SCHEDULINGTASKSWITHAND/ORPRECEDENCE

Transcription

1 SCHEDULINGTASKSWITHAND/ORPRECEDENCE inthesecondtypeofproblem,someorpredecessorsmaybeleftunscheduled.weshowthatmost itspredecessorsarecomplete.wecallsuchataskanandtask.inthispaperweallowcertain thersttypeofproblem,allthepredecessorsofeveryortaskmusteventuallybecompleted,but tasks.weanalyzethecompleityoftwotypesofreal-timeand/ortaskschedulingproblems.in taskstobereadywhenjustoneoftheirpredecessorsiscomplete.thesetasksareknownasor Abstract.Intraditionalprecedence-constrainedschedulingataskisreadytoeecutewhenall D.W.GILLIESyANDJ.W.-S.LIUz CONSTRAINTS problemsinvolvingtaskswithindividualdeadlinesarenp-complete,andthenpresenttwoprioritydrivenheuristicalgorithmstominimizecompletiontimeonamultiprocessor.thesealgorithms providethesamelevelofworst-caseperformanceassomepreviouspriority-drivenalgorithmsfor analysis,multiprocessorsystems,np-completeproblems. Keywords.non-preemptivescheduling,listscheduling,minimallengthschedules,algorithm AMSsubjectclassications.68M20,68Q25,90B35,90C90. schedulingand-onlytasksystems. taskmaybegineecutionwhensomebutnotallofitspredecessorsarecompleted. WecallsuchataskanORtask.Theresultingtasksystem,containingbothAND describemanyreal-timeapplicationsencounteredinpractice.intheseapplicationsa andortasks,issaidtohaveand/orprecedenceconstraints. themisknownasand-onlyprecedenceconstraints.thistraditionalmodelfailsto betweentasksarerepresentedbypartialordersknownasprecedenceconstraints. predecessorsarecompleted.wecallsuchtasksandtasks,andthepartialorderover Eachtaskmayhaveseveralpredecessorsandmaynotbegineecutionuntilallits 1.Introduction.Inthetraditionalmodelofreal-timeworkloads,dependencies denceconstraintstomeetdeadlines.weinvestigatetwovariantsofthisproblem, calledtheunskippedandtheskippedvariants. pleted,thatis,theycannotbeskipped.wecallthemodelforthistypeofapplication theand/or/unskippedmodel.foreample,inroboticassembly[1],oneoutof fourboltsmaysecureanengineheadwellenoughtoallowfurtherworkonother partsoftheenginehead.however,theremainingthreeboltsmusteventuallybe InsomeapplicationsallthepredecessorsofanORtaskmusteventuallybecom- InthispaperweareconcernedwithhowtoscheduletaskswithAND/ORprece- needaresourcefromoneofseveralpredecessorsinordertoeecuteandhenceis installed.theunskippedvariantalsomodelstasksthatshareresources.ataskmay eledasanortask.again,theotherpredecessorsmusteventuallybecompleted. readytoeecutewhenanyonepredecessoriscomplete.suchataskcanbemod- J-1181andNVYN J-1146) precedenceconstraintsaretoostrictfortaskstomeettheirdeadlines.byrelaing theprecedenceconstraintsofsometasks,andrestructuringtheapplicationcodeto TheAND/OR/unskippedproblemalsoarisesinhardreal-timeschedulingwhenthe deadlines. accommodatetherelaedconstraints,itmaybepossibleforthetaskstomeettheir ThisresearchwassupportedbytheOceofNavalResearch(ContractsNo.NVYN zdepartmentofcomputerscience,universityofillinois. ydepartmentofelectricalengineering,universityofbritishcolumbia. 1

2 2 D.W.GILLIESANDJ.W.-S.LIU InotherapplicationssomepredecessorsofanORtaskmaybeskippedentirely. WecallthistheAND/OR/skippedmodel.Oneeamplecanbefoundintheproblemofinstructionschedulingonsuperscalar,MIMD,orVLIWprocessors.Onsuch processors,severaldierentinstructionsequencesmaybeusedtocomputethesame arithmeticepression.thesedierentsequencesarisefromalgebraiclawssuchasassociativityanddistributivity.onlyonesequenceneedstobeeecuted,andtheother sequencesmaybeskipped.anotherapplicationthatcanbecharacterizedbythis modelismanufacturingplanning[5]becausecertainmanufacturingstepsobeyassociativeanddistributivealgebraiclaws.theand/or/skippedproblemalsoarisesin hardreal-timescheduling.whenthereisinsucienttimeforatasksystemtomeet itsdeadlines,wemayconvertappropriatetaskstoimprecisecomputations[3],which maybemodeledasortaskswhosepredecessorsmaybeskipped. WeareconcernedwithwaystoscheduleAND/ORprecedence-constrainedtasks tomeetdeadlinesortominimizecompletiontime.mostoftheseproblemsaregeneralizationsoftraditionaldeterministicschedulingproblemsthatarenp-hard.inthis paperweanalyzethecompleityoftheproblemsthatarenotknowntobenp-hard. FortwoproblemsthatareknowntobeNP-hard,wegiveheuristicalgorithmstominimizecompletiontime.Thealgorithmshavesmallrunningtimeandgoodworst-case performance. Ourworkisrelatedtosomepreviousworkondeterministicschedulingtomeet deadlines[6][8]andtominimizecompletiontime[9][10][13][14].wewereinspiredby anand/ormodelthatwasproposedasameansofmodelingdistributedsystems forreal-timecontrol[18].tworecentsystemsincorporatedand/orprecedence constraintsofsomesortintheirimplementation[16][19]. Theremainderofthispaperisorganizedasfollows.Section2describesour assumptionsabouttheand/orschedulingproblemandintroducestheterminology usedinlatersections.section3investigatestheunskippedproblemwithmultiple deadlinesandanalyzesanalgorithmtominimizecompletiontime.insection4,we investigatetheskippedproblemandgiveasecondalgorithmtominimizecompletion time.section5drawsconclusionsanddiscussesfuturework.theappendicontains proofsofthetheoremsstatedinsections3and4. 2.TheAND/ORModel.Alltheschedulingproblemsconsideredhereare variantsofthefollowingproblem.therearemidenticalprocessorsandasetoftasks T=fT1;T2;:::;Tng.EachtaskTimusteecuteononeprocessorforpiunitsoftime andissaidtohaveprocessingtimepi.thereisapartialorder<denedovert. IfTi<Tj,thenTiisapredecessorofTj,andTjisasuccessorofTi.ThetaskTi isadirectpredecessoroftjifthereisnotksuchthatti<tk<tj.thetasktj isanandtaskifitseecutionmaybeginonlyafterallitsdirectpredecessorshave completed.thetasktjisanortaskifitseecutionmaybeginafteronlyoneof itsdirectpredecessorshascompleted.thepartialorder<isanin-forestifwhenever Tk<TiandTk<Tj,wehaveeitherTi<TjorTj<Ti;thepartialorder<isan in-treeifithasauniqueelementwithnosuccessors.ataskfollowedbyaseriesof directsuccessorsti1<ti2<iscalledataskchain. ThepartialorderisalsorepresentedbyaweightedandtransitivelyreduceddirectedgraphG=(T;A;P)calledthetaskgraph.InthisgraphthereisaverteTi foreverytaskinthesett.thesetaisknownasthesetofarcs.iftiisadirect predecessoroftjinthepartialorderthen(ti;tj)2a.thesetp=fp1;;png denotesthesetofprocessingtimes.ataskgraphtogetherwithasetofdeadlines D=fd1;;dngisa2-tuple(G;D)thatcharacterizesaschedulingproblem;itis

3 (a)tasksystem AND/ORSCHEDULING (b)and/or/unskippedschedule 3 calledatasksystem.whenseveralgraphsg1;g2;arepresent,thefunctions T(Gi),A(Gi),andP(Gi)willbeusedtoetractthesetsT,A,andPfromthe cessorsofti,andletp(g;ti)=ftjj(tj;ti)2a(g);ti2t(g)gdenotethesetof graphgi. directpredecessorsofti.letl(g;tj)bethelengthofthelongestdirectedpath LetS(G;Ti)=fTjj(Ti;Tj)2A(G);Ti2T(G)gdenotethesetofdirectsuc- Fig.1.Sampleproblemandsolution. ingendingattj.moreprecisely,l(g;tj)=pjiftjhasnopredecessorsing, schedules. timeofthetasksonthelongestchain.lateritwillbeshownthatand-onlygraphs withminimall*(g)ande*(g)canbeusedtoproducenear-optimalpriority-driven ingtimeofanand-onlygraph,i.e.thetotalprocessingtimeminustheprocessing predecessorsoftiing.lete*(g)=ppi?l*(g)denotethe\residual"process- andl(g;tj)=pj+makfl(g;tk)j(tk;tj)2a(g)giftjhaspredecessors.let graphg.lete(g;ti)=ptj<tiingpjdenotethetotalprocessingtimeofallthe L*(G)=mafL(G;Tj)jTj2T(G)gbethelengthofthelongestdirectedpathina algorithmmustchooseonedirectpredecessortitobeessentialandtheprecedence meansthattheymustappearinavalidschedule.ifanandtaskisessential,thenall constraintti<tjmustbeobeyedinschedulingthetasksystem.ifataskisnot itsdirectpredecessorsareessential.ifanortasktjisessential,thenthescheduling classiedasessential,thenitisinessential.wedistinguishbetweentwoproblems inessentialtasksmustbeeecuted. referredtoasskippedandunskippedproblems,respectively.inaskippedscheduling problem,inessentialtasksmaybeleftuneecuted.however,inanunskippedproblem, Allthetaskswithnosuccessorsinataskgraphareclassiedasessential;this T5requireseunitsofprocessingtime.Wherenecessary,deadlineswillbewritten separately,nettotheassociatedtasks.ifthedeadlinesareomittedfromagure, labeledbytheirname,orbytheir(name,length),so(t5;e)wouldindicatethattask thenthereadershouldassumethatallthedeadlinesareidentical.everytaskinthis thelengthsanddeadlinesareomittedfromthisgure.figure1(b)depictsaschedule inwhicht3isanessentialtask,andt2isaninessentialtask.figure1(b)showsa pictedbycirclesandortasksaredepictedbycircleswithinboes.tasksaregenerally eamplehasaprocessingtimeofoneandallthetaskshavethesamedeadline,hence, Figure1(a)depictsanAND/ORtasksystem.InthegureANDtasksarede- taskgraph,thenaskippedschedulecouldbeobtainedbydeletingt2fromtheendof scheduleoftheunskippedtaskgraphfromfigure1(a).iffigure1(a)wereaskipped thescheduleinfigure1(b). tionallyleaveprocessorsidle.thesealgorithmsareknownaspriority-drivenorlist- schedulingalgorithms.wheneveraprocessorisavailable,alist-schedulingalgorithm schedulesthereadytaskwiththehighestpriorityaccordingtoaprioritylist.because Theschedulingalgorithmsinthispaperaresimpleheuristicsthatneverinten- T 1 T4 T 2 T 5 T 3 T 3 T 5 T 1 T 4 T 2 time

4 4 D.W.GILLIESANDJ.W.-S.LIU algorithmsarealsocalledgreedyalgorithms.ascheduleproducedbyalist-scheduling theytrytomakethebestlocalchoiceateachschedulingdecisionpoint,list-scheduling algorithmisknownasalistscheduleandthetimeatwhichallthetasksintare completeisthelengthoftheschedule. (a)eact3-coverproblem e 1 Fig.2.Eact3-covertransformation. (b)and/ortasksystem taskmaybegineecutionassoonasanessentialpredecessoriscompleted.insome situationseachtasktihasadeadlinedi;timustbecompletedatorbeforetimedi. thathasafeasiblescheduleiscalledfeasible.givenatasksystemourobjectiveisto afeasiblescheduleinthesesituationsisequivalenttotheproblemofminimizingthe ndafeasiblescheduleordeterminethatnofeasiblescheduleeists. Ascheduleiscalledfeasibleifeverytaskcompletesbyitsdeadline.Atasksystem WeassumethateverytaskinThasreadytimeequaltozero,thus,anOR heuristictominimizecompletiontimeonmprocessors.wetheneplainwhyno AND/OR/unskippedschedulingproblem.Aftershowingthatmostnaturalproblems withdeadlinesarenp-completeonasingleprocessor,wepresentapriority-driven overallcompletiontime,i.e.thetimeatwhichthelasttaskcompletes. priority-drivenheuristiccanprovideabetterworst-caseperformanceboundthanthe Inothersituationsallthetasksshareacommondeadline.Theproblemofnding onepresentedhere. knownpolynomial-timealgorithms[6][8]forschedulingtaskswithand-onlyprece- denceconstraints,identicalprocessingtimes,andarbitrarydeadlinesononeortwo processors.itisnaturaltoaskwhetherthecorrespondingand/orschedulingproblemsmaybesolvedinpolynomialtime.unfortunately,thisetendedproblemis NP-complete,evenwhenallthedeadlinesarethesame.Thisfactisepressedinthe followingtheorem. 3.1.SchedulingtoMeetDeadlinesonaSingleProcessor.Therearewell- 3.Unskippedproblems.Inthissectionwediscussthecompleityofthe Theproofisbasedonareductionfromeact3-cover(X3C).GivenahypergraphH tasksysteminwhichalltheortasksmustmeetacommondeadlineisnp-complete. =(V;E)of3nverticesandasetofhyper-edges,eachofwhichisincidenttothree vertices,theproblemistondasetofeactlynedgesthatcoversallthevertices withnooverlap.thisproblemisnp-complete[7]. Theorem3.1.TheproblemofAND/ORskippedorunskippedschedulingofa lemasfollows.createatasksystem(g;d)composedentirelyofunitprocessing-time Proof.ItsucestoprovethattheproblemisNP-completeonasingleprocessor. Theeact3-coverproblemcanbetransformedintoanAND/ORschedulingprob- v 1 v 3 v 5 v 2 v 4 v 6 e 2 e 4 e 3 T 1,2,3 T 4,5,6 T T T T T T T 1,3,5 T 2,3,4

5 AND/ORSCHEDULING 5 tasks.thereisanortasktiinthetasksystemforeachhypergraphverteviin H.Inthetasksystemall3nORtaskshavedeadline4n.CreateanANDtaskTi;j;k foreachhyper-edgethatconnectsvi,vj,andvk.thesuccessorsoftaskti;j;karethe ORtasksTi,Tj,andTk.Figure2isaneampleofthistransformation.Nowweask ifthereeistsascheduleinwhicheveryortaskmeetsitsdeadline.clearly,ifthe givenhypergraphhhasaneact3-cover,nandtaskscorrespondingtothecover mayeecuteinthetimeinterval[0;n],therebyallowingall3nortaskstocomplete bytime4n.ifnosuchcovereists,thenatleastn+1edgesmustbeusedtocoverthe hypergraph.henceatleastn+1+3ntimeunitsmustelapsebeforealltheortasks arecompleteregardlessofwhetherthisaskippedoranunskippedproblem.thus, ifaschedulerproducesafeasibleschedule,thenthereisaneact3-cover,andifthe schedulerfails,thennosuchcovereists. TheproofofTheorem3.1indicatesthatthisschedulingproblemisatleastashard asthen-dimensionalcoverproblem,ageneralizedversionofn-dimensionalmatching. Aboutthirtyyearsago,T.C.Hugaveapolynomial-timealgorithmtoschedulean AND-onlytasksystemwithin-treeprecedenceconstraintsonmprocessors[14].Thus, thereissomehopethatifwerestricttheand/or/unskippedtasksystemtohave in-treeprecedenceconstraints,theremayeistapolynomial-timealgorithm.unfortunately,thefollowingtheoremshowsthatthisand/orschedulingproblemis NP-complete. Theorem3.2.TheproblemofAND/OR/unskippedschedulingtomeetdeadlines, wheretaskshaveidenticalprocessingtimes,arbitrarydeadlines,andin-treeprecedence constraints,isnp-complete. Proof.Theproofiscontainedintheappendi. Corollary3.3.TheproblemremainsNP-completefortasksystemsinwhich onlytheortaskshavedeadlines. Proof.Theproofiscontainedintheappendi. TheproofsofTheorems3.2andCorollary3.1intheappendimakeuseoflong chainsofandtaskswithdieringdeadlines.wenowconsideraclassoftasksystems whereonlytwotasksinachainmayhavedeadlines.inasimplein-forest,(1)eachintreeconsistsofanortaskwithadeadline,nosuccessors,andtwodirectpredecessors, and(2)eachdirectpredecessorofanortaskhasadeadlineandistherootofan in-treeofandtaskswithnodeadlines(i.e.thedeadlinesareinnite).asimpleinforestrestrictstheallowableprecedenceconstraintsandallowabletaskswithdeadlines inatasksystem.wehavefoundnosimplernon-trivialcombinationofprecedence constraintsanddeadlines.surprisingly,eventhissimpliedand/orscheduling problemisnp-complete Theorem3.4.TheproblemofAND/OR/unskippedschedulingtomeetdeadlines, wherethetasksystemisasimplein-forestwithidenticalprocessingtimes,isnpcomplete. Proof.Theproofmaybefoundin[11]. Theorems allowustoarriveatthefollowingconclusion.EveryAND/OR taskgraphwithkortasks,eachofwhichhasldirectpredecessors,correspondsto asetoflkdierentand-onlytaskgraphs.afeasiblescheduleofsuchatasksystem correspondstoanimplicitselectionofoneoftheselkand-onlytaskgraphs.therefore,whenthereareo(logn)ortasksintheand/ortasksystem,itispossibleto enumerateinpolynomialtimethesetofallpossibleand-onlytaskgraphsandapply anoptimaland-onlyschedulingalgorithmsuchastheonedescribedin[8].onthe otherhand,theorems showthatmanynaturalschedulingproblemswitho(n)

6 6DeadlineLocationGeneralGraph OnAllTasks OnORTasksOnlyNP-C(Theorem3.1)NP-C(Corollary3.1)Trivial (a)schedulingtomeetdeadlineswithidenticalprocessingtimeson1processor. 2Deadlines CompleityofAND/OR/unskippedproblems. D.W.GILLIESANDJ.W.-S.LIU Table1 TaskProcessingTimeGeneralGraph Identical Arbitrary (b)schedulingtominimizecompletiontimeonmprocessors. NP-C(Theorem3.1)NP-C(Theorem3.2)NP-C(Theorem3.3) NP-C[15]forAND-onlyNP-C(Theorem3.4) Minimum-PathHeuristicMinimum-PathHeuristic In-Tree O(n)Deadlines In-TreeSimpleIn-Forest Input:TaskgraphG=(T;A;P) Step1:ForeachORtaskTiwithnoORpredecessors: Step2:TheresultingtasksystemhasonlyANDtasks.Schedulethistasksystemusinga priority-drivenalgorithmandanarbitraryprioritylist. (b)converttiintoanandtaskwhoseonlydirectpredecessoristk. (a)lettkbeadirectpredecessoroftithatminimizesthelongestpathendingattk. Inotherwords,Tk2P(G;Ti)andforallTj2P(G;Ti)withj6=k,L(G;Tj) L(G;Tk). temandthecompleityofthecorrespondingand-onlyschedulingproblem.these ORtasksareNP-complete.ItfollowsthatthecompleityoftheAND/OR/unskipped problemisdeterminedalmosteclusivelybythenumberofortasksinthetasksys- resultsaresummarizedintable1(a). worst-caseperformance.forthesimpleproblemstudiedintheorem3.3,wehave Itappearsdiculttodesignapriority-drivenschedulingheuristicwithgood Fig.3.Theminimumpathheuristicforgeneralgraphs. theworstcasethesealgorithmsmaymeetonlypndeadlineswhenitispossibleto ofthealgorithmsin[4]neglecttocomparethedeadlinesamongdierentin-trees.in algorithmssuchasfewestpredecessorsrst,leastslackrst,andsomegeneralizations meetnoutofn+1deadlines.formoreinformationthereaderisreferredto[11][12]. deadlinesandnon-deadlineinformation,oneisolatedin-treeatatime,mayperform pntimesworsethananoptimalalgorithm.someobviouspriority-drivenscheduling producedeamplestoshowthatanyalgorithmthatonlyconsidersslacksbetween processingtimes.however,hu'salgorithmsolvesthisprobleminpolynomialtime forin-treeprecedenceconstraints.unfortunately,theproblembecomesnp-complete benp-complete[15]forand-onlytasksystemswhereallthetaskshaveidentical ingtominimizetheoverallcompletiontime.ullmanhasshownthisproblemto whenortasksareallowed. lemofschedulingand/or/unskippedtaskswitharbitraryprocessingtimesonm processorstomeetacommondeadline.thisproblemisequivalenttothatofschedul- Theorem3.5.TheproblemofschedulinganAND/OR/unskippedtasksystem 3.2.SchedulingtoMinimizeCompletionTime.Wenowconsidertheprob- tominimizecompletiontimeonmprocessors,wheretaskshaveidenticalprocessing timesandin-treeprecedenceconstraints,isnp-complete. Proof.Theproofiscontainedintheappendi. InFigure3,wepresentaheuristicthatminimizesthecompletiontimeofan

7 andthecompletiontimeofthetasksystemaccordingtoascheduleproducedbythe canbeimplementedtorunintimeo(n+jaj)byreversingthedirectionofthearcs implicitand-onlygraphandthecompletiontimeofthetasksystemaccordingtoan optimalschedule.letg0=(t0;a0;p0)andw0denotetheimplicitand-onlygraph ingandemployingdepth-rstsearch.letgo=(to;ao;po)andwodenotethe AND/OR/unskippedtasksystemwitharbitraryprocessingtimes.Thebasicideais tochooseanand-onlygraphthatminimizesthelongestpathing.theheuristic AND/ORSCHEDULING 7 MinimumPathHeuristic,respectively.Theworst-caseperformanceoftheMinimum PathHeuristicdependsonthefollowinglemma. decessorsdierbetweentheoptimalgraphandthegraphproducedinstep1ofthe MinimumPathHeuristic.IfH=;,thentheAND-onlytaskgraphsareidenticaland Tj2HwithTj<TiinGo.BytheconstructionofG0,jP(G0;Ti)j=jP(Go;Ti)j=1. isusedinductivelytotransformgointog0withnoincreaseinthemaimumpath thelemmaisestablished.otherwise,letti2hbeataskforwhichthereeistsno WechangeAo,replacingthearc(P(Go;Ti);Ti)by(P(G0;Ti);Ti)andobtainnoincreaseinthelongestpath(bysteps1(a)and1(b)oftheheuristic).Thisargument Lemma3.6.L*(G0)L*(Go). length.thisestablishesthelemma. Proof.LetH=fTijP(G0;Ti)6=P(Go;Ti)gdenotethesetoftaskswhoseprecutesduringalltheidleperiods(whenoneormoreprocessorsarenotinuse),and thischainisnotlongerthanthecompletiontimeofanoptimalschedule. theminimumpathheuristic,thenwpl*(g0)l*(go)wobylemmas3.1 and3.2. givenbyw0=wo2?1=m.moreover,thisboundistight. Thefollowingfactisprovedinthewell-knownpaper[13]. Lemma3.7.Inanypriority-drivenschedule,thereisachainoftasksthateedrivenschedule.LetWpdenotethetotallengthofalltheidleperiodsinaprioritydrivenschedule.Duringtheidleperiodsatleast1andnomorethanm?1tasks formulatedasalinearprogram: Theorem3.8.Theworst-caseperformanceoftheMinimumPathHeuristicis Proof.LetWbdenotethetotallengthofallthebusyperiodsinapriority- eecute,andduringthebusyperiodseactlymtaskseecute.itshouldbeclear IfWpdenotesthetotallengthofalltheidleperiodsinascheduleproducedby thatw0=wp+wb.hence,theworst-casecompletiontimeofthisheuristicmaybe and[10].itisknown[10]thatnoand-onlypriority-drivenheuristiccanavoid2?1=m MaimizeWp+Wb=W0 processor,andsometimesintentionalidlingisneeded).ourpriority-drivenheuristic worst-caseperformance(becausepriority-drivenheuristicsneverintentionallyidlethe SolvingtheprogramyieldsWp=Wo,Wb=(1?1=m)Wo,i.e.W0=Wo2?1=m. EamplesofAND-onlytasksystemsthatachievethisboundmaybefoundin[2] subjecttomwb+1wpmwo WpL*(G0)L*(Go)Wo willscheduleand-onlytasksystemsasaspecialcase.hence,itisnotpossibleto getbetterworst-caseperformancefromanand/orschedulingalgorithmwithout abetterand-onlyschedulingalgorithm.infact,ithasbeenalong-standingopen problemtondabetterand-onlyschedulingalgorithm[15].

8 8DeadlinesLocationGeneralGraph OnAllTasks ONORTasksOnlyNP-C(Theorem3.1)NP-C(Theorem4.1)[17]Algorithm (a)schedulingtomeetdeadlineswithidenticalprocessingtimeson1processor. 1Deadline CompleityofAND/OR/skippedproblems D.W.GILLIESANDJ.W.-S.LIU Table2 TaskProcessingTimeGeneralGraph Identical Arbitrary (b)schedulingtominimizecompletiontimeonmprocessors. NP-C(Theorem3.1)NP-C(Theorem4.1)NP-C(Theorem4.2) NoAlgorithm NP-C[15](3=2OPT)NP-C(Theorem4.3) In-Tree O(n)Deadlines In-Tree Path-BalancingHeuristic SimpleIn-Forest whentheproblemsofsection3areformulatedintheskippedmodeltheyremain essentialpredecessorsofanortaskmaybeskippedentirely.werstshowthat NP-complete.Thenwepresentaheuristicalgorithmforschedulingtominimizecompletiontimeonmprocessors.Thisheuristicalgorithmworksforin-treeprecedence constraints,butnotforarbitraryprecedenceconstraints. 4.SkippedProblems.InanAND/OR/skippedschedulingproblem,thein- wheretaskshaveidenticalprocessingtimesandin-treeprecedenceconstraints,isnp- isnp-completeonasingleprocessor,therefore,weimmediatelyconsidersimplifying theprecedenceconstraints. AND/OR/skippedschedulingwithonedeadlineandarbitraryprecedenceconstraints Proof.Theproofiscontainedintheappendi. Theorem4.1.TheproblemofAND/OR/skippedschedulingtomeetdeadlines, 4.1.SchedulingtoMeetDeadline.Theorem3.1showedthattheproblemof wherethetasksystemisasimplein-forestwithidenticalprocessingtimes,isnpcomplete. ORtaskshavedeadlines.Forthistypeoftasksystem,analgorithmtondafeasible schedulecaneamineeachortaskandchooseasitsdirectpredecessortheandtask Proof.Theproofmaybefoundin[11]. Theorem4.2.TheproblemofAND/OR/skippedschedulingtomeetdeadlines, graphisscheduledusingtheearliestdeadlinerstrule.thismethodalwaysproduces afeasiblescheduleifthetasksystemisfeasible.ifthetasksystemisinfeasibleitis stillpossibletomaimizesthenumberofortasksthatsimultaneouslymeettheir anortasktogetherwithonepredecessorsubtreeconsistingofkiandtasksmaybe deadlinesandhaveessentialpredecessors.toproducesuchaschedule,wenotethat whichhasthefewesttotalpredecessors.afterthesechoicesaremade,theand-only Nowweconsiderthecasewherethetasksystemisasimplein-forestandonlythe thoughtofasonelargetaskwithprocessingtimeki+1.thenthealgorithmof[17], ashighasthecompleityoftheunskippedproblem.thisfactissummarizedintable withprocessingtime(ki+1),tomaimizethenumberofortasksthatmeettheir whichminimizesunitpenaltyonasingleprocessor,maybeusedtoscheduletasks deadline. 2(a). Insummary,wendthatthecompleityoftheskippedproblemisalwaysatleast

9 AND/ORSCHEDULING 9 Input:TaskgraphG=(T;A;P) Step1:ConverttheORtasksinthein-treeGintoANDtasks,toobtainanAND-onlygraph G0thatminimizesf(G0),asfollows. ForeachpathCi=fT1<T2<:::<TkgfromtheroottoaleafinGdobegin (a)[copyg]gc G. (b)[freezeortasksalongpathci]foreachortasktj2ciletac=(ac? P(Gc;Tj))[f(Tj?1;Tj)g(i.e.makeTjanANDtaskinGc). (c)[truncateallpathslongerthanci]letcj6=cibealongerpathingc.ifnosuch Cjeists,gotoStep(d).Otherwise,letTkbetheleastORtaskonCj.Ifnosuch TkeiststhengotoStep(f).ForeachTl2P(Gc;Tk)onapathlongerthanCi,do beginremovethearc(tl;tk)fromgcend.ifjp(gc;tk)j=0noand-onlygraph eistswithciasthelongestpath,sogotostep(f).elserepeatstep(c). (d)[minimizeprocessingtime]foreachortasktkwith2ormoredirectpredecessors andnoorpredecessorsinthegraphgc,pickasasolepredecessoroftkthetask Tj2P(Gc;Tk)suchthatforallTi2P(Gc;Tk)withi6=j,E(Gc;Ti)E(Gc;Tj). (e)iftheresultingand-onlygraphyieldsalesservalueoff(gc)thenletg0 Gc. (f)end. Step2:TheresultingtasksystemG0containsonlyANDtasks.Schedulethistasksystem usingapriority-drivenheuristicandanarbitraryprioritylist. Fig.4.Thepath-balancingheuristicforin-trees. 4.2.SchedulingtoMinimizeCompletionTime.Table2(b)givesthecompleityofschedulingmprocessorstominimizecompletiontime.Thenettheorem concludesourinvestigationintothecompleityofand/orscheduling. Theorem4.3.TheproblemofschedulinganAND/OR/skippedtasksystemto minimizecompletiontimeonmprocessors,wheretaskshaveidenticalprocessingtimes andin-treeprecedenceconstraints,isnp-complete. Proof.Theproofiscontainedintheappendi. Nowwepresentaheuristicalgorithmthatminimizesthecompletiontimeofan AND/OR/skippedtasksystemwithin-treeprecedenceconstraints.Letf(G)= E*(G)=m+L*(G)denoteafunctionofanAND-onlyprecedencegraph.Thisfunctionisanestimateoftheworst-casecompletiontimeofapriority-drivenschedule. OuralgorithmconvertsanAND/ORin-treeintoanAND-onlyin-treethatminimizes thisfunction.inageneralgraphitisdiculttominimizethisfunctionquickly.if m=1,apolynomial-timealgorithmtominimizef(g)couldbeusedtosolveany eact3-coverproblem(refertotheorem3.1),implyingp=np.becauseofthisthe PathBalancingHeuristicdescribedbelowisrestrictedtoin-treetaskgraphs.The algorithmappearsinfigure4. Thecompleityofthealgorithmcanbedeterminedasfollows.TheO(n)possible pathsfromtheroottotheleavescanbeenumeratedintimeo(n)usingdepth-rst search.eachiterationofthesteps1(a)-1(e)canbecarriedouttogetherino(n) timeusingarecursivedepth-rstsearch.mostoftheworkisdonewhenreturning fromprocedurecalls.hence,theoverallcompleityofthisheuristiciso(n2). Toderivetheworst-caseperformanceofthePath-BalancingHeuristicwebeginby showingthatstep1ofthisheuristicminimizesf(). Lemma4.4.f(G0)f(Go). Proof.ConsiderthelongestpathoflengthL*(Go)inGoThispathstartsatthe treerootandendsataleafverte.clearly,thepathbalancingheuristicconsiders thispathinsomeiterationofstep1.step1(c)oftheheuristicensuresthatno otherpathsarelongerthanthislongestpath,withoutincreasinge*(g0)morethan isnecessary.step1(d)oftheheuristicchoosesthedirectpredecessorsofeachor

10 givenby: tasktominimizee*(g0),thus,theheuristiccannotfailtondagraphforwhich 10 f(g0)isatmoste*(go)=m+l*(go). Theorem4.5.Theworst-caseperformanceofthePathBalancingHeuristicis D.W.GILLIESANDJ.W.-S.LIU (1)Moreover,thisboundistight. thetasksystemdividedbymprocessors,andalsonoearlierthanl*(go).hence Proof.Anyoptimalschedulecompletesnoearlierthanthetotalprocessingtimeof WomaE*(Go)+L*(Go) W0 Wo2?1m: AndbyLemmas3.2and4.1,wehave Hence W0E*(G0)=m+L*(G0)E*(Go)=m+L*(Go): ;L*(Go): WesimplifyEquation(2)intwocases. Case1.Themafgin(2)evaluatestoitsrstargument.Thenwehave WoW0 WE*(Go)+L*(Go)m mae*(go)+l*(go) E*(Go)+L*(Go)=B: E*(Go)=m+L*(Go) m ;L*(Go): E*(Go)=(m?1),sowehaveanupperboundonL*(Go).Thederivativeofthe Notethatthemafgin(2)evaluatestoitsrstargumentifandonlyifL*(Go) thus boundin(3)is Becausethederivativeof(4)isnonnegativeforallm1andE*(Go)0,amaimum of(3)occurswhenl*(go)isasgreataspossible,i.e.l*(go)=e*(go)=(m?1), W0 WE*(Go)(m?1)+E*(Go)m dl*(go)=e*(go)(m?1) dbe*(go)(m?1)+e*(go)=2?1m: 2(E*(Go)+L*(Go))0: onlyife*(go)l*(go)(m?1).wesubstitutee*(go)l*(go)(m?1)intothe numeratorof(2)toobtain(1). T1=fT1;T2;T4;1;:::;T4;m(m?1)=e+1gandletT2=fT2;T3;1;:::;T3;m;T5;1;:::;T5;mg. Case2.Themafgin(2)evaluatestoitssecondargument.Thisoccursifand TheeampleinFigure4demonstratesthatthisworst-caseboundistight.Let

11 AND/ORSCHEDULING 11 Fig.5.Aworst-caseAND/OR/skippedin-tree. (T 1,m ε) ThePathBalancingHeuristicchoosesbetweenthein-treesG1=(T1;A1;P1)and G2=(T2;A2;P2),whereA1andA2denotetheassociatedarcsets.Thelengths Furthermore,E*(G1)=E*(G2)=m2?m.Thus,thePathBalancingHeuristic ofthelongestpathsinthesein-treesarel*(g1)=l*(g2)=m+,respectively. isascheduleoflengthm+2forg2,buttheshortestpossiblescheduleforg1has lengthm+m(m?1)=m+wheneveredivides(m?1)evenly.asd!0,theratio oftheseschedulelengthsapproaches2?1=m. choosesarbitrarilybetweenthesetwotrees,sinceeitheroneminimizesf(g0).there tasksystemsismuchharderthantheproblemofschedulingand-onlytasksystems. problem,asdescribedintheproofoftheorem3.1,ona(3n+1)-processorsystem. WeaddtothetasksystemanANDtaskwith2n+1directpredecessors,andask ConsiderschedulinganAND/OR/skippedtasksystemderivedfromaneact3-cover correspondingtohypergraphverticestogetherwiththeotheraddedandtaskbegin tasksystemisfeasibleifntaskscorrespondingtoedgesinaneact3-covertogether ifthereisaschedulethatcompletesin2unitsoftimeon3n+1processors.the withtheadditional2n+1andtasksbeginprocessingattime0,andallthetasks WenowoeradditionalevidencethattheproblemofschedulingAND/OR/skipped ifthetasksystemisand-only,itisknown[15]thatnopolynomial-timeheuristic theirprocessingattime1.hence,thereisaschedulewithacompletiontimeof2if andonlyifthereisaneact3-cover.itfollowsthatunlessp=npnopolynomialtimeand/or/skippedschedulingheuristiccanguaranteeaworst-casecompletion AND/ORschedulingproblemwithdeadlines.Intheskippedvariant,sometasks timeoflessthan3/2timesthelengthofanoptimalschedule.incontrasttothis, canguaranteeaworst-casecompletiontimeoflessthanof4/3timesthelengthofan optimalschedule. theproblemwasshowntobenp-complete,evenfordrasticallysimpliedprecedence timeonmprocessors,andshowedthatitsworst-caseperformanceboundcannotbe Whentaskshaveidenticalprocessingtimes,deadlines,andthereisasingleprocessor, maybeleftunscheduled,butintheunskippedvariantalltasksmustbescheduled. constraints.wepresentedanecientpriority-drivenheuristictominimizecompletion 5.Conclusion.Wehaveanalyzedtheskippedandunskippedvariantsofthe { G 1 (T G { 2 (T 4,1,ε) 4,m(m 1)/ε + 1,ε) (T 5,1, m δ) (T 5,m,m δ) (T 3,2,δ) (T 3,m,δ) (T 3,1,δ) (T 2,δ)

12 12 D.W.GILLIESANDJ.W.-S.LIU improvedbyusingadierentpriority-drivenheuristic.wealsopresentedaheuristic tominimizethecompletiontimeofanand/or/skippedtasksystemwithin-tree precedenceconstraints.wederivedtheworst-caseperformanceforthisalgorithmand eplainedwhythealgorithmcannotbeetendedtohandlegeneraltaskgraphswith thesameperformanceunlessp=np. Throughoutthispaperweassumedthatonlyonedirectpredecessortaskhadto becompletedbeforeanortaskcouldbegin.underamoregeneralassumption, ORtaskTicanbeginoncekipredecessortasksarecomplete.Thealgorithmsand theoremsinthispaperrequireminormodicationstohandlethismoregeneralcase. ThereisalsoasimilarAND/ORmodelwhereindividualarcs(andnottasks)can beandarcsororarcs.byusingtaskswithaprocessingtimeofzero,ourmodel cansimulatethisothermodel.therearealsosituationswherebothor/skippedand OR/unskippedtasksarepresentinasinglein-tree.Withslightmodicationsour AND/OR/skippedheuristiccanbeusedtohandlesuchmiedtasksystems.Details ofthesetransformationsandalgorithmsappearin[12]. Duringthisinvestigationwereachedseveralconclusionsaboutthecompleityof AND/ORscheduling.Contrarytoourintuition,theskippedproblemsweconsidered weregenerallyofhighercompleitythanthecorrespondingunskippedproblems.this canbeseenbycomparingtable1andtable2,andtheproofsintheappendi.in theproblemofschedulingtomeetdeadlines,wehaveseveralobservations.itwas generallynothelpfultorestrictthein-degreeofortasksinthetaskgraph.itwas alsonothelpfultorestrictdeadlinestoonlytheortasks,ortorestrictthetaskgraph tobeanin-treeoranin-forestorevenasimplein-tree,thesimplestrelationpossible forthistypeofproblem. A.Appendi.ThisappendipresentstheproofsofTheorems3.2,3.4,4.1,4.3, andcorollary3.1.proofsoftheorems3.3and4.2maybefoundinboth[11]and [12].Eceptwherenoted,allproofsrefertotheschedulingofasingleprocessor. Theorem3.2.TheproblemofAND/OR/unskippedschedulingtomeetdeadlines, wheretaskshaveidenticalprocessingtimes,arbitrarydeadlines,andin-treeprecedence constraints,isnp-complete. Proof.Ourproofisbasedonareductionfrom3SAT.Givenaninstanceofa3SAT problem,withkbooleanvariablesandnclauses,wewillcreatekortasks.for eachvariableiwhichoccursinliclauseswecreateanin-treecontainingoneor taskandtwochainsoflengthli.onechaincorrespondstotruthfortheassociated variable,andtheothercorrespondstofalsity.therefore,thereare3ntasksinall chainscorrespondingtotruth,and3ntasksinallchainscorrespondingtofalsity. TheORtasksaregivendeadlinesofe=3n+k.AneampleisshowninFigure6. Thiseampleisanin-treeforavariablethatappearsin4clauses.Deadlinesare depictedaboveorbelowthetasks.becauseofthedeadlinesoftheortasks,inany feasibleschedulekortasksandkchainseecutethroughoutthetimeinterval[0;e], andnoothertasksmayeecuteinthisinterval.thisleavesktaskchainstoeecute inthetimeperiod[e;e+3n]inafeasibleschedule. Foreach3SATclauseweassignanintervalofthreetimeunitsstartingattimee. Hencethetimeintervals[e;e+3];[e+3;e+6];:::;[e+3n?3;e+3n]correspond toclause1,clause2,:::,clausen,respectively.eachintervaloftimeisdivided intotwoparts.inthersttwotimeunits,tasksinleftoverchainscorresponding totruthorfalsityinaclausemayeecute.inthethirdtimeunit,onlyatask correspondingtotruthmayeecute.toenforcethisrule,wegivelaterdeadlinesto thetasks/termsthatwouldmakeeachclausetrue.infigure6,variableoccursin

13 AND/ORSCHEDULING 13 e+12 Falsity e+5 e+8 e+11 e e+3 Truth e+6 e+9 e+2 Fig.6.Anin-treeforavariableappearingintherst4clauses. therst4clausesofthe3satepression.itappearsuncomplementedinclauses1 and4,andcomplementedinclauses2and3.ifappearsinthe3satepression forthei'thtimeasanuncomplementedvariableinclausej,thedeadlineforthei'th taskinthetruthpredecessorchainise+3j,andise+3j?1forthei'thtaskinthe falsitypredecessorchain.thesedeadlinesareechangedifthei'thappearanceof isasacomplementedvariableinclausej.wegivealltheortasksacommonand successorwithadeadlineofinnity,toformasingletree. Ifaschedulingalgorithmndsafeasibleschedule,theneachtaskthateecutesin theinterval[e+3j?1;e+3j]correspondstoavariable(oracomplementedvariable) thatistrueinclausej.ifthevariablewerenottrue,thenthedeadlineofthetask wouldhaveepiredonetimeunitearlier.furthermore,thetaskchainsguaranteethat thetruthorfalsityofavariableisconsistentamongdierent3satclauses.thus,a scheduleisfeasibleifandonlyifthereisasatisfyingtruthassignment. Alltheotherproofsinthisappendiandin[11]and[12]aremodicationsofthe proofoftheorem3.2.inparticular,theorems3.3and4.2requirealargesimple in-treeforeachtermina3satepression,andhavebeenomittedforbrevity. Corollary3.1.TheproblemremainsNP-completeifonlytheORtaskshave deadlines. Proof.WemakethefollowingchangestotheproofofTheorem3.2:replacethe in-treesofthetypedepictedbyfigure6bynewin-treessuchastheoneinfigure7. ThisisdonebyaddinganANDtaskwithadeadlineofetothebeginningofeach truthandfalsitychain,convertingeachandtaskwithadeadlineintoanortask withoneortwoetraandpredecessortasks,andsettinge=3n+5k.asinthe previousproof,thelastdeadlineassociatedwithavariableinaclauseise+3n.note thatthereareeactlye+3ntasksofthetypethatareshadedinfigure7,inthe entiretasksystem.becuaseoftheirdeadlines,theshadedtasksmusteecuteinthe timeinterval[0;e+3n],andtheunshadedtasksmusteecuteaftertimee+3nin anyfeasibleschedule. Itisnotdiculttoverifythatinafeasibleschedulethetasksthateecuteinthe timeinterval[e;e+3n]correspondtoasatisfying3sattruthassignment. Theorem4.1.TheproblemofAND/OR/skippedschedulingtomeetdeadlines, wheretaskshaveidenticalprocessingtimes,arbitrarydeadlinesontheortasksonly, andin-treeprecedenceconstraints,isnp-complete. Proof.Thistheoremetendsthepreviouscorollarytoskippedtasks.Weuse nearlythesamein-treesasincorollary3.1.however,wesete=2k,wegivethe ORtasksattherootofeachin-treeadeadlineofe+3n+kratherthane,andwe replaceeachunshadedtaskbyachainofeandtasks.itisnotdiculttocheck thatthetaskchainsthateecuteinthetimeintervals[e+3j?1;e+3j],1jn, correspondtoatruthassignmentsatisfyingthe3satclauses.

14 14 D.W.GILLIESANDJ.W.-S.LIU Fig.7.Anin-treeforschedulingwithdeadlinesonORtasksonly. e+5 e+8 minimizecompletiontimeonmprocessors,wheretaskshaveidenticalprocessingtimes andin-treeprecedenceconstraints,isnp-complete. asystemwithm=k+1processors.foreachvariableiwecreateanin-treewith Theorem4.3.TheproblemofschedulinganAND/OR/skippedtasksystemto Proof.Givena3SATproblemwithkbooleanvariablesandnclauses,wespecify Fig.8.In-treetasksystemforAND/OR/skipped/unskippedtasksonmprocessors. acommonanddirectsuccessortk+1.foreach3satclauseweassignanintervalof correspondstotruth,andtheothercorrespondstofalsity.allthetaskst1:::tkhave 3unitsoftimestartingattimezero.Hencetheintervals[0;3];[3;6];:::;[3n?3;3n] oneortasktiattherootandtwopredecessorchainsoflength3n+1.onechain correspondtoclause1,clause2,:::,clausen.ifavariableiappearsuncomplemented of3n+3,thenbyinterchangingtasksamongdierentprocessors,wecantransform (complemented)inclausej,wecreatetwoandtaskst*i;jandt** multipledeadlines,whicharenotallowedbytheproblemstatement. successorsti;3j+1andt0i;3j(ti;3jandt0i;3j+1)respectively.figure8illustratesthe inthesecondclause.thepredecessorchainsoflength3n+1areusedtosimulate rst,second,andfourthclausesofthe3satprobleminstance,andiscomplemented transformationfora3satproblemwithn=4clauses.thevariableappearsinthe Ifaschedulingalgorithmndsafeasibleschedulewithanoverallcompletiontime i;jandmaketheir theschedulesothatprocessorsonethroughkeecuteatruthorfalsitychainoflength 3n+1inthetimeinterval[0;3n+1],andprocessork+1eecutesonlytasksoftypeT*i;j ort** i;jinthesametimeinterval.theneachtaskthateecutesinthetimeinterval Truth T 1,1 T 1,2 Falsity T' 1,1 T' 1,2 T * 1,3 T ** 1,3 Truth Falsity _ e e e+3 e+2 [0,3] [3,6] [6,9] e+6 e+9 [9,12] e+12 e+11 T 1,3n T' 1,3n e T 1 T 2 T k T k+1

15 AND/ORSCHEDULING 15 [3j?1;3j];1jn,onprocessork+1correspondstoavariableorcomplemented variablethatistrueinclausejofthe3satprobleminstance.becauseonlyonetruth orfalsitychainforeachortaskeecutesinthetimeinterval[0;3n+1],thetruth orfalsityofavariableisconsistentamongdierent3satclauses.thus,afeasible schedulecanbefoundifandonlyifthereisasatisfyingtruthassignment. Theorem3.4.TheproblemofschedulinganAND/OR/unskippedtasksystem tominimizecompletiontimeonmprocessors,wheretaskshaveidenticalprocessing timesandin-treeprecedenceconstraints,isnp-complete. Proof.TheproofisnearlyidenticaltotheproofofTheorem4.3.Givena3SAT problem,wegeneratethesamein-treeasintheproofoftheorem3.4,eceptweadd achainof6n+6andsuccessorstotasktk+1.thenweaskifthereisaschedule withanoverallcompletiontimeof9n+9.insuchaschedulektaskchainswithout essentialtaskshaveplentyoftimetocompleteinthetimeinterval[3n+3;9n+9].it isnotdiculttoseethattherearetasksthateecuteinthetimeintervals[3j?1;3j], 1jn,thatcorrespondtoasatisfyingtruthassignment. Acknowledgement.WewishtoacknowledgeMohlaSeka,whoimplemented thepathbalancingheuristicandimproveditsdescription,andalsosandrabroadrick- Allen,whohelpedtoimprovelaterversionsofthispaper. REFERENCES [1]P.-R.Chang,ParallelalgorithmsandVLSIarchitecturesforroboticsandassemblyscheduling, Ph.D.thesis,PurdueUniversity,WestLafayette,IN,1988. [2]E.G.Coffman,Jr.,ed.,ComputerandJobShopSchedulingTheory,JohnWiley,NewYork, NY,1976. [3]J.Y.Chung,J.W.-S.Liu,andK.J.Lin,SchedulingPeriodicJobsThatAllowImprecise Results,IEEETrans.Computers,39(1990),pp [4]E.G.Coffman,Jr.,J.Y.Leung,andD.W.Ting,Binpacking:maimizingthenumberof piecespacked,actainformatica,9(1978),pp [5]L.S.HomemdeMelloandA.C.Sanderson,AND/ORgraphrepresentationofassembly plans,proc.aaai(1986)pp [6]M.R.GareyandD.S.Johnson,Two-processorschedulingwithstart-timesanddeadlines, SIAMJ.Comput.,6(1977),pp [7]M.R.GareyandD.S.Johnson,ComputersandIntractability:aGuidetotheTheoryof NP-completeness,W.H.FreemanandCo.,SanFrancisco,CA,1979. [8]M.R.Garey,D.S.Johnson,B.B.Simons,andR.E.Tarjan,Schedulingunit-timetasks witharbitraryreleasetimesanddeadlines,siamj.comput.10(1981),pp [9]D.W.GilliesandJ.W.-S.Liu,Greedinresourcescheduling,Proc.IEEEReal-TimeSystems Symposium,10(1989),pp [10]D.W.GilliesandJ.W.-S.Liu,Greedinresourcescheduling,ActaInformatica,28(1991), pp [11]D.WGillies,andJ.W.-S.Liu,SchedulingTaskswithAND/ORPrecedenceConstraints, Rep.No.UIUCDCS-R (UIUC-ENG-1766),DepartmentofComputerScience, Univ.ofIllinois,Urbana,1991. [12]D.WGillies,AlgorithmstoscheduletaskswithAND/ORprecedenceconstraints,Ph.D.Thesis,DepartmentofComputerScience,Univ.ofIllinois,Urbana,1993. [13]R.LGraham,Boundsonmultiprocessingtiminganomalies,SIAMJ.Appl.Math.,17(1969), pp [14]T.CHu,Parallelsequencingandassemblylineproblems,OperationsRes.,9(1961),pp [15]E.L.Lawler,J.K.Lenstra,A.H.G.RinnooyKan,andD.B.Shmoys,Sequencingand Scheduling:AlgorithmsandCompleity,Rep.No.BS-R8908,CentreforMathematicsand ComputerScience,Amsterdam,Holland,1989. [16]M.C.McElvany,GuaranteeingdeadlinesinMAFT,Proc.IEEEReal-TimeSystemsSymposium,9(1988),pp

16 16 [17]J.M.Moore,Annjob,onemachinesequencingalgorithmforminimizingthenumberoflate [19]V.SaletoreandL.V.Kale,ObtainingrstsolutionfasterinANDandORparalleleecution [18]D.PengandK.G.Shin,Modelingofconcurrenttaskeecutioninadistributedsystemfor oflogicprograms,northamericanconferenceonlogicprogramming,1(1989),pp.390- jobs,managementsci.,15(1968),pp real-timecontrol,ieeetrans.computers,36(1987),pp D.W.GILLIESANDJ.W.-S.LIU