In multi-tasking systems when it is not possible to guarantee completion of all activities by specified

 times, the scheduling problem is not straightforward.

Examples of this situation in real-time programming include the occurrence of alarm conditions and the buffering of output to peripherals in on-line facilities. The latter case is studied here with the hope of indicating one solution to the general problem

Haworth, Guy McCrossan

English

Central Archive at the University of Reading

ComputingwithREAL-TIME SYSTEMSVolume2Proceedingsof theSecond European SeminarUniversityof Erlangen- NurnbergEditedbyI.C. PYLEAERE HarwellandP. ELZER.. Universityof Erlangen@TRANSCRIPTA BOOKS30CffwenStreet,Strand, London WC2~Process schedulingbyoutputconsiderationsG.McC.HAWORTHUniversityof Cambridge,EnglandIntroductionIn multi-taskingsystemswhenit is notpossibleto guaranteecompletionof all activitiesby speci-fied times, theschedulingproblemis notstraight-forward. Examplesof this situationin real-timeprogrammingincludetheoccurrenceof alarmconditionsandthebufferingof outputoperipher-als in on-linefacilities. The latter caseisstudiedherewiththehopeof indicatingonesolu-tionto thegeneralproblem.Threeparametersare associatedwitheachjobJi (i =1,... ,n). Thesearetheprocessingtimeestimatep., thedue-datedi' anda positiveweighting"/..wi' Threeproblemareasaredistin-guished: .1. Theunweightedeterministiccase(UDP)where Pi is theknownprocessingtimeandwi =1 for each Ji'2. Theweighteddeterministiccase(WDP)wherewehavegeneralweightswi andknownprocessingtimes Pi'3. The stochasticcase(SP), wherethepro-cessingtimeof Ji is a randomvariablewithanexponentialdistributionandmeanPi' Generalweightscausenodifficultyinthis case, butweareparticularlyinterestedin sub-optimalschedulesas theoptimalschedulecannotbederivedentirelyinalgebraicterms.Givena schedule5, Ji will becompletedattime ci (5): thecostwhichweseekto minimiseis nT =Expectation !~w.x max[O,I i=1 "/..andjobsareindexedsothatc.(5) - d.] ~"/.. "/.. ~. . => -1 -1"/..<J p.W. <p.w, or"/.."/.. J J~1 -1(p .W. =p .W" d. <d.)"/.. "/.. J J "/.. JThe full derivationof thealgorithms,whichare original, canbefoundin [4].TheunweightedeterministicproblemThecostin thedeterministiccasesis minimisedby sequencingthe Ji withoutpre-emptionor idle-time [1, p.24L Whenwi=1, theiterativealgo-rithm onp.76 computesthebestsequence.Thismethodis moreconvenientthanthebranch-and-boundapproachof [2] butspringsfrom thesamesource, namelyTh. UDld1<max(d2,P2) <==>J 1(Fig. 1)precedes J 2 when n =2Th. UD2i <j, d. <max(d.,p.) =>J. precedesJ."/.. J J "/.. J(for any 'n')dZ.'.'..'..'.'t'run Jl run JzdlFig.1An 'eligible' job is onewhichis notexcludedfrom thenextpositionby TheoremUD2. ThealgorithmformstheLEJ (longesteligiblejob)sequenceby executingthelongesteligiblejobateachstage. The LEJ sequenceis certainlyasgoodas theEDD (earliestdue-date)andLST(leastslack-time)sequences,andconnotbeimprovedbyinterchangingjobsadjacentin thesequence.75Algorithmfor theUDPglobal proc, due, best sequence [l:n] ;comment 'procl and 'duel hold the processing time and due datesrespectively of the jobs l, ,n;{commentthis block computesthe best sequence [l:n] fromproc, due [l:n] for the unweighted deterministic problem;~ tard, done, seq [I :n] , best cost~.deferredjob,start of tai I, in doubt;form LEJ sequence (l,n); best sequence := seq; best cost := cost;checktail;while in doubt ~ {try to improve best sequence; checktail}}form LEJ sequence (start, limit) ~{commentthis fills places start,...,n with job indices ~ limit;local i, 1, pr, place; pr:= 0;if start 'f 1 then i2.!: i := 1.!2 start-l ~ pr := pr+proc[seq[i]];for place := start to n do---{comment this p~s the longest eligible job in place;i :=1; whi 1e done[ i] >0 do i :=i+1; 1: =i+1;whi le 1~t and proc[l]< due[i] - pr do--rIfdone[l]~ and due[IJ<dueIi] th;n i := 1; 1 := 1+1};seq[PJace]:= i; pr := pr + procIi]; -tarde;] := max(0, pr-due i);.!!. donee;]=-1 then limit:= n; dor.e[i]:=l}}cost ~ ~ ta rd[ ;]- i=lcheck tail ~{commentthis looks for the shortest tail sequencethat mightnot be in the optimal order;local place, prefer;in doubt := false; place:= n; prefer:= n+l;while not in doubt and place>O do~one[seq[place]] := 0;if seq[place] >prefer then--{in doubt := true; deferred job := seq[place];start of tai~ place}else if tard[seq[place]] > proc[seq[place]] then- -- {done[seq[place)):=-1; -if seq[place] < prefer then prefer := seq[place]}place := p];ce - I}} -try to improvebest sequence ~{commentthis defers the first job in the doubtful tail sequence;form LEJ sequence (start of ta i I, deferred job -1);.!!. cost <best cost then {best cost := cost; best sequence := seq}}76> 2."1..""" ,.'_2!.3~,.::::"~2~Y-- - - _2~3S- - - - - - - - - - -211y.".,' /./;..://2221/6 STOCHASTICPOLICIESIIIIIIIIIIIIII 1131IIL.----33314dZ111120 4///2If W= {i!c.(LEJ) - p. >d.. 3j > i s.t. J.1- 1- 1- JprecedesJ. in LEJ }1-then the minimal cost is not less thanT (LEJ) - 2: [e.(LEJ) - p'.- d.].iEW 1- 1- 1-Hence, W=q,> is a sufficientbutnotnecessarycondition for the LEJ sequenceto be optimal, andit is usually satisfied.If ei(LEJ) - p...;; di for all i, then LEJcoincides with EbD andis optimal. Thus the EDDrule, which is knownto be best if each Ji can becompletedby di' is seen to havefar wider opti-mal properties.The algorithm has beenrun on collections of100 job-sets for various 'n',with Pi , di randomin [0,1] and IO, in] respectively. For n = 4,W=q, 95times (outof 100)andLEJ wassub-opt.3 times.For n = 8,W= q, 77 times and LEJ was sub-optimal 7 times.. > For n = 16,W==q, 28times, LEJ was sub-optimal 26 times,and on average, the algorithm performed onlytwo iterations after the initial LEJ. .The weighteddeterministicproblemThe addition of general weights 1iJi complicatesi>3333,..../3332/2222KEYOPTIMAL,""h 'h"""''''''' SUB-OPT.A11 B11 CVALUESPl=l pz=eP3=ez wi=l d3=Odl6 8the problem considerably, andno methodmatch-ing theefficiencyof theOOP algorithmis possible.Onemethod[5] hasbeenproposed,butit isunwieldyfor our purposesandhas,beenshowntobe sub-optimalbya three-jobcasestudy [1,pp.46-8L .'.Correspondingto Th. 001, wehave,whendI' dz ..;;PI + PzTh. WDl -1dI ..;;max[P2+ (l-1iJI 1iJz)PI .1iJ~I"'2dz+ 1iJ~I("'1-"'2)(PI +pz)]-- JI precedes J2 when n=2 (see Figs.2,3)butthis doesnot leadto theanalogousform ofTh.OO2.dZrunJI run J2N~1\....~ '1dIFig. 277dZrun Jl run JzN:;tv.....:;tdlFig.3The currentgoalis tofindanalgorithmwhichwill reduceto theUDP algorithmof p.76whenwesetthegeneralweightsto 1. In this directionwehave:i<j<k: d. 0;;;p. + (l-lV-:IlV.)p.'Z- J 'Z- J 'Z-=> d. 0;;;Pk + (l-lV-:IlV k)P''Z- 'Z- 'Z-analogousto d. <p. =>d.<Pk'Z- J 'Z-It is a remarkablefactthatgeneralweightsaremoreeasily includedin a stochasticstudythaninthedeterministicstudy,andwecanfor themomentonlyapetheLEJ sequence,leavingthequestionof suitableoptimalitycriteria andaniterativeprocedureopen.Th. WD2dZpreferencecyClicWi = 1d3 = 0dlFig. 4The dynamicprogrammingprincipleofoptimal-ity givestheequation- C(S,t) = es(t) + min p:I[C(Si,t)-C(S,t)]iES 'Z-where SC{I, ,n}; S. =S-{i};Z = {iliES, d. o;;;t; es(l;)= 1; lV. and C(S,t) is'Z- iE Z 'Z-theexpectedadditionalfuturecostat time t,giventhatwefollowtheoptimalpolicyandhaveto schedulethe jobs Ji, i ES .It is usefulto knowthattheform of thecostfunctionisnC =Z(t) + L: a.(t).exp(t/p.)i=l 'Z- 'Z-wherethecoefficientsZ(t), ai(t) are constantin half-openintervals [c,d). There is onlyafinitenumberof theseintervals.global p, d, w, seq[l:n];local done[l:n], i, £, place, pr ;pr := 0; for i :=1to n do done[i] :=~;for place := 1 to n do '{i ':= 1; while done[i] do i := i+1; £ := i+1;while £~n and W.Po + (w.-wo)p. <w.(d.-pr) do- -I" ,,, I I' -{if not done[£]and wo(do-pr)+(w.-wo)(P.+Po)<w.(d.-pr)-- -"" I"'" 'Ithen i :=£;£,:=£+1}seq[place] := i; pr := pr +proc[;]; done[i] :=~ }}ThestochasticproblemWeaddconditionsofuncertaintyto theprobleminthemostconvenientwaybyassumingthattheprocessing~timeof Ji is a randomvariableexponentiallydistributedwithknownmeanPi'Theprobabilityof Ji terminatingdoesnotthendependonthetimefor whichit hasalreadyrun,andso this simplificationleadsto a Markovsys-tem. More generalprobabilitydistributionsandcostfunctionsare treatedin [3] for thecasen =2.78Theresult for n=2 isTh.S1dl 0;;;dz + PI loge(lVIPZ)/(lVzPI) <= run JI(J I < J z)Notethatwiththreejobs, thesituation J I <Jz,Jz < J 3 andJ3 < JI mayarise! (SeeFig. 4.)Thebest-decisionareasfor wi = 1, d3=0and Pi=exp(i-l) are depictedin thegraphonp.77. It is conjecturedthatthedecisionattimet';;; d3 is the decision at t =d3 for n =3, inwhichcasethegraphis sufficientrepresentationof thesolution.Sincenumericaltechniquesare involvedinfindingthedecisionareas, weturnto morecon-venientsub-optimalpoliciesfor implementation.Sub-optimalstochasticpoliciesThe graphonp.77 compare~theoptimaldecisionrule withthreesub-optimalrules. Thequartet'ijkZ'in eachregionmeans"choice'i' withopti-mal rule, choice'j' withrule A, 'k' withBand'z'withC"...A. If weinsistthatwhenweswitchfromJj to Jiusingthis rule, Jj is thebestchoicein theabsenceof Ji' andvice versa, thentheboundariescanbecalculatedalgebraically. Wheretheydiffer from thoseof theoptimalpolicyand.thoseof policyB below,theyare markedbydottedlines. Figure 5 indicateshowtheorder ofpreferenceof thejobsat anyonetimechangesina simplecase.B. With i < j, define Ji > Jj asd. - d.>p. . log [(w.p.)/(w.p.)]'Z- J 1.- e 1.- J J 1.-Z={jIJ.>J.'rJi<j}3kEZ s.t. k;;;'j 'rJjEZ1.- JThedecisionof rule B is to run J k' andthisis analogousto theLEJ sequencein thedetermin-istic case..C. Pi+di actsas anindexwhenall wi = 1.With rule C, werun Jk wherePk +dk.;;;Pi +di 'rJi.D. Also whenthe 1J)i= 1,wemightapplythedeterministicrule. This is not shownonthegraphas it simplydividestheareadepictedbetweenJ 1 and J2.Rule B is thebestfor thepurposesof schedul-ing in real-time systemsas it is applicablewithgeneralweights,quickto calculateandin closeagreementwiththeoptimaldecisionrule. Inmanyapplications,includingtheonestudiedinthenextsection,theothermodellingassumptionsbeingmadewill makethedifferencebetweenrule B andtheoptimalrule insignificant.Oneapplication:anon-linefacilityTheprimarypurposeof processorschedulinginanon-line systemis to maintaintheillusion foreachuser thatthecomputeris dedicatedto him.Mostof theschedulingalgorithmsproposedandimplementedare basedonresourceconsidera-tionsonlysuchas timequanta,timealreadyspentona process, or - if theprocessis notrepresentedin core- storagerequirements.The illusion mentionedabovewill beachievedif all outputchannelsare keptbusy,andtheproposalof this sectionis thatwecancomecloser to thatgoalbymonitoringtheoutputper-formanceof eachprocess.The on-line systemmodelledhereis oneinwhich 'n'users Ui at separateconsoleshaveeachsubmitteda task J.. Thenumber'n' varieswithtime, butwemake'Z-noattempto anticipatethearrival of newtasks: later arrivals pre-emptthecurrentlyrunningprocessif necessary. Anamountdi (t) of outputhasbeencreatedby Jibutnot receivedby ui, andthis backlog,meas-uredbythetime it will taketo clear, givesus adue-datefor J.. A furthertime d.(t) mayelapsebefore1.-JiproducesmoreoJtputto main-tainthecontinuityof thedatastream. Pi is anestimateof thetimerequiredby Ji to produceaunit of output,anda varietyof simpleheuristicsare availableto decidethis parameter. Theweightwi mightrepresentanamalgamationoffactorssuchasthestatusof Ui' thetimealreadyreceivedby Ji andtheoutputalreadyreceivedby U..The stoch~tic modelwithsub-optimalpolicy'B' seemsappropriategiventheapproximationsabove. Since P, !!1,4 and 'n' are all functions oftime, thealgorithmmustbequantisedin somewayto maintainCPU utilisationandcontrolhousekeepingoverheads.Fig. 5 Illustratinghow the preferenceordercan changewith sub-optimalstrategyA in thestochasticcase.The calculationsaredone in backwardstime.790 d4 d3 dz dlI I I I I I I I I I I I )04 3 3 3 2 2 2 2......1 13 4 42/3 3 3At"1 2 22 2 2 4 4 4.,..1 3 3 31 1 1 1 1 1 4 4 4 4Acknowledgement DiscussionI shouldliketothankProfessorJ. F. CoalesandDr J. D. Robertsfor supportingthiscontributiontotheSeminar,andP. Nashfor thestimulusofa continuingdialogueontheramificationsofthestochasticproblem.Q. Is thealgorithmfor theWeightedDeterminist-ic Problemlinear?~. No, theexecutiontimeis proportionaltoN logN. (Similarlyfor theUnweightedDetermi-nistic Problem.) .1. CONWAY,R.W.,MAXWELL,W.L.,andMILLER,L.W., Q. Whathappenswhena newjob is added?'Theory of scheduling',Addison-WesleyPublishingCorporation (1967).2. EMMONS, H., 'Onemachinesequencingto minimisecertain functionsof job tardiness', OperationsResearch,V17, pp.701-715(1969).3. GITTINS, J.C., 'Optimalresource allocation in chemicalresearch', Advancesin Applied Probability, V1, Q. Howdoesyourmethodcomparewithapurepp.238-270(1969). ...t h ?4. HAWORTH, G.McC., 'Tardiness schedulingandcomputerpnorl y sc eme.systems', CambridgeUniversity EngineeringDepartment ..TechnicalReportCUEDjB-ControljTR22(1972). A. It works as thougheachJob has a gradually5. SClfiLD, A., andFREDMAN,I.J., 'Schedulingtaskswith increasing priority.deadlinesandlinearldssfunctions',ManagementScience,V7, pp.280-285(1961).A. Its priority hasto bedecided,andthenthealgorithmusedto determinetheschedulefor thenewsetof jobs.80- ~---.-----C

Process scheduling by output considerations

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