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An Efficient Fault-tolerant SchedulingAlgorithm for Real-time Tasks with Precedence Constraintsin HeterogeneousSystem s*

XiaoQin,

Hong Jiang andDavidRSwanson .

DepartmentofComputerScienceandEngineering UniversityoNebraska-Lincoln f {xqin,jiang,dswanson}@cse.unl.edu

TechnicalReportNo. TR-UNL-CSE2002-0501 February2002 yoNebraska-Lincoln f Lincoln,NE68588-0115

T * hisworkwassupported bayN n SF grant(EPS-0091900)and Nebra a grant(26-0511-0019). A shortversion othis f paperwasaccepte of the2002International ConferenceoP n arallel Processing Vancouver,British Columbia,Canada 1

skaUniversity Foundation dforpublicationitnheproceedings (ICPP-02), August18-21,2002,

An Efficient

Fault-tolerantSchedulingAlgorithmfor Real-time Taskswith PrecedenceConstraintsin HeterogeneousSystems XiaoQin Hong David Jiang RSwanson . DepartmentofComputerScienceand Engineering UniversityoNebraska-Lincoln f Lincoln,NE68588-0115,{xqin,jiang,dswanson}@cse.unl.edu

Abstract Inthispaper,weinvestigateanefficientoff-lineschedulingalgorithm

inwhichreal-timetasks

withprecedenceconstraintsinaheterogeneousenvironment.Itprovidesm

orefeaturesand

capabilitiesthanexistingalgorithmsthatscheduleonlyindependenttasksinr

eal-timehomogeneous

systems. In addition, the proposed algorithm takes the heterogeneities of communicationandreliabilityintoaccount,therebyimprovingthereliabili tolerantcapability,thealgorithmemploysaprimary-backupcopyschemethat toleratepermanentfailuresinanysingleprocessor.Inthisscheme, overlapwithotherbackupcopiesonthesameprocessor,aslongastheir copiesareallocatedtodifferentprocessors.Tasksarejudiciouslyall reducetheschedulelengthaswellasthereliabilitycost,defined failurerateandtaskexecutiontime.Inaddition,thetimefordetec faultisincorporatedintotheschedulingscheme,thusmakingthealgorithmmor

computation, ty.Toprovidefaultenablesthesystemto abackupcopyisallowedto correspondingprimary ocatedtoprocessorssoasto

tobetheproductopf rocessor tingandhandlingoapf ermanent epractical.To

quantifythecombinedperformanceofault-tolerance f andschedulability,thepe

rformabilitymeasure

isintroduced.Simulationresultsshowthattheproposedschedulingalgorithmsi

gnificantlyimproves

theschedulability,thereliability,andperformabilityoverexist

ing,comparablealgorithmsinthe

literature.

Keywords: Real-timetasks,off-linescheduling,fault-tolerance,he precedenceconstraints,reliability cost, performability

2

terogeneousdistributedsystem,

1.INTRODUCTION Heterogeneousdistributedsystemshavebeenincreasingly applications,includingreal-timesafety-criticalapplic

usedforscientificandcommercial

ations,inwhichthesystem dependsnotonlyon

theresultsoacomputation, f butalsoonthetimeinsta

ntsaw t hichtheseresultsbecomeavailable.

Examplesosuch f applicationsincludeaircraftcontrol,

transportationsystemsandmedical electronics.

Toobtainhighperformanceforreal-timeheterogeneoussy importantrole.While

stems,schedulingalgorithmsplayan

aschedulingalgorithmmapsreal-timetaskstoprocessorsin

thatdeadlinesandresponsetimerequirementsaremet[29], functional andt imingcorrectnessevenitnhepresence

thesystemmustalsoguaranteeits of faults.

Theproposedalgorithm,referredtoaseFRCD

(efficientFault-tolerantReliabilityCostDriven

Algorithm), endeavorstocomprehensivelyaddresstheissuesofaultf taskprecedenceconstraints,andheterogeneity.

thesystemsuch

tolerance,reliability,real-time,

Totolerateanyoneprocessor’spermanentfailure,the

algorithm uses P a rimary/Backuptechniquetoallocate twocopie Tofurtherimprovethequality othe f schedule, baackupcop copiesonthesameprocessor,aslongatsheircorrespo

of seach task todifferentprocessors. iyallowed s toverlapwith otherbackup ndingprimarycopiesareallocatedtodifferent

processors.Asanaddedmeasureoffault-tolerance,the

proposedalgorithmalsoconsidersthe

heterogeneitiesofcomputationandreliability,therebyi

mprovingthereliabilitywithoutextra

hardwarecost.Moreprecisely,tasksarejudiciouslyall schedulelengthaw s ellasthereliabilitycost,define taskexecutiontime.Inaddition,thetimefordetectin

ocatedtoprocessorssoastoreducethe dtobteheproductofprocessorfailurerateand gandhandlingofapermanentfaultis

incorporatedintotheschedulingscheme,thusmakingthealgor

ithm morepractical.

algorithmsstudiedin[1-11,13-29]shareoneotwo r features

witheFRCD,intermsotfheassumed

operationalconditions,asexplainedinSection2,the

3

Whilethevarious

latterisarguablythemostcomprehensive,in

termsothe f numberofdifferentschedulingissuesaddress

ed,andoutperformsseveralquantitatively

comparablealgorithmsintheliterature.Extensivesimula algorithm significantly outperformsallthreerelevantan

tionstudiesshowedthattheeFRCD dquantitativelycomparablealgorithmsfound

in theliterature. Inthesectionthatfollows,relatedworkintheliter

atureisbrieflyreviewedtopresenta

backgroundfortheproposedalgorithmandtocontrasteFRCD

withotheralgorithmstoshowits

relevance,similarity anduniqueness.Therestof thepape

is rorganizedafsollows.Section3presents

theworkloadandthesystemcharacteristics.Section4

describestheeFRCDalgorithmandthemain

principlesbehindit,includingtheoremsusedforpresentingt giveniS n ection 5Section . 6concludesthepaperby summa

healgorithm.Performanceevaluationis rizingthemaincontributionsof thispaper.

Finally,theappendixcontains saimpleexampletoelucida

te theproposedalgorithm.

2.RELATED WORK Theissueofschedulingonheterogeneoussystemshasbeen years.Aschedulingscheme,STDP,forheterogeneoussyst

studiedintheliteratureinrecent emswasdevelopedin[25].In[8,28],

reliabilitycostwasincorporatedintoschedulingalgori

thmsfortaskswithprecedenceconstraints.

However,thesealgorithmsneitherprovidefault-tolerance

norsupportreal-timeapplications.

Previousworkhasbeendonetofacilitatereal-timecom

putinginheterogeneoussystems.In[13],a

solutionforthedynamicresourcemanagementproblemin proposed.Aprobabilisticmodelforaclient/serverhete

real-timeheterogeneoussystemswas rogeneousmultimediasystemwaspresented

in[26].Thesealgorithms,however,cannottolerateanypro

cessorfailure.Fault-toleranceis

consideredinthedesignoreal-time f schedulingalgorithms

tomakesystemsmorereliable[15][18].

Liberatoetal.proposedafeasibility-checkalgorithmfor

fault-tolerantscheduling[16].Thewell-

knownRate-MonotonicFirst-Fitassignment

4

algorithmwasextendedin[5].Adelayedscheduling

algorithm using passive a replicamethodwasdevelopedi[n2].[ hybridtasksetsconsistingoffirmandhardperiodictasks. assumethattheunderlyingsystem eitherishomogeneousor Thealgorithmin[1]is real-time a schedulingalgorithmfo doesnotsupportfault-tolerance.Dimaeal. tdevisedanof algorithm usingreplicationooperations f anddatacommunicat execute thebackupcopy otafasksimultaneously withits Manimaranetal.[17]andMosseeat l.[9]haveproposed timetaskswithfault-tolerancerequirementsonmulti theiralgorithmsareindependentoof neanotherandares algorithm onthesamesystem andtaskmodel asthatin[9].Ohan tolerantschedulingalgorithmthatstaticallyschedules featuresamongthesealgorithms[9,16,18,20,21]arethat(1)tas and(2)theyaredesignedonlyforhomogeneoussystems.Al consideredinboth[28]and

6]presentedaanlgorithm toschedule However,bothotfheabovealgorithms consistsof saingleprocessor. tasks r withprecedenceconstraint,butit flinereal-timeandfault-tolerantscheduling ions[7].However,thisalgorithmmust primary copy. dynamicalgorithmstoschedulerealprocessorsystems,butthetasksscheduledin cheduledon-line.Martin[19]devisedan dSonstudied real-time a andfaultasetofindependenttasks[21].Twocommon ksareindependentfromoneanother thoughheterogeneoussystemsare

eFRCDthe , latterconsidersfault-toleranceandreal-time

taskswhilethe

formerdoesnotconsidereither. Veryrecently,Giraultetal.[10,11]proposedareal-tim systemsthatconsidersfault-toleranceandtaskswithpr closesttoeFRCDthattheauthorshavefoundinthelite andeFRCDarefour-fold:(a).theformerdoesnotconside does;(b).eFRCDconsidersheterogeneitiesincomputatio

eschedulingalgorithmforheterogeneous ecedenceconstraints.Thisstudyibs yfarthe rature.Themaindifferencebetween[10,11] traskdeadlinesexplicitly,whileeFRCD n,communicationandreliabilitywhilethe

formeronlyconsiderscomputationalheterogeneity;(c).t

heformerdoesnottakereliabilitycostinto

consideration,whereaseFRCDisreliability-costdrive

n;and(d).theformerallowstheconcurrent

5

executionoprimary f andbackupcopiesotafaskwhileeFRC

Dallowsbackupcopiesotasks f whose

primary copiesarescheduledodnifferentprocessors to Intheauthors’previouswork,

overlaponeanother.

both static[23,24]anddynamic[22]real-timeschedulingschemes

forheterogeneoussystemsweredeveloped.Onesimilarity ReliabilityCostDrivenScheme

amongthesealgorithmsisthatthe

isapplied.WiththeexceptionoftheFRCDalgorithm[24],

algorithmsproposedin[22,23]cannottolerateanyfailure.

other

Inthispaper,theFRCDalgorithm[24]is

extended by relaxingtherequirementthatbackupcopiesof tasksbenot

allowedtoboeverlapped.

3.WORKLOAD ANDSYSTEMCHARACTERISTICS Areal-timejobwithdependenttaskscanbm e odelledby E},where

V={v

amongtasks.

1v,

DirectedAcyclicGraph(DAG),T={V,

as ofreal-timetasks,andset a ofedges 2,...,vni}set

Erepresentscommunication

viv, j) ∈ Eindicatesamessagetransmittedfromtask

e=ij(

volumeodata f beingsent.Totolerantpermanentfaults

vto i

vand j,

|eijdenotes | the

inoneprocessor, parimary-backuptechnique P vand

isapplied.Thus,eachtaskhastwocopies,namely, processors.Withoutlossogf enerality,iitsassumedt

vBexecuted , sequentiallyontwodifferent

hatprimaryandbackupcopiesoaftaskare

identical.Theproposedapproachcanalsobaeppliedwhentw

ocopiesoeach f taskaredifferent.Fig.

13inAppendix(F)showsan exampleDAG. Theheterogeneoussystemconsistsofaset

P={p

connectedbyanetwork.Aprocessorcommunicateswitho andthecommunication t imebetween twotasksassignedt Ameasureof

m}ofheterogeneousprocessors

therprocessorsthroughmessagepassing, othesameprocessorisassumedtobzeero.

von i processor

pA of j.measure

essor.Thus,

C:V ×P→Z

ferom

+

,which

cj(vid) enotestheexecutiont ime

communicationalheterogeneity

E×P×P →Z +Communication . timeforsendingmessage a

6

2,...,p

computationalheterogeneity ismodeledbyafunction,

representstheexecutiont imeoeach f taskoneachproc oftask

1,p

von s

ismodeledbyafunction ptoi

vr on pisdj eterminedby

M:

wij*|e|,where

|e|isthecommunicationcostand

wijistheweightontheedgebetween

representingthedelayinvolveditnransmitting message a o Givenatask

v ∈V , d(v),s(v)

if forall

v∈ V,

and i(v)

itsatisfiesboth f(v

pj,

unit f lengthbetween thetwoprocessors.

andf(v) denotethedeadline,scheduledstarttime,andfinisht

respectively. p(v)denotestheprocessortowhich constraints: f(v)=s(v)+c

pand i

ime,

visallocated.Theseparametersaresubjectto

≤ f(v)d(v)

where ,

)d(v) ≤

P

p(v)=p A jobhasa i.real-time

feasibleschedule

f(vB) ≤d(v).

and ,

A k-timely-fault-tolerant (k-TFT)schedule[21]isdefinedasthescheduleinwhichnotask deadlinesaremissed,despiteof

karbitraryprocessorfailures.ThegoalofeFRCDistoa

TFT.Iitsassumedthatprocessorfailures, whichare

duringthetimeinterval

independentwithoneanother,followinga

λThus, .

PoissonProcesswithanarrivalrate τ[12].Let

where λi (1 ≤i

≤m) is

t.Therefore,

pi'sfailurerateinavector

X=(i1

(1)

λ1, λ2…, ,

R=(

λm).Thestateotfhesystemis

{0,1,2, …,m}

≤i

≤m)

More . precisely,

signifiesthat

X=0

pencounters permanent i p ( v P ) =i

pi

mary copies of tasks assignedto

m

m

i =1

i =1

∏ NLi (τ i ) =∏ e −λiτ i

= x0 (2)

(1 − NLi (τ i )) ∏ NF j (τ j ) = (1 − e −λ τ ) ∏ e −λ τ m

j j

i i

j =1, j ≠ i

means

τ i = MAX { f (v )} istheschedule

m

7

nd NFi(t) be

e − λit

Xisdeterminedbyequation(2),where

length of pwhich ithe s latestof finish times amongall pri i,

Pr[X= x]

nevents

NFi(t) isbexpressedbtyheequationbelow,

X which takes valuein

thatnoprocessorpermanentlyfails,and

for

e − λτ (λτ ) n givestheprobabilityof n!

=Pr[K=0] = i(t)

representedbrandom ay variable

failures.Theprobabilityfor

Prn (τ ) =

Kbetherandomvariableforthenumberoffaults,a

theprobability that pwill failby t ime i not NF

chieve1-

j =1, j ≠ i

for = xi,(1

≤i ≤m)

Thereliabilitycostof task

von i processor

on pjIt.shouldbenotedthat heterogeneityin

vi'sexecutiont ime λand j

pj isdefinedatsheproductof

reliabilityheterogeneity isimpliedinthereliabilitycostbyvirtueof RC(R, Ψ)bethereliability costof gaivenschedule λLet j.

cj(viand )

Ψfor gaiven

R.

m

Undertheassumptionthatnomorethanoneprocesso

∑ Pr( X = i) = 1,

pr ermanentlyfails,thatis,

i =0

theexpectedvalueof

itcanbexpressedas, m

(3)

E (RC ( R, Ψ )) = ∑ (RCi ( R, Ψ ) × Pr[ X = i]) i =0

d RCi(R,Ψ) (1 ≤i

where RC0 (R, Ψ)isthereliabilitycostwhennoprocessorfailsan reliabilitycostwhen

pfails. i

schedule Ψ. RCand 0

RCare beyquation (4) and(5), respective i determined

RC

0

RCand 0

RCisti hesummationoveralltasks'reliabilitycosts

m

∑

(R,Ψ ) =

∑

i =1

RC i ( R , Ψ ) = =

p (v

m

∑

P

   j =1, j ≠ i 

∑

ly. (4)

)=i

j =1, j ≠ i p ( v P ) = j m

basedon

λ ic i (v )

∑

∑

≤m) isthe

λ j c j (v ) +

λ j c j (v ) +

p (v P )= j

m

∑

∑

∑

(5)



λ j c j ( v ) 

p ( v P ) = i , p ( v B )= j

In equation (5), thefirstsummation term on theri

λ c (v )

j j j =1, j ≠ i p ( v P ) = i , p ( v B ) = j



ghthandsiderepresents thereliability costdue t

taskswhoseprimarycopiesresideinfault-freepro

o

cessors,whilethesecondsummationterm

expressesthereliability costdue tothebackupco

piesothe f taskswhoseprimary copiesresideotnh

failedprocessor.Usingequations (3)-(5), we obtai

tnheexpectedvalueoreliability f costas follows

e ,

m m  m −λ τ  E (RC ( R, Ψ )) = RC 0 ( R, Ψ ) × ∏ e −λiτ i + ∑  RCi ( R, Ψ ) × (1 − e −λiτ i ) (6) e j j ∏ i =1 i =0  j =1, j ≠ i 

Reliability,giveninthefollowingexpression,cap parallel jobs

in thepresenceosafingleprocessor failure. RL( R, Ψ ) = e − RC ( R ,Ψ )

8

turestheabilityofthesystemtocomplete

(7)

Obviously,reliabilityincreasesasthesystem’sre

liabilitycostdecreases.Thissuggeststhat

schedulingtaskstoprocessorsinawaywhichgives

risetoaminimumsystemreliabilitycostwill

resultin high a system reliability. 4. SCHEDULING ALGORITHMS

Inthissection,wepresenteFRCD,anefficientalg

orithmthatschedulesreal-timejobswith

dependenttasksact ompiletime.Theobjectivesof schedulelengthisreduced

thealgorithmaremultiple,namely,(1)Total

sothatmoretaskscancompletebeforetheirdeadli

failuresinoneprocessorcanbteolerated;and(3)

nes;(2)Permanent

Thesystemreliabilityiesnhancedbyreducingthe

overall reliability costof theschedule. 4.1 An Outline

Thekeyfortoleratingasingleprocessorfailurei

tsoallocatetheprimaryandbackupcopiesoaf

tasktotwodifferentprocessorssuchthattheback

upcopysubsequentlyexecutesiaf ndonlyitfhe

primarycopyfailstocompleteduetoitsprocessor

failure.Notallbackupcopiesneedtoexecute,

eveninthepresenceoasingle f processorfailure.

Sinceonlytasksallocatedtothefailedprocessor

areaffectedandneedtheirbackupcopiestobexe overlapwithoneanother.

Moreprecisely,a

cuted,certainbackupcopiescanbescheduledto B vis allowedto verlapwithotherbackupcopieson

sameprocessor,if thecorrespondingprimary copies vi

areallocatedtothedifferentprocessors towhich P the vis notallocated.Thus,in feasible a schedule,

P

p1

the

time vjB

theprimarycopiesoaf nytwo

vjB

tasksmustnotbe

p2

vjP p3

allocatedtothesameprocessoriftheirbackup overlap

Fig.1Primarycopiesofv and v are to i j allocated p1andp 3r, espectively,andbackupcopiesofv andv are allocatedtop . twobackup j both 2These copiescanbeoverlappedwitheachother.

copiesareonthesameprocessorandthereisan i

overlapbetweentwothebackupcopies.This statementisformally describedabselow.

9

Proposition 1.

(

) ((

) (

p3respectively, , andbackupcopiesof

vand i

))

∀vi , v j ∈ V : p (viB ) = p (v Bj ) ∧ s(viB ) ≤ s(v Bj ) < f (viB ) ∨ s(v Bj ) ≤ s(v iB ) < f (viB ) → p(v iP ) ≠ p (v Pj ) Fig.1showsanexampleillustratingthiscase.In thisexample,primarycopiesof vand vare i j allocatedto

pand 1

backupcopies canboeverlappedwith each otherbec

vare allocatedto j both

auseamost t one

p2These . two

of them will everexecuteitnhe

single-processorfailuremodel. Thealgorithmschedulestasksinthefollowingthre by theirdeadlinesinon-decreasingorder,such th Second,theprimarycopiesarescheduledtosatisfy schedulelengthandtheoverallreliabilitycost.F mannerastheprimarycopies,exceptthattheymay reduceschedulelength. Morespecifically,inthe mustsatisfy thefollowingthreeconditions:(1)it

emainsteps.First,real-timetasksareordered attasks with tighterdeadlineshavehigherpriorit theprecedenceconstraintsandtoreducethe inally,thebackupcopiesarescheduledinasimila

condition(1);and(3)ishould t beabletoreceive

secondandthirdsteps,theschedulingoeach f task

abilitycostamongallprocessorssatisfying messagesfromallitspredecessors.Inadditionto

theseconditions,eachbackupcopyhastwoextraco theprocessorthatisdifferentthantheoneassign

nditionstosatisfy,namely,(i)iis tallocatedon edforitsprimarycopy,and(ii)iitsallowedto

overlapwithotherbackupcopiesonthesameproces differentprocessors. Proposition2.

soriftheirprimarycopiesareallocatedto

Condition (i)and(ii)canbfeormally describedby

Aschedulei1-TFT s

thefollowingproposition.

→ ∀v ∈ V : ( p (v P ) ≠ p (v B ) ) ∧ (s(v B ) ≥ f (v P ) + dt ) .

Inthesubsectionthatfollows,theeFRCDalgorithm of andrelationshipsbetween tasks andtheirprimar

ispresented,alongwithsomekeyproperties ayndbackupcopies.

4.2 TheeFRCD Algorithm

Tofacilitatethepresentationothe f algorithm,so 10

r

beoverlappedonthesameprocessorstofurther

deadline s shouldbm e et; (2) theprocessor allocat

shouldleadtotheminimumincreaseinoverallreli

ies.

meothe f conditionslistedabove,(1)-(3)and(i)-

ion

(ii),andothernecessary notations andproperties NOTATION D(v) S(v) F(v)

B(v) VQi VQi’(v)

Table1Definitions . oNotation f DEFINITION ∈E} The setof predecessorsof task v. D(v) ={v (v | i v) i, The setof successorsof task v, S(v) ={v (v, |vi E} i) ∈ B The setof feasible processorsto which vcan baellocated,determinedinpart by Theorem2. The setof predecessorsof v’s backupcopy,determinedbE y xpression(11). , f(v0= 0) The queue inwhichall tasks are scheduled to pi,s(v q+1=) ∞and The queue inwhichall tasks are scheduled to pand overlap withthe backupcopy of i, cannot task v,where s(vq+1=) ∞and , f(v0= 0) vi is schedule-preceding v if , only if s(v j) ≥f(v i ). jand vi is message-preceding v j, if andonly if v i sends m a essage to v j. Note that vi ⇒ v j implies vi f v but notinversely . v execution-preceding v and only iboth f tasksexecute and vi ⇒ v Note vi a v implies i j, if j that j v and v f v vi ⇒ j i j The earliest available time forthe primary obackup r copy otask f ivmessage f seentfrom vj∈ D(v) representsthe only precedence constraint.

vi f v j vi ⇒ v j vi a v

arelisteditnhefollowingtable.

j

EATi (v,v j) EATi P(v)

j

MIN v P∈D ( v P ) {EATi (v P , v Pj )} j

B

EATi (v)

MIN v ∈D ( v B ) {EATi ( v B , v j )} j

P

ESTi (v) ESTi B(v) ESTP(v)

The earlieststart time forthe primary copy of The earlieststart time forthe backupcopy of

ESTB(v)

MIN Pi∈P {ESTi (v B )}

InTable1,

ovnprocessor ovnprocessor

pi . pi .

MIN Pi∈P {ESTi (v P )}

EATiP(v)itshe

earliestavailabletime

timeforitoreceivemessagesfromallitspredec availablet imeonprocessor

pfor i

pfor i

essors.Similarly,

ESTiB(v) overall p

i

∈P.

system aregiven amongexpressions (8)through (13) accompaniedbyexplanations,ispresentedbelow.

ESTB(v)itsheearlieststarttimefor

vPtaking , intoaccountthe

EATiB(v) denotesthe

vB. ESTP(v),determinedbytheminimalvalueo f

∈P , isthe earlieststarttime for vPLikewise, . theminimalvalueof

onprocessor

earliest

ESTiP(v) forall

pi

vBand , iesqualto

Theseessentialvaluesforagivenheterogeneous .A detailedpseudocode othe f eFRCDalgorithm, Anexampletoillustratehowtheproposed

algorithm works is giveninAppendix (F). The eFRCDAlgorithm: Input: T={V,E} , P,C,M,R /*DAG,DistributedSystem,Computational,Communicatio nalandReliability Heterogeneity */ Output: Schedule feasibility of T,and vaiable schedule Ψif it is feasible . 1.Sort tasks by the deadlinesin non-decreasing order,subj ectto precedence constraints, and generate anordered list OL; P 2. for eachtask v in OL, following the order schedule , the primary copy vdo /* Schedule primary copiesof tasks*/

11

s(vP) ← ∞rc ; ← ∞VQ ; =i ∅; for eachprocessor pdo /* Determine whethertask v should allocated be tporocessor p*/i i * Calculate ESTPi (v),where VQ= {i v 1v, 2…, , vi}the inwhichall */ qs queue , f(v0= 0) */ ks are scheduledto pi,s(v q+1=) ∞and 2.2.1 for ( = t0j+ qo1 /* Compute ) do the earlieststarttime of ovn p*/i if s(vj+1 ) MAX{f(v EAT i P(v)}c≥ i (v) then/*checkithe f unoccupiedtime intervals,interspersed*/ j), P ESTPi (v) /* bcyurrently scheduled tasks,canaccommodate *v/ = MAX{f(vj),EAT i (v)}; endfor P executing at ESTPi (v) the earliest EST*/i andcan be completedbefore d(v) then /*Determine 2.2.2 if vstarts P Determine reliability costof von pi; /* basedon Equation(10) */ if (( rci