Inthisscenario,eachtaskhasanominalWCETCi=10andthereisnouncertainty,soitscomputationindeedtakes10timeunitstonish.ThereareUs=8processors.Udarevariedfrom4to10todemonstratethatthetotalresourceutilizationsallocatedtotasksbybothalgorithmsaregovernedbyUdsinceUagrowsasUdincreases.WhenUd>8,themode-transitionproceduresofbothalgorithmsdonotallowhavingtotalresourceallocationthatexceedsthecapacityofresourcesinordertoavoiddeadlinemisses.Hence,UaremainsthesameafterUdexceedsUsasshowninFigure 6-15 .Bothalgorithmshavethesamebehaviorinthisscenario. Scenario2:OverestimatedWCETs Similarlytothepreviousscenario,Ci=10andUs=8.However,thetasks'actualcomputationstakeonly8timeunitstocomplete,sotheWCETsareoverestimated.Figure 6-16 showstheresultsfrombothalgorithms.WhilethealgorithmsexpectUa=8becauseofthetasks'nominalWCETs,theactualresourceutilizationsUainbothcaseareonly6.4duringtherst10timeunits.AsUdincreases,EAGLE-Tcannotmakeanyimprovementinutilizingmoreresources,whileXEAGLE-TgetsUa=7.2,whichismuchclosertoUs.ThisvalidatesthattheextensionsofXEAGLE-Tcanefcientlyutilizeslacktimesifanyexists. 178

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TodemonstratethebenetsofXEAGLE-Tinmoredetail,simulationsofthecasewhenWCETsareoverestimatedwereconductedwithvariousvaluesofUsandtheactualexecutiontime.Usvariesbetween5to10processors,whiletheactualexecutiontimeofeachtaskrangesfrom5to10timeunits.Alltaskshavethesameexecutiontimeineachsimulationsetting.Figures 6-17 6-18 6-19 and 6-20 showtheimprovementpercentageofXEAGLE-ToverEAGLE-Tinthemeannumberofexecutedinstancespertimeunit,thetotalnumberofcompletedinstances,meanerrorandmeanjUa)-245(Usj,respectively.WhencomparedtoEAGLE-T,XEAGLE-Texecutesmoreinstancespertimeunit,completesmoreinstances,enablestheensemblesystemtoproduceoutputswithlowererrorandmaintainsthesystem'sresourceutilizationclosertotheavailableresourcecapacity.TheXEAGLE-T'simprovementsforallfourmeasurementsaresignicantwhentheuncertaintyishighandresourcesarehighlylimited.Thiscorrespondstotheleft-mostbarintheleft-mostgroupineverygraph.Themaximumimprovementsforallfourmeasurementsareapproximatelyat66%.Therearenotbigdifferencesinmeanerrorassoonas10expertscanbeexecuted(i.e.,whentheactualexecutiontimeis5to8unitsonthex-axis).ThisisconsistentwiththeresultinFigure 6-14 inSection 6.6 Scenario3:UnderestimatedWCETs Inthisscenario,Ci=5andUs=13.Figure 6-21 presentsthecasewhentheactualcomputationtakes8timeunitstocomplete,sotheWCETsareunderestimatedandthesystembecomesoverloaded.Duringtherst10timeunits,bothschedulerexpectstoscheduleandcomplete10taskinstances.TheactualresourceutilizationforbothEAGLE-TandXEAGLE-Tendupat8,whichalreadyexceedsUd,butstilldoesnotcauseanoverloadsincethereareUs=13processors.NoticethatEAGLE-Tonlyexecutestaskinstancesinalternatetimeunitssinceitbelievesthattheseinstancesonlyrequire5unitseachandthereisnoneedtoexecutethemineverycycle.Ontheotherhand,XEAGLE-Tusesaslackreclamationschemetollinidleprocessorsas 179

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muchaspossible,sothereare10executedinstancesineachtimeunituntillaterineachensemblecyclewhenallinstanceshavecompleted.Becauseofthisdifference,XEAGLE-TisabletonishalargernumberofjobswhileEAGLE-Tdoesnone.AsUdincreases,XEAGLE-Talsocankeepupitsperformanceandlimitthesystemloadatthepointwhennodeadlinemissoccurs,unlikeEAGLE-Twhichjustkeepsthesysteminanoverloadedcondition(fromt=40to90). Figures 6-22 6-23 6-24 and 6-25 showthepercentagesofimprovementofXEAGLE-ToverEAGLE-Tonthefourmeasurements(similarlytothepreviousscenario).Usrangesfrom10to15processors.Theactualexecutiontimevariesfrom5to10timeunits.Again,forthemeannumberofexecutedinstanceperunitandmeanjUa)-232(Usj,theimprovementsofXEAGLE-Taredrasticastheactualexecutiontimeshiftsgreatlyfromthenominalvalueandtheresourcesaremostlimited(correspondingtotheleft-mostbarsintheright-mostgroupofthosegraphs).Themaximumimprovementsforthemeannumberofexecutedinstancespertimeunit,themeanerrorandthemeanactualutilizationareroughly27%,94%and44%,respectively.ThetotalnumbersofcompletedtaskinstancesarenotreportedaspercentagesheresincetheEAGLE-Tcannotnishanyinstance.Atthebestcase,XEAGLE-Tcannish304taskinstancesmorethanEAGLE-T.Unfortunately,asactualexecutiontimebecomesmoreuncertain,smallernumbersofjobscanbecompletedwiththeavailableresources,sotheimprovementonmeanerroralsodropsforthosecasesontherighthalvesoftheseplots. Scenario4:TemporarilyuncertainWCETs Previously,thecaseswhenWCETsarealwaysoverestimatedorunderestimatedareconsidered.Next,thecaseswhenthetasks'actualexecutiontimesmaydecreaseorincreaseforanintervaloftimearestudied.First,letconsiderwhentheactualexecutiontimesdropfromCi=10to5timeunitsduringt=30tot=100.Fig. 6-26 presentsEAGLE-T'sandXEAGLE-T'sresults.AscanbeseenfromFig. 6-26 ,XEAGLE-Tisable 180

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toadjusttheactualresourceutilizationofthesystemaccordingtoUdandbacktothelevelthatiscloseenoughtoUs=8.Uais7.5duringt=70tot=100.AlsotheaveragenumberofexecutedinstancesofXEAGLE-TarehigherthanthatofEAGLE-T;thiscanbeobservedfromthenExecutedlinesinbothplots.InFig. 6-27 ,thecomputationtimesareincreasedfromtheirnominalvaluesCi=5to10timeunitsduringthesameperiodasbefore.ThegureshowsthatXEAGLE-TkeepstheoverloadconditionminimalandclosertoUsthanEAGLE-T. SimulationresultsshowninallscenariosaboveconrmthatXEAGLE-TcanbeusedbyFuzzyEEStoyieldanimprovementinhavingtheactualresourceutilizationthattrackscloselytheavailableresourcecapacity,givenUdthatisdynamicallycongurablebytheFZandTUA. 6.7.2UncertaintyinAvailableResourceCapacity GiventheresultsshowninSection 6.7.1 ,itisclearthattheclosertheactualsystemutilizationistothereferencevalue(i.e.,theavailableresourcecapacityUs),thelesslikelyisthatthesystem'serrorwillgetworse(andcangenerallybemuchimproved).SincethecaseforuncertainWCETshasbeendemonstratedindetailearlier,thisandthefollowingsubsectionspresenttheevaluationresultsoftheentireFuzzyEESforthecaseswhenonlyresourcecapacityisimpreciseandwhenbothresourcecapacityandWCETsareuncertain. Inthisstudy,thenominalvalueofavailableprocessorsMis13andtheuncertainresourcecapacityUscanincreaseordecreasefromthisvaluebynomorethan5processorunits.Aresource-capacitytraceleisgeneratedforasimulationofthecase-studyensemblesystemwith1967ensemblecycles,whereeachensemblecyclelasts10timeunits(totalof19670timeunitspersimulation).UsinthetraceleisrandomlyselectedfromanormaldistributionwithmeanvalueMandavariancevaryingfrom1to5.EachchangeinUsalsolastsforanintervaloftime,whichisassignedbyauniformly-distributedrandom-numbergeneratorwithameanthatrangesfrom50to 181

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500cycles(50-cycleincrement),andavariancethatisone-fthofthespeciedintervalmean.OnehundredtracelesaregeneratedforeachcombinationofUs'svariancesandtheinterval'smeans.Hence,traceleswithahighUs'svarianceandasmallintervalmeanrepresentthecaseswhenthedegreeofuncertaintyislarge.Ontheotherhand,caseswithsmalleruncertaintycanbegeneratedusingasmallUs'svarianceandahighintervalmean.Themeanerrorandtotaldeadlinemissesfrom100simulationtrials(i.e.,oneforeachtracele)areaveraged. Thecomparisonsoftheopen-loopsystem(i.e.,whentheEESmanagerisused)andtheclosed-loopsystem(i.e.,whentheFuzzyEESmanagerisused)areshowninTable 6-3 .Thegraphsofthepercentageoftheclosed-loopimprovementsinmeanerrorareprovidedinFig. 6-28 .TheresultsareconsistentwiththecasewhenconsideringonlyuncertaintyofWCETs.TheFuzzyEESmanageralwaysoutperformstheEESmanagerasresourceavailabilityvaries.Thepercentagesofimprovementinmeanerrorgrowsasuncertaintyincreases(i.e.,meaninterval=50andvariance=5).Fromthisstudy,FuzzyEEScanimprovetheopen-loopsystembyalmost78%inseveralcases.Fromthetable,largeerrorsfortheopen-loopsystemareduetothefactthatithasagreaternumberofdeadlinemisses.However,noteverysingledeadlinemissdirectlyaffectsthequalityofthesystem'soutputs.Thosebelongingtoirrelevantexperts(i.e.,notoneofthetopthreefromeachgroupmentionedinSection 6.6 )areinsignicant.Inensemblesystemswherenon-winnershaveabilitiestolearnovertime,thiscouldimpactthesystem'soutputerrorsignicantly,sothebenetoftheFuzzyEESisanticipatedtobeevengreater.Whencomparingthemeanerrortotheidealmeanerror(ofthelimited-resourcebutno-uncertaintysystem),whichis0.0472fromFig. 6-14 atM=13,theFuzzyEESperformsmuchcloser(approximately0.0761intheworst-case)totheidealmeanerrorthantheopen-loopsystem,whichhasanaverageerrorof0.3445. 182

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6.7.3UncertaintyinBothResourceCapacityandTaskWCETs Thissimulationstudyconsidersacoexistenceofresource-capacityandWCETsuncertainty.Anothersetoftracelesaregeneratedwiththefollowingparameters:M=10,5Us15(themean=8andavariancevaryingbetween2to4),andintervalwithameanvaryingfrom100to500(100-cycleincrement)andavarianceisone-fthofthemean.FiftyUs-tracelesaregeneratedpercombination.FiveeCi-tracelesarealsocreatedtospecifywhenthetasks'computationtimecouldchange.Asimilarapproachforcreatingthetraceleisused,howeverthemeanoftheactualexecutiontimesisxedat8andtheminimumandmaximumvaluesoftheactualexecutiontimesare5and10,respectively.Theintervalofchangesisrandomlyselectedfromauniformdistributionwiththemeanequals500timeunitsandthevarianceequals2.Eachtaskdoesnotnecessarilyhavethesameactualexecutiontimeinthisstudy.EacheCi-traceleisusedforagroupof10simulations.)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(oand)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(uaresetto0.3. Table 6-4 providestheaveragedeadlinemissesandmeanerrorfromtheopen-loopandclosed-loopsystemsandFig. 6-29 showsthepercentageofimprovementinmeanerrorofFuzzyEES.Fromthegureandthetable,FuzzyEESyieldsuptoabout55%improvementinmeanerrorandhassignicantlysmallernumberofdeadlinemissesinallsimulations.Consistentwithotherformerstudiesinthissection,themaximumimprovementhappenswhenuncertaintyarisesveryfrequently. 6.8Summary ThischapterpresentsanextensionoftheEESmanager,calledFuzzyEES,tohandleuncertaintyinresourcecapacityandtaskWCETs.Byincorporatingaclosed-loopfeedbackcontrolbasedonFuzzy-LogicinferenceintotheEESmanager,thetotalworkloadoftheensemblesystemwithlimitedresourcescanbeadjustedincorrespondencetochangingsystemresourcecapacityandvaryingtaskexecutiontime.TheEAGLE-Talgorithmhasbeengreatlyextendedtosupporttheensembletaskandreactproperlytothechangesintheoveralldesiredresourceutilization(calculatedbythe 183

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TUA).TheperformanceevaluationresultsvalidatethebenetsofFuzzyEESovertheopen-loopEESmanagerwhenvariousdegreesofuncertaintyarepresent.Onaverage,FuzzyEEScanreducemeanerrorbymorethan50%ifalargeextentofuncertaintyoccurs. Otherkindsofuncertaintycanalsobeinvestigatedasfuturework.Forexample,insomeensemblesystems,theexpertresponsibilitiesusedinmakingschedulingdecisionsmightnotbepreciselydeterminedduetomanyreasons,suchaserrorfrompredictionorensemble'sconceptdrifts.Insuchcase,itcouldturnoutthatwinnerscanbemisidentied.EarlystudyhasbeenmadetoapplytherecentlyproposedtechniquesinrobustoptimizationtotheTUA'soptimizationproblems(bothformulationsinChapter 4 andSection 6.3 ,inordertoaccountforthecasewhenresponsibilitiesareuncertain.However,athoroughstudyoftherelationshipbetweenrobustnessandsystemperformanceneedstobecarefullydesignedandconductedtogainunderstandingofthetradeoffbetweenthemfortheapplicationsofensemblesystems.Hence,thisisleftasfuturework. 184

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Figure6-1. AfeedbackschedulingarchitectureoftheFuzzyEESmanager. Figure6-2. AnexampleillustratingtwokindsofWCETsinensemblesystemswithlimitedresources. 185

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Figure6-3. ThecasewhenWCETsareuncertain. Figure6-4. Astructureofafuzzy-logicsystem. 186

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Figure6-5. Themembershipfunctionsoftheinput/outputlinguisticvariablesofuDi,uRateand. Figure6-6. IllustrationofhowcrispscalarvaluesofuDianduRatecanbefuzziedtofuzzyvalues. 187

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Figure6-7. Anexampleofaruleinference. Figure6-8. Defuzzicationusingthecentroidcalculationmethod. Figure6-9. TheoverallstructureoftheFZ. 188

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Figure6-10. TheoutputsurfaceoftheFZ. Figure6-11. Anexampleofatwo-jointroboticarm. 189

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Figure6-12. Expertresponsibilitiesofthecase-studyensemblesystem.Eachplotshowsresponsibilitiesovertheinputspaceof8expertsatatime. 190

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Figure6-13. Input-spacecoverageofexpertsinthecase-studyensemblesystem.Eachplotshows8expertsatatime. 191

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Figure6-14. Performanceofthecase-studyensembleasafunctionofnumberofavailableprocessors. 192

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Figure6-15. PerformanceofEAGLE-TandXEAGLE-TwhenWCETsarecertain.Inthisscenario,thecase-studyensemblesystemisscheduledonUs=8processorsusingEAGLE-TandXEAGLE-T.ThetaskWCETsCi=10andthereisnouncertainty.UaincreasesafterUduntilUd>8. Figure6-16. PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareoverestimated.Inthisscenario,theactualtaskexecutiontimereducesfromCi=10to8timeunits. 193

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Figure6-17. ThepercentageofimprovementinthenumberofexecutedinstancespertimeunitwhenWCETsareoverestimated. Figure6-18. ThepercentageofimprovementinthetotalnumberofcompletedjobswhenWCETsareoverestimated. 194

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Figure6-19. ThepercentageofimprovementinthemeanoutputerrorwhenWCETsareoverestimated. Figure6-20. ThepercentageofimprovementinjUa)-221(UsjwhenWCETsareoverestimated. 195

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Figure6-21. PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareunderestimated.Inthisscenario,theactualtaskexecutiontimeincreasesfromCi=5to8timeunits. Figure6-22. ThepercentageofimprovementinthenumberofexecutedinstancespertimeunitwhenWCETsareunderestimated. 196

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Figure6-23. ThepercentageofimprovementinthetotalnumberofcompletedjobswhenWCETsareunderestimated. Figure6-24. ThepercentageofimprovementinthemeanoutputerrorwhenWCETsareunderestimated. 197

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Figure6-25. ThepercentageofimprovementinjUa)-221(UsjwhenWCETsareunderestimated. Figure6-26. PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareoverestimatedtemporarily.Inthisscenario,theactualtaskexecutiontimedropsfromCi=10to8timeunitsduringt=30)]TJ /F6 11.955 Tf 11.95 0 Td[(100. 198

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Figure6-27. PerformanceofEAGLE-TandXEAGLE-TwhenWCETsareunderestimatedtemporarily.Inthisscenario,theactualtaskexecutiontimeincreasesfromCi=5to8timeunitsduringt=30)]TJ /F6 11.955 Tf 11.96 0 Td[(100. Figure6-28. Thepercentagesofimprovementinmeanoutputerrorundervariousdegreesofresource-capacityuncertainty. 199

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Figure6-29. Thepercentagesofimprovementinmeanoutputerrorundervariousdegreesofresource-capacityandWCETuncertainty. 200

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Table6-1. Thelistofimportantnotations. NotationsDescriptions tThecontrolsamplingperiod,whichindicatesthenesttimegranularityinthescopeofthiswork cTheensemblecyclewherect NNumberofexpertsintheensemblesystem MTotalnumberofprocessorsavailableintheenvironmentinwhichtheensemblesystemisdeployed Us(t)Totalresourcecapacity(i.e.,thenumberofunit-capacityprocessors)thatcanbeusedtodeployanensemblesystemattimet UiminMinimumutilizationneededbyanexperti UimaxMaximumutilizationrequestedbyanexperti Ud(t)theTUA'sparameterindicatingthemaximumamountoftotalresourceutilizationthatshouldbeallocatedtoexpertsattimet Ui(t)Amountofresourceutilizationallocatedtoanexpertiattimet Ua(t)Actualresourceutilizationconsumedbyallexpertsattimet U(t)AvectorofUi(t),i.e.,[U1(t),U2(t),...U3(t)] ri(t)Theexperti'sresponsibilityattimet(Note:theresponsibilityonlygetsupdatedonceineachensemblecycle) R(t)Avectorofri(t),i.e.,[r1(t),r2(t),...r3(t)] ei(t)Theexperti'sperformanceindicator(e.g.,outputerror)attimet E(t)Avectorofei(t),i.e.,[e1(t),e2(t),...e3(t)] 201

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Table6-2. TherulebaseoftheFZ. r,cuRatec FDDSIFI uDirEOSVSVSESES VOMRSSVSES OMMRSSVS XBRBMMRS UEBVBBRBM VUEBVBBRBRB EUEBEBVBVBB 202

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Table6-3. PerformancecomparisonbetweenEESandFuzzyEESwhenUsvariesbetween8and18. VarianceMean EES(open-loop)FuzzyEES(closed-loop)ofUsof DeadlinemissesMeanerrorDeadlinemissesMeanerror intervalAvg.Stdev.Avg.Stdev.Avg.Stdev.Avg.Stdev. 15038.3212.870.110.020.000.000.050.00 10020.4411.960.140.040.000.000.050.00 15013.0410.230.130.050.000.000.050.00 2009.167.810.130.060.000.000.050.00 2508.986.700.140.080.000.000.050.01 3005.985.880.150.060.000.000.050.00 3507.708.010.190.100.000.000.060.01 4005.786.540.170.090.000.000.060.01 4503.965.360.160.080.000.000.050.00 5003.925.340.200.100.000.000.050.01 250100.6244.250.150.040.000.000.060.01 10058.8826.480.160.050.000.000.060.01 15040.4825.860.170.070.000.000.060.01 20026.8418.680.200.110.000.000.060.01 25022.8819.180.190.090.000.000.060.01 30023.8823.760.210.120.000.000.060.01 35021.3616.800.220.130.000.000.060.01 40017.3019.080.220.110.000.000.060.01 45011.9211.910.200.120.000.000.060.01 50017.8214.540.230.140.000.000.070.01 203

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Table 6-3 .Continued. VarianceMean EES(open-loop)FuzzyEES(closed-loop)ofUsof DeadlinemissesMeanerrorDeadlinemissesMeanerror intervalAvg.Stdev.Avg.Stdev.Avg.Stdev.Avg.Stdev. 350165.2854.920.180.030.000.000.070.01 10086.2634.860.180.060.000.000.070.01 15062.2427.380.210.090.000.000.070.01 20045.5026.640.240.110.000.000.070.02 25038.4828.080.210.110.000.000.070.01 30036.3430.840.250.120.000.000.070.02 35038.1434.100.260.130.000.000.070.02 40029.4427.400.330.180.000.000.080.02 45027.1426.640.230.170.000.000.050.02 50021.5621.940.250.170.000.000.050.02 450211.1655.910.170.040.000.000.050.01 100107.9651.970.190.060.000.000.050.01 15080.3642.000.220.100.000.000.050.01 20060.9441.350.250.100.000.000.050.01 25053.4841.510.270.150.000.000.050.02 30047.0637.610.300.180.000.000.050.02 35035.4229.300.260.190.000.000.050.02 40040.8227.530.300.190.000.000.050.02 45028.7829.190.210.140.000.000.050.02 50031.3025.660.250.210.000.000.050.02 204

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Table 6-3 .Continued. VarianceMean EES(open-loop)FuzzyEES(closed-loop)ofUsof DeadlinemissesMeanerrorDeadlinemissesMeanerror intervalAvg.Stdev.Avg.Stdev.Avg.Stdev.Avg.Stdev. 550247.2668.060.200.040.000.000.060.01 100133.5653.570.210.080.000.000.060.01 15083.1843.930.190.090.000.000.050.01 20064.0437.790.260.160.000.000.050.02 25056.2638.350.240.180.000.000.050.02 30055.3037.800.300.190.000.000.060.02 35046.4435.670.290.180.000.000.050.02 40040.1030.560.340.200.000.000.060.02 45040.6830.080.280.150.000.000.060.02 50036.3230.110.260.210.000.000.060.02 205

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Table6-4. PerformancecomparisonbetweenEESandFuzzyEESwhenUsvariesbetween8and18andtheactualtaskexecutiontimevariesbetween5to10. VarianceMean EES(open-loop)FuzzyEES(closed-loop)ofUsof DeadlinemissesMeanerrorDeadlinemissesMeanerror intervalAvg.Stdev.Avg.Stdev.Avg.Stdev.Avg.Stdev. 21006147.22791.600.320.030.020.080.140.01 2003305.04679.700.280.030.010.050.140.01 3002426.58615.580.260.030.000.000.140.01 4001938.56651.860.280.040.000.000.140.02 5001563.46810.720.260.050.000.000.140.02 31009187.001010.080.390.030.070.110.180.01 2005685.181276.740.350.040.030.080.170.02 3004295.681055.850.330.040.000.040.170.02 4003340.641081.880.330.060.000.030.170.02 5003107.041087.010.340.060.000.000.180.03 410011618.661262.890.450.030.120.160.210.02 2007305.741201.170.400.040.080.110.200.02 3005426.681163.350.380.050.030.070.200.02 4004617.021340.780.370.060.030.020.200.03 5003604.621000.050.370.050.000.040.200.03 206

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CHAPTER7CONCLUSIONSANDFUTUREWORK Thisdissertationaddressestheproblemofschedulingexecutionofexpertsinensemblesystemsthataresubjecttosize/weightconstraints,powerlimitationsorcostfactors,whichpreventtheavailabilityofallthecomputationalresourcesrequiredtosupportconcurrentexecutionofalloftheirexperts.Insuchsystems,anefcientresource-managementschemeisnecessarytominimizetheimpactoflimitedresourcesontheoverallsystemperformance. 7.1SummaryofContributions Toachievetheabovegoal,thecontributionsprovidedbythedissertationarethefollowing: TheElasticEnsembleScheduling(EES)manager:TheEESmanageristheproposedgeneralizedschedulingarchitectureconsistingofaTaskUtilizationAdaptor(TUA)andanadaptiveReal-TimeScheduler(RTS).TheTUAutilizesexpertresponsibilitiesandoptimizationtechniquestodynamicallydetermineoptimalornearly-optimalresourcedemandsoftasksaccordingtocapacity,executionandlearningconstraints.Criticalexpertswhoseexecutionsareessentialforthefunctionalityofthesystemareguaranteedtoexecuteintimetoproduceresultsneededforgeneratingthesystem'soutput,whileotherexpertsreceiveatleasttheirminimumresourcedemands.UsingtheallottedresourcedemandsfromtheTUA,theRTSdecidestheamountofresourcestobeallocatedtoeachtaskandimplementsanadaptivereal-timetaskscheduleforexpertexecutiononmultiprocessorswhileensuringthatnotemporalandapplication-specicconstraintsareviolated.ThemodulararchitectureoftheEESmanagerdecouplesthedevelopmentofalgorithmsfortheTUAandRTSand,asaresult,allowsgreatexibilityintheirimplementations. Task-Throttling(TT)algorithms:TwoTTalgorithmsareproposedforanimplementationoftheTUA,namelyTT-TC*andTT-Top.Giventheneedtopotentially 207

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adapttaskresourcedemands(i.e.,TT)ineverycycleandthelimitedtimeavailableforitscomputation,aO(N)optimal-solutionsensitivityanalysisprocedureisusedinbothalgorithmstotesttheprevioussetandapredictedsetofoptimalresourcedemands.Ifoneofthemremainsoptimalfortheupdatedexpertresponsibilities,aO(N2)demandreadaptationinthecriticalpathofexpertexecutioncanbeavoided.Whenareadaptationisneeded,TT-TC*usestheTask-Compression(TC)algorithmtorecomputeanewsetofoptimalresourcedemands,whiletheTT-Topalgorithmapproximatestaskresourcedemandsbygivingpreferencetothetop-responsibilityexperts.Dependingonensembleapplications,theoccasionalviolationofexperts'deadlines(inTT-TC*)ordecreaseinaccuracy(inTT-Top)maybeacceptableorcauselittleperformancedegradation.Experimentalperformanceevaluationindicatesthatoverheadcausedbyreadaptationisgreatlyreduced(upto85%forTT-TC*and100%forTT-Top)duetotheuseofaresponsibilitypredictorandsensitivityanalysis. TheEPOCscheduler:Anadaptivereal-timemultiprocessorschedulerimplementedbytheRTS,calledEPOC,createsafeasiblescheduleoftasksaccordingtotheadaptedresourcedemands.ItisproventhatEPOCschedulesneverviolateanyensembleconstraint,andinmanycasesyieldprovablyoptimalutilizationoftasks.Atestlimited-resourcesystemscheduledwiththeEESmanagerimplementingTTalgorithmsandEPOCisshowntoproducesystemoutputscloselysimilar(8%error)tothesystemwithsufcientresources,whilereducingupto90%oftheexpensivetaskcompressionexecutionstypicallycomputedatthebeginningofeverycycle. TheEAGLE-Tscheduler:Forensemblesystemsinwhichalltasksdonotnecessarilyhavethesameworst-caseexecutiontime(=1ensemblecycle)andthesetasksaretransitional(orprogressivelyadaptable),ratherthanstep-wiseadaptable,anewschedulingalgorithm,calledEfcientApproximationofGradualLoadExecutionforTransitionaltasks(EAGLE-T),canbeusedtoimplementtheRTS.EAGLE-Tincorporatesabasealgorithm,i.e.,EAGLE,andamodetransitionprotocol 208

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derivedfromEAGLE'sschedulabilityanalysisundertaskperiodorutilizationchanges.EAGLEisoptimalandgreatlyreducesschedulingoverheads(i.e.,preemptionsandmigrations)overtheotherknownoptimalalgorithmsby13.35%to39.24%asshownintheevaluationresults.EAGLE-TinheritsEAGLE'soptimalityandefciencyand,atthesametime,averagemaximal-utilizationdriftanddelayduringmodetransitionofEAGLE-Tarereducedfromthoseofastep-wiseapproachbyupto68.75%and32.16%,respectively.ProgressiveadaptationprovidedbyEAGLE-Talsoallowstaskstoachieveresourceutilizationwithin56%to90%oftheirdemands,comparedto11%-81%whenthestep-wiseschemeisused. TheFuzzyEESmanager:TheEESmanagerisextendedusingfeedback-schedulingco-designinordertohandlinguncertaintyofresourcecapacityandtaskWCETs.TheextendedEESmanager,calledFuzzyEES,incorporatesafuzzy-feedbackclosed-loopcontrolwiththeothercomponentsoftheEESmanager.Fuzzy-logicinferencetechniqueisemployedtointelligentlydeterminetheamountoftotalresourcedemands(i.e.,Ud)usedintheconstraintoftheTUA'soptimization.TheEAGLE-Talgorithmisalsoextendedtoutilizeslacktimeoccurredwhensometasksnishtheircomputationssoonerthanoriginallyrequested,anditsmode-transitionprocedureusesaprobabilisticapproachtoanticipatethetasks'actualcomputationtimeswhenprocessingmode-transitionrequestsissuedbytheTUA.Withtheseextensions,adjustingthevalueofUdcanbeusedincontrollingtheensemblesystemtoachieveresourceutilizationascloseaspossibletotheactualresourcecapacityavailabletoitdespitetheexistenceofuctuatedresourceavailabilityandimprecisetaskWCETs.Whenthedifferenceinutilizationislow,thenumberofexecutedinstancespertimeunitandthenumberofcompletedinstancesarehighandthemeanoutputerrorislow.Hence,fullutilizationofavailableresourcesdirectlyresultsinahightolerancetothesystem'suncertainty.Incomparisontotheopen-loopEESmanager,evaluationresultsshowthatFuzzyEES 209

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canachievetheimprovementinuncertaintytolerancebyatleast45%whenschedulingensemblesystemsunderco-existenceofresource-capacityandWCETuncertainty. 7.2FutureWork TheresultsreportedinthisdissertationprovideabasisuponwhichfurtherresearchcouldbepursuedtoenablefulldeploymentoftheEESmanagerinrealensemblesystems,suchasaresourcemanagerinacollaborativeresearchplatforminBrain-MachineInterfacesorswitching-controlsystems.Infuturework,switchingcosts(whenmigratingexpertsbetweenprocessingcores)andothersourcesofuncertainty(e.g.,responsibilitypredictionerrorandinaccuracyofsystemtimekeeping)shouldbetakenintoconsiderationwhenadaptingresource-utilizationdemands,allocatingresourcesandcreatingschedules. 210

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BIOGRAPHICALSKETCH PrapapornRattanatamrongreceivedherbachelordegreeinComputerEngineeringwithhonorsin2001fromKasetsartUniversityinThailand.ShehadworkedasanetworkengineeratShinSatellite,aleadingcommercialsatelliteoperatingcompanyinThailand,foroneyear.In2004,shecompletedhermasterdegreeinComputerScienceswithaspecializationinComputerNetworksfromUniversityofSouthernCalifornia,LosAngeles.CurrentlysheisaPh.D.candidateintheElectricalandComputerEngineeringDepartmentatUniversityofFlorida.ShehasbeenaresearchassistantattheAdvancedComputingandInformationSystemsLaboratoryundersupervisionofDr.JoseA.B.Fortessince2005andherresearchfocusesonoptimizationandresourcemanagementinreal-timeandembeddedsystems.ShehasextensivelyparticipatedintheDynamic,DataDrivenAdaptiveSystemsBMIprojecttodesignanddevelopedawebportalandmiddlewaremanagementfortheCyberWorkstation,acyberinfrastructuretoefcientlyprovidethecomputingresourcesfortheBrain-MachineInterfaceresearchwithreal-timeperformanceguarantee.Herresearchcontributionshavebeenrecognizedandpublishedinvariousconferencesheldbywell-respectedorganizationsandcommunitiesinhereld.Afterreceivingherdoctoraldegree,sheisgoingtobeafacultyattheDepartmentofComputerScienceandTechnologyatThammasatUniversityinhercountry. 224