University of Southern California Dissertations and Theses

Resource allocation in dynamic real-time systems

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Resource allocation in dynamic real-time systems

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Content RESOURCEALLOCATIONINDYNAMICREAL-TIMESYSTEMS
by
SethavidhGertphol
ADissertationPresentedtothe
FACULTYOFTHEGRADUATESCHOOL
UNIVERSITYOFSOUTHERNCALIFORNIA
InPartialFulfillmentofthe
RequirementsfortheDegree
DOCTOROFPHILOSOPHY
(ELECTRICALENGINEERING)
December2006
Copyright 2006 SethavidhGertphol
ii
Acknowledgments
Firstandforemost,Iwouldliketothankmyadvisor,ProfessorViktorK.Prasanna,
forguidingmethroughallthelearning,researching,collaborating,paperwriting,and
presentingprocesses.Healsohasgreatpatienceinhisadviceandalwaysfindsoppor-
tunitiestomotivate,challenge,andencouragemeasaresearcher. Ialsowouldlike
tothankmembersofmyqualifyingexaminationanddefensecommittees,Professor
MonteUng,ProfessorCauligiRaghavendra,ProfessorPetrosIoannou,andProfes-
sor Roger Zimmerman for their feedback and advices. I also benefit greatly from
collaborationswithProfessorH.J.SiegelandProfessorAnthonyMaciejewski
Iamindebtedtomyco-authorsYangYu,ShoukatAli,andJong-KookKimfor
theirinsightfuldiscussions,innovativeideas,enthusiasticcollaborations,andnumer-
ous contributionsthat make this Thesis possible. I also want to show my appreci-
ationto AmmarAlhusaini, SumitMohanty, Bo Hong, AmolBakshi, MitaliSingh,
JingzhaoOu,ZacharyBaker,andAnimeshPathakfortheircamaraderieandmutual
iii
supportinourcommonpursuitofknowledge(andaPh.D.degree).IalsothankHen-
ryk Chrostek and Aimee Barnard, who make administrativeprocedures and paper-
workpainlessandallowmetofocusonmyresearchandstudy.
To Somsak, Matt, Eddy, Tom, Pawat, Kor, and the rest, I am grateful for your
friendship and support during my long years in this foreign country. Last but not
least, I thank mymotherThippaporn, my father Sakdiprasert, and my sisterSopit-
jariya; I will always be in your debt for your support, understanding, and patience
duringalltheseyears.
iv
Contents
Acknowledgments ii
ListofFigures vi
Abstract x
Chapter1: Introduction 1
1.1 Dynamicreal-timesystemanditsenvironment . . . . . . . . . . . 2
1.2 Targetapplicationandhardwareplatform . . . . . . . . . . . . . . 4
1.3 Qualityofservicerequirements. . . . . . . . . . . . . . . . . . . . 7
1.4 Real-worldexample . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Resourceallocationproblem . . . . . . . . . . . . . . . . . . . . . 11
1.6 Thesiscontributions. . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6.1 Systemandapplicationmodel . . . . . . . . . . . . . . . . 16
1.6.2 Performancemetric . . . . . . . . . . . . . . . . . . . . . . 16
1.6.3 Threeapproachestosolvetheinitialallocationproblem . . 17
1.7 Thesisorganization . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter2: RelatedWork 21
Chapter3: Problemdefinition 31
3.1 Systemandapplicationmodels . . . . . . . . . . . . . . . . . . . . 31
3.2 Run-timeparameters . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Performancemetric . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Operatingregionandconformingregion. . . . . . . . . . . 38
3.3.2 Runtimebehaviorofthesystem . . . . . . . . . . . . . . . 40
3.3.3 Definingperformancemetricforinitialresourceallocation . 43
3.4 Formalproblemdefinition . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Mathematicalformulation . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Objectivefunction. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.7 Formalmathematicalformulation . . . . . . . . . . . . . . . . . . 55
v
Chapter4: Mixedintegerprogrammingwithpre-selectionapproach 57
4.1 Pre-selectiontechniqueforlinearization . . . . . . . . . . . . . . . 57
4.2 MIP(*)formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Experimentalprocedure . . . . . . . . . . . . . . . . . . . . . . . . 60
Table4.1:Characteristicsof and matrices. . . . . . . . . 62
4.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter5: Mixedintegerprogrammingwithsubstitutionapproach 81
5.1 Variablesubstitutiontechniqueforlinearization . . . . . . . . . . . 81
5.2 SMIPformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3 ExperimentsandResults . . . . . . . . . . . . . . . . . . . . . . . 90
5.3.1 ProblemGeneration . . . . . . . . . . . . . . . . . . . . . 91
5.3.2 OtherHeuristicsforComparison . . . . . . . . . . . . . . . 92
5.3.3 ExperimentalProcedure . . . . . . . . . . . . . . . . . . . 92
5.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Chapter6: Iterativeintegerprogrammingapproach 103
6.1 ExperimentsandResults . . . . . . . . . . . . . . . . . . . . . . . 107
6.1.1 Problemgeneration . . . . . . . . . . . . . . . . . . . . . . 108
6.1.2 Otherapproachesforcomparison . . . . . . . . . . . . . . 109
6.1.3 ExperimentalProcedure . . . . . . . . . . . . . . . . . . . 109
6.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.1 CharacteristicsofIIP . . . . . . . . . . . . . . . . . . . . . 119
6.2.2 ComparisonwithSMIPandMIP(CBH) . . . . . . . . . . . 121
Chapter7: Conclusion 123
7.1 Thesissummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.1.1 Systemandapplicationmodels. . . . . . . . . . . . . . . . 123
7.1.2 Performancemetricandobjectivefunction . . . . . . . . . 124
7.1.3 Approachestosolvetheallocationproblem . . . . . . . . . 125
Chapter8: Futurework 129
8.1 Improvingthesystemandapplicationmodels . . . . . . . . . . . . 129
8.2 ImprovingMIP(*)andIIP . . . . . . . . . . . . . . . . . . . . . . 130
8.3 Investigatingrelationshipbetweenqualityofresults,executiontime,
andmachine-taskheterogeneityenvironments . . . . . . . . . . . . 132
8.4 Developingadaptationalgorithms . . . . . . . . . . . . . . . . . . 132
Bibliography 134
vi
ListofFigures
1.1 Interactionsbetweenadynamicreal-timesystemanditsenvironment. 3
1.2 Angeneraldataflownetwork. . . . . . . . . . . . . . . . . . . . . 5
1.3 Commercialoff-the-shelfhardwareplatformforreal-timesystem. . 7
1.4 HiPer-Dhardwaresystem. . . . . . . . . . . . . . . . . . . . . . . 9
1.5 LandAttackapplicationforHiPer-Dtestbed. . . . . . . . . . . . . 11
3.1 Anexamplesensor-actuatornetwork. . . . . . . . . . . . . . . . . 32
3.2 Anexampleetcfmatrix. . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Anexample2-dimensionQoSspace . . . . . . . . . . . . . . . . . 39
3.4 Anexample2-parametersystemstatespace . . . . . . . . . . . . . 40
3.5 Anexampleruntimebehaviorofthereal-timesystem. . . . . . . . 42
3.6 AnexampleXmatrix.. . . . . . . . . . . . . . . . . . . . . . . . . 48
3.7 Calculatingbaselatency. . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 Variationoftheactuallatencyoftasksandprimaryrouteswithrespect
to . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.9 MathematicalFormulation . . . . . . . . . . . . . . . . . . . . . . 56
4.1 CBHheuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 MIP(*)approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 IPapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 PseudocodeforGreedyheuristic . . . . . . . . . . . . . . . . . . . 64
4.5 PseudocodeforMin-Minheuristic . . . . . . . . . . . . . . . . . . 65
vii
4.6 Simulationresultsforproblemswith3machinesand12tasks. . . . 69
4.7 Simulationresultsforproblemswith4machinesand12tasks. . . . 70
4.8 Simulationresultsforproblemswith5machinesand12tasks. . . . 71
4.9 Simulationresultsforproblemswith40tasksand10-20machines
inLo-Loenvironment . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.10 Simulationresultsforproblemswith40tasksand10-20machines
inLo-Hienvironment . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.11 Simulationresultsforproblemswith40tasksand10-20machines
inHi-Loenvironment . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.12 Simulationresultsforproblemswith40tasksand10-20machines
inHi-Hienvironment . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.13 Simulationresultsforproblemswith12machinesand30-60tasks
inLo-Loenvironment . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.14 Simulationresultsforproblemswith12machinesand30-60tasks
inLo-Hienvironment . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.15 Simulationresultsforproblemswith12machinesand30-60tasks
inHi-Loenvironment . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.16 Simulationresultsforproblemswith12machinesand30-60tasks
inHi-Hienvironment . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.17 MissratioofGreedyandMin-Min . . . . . . . . . . . . . . . . . . 76
4.18 ExampleproblemforTheorem4.5.2 . . . . . . . . . . . . . . . . . 79
4.19 Constructionofanew matrix . . . . . . . . . . . . . . . . . . 80
5.1 MIP(CBH)searchspace. . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Relationshipbetween and . . . . . . . . . . . . . . . 85
5.3 SMIPapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 TheaverageMAILvalueofallocationsfromeachapproachinHi-Hi
environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.5 ExecutiontimeofeachapproachinHi-Hienvironment . . . . . . . 94
viii
5.6 TheaverageMAILvalueofallocationsfromeachapproachinLo-Lo
environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.7 ExecutiontimeofeachapproachinLo-Loenvironment . . . . . . . 96
5.8 TheaverageMAILvalueofallocationsfromeachapproachinLo-Hi
environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.9 ExecutiontimeofeachapproachinLo-Hienvironment . . . . . . . 97
5.10 TheaverageMAILvalueofallocationsfromeachapproachinHi-Lo
environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.11 ExecutiontimeofeachapproachinHi-Loenvironment . . . . . . . 98
5.12 The and matrix where MIP(CBH) will not find a feasible
allocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.13 ExtendedproblemwhereMIP(CBH)willnotfindafeasibleallocation.101
6.1 Relationshipbetween values, and . . . . . . . . . 105
6.2 IIPFormulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3 The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinHi-Hienvironment. . . . . . . 111
6.4 The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinHi-Loenvironment. . . . . . . 112
6.5 The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinLo-Hienvironment. . . . . . . 112
6.6 The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinLo-Loenvironment.. . . . . . 113
6.7 ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inHi-Hienvironment.. . . . . . . . . . . . . . . . . . . . . . . . . 114
6.8 ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inHi-Loenvironment. . . . . . . . . . . . . . . . . . . . . . . . . 114
6.9 ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inLo-Hienvironment. . . . . . . . . . . . . . . . . . . . . . . . . 115
6.10 ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inLo-Loenvironment. . . . . . . . . . . . . . . . . . . . . . . . . 115
ix
6.11 TheaverageMAILvalueofallocationsfromtheIIPapproachwith
varying valueinHi-Hienvironment. . . . . . . . . . . . . . . . . 116
6.12 TheaverageMAILvalueofallocationsfromtheIIPapproachwith
varying valueinHi-Loenvironment. . . . . . . . . . . . . . . . . 116
6.13 TheaverageMAILvalueofallocationsfromtheIIPapproachwith
varying valueinLo-Hienvironment. . . . . . . . . . . . . . . . . 117
6.14 TheaverageMAILvalueofallocationsfromtheIIPapproachwith
varying valueinLo-Loenvironment. . . . . . . . . . . . . . . . . 117
6.15 Execution time of the IIP approach with varying value in Hi-Hi
environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.16 Execution time of the IIP approach with varying value in Hi-Lo
environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.17 Execution time of the IIP approach with varying value in Lo-Hi
environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.18 Execution time of the IIP approach with varying value in Lo-Lo
environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
x
Abstract
Dynamic real-time systems usually operate in an environment that is continu-
ously changing. These changes in the environment cause workload of the system
to vary during run time, and in effect causing system performance to fluctuate. In
addition,certainQualityofService(QoS)requirementsareimposedonthereal-time
systemandmustbesatisfied.Allocatingsystemresourceinthisenvironmentischal-
lengingbecauseagoodandefficientallocationmaybecomeinvalidwhenvariations
inworkloadcauseQualityofServiceviolations.
Thisresearchexaminestheproblemofinitialresourceallocationforareal-time
system that operates in changing environment. An initial allocation is calculated
beforesystemstartsupusingexpectedvaluesandvariationsthatmayhappenduring
runtime. Inthisresearch, resourceallocationreferstotheassigningofapplication
taskstoprocessingnodes.Theallocationshouldbe“robust”withrespecttothevari-
ationinworkloadduringruntime. Thatis,theallocationshouldbeableto“absorb”
thelargestamountofworkloadvariation,andthusdelayingtheinstancethatQuality
ofServiceviolationoccurs.
xi
A novelperformance metricisproposedtoevaluatetherobustnessof aninitial
allocationwithrespecttovariationsduringruntime.Theproblemofinitiallyallocat-
ingsystemresourcestomaximizetheproposedperformancemetricisthentranslated
intoamathematicalformulation.Itshouldbenotedthattheresourceallocationprob-
lemitselfisnon-linear,butcanbelinearizedbytwoheuristicsthataredevelopedto
removenon-linearityfromthemathematicalformulationoftheproblem.Threealgo-
rithms,allbasedonthewell-researchedlinearprogrammingapproach,aredeveloped
tosolvethemathematicalproblem. The firstapproachisfast,butcannotgaurantee
the quality of resulting allocations. The second approach always finds an optimal
solution,butisrelativelyslowerthanthefirstapproach.Inthethirdapproach,trade-
off between the quality of results and execution time can be controlled through a
user-providedparameter. Togetherwiththetwolinearizationheuristics,thesethree
approaches provide a full spectrum of solutions for the initial resource allocation
problem.
1
Chapter1
Introduction
Dynamic real-time systems such as embedded systems [18, 25] have recently
gainedimportanceinseveralapplicationdomainssuchasautomobilecontrol,avion-
ics, and defense. Embedded system is usually enclosed in a larger platform and
must co-exist and cooperate with other various subsystemsin that machinery. For
example,anreal-timesysteminanautomobilecontrolapplicationoperatestogether
withengineandtransmissionsubsystemsthatactuallypowerthecar. Allthesesub-
systems,whetherelectrical,computation,ormechanical,mustoperateandcommu-
nicate with one another to ensure the correct functionality of the whole platform.
Furthermore,theremaybeseveralreal-timesystemsexecutingdifferentapplications
required for the operation of the larger hardware platform. For instance, in a war-
ship, there may be one real-time system monitoring enemy threats on the surface,
onesystemwatchingenemythreatscomingfromtheair,andonesystemcontrolling
themaneuveringoftheship.Onecharacteristicofdynamicreal-timesystemsisthat
they interact extensively with the environment in which they operate. In addition,
theenvironmentthesesystemsoperateinisusuallyverytransitoryandcontinuously
changing.Thesechangesintheenvironmentoftenaffecttheworkloadofthedynamic
real-timesystems. Anotherimportantcharacteristicofthesesystemsisthereal-time
2
requirementsimposeduponapplicationtasks.Therequirementisoftenexpressedin
termsoftaskdeadline,i.e.,theamountoftimeinwhichthetaskmustbecompleted.
Akeychallengeindynamicreal-timesystemsistheallocationofsystemresources
tosatisfyapplicationrequirementsinspiteofcontinuouslychangingoperatingenvi-
ronment.
1.1 Dynamicreal-timesystemanditsenvironment
Onecommoncharacteristicofthedynamicreal-timesystemsisthattheyinteract
extensively with the environment in which they are operated. Depending on each
system,theenvironmentmayencompassseveralentities. Forexample,inthewar-
ship,anenvironmentincludesimmediateair,surface,andunderwaterspacearound
theship. Inthisenvironment,theremaybeseveralentities,suchasafriendlysub-
marine,enemyaircrafts,orneutralfishingboats. Interactionsbetweentheseentities
andthedynamicreal-timesystemsareaccomplishedthroughasetofmechanicalor
electronicsensorsandactuatorslocatedontheplatform. Sensorsareutilizedtocol-
lectreal-timeinformationabouttheentitiesintheenvironment.Informationincludes
headingandspeedofanairplane,depthofasubmarine,andIFFsignalwhichhelps
identifywhethertheentityisfriendorfoe,etc. Theremaybeseveraltypesofsen-
sorsworkingtogethertogathertheinformation. Forexample,thewarshipmayuti-
lizessurfaceradarstodetectshipsatsea,bothpassiveandactivesonarstolistento
3
Figure1.1:Interactionsbetweenadynamicreal-timesystemanditsenvironment.
enemysubmarines,airradarstodetectaircrafts,andIFFreceiverstoreceiveIFFsig-
nals. Actuatorsareasetofmechanicalorelectronicdevicesthatreceivecommands
fromthedynamicreal-timesystemsandactontheentitiesintheenvironment. For
instance,oneactuatorinthewarshipexample-anairdefensesystem-mayreceive
orderto fireuponanenemyaircraft. Thecommandsreceivedincludethedirection
offire,thetypeofweapontobeemployed,andthetarget.
Theenvironmentinwhichthereal-timesystemoperatesisdynamic,oftenchang-
ing due to various factors outside the control of system operators. In the warship
example,enemyaircraftsmaylaunchmissilestowardtheship,whichisdetectedby
the ship’s radar. An automobile may suddenly travel through a rainstorm, causing
the road to become slippery. Some actions taken by the system through actuators
mayalsoalterthestateoftheenvironment. Amissilefiredinthewarshipexample
4
maycatchupanddestroyanenemyairplane,resultinginonelessaircraftinthesky.
The newinformationof the environment, whethercausingby thesystemor not, is
gathered and processed, and may result in a new action taken by the system. This
dynamic behavior between the real-time system and the environment is shown in
Figure1.1.
1.2 Targetapplicationandhardwareplatform
Raw data collected from the environment through sensors are transferred to a
hardwaresubsysteminthedynamicreal-timesystemforprocessing.Theprocessing
subsystem may consist of more than one computing machines connected together
through a communication network. These machines execute a collection of tasks
that provides functionalities for target applications designed for the platform. In
thewarshipexample,ashipmaybedesignedforthreeapplications: shootingdown
enemyairplanes,destroyingenemyships,andsinkingenemysubmarines. Inorder
to shoot down an enemy airplane, the raw return signal from air radar must first
be processed and the airplane detected. Its altitude, heading, and speed are also
calculatedfromtheairradarsignal.Theplaneisthenidentified,usingtheIFFsignal,
whetheritisafriendorafoe.Iftheplaneisdeterminedtobeanenemy,anintercept
course for a surface-to-air missile is calculated from the current altitude, heading,
and speed of the plane. Finally, after thecomputationis finished, thecommandto
fireamissiletogetherwiththeinterceptcoursearesenttotheactuators. Ascanbe
5
Figure1.2:Angeneraldataflownetwork.
seenfromthisexample,theapplicationofshootingdownanairplanerequiresseveral
differenttasks(detection,identification,interceptcalculation,firing).Thesetasksare
executedinsequencewithsomeofthetasksalsorequireresultsfromanothertasks
inadditiontothedatafromsensors. Thus,theapplicationcanbeviewedasflowsof
data(bothfromsensorsandothertasks)throughaseriesoftasks. Figure1.2shows
agenericdata flownetworkinadynamicreal-timesystemwithblackarrowsasthe
directionofdatapropagation.Theflowsstartfromsensorsandstreamthroughtasks
thatmaymanipulateonthem. Some flowsmayalsocombinedtogetherordiverged
data out to different tasks. Finally, the flows of data end at various actuators as
commandstodirectthemtomanipulatetheenvironment.
6
As stated previously, the computing subsystem of the dynamic real-time sys-
temconsistsofseveralmachinesconnectedbycommunicationnetworks. Tradition-
ally,requirementsforreal-timeapplicationsnecessitatetheuseofcustom-madesys-
tems. Thesesystemsusuallyutilizespecialpurposeoperatingsystems,application
programming languages and interfaces, customized interconnection networks, and
special-builtprocessors. Developing, deploying, and maintainingall these compo-
nentsisgreatlytime-consuming,complex,andexpensive.Inaddition,thesesolutions
offer little flexibility in terms of software reuse and hardware upgrade. Recently,
powerful commercial off-the-shelf (COTS) products have gained widespread and
favorable reception as one possible substitutefor custom-madesystems as a target
platform for real-time systems. COTS-based systems provide high flexibility due
toheterogeneityinprocessors,operatingsystems,memorystructure,andcommuni-
cation network. Standard application languages and interfaces also facilitate rapid
developmentoftheapplicationtasks,easemaintenanceissues,andprovidecompat-
ibilityforhardware upgrade. Thereisalsoawell-researched areacalledHeteroge-
neousComputing(HC),whichstudiesaco-ordinatedmanagementofdifferenttypes
ofprocessorsandnetworkstomeetrequirementsofgreatlyvaryingapplicationsand
tomaximizeadefinedperformancemetric. Bybuildingonthisvastresource,man-
agementandoptimizationsystemsforCOTS-based real-timesystemscanbeprop-
erly and swiftly developed. Figure 1.3 shows the COTS-based real-time systems
togetherwithconnectionstosensorsandactuatorsinthehardwareplatform.
7
Figure1.3:Commercialoff-the-shelfhardwareplatformforreal-timesystem.
1.3 Qualityofservicerequirements
In real-time systems, the correctness of the computations not only depends on
theirlogicalexactness,butalsoonthetimeatwhichtheresultisproduced. Inother
words,therearequalityofservices(QoS)thattheapplicationsrequirefromthehard-
wareplatformandthemanagementsystem. Mostofthesequalityofservicescome
intheformofdeadlineimposedonsometasksoraseriesoftaskscomprisingsome
criticalfunctionalities. Forexample,theapplicationdesignermaydemandthatthe
friend-or-foe identification task in the warship example must finish execution and
determinewhetheraradarcontactisathreatin3secondsfromthetimethedatais
received from the sensors. Another example quality of service requirement is that
the whole process chain of contact detection, threat identification, intercept course
8
calculation, and weapon firing must complete in 15 seconds. Lookingat the latter
requirementinthedataflowpointofview,itmeansthatthedatafirstreceivedfrom
asensormustbeprocessedandreachthedesignatedactuatorinthespecifiedamount
of time. Another type of QoS requirements insists that a task or a series of tasks
processesatleastaspecifiednumberofinformationwithinthespecifiedamountof
time.Forinstance,inorderforawarshiptoengagemultipletargets,themilitarymay
demandthatafiringtaskmustbeabletolaunch3missilesin12seconds.Thisisdif-
ferentthanrequiringafiringtasktofinishexecutionin4seconds,becauseaccording
tothisrequirementonemissilecantakelongerthan4secondsto fireaslongasall
threelaunchwithin12seconds.
Failing to meet quality of service requirements indicated by the application
designerresultsinan undesirableoutcome. Theseriousnessofthefailure depends
onthetypeofQoSrequirementsandontheapplication.Aqualityofservicerequire-
mentiscategorizedashardiftheresultoffailingtherequirementmaycausecatas-
trophetothehardwareplatformorhumanlives.Failingtolaunchmissilesintimeto
shootdownenemyaircraftscomingtowardthewarshipmaycausedeathtothecrews
anddestructiontopartsoftheshiporevensinkingofthewholeship. Ontheother
extreme,asoftQoSrequirement,ifviolated,maybeundesirable,butnothazardous.
Takinglongertowarmupseatsinanautomobilemaybeuncomfortabletothedriver
andpassengers,butdoesnotposeathreattothemortotheautomobile.
9
Figure1.4:HiPer-Dhardwaresystem.
1.4 Real-worldexample
HiPer-D [13] is a COTS-based test environment for High Performance Dis-
tributedComputingProgramattheNavalSurfaceWarfareCenter,DahlgrenDivision
(NSWCDD). HiPer-D is a main testbed for developing and evaluating distributed
applicationsfor AEGIS combat platform. One configuration of HiPer-D system is
shown in Figure 1.4 and consists of 4 Sun machines, 1 Windows NT-based Intel
machine, and 3 SGI machines fully connected together by a switch. The AEGIS
platform is designed to accomplish 3 main missions: 1) anti-air warfare (AAW),
2)LandAttack, and3)shootingdownTheaterBallisticMissile(TBM).Figure1.5
shows the Land Attack application for the HiPer-D testbed. Sensors are shown in
10
diamondandtheonlyactuatorisshowninsquare. Applicationtasksthatneedtobe
allocatedtomachinesareshownincircle. Labelsnexttothedata flowdirectionis
thecommunicationprotocolusedinthedatatransfer. Thetwowaycommunication
representsmostlyacknowledgmentforTCPprotocol.Becausetheapplicationisrun-
ningona testbed, notanactualAEGIS platform, inputfromsensorsare simulated
by three components: FIRESIM 21, OTH Server, and ALTDS Track Data Server.
FIRESIM 21 is an army software tool that provides land target engagement infor-
mationtotheLandAttackapplication. Theengagementorderisissuedatarateof
about20engagementsper10minutes. OTH(Overthehorizon)ServerandALTDS
(AirLocalTrackDataSource),periodicallyevery1-4seconds,providesinformation
aboutlandandairtracksthattheAEGISsystemdetects.TACFIREisaneventdriven
taskthatdetermineswhatisbeingengaged,reformatstheinformationintoCFFfor-
mat, and issuescommandto fire the gun. CFF Broker is also an event driventask
thatcorrelatesbetweenthefireorderandtheactuallandtargetoverthehorizon.Land
AttackEngagementServeroverseestheengagement.DeconflictionServerexamines
thelandtrackbeingengagedandairtrackpicture(suchasanyplanecrossingthefire
path). It may send data (e.g. interrupting the firing if a friendly plane crosses the
firepath)totheLandAttackEngagementServer. GunSIMsimulatesthecommand
sendingtotheactualgun. Finally,thedisplaycomponentmustrunontheWindows
NTmachineandshowsthegraphicaldisplayoftheengagement.Therearetasksthat
11
Figure1.5:LandAttackapplicationforHiPer-Dtestbed.
mustalsobeallocatedbutnotshowninFigure1.5,suchasENSEMBLEGossipDae-
monrequiredforENSEMBLEcommunicationandNDDSagentrequiredforNDDS
datatransfer.
1.5 Resourceallocationproblem
AsdiscussedinChapter1.1,thestatusoftheenvironmentinwhichthereal-time
systemoperatesusuallyfluctuatesduetobothactionsfromthesystemanduncontrol-
lablefactors.Thesechangesintheenvironmenthavetobeprocessedbythecomput-
ingsubsystem,andoftenaffectworkloadofthesystem.Forexample,anautomobile
control system receives more workload calculating the distributionof transmission
12
power when the car travels from a straight road into a winding road. A friend-or-
foe identificationsystemina warshipconsumesadditionalcomputingpowerwhen
more aircrafts appear in the sky. Because system resources are limited, this addi-
tionalworkloadmustdrainresourcesfromotherapplicationtasksthatarecurrently
running. Thisdepletionof resources directly disruptsthe abilityof the application
taskstomeetthequalityofservicerequirementsimposeduponthem. Onekeychal-
lenge in dynamic real-time systems is the allocation of system resources to satisfy
QoSrequirementsinspiteofthecontinuouslychangingoperatingenvironment.
Resourceallocationistheprocessofdistributingsystemresourcestoapplication
tasksinordertoobtainsomedefinedobjectives. Thesystemresourcesmayinclude
computingpower,communicationbandwidth,mainmemory,diskspace, orspecial
hardware required for some tasks. There are various objectives proposed for het-
erogeneouscomputingsystem,suchasmaximizingresourceutilization,minimizing
executiontime,etc. Whilemanyoftheseobjectivescanbeeasilymodifiedforuse
withreal-timesystems,themostimportantobjectivesinthiscontextismeetingthe
qualityofservicerequirementsimposedontheapplicationtasks. Inthisthesis,allo-
cationreferstoassigningeachtasktoacomputingmachinesuitabletoachievethe
desiredobjective.Inmanystate-of-the-artreal-timesystems,allocationalsorefersto
theprocessofdeterminingtheexactamountofresourceavailabletoeachtask(such
asexact percentageof CPU power)or theexactscheduleoftaskexecution. These
proceduresrequirespecialsoftwarecontrolandinterfacesthatmaynotbeavailable
13
inthetargetcommercialoff-the-shelfproducts. Inthisthesis,alltasksassigningto
thesamecomputingmachinearescheduledbyasimpleround-robinschemeandthus
shareallmachineresourcesequally.
Resource allocationcan beclassifiedintotwocategories: initial allocationand
resourceadaptation.Initialresourceallocationdeterminesthedistributionoftasksto
machinesbeforethedynamicreal-timesystemstartsup.Theallocationiscalculated
based on known systemresources and estimatedworkload the systemmay receive
when the systemstarts up. In a sense, thistype of allocationscan be called static,
because resources are allocated based on pre-determined values and the allocation
doesnotchangewithtime. However,sincetheallocationiscalculatedoff-line(i.e.,
withoutusingsystemresourcesduringruntime),complexalgorithmswithlargeexe-
cutiontimecanbeusedtodeterminethebestallocation.Ontheotherhand,resource
adaptationisusedtoadjustorallocatesystemresourceson-the-flyduringruntime.It
isofteninvokedwhenthereisapossiblequalityofserviceviolationinthesystemdue
tochangingworkloadorunpredictedeventssuchaspartialsystemfailure. Resource
adaptation a dynamic allocation because the allocation is calculated based in the
mostcurrentinformationavailableaboutsystemresourcesandworkload. However,
because resources are adjusted during run time, the adaptation algorithm must be
relativelysimpleandfast,butstillefficient. Inaddition,theadaptationprocessmay
drainsystemresourcesfromotherapplicationtasksunlesssomeresourcesarespecif-
icallyreservedforcalculatingtheadaptation.
14
Inthisthesis,wefocusontheinitialallocationofsystemresources. Thegeneral
goalofourinitialresourceallocationistoaccommodateworkloadvariationsforthe
longestperiodoftime.Theallocationgeneratedbyourtechniqueswillberobustwith
respecttochangesintheenvironment[4,5,27,28,29,36].Asdiscussedpreviously,
resource adaptation occurs to deal withany QoS violationthat may happen due to
thevariationsofsystemworkload.Byinitiallyallocatingresourcestoaccommodate
thesevariationsbeforehand,therun-timeadaptationwillbepostponedtothefarthest
possibletimeinthefuture. Themotivationbehindthegeneralgoalisthatresource
adaptationisacostlyoperationintermofcomputingpowerandtime.Becauseadap-
tationoccurs whena real-timesystemisinoperation, on-linecomputingresources
areusedtodetermineanewallocation. Thesecomputingresourcesarealsoshared
bythetasksexecutinginthesystem. Thus,inordertoperformresourceadaptation,
computingresourcesmustbetakenawayfromothertasksduringruntime. Incon-
trast,aninitialallocationiscomputedbeforesystemstartsupusingoff-lineresources,
thusthecomputationdoesnotinterferewiththeexecutionofothertasks.Inaddition,
resource adaptation occurs when tasks do not have enough resources to meet their
QoSrequirements. Thus,whenthesystemperformsadaptation,resourcesaretaken
awayfromtaskswhentheyareneededmost.Calculatinganewallocationalsotakes
someamountoftime,whichisalsoapreciousresourcewhenthereisaQoSviolation
inthesystem.
15
Notethatresourceadaptationisnecessaryevenwithvariousdisadvantagesdis-
cussedearlier.Theidealinitialresourceallocationistheonethatwillneverhaveany
QoSviolationfromthetimesystemstartsuptotheendofoperation. However,itis
veryunlikelythatsuchanallocationcanbeobtainedinrealityduetounpredictable
changesintheoperatingenvironmentduringruntime.Thus,thegeneralgoalofini-
tialallocationistopostponethe firstadaptationthatwilloccur. Afterthe firstQoS
violationhappens,theperformanceofthesystemdoesnotdependonlyontheinitial
allocation,butalsoontheadaptationpolicyemployed.
1.6 Thesiscontributions
Inthisthesis,weproposeamethodforinitiallyallocatingdynamicreal-timesys-
temresourcessuchthattheallocationcansustainthesysteminspiteofcontinuously
changingenvironment. Indoingso,moresystemresourcesareavailabletoapplica-
tiontasksforexecution,ratherthanbeingusedforperformingadaptation. Anovel
performance metric is developed to evaluate an initial allocation for its robustness
withregardtovariationofworkloadduringruntime.Taskallocationinreal-timesys-
temsinordertomeetcertaindeadlinesisknowntobeanNP-hardproblem[41,50].
Theproblembecomesmoredifficultwhenchangesintheenvironmentmustalsobe
takenintoaccount. Severalalgorithmsandheuristics(e.g., [45,64,81])havebeen
developed to allocate resources using system parameters that are known a priori.
16
However,theydonotconsiderthechangingworkloadthatmaycauseviolationsin
qualityofservicerequirementsimposedonthesystem.
1.6.1 Systemandapplicationmodel
ThemodelsthatdescribeCOTS-basedsystemanddataflowapplicationtasksare
developedfortheproblemofresourceallocation.Thehardwaremodelisbasedona
heterogeneoussystemmodelwheremachinescan(butnotnecessary)havedifferent
computingpowerandcommunicationbandwidth. Systemresourcesareassumedto
befairlysharedduetotheround-robinschedulingusedbythemachines. Theappli-
cation modelisbased on the asynchronousdataflow(ASDF) processnetwork [51]
whichdescribesthe applicationtasksas adirected acyclicgraph (DAG).However,
theASDFprocessnetworkdoesnotspecifyanyreal-timequalityofservicerequire-
mentsorvariationsofworkloadduringruntime. WeextendtheASDFprocessnet-
worktocapturethechangingworkloadbydefiningparametersthatcanchangedur-
ingruntime. Themannersthattheserun-timeparameterscanchange,aswellasthe
effects these variations have on the system, are also described. Finally, quality of
servicerequirementsareassociatedwithataskoragroupoftasks.
1.6.2 Performancemetric
In order to evaluate different initial allocations, a novel performance metric is
proposedtocapture therobustnessof an initialallocationwithrespect tochanging
17
workloadduringruntime.Themeasureofrobustnessofaninitialallocationiscalled
maximum allowable increase in workload (MAIL),whichrepresentsthemaximum
amountofworkloadthataninitialallocationcanaccommodatebeforeQoSviolation
occurs. Basedonthesystemandapplicationmodels,theinitialallocationproblem
isdescribedmathematicallyandanobjectivefunctionforthemathematicalproblem
is proposed. We give proof that by optimizing this objective function, a resulting
solution of the mathematical problem will be an initial allocation with the highest
valueoftheproposedperformancemetric.
1.6.3 Threeapproachestosolvetheinitialallocationproblem
Threealgorithms,allbasedonthewell-researchedlinearprogrammingapproach,
aredevelopedtosolvethemathematicalproblem. Becausethemathematicalformu-
lation of the resource allocation problem is non-linear, two techniques are devel-
oped to linearize the problem so that it can be solved by the linear programming
approaches. The first algorithm, called MIP(*), is a mixed-integer-programming-
based approach, which utilizesa user-defined heuristic to pre-select the number of
tasksallocatedtoeachmachine,andaccordinglylinearizetheproblem. Inthisthe-
sis,theheuristicusedtopre-selectthenumbersisbasedontherelativecapabilityof
machinewithrespect to othermachines, and thuscalled capability based heuristic
orCBH.Because thenumberofvariablesarereduced, MIP(CBH) isveryfast, but
thequalityofresultingallocationdependsontheheuristicused. Experimentresults
18
showthatMIP(CBH),producesontheaverage,anallocationwiththeperformance
metricupto86%oftheoptimalvalue,usingonly0.075seconds.Itcanalsobeshown
thatifthepre-selectednumbersmatchtheactualnumbersintheoptimalallocation,
the mixed-integer-programming approach will also produce an optimal allocation.
Ontheotherhand,iftheheuristicpre-selectsthenumberunreasonably,theapproach
maynotfindavalidallocation,evenifoneexists.
Thesecondapproach,calledSMIP,isalsoamixedintegerprogram,butitutilizes
avariablesubstitutiontechniquetolinearizetheproblemwithoutfixinganyvariable
to some arbitrary values. The substitution technique replaces a multiplication of
twovariableswithonenewvariableandaddconstraintsbindingthethreevariables
together. Because no variable is fixed to any arbitrary value, the solutionspace of
thisapproachisnotrestricted.Incomparison,thesolutionspaceofMIP(*)islimited
toallocationsthathave thenumbersof taskson allmachineequal the pre-selected
values. Thustheresultingallocationproduced bySMIP isalwaysoptimal(having
highestvalueoftheperformancemetricamongallvalidallocations). However,the
substitutiontechniqueintroducesadditionalvariablesandconstraintsintothemixed
integer program, making SMIP significantly slower than MIP(*) when solving the
same problem. Experiment results confirm that SMIP always produce an optimal
solutionbutusing464seconds. ItshouldbenotedthattheexecutiontimeofSMIP
ismuchlongerthanthatofthefirstapproachevenonasmallsizeproblem,butstill
acceptableforaninitialallocationproblem.
19
Thethirdapproach,calledIIP,alsousesthesamevariablesubstitutiontechnique
asthesecondapproach.However,thecomplexityoftheproblemisreducedbysolv-
ingapureintegerprogramiteratively,insteadofsolvingamorecomplicatedmixed-
integer-program.Betweeniterations,theacceptablevalueoftheperformancemetric
isadjustedbyaspecific, user-definedamount,suchthattheintegerprogramofthe
nextiterationismoreconstrained. Whenanintegerprogramfailsto findafeasible
solution,thelastknownvalidsolutionisusedastheinitialallocation.Thisapproach
guaranteestoproduceanallocationwiththeobjectivevaluethatfallswithintheuser-
defined amount (the adjustment value between iterations) from the optimal value.
In addition, the execution time of IIP also depends on this user-defined value; the
smallertheadjustmentvalue,thelongertheexecutiontime.Thus,thereisatrade-off
betweenthequalityoftheresultingallocationandtheexecutiontimeoftheapproach,
whichcanbe decidedbytheuser-definedparameter. Experimentresultsshowthat
theIIPwiththeadjustmentvalueof1%canfindanallocationwiththeperformance
metricat99%oftheoptimalvalueusing56seconds.
1.7 Thesisorganization
Chapter2describespreviousresearchesrelatedtotheproblemofinitialresource
allocation in dynamic real-time systems. These works include resource allocation
problems in both heterogeneous systems and real-time systems. Several existing
heuristicsforbothproblemsareexaminedandevaluatedwhethertheycanbeadapted
20
to our problem. In addition, we investigate the linear programming algorithm and
exploreitsprevioususesinvariousfieldofresearches.
A formal problem definition is described in Chapter 3, including system and
application models (Chapter 3.1), run-time parameters (Chapter 3.2), and the per-
formance metric (Chapter 3.3). The mathematical problem and objective function
foroptimizationaredefinedinChapter3.5andChapter3.6,respectively.Theformal
mathematicalproblemdefinitionisgiveninChapter3.7.
The pre-selection and substitution techniques used to linearize the mathemat-
ical problem are described in Chapter 4.1 and Chapter 5.1, respectively. The
three approaches to solvethemathematicalproblem- MIP(*), SMIP, andIIP - are
explainedinChapter4,Chapter5,andChapter6,respectively. Thelayoutsofthese
threechaptersaresimilar.Firsttherespectiveproblemformulationisdescribed,fol-
lowedbyexperimentsandresults.Finallyeachchapterendswithadiscussionofthe
characteristicsofeachapproaches.
Chapter7concludesthethesisandoutlinespossiblefuturework.
21
Chapter2
RelatedWork
Theresourceallocationproblemdealswiththedistributionoflimitedresources
among consumers who compete for those resources. This kind of problems exists
inmanyfields,suchasinOperationResearchwhichmanagestheallocationoftime
tousespecificmachinesforvariousjobsinafactory,orinEconomicswhichdeals
withthedistributionofcapitalmoneytovariousinvestments. InComputerScience,
resource allocation usually means the distribution of computing resources such as
CPU time or memory space to various jobs in the system. The management of
resources normally is conducted so that some specific goals are achieved. These
goalsvaryfromonesystemtoanotherandreflectthedemandofusersfromthatsys-
tem. Forexample,onefactorymanagermaywanttominimizewearandtearonhis
machines, while another may want to maximize the number of products made. A
goalmayalsochangeovertime,suchasaUNIXadministratormayrequireaserver
to minimize user response time during the day, but want the same server to maxi-
mizethethroughputofbatchprocessingatnightwhentherearefewusersusingthe
machine. Howwelltheresourceallocationaccomplishthegoalsetbythemanager
ismeasuredbyaperformancemetricrelevanttothatgoal.
22
The performance metric proposed in this Thesis is designed to measure the
robustness of initial resource allocations with respect to future variations of work-
load.Thedefinitionofrobustnessingeneralisstillanongoingresearcharea[1]and
encompassesseveralfieldofresearchesincludingecology[33,38],genetics[9,53],
biology[35, 65], computer science [6, 32], control theory [42], and statistics[39].
In our work, robustness is defined as the ability of the system to maintain correct
operations (no QoS violations) even when there are variations of workload during
run time. The degree of robustness is the degree of variations that the allocation
being measured can accommodate before failing. In ecology, ecological resilience
is defined as the amount of disturbance that an ecosystem could withstand with-
out changing its structures [33]. If the disturbance encountered by the ecosystem
is greater than some threshold, it must adapt its internal structure and transforms
into aother stable system. On the other hand, engineering resilience is defined as
the rate at which an ecosystem returns to a steady state following a perturbation
assumingthatthesystemstayswithinastabledomain[38].Inotherwords,theengi-
neering resilience measures how well an ecosystem can tolerate perturbations, and
the ecological resilience measures howmuch perturbationsan ecosystem can take.
Our definition of robustness closely resembles ecological resilience which focuses
onuncertainties,variations,andadaptability,whileengineeringresiliencecenterson
thedifferentparadigmofcontrol,stability,andpredictability. In[44],robustnessis
differentiatedfromstabilityashavingabroadercontextwhichincludestopicssuch
23
as organization structure of the system in question, dynamic relationship between
systemsandenvironments,abilityofthesystemtoadapt,anticipationoffutureper-
turbations (variations) in multiple dimension, etc. In biology, a robust system is a
systemthatitsbehaviorisqualitativelynormalinthefaceofsubstantialchangesin
systemcomponents(suchasgenes) [65]. In[32], examplesare shownwhere even
small changes in complex and coupled computer systems can lead to large, unex-
pected changes in system behavior that can result in failure of the systems in the
behaviorcalledthebutterflyeffect[54].Theauthoralsooutlinedapproachestobuild
robustcomplexsystemsbyover-provisioning,admissioncontrol,monitoring,adap-
tation,andplanningforfailure. However,robustnessisnotexplicitlydefinedandis
describedgenerallyastheabilityofthesystemtooperatecorrectlyinwiderangeof
operationalconditionsandandtodegradegracefullyoutsideoftheseconditions.
Inseveralfields,robustnessisdefinedastheresilienceofthesystemagainstunex-
pectedorabnormalconditions,bothfromtheoutsideenvironmentandinternalstruc-
turesormodels. Instatistics,robustnessisthedegreeofinsensitivityagainstsmall
deviationsinassumptionsormodels,suchascontaminationofdataordeparturefrom
assumedsampledistribution[39]. Onemeasureof robustnessinstatisticsiscalled
abreakdownpointofanestimator,whichintuitivelyisthefractionofincorrectdata
(suchasarbitrarylargevalue)anestimatorcanhandlebeforetheestimatorwillalso
giveanarbitrarylargeresult. Forexample,themeanof randomvariables to
hasabreakdownpointof0;ifweincreasejustonlyonevariabletoanarbitrarylarge
24
value, the mean will also be arbitrary large. On the other hand, the median has
abreakdownpointof0.5becausemorethanhalfofthevariablesmustbearbitrary
largebeforethemedianwillalsobecomearbitrarylarge. Othermeasuresofrobust-
nessofanestimatorinstatisticsinclude influence functionand sensitivitycurve. In
controltheory,arobustcontrolsystemisonethatisinsensitivetouncertaintiesinthe
system[42]. Theseuncertaintiesstemsfromtwosources: (a)discrepancybetween
realsystemandthemathematicalmodelofthesystemusedtodesignthecontroller
and(b)noisesanddisturbancesthatarenotmodelledinthe firstplace. Insoftware
engineering, a software is robustif it stilloperates correctly in abnormalbehavior,
suchasincorrectuserinputsorincorrectoperationsfromothersoftware(e.g.,bugs
inthesoftware).
ThetargetplatformofourresearchisaCOTS-based(Commercial-off-the-shelf)
heterogeneous system and application [12, 73, 74] similar to one in the HiPer-D
project [13]. COTS-based heterogeneous platforms offer several advantages over
custom-made systems including ease of implementation, flexibility due to hetero-
geneity,easeofupgrade,standardizedsoftwarelibraryandinterface.However,there
are several challenges in building systems from COTS components, especially the
decreasingcontrolovercomponentssuchasschedulingandsecurity. Developinga
fault-tolerance,dependablereal-timesystemfromCOTScomponentisanactivefield
ofresearch[21,30,66].
25
In traditional heterogeneous systems, the goal of resource allocation is usually
to minimize total execution time [24, 41, 43, 49, 59, 75]. These allocation strate-
gies do not consider tasksthat have inherent Quality of Service requirementssuch
asdeadlineofthetask. Theyalsodonotconsidervariationsinworkloadduringrun
time. Inourwork,bothQualityofServicerequirementsandfutureworkloadvaria-
tionsduringruntimeareveryimportantpartsoftheapplicationmodelandtasksare
allocated in accord with bothaspects. Some resource allocationtechniquesin het-
erogeneoussystemsaimtobalancetheworkloadinthesystembydistributingtasks
across machines based on machine affinity [10, 16, 48, 69]. The intuition behind
balancingtheworkloadacrossallmachinesisthateverymachinewillthenhaverel-
ativelythe same amount of unused power left for future workload. However, load
balancingparadigmimplicitlyassumesthatthefutureworkloadincreasewillbethe
sameforallmachines.Incontrast,amethodtocharacterizefutureincreaseinwork-
loadisdevelopedinthisThesis,sothattheincreaseononemachinecanbedifferent
fromanother.
In an area of dynamic load balancing, resources are allocated during run
time based on various indices such as CPU utilization, stretch factor (the ratio
betweentaskexecutiontimeonaloadedandanunloadmachine.),andCPUqueue
length [8, 23, 26, 34, 70]. In effect, these techniques allocate resources based on
currentinformationaboutthesysteminsteadofallocatingwithregardtothefuture.
Interestingly, in [31] dynamic load sharing heuristics that use prediction of future
26
workload are proposed. The prediction is based on gathered data about the task
resource usage (such as CPU utilization, memory, I/O) which form resource usage
statessimilartothestatespacedescribedinChapter3.3.1.Transitionsbetweenthese
resource usage states are also recorded during data collection and reconstructed as
transitionsequencesofthetask.Theprobabilitiesofeachstatetransitionisthencal-
culatedandusedaspartofthepredictionalgorithm. Inourwork,boththeresource
usage(informoftheexecutiontimeofthetask)andfutureworkloadvariationsare
describedmathematicallyinsteadofcollectedastracedata. Inaddition,thereisno
definitionofrobustnessin[31];theresourcesareallocatedtominimizeatraditional
performancemetric,i.e.,meanresponsetimeofeachtask.
In MSHN (Management System for Heterogeneous Networks) project [37], a
resourcemanagementsystemisdevelopedtodeliverrequiredQualityofServiceto
variousapplicationsinaheterogeneoussystems.MSHNincludesnotonlyaresource
managerthatallocatestaskstomachines[3,11],butalsoresourcestatusmonitorand
library that contains information about resource usage of various tasks. In [3], a
frameworkfordevelopingresourceallocationandschedulingalgorithmsisproposed
that considered both computation and communication time in a unified methodol-
ogy. Whiletheperformancemetricproposedin[3]andinMSHNingeneraldonot
considervariationofworkloadduringruntime,techniquesandframeworksusedby
MSHNresourcemanagercanbeusedasabasistodesignallocationalgorithmsfor
dynamicreal-timesystems[4,5,27,28,29].
27
Intraditionalreal-timesystems[67],resourcesareallocatedsuchthattheQuality
ofServicerequirementssuchasdeadlinesaresatisfied. Inourwork,theconsequent
of violationof a QoS requirement is catastrophic, and thus these requirements can
becategorizedasharddeadlinesinreal-timesystems.Themostimportantdifference
between our work and classic studies in real-time system is the non-predictability
of the environment in which our systems operates. Consequently, it is difficult to
accurately estimatetaskscharacteristicsduringruntime, suchasexecutiontimeor
workload.Theabsenceoftheseaprioriinformationcanleadtoascenariowherethe
trueworst-casevaluesofthesecharacteristicssurpassthepredictedworst-caseval-
ues,whichiscalledaspecificationviolationin[15]. InthisThesis,theenvironment
isexpectedtochangeduringruntimeandresourcesareallocatedsuchthatthosevari-
ationscanbeabsorbedwithoutcausingQoSviolationsinthesystemforthelongest
periodoftime.Manyalgorithmsandheuristics[47,56,58,57,60,61,64,71,72]that
have been developedto solvetheproblem of resource allocationinthese real-time
systemsdonotconsiderthechangingenvironmentinwhichthesystemoperates,and
thusaresusceptibletothespecificationviolation. Inadditiontoallocatingresources
to task, most of these algorithms also schedule when each task will be executed.
In our work, only resource allocation is considered; task scheduling is determined
by the operating system of each machine. This is because our target platform is
COTS-based system, which may not provide much control over several aspects of
thesystemsincludingscheduling.In[58],theobjectivefunctionforminimizationis
28
calledsystemhazardwhichissimilartoour describedinChapter3.6. Thedif-
ferenceisthatourapproachexplicitlyconsidersworkloadvariationinmathematical
equationsconstitutingtheobjectivefunction,whilethesystemhazard iscalculated
withnoexplicitassumptionaboutchangingworkload.
Resourceadaptation[2,62,63,77]isoftenusedtore-allocateresourcesduring
run time in order to cope with uncertainties in the environment. Several adapta-
tiontechniqueshavebeendeveloped,suchasre-allocatingtaskfromheavilyloaded
machine to a more lightlyloaded one, replicating task to share workload from the
originaltask,andre-negotiatingtheQualityofServicerequiredbysometasks. Our
work focuses on an initial allocation and is complementary to these studies which
centers on run time resource management. DeSiDeRaTa [76, 77] is a QoS man-
agement tools for dynamic, scalable, dependable, real-time systems that focus on
definingQoSmetrics[14,40],dynamicresourcemanagement[79,80],andbench-
marking[68,78]. Inaddition,theapplicationpathmodelinDeSiDeRaTaissimilar
totheprimaryroutedescribedinChapter3.1. TheDeSiDeRaTa runtimeresource
manageranalyzespastworkloadvariationstodeterminewhichtaskisunhealthyand
whatactionistobetaken.However,theallocationitself(wheretoexecutethetasks)
isdeterminedbasedonthecurrent“health”ofthetargetedmachines,notfuturework-
loadofthetroubledtask.
Recently,severalresourceallocationheuristicsthatexplicitlyconsiderworkload
variation during run time have been developed [4, 36, 52]. Heuristics presented
29
in [4] are based on the Greedy approach and try to maximize an objective func-
tion similar to . In [36], dynamic resource allocation algorithmsproposed are
consideredproactiveinasensethatuserscantriggeradynamicallocationbasedon
user-specifiedworkloadpatternoftasks. Theexecutiontimeoftasksdependonthe
number of data objects (workload) and is characterized by profile-functions which
issimilartoour and describedinChapter3.2. Anticipatedworkloadfor
eachtaskiscapturedinanadaptationfunctionofthattaskwhichalsospecifywhen
resourceadaptationwilloccur. Theactualadaptationmechanismreplicatesheavily
loadedtaskssothatworkloadisdistributedequallyamongalltaskreplicas.However,
theQualityofServicerequirementsoftasksin[36]arenegotiableandthusarenot
harddeadlinesassumedinthisThesis.Inaddition,heuristicsproposedin[36]dono
guaranteethatalltaskswillfinishbytheirdeadline,buttrytominimizetheaggregate
misseddeadlineratioinstead.Ourworkfocusesonaninitialallocationanddoesnot
assumethattaskscanbereplicated.Becauseexecutiontimeofthealgorithmisnota
criticalfactorininitialresourceallocation,ourallocationapproachescanguarantee
thatallQoSrequirementsaresatisfied.
AllthreeapproachespresentedinthisThesisforsolvingtheresourceallocation
problem are based on a well-researched optimization technique called Linear Pro-
gramming(LP)[17],whichhasbeenusedextensivelyinvarious fieldsofresearch,
suchasOperationResearch[55],economics[22],transportation[19],andresource
allocations[29,60]. Alinearprogramdescribesthewholeallocationproblemasa
30
mathematical problem and solves it using standard method in linear programming
literature,suchasthesimplexmethod[20]ortheinteriorpointmethod[46]. Even
thoughourresourceallocationproblemisnon-linear,thereareseverallinearization
techniquesthatcanbeutilizedtoconvertthemathematicalrepresentationintolinear
form. Oneadvantageofusinglinearprogrammingtosolveourresourceallocation
isthat severalperformance attributesand requirementsof the real-timesystemcan
easilybetranslatedintolinearequations,forexample,thedeadlineofeachtasks,the
throughput, and the end-to-end latency (from sensor to actuator). For comparison,
in[4]agreedyheuristicisdevelopedtosolvethesameresourceallocationusingthe
sameperformancemetricaspresentedinthisThesis. However,becausethegreedy
heuristicallocatesonetasktoonemachineatatime,itcannotguaranteethatrequire-
mentsinvolvinginterrelatedtasks,suchasend-to-endlatencyorwhenresourcesare
shared, will be satisfied. One advantage of our three approaches over the greedy
heuristicin[4]isthatwecanguaranteethatanysolutionprovidedbyourapproaches
will be a valid allocation. Obvious disadvantages of the linear programming are
highercomplexityandlongerexecutiontime.
31
Chapter3
Problemdefinition
Infollowingsections,technicalapproachofthethesisisexplainedindetail.
3.1 Systemandapplicationmodels
We consider a dynamic real-time system consisting of sensors, actuators, and
multitasking-capablemachines. Let denote the set of sensors, and denote the
set of actuators. Sensors and actuators are part of the system hardware, and thus
cannotbeallocated.Letaset represents multitasking-capable
machinesinthesystem. Eachmachineisconnectedtoanetworkswitchviaafull-
duplexcommunicationlink. Thecapacityofeachcommunicationlinkmaybedif-
ferentfromoneanother.
The application tasks in such systems is modeled using the asynchronous
dataflow (ASDF) process network [51]. Each task in the ASDF process network
has associated with it a firing rule, which determines when a task starts execution
basedontheavailabilityofitsinputs.However,theASDFprocessnetworkdoesnot
specifyanyreal-timerequirementsorparametersthatcanvaryatruntime.Weextend
theASDFprocessnetworktocapturethevariationsofrun-timeparametersaswell
32
Figure3.1:Anexamplesensor-actuatornetwork.
astoassociatereal-timerequirementswitha taskoragroupoftasks. Let
denotes an ASDF process network, where is a set of all edges and is a set of
nodes. representsthesetoftaskstobeallocated,andeachedgein connecting
twonodesrepresentsadirecteddataflowfromonetasktotheothertask.
Suchadynamicreal-timesystemandapplicationtasksareshowninFigure3.1.
Thesensorscontinuouslysendinformationabouttheenvironmenttothetasks.These
tasksprocessthedatafromthesensorsandissuecommandstotheactuators.
33
IntheASDFprocessnetwork,asourcenoderepresentsataskthatreceivesdata
fromasensorthroughitsprimaryedge. Notethatonesensorcansenddatatomul-
tiple source nodes. A sink node represents a task that sends data to actuators. An
actuatorcanreceivedatafromexactlyonesinknode. Thisisanaturalassumption,
whichallowsanactuatortobecontrolledbyonlyonetaskinordertoavoidpotential
conflicts. Each sensor outputs data to the corresponding source node(s) at a fixed
rate. Thisratecanbedifferentfordifferentsensors. Thesetofsourcenodesisrep-
resentedby , andthesetofsinknodesby . A routeisdefinedasaset
ofnodesconstitutingapathfromasourcenodetothegivennodethroughaseriesof
primaryedges. Fromthedefinitionofprimaryedge,itfollowsthatarouteforeach
nodeisunique. Therouteforasinknodeiscalleda primary route(PR).Thereare
exactly primary routes(shownenclosed byboxesin Figure 3.1). Note that
datafromasensorcanreachanactuatorthroughaprimaryrouteonly.Let denote
aprimaryroute,where . Thesetofallprimaryroutesisrepresented
by .
A task , where , may be associated with a throughput requirement,
.Thatis,theoutputdatarateoftask isrequirednottobeslowerthanits
throughputrequirement, . Foreachsourcenode ,itsthrough-
put requirement is set to be the output data rate of the sensor feeding the data to
. This throughput requirement is imposed on all the nodes along the routes that
originatefrom . Thus, ,thethroughputrequirementoftask ,isequal
34
to if there is a route from to . A primary route connecting a
sensor-actuatorpair ,where and ,maybeassociatedwithan
end-to-endlatencyrequirement, .Thatis,thetimebetweenthesensorsend-
ing data out and the actuator receiving a message resultingfrom the processing of
thatdatacannotexceed .Let and bevectorsthatrepresent
allend-to-endlatencyandthroughputrequirements,respectively.
3.2 Run-timeparameters
Changes in the environment during run time are captured in parameters called
run-timeparameters.Foreveryprocessingtaskthatreceivesdata,exactlyoneincom-
ingedgeissettobeaprimaryedge(dark,solidedgesinFigure3.1). Ataskstarts
executiononlyafteritreceivesdatafromitsprimaryedge.IntheASDFterminology,
thefiringruleofataskcanbestatedas:ataskfires(startsexecution)onlywhendata
isreceivedfromtheprimaryedge. Theamountofdatathattask receivesfromits
primaryedge,calledloadleveloftask anddenotedas ,canvaryduringruntime
andisconsideredtobearun-timeparameter. Dataarrivingfromnon-primaryedges
(gray, dashed edges in Figure 3.1) is used for informationupdating (e.g., updating
theinternaldatabaseofthetask). ThisisconsideredtoconsumesomeCPUcycles
independentoftheloadlevelofataskanddoesnotvaryduringruntime. Let be
thesetofallprimaryedges.Letvector denotetheloadlevelofalltasks,with
denotingtheinitialloadlevel.
35
Ifweallowrun-timeparameterstovaryindependentlyofoneanother,therewill
betoomanypossibilitiesoftherun-timeparametervariations.Thus,weassumethat
thereisarun-timeparameter,called ,whichgovernsthechangesintheloadlevel
ofallthetasks. Duringruntime,loadlevelofataskvariesaccordingtothechanges
in anditsinitialloadlevel. Mathematically, ,forall .
Notethatwhenthesystemstartsup, isequaltozero,andthecurrentloadlevelof
eachtaskisequaltoitsinitialloadlevel.
Thechangeintheloadlevelofataskisreflected inthevariationofitscompu-
tationandcommunicationlatencies.Anestimated-time-to-compute-functionmatrix,
,givesafunctionforeachtask-machinepair( , )thatmaps toanestimated
computationtimeof when isallocatedontomachine .Anestimated-time-to-
c(K)ommunicate-functionmatrix, ,givesafunctionforeachtask-machinepair
( , )thatmaps toanestimatedcommunicationtimebetweentask allocated
ontomachine tothe networkswitch. Forthisstudy, each entry inthe and
matrices, denoted as and , is assumed to be a linear function of
,withapositiveslope, ,andzerointercept. Theslopedeterminestherateat
whichtheestimatedcomputationlatencyvarieswithrespecttothe variationofthe
loadlevelontheprimaryedgeofthetaskduringruntime. Theinterceptdetermines
theCPUtimeoverheadforatasktoprocessthedataarrivingfromitsnon-primary
edges(whichweignoreinthispaper).Figure3.2showsanexample matrixand
36
Figure3.2:Anexampleetcfmatrix.
theentry .An matrixwillhaveasimilarlayoutasthe matrixshown
inFigure3.2.
Giventhecurrentloadlevelofatask,thetaskcomputationandcommunication
latencies can be calculated using the and matrices. These latencies are
calledbaselatencies,whicharethelatenciesofataskwhenresourcesarenotshared.
Figure3.2showsthebasecomputationlatencyoftask onmachine ,asaresult
oftranslatingthecurrentloadleveloftask usingthe matrix. Thebasecom-
municationlatency of task on machine can be similarlycalculated usingthe
currentloadleveloftask andthe matrix. Whenmultipletaskscompetefor
thesameresource,itisassumedthattheresourceisfairlysharedamongthosetasks.
37
Thus,theactuallatency—thelatencyofataskwhenaresourceisshared—isthe
base latency multiplied by the number of tasks competing for that resource. Note
thattheactuallatencyofataskdependsonbothitsloadlevel(whichcanvarydueto
changesintheenvironment)andonthecurrentresourceallocation.
3.3 Performancemetric
InChapter1,thegeneralgoalofaninitialresourceallocationinadynamicreal-
timesystemisspecifiedas“delaying”thefirstdynamicre-allocationduringruntime.
Aperformancemetricisneededtoevaluateanallocationforitsabilitytopostpone
thefirstdynamicre-allocation. Atfirstglance,therequiredmetriccanbethetimeit
takesfor theallocationtobecomeinvalidwhensubjectedtovariationsintheenvi-
ronment.However,determiningthefirstinstanceofre-allocationisdifficult,because
thevariationsduringruntimecannotbepre-determined.Thus,anotherperformance
metricthatindirectlymeasuresthedelaymustbespecified.Intuitively,anallocation
thatcan“absorb”largeramountofvariationsinrun-timeparametersismoredesir-
able.Thereasonisthat,givenaspecificpatternofvariationsinrun-timeparameters,
anallocationthatcanabsorblarger amountofvariationshasabetterchancenotto
becomeinvalid. Inthissection,weproposeaperformancemetricthatcanmeasure
howwell-prepared adynamicreal-timesystemisfor absorbingtherun-timevaria-
tionsintheenvironmentwithoutviolatingQoSconstraints.
38
3.3.1 Operatingregionandconformingregion
Suppose that a dynamic real-time system must satisfy QoS attributes. An
-dimensionQoSspacecanbedefined,suchthateachdimensionofthespacecor-
respondsto each QoS attribute. The QoS requirementsof the systemwilldefine a
regionwheretherequirementsaresatisfied. Asanexample,considerasystemwith
, sothere are 2 QoSattributesdefined inthesystem. Furthermore, suppose
thattheQoSattributesaredenotedasQoS andQoS ,andtheQoSrequirements
ofthesystemareQoS andQoS . Figure3.3showstheexampleQoS
space,togetherwiththeQoSrequirements.TheshadedareainFigure3.3represents
thespacewherethesystemQoSvaluesconformwithallQoSrequirements,andis
calledconformingregion.
Supposethatthereare run-timeparametersthatcharacterizethecurrentstateof
theenvironmentinwhichthedynamicreal-timesystemoperates. An -dimensional
systemstatespace,whereeachdimensioncorrespondstoarun-timeparameter,can
bedefined. Anexamplesystemstatespacefor isshowninFigure3.4. Each
pointinthestatespace,denotedasanoperatingpoint,representsastateoftheenvi-
ronmentasdescribedbythevaluesofrun-timeparameters.Aninitialoperatingpoint
isspecifiedbytheinitialvaluesofallrun-timeparameters.Weassumethattheseini-
tialvaluesareknownandgivenaspartoftheproblem,ortheirexpectedvaluescan
bedeterminedbysomemethods.
39
Figure3.3:Anexample2-dimensionQoSspace
Given a resource allocation, denoted as , the values of QoS attributes can be
determinedforanoperatingpoint. TheseQoSvaluesaremerelyresultsofrunning
a system with an allocation at that operating point. From this point of view, an
allocation isafunctionthattranslatesapointinastatespace(Figure3.4)toapoint
inaQoSspace(Figure3.3). Anoperatingpointrepresentingthecurrentstateofthe
environmentistheinputofthefunction,andtheresultingoutputistheQoSvalues
ofthesystem.
ThesystemmustsatisfysomespecifiedQoSrequirements.Anoperatingpointis
saidtobevalidiftheresultingQoSvalues,calculatedbyfunction (theallocation),
satisfiesallQoSconstraints.Forthesakeofdiscussion,allthevalidoperatingpoints
areassumedtoformaclosedandcontinuousspace,calledanoperatingregion. The
40
Figure3.4:Anexample2-parametersystemstatespace
shadedspaceinFigure3.4specifiestheoperatingregionofallocation . Thus,the
operatingregioninFigure3.4correspondswiththeconformingregioninFigure3.3
throughtheallocation .Notethatoperatingregionsofdifferentresourceallocations
mayhavedifferentlocation,shape,andsizeinthestatespace. However,intheQoS
spacetherecanbeonlyoneconformingregion,whichisspecifiedbyQoSrequire-
mentsofthesystem. Theinitialoperatingpointisalsothesameforallallocations.
Inaddition,aninitialallocationisfeasibleifandonlyifitsoperatingregioncovers
thetheinitialoperatingpoint.
3.3.2 Runtimebehaviorofthesystem
Figure 3.5 shows the run time behavior of the system in both state space and
QoSspace.Afterstartingareal-timesystemupwithinitialresourceallocation and
41
initialvaluesofrun-timeparameter(number1inFigure3.5),thesystemissubjected
tochangesintheenvironment. Thesechangesarecapturedinthevariationofrun-
timeparameters, whichcausesthesystemtotransitfromtheinitialoperatingpoint
toanother. TransitiontothenewoperatingpointalsoaffectstheQoSvaluesofthe
system, causing the current QoS values to move to a new co-ordinate in the QoS
space. These transitions are shown as number 2 in Figure 3.5. When the system
transitsoutoftheoperatingregionofthecurrentallocation(number3inFigure3.5),
theQoSvaluesalsotraversestoaco-ordinateoutsideoftheconformingregion. At
thisinstance,aQoSviolationoccurs,andthecurrentresourceallocationisnolonger
valid.Thedynamic(on-line)resourcere-allocationalgorithmisinvokedtofindanew
resourceallocation(arbitrarilydenotedasallocation )forthesystem.Theoperating
region of allocation mustcover the current operating point of the system for the
allocationto be valid. The new, validallocationalso movesthe QoS valuesof the
systembackintotheconformingregionintheQoSspace.Theeffectsofre-allocating
resourcesareshownasnumber4inFigure3.5. Ifresourceallocationisviewedas
afunctionwithinputanoutput,aQoSviolationoccurswhentheinputoffunction
changessuchthattheoutputexceedsacceptable value. Resource re-allocationis
then utilizedto find a new function suchthat its outputfalls back to withinQoS
requirementsatthesameinputlevel.
42
Figure3.5:Anexampleruntimebehaviorofthereal-timesystem
43
3.3.3 Definingperformancemetricforinitialresourceallocation
Thegeneralgoalofaninitialallocationistodelaythefirstdynamicre-allocation
ofresourcesduringruntimeduetovariationsintheenvironment.Therefore,thegoal
istofindanallocationwhoseoperatingregioncovers,forthelongestperiodoftime,
alltheoperatingpointstowhichthesystemtraverseduringruntime. However,the
exacttransitpathsofoperatingpointandalsoofQoSco-ordinatevalues(bothofthem
shownasnumber2inFigure3.5)cannotbedeterminedbeforethesystemactually
startsrunning.Thus,determiningthetimethatQoSviolationoccursduringruntime
(number 3 in Figure 3.5) is difficult during off-line stage that initial allocation is
calculated.
Intuitively,anallocationthatcanaccommodatelargeramountofrun-timevaria-
tionswillbeabletosustainthesystemforalongerperiodduringruntimewithout
anyQoSviolation.Thereasonisthat,givenatransitpathofthesystem,ittakesmore
timetotraversefromtheinitialoperatingpointtotheedgeofanoperatingregionthat
coversalargerareaaroundtheinitialoperatingpoint. Thisistrueeveniftheactual
path is not known during initial resource allocation at the off-line stage. The only
assumption is that interactions between the real-time system and the environment
doesnotchangewithdifferentresourceallocations(i.e.,ifthereal-timesystemmod-
ifiestheenvironment,differentresourceallocationswillaltertheenvironmentinthe
sameway.)
44
Oneperformancemetricthatcanbeusedtodeterminetheamountofvariations
an allocation can accommodate is the sum of QoS values. The lower the sum, the
larger the aggregated amount the QoS values can increase due to variations in the
environmentbefore theQoS requirementsisviolated. The benefit of thismetric is
thatitiseasytoevaluateandoptimize,becausetheconformingregioncanbereadily
determined from the given QoS requirements. The initial QoS values can also be
calculated from the given initial operating point and the resource allocation which
is to be evaluated. On the other hand, an operating region of the allocation is not
known until every operating point is tested for validity, or an inverse function that
translatesQoS co-ordinatetoitscorrespondingoperatingpointisdetermined. The
problemisthatthismetricisacollectiveamountofallQoSvalues,andthuscannot
judgetheamountthatanindividualQoSattributescansafelybeincreased.Figure3.3
showsthesumofallQoSvaluesasaperformancemetric,togetherwiththeinherent
problem. Supposethat2resourceallocations and aretobeevaluated,andboth
allocations have the sum of QoS values equal to . The initial QoS co-ordinates
of allocation and are shown as and in Figure 3.3, respectively. Because
the sumof QoS valuesin both allocationsisequal, and mustlie on the same
straightline . However, lies closer to the edge of conforming
regionthan ,andthushaslessspacetoforQoS toincrease.Accordingtothesum
metric,allocation isasgoodasallocation ,butfromFigure3.3allocation ismore
preferable. Becauseofthisintrinsicproblem,thesummetricwillnotbeutilizedas
45
theperformancemetricforinitialresourceallocationindynamicreal-timesystems.
Interestedreaderscanreferto[28]wherethesummetricisdiscussedinmoredetail.
Inthispaper,theproposedperformancemetricusedtoevaluateinitialallocations
isa norm metric. A normisdefined in thestate space as thedistancebetween the
initialoperatingpointandanotheroperatingpoint. Let denoteanoperatingpoint,
with denotingthe initialoperatingpoint. Also let denote a set of operating
pointsthatlieontheboundaryoftheoperatingregion. Let denoteanormonthe
spaceofvaluesspecifiedbyrun-timeparameters.Thus, ,where
, is the minimumdistance between the initial operating point and the boundary
of the operating region. This value corresponds to the maximum amount of run-
timevariationsthattheallocationcanaccommodateintheworstcase(i.e.,whenthe
variationsoccursuchthatthesystemtraversealongtheshortestpathtotheoperating
regionboundary.)Aninitialallocationthathasahighervalueof can
accommodatelargeramountofvariationsduringruntime,andcorrespondinglyisa
betterallocation.
In our application model, the load levels of all tasks are controlled by a single
run-timeparameter .Let bethemaximumvalueof ,undertheconditionthat
thereisnoQoSviolationinanytasksorprimaryroutes. Notethat dependson
bothQoSrequirementsandtheresourceallocationofthesystem.Thevalueof
isthusthe valueand canbe usedas theperformance metric ofthe
46
allocation. Thisspecific performance metric is called MAIL, MaximumAllowable
IncreaseinLoadlevel.
3.4 Formalproblemdefinition
Based on the models described in Chapter 3.1, and the performance metric
explainedinChapter3.3,aformaldefinitionoftheresourceallocationproblemcan
bestatedasfollows:
Given:
1. Asetofmachines
2. Asetofsensors, ,andasetofactuators,
3. AnASDFprocessnetwork, andasetofallprimaryedges
4. and matrices
5. Vectors , and
Find:
Anallocationofalltasksin ontomachinesin thathasthehighestMAILvalue
47
3.5 Mathematicalformulation
Amathematicalformulationfortheproblembasedonourmodelisdescribedin
thissection.Anobjectivefunctionisdefined,whichleadstoanoptimalMAILvalue
undersomeassumptions.
Let beamatrixthatrepresentsanallocationoftasksontomachinessuchthat
if isallocatedonto
otherwise
where and . Figure3.6showsanexample matrixinareal-time
systemwith5machinesand10tasks. Therowsofthe matrixrepresenttasksand
the columns represent the machines in the real-time system. In this example, task
is allocatedontomachine , task is allocatedontomachine , and soon.
A task can be allocated to only one machine. Consequently, for any task , there
isexactlyone thatisequalto1forall . Asshowninthe matrixin
Figure 3.6, each row (representing a task) can have only one element equal to “1”
andtherestareequaltozero.Givenanallocation,let bethetotalnumberoftasks
thatexecuteonmachine ,i.e.,
(3.1)
48
Figure3.6:AnexampleXmatrix.
which is the sum of the value of all elements in column in the matrix. For
example, inFigure3.6, (thetotalnumberoftasksrunningonmachine )is2
and (thetotalnumberoftasksrunningonmachine )is3.
The base latency of a task on a machine is calculated from the load level of
that task using the (estimated-time-to-compute-function) or (estimated-
time-to-c(k)ommunicate-function) matrix (refer to Chapter 3.2). Figure 3.7 shows
theprocessofcalculatingthebasecomputationlatencyfromtheloadlevelandthe
matrix. Thebasecommunicationlatencyiscalculatedinasimilarmanner,but
usingthe insteadofthe matrix. FromFigure3.7,itisclearthatthebase
49
Figure3.7:Calculatingbaselatency.
latency can be calculated as the product of the slope of the linear function of that
task-machinepairandtheloadlevelofthetask. Mathematically,thebaselatencyof
task withload onmachine is .
Recallthatthebaselatencyisthelatencywhenthereisonlyonetaskallocatedto
themachine,i.e.,thereisnosharing(refertoChapter3.2). Whenresourceisfairly
shared, the actual latency of a task is the base latency of that task multiplyby the
totalnumberoftaskssharingthesameresource. Mathematically,theactuallatency
ofatask withload onmachine is .
At the time of resource allocation, the allocation of tasks to machines is not
known. However,byusingthe matrix,theequationdescribingtheactuallatency
oftask canbegeneralizedasthesumofthelatencyofthetaskoneachmachine,
50
weightedby ofeachtask-machinepair. Thus,theactualcomputationlatencyof
task fortheinitialloadlevel,denoted ,canbecalculatedas
(3.2)
Recallthatonlyone intherowinthe matrixrepresenting canbeequalto
one.Also, isafunctionof .Thus,Equation(3.2)willprovidethecorrectactual
computationlatencyoftask ,nomatterwhichmachine isallocatedtoandhow
manyothertaskssharethesamemachine.
The equation describing the actual communication latency of task can also
begeneralizedinasimilarmanner. However,assumethatcommunicationofatask
occurs twice for each computation: receiving data from the network, and sending
resultsbackaftercomputation.Theactualcommunicationlatencyoftask , ,is
(3.3)
,theactualend-to-endlatencyofaprimaryroute isthesumofalllatencies
ofalltaskscomprisingtheroute.Thus,
(3.4)
51
Givenatask ,itsthroughputrequirementissatisfiedwhenbothitscomputation
latencyandcommunicationlatenciesdonotexceedthereciprocalofitsthroughput
requirement.Mathematically,
(3.5)
Given a primary route , its end-to-end latency requirement is satisfied
whentheactuallatencyoftheroutedoesnotexceeditsrequirement.Mathematically,
(3.6)
3.6 Objectivefunction
InChapter3.2, isdefinedasarun-timeparameterthatgovernsthevariations
oftheloadlevelofalltasksinthereal-timesystem.Inaddition,themaximumvalue
of thatcausesnoQoSviolationinanytasksorroutesiscalled . This
is used as the performance metric for evaluatingthe initial allocation to determine
its robustness with respect to load variations during run time. In this section, an
objectivefunctionforamathematicalformulationoftheresourceallocationproblem
isdeveloped.Itisalsoshownthatanallocationthatpossessestheoptimizedvalueof
theobjectivefunctionwillalsohasthehighestvalueof ,andthusistheoptimal
initialallocation.
52
Let bethenormalizedcomputationslacknessoftask .Itiscalculatedas:
(3.7)
Similarly, ,thenormalizedcommunicationslacknessofatask isdefined
as:
(3.8)
Foraroute ,thenormalizedslacknessis
(3.9)
Conceptually,thenormalizedslacknessrepresents,asapercentage,theslackoravail-
ableroomforthelatencyofataskoraroutetoincrease,beforethethroughputand/or
end-to-endlatencyrequirementisviolated.Notethatanallocationsatisfiesallperfor-
mancerequirementsiffthenormalizedslacknessforalltasksandallprimaryroutes
isnon-negative.
Givenanallocation,let
53
Figure3.8:Variationoftheactuallatencyoftasksandprimaryrouteswithrespectto
.
(3.10)
isthustheminimumnormalizedslacknessofalltasksandprimaryroutesof
thegivenallocation.
Claim1:Givenafeasibleallocation,
Proof: Fora givenfeasibleallocation,Figure 3.8showsthevariationoftheactual
computationorcommunicationlatencyofalltasksandtheactualend-to-endlatency
of all primary routes with respect to . In Figure 3.8, actual latency refers to
the actual computation or communication latency of a task, or the actual end-
to-end latency of a primary route. Similarly, refers to the constraint on
54
task or primary route ( for a task or for a primary route).
Note that when equals -1, the actual latency of any task or route becomes zero.
The slope of the line that corresponds to the actual computation latency of task
can be calculated as . By replacing with
in the previous expression, we get the slope of the line that corresponds
to the actual communication latency of task . The slope of the line that corre-
sponds to the actual end-to-end latency of primary route can be calculated as
.
Let representthetotalnumberoflinesinFigure3.8. Wehave .
The actual latency for a task or a primary route increases when increases (i.e.,
theloadlevelofeachtaskincreases). Foratask (primaryroute ),let bethe
valueof suchthattheactuallatencyof ( )reaches . Givenanyfeasible
allocation,wehave .
Basedonthedefinitionof (theMAILvalueofanallocation)inChapter3.3,
wehave . Thus,thereexistssome , ,suchthat
.Notethat,
FromFigure3.8,wehave .Itfollowsthat
55
Thus,wehave
Claim 2: The allocationthat givesthe highestvalueof among allallocations
willalsoresultinthehighestvalueof .
Proof: FromClaim1,wehave . Because ismaximizedwhen
ismaximized,maximizing maximizes
Thus, we can use as the objective function for maximization, instead of
tryingtofind directly.
3.7 Formalmathematicalformulation
The mathematical formulation of the resource allocation problem is shown in
Figure 3.9. The auxiliary variable is a real number, while , an entry in ,
is either 0 or 1. The objective is to maximize , but is limited by the first three
constraintscorrespondingtothenormalizedslacknessofalltasksandprimaryroutes.
Thelastsetofconstraintsenforcesatasktobeallocatedtoonlyonemachine. After
solvingtheformulation,theauxiliaryvariable willbeequalto . Furthermore,
56
Given:
Find:
to
Maximize:
Subjectto:
Figure3.9:MathematicalFormulation
willrepresentaresourceallocationthatgivesthehighestvalueof andthus
thehighestvalueof .
57
Chapter4
Mixedintegerprogrammingwith
pre-selectionapproach
Inthischapter,anapproachtosolvethemathematicalformulationinFigure3.9,
calledMIP(*),isdescribed.Inthisapproach,theformulationislinearizedbyatech-
nique called pre-selection. Then, the problem is re-formulated as a mixed-integer-
programmingproblemandsolvedusingstandardlinearprogrammingtechnique.
4.1 Pre-selectiontechniqueforlinearization
Direct mathematical formulation of the problem shown in Figure 3.9 contains
non-linearequations. Specifically, inordertocalculate and , twovariables
and aremultipliedtogetherinEquation(3.2)andEquation(3.3).Recallthat
.Tolinearizetheformulation,thenumberoftasksallocatedontoamachine
ispre-selectedusingaheuristic.Thisisspecifiedasavector ,where isthepre-
selected numberof taskson machine . Anexampleofa heuristicusedtoselect
valuesinaheterogeneoussystem,calledthecapability-basedheuristic(CBH),is
showninFigure4.1.
58
Let be the reciprocal of the aggregated slopeof the and matrices of
task onmachine :
Let betherelativespeedoftask onmachine :
Thecapabilityofmachine ,called ,iscalculatedas:
Thepre-selectednumberoftasksallocatedontoamachineisthen:
Figure4.1:CBHheuristic
By usingthe pre-selectiontechnique, Equation(3.2) and Equation(3.3) can be
rewrittenasfollow.Firstlet and representthecomputationandcommunication
latencyoftask inthepre-selectiontechnique.Thus,
(4.1)
(4.2)
59
and
(4.3)
(4.4)
Theend-to-endlatencyofaprimaryroute inthepre-selectiontechniqueisthus
(4.5)
Also,thenormalizedcomputation,communication,andprimaryrouteslackness
are
(4.6)
(4.7)
(4.8)
60
4.2 MIP(*)formulation
Theproblemcannowbeformulatedusingmixed-integer-programming,asshown
inFigure4.2. Theauxiliaryvariable isarealnumber,while ,anentryin ,is
either0or1. Attheendoftheoptimization,theauxiliaryvariable willbeequalto
.Furthermore, willrepresentaresourceallocationthatgivesthehighestvalue
of andthusthehighestvalueof . TheheuristiciscalledMIP(*),where*
canbesubstitutedbythenameoftheheuristicusedtopre-select values. Thus,if
CBHisusedtopre-select valuesforthemixed-integer-programmingformulation
(step1inFigure4.2),theentireapproachwillbecalledMIP(CBH).Thedifferences
betweenMIP(*)formulationandthemathematicalformulationinFigure3.9are(1)
MIP(*)requires ,thepre-selectednumberoftasksoneachmachine,asinputand
theequationsdescribingthenormalizedslacknessaremodifiedaccordinglyand(2)
additionalconstraints(shownasthelastsetofconstraintsinFigure4.2)areincluded
sothatthenumbersoftasksoneachmachineareequaltothepre-selectednumbers.
4.3 Experimentalprocedure
AsimulatorbasedonthemathematicalformulationpresentedinChapter3.5was
developedtoevaluatetheperformanceofourresourceallocationtechniques. Given
anallocation,thesimulatorcalculatesthetaskexecutionlatenciesusingtheequations
presented in Chapter 3.5. The performance attributes (i.e., , , and ) are then
61
Begin
1.Select ,forall ,byusingaheuristic.
2.Formulatetheproblemusingmixed-integer-programming,asfollows:
Given:
Find:
to
Maximize:
Subjectto:
End
Figure4.2:MIP(*)approach
calculated and compared with the corresponding requirements. If all performance
requirementsaresatisfied,thenormalizedslacknessvaluesofalltasksandprimary
routesarecalculated,whichconsequentlydetermine .The valueforthis
allocationcanthenbecalculatedusingtheequation ,derivedfrom
Claim1inChapter3.5
Theexperimentsweredividedintotwosetsbasedontheproblemsize. Thefirst
setconsistedofsmallproblemsthatconsistofthreeto fivemachinesand12tasks.
62
Table4.1:Characteristicsof and matrices
heterogeneity inputparameters
Lo-Lo 50 ** 0.1 0.1
Hi-Lo * 30 0.1 0.5
Lo-Hi 50 ** 0.5 0.1
Hi-Hi 50 ** 0.5 0.5
*iscomputedforeachtaskasafunctionof and
**iscomputedforeachmachineasafunctionof and
Thesecondsetrangedfrom10to20machinesand30to60tasks. AnASDF pro-
cessnetwork, and matrices,theloadlevelofeachtask,andperformance
requirements,weregeneratedforeachprobleminstance. The and matri-
cesweregeneratedtocapturethemachineandtaskheterogeneities[7]. Specifically,
each matrixwas characterized by two parameters: machine heterogeneityand task
heterogeneity.Bothheterogeneitiescanbemodeledas“high”or“low.”Gammadis-
tributionswereusedtogeneratethematrices.Thefourcategoriesofthematricesand
thecorrespondinginputparametersareshowninTable4.1.Inthetable, quantifies
theaveragevalueoftheslopesinthematricesand quantifiestheheterogeneities.
To investigatethe performance of MIP(CBH) under various task and machine het-
erogeneities,experimentswereconductedusingallfourcategories. Eachelementin
vector isgeneratedbysamplingauniformdistributionofvaluesrangingfrom
10to100.
Foreachtask,theaveragevaluesofitscomputationandcommunicationlatencies
over all machines were calculated from the and matrices, and the
values. For each primary route, the sum and the maximum value of these average
63
valuesofallnodesalongtheroutewerecalculated,denotedas and ,respectively.
Let beequalto . Theend-to-endlatencyrequirementofthePRwasthen
set to . The factor is used to adjust the tightness of the constraints.
ThethroughputrequirementofeachtaskalongthePRwassettobe .
Due to space limitations, we emphasized only computation intensive applications
– the average communication latency for each task is around 1/100 of its average
computationlatency.
We also implemented three other approaches: integer-programming-based
(referred to as IP in the following text), Min-Min, and Greedy. CBH is used to
pre-selectthenumberoftasksrunningoneachmachineintheIPapproachalso.The
problemisthenformulated,usinginteger-programming,tominimizethesumofthe
end-to-endlatenciesofallPRs,whichissimilartotheobjectivefunctionpresented
in[28].TheGreedyheuristicmapstasksinarandomorder,andeachtaskismapped
onto the machine that gives the shortest computation and communication latency
based on the mapping information so far. The Min-Min heuristic is a variation of
algorithm D in [41] that orders tasks using their computation and communication
latencies on the best machines. In all the three approaches, the computation and
communicationlatenciesarecalculatedwhileconsideringmultitaskingoftasks.The
formulationforIPisillustratedinFigure4.3.ThepseudocodefortheMin-Minand
GreedyheuristicsareshowninFigures 4.4and 4.5,respectively. Theperformance
ofMIP(CBH)wascomparedwithallthesethreeapproaches.
64
Begin
1.Select ,forall ,byusingCBH.
2.Formulatetheproblemusinginteger-programming,asfollows:
Given:
Find:
to
Minimize:
Subjectto:
End
Figure4.3:IPapproach
Begin
1.Arbitrarilyorderalltasksin
2.Foreachmachine ,set (thenumberofallocatedtasks)to0.
3. foreachtask
4. Findamachine thatgivesthesmallestvalueof
5. Allocatetask tomachine
6. Increment by1
End
Figure4.4:PseudocodeforGreedyheuristic
65
Begin
1.Foreachmachine set (thenumberofallocatedtasks)to0
2. whilethereexistunallocatedtasks
3. foreachunallocatedtask
4. Findamachine thatgivesthesmallestvalueof
;recordthisvalueas
5. Findatask thathasthesmallest value
6. Allocatetask tothemachine thatgivestheabove value
7. Increment by1
End
Figure4.5:PseudocodeforMin-Minheuristic
4.4 Results
For small problem instances, an optimal allocation that results in the highest
valuewasfoundbyenumeratingallpossibleallocations.Becausetheexecu-
tiontimeforenumerationisquitelarge,thenumberofmachineswaslimitednotto
exceed fiveandthenumberoftaskswas fixedat12(twosources, threesinks, out-
degree ) inall smallproblem instances. To studythe effect of the tightnessof
constraints, wassettotwodifferentvalues(1.5and2).
The simulationresults are shown in Figure 4.6, Figure 4.7, and Figure 4.8. In
thesefigures,theratiosofthe valuesobtainedfromvariousapproachestothe
optimum valuearepresented. Eachbarrepresentstheaveragevalueofthe
ratio over40 instances (samplingsof the gammadistributionsused togenerate the
and matrices),witha90%confidenceintervalanda20%(orbetter)
precision.Theconfidenceintervalsareindicatedbylinesatthetopofeachbar.Ifany
66
approach fails in an instance (i.e., the allocationfrom that approach does notmeet
allperformancerequirements),thatinstanceisexcludedfromthecalculationofthe
averageratioandtheconfidenceintervalforallapproaches. Theplotsclearlyshow
thatontheaverage,MIP(CBH)outperformstheotherthreeapproachesforallmatrix
characteristics. IntheLo-HienvironmentofFigure4.8,theaverageperformanceof
MIP(CBH)is4.8timesbetterthanthatofMin-Min.Also,forMin-MinandGreedy,
amissratio(thenumberofinstancesthatanapproachfailsnormalizedwiththetotal
number of instances) of up to 57% was observed in the simulations, and this miss
rateincreaseswhenthevalueof decreases. Detaileddiscussionofthemissratiois
presentedlaterinthissection.
Resultsfor large problemsare showninFigure 4.9-4.16. Due tospace limita-
tions,resultsforonlyasinglevalueof arepresented.Becauseitisimpracti-
caltofindanoptimalallocationbyenumerationforlargeproblems,theactual
valuesfor allthe four approaches are compared with one another. Each bar repre-
sentstheaverage valueover40instanceswitha90%confidenceintervaland
a 15% (or better) precision. The confidence intervals are indicated by the lines at
the top of each bar. If any approach fails in an instance, that instance is excluded
fromthecalculationoftheaverage valueandtheconfidenceintervalforall
approaches. By fixing the number of machines (tasks) and varying the number of
tasks(machines),twosetsofsimulationswereperformedforlargeproblems.
67
In the first set, the number of tasks was fixed at 40 (7 sources, 7 sinks, out-
degree )whilethenumberofmachinesrangedfrom10to20(resultsshownin
Figure4.9-4.12). Inthesecondset,thenumberofmachineswas fixedat12,while
thenumberoftasksrangedfrom30(fivesources, fivesinks,out-degree )to60
(10 sources, 10 sinks, out-degree ). The results of the second set are shown
inFigure4.13-4.16. Inallcases, MIP(CBH) providedthehighestaverage
values,andperformedupto4.7timesbetterthanMin-Min.
Notethatthe valueofallapproachesinFigure4.9-4.11doesnotincrease
withtheincreaseinthenumberofmachines.Thisisbecausethevalueof decreases
when the number of machines increases (for a fixed number of tasks) and conse-
quently, the throughput and end-to-end latency requirements of the tasks become
tighter. In otherwords, the increase inthe numberof machinesisbalanced outby
thetighterconstraints.However,intheHi-HienvironmentofFigure4.12,thediffer-
encesbetweenslopeofthefunctionindifferententriesofthe and matrices
are verylarge, allowingfor a greater degree of optimizationthan in otherenviron-
ments.Thus,thehighdegreeofoptimizationintheHi-Hienvironmentresultsinthe
higher valueofeachapproach,comparedwiththe valueofthesame
approachinotherenvironments.Inaddition,whenthenumberofmachinesincreases,
theincreaseinthedegreeofoptimizationoutweighsthetighterperformancerequire-
mentsintheHi-Hienvironment.Thus,the valueofMIP(CBH)inFigure4.12
increases when the number of machines increases. Because other approaches do
68
nottakeadvantageofthehigherdegreeofoptimizationasgoodasMIP(CBH),the
value of their allocations does not increase as much when the number of
machinesincreases.
Asimilar(butreverse)explanationholdsforFigure4.13-4.16,wherethenumber
ofmachineis fixedandthenumberoftasksincreases. Inthiscase, theconstraints
arelooserwiththeincreaseinthenumberoftasks.
To study the miss ratios of MIP(CBH), IP, Greedy, and Min-Min approaches,
experiments were conducted with four machines, 12 tasks, and values ranging
from1.5to2,inallcombinationsoftaskandmachineheterogeneities. Figure4.17
showsthemissratioofGreedyandMin-Min. Inboththeapproaches,themissratio
increaserapidlyas valuedecreases.When equals1.5,Greedyshowsa0.575miss
ratiointheLo-Loenvironment.Ontheotherhand,themissratiosofMIP(CBH)and
IParezeroinallexperimentsfor rangingfrom1.5to2.
In theory, the integerand mixedinteger programmingformulationsproduce an
optimalsolutionaccordingtotheirobjectivefunctionssubjecttotherespectivecon-
straints.However,inthesimulations,theamountoftimethatthesolverwasallowed
toexecuteimpactsthequalityoftheresultingallocationofMIP(CBH)andIP.Inthe
experiments,thesolverwasexecutedfor15secondsforeachprobleminstance. For
comparison,Min-MinandGreedytooklessthan1secondtofinishexecutioninany
probleminstancesimulated.
69
3 machines, 12 tasks, f = 1.5
0
0.2
0.4
0.6
0.8
1
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
etcf and etkf matrix characteristics
ratio over optimum MAIL value
MIP(CBH) IP Min-Min Greedy
3 machines, 12 tasks, f = 2
0
0.2
0.4
0.6
0.8
1
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
etcf and etkf matrix characteristics
ratio over optimum MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.6:Simulationresultsforproblemswith3machinesand12tasks
70
4 machines, 12 tasks, f = 1.5
0
0.2
0.4
0.6
0.8
1
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
etcf and etkf matrix characteristics
ratio over optimum MAIL value
MIP(CBH) IP Min-Min Greedy
4 machines, 12 tasks, f = 2
0
0.2
0.4
0.6
0.8
1
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
etcf and etkf matrix characteristics
ratio over optimum MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.7:Simulationresultsforproblemswith4machinesand12tasks
71
5 machines, 12 tasks, f = 1.5
0
0.2
0.4
0.6
0.8
1
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
etcf and etkf matrix characteristics
ratio over optimum MAIL value
MIP(CBH) IP Min-Min Greedy
5 machines, 12 tasks, f = 2
0
0.2
0.4
0.6
0.8
1
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
etcf and etkf matrix characteristics
ratio over optimum MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.8:Simulationresultsforproblemswith5machinesand12tasks
72
40 tasks, f = 1.4, Lo-Lo environment
0
0.5
1
1.5
2
10 12 14 16 18 20
number of machines
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure 4.9: Simulationresultsforproblemswith40tasksand 10- 20machinesin
Lo-Loenvironment
40 tasks, f = 1.4, Lo-Hi environment
0
0.5
1
1.5
2
10 12 14 16 18 20
number of machines
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.10: Simulationresultsforproblemswith40tasksand10-20machinesin
Lo-Hienvironment
73
40 tasks, f = 1.4, Hi-Lo environment
0
0.5
1
1.5
2
10 12 14 16 18 20
number of machines
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.11: Simulationresultsforproblemswith40tasksand10-20machinesin
Hi-Loenvironment
40 tasks, f = 1.4, Hi-Hi environment
0
0.5
1
1.5
2
10 12 14 16 18 20
number of machines
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.12: Simulationresultsforproblemswith40tasksand10-20machinesin
Hi-Hienvironment
74
12 machines, f = 1.4, Lo-Lo environment
0
0.5
1
1.5
2
30 36 42 48 54 60
number of tasks
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.13: Simulationresultsforproblemswith12machinesand30-60tasksin
Lo-Loenvironment
40 tasks, f = 1.4, Lo-Hi environment
0
0.5
1
1.5
2
30 36 42 48 54 60
number of machines
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.14: Simulationresultsforproblemswith12machinesand30-60tasksin
Lo-Hienvironment
75
40 tasks, f = 1.4, Hi-Lo environment
0
0.5
1
1.5
2
30 36 42 48 54 60
number of machines
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.15: Simulationresultsforproblemswith12machinesand30-60tasksin
Hi-Loenvironment
40 tasks, f = 1.4, Hi-Hi environment
0
0.5
1
1.5
2
30 36 42 48 54 60
number of machines
MAIL value
MIP(CBH) IP Min-Min Greedy
Figure4.16: Simulationresultsforproblemswith12machinesand30-60tasksin
Hi-Hienvironment
76
4 machines, 12 tasks, Greedy approach
0
0.2
0.4
0.6
1.5 1.6 1.7 2
value of f
miss ratio
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
4 machines, 12 tasks, Min-Min approach
0
0.2
0.4
0.6
1.5 1.6 1.7 2
value of f
miss ratio
Lo-Lo Hi-Lo Lo-Hi Hi-Hi
Figure4.17:MissratioofGreedyandMin-Min
77
4.5 Discussion
Inthissection,wecomparetheperformanceofMIP(CBH)withthatofGreedy
and Min-Min defined in Chapter 4.4 to show the effectiveness of the MIP(CBH)
approach. Givenaprobleminstance,let denotethe value
obtained by MIP(CBH); , the value obtained by Greedy; and
,the valueobtainedbyMin-Min.
ThefollowingtheoremshowsthattheperformanceofMIP(CBH)canbearbitrar-
ilybetterthantheperformanceofGreedyandMin-Min.
Theorem4.5.1 For any integer , where is the number of tasks to be allo-
cated, there are problem instances such that both MIP(CBH) and greedy heuristic
succeedin findingafeasibleallocationand ,forany .
Proof: Weconsiderproblemsconsistingof tasksand machineswithallcom-
munication latencies set to 0. An example problem for is shown in Fig-
ure 4.18. Let be the reciprocal of the throughput requirement of task . For
simplicity,weassumethattheend-to-endlatencyrequirementoftheprimaryroute
connecting to isverylarge. Thus, when increases, thethroughputrequire-
ment of task , for some , will be violated before the end-to-end
latency requirement is violated. For this example problem, let (refer to
Figure 4.18). It is easy to verify that the allocation obtained by MIP(CBH) has
. Because , of this allocation is
78
. Similarly, the allocation obtained by Greedy has , and thus
. Clearly, when , bothMIP(CBH) and the greedy
heuristicsucceedinfindingafeasibleallocation.Then .
Let bethe lowerboundof theratiobetween and
when .Thislowerboundcanbeachievedwhen approaches .Thus,we
have . Givenanarbitrary
,if ,wealwayshave forany . Otherwise,
thereexists where ,suchthat .
From the above example problem, a new problem consisting of tasks and
machines can be recursively defined using the construction method shown in
Figure 4.19. For each such larger problem, it is easy to verify that MIP(CBH)
producesasimilarallocationastheoriginalproblem.Similarstatementholdsforthe
Greedyapproach. Consequently,thesameperformanceboundbetweenMIP(CBH)
and Greedy holds. The construction can be easily extended to construct problems
where numberoftasks
Thefollowingtheoremshowsthat can findafeasiblesolutionfor
someproblems,whileGreedyandMin-Minfail.
Theorem4.5.2 Foranyinteger ,wherenisthenumberoftaskstobeallocated,
thereareprobleminstancesforwhichGreedyfailstofindafeasibleallocation,while
MIP(CBH)succeeds.
79
Figure4.18:ExampleproblemforTheorem4.5.2
Proof: FortheexampleexplainedinTheorem4.5.1,if ,itcanbe
verifiedthatGreedyfailstofindafeasibleallocation,whileMIP(CBH)succeeds. A
similarconstructionmethodasexplainedintheproofofTheorem4.5.1canbeused
toextendtheexampletoprobleminstanceswhere .
AsimilarclaimandproofcanbeshownforMIP(CBH)andMin-Min.
TwointerestingvariationsofMIP(*)areMIP(Greedy)andMIP(Min-Min),which
utilizetheGreedyandMin-Minheuristicstopre-selectthenumberoftasksoneach
machineforMIP(*),respectively.
Claim: For any instance of the allocation problem solved by Greedy and
MIP(Greedy),wehave .
80
Figure4.19:Constructionofanew matrix
Proof: To solve this problem instance using MIP(Greedy), is obtained from
the resource allocation given by the Greedy heuristic (see Step 1 of MIP(*) in
Figure4.2). Specifically,attheendofStep1, , (seealso
Figure4.4).DuringStep2inFigure4.2,MIP(Greedy)willfindanoptimalallocation
thathastheactualnumberoftasksrunningonmachine equalto .
Becausethenumberoftasksoneachmachineisthesameforbothapproaches,the
allocationgivenbytheGreedyheuristicwillalsobeconsideredduringoptimization
inMIP(Greedy)approach. Thus,wemusthave .
AsimilarclaimandproofcanalsobeshownforMIP(Min-min)andMin-Min.
81
Chapter5
Mixedintegerprogrammingwith
substitutionapproach
In thischapter, anotherapproach tosolvethemathematicalformulationinFig-
ure 3.9, called SMIP, is described. In this approach, the formulation is linearized
byatechniquecalledvariablesubstitution. Then,theproblemisre-formulatedasa
mixed-integer-programmingproblemandsolvedusingstandardlinearprogramming
techniques.
5.1 Variablesubstitutiontechniqueforlinearization
In Chapter 4.1, an approach to linearize the mathematical formulation by pre-
selecting values(thenumberoftaskstobeexecuteoneachmachine )ispre-
sented.TheoverallapproachiscalledMIP(*),where*issubstitutedbythenameof
theheuristicusedtopre-selectthe values. However,intheMIP(*)approach,the
searchspaceofthelinearizedMIPformulationislimitedtotheallocationswiththe
actualnumberoftasksoneachmachineequaltothepre-selectedvalue. Thislimi-
tationismathematicallydescribedasthelastsetofconstraintsshowninFigure4.2.
82
Figure5.1:MIP(CBH)searchspace.
Figure5.1showsanexampleoftherelationshipbetweenthesolutionspaceandthe
spacesearchedbyMIP(CBH).Recallthat meansthattask isallocatedto
machine . Inthisexample,3tasksarebeingallocatedto2machines. Thereare8
possiblesolutions,whichareshowninthelargeboxinFigure5.1. Supposethatthe
CBHheuristicpre-selects (thenumberoftasksonmachine )equalto2and
(thenumberoftasksonmachine )equalto1.ThesearchspaceofMIP(CBH)thus
islimitedtothe3solutionsshowningreyareainFigure5.1.These3solutionshave
2tasksallocatedontomachine and1taskallocatedontomachine .MIP(CBH)
willneverconsidertheother5solutionsoutsideofthegreyarea.
83
Thequalityoftheresultingallocationdependslargelyonthepre-selectednumber
oftasksallocatedontoeachmachine. Ifthesepre-selectednumbersareequaltothe
actualnumberoftasksoneachmachineoftheoptimalsolution,MIP(CBH)willalso
produceanoptimalsolution.InFigure5.1,MIP(CBH)willfindtheoptimalsolution
onlyifitisoneofthe3solutionsinsidethegreyarea. Ontheotherextreme,ifthese
pre-selected numbers are unreasonable (e.g., if all tasks are allocated to the same
machine), it is very likely that MIP(CBH) cannot find a feasible allocation at all.
Thatis,ifallsolutionsinthegreyareainFigure5.1arenotvalid,MIP(CBH) will
notfindafeasibleallocation.
Byusingthepre-selectionmethodtolinearizetheMIPformulation,theresulting
allocationwillnotbea“globally”optimalallocation. Feasibleallocationswiththe
numbersof tasksoneach machinedifferentthanthepre-selectednumberswillnot
beconsideredtobevalidsolutions,eveniftheyprovidebetterresults(higher
values). Inordertoconsidertheseallocations,otherlinearizationtechniquesthatdo
notrestrictthe numberof tasksallocatedtoeach machinemustbeused. One way
to linearize the formulation is by replacing the product of two variables with one
auxiliaryvariable,togetherwithadditionalconstraints.
Mathematically,
84
where and arebinary(0-1)variables, ,and issomeconstant,can
bereplacedwith
(5.1)
where isalsoabinaryvariable,withadditionalconstraints
(5.2)
(5.3)
(5.4)
Thesethreeconstraintstietheauxiliaryvariable tothetwobinaryvariables
and . The first two constraints put an upper bound on the auxiliary variable so
that willalwaysbelesserorequaltothelesserofthetwo and . Thelast
constraint puts a lower bound on . When both and we have
and , making . If either or , we have
and , causing . If both and , we still
have ,but . However,avariableinalinearprogrammustbe0or
positive(mathematically, and forall ). Thus,inthiscase,
reducesto andresultsin . Figure5.2showsallfour
85
Figure5.2:Relationshipbetween and .
possibilitiesof anddemonstratesthatbyconformingtothethreeconstraints
theresulting iscorrect.
Notethatif , then andnosubstitutionisneeded. Also,
and represent the same product, and thus only one additional variable is
used.Thistechniqueisaspecialcaseofatechniqueshownin[55].
5.2 SMIPformulation
In the mathematical formulation of the resource allocation problem, two vari-
ables, and , are multiplied together. However, can be expanded into
. Thus,Equation(3.2)andEquation(3.3)canberewrittenasfollow.First
let and representthecomputationandcommunicationlatencyoftask inthe
substitutiontechnique.Then,
86
(5.5)
(5.6)
and
(5.7)
(5.8)
Supposethat and ,wecanrewrite asfollow
ByusingEquation5.1,wehave
87
Thus,wehave
(5.9)
ReplacingtheEquation(5.9)intoEquation(5.5)andEquation(5.7),wehave
(5.10)
and
(5.11)
Let denote the end-to-end latency of a primary route in the substitution
technique.Wehave
(5.12)
88
Also,thenormalizedcomputation,communication,andprimaryrouteslackness
are
(5.13)
(5.14)
(5.15)
After the non-linear terms are replaced by the auxiliary variables, the formu-
lation is re-formulated as a mixed-integer-programmingformulation. This specific
formulationiscalledSMIPandisshowninFigure5.3.TheSMIPformulationisnot
restricted by anypre-selected numbersof tasks on each machine, thus, solvingthe
formulationwillproducea“globally”optimalallocation.
BecauseSMIPapproachdoesnotpre-selectthenumberoftasksoneachmachine,
it does not require vector as input. SMIP formulation also does not constrain
89
the actual number of tasks on each machine to be equal to the pre-selected num-
ber. However,theadditionalvariables(thesetof ’s)mustberelatedtothemain
variables (the set ) as described in Chapter 5.1. These restrictions are reflected
in the last 3 sets of constraints in Figure 5.3. The other constraints are the same
asinthemathematicalformulationshowninFigure3.9. Namely,the first3setsof
constraints restrict the normalized slackness of all tasks and primary routes of the
resultingallocationtobegreaterorequaltothe value. Theforthsetofconstraints
enforceseachtasktobeallocatedtoonlyonemachine. Becauseonesubstitutevari-
able isneededfor each productof and and3constraintsare addedper one
substitutevariable, additionalvariablesand additional
constraintsareintroducedintotheformulation.Inpractice,thenumberofadditional
variables and constraints can be reduced as follow. If , then , and
thus (because and arebinaryvariables).Consequently,when
, there is no need for substitution. In addition, and both represent
,andthusonlyoneofthetwoadditionalvariablesisneeded.Thus,inprac-
tice,only additionalvariablesand additional
constraints are introduced into the formulation. After solving the formulation, the
auxiliaryvariable willbeequalto .Furthermore,the matrixwillrepresenta
resourceallocationthatgivesthehighestvalueof andthusthehighestvalueof
.
90
Given:
Find:
to
Maximize:
Subjectto:
End
Figure5.3:SMIPapproach
5.3 ExperimentsandResults
AsimulatorbasedonthemathematicalformulationpresentedinChapter3.7was
developedtoevaluatetheperformanceofourresourceallocationtechniques. Given
an allocation, the simulator calculates the actual computation and communication
latencies of a task using the equations presented in Chapter 3.7. If all real-time
requirementsaresatisfied,thenormalizedslacknessvaluesofalltasksandprimary
routesare calculated, whichconsequentlydetermine . The valueforthis
allocationcanthenbecalculatedusingtheequation .
91
5.3.1 ProblemGeneration
The and matricesweregeneratedtocapturethemachineandtaskhet-
erogeneities [7]. Specifically, each matrix was characterized by two parameters:
machine heterogeneity and task heterogeneity. Both heterogeneities can be mod-
eledas“Hi”or“Lo.” Gammadistributionswereusedtogeneratethematrices. The
experimentwasconductedforeachmachine-taskheterogeneitycategory:Hi-Hi,Hi-
Lo, Lo-Hi, and Lo-Lo. The first word in the category description corresponds to
the heterogeneity of machines in the system, while the second word explains the
heterogeneity of tasks. Thus, the Hi-Lo category means that the computingpower
and communication link capacity vary greatly from one machine to another (high
machine heterogeneity), while the tasks in the system are quite similar (low task
heterogeneity). Consequently,theexecutiontimeof task onmachine isvery
differentfromtheexecutiontimeof on ,butclosetotheexecutiontimeof
on machine . Each element in vector is generated by sampling a uniform
distributionofvaluesrangingfrom10to100.
Foreachtask,theaveragevaluesofitscomputationandcommunicationlatencies
over all machines were calculated from the and matrices and the
vector. ForeachPR,thesumandthemaximumvalueoftheseaveragevaluesofall
nodesalongtheroutewere calculated, denotedas and , respectively. Let be
equal to . The end-to-end latency requirement of the PR was then set to
. isaspecifiedfactorthatisusedtoadjustthetightnessoftheconstraints.
92
ThethroughputrequirementofeachtaskalongthePRwassettobe .
Due to space limitations, we emphasized only computation intensive applications
– the average communication latency for each task is around 1/100 of its average
computationlatency.
5.3.2 OtherHeuristicsforComparison
Wealsoimplementedtwootherapproachestosolvetheresourceallocationprob-
lemcalledMin-MinandGreedy.TheGreedyheuristicmapstasksinarandomorder,
and each task is mapped onto the machine that givesthe shortestcomputationand
communication latency based on the mapping information so far. The Min-Min
heuristic ranks tasks using their computation and communication latencies on the
bestmachines,thenallocatesthetasksinorder.Inbothapproaches,thecomputation
andcommunicationlatenciesarecalculatedwhileconsideringmultitaskingoftasks.
oftheresultingallocationsfromthesetwoapproachesiscomparedwith
oftheallocationsgivenbyMIP(CBH)andSMIP.
5.3.3 ExperimentalProcedure
For each machine-task heterogeneity, the experiment was conducted in 3 sets
with40probleminstanceseach.Aprobleminstanceisgeneratedwith12processing
tasks(3sources,3sinks,out-degree ). Thenumberofmachinesvariesfrom3to
5betweensets. The (tightnessofconstraint)factorissettobe1.5forallproblem
93
MAIL value from various approaches in Hi-Hi machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
SMIP
MIP(CBH)
Min-Min
Greedy
Figure 5.4: The average MAIL value of allocations from each approach in Hi-Hi
environment
instances.Foreachprobleminstance,anoptimalallocationthatresultsinthehighest
valuewasfoundbyenumeratingallpossibleallocations. Then,theproblemis
solvedusing4differentapproaches:SMIP,MIP(CBH),Greedy,andMin-Min.SMIP
isamixed-integer-programming-basedapproach,whichislinearizedusingthevari-
ablesubstitutiontechnique. MIP(CBH)isalsoamixed-integer-programming-based
approach,butislinearizedbypre-selecting valuesusingtheCBHheuristic. The
MAIL value of the resultingallocation from each approach is then compared with
the optimalvalue foundby enumeration. In everyprobleminstance, the execution
timeofeachapproachincludingenumeration,isalsorecorded.
94
Execution time of various approaches in Hi-Hi machine-task
heterogeneity environment
0
500
1000
1500
2000
2500
3 4 5
number of machines
seconds
Enumeration
SMIP
MIP(CBH)
Figure5.5:ExecutiontimeofeachapproachinHi-Hienvironment
5.3.4 Results
Figure5.4showstheaverageMAILvalue(asapercentageoftheoptimalvalue
found by enumeration) of the allocations produced by each approach in the Hi-Hi
environment.Asexpected,SMIP(MIP-basedapproach,usingsubstitutiontechnique
forlinearization)alwaysproducesanallocationwiththeoptimalMAILvalue. Allo-
cationsgivenbytheMIP(CBH)approachhasMAILvaluearound80%oftheopti-
malvalueontheaverage.Min-MinandGreedygenerateallocationswiththeaverage
MAILvaluearound55%and50%oftheoptimalvalue,respectively.
Figure 5.5 shows the average execution time of enumeration, SMIP, and
MIP(CBH) on a system with a 400MHz Ultrasparc-II processor and 1GB of main
95
MAIL value from various approaches in Lo-Lo machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
SMIP
MIP(CBH)
Min-Min
Greedy
Figure 5.6: The average MAIL value of allocations from each approach in Lo-Lo
environment
memory.Theexecutiontimeofenumerationincreasesveryrapidlywhenthenumber
ofmachinesintheexperimentincrease,andisprovidedasthebaselineforcompar-
ison. SMIP’sexecutiontimealsoincreasewiththenumberofmachines,butnotas
quicklyasenumeration’s. SMIP takeslongertoexecutethanenumerationinsmall
problems(3and4machines), butwhentheproblembecomeslarger, theexecution
timeofenumerationincreasesrapidly,overtakingthetimeusedbySMIP.Forexam-
ple,todetermineanallocationforasystemwith5machines,SMIPtakes500seconds
onthe average , compared with1900secondsused byenumeration. MIP(CBH) is
veryfast,usinglessthan1secondsinall3experimentsets(3,4,and5machines).
96
Execution time of various approaches in Lo-Lo machine-task
heterogeneity environment
0
1000
2000
3000
4000
5000
6000
7000
3 4 5
number of machines
seconds
enumeration
SMIP
MIP(CBH)
Figure5.7:ExecutiontimeofeachapproachinLo-Loenvironment
Figure 5.6, Figure 5.8, and Figure 5.10showaverage MAIL values of the four
approaches in Lo-Lo, Lo-Hi, and Hi-Lo machine-task heterogeneity environments
respectively. TheresultsaresimilartoonesshowninFigure5.4fortheHi-Hienvi-
ronment.Namely,theSMIPformulationalwaysproducesanallocationwithoptimal
MAILvalue.Interestingly,MIP(CBH)alsoproducesverygoodresults,withaverage
MAILvalueatalmost100%oftheoptimalvalueintheLo-LoandLo-Hienviron-
ments. In the Hi-Lo machine-task heterogeneity environment, the average MAIL
valueoftheMIP(CBH) dropssomewhatinto90%range, butthisresultisstillbet-
ter than one in the Hi-Hi environment. Results produced by Min-Minand Greedy
97
MAIL value from various approaches in Lo-Hi machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
SMIP
MIP(CBH)
Min-Min
Greedy
Figure 5.8: The average MAIL value of allocations from each approach in Lo-Hi
environment
Execution time of various approaches in Lo-Hi machine-task
heterogeneity environment
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
3 4 5
number of machines
seconds
enumeration
SMIP
MIP(CBH)
Figure5.9:ExecutiontimeofeachapproachinLo-Hienvironment
98
MAIL value from various approaches in Hi-Lo machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
SMIP
MIP(CBH)
Min-Min
Greedy
Figure 5.10: The averageMAIL valueof allocationsfrom each approachin Hi-Lo
environment
Execution time of various approaches in Hi-Lo machine-task
heterogeneity environment
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
3 4 5
number of machines
seconds
enumeration
MIP(CBH)
SMIP
Figure5.11:ExecutiontimeofeachapproachinHi-Loenvironment
99
approaches in Lo-Lo, Lo-Ho, and Hi-Lo environments are generally better than in
Hi-Hienvironmenttoo.
Figure 5.7, Figure 5.9, and Figure 5.11 show the execution time of SMIP,
MIP(CBH),andenumerationapproachesinLo-Lo,Lo-Hi,andHi-Loenvironments
respectively.Intheseenvironments,SMIPtakesalotoftimetoproducetheoptimal
allocation. EventhoughtheexecutiontimeofSMIPinsmallproblemsof3,4,and
5machinesislongerthantheenumerationapproach,thedifferencebetweentheexe-
cutiontimeofSMIPandenumerationgrowssmallerwhenthenumberofmachines
increases. ConsequentlyitispossiblethatinlargerproblemsinLo-Lo,Lo-Hi,and
Hi-Loenvironments,SMIPmayexecutefasterthanenumeration.
FromtheseresultsshowninFigure5.4toFigure5.11,SMIPisobviouslythebest
approach;italwaysprovidesallocationswiththeoptimalMAILvalue,andproduces
resultsupto1.25timesbetterthanallocationsfromMIP(CBH).Theexecutiontime
of SMIP is also acceptable for an initialallocation problem. MIP(CBH) runsvery
fast and produces relatively good results, especially in Lo machine heterogeneity
environments(i.e.,whenmachineshavesimilarcapabilities.)Itisapossiblealterna-
tiveapproachintheseenvironmentsorwhenthesizeoftheproblembecomeshuge
orforrun-timeresourcere-allocation.
100
Figure5.12: The and matrixwhereMIP(CBH)willnot findafeasibleallo-
cation.
5.4 Discussion
MIP(CBH) can be regardedas a two-phase approach. The first phase allocates
resourcesonamachine-levelbasisbyusingtheCBHheuristictopre-selectthenum-
beroftaskstobeallocatedontoeachmachine. Thesecondphaseutilizesthemixed
integerprogrammingoptimizationtechniquetomatchtaskstomachinestomaximize
,giventhepre-selectednumberfromthefirstphase.Fromthisrespect,theactual
allocationofresources isdeterminedin the first phase; theMIP formulationinthe
secondphaseisusedonlytooptimizetheallocation.Theresultingallocationwillbe
optimal,onlyifthepre-selectednumbersequalthenumberoftasksoneachmachine
ofanoptimalallocation. However,itispossiblethatthepre-selectednumberswill
constraintheMIPformulationsuchthatthereisnofeasibleallocation.Wecanshow
thefollowingresult.
101
Figure5.13:ExtendedproblemwhereMIP(CBH)willnotfindafeasibleallocation.
Theorem5.4.1 For all , where is the number of tasks to be allocated, there
are problem instances for which MIP(CBH) fails to find a resource allocation that
satisfies all performance requirements, while SMIP succeeds (in finding a feasible
allocation).
Proof: Figure5.12showsthe and (refertoFigure4.1)matricesforasystem
with2tasksand2machines.Recallthatanentryinthe matrixcontainstheslope
ofafunctionthattranslatesthecurrentloadleveltoanexecutionlatency. Suppose
that . This configuration of the matrix is possible in a high
machine-andhightask-heterogeneityenvironment. Forsimplicity,the matrix
is ignored, and the initial load level of all tasks is assumed to be 1. Assume also
102
thatbothtasksmust finishexecutionin seconds. Fromthe matrix,theCBH
heuristic will generate the matrix in Figure 5.12, where corresponds to the
relativespeedoftask onmachine . Because , , ,
, and , respectively. Thus, the relative power of machine and
is2and0,respectively. Itisobviousfromthe matrixthattheCBHheuristic
willpre-select the numberof taskson machine to be 2, and on machine to
be0. Theactualexecutionlatencyoftask and willbe and ,respectively.
The resulting allocation will violate the real-time requirement. Thus, MIP(CBH)
failstofindafeasibleallocation,whereoneexists(task onmachine andtask
onmachine ). Ontheotherhand, SMIP,whichusesthevariablesubstitution
techniquetolinearizetheformulation,willfindthefeasibleallocation.
The problem can be extend to tasks and machines as in Figure 5.13. The
latencies of tasks and on machine , , and the latencies of task
, onmachine and , aresetarbitraryhigh. Thus, ,when
and ,orwhen and . Consequently,inFig-
ure5.13, and ,unchangedfromFigure5.12. Theresultisthatno
taskwillbeallocatedtomachine byMIP(CBH).Becausethefeasibleallocations
inthisproblemmusthavetask onmachine ,itfollowsthatMIP(CBH)willnot
findafeasibleallocation.
103
Chapter6
Iterativeintegerprogramming
approach
In thischapter, anotherapproach tosolvethemathematicalformulationinFig-
ure 3.9, called Iterative Integer Programming(IIP), isdescribed. In thisapproach,
the formulation is linearized by the variable substitution technique described in
Chapter 5.1. However, instead of re-formulating as a mixed-integer-programming
problem(asintheSMIPapproach), theproblemissolvediterativelyasaninteger-
programmingproblem.Theideabehindthisiterativeintegerprogrammingapproach
is that an integer program is easier to solve than a mixed-integer program. How-
ever, to find an initialresource allocationby thismethod, severalintegerprograms
mustbeformulatedandsolved,whileSMIPapproachformulatesandsolvesonlyone
mixed-integerprogram.
RecallthatinbothMIP(*)andSMIPapproaches, themixedintegerprogramis
being solved for two sets of unknown variables. The first is a set of binary (0-1)
variables ’s,whichrepresentstheallocationontasksontomachines. Thesecond
variable is a real number , which will be equal to (the minimumnormalized
slackness)oftheresultingallocationwhenthemixed-integer-programissolved. In
104
IIPapproach,thecomplexityoftheallocationproblemisgreatlyreducedbyreplac-
ingtherealvariable withavaluecalled . Theproblemformulationthus
consistsofonlyintegervariables(thesetof )andreducestoapureintegerpro-
gram (IP). The IIP approach finds a solution to the allocation problem by solving
the integer program iteratively. In each iteration, the IIP approach tries to find an
allocationsuchthatits isgreaterthanthevalue . Fromthispointof
view, is the minimum value that is acceptable in that iteration. If
theintegerprogramofthecurrentiterationisfeasible(i.e.,thereexistsanallocation
with its greater than value), the value of is increased in
the subsequent iteration, resulting in more strict IP formulation. The amount that
increasesineachiterationcanbespecifiedbytheuserusingaparameter
called . Namely,the ofthenextiterationis (minimumnormalized
slackness)ofthecurrentsolutionplus ,assumingthatthecurrentintegerprogram
has a solution. Figure 6.1 showsthe relationshipbetween the values of
the current and next iterations, of the current iteration, and . The black ver-
ticalbarsrepresentthenormalizedcomputationslacknessofalltasks( ), while
thegreybarsshowthenormalizedcommunicationslackness( ). Thewhitebars
describethenormalizedslacknessofallprimaryroutes( ).Thelowesthorizontal
lineisthe valueofthecurrentiterationoftheIIPapproach. Anyfeasible
solutionofthecurrentIPformulationmusthavethetopoftheshortestverticalbar
inFigure6.1(its )higherthanthiscurrent line. Ifthereisafeasible
105
Figure6.1:Relationshipbetween values, and
solution,the valueofthenextiterationissetto ,whichisthetop
horizontallineinFigure6.1.Becauseofthehighervalueof ,therewillbe
fewerfeasibleallocationsinthenextiteration. Notethatthesolutionofthecurrent
iteration will not be valid in the next iteration. When is increased to a
pointthat the IP formulationcannot produce any feasible allocation, the algorithm
terminatesandtheallocationofthelastiterationisusedasthesolution.
Figure6.2showstheIterativeIntegerProgrammingapproach. Asshowninstep
1ofFigure6.2,the firstvalueof issetto0,thusthe firstIPformulation
is not restricted by the value. If the IP formulation in the first iteration
fails to produce a feasible allocation, then the resource allocation problem itself is
106
Begin
1.Set .
2.SolvethefollowingIntegerProgrammingProblem:
Given:
Find:
to
Minimize:
Subjectto:
3. If the Integer Program is infeasible, stop and use the solution of the previous
iteration.
4.IftheIntegerProgramhasasolution,calculate andsavetheallocation.
5.Set andgotostep2.
End
Figure6.2:IIPFormulation.
infeasible.Theintegerprogramisformulatedandsolvedinstep2ofFigure6.2.The
newvalueof iscalculatedinstep4and5ofFigure6.2,andisequaltothe
(oftheallocationfromstep2)plus ,whichisrealnumberbetween0and1.In
effect,theIPformulationofthenextiterationwilltrytofindanallocationwith
atleast percentbetterthanthebest valuefoundsofar.
107
IntheIPformulation,thefirst3setsofconstraintsrestrictthenormalizedslack-
nessofalltasksandprimaryroutesoftheresultingallocationtobegreaterorequal
tothe value.Thenextsetofconstraintsenforceseachtasktobeallocated
toonlyonemachine. Thelast3setsofconstraintsrelateandrestricttheadditional
variables( )tothemainvariables( ).TheobjectivefunctionoftheIPformula-
tionisnotcrucialtothecorrectnessofthealgorithm(tobediscussedinChapter6.2)
because maximum is determined indirectly. However, the objective function
ofIPformulationshouldbeeasytocalculatesothattheformulationcanbesolved
quickly. Inthisresearch, theobjectivefunctionoftheIPformulationisthesumof
thelatenciesofalltasks.
6.1 ExperimentsandResults
AsimulatorbasedonthemathematicalformulationpresentedinChapter3.7was
developedtoevaluatetheperformance ofourresource allocationtechnique. Given
an allocation, the simulator calculated the actual computation and communication
latencies of a task using the equations presented in Chapter 3.7. If all real-time
requirementsweresatisfied,thenormalizedslacknessvaluesofalltasksandprimary
routeswerecalculated,whichconsequentlydetermined .The valueforthis
allocationcouldthenbecalculatedusingtheequation .
108
6.1.1 Problemgeneration
The and matrices were generated to capture the machine and task
heterogeneities[7]. Specifically,eachmatrixwascharacterizedbytwoparameters:
machineheterogeneityandtaskheterogeneity. Bothheterogeneitiescouldbemod-
eledas“Hi”or“Lo.” Gammadistributionswereusedtogeneratethematrices. Due
totimelimitation,theexperimentwasconducted onlyinHi-Hi(highmachineand
hightaskheterogeneity)environment.Eachelementinvector wasgeneratedby
samplingauniformdistributionofvaluesrangingfrom10to100.
Foreachtask,theaveragevaluesofitscomputationandcommunicationlatencies
over all machines were calculated from the and matrices and the
vector. ForeachPR,thesumandthemaximumvalueoftheseaveragevaluesofall
nodesalongtheroutewere calculated, denotedas and , respectively. Let be
equal to . The end-to-end latency requirement of the PR was then set to
. isaspecifiedfactorthatisusedtoadjustthetightnessoftheconstraints.
ThethroughputrequirementofeachtaskalongthePRwassettobe .
Due to space limitations, we emphasized only computation intensive applications
– the average communication latency for each task is around 1/100 of its average
computationlatency.
109
6.1.2 Otherapproachesforcomparison
We compared the iterative integer programming approach with two other
approaches based on mixed integer programming. Both integer programming (IP)
and mixed integer programming (MIP) are a derivative of an optimizationmethod
called linear programming. The difference between the two derivatives is that an
integerprogramhasonlyintegervariableswhilea mixedintegerprogramcontains
bothrealandintegervariables. oftheresultingallocationfromeachapproach
wascomparedwith oftheallocationsgivenbyIIP.
The firstMIP-basedapproachusedforcomparisonistheSMIP,whichencodes
the value(arealnumber)intoanMIPformulationandoptimizes directly.
TheSMIPformulationderivedfromtheresourceallocationproblemissolvedonly
once,andisexpectedtoalwaysproduceanoptimalallocation.
TheotherMIP-basedapproachistheMIP(*),whichalsoencodesandoptimizes
valuedirectly. However,MIP(*)linearizestheproblembypre-selecting ,the
numberoftasksonmachine ,foreverymachine . Because valuesarepre-
selectedusingthecapability-basedheuristic(CBH),theoverallapproachisreferred
toasMIP(CBH).
6.1.3 ExperimentalProcedure
ThefirstexperimentwasdesignedtoevaluatetheeffectivenessofIIP,compared
withother2approachesoutlinedinChapter6.1.2. Theexperimentwasdividedinto
110
3sets,eachwith40probleminstances.Duetoalackofspace,aprobleminstancein
eachsetwasgeneratedwithafixednumberof12processingtasks(3sources,3sinks,
out-degree ). Notethatallapproachestestedinthispapercanhandleproblems
withvaryingnumberoftasksaswell. Thenumberofmachinesvariedfrom3to5
betweensets. The (tightnessofconstraint)factorwassettobe1.5forallproblem
instances.Foreachprobleminstance,anoptimalallocationthatresultsinthehighest
value was found by enumerating all possibleallocations. Then, the problem
wassolvedusing3differentapproaches: IIPwith ,SMIP,andMIP(CBH).
The MAIL ( ) value of the resulting allocation from each approach was then
comparedwiththeoptimalvaluefoundbyenumeration. Ineveryprobleminstance,
theexecutiontimeofeachapproachincludingenumeration,wasalsorecorded.
Thelastexperimentwasdesignedtodeterminetheeffectof valueonthequality
of solution and the execution time of IIP. In this experiment, problem generation
proceededexactlythesameasthe firstexperiment. Each problemwasthensolved
3 times by the IIP approach with value at 0.01, 0.05, and 0.1, respectively. The
MAILvaluesfromthe3solutionswerecomparedwiththeoptimalvaluefoundby
enumeration.Thetimeusedtofindeachsolutionwasalsorecordedandcompared.
6.1.4 Results
Figure 6.3 to Figure 6.6 show the average MAIL value (as a percentage of
the optimal value found by enumeration) of the allocations produced by IIP with
111
MAIL value from various approaches in Hi-Hi machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal
value found by enumeration)
IIP(0.01)
MIP(CBH)
SMIP
Figure 6.3: The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinHi-Hienvironment.
, MIP(CBH), and SMIP approaches in Hi-Hi, Hi-Lo, Lo-Hi, and Lo-Lo
machine-taskheterogeneityenvironmentsrespectively. TheIIP approach resultsin
allocationswith MAIL value close to optimal, around 99% on the average in all 3
sets of experiments in all environments. Allocations given by the MIP(CBH) has
MAILvaluearound83%, 96%, 93%and98%oftheoptimalvalueontheaverage
inHi-Hi,Hi-Lo,Lo-Hi,andLo-Loenvironments,respectively. Asexpected,SMIP
alwaysproducesanallocationwiththeoptimalMAILvalue.
Figure6.7toFigure6.10showstheaverageexecutiontimeofenumeration,IIP
with ,SMIP,andMIP(CBH)onasystemwitha400MHzUltrasparc-IIpro-
cessorand1GBofmainmemory.Theexecutiontimeofenumerationincreasesvery
112
MAIL value from various approaches in Hi-Lo machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
IIP(0.01)
MIP(CBH)
SMIP
Figure 6.4: The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinHi-Loenvironment.
MAIL value from various approaches in Lo-Hi machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
IIP(0.01)
MIP(CBH)
SMIP
Figure 6.5: The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinLo-Hienvironment.
113
MAIL value from various approaches in Lo-Lo machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
IIP(0.01)
MIP(CBH)
SMIP
Figure 6.6: The average MAIL value of allocations from IIP with ,
MIP(CBH),andSMIPapproachesinLo-Loenvironment.
rapidly when the number of machines in the experiment increase, and is provided
as the baseline for comparison. SMIP takes longer toexecute thanenumerationin
smallproblems(3and4machines),butwhentheproblembecomeslarger,theexe-
cutiontimeofenumerationincreasesrapidly,andactuallyovertakingthetimeused
bySMIPintheHi-Hienvironment.IIP’saverageexecutiontimeisbetterthanSMIP
in all cases and is 6 times faster than SMIP for a systemwith 5 machines and Hi-
Himachine-taskheterogeneity. MIP(CBH)isthefastestapproach,usinglessthan1
secondinall3experimentsets.
Figure6.11toFigure6.14showtheeffectof valueontheMAILvalueofthe
resulting allocation produced by IIP. The MAIL value shown is an average over
114
Execution time of various approaches in Hi-Hi machine-task
heterogeneity environment
0
200
400
600
800
1000
1200
1400
3 4 5
number of machines
seconds
enumeration
IIP(0.01)
MIP(CBH)
SMIP
Figure6.7: ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inHi-Hienvironment.
Execution time of various approaches in Hi-Lo machine-task
heterogeneity environment
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
3 4 5
number of machines
seconds
enumeration
IIP(0.01)
MIP(CBH)
SMIP
Figure6.8: ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inHi-Loenvironment.
115
Execution time of various approaches in Lo-Hi machine-task
heterogeneity environment
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
3 4 5
number of machines
seconds
enumeration
IIP(0.01)
MIP(CBH)
SMIP
Figure6.9: ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inLo-Hienvironment.
Execution time of various approaches in Lo-Lo machine-task
heterogeneity environment
0
1000
2000
3000
4000
5000
6000
7000
3 4 5
number of machines
seconds
enumeration
IIP(0.01)
MIP(CBH)
SMIP
Figure6.10:ExecutiontimeofIIPwith ,MIP(CBH),andSMIPapproaches
inLo-Loenvironment.
116
MAIL value from various approaches in Hi-Hi machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal
value found by enumeration)
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure 6.11: The average MAIL value of allocations from the IIP approach with
varying valueinHi-Hienvironment.
MAIL value from various approaches in Hi-Lo
machine-task heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal
value found by enumeration)
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure 6.12: The average MAIL value of allocations from the IIP approach with
varying valueinHi-Loenvironment.
117
MAIL value from various approaches in Lo-Hi machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal value
found by enumeration)
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure 6.13: The average MAIL value of allocations from the IIP approach with
varying valueinLo-Hienvironment.
MAIL value from various approaches in Lo-Lo machine-task
heterogeneity environment
0
20
40
60
80
100
120
3 4 5
number of machines
MAIL value (as % of optimal
value found by enumeration)
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure 6.14: The average MAIL value of allocations from the IIP approach with
varying valueinLo-Loenvironment.
118
Execution time of various approaches in Hi-Hi machine-task
heterogeneity environment
0
10
20
30
40
50
60
3 4 5
number of machines
seconds
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure6.15: ExecutiontimeoftheIIPapproachwithvarying valueinHi-Hienvi-
ronment.
40 problem instances. As clearly seen in these Figures, the average MAIL value
decreases with increasing . However, at a specific value of , the average MAIL
value does not vary significantly with varying number of machines. For example,
at inHi-Himachine-taskenvironments,theaverageMAILvalues(asper-
centage of optimal value) are 94%, 94%, and 93% for problems with 3, 4, and 5
machines,respectively.
Figure 6.15 to Figure 6.18 show the effect of value on the average execution
time of IIP over 40 problem instances. It is obvious from these Figures that the
averageexecutiontimeofIIPdecreaseswithincreasing .
119
Execution time of various approaches in Hi-Lo machine-task
heterogeneity environment
0
200
400
600
800
1000
1200
1400
1600
3 4 5
number of machines
seconds
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure6.16:ExecutiontimeoftheIIPapproachwithvarying valueinHi-Loenvi-
ronment.
6.2 Discussion
6.2.1 CharacteristicsofIIP
Theiterativeintegerprogrammingapproachhasaninterestingcharacteristicthat
itcanbeeasilyextendedtooptimizeasecondaryobjective. IIPfindsthesolutionof
theresourceallocationproblembyiterativelyadjustingtheconstraintsofaninteger
program. In effect, the objective function of the integer program does not have a
directeffectonthequalityofthesolution. Thus,theobjectivefunctioncanbeused
tooptimizeasecondaryobjective,forexample,minimizingtheexecutiontimeofan
importanttaskoracriticalprimaryroute.Notethattheobjectivefunctiondoesaffect
120
Execution time of various approaches in Lo-Hi machine-task
heterogeneity environment
0
500
1000
1500
2000
2500
3000
3 4 5
number of machines
seconds
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure6.17:ExecutiontimeoftheIIPapproachwithvarying valueinLo-Hienvi-
ronment.
Execution time of various approaches in Lo-Lo machine-task
heterogeneity environment
0
500
1000
1500
2000
2500
3000
3500
4000
3 4 5
number of machines
seconds
IIP(0.01)
IIP(0.05)
IIP(0.1)
Figure6.18:ExecutiontimeoftheIIPapproachwithvarying valueinLo-Loenvi-
ronment.
121
the execution time of IIP, i.e. a more complex objective function will take longer
timetoevaluate.
AnotherinterestingcharacteristicofIIPisthetrade-offbetweenexecutiontime
andqualityofsolutionthroughtheparameter .Because isthevalueusedtoadjust
theconstraintsoftheintegerprogram,ineffectthereisagapwiththesize between
the current (best known) value and the acceptable minimum of the next
iteration. Iftheoptimalvalueof fallswithinthisgap,thentheintegerprogram
ofthenextiterationwillbeinfeasible. Inotherwords, ofthesolutionproduced
byIIPmustbewithin fromtheoptimalvalueof .Thus,thesmaller willresult
inabettervalueof ofthesolutionandabetterMAILvalue. However,witha
smaller ,theadjustmentonconstraintsissmaller. Thus,moreiterations(andthus
moretime)mayberequiredduringIIPexecution. TheresultsshowninFigure6.11
andFigure6.15clearlyagreethatincreasing willresultindecreasingMAILvalue
(qualityofresultbecomesworse)anddecreasing(better)executiontime.
6.2.2 ComparisonwithSMIPandMIP(CBH)
BothIIPandSMIPusethesubstitutiontechniquetolinearizetheresourceallo-
cation formulation. However, SMIP formulation contains one additional variable
( ),whichisarealnumberrepresentingtheminimumnormalizedslacknessvalue
tobeoptimized. Thus,SMIPisamixedintegerprogrammingformulation,whichis
moredifficulttosolvethanapureintegerprogram. EventhoughIIPhastosolvean
122
integerprogramseveraltimes,thetotalexecutiontimeofIIPisstillbetterthanSMIP
(as shown in Figure 6.7). SMIP, however, guarantees to find an optimal solution,
becausethemaximizationof isencodeddirectlyintotheformulation. Incom-
parison,IIPguaranteesonlythatthe valueofthesolutionfallswithin fromthe
optimalvalueof .
MIP(CBH) uses the pre-selection technique to linearize the formulation, while
IIP(andSMIP)usesthesubstitutiontechnique,whichintroducesadditionalvariables
andconstraintsintotheformulation.Infact,IIPhas additionalvari-
ablesand additionalconstraintsintroducedintotheformulation,
where is the total number of tasks to be allocated and is the total num-
ber of machines in the system. In effect, MIP(CBH) executes a lot faster than IIP
becauseitcontainsfarlessconstraintsandvariables(asshowninFigure6.7). How-
ever,thequalityofthesolutionproducedbyMIP(CBH)dependslargelyonthepre-
selectednumberoftasksallocatedontoeachmachine.Ifthesepre-selectednumbers
are unreasonable, it is very likely that MIP(CBH) cannot find a feasible allocation
atallevenifoneexists. Incontrast,IIPwill findafeasibleallocationifoneexists,
becausethefirstiterationofIIPisnotrestrictedbythevalueofacceptableminimum
normalizedslacknessatall.
123
Chapter7
Conclusion
In this Chapter, we summarize the contributions and findings of the thesis. In
addition,possiblefutureresearchdirectionsarealsooutlined.
7.1 Thesissummary
The focus of this thesis is to find methodsto allocate resources for application
tasksbeforethesystemstartsup,suchthattheinitialallocationisrobustwithrespect
tochangingworkloadduringruntime.Wesummarizethemaincontributionsofthis
thesisasfollows.
7.1.1 Systemandapplicationmodels
Chapter 3describesthesystemandapplicationmodelsusedinthisthesis. The
modelsdefinesloadlevelofeachtasksasparametersthatcanvaryduringruntime.A
run-timeparametercalled isusedtocontroltheamountofvariationoftheloadlevel
ofeachtask.Loadlevelvariationofataskaffectsthetaskexecutionandcommunica-
tiontime,andisshownaslinearfunctionsinthe and matricesrespectively.
Thebasecomputationandcommunicationlatencyofataskcanbedeterminedfrom
124
the and matricesandthecurrentloadlevelofthattask(whichcanbecal-
culated from ). Because machine resources (compute power and communication
bandwidth) are fairly shared among all tasks executed on the machine, the actual
computationand communicationlatency of a task is the multiplicationof the base
valueswiththenumberoftasksrunningonthesamemachine. Inorderforanallo-
cationtobevalid,thelatenciesofallindividualtasksandalldefinedgroupoftasks
mustnotexceedthespecifiedqualityofservicerequirements. Insummary,thesys-
temandcommunicationmodelscapture: 1)parametersthatvaryduringruntime,2)
theactualvariationsofthoseparameters,3)theeffectofthevariationsontasklaten-
cies,4)theeffectofresourcessharingamongtasksontasklatencies,and5)quality
ofservicerequirementsofatasksorgroupoftasks.
7.1.2 Performancemetricandobjectivefunction
Duringruntime,theamountofloadlevelvariations( )mayincreaseordecrease
overtimedependingonthechangesintheenvironmentinwhichthesystemoperates.
Givenanallocation,thehighestvaluethat canincreasewithoutcausinganyQoS
violationiscalled ofthatallocation.Anallocationwithahighervalueof
canabsorbmoreincreaseinloadlevel,andconsequentlyismorerobustthananallo-
cationwithlower value. Thus,wecanuse astheperformancemetricfor
robustnessofaninitialallocation. ThisspecificperformancemetriciscalledMAIL
125
(MaximumAllowableIncrease inLoadlevel). Chapter 3.3explainstheprocessof
derivingthisperformancemetricinmoredetail.
Because iscalculatedfromacompleteallocationoftaskstomachines,itis
notmeasurableduringtheallocationprocessifthealgorithmdoespartialallocation
(allocating one task at a time). Thus in Chapter 3.6 an objective function called
(minimum normalized slackness of all tasks and primary routes) is proposed
toguidetheallocationprocess. Conceptually,thenormalizedslacknessrepresents,
as a percentage, the slack or available room for the latency of a task or a route to
increase, before the throughput and/or end-to-end latency requirement is violated.
ProofisgiveninthesameChapterthattheallocationthatgivesthehighestvalueof
amongallallocationswillalsoresultinthehighestvalueof .
7.1.3 Approachestosolvetheallocationproblem
Three approaches to solve the initial allocation problem are proposed: MIP(*)
(Chapter4),SMIP(Chapter5),andIIP(Chapter6). Allthreeapproachesarebased
onawell-researchedmathematicaloptimizationmethodcalledLinearProgram(LP).
The LP method is selected because it is well-suited to optimize an objective func-
tion, while maintaining constraints imposed on the program. Specifically, we can
optimize whileobservingthequalityofservicerequirementsimposedonatask
or a groupof tasks. Because themathematicalformulationof the initialallocation
126
problem is non-linear, two techniques called pre-selection (Chapter 4.1) and vari-
ablesubstitution(Chapter5.1)aredevelopedtolinearizetheproblemsothattheLP
methodcanoptimizedtheproblem.Inthepre-selectiontechnique,aheuristicisused
topre-selectasetofunknownvariablessothatthereisnomultiplicationofvariables
thatcausesnon-linearity. Inthevariablesubstitutiontechnique, amultiplicationof
variablesarereplacedbyasubstitutedvariable,togetherwithadditionalconstraints
toassociatedthereplacedvariableswiththesubstitutedone.
MIP(*) is a mixed-integer-programming-basedapproach tosolvetheallocation
problem. In a mixed integer program, the unknown variables are a mix of integer
andrealnumbers. MIP(*)usesthepre-selectiontechniquetolinearizetheproblem,
resulting in a reduced number of variables in the program. In MIP(*) approach,
the*willbereplacedbythenameoftheheuristicusedtolinearizetheproblem. In
thisthesis,theheuristicusediscalledCBH(capability-basedheuristic)whichpicksa
numberoftasksco-locateonthesamemachinebasedontherelativecapabilityofthat
machine compared with others. Thus, the approach is called MIP(CBH). Because
MIP(CBH)containsreducednumbersofvariable,itisveryfast.However,becausea
setofvariablesispre-selected,theoptimizationisnotglobal. Inthisregard,MIP(*)
canbe viewedasa2-phaseapproach. Inthe firstphase, thepre-selectionheuristic
selectsthenumberoftaskstobeexecutedoneachmachine. Thelinearprogramis
then used to find the optimal allocation of tasks to each machine according to the
127
pre-selectednumbers.Thus,thequalityoftheresultingallocationdependssolelyon
thefirstpre-selectionphase.
SMIP isalsoamixed-integer-programming-basedapproachtosolvethealloca-
tionproblem. However,itisdifferentfromMIP(*)becauseitusesthevariablesub-
stitutiontechniquetolinearizetheproblem. BecauseSMIPdoesnotpre-selectany
variables,theoptimizationisglobalandwillalwaysresultinanoptimalallocation.
However,theapproachintroducesadditionalvariablesandconstraintsintothelinear
program,andthustakeslongertimetoexecute.
IIP uses the same variable substitution technique as SMIP. However, the com-
plexityoftheproblemisreducedbysolvingasetofintegerprogramsiteratively.An
integer program consistsof only integer variables, and thus is less complexthan a
mixedintegerprogram. AnIPformulationintheIIPapproachtriesto findanallo-
cationwith greater thana valuecalled . If there isa validsolution,
value of IP formulation of the next iteration is increased from the
ofthevalidallocationbyauser-definedamountcalled . Whenanintegerprogram
fails to find a feasible solution, the last known valid solution is used as the initial
allocation.InthisIIPapproach, ofthefinalsolutionisguaranteedtobewithin
percentofthe oftheoptimalallocation.However,alower willresultinlonger
executiontime of the IIP approach. Thus, there is an adjustabletrade-off between
thequalityofsolutionandtheexecutiontime. Because ofthesolutionisfound
128
byadjustingthe value,theobjectivefunctionoftheIPformulationisnot
crucialforthecorrectnessoftheIIPapproach.
MIP(CBH), SMIP, andIIP togetherprovideafullspectrumofsolutionsforthe
problemofinitialresourceallocation. MIP(CBH)isfastandefficient,butthequal-
ityofthesolutionisrelativelylowerthanotherapproaches. SMIPprovidesthebest
solution,buttakesalongtimetoexecute.IIPprovidessolutionsbetweenMIP(CBH)
andSMIP,duetoitsadjustabletrade-offbetweenthequalityofsolutionandexecu-
tiontime.However,IIPcannotguaranteethatitwillfindanoptimalsolution,andits
executiontimecannevermatchMIP(CBH).
129
Chapter8
Futurework
Inthischaptertheoutlineforfuturedirectionsofresearchisdescribed.
8.1 Improvingthesystemandapplicationmodels
ThesystemandapplicationmodelsdescribedinChapter3cancapturetherun-
timeparametersandtheirvariationswell. However,theeffectofarun-timeparam-
etervariationontasklatenciesismodeledasasimplelinearfunction. Futurework
willinvestigateacasewheretherelationshipbetweenrun-timeparametersandtask
latenciesisnon-linear(suchaspolynomial). Indoingsothemodelswillbeableto
capturemorediversetypesofapplicationtasks. However,theclaimthatoptimizing
(theminimumnormalizedslackness)ofalltasksandprimaryroutesleadstoan
allocationwiththehighestMAILvaluewillnotbevalidanymore. Anewobjective
functionmustbedevelopedtoguidetheoptimizationprocess.
In the current applicationmodels, the information arrivingfrom a non-primary
edgeisassumedtouseaconstantamountofresourcesanddoesnotvaryduringrun
time. However, all entries in the and matrices have a positiveslope but
zerointercept. Ineffect,themodelscapturetheloadvariationofdatafromprimary
130
edgebuteffectivelyignoringtheconstantutilizationofresourcesofdatafromnon-
primaryedge.Thisisbecauseoptimizing willnotleadtoanallocationwiththe
highestMAILvalueifthetasksandprimaryrouteshavedifferentinterceptvalues.
Improved models can circumvent this problem by dividing a task into two inter-
relatedtaskswithtwocorresponding and entries. Oneentryspecifiesthe
loadvariationofdatafromprimaryedgewithpositiveslopeandzerointercept. The
otherentrydescribestheconstantloadfromnon-primaryedgewithzero slopeand
positive intercept. The requirements that these two tasks must be allocated to the
samemachinecaneasilybeaddedintoconstraintsofthelinearprogram.
8.2 ImprovingMIP(*)andIIP
MIP(*) is shown to be a 2-phase optimization technique where the quality of
theallocationdependssolelyonthe firstpre-selectionheuristic. Ifthepre-selected
numbers from the heuristic match the actual numbers of tasks on each machine of
theoptimalallocation,MIP(*)willalsofindanoptimalallocation. Itisthusnatural
todevelopheuristicsotherthanCBHsuchthatthepre-selectednumbersmatchthe
optimalnumbers. Itisalsointerestingtoinvestigateifdifferencesbetweenthepre-
selected numbers and the optimal numbers will lead to some bounds between the
optimal valueandthe valuefromMIP(*).
In the IIP approach, informationfrom the current iteration is not transferred to
the nextiterationexceptthe new value. Allallocationswith lower
131
thanthe current valueis consideredinvalidin thiscurrent iteration, and
thuswillstillbeinvalidinthenextiterationwherethe valueisincreased.
However, this information is not passed to the IP formulation of the next iteration
which may have to go through these invalid allocations again, resulting in higher
executiontimethannecessary. Inotherwords,theIIPapproachisnotadivide-and-
conquer approach; the IP formulationin all iterations is solvedindividually. In an
improved IIP approach, a technique called Benders Decomposition can be used to
dividetheresource allocationproblemintoa masterproblemand subproblems. In
a subproblem, some variables of the master problem can be fixed, and the master
problem is iteratively restricted by adding inequalities which are the results of the
subproblem. ThistechniqueisverysimilartotheIIP approach, withan advantage
thatinformationfromthesubproblemsisretainedinthemasterproblemintheform
ofaddedconstraints.ByusingBendersDecomposition,itmaybepossibletofindan
optimalallocationfasterthanthecurrentIIPapproach.
The objective function of the IIP approach is currently not utilized in a mean-
ingful method. Recall that the solution of IIP approach is found through iterative
adjustmentof value. Infuturework, theobjectivefunctionwillbeused
either to optimize a secondary objective or help the IIP approach reach a solution
faster.
132
8.3 Investigating relationship between quality of
results, executiontime, andmachine-taskhetero-
geneityenvironments
MIP(CBH),whilecannotguaranteeanyboundonthequalityofresults,provides
very good solutions especially in Lo machine heterogeneity environments (Lo-Hi
andLo-Lo).ThecauseofthishighaverageMAILvaluemaybethatthecurrentpre-
selectionalgorithmoperatesverywellintheseenvironments.Ontheotherhand,the
executiontimeofSMIPandIIPissignificantlyfasterinHi-Hienvironmentthanin
otherenvironments. Thismaybebecausethebestsolutionismoreobvioustothese
approaches in the Hi-Hi environment where machines and tasks are greatly differ
fromoneanother. Theeffectofmachine-taskheterogeneityonvariousapproaches
shouldbefurtherinvestigated,sothatasuitableapproachcanbeselectedtomatch
theenvironmentinwhichitisoperated.
8.4 Developingadaptationalgorithms
Thegoalofourinitialallocationofresourcesistodelaythefirstresourceadap-
tationduringruntime,however,resourceadaptationcannotbeentirelyavoided.The
workinthisthesisisonlyhalfthesolutioninthebigpictureofresourceallocation
indynamicreal-timesystems. Resourceadaptationconstitutestheotherhalfofthe
133
solution. Resourceallocationalgorithmsthatshareinformationwiththeinitialallo-
cationshouldbedeveloped,sothattheycanbettermanagesystemresourcesduring
runtime. Forexample,theinitialallocationalgorithmcannotifytheresourceadap-
tationalgorithmaboutthebottleneck(thetaskorprimaryroutethathastheminimum
normalizedslackness) of the initialallocation. Thus, theresource adaptationalgo-
rithmcanprovidespecialattentiontotaskorprimaryroutethatisexpectedtoviolate
itsrequirement first. Inaddition,theresourceadaptationalgorithmcanmonitorthe
workloadanddetectanydiscrepancybetweentheactualandexpectedloadlevelvari-
ationsassumedbytheinitialallocation.
134
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Abstract (if available)

Abstract Dynamic real-time systems usually operate in an environment that is continuously changing.These changes in the environment cause workload of the system to vary during run time, and in effect causing system performance to fluctuate. In addition, certain Quality of Service (QoS) requirements are imposed on the real-time system and must be satisfied. Allocating system resource in this environment is challenging because a good and efficient allocation may become invalid when variations in workload cause Quality of Service violations.

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Creator Gertphol, Sethavidh (author)

Core Title Resource allocation in dynamic real-time systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/26/2006

Defense Date 12/07/2005

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,real-time systems,resource allocation,robust

Language English

Advisor Prasanna, Viktor K. (committee chair), Raghavendra, Cauligi S. (committee member), Zimmermann, Roger (committee member)

Creator Email gertphol@halcyon.usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m112

Unique identifier UC1210062

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Legacy Identifier etd-Gertphol-20061026.pdf

Dmrecord 27600

Document Type Dissertation

Rights Gertphol, Sethavidh

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu