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SchedulingFixed-Priority Taskswith PreemptionThr eshold�
Yun Wang
Departmentof ComputerScience
ConcordiaUniversity
Montreal,QC H3G1M8, Canada
ManasSaksena
Departmentof ComputerScience
Universityof Pittsburgh
Pittsburgh,PA 15260,USA
Abstract
While it is widely believedthat preemptabilityis a necessaryrequirementfor developingreal-time
software, there are additional costsinvolvedwith preemptivescheduling,as comparedto non-preemptive
scheduling. Furthermore, in the context of fixed-priority scheduling, feasibility of a task set with non-
preemptiveschedulingdoesnot implyfeasibilitywith preemptivescheduling(andvice-versa).In thispaper,
weusethenotionof preemptionthresholdto developa new schedulingmodel,which unifiestheconceptsof
preemptiveandnon-preemptivescheduling,andsubsumesbothasspecialcases.
Thenotionof preemptionthresholdwasintroducedby ExpressLogic, Inc. in their ThreadXreal-time
operatingsystem.It allowsa taskto disablepreemptionfromtasksupto a specifiedthresholdpriority. Tasks
havingpriorities higherthanthethresholdare still allowedto preempt.In our new model,each taskhasa
regular priority anda preemptionthreshold.We showhowthenew schedulingmodelcanbeanalyzedfor
responsetimes.Wealsodevelopalgorithmsfor optimalassignmentof priorities andpreemptionthreshold.
In thispaperwealsoprovideevidencethatthisnew schedulingmodelprovidessubstantialquantitative
benefitsover both preemptiveand non-preemptiveschedulingmodels. We showthat the new modelcan
resultin substantialimprovementof schedulabilityby takingadvantageof aspectsof bothpreemptiveand
non-preemptivescheduling. Furthermore, we showthat the new modelprovideslower run-timecostsby
eliminatingunnecessarypreemptions.�This work waspresentedin partat RTCSA’99, Hongkong,December1999anda brief versionof this work waspublishedin
theconferenceproceedingsof RTCSA’99 by theIEEE ComputerSocietyPress.
1 Intr oduction
Sincethepioneeringwork of Liu andLayland[14], muchwork hasbeendonein theareaof real-time
scheduling,andin the analysistechniquesto a priori predictschedulabilityof tasksetsundera particular
schedulingdiscipline. In particular, significantprogresshasbeenmadein schedulabilityanalysisof task
setsunderfixed priority preemptive scheduling(e.g., [8, 9, 11, 12, 18]). The benefitsof fixed priority
preemptiveschedulingincluderelatively low run-timeoverheads(ascomparedto dynamicpriority schemes,
suchasearliestdeadlinefirst) andability to supporttighterdeadlinesfor urgenttasks(ascomparedto non-
preemptivescheduling).
Whilepreemptabilityis oftennecessaryin real-timescheduling,it is fallaciousto assumethatit always
resultsin higherschedulability. Indeed,it canbe shown that, in the context of fixed priority scheduling,
preemptiveschedulersdonotdominatenon-preemptiveschedulers,i.e.,theschedulabilityof atasksetunder
non-preemptive schedulingdoesnot imply the schedulabilityof the tasksetunderpreemptive scheduling
(andvice-versa). Moreover, preemptive schedulershave higherrun-timeoverheadsascomparedto non-
preemptiveschedulers.
In thispaper, weproposeageneralizedmodelof fixedpriority schedulingthatintegratesandsubsumes
bothpreemptiveandnon-preemptiveschedulers.Themodelusesthenotionof preemptionthreshold, which
wasintroducedby ExpressLogic, Inc. in their ThreadXreal-timeoperatingsystemto avoid unnecessary
preemptions[10]. In our new schedulingmodel,eachtaskhasa preemptionthreshold,in additionto its
priority. In essence,this resultsin a dual priority system. Eachtaskhasa regular priority, which is the
priority at which it is queuedwhenit is released.Oncea taskgetsthe CPU, its priority is raisedto its
preemptionthreshold.It keepsthis priority, until theendof its execution.For recurringtasks,this process
repeatseachtime thetaskis released.
Thepreemptionthresholdschedulingmodelcanbeusedto getthebestaspectsof bothpreemptiveand
non-preemptive scheduling.By choosingthepreemptionthresholdof a taskthat is higherthanits priority,
a taskavoidsgettingpreemptedfrom any taskthathasa priority lower thanits preemptionthreshold.By
varyingthepreemptionthresholdsof tasks,thedesiredamountof non-preemptabilitymaybeachieved. In
thisway, thepreemptionthresholdmodelmaybeviewedasintroducingnon-preemptabilityin acontrollable
manner. A suitablesettingfor thepreemptionthresholdscanthusbeusedtoget“just enoughpreemptability”
neededto meetthereal-timeresponsivenessrequirements;therebyeliminatingrun-timeoverheadsarising
from unnecessarypreemptabilityin theschedulingmodel.
It is alsoeasyto seethatbothpreemptiveandnon-preemptiveschedulingarespecialcasesof schedul-
Task Comp. Time Period Deadline��� ��� ���� 20 70 50 � 20 80 80�� 35 200 100
Table 1. An Example Task Set.
ing with preemptionthreshold. If the preemptionthresholdof eachtask is the sameas its priority, then
themodelreducesto purepreemptive scheduling.On the otherhand,if thepreemptionthresholdof each
taskis setto the highestpriority in thesystem,thenno preemptionsarepossible,leadingto fixedpriority
non-preemptivescheduling.
1.1 A Motivating Example
Beforewe delve into a theoreticaltreatmentof the new schedulingmodel, it is instructive to takea
look at a simpleexamplethatshows how schedulabilitycanbeimprovedwith this new schedulingmodel.
Weconsidera tasksetwith 3 independentperiodictasks,asshown in Table1. Eachtaskis characterizedby
a period( ��� ), adeadline( ��� ) anda computationtime( ��� ).Theschedulingattributesfor eachtaskincludeits priority ( ��� ) andits preemptionthreshold( ��� ). As-
sumingfixed-priorityscheduling,theoptimalpriority orderingfor this tasksetis deadlinemonotonicorder-
ing with bothpreemptivescheduling[13] andnon-preemptivescheduling[5]. Underthispriority ordering1,
theworst-caseresponsetimesfor thetasksareshown in Table2. We canseethat �� missesits deadlineun-
derpreemptivescheduling,while � missesits deadlineundernon-preemptivescheduling.Sincethepriority
orderingis optimal, this implies that the tasksetis not schedulableundereitherfixed-prioritypreemptive
scheduling,or fixed-prioritynon-preemptivescheduling.2
Whenwe usepreemptionthreshold,we canmakethe tasksetschedulableby settingthepreemption
thresholdfor � as3, and � as2. By settingthepreemptionthresholdof �� to 2, weallow it to bepreempted
by � , but not by � . This effectively improvestheresponsetimeof � (ascomparedto thenon-preemptive
case)sinceit canno longerbe blockedby �� . At the sametime, it improvesthe responsetime of �� (as
1Throughoutthispaper, weusehighernumbersto denotehigherpriorities.2Wealsonotethataslightmodificationto thisexamplewill show thatthefeasibilityunderpreemptiveschedulerdoesnot imply
feasibilityundernon-preemptivescheduler(by changingthedeadlineof ��� to 120),andviceversa(by changingthedeadlineof ���to be60).
Task � WCRT WCRT ! � WCRT
Preemptive Non-Preemptive Preemption-Threshold� ��" � � � ��"$#� 3 20 55 3 40 � 2 40 75 3 75 � 1 115 75 2 95
Table 2. Response Times for Tasks under Different Scheduler s.
comparedto the preemptive case)sinceit cannotbe preemptedby � once � hasstartedrunning. The
resultantresponsetimesarealsoshown in Table2.
Figure1 illustratestherun-timebehavior of thesystemwith preemptionthreshold,andwhy it helpsin
increasingschedulability. In thefigure,thearrows indicatearrival of taskinstances.Thefigureshows the
schedulingof tasksstartingfrom the critical instantof �� , which occurswheninstancesof all tasksarrive
simultaneously(time 0). Thescenarioleadsto theworst-caseresponsetime for �� . We canseethatat time
70, a new instanceof � arrives. Sincethe priority of � is higherthanthepreemptionthresholdof �� , ��is preempted.At time 80, a new instanceof � arrives. It cannot preempttheexecutionof � . However,
at time 90, when � finishes,a purepreemptive schedulerwould have run � , delaying �� . In contrast,by
settingthepreemptionthresholdof �� to 2, we have �� scheduledat time 90 underour schedulingmodel.
Thiseffectively improvestheworst-caseresponsetimeof �� , makingit schedulable.Notehoweverthatthis
alsoaddsblockingtime to � (ascomparedto thepreemptivecase),which increasesits worst-caseresponse
time,but doesnotaffect its schedulabilityin thisexample.
The useof preemptionthresholdsalsoreducesthe run-timeoverheadsassociatedwith preemptions
andthe associatedcontext-switches. This is dueto the introductionof somenon-preemptabilityinto the
schedulingmodel.We simulatedtheexecutionof tasksin thisexamplefor oneLCM length(i.e.,2800time
units).Whenall taskswerereleasedsimultaneouslyat time0, we find thatpreemptiveschedulingresultsin
17 preemptions,while with preemptionthresholdswe get 8 preemptions.If we staggerthe releasetimes,
suchthat � is releasedat time 0, � at time 1, and at time 2, then the numberof preemptionswith
preemptiveschedulingis 30,while usingpreemptionthresholdsreducesit to 10.
200 40 70 80 9590
PSfragreplacements
� ���
Worst-CaseResponseTime for ��Figure 1. Run-time Behavior with Preemption Threshold
1.2 Contributions
Themaincontributionof thispaperis in developingtheproposedschedulingmodelusingpreemption
threshold,showing how it improvesover purepreemptive andnon-preemptive scheduling,andsystemat-
ically allocatingfeasibleschedulingattributes(priority andpreemptionthreshold)to tasksets. We begin
with developingtheresponse-timeanalysisfor thenew modelbasedon well-know techniqueusingcritical
instantsandbusyperiods [8, 9, 11,12, 18]. We thendevelopanalgorithmfor assignmentof taskpriorities
andpreemptionthresholds.Our solutioncomesin two steps.First, givenan assignmentof priorities,we
developanefficient algorithmthatcomputesanoptimalsetof preemptionthresholdsusinga searchspace
of %'&)( � * (insteadof %'&)(,+ * for anexhaustivesearch).Thealgorithmis optimalin theusualsenseof finding
a feasibleassignment,i.e., onethatmakesthetasksetschedulable,if oneexists. Usingthis algorithmasa
“sub-routine”we thenproposeanoptimalsearchalgorithmto find anoptimalassignmentof bothpriorities
andpreemptionthresholdsby extendingAudsley’soptimalpriority assignmentalgorithm[1, 18]. We also
presentanefficient (but non-optimal)greedyheuristicalgorithmthatis usedin oursimulations.
The proposedschedulingmodelprovidessomeinterestingfeaturessuchasimproved schedulability
(comparedtobothpreemptiveandnon-preemptiveschedulers)andreducedpreemptionoverhead(compared
to preemptive schedulers).We quantifythebenefitsof theproposedschedulingmodelthroughsimulations
overrandomlygeneratedtasksets.To quantifytheimprovementin schedulability,weusebreakdownutiliza-
tion [11] asameasureof schedulability. Oursimulationsshow thatdependingonthetasksetcharacteristics,
usingpreemptionthresholdcanincreasethe schedulabilityby asmuchas -/.�0214365 of processorutiliza-
tion ascomparedto preemptivescheduling,andevenmoreascomparedto non-preemptivescheduling.We
alsopresentsimulationresultsthat quantify the reductionin preemptions.For this purpose,we develop
anotheralgorithmthatgivenafeasibleassignmentof prioritiesandpreemptionthresholds(asgenerated,for
instance,by our greedyalgorithm)attemptsto assignlarger preemptionthresholdswhile still keepingthe
tasksetschedulable.Usingthis new assignment,we thensimulatetheexecutionof a tasksetandmeasure
thenumberof preemptions,andcomparethatwith purepreemptivescheduling.Ourresultsshow that,again
dependingon the tasksetcharacteristics,we cangeta significantreduction(up to 30%) in thenumberof
preemptions(averagedover multiple tasksetswith thesamecharacteristics).
1.3 PaperOrganization
The remainderof this paperis organizedas follows. Section2 describesthe proposedscheduling
model,andthe problemsthat arebeingaddressedin this paper. In Section3, we develop schedulability
analysisequationfor ournew model.Section4 focusesondevelopinganoptimalalgorithmfor assignment
of priorities andpreemptionthresholdsfor a given taskset. In Section5, we quantify the improvement
in schedulabilityusingthe preemptionthresholdschedulingmodel. Then,in Section6, we show how to
systematicallyreducethenumberof preemptions.Finally, wepresentsomeconcludingremarks.
2 ProblemDescription and Solution Overview
Theexamplepresentedin Section1.1showedthatby usingpreemptionthresholdswe canpotentially
increaseschedulabilityandat thesametime reducerun-timeoverheads.This raisesthequestionof how to
makeeffective useof the new schedulingmodelto get thesebenefits.In this section,we first presentthe
taskandrun-timemodelassumedin thispaper. Then,we givea statementof theproblemsbeingaddressed
in thispaper, followedby anoverview of oursolutionapproach.
2.1 Taskand SchedulingModel
We considera setof ( independentperiodicor sporadictasks 7 "98 : ��:/;/;�;<: �=�> . Eachtask �is characterizedby a 3-tuple ?)��� : ��� : ���A@ , where �B� is its computationtime, �C� is its period(or minimum
inter-arrival time), and � � is its relative deadline.We assumethat (a) the tasksareindependent(i.e., there
is no blockingdueto sharedresources),(b) tasksdo not suspendthemselves,otherthanat theendof their
computation,and(c) theoverheadsdueto context switching,etc.,arenegligible (i.e.,assumedto bezero).
Eachtaskwill alsobegivenapriority � �ED�F - : (HG andapreemptionthreshold� �IDJF � � : (HG . Thesevalues
areassignedoff-line andremainconstantat run-time.Weassumethatlargernumbersdenotehigherpriority,
andthatnotwo taskshavethesamepriority. Weassumethattherun-timedispatchingis preemptive,priority
based,wheretasksaredispatchedfrom anotionalreadyqueuesortedby priorities.Whena taskis released
it is insertedinto the notional readyqueueat its priority � � . When the task is dispatched,its priority is
effectively raisedto its preemptionthreshold� � , andit keepsthispriority until it finishesexecution.In other
words,it canonly bepreemptedby anothertask LK if � KNM � � . Notethatsincetasksarerecurring,they will
go throughthesedualprioritiesfor eachreleaseof a task.
2.2 ProblemStatement
In therestof thepaperourobjectiveis toshow how to effectivelyusethepreemptionthresholdschedul-
ing modelto realizeits potentialbenefits.Accordingly, we addressthefollowing inter-relatedproblems.
2.2.1 Schedulability Analysis
First,weconsidertheproblemof assessingschedulabilityof a tasksetunderthepreemptionthreshold
schedulingmodel,assumingthat the schedulingattributesarealreadyknown. For this, we develop the
worst-caseresponsetime analysisfor a tasksetby extendingthe well-known critical instant/busy-period
analysisusedin fixed-priorityscheduling.Let O bea priority assignment,and P bea preemptionthreshold
assignment.Let QR��&)O : P * denotethe worst-caseresponsetime of a task � underthe givenassignments.
Then,theschedulabilityof a tasksetis givenby thefollowing booleanpredicate:
S/T UWV/X &)7 : O : P *ZY [)\" &^]�_a`b`�-Nc2_ac2( * Q � &)O : P * c2� � (1)
2.2.2 Attrib ute Assignment
We thenconsidertheproblemof assigningschedulingattributes,i.e., priority andpreemptionthresh-
olds, to tasks.Theprimarygoalof finding the schedulingattributesis to determineif thereexistsa setof
schedulingattributesthat will makethe tasksetschedulable.That is, we wish to determinewhetherthe
following predicateis true:
&)d�O * &)d�P * `^` S�T UeV�X &)7 : O : P * (2)
If it isdeterminedthatatasksetisschedulable,thenweadditionallywanttofindasetof schedulingattributes
thatwill minimizerun-timeoverheadsby eliminatingany unnecessarypreemptions.
2.2.3 Quantitative Assessment
Sincethe preemptionthresholdschedulingmodelgeneralizesboth preemptive andnon-preemptive
schedulingmodels,it will do no worsethaneitherof themin termsof schedulability. However, we still
needto determinewhetherthereare any significantgains in schedulabilityif the preemptionthreshold
schedulingmodelis employed.To thisend,wewish to makea quantitativeassessmentof theimprovement
in schedulabilitythroughtheuseof preemptionthresholds.
Likewise, we are interestedin quantitative assessmentof run-timeoverheadsfor feasibletasksets.
Unlike schedulability, a non-preemptiveschedulerwill resultin lower run-timeoverheads.Moreover, since
non-preemptive schedulingis a specialcase,our schedulingmodelwill incur thesameoverheadasa non-
preemptive schedulerfor tasksetsthat areschedulablewith non-preemptive scheduling. Therefore,our
interestis in assessingrun-timeoverheadsdueto preemptionsandcomparethemwith a preemptivesched-
uler.
2.3 Solution Overview
Oursolutionconsistsof thefollowing parts:
(1) First, we show how givenboth the prioritiesandthe preemptionthresholds,we canfind the worst-
caseresponsetimesfor thetasks,andhencedeterminefeasibilityof aparticularpriority andthreshold
assignment.
(2) We thenconsidertheproblemof determininga feasiblepreemptionthresholdassignmentwith pre-
definedpriorities. We develop an optimal andefficient algorithmthat hasa searchspaceof %'&)( ��*(insteadof %f&)(,+ * for bruteforcesearch)to solve thisproblem.
(3) Weusetheoptimalpreemptionthresholdassignmentalgorithmto developabranchandboundsearch
algorithmto find a feasiblesetof schedulingattributes.Sincethesearchalgorithmmayresultin an
exponentialsearchspace,wealsodevelopanefficientbut non-optimalgreedyheuristicalgorithm.
(4) We usethegreedyalgorithmfor quantifyingtheimprovementin schedulability(ascomparedto both
preemptiveandnon-preemptiveschedulingmodels)usingrandomlygeneratedtasksets.Thisensures
thattheoff-line costsof usingdifferentschedulingmodelsareapproximatelysimilar resultingin fair
comparisons.
(5) To eliminateany unnecessarypreemptions,we presentan algorithm that takesa feasiblepriority
andpreemptionthresholdassignmentas input and refinesit (by increasingpreemptionthresholds
of tasks)to eliminateunnecessarypreemptionwhile maintainingfeasibility. Again, we quantifythe
improvementsusingsimulationsover randomlygeneratedtasksets.
3 Schedulability Analysis
We begin with schedulabilityanalysisof tasksetswhen the priority and the preemptionthreshold
of eachtaskarealreadyknown. The schedulabilityanalysisis basedon computationof the worst-case
responsetime of eachtask. If the worst-caseresponsetime of a task is no more than its deadline,then
the taskis schedulable.If all the tasksin a systemareschedulable,thenthesystemis calledschedulable.
The responsetime analysisemployedin this paperis an extensionof the well-known level-i busy period
analysis[8, 9,11,12,18], in whichtheresponsetimeis calculatedby determiningthelengthof busyperiod,
startingfrom a critical instant.Thebusyperiodat level-i is definedasa continuousinterval of time during
which theprocessoris continuouslyexecutingtasksof priority _ or higher.
3.1 Review of Level-i BusyPeriod Analysis
To calculatetheworst-caseresponsetime of a task,thebusyperiodanalysisessentiallysimulatesthe
effect of schedulingundera worst-casescenariofor the task. The busy periodfor task � is constructed
by startingfrom a critical instant(time0). Thecritical instantoccurswhen(1) an instanceof eachhigher
priority taskcomesat thesametime (time 0), and(2) thetaskthatcontributesthemaximumblockingtimeg & � * hasjust startedexecutingprior to time 0. Furthermore,to gettheworst-caseresponsetime,all tasks
areassumedto arrive at theirmaximumrate.
3.1.1 PreemptiveScheduling
For our taskmodeldescribedearlier, with arbitrarytaskdeadlines,the busyperiodof task � in tra-
ditional fixed-priority, preemptive schedulingcan be iteratively computedby using the following equa-
tion [18]:
h � &)i * " ikj � ��l mn K�o p�q<r�p/s t h ��&)i *� Kvu j � K (3)
In this equation,h � &wi * denotesthe lengthof a busyperiodfor task � , where i instancesof � areincluded
in thebusyperiod.Thelengthof thebusyperiodfor task � is givenby:x �y" z�{b|}�~4� <o ��o � o������ � h � &wi * `^` h � &)i * c2ikj � � (4)
where h � &)i * is the smallestvalueof h � &)i * that satisfiesEquation3. Essentiallythe equationstatesthat
duringthebusyperiodfor � , all instancesof � andhigherpriority tasksthatarrive within thebusyperiod
mustalsobeexecutedwithin thebusyperiod.Thebusyperiodlengthis computedby iteratively computingh � &)i * for i " - : 1 : # :�; ; ; usingEquation3 until wereacha i , i "�� suchthat h � & � * c � j � �The worst-caseresponsetime of � will be the longestresponsetime of all instancesthat arrive and
finish in thebusyperiod.Let usdenote�B��&)i * (q-th finish time) asthesmallestvalueof h ��&wi * thatsatisfies
equation3. Sincetheq-th instanceof � arrivesat &)ik0�- * j���� , theworst-caseresponsetime for � is given
by: Q ��" z����}�~�� <o������ o �E� &�� � &)i * 02&)ik0�- * j � � * (5)
3.1.2 Non-PreemptiveScheduling
Even thoughthe analysisdescribedabove wasdonefor preemptive scheduling,the sametechnique,
with minor modifications,canbeusedfor non-preemptive schedulingaswell [5]. However, therearetwo
importantdifferences.First, a taskmay be blockedby any lower priority task, if the lower priority task
startedexecutingjust prior to thecritical instant.Second,oncea taskgetstheCPU,it cannotbepreempted
by any higherpriority taskuntil it finishesexecution.
Theblockingtimefrom lowerpriority tasksis easyto incorporate.A taskmaybeblockedby only one
lowerpriority task.In theworstcase,this taskwouldhavestartedexecutingjustprior to thecritical instant.
Thus,theworst-caseblockingtime for task � , denotedasg � is givenby:g ��" z����n K�o p�q<��p/s � K (6)
Equation3 is modifiedfor non-preemptiveschedulingas:h � &)i * " g �el ikj � ��l mn K�o p�q<r�p/sZ� - l�� h � &)i * 0�� �� K ��� j � K (7)
Notethatthetermfor interferencefrom higherpriority tasksis modifiedto includetaskarrivalsup to (and
including)time h �<&)i * 0���� , i.e.,whentheq-th instancetask � startsexecuting.
3.2 ResponseTime Analysiswith PreemptionThresholds
Our responsetime analysiswith preemptionthresholdsfollows the sameprinciplesof the analysis
presentedin previoussection.As in thenon-preemptive case,we needto worry aboutblockingtime from
lower priority tasks.Also, asin preemptive scheduling,higherpriority tasksmaycauseinterferenceeven
aftera taskstartsexecuting.However, the interferenceis differentfrom purepreemptive scheduling,since
thetask’seffective priority changesto its preemptionthresholdonceit startsexecutingandonly taskswith
higherpriority thanthis effectivepriority maycauseinterference.Therefore,weexplicitly defineandcom-
putebothq-th starttime for a task � (denotedby � &)i * ) andtheq-th finish time (denotedby � � &)i * ). After
we gettheq-th finish time by themodifiedbusyperiodanalysis,theworst-caseresponsetime is calculated
asbefore,following Equation5.
3.2.1 Computing Blocking Time
We begin with determiningthe blocking time for a task. The blocking is causeddueto preemption
thresholds,whena lowerpriority taskhasa preemptionthresholdhigherthanthepriority of thetaskunder
consideration.Then, if the lower priority task is alreadyrunning, it cannotbe preempted(due to higher
preemptionthreshold)by thetaskunderconsideration,leadingto blockingtime.
Definition 3.1(Blocking Range) Theblockingrangeof a task � is definedastherangeof priorities given
by F � � ,� � G .Definition 3.2(Active Task) A task(instance)is calledactiveif it hasstartedrunning,but is not finished
yet.
Proposition3.1 There is nooverlappingof blockingrangesbetweenthesetof activetasksat anyinstantof
time.
Proof: By contradiction.Suppose � and K areactive at sometime, andtheir blockingrangesoverlap.
Without lossof generalityassumethat � startedexecutionfirst. Then, for K to start runningbefore �finishes,it mustbe the casethat ���k¡¢� K . That is ���£c¤���k¡¥� K c¦� K . Thus,the blockingrangesdo not
overlap.
Proposition3.2 A task � canbeblockedbyat mostonelowerpriority task LK . Furthermore,it mustbethe
casethat � K¨§ � � .Proof: A new arriving taskinstanceof � cannotpreempta lowerpriority activetask LK only if � � falls in
theblockingrangeof LK , i.e., � K ¡�� � c©� K . FromProposition3.1,we know thatblockingrangesof active
taskswill notoverlap.Therefore, � will only fall into atmostoneof theseblockingranges,i.e.,beblocked
by theownerof thatblockingrange.Furthermore,it is easyto seethatany lowerpriority tasksthathavenot
startedexecutionbeforethearrival of � will notblock � .
Proposition3.2shows thatin computingtheblockingtime for a task � , we needto considerblocking
from only onelower priority task LK suchthat � K�§ � � . Furthermore,this is true for theentirebusyperiod
of a task � sinceduring the busy perioda lower priority taskcannotstartexecution(dueto the pending
instancesof � ). Therefore,themaximumblockingtimeof a task � , denotedbyg & � * , is givenby:g & � * " z����n K�o ª<q<«�p/swr�p�q � K (8)
3.2.2 Computing q-th Start Time:
Beforea task � startsexecution,thereis blocking from lower priority tasksand interferencefrom
higherpriority tasks.Amongall lowerpriority tasks,only onelowerpriority taskcancauseblockingaswe
showedin Proposition3.2. This taskmusthave arrivedandbegunexecutingbeforethebusyperiodstarts.
In theworstcase,it just startsexecutingbeforetime 0. All higherpriority tasksthatcomebeforethestart
time � &)i * andany earlierinstancesof task � beforeinstancei shouldbefinishedbeforetheq-thstarttime.
Therefore, � &)i * canbecomputediteratively usingthefollowing equation.
� &wi * " g & � * l &wi�02- * j�� ��l mn K�o p�q<r�p/s � - l � � &)i *� K ��� j � K (9)
3.2.3 Computing q-th Finish Time:
Oncethe q-th instanceof taskstartsexecution,we have to considerthe interferenceto computeits
finish time. Fromthedefinitionof preemptionthreshold,we know thatonly taskswith higherpriority than
thepreemptionthresholdof � canpreemptandgettheCPUbefore � finishes.Furthermore,we only need
to considernew arrivalsof thesetasks,i.e.,arrivalsafter � &)i * . Basedonthis,wegetthefollowingequation
for computing� � &)i * :����&)i * " ��<&)i * l ��� l mn K�o p�q�r�ª�s¬� t �B��&)i *� K u 0 � - l � ��&)i *� K¦���k� j � K (10)
4 FeasibleAssignmentof Priority and PreemptionThr eshold
In the previous section,we showed how to computethe worst-caseresponsetimeswhenpriorities
andpreemptionthresholdsof tasksareknown. In this section,we developanalgorithmto systematically
assignprioritiesandpreemptionthresholdsto tasks,suchthattheassignmentensuresschedulability. Wefirst
presentsometheoreticalresultsaboutourmodel.Then,wepresentanalgorithmmotivatedby theseresults
for optimalpreemptionthresholdassignmentto tasksetswith predefinedpriority. After that,we proposea
systematicapproachto find theoptimalpriority andpreemptionthresholdassignmentusingthis algorithm
asasub-routine.Finally, wemodify theoptimalalgorithmandpresentanefficient,but approximate,greedy
algorithm.Thegreedyalgorithmis usedin oursimulationsto enablea fair comparisonwith preemptiveand
non-preemptiveschedulingstrategies.
4.1 Propertiesof the Model
With the responsetime analysiswe have doneabove, we notice that this generalizedfixed-priority
schedulingmodelhassomeinterestingproperties. Assumingthat the priorities of tasksarefixed, these
propertieshelpus reasonabouttheeffect of changingpreemptionthresholdsof tasks.Furthermore,these
propertieswill helpusto reducethesearchspacefor finding theoptimalpreemptionthresholdassignment.
Therefore,wewill usethesepropertiesasguidelineswhile designinganalgorithmfor preemptionthreshold
assignment.
Lemma 4.1 Changingthe preemptionthresholdof a task � from � to � � mayonly affect the worst-case
responsetimeof task � andthosetaskswhosepriority is between� and � � .This canbeseenby examiningthe equationsdevelopedfor calculationof responsetimes. The pre-
emptionthresholdof a task � determineswhich (higherpriority) tasksmaybeblockedby � . Thesetasks
arethosewhoseprioritiesfall in therangeF � � : � � G . Therefore,theresponsetimeof all thesetasksmaybeaf-
fectedwhen � ’spreemptionthresholdis modified.It alsomayaffect its own responsetimesinceit changes
the setof tasksthat canpreemptit onceit hasstartedrunning. Note that, the preemptionthresholdof �doesn’t affect theinterferencefrom � on any lower priority task LK (which dependson LK ’s threshold,and � ’spriority). Therefore,changingthepreemptionthresholddoesn’t affect any lowerpriority task.A useful
propositiondirectly followsfrom this lemma,andis presentedbelow.
Proposition4.1 Theworst-caseresponsetime of task � will not be affectedby the preemptionthreshold
assignmentof anytask LK with � K®M � � .Proposition4.1 is usefulin developinga strategy for optimalassignmentof preemptionthresholds.It
shows that theschedulabilityof a taskis independentof thepreemptionthresholdsettingof any taskwith
higherpriority. Therefore,this suggeststhat thethresholdassignmentshouldstartfrom thelowestpriority
taskto highestpriority task.
Furthermore,Theorem4.1,presentedbelow, helpsusdeterminetheoptimalpreemptionthresholdas-
signmentfor aspecifictask.Thepreemptionthresholdof ataskcanrangefrom its ownpriority to thehighest
priority in thetaskset.Fromtheequationsfor worst-caseresponsetimeanalysis,wecanseethatataskmay
reduceits worst-caseresponsetimeby increasingits preemptionthreshold,whichrestrictsthesetof (higher
priority) tasksthatcanpreemptit. However, this is doneat thecostof a possibleincreasein theblocking
time of higherpriority taskswhich may leadto increasedworst-caseresponsetime of the higherpriority
task.Therefore,if thereis a setof preemptionthresholdvaluesthatcanmakea taskschedulable,choosing
theminimumof themwill maximizethechancesof findinga feasiblepreemptionthresholdassignment.
Theorem 4.1 Considera set of ( tasks 7 "¯8 � : � :/;�;/;<: = > , and a set of schedulingattributes O "?°� :�;/;/;L: � = @ and P " ?w� :/;�;/;<: � = @ , such that thetasksetis schedulablewith O and P (i.e., S�T UeV/X &)7 : O : P *is true). Then,if changingonly the preemptionthresholdof LK from � K to ��±K ( �C±K ¡¦� K ), canstill make LKschedulable,thenthewholesystemis alsoschedulablebysetting� ±K asthepreemptionthresholdof LK . That
is, Q K &)O : P²&�� K�³ � ±K *L* c�´ K¶µ·S�T UeV�X &)7 : O : P²&¸� K�³ � ±K *<*Proof: Whenthepreemptionthresholdof LK changesfrom � K to � ±K ( � KRM � ±K ), theworst-caseresponse
timeof any task �¹ with � ¹ ¡2� K or � ¹�M � K will notchange.Theworst-caseresponsetimeof atask �¹ such
that � ±K § � ¹fM � K will alsostaythesame.Furthermore,any task �¹ with priority � ±K ¡�� K c©� K will have
noworseworst-caseresponsetime with � ±K thanwith � K . Moreover, we alreadyknow that LK is schedulable
with � ±K . Therefore,if thewholesystemis schedulablewith � K , it is alsoschedulablewith � ±K .A given tasksetmay be unschedulablewith any preemptionthresholdassignment.The following
theoremgivesasufficientconditionto claimatasksetto beunschedulable.
Theorem 4.2 For any givenpriority assignmentif there existsa task � , such that settingthe preemption
thresholdof a task � equalto thehighestpriority in thesystemcannotmakea specifictask � schedulable,
thenthetasksetis unschedulable.
Proof: ComparingEquation10with Equation3, wecanseethatpreemptionthresholdreducestheworst-
caseresponsetime of a task � by preventingtheinterferencefrom somehigherpriority tasksafter � starts
execution.By settingthepreemptionthresholdof � to thehighestpriority in thesystemgivesthemaximum
reductionto theworst-caseresponsetimeof � . If � is still notschedulable,sincethepriority is predefined,
thereis nowayto makeit schedulable.Furthermore,thetasksetwill alsobeunschedulable.
Algorithm: AssignThresholds
// Assumesthat taskpriorities are alreadyknown
(1) for ( º := 1 to » )
(2) ¼/½C¾À¿6½ // start fromlowestvalue
// Find worst-caseresponsetimebasedon thetentativeassignment
(3) ÁN½ = WCRT( Â�½ ,¼�½ ) ;
(4) while ( Á ½CÃÅÄB½ ) do // whilenotschedulable
(5) ¼ ½ ++ ; // increasepreemptionthreshold
(6) if ¼ ½CÃ » then
(7) return FAIL; // systemnotschedulable.
(8) endif
(9) ÁN½e¾ WCRT( Â�½ ,¼�½ ) ;
(10) end
(11) end
(12) return SUCCESS
Figure 2. Algorithm for Preemption Threshold Assignment
4.2 Optimal PreemptionThresholdAssignmentwith Given Priorities
Basedon theresultsdevelopedin theprevioussection,we have developedanalgorithmthatfindsan
optimalpreemptionthresholdassignmentassumingthat theprioritiesareknown andfixed. Thealgorithm
is optimalin thesensethatif thereexistsafeasiblepreemptionthresholdassignmentthenthealgorithmwill
alsofind afeasibleassignment.
Figure2 gives the pseudocodeof the preemptionthresholdassignmentalgorithm. The algorithm
assumesthat the tasksarenumbered- : 1 :/;/;�;�: ( , andthat � �²" _ . Thealgorithmconsidersthepreemption
thresholdassignmentof onetaskat a time startingfrom thelowestpriority task. For eachtaskconsidered,
it finds the lowestpreemptionthresholdassignmentthat will makethe taskschedulable.This is doneby
computingtheresponsetimeof thetaskusingthefunctionWCRT(task, threshold), andcomparingit with
its deadline.Note that, the responsetime calculationis possibleeven with a partial assignmentsincewe
considertasksfrom low priority to highpriority.
It is easyto seetheworst-casesearchspacefor this algorithmis %f&)( � * . Thealgorithmis optimal in
thesensethat if thereexistsa preemptionthresholdassignmentthatcanmakethesystemschedulable,our
algorithmwill alwaysfind anassignmentthatensurestheschedulability. Theoptimalityof thealgorithmis
givenin thefollowing Theorem4.3.
Theorem 4.3 Givena fixedpriority assignment,thealgorithmAssignThresholdswill find a feasiblepre-
emptionthresholdassignment,if thereexistsone.
Proof: Assumea setof ( tasks 7 "Æ8 �ÈÇ -fcÉ_¨c¢( > , with a priority assignmentO " ?°� � :�;/;�;�: � = @ .Without lossof generality, assumethat the taskshave beenlabeledsuchthat � �Ê" _ . Furthermore,let the
tasksetbeschedulablewith asetof preemptionthresholdvaluesP "Ë8 � �EÇ &�-¨c2_Ec2( * > .Thealgorithmassignspreemptionthresholdsto thetasksstartingfrom � andgoingupto = . Let the
preemptionthresholdsfound by the algorithmbe labeled� ± : � ±� , etc. Assumethat task � is the first task
suchthat thepreemptionthresholdfoundby thealgorithmis differentfrom thegivenfeasibleassignment,
thatis, � ±�ÊÌ" � � . Then,it mustbethecasethat � ±� ¡�� � , otherwise,ouralgorithmwill find � � ratherthan � � ’.Basedon Theorem4.1,we know that thetasksetwill still beschedulableif we use � ±� to replace� � in the
above feasiblepreemptionthresholdassignment.
By repeatedlyusingtheabove argument,we canseethat thealgorithmwill alsofind a feasiblepre-
emptionthresholdassignment8 �C±� Ç �C±� c���� : -¨c2_Ec2( > .4.3 Optimal Assignmentof Priorities and PreemptionThresholds
In thissection,weaddressthegeneralproblemof determininganoptimal(i.e.,onethatensuresschedu-
lability) priority andpreemptionthresholdassignmentfor a given taskset. We give a branch-and-bound
searchalgorithmthatsearchesfor theoptimalassignment.Whethermoreefficientalgorithmscanbefound
for this problem,remainsan openquestionat this time. Our algorithmborrows the basicideasfrom the
optimalpriority assignmentalgorithmpresentedin [1, 18] for preemptive priority schedulingof a taskset.
Unfortunately, the introductionof preemptionthresholdsnot only addsanotherdimensionto the search
spacebut alsobringsmorebranchesinto thesearch,makingthesearchspaceexponentialin size.
Our searchalgorithm, presentedin Figure 3, proceedsby performinga heuristicguidedsearchon
“good” priority orderings,andthenwhena priority orderingis complete,it usesthealgorithmpresentedin
theprevioussectionto find afeasiblethresholdassignment.If a feasiblethresholdassignmentis foundthen
wearedone.If not,thealgorithmbacktracksto find anotherpriority ordering.
The algorithmworks by dividing the taskset into two parts: a sortedpart, consistingof the lower
priority tasks,andanunsortedpart, containingthe remaininghigherpriority tasks. The priorities for the
tasksin thesortedlist areall assigned.Theprioritiesfor thetasksin theunsortedlist areunassigned,but are
Algorithm: AssignSchedAttributes(Í , ¿ )
/* TerminatingCondition;assignpreemptionthresholds*/
(1) if ( Í == Î�Ï ) then
/* Usealgorithmin Figure2 for preemptionthresholdassignment*/
(2) return AssignThresholds()
(3) endif
/* Heuristicallygeneratea priority assignment*/
/* AssignHeuristicValueto Each Task*/
(4) L := Î/Ï ;
(5) foreach Â�ÐkÑ�Í do
(6) ¿6Ð := ¿ ; ¼�ШÒÓ¾�» ; ÁNÐ := WCRT( Â�Ð );(7) if ÁNÐ Ã�Ä Ð then Continue ; /* prune*/
(8) ¼/ÐkÒÓ¾Ô¿4Ð ; ÁNÐ := WCRT( Â�Ð );(9) if ÁNÐkÕ Ä Ð then
(10) Ö Ð ÒÓ¾ GetBlockingLimit(Â Ð ); /* positivevalue*/
(11) else
(12) Ö Ð ÒÓ¾Å× ÐÙØ Á Ð ; /* negativevalue*/
(13) endif
(14) L := L + Â�Ð ;(15) ¿6Ð := » ; /* reset*/
(16) end
/* Recursivelyperformdepthfirst search */
(17) while (L != Î/Ï ) do
/* Selectthetaskwith thelargestheuristicvaluenext */
(18) Â�ÐkÒÓ¾ GetNextCandidate(L);
(19) ¿6УÒÓ¾�¿ ;
(20) if AssignSchedAttributes(Í Ø Â�Ð , ¿ +1) == SUCCESSthen
(21) return SUCCESS;
(22) endif
(23) L := L - �Р;(24) end
(25) return FAIL
Figure 3. Search Algorithm for Optimal Assignment of Priority and Preemption Threshold
all assumedto behigherthanthehighestpriority in thesortedlist. Initially, thesortedpart is emptyandall
tasksarein theunsortedpart.Thealgorithmrecursively movesonetaskfrom theunsortedlist to thesorted
list, by choosinga candidatetaskbasedon heuristics,asdescribedbelow. Whenall tasksarein thesorted
list, a completepriority orderinghasbeengenerated,andthethresholdassignmentalgorithmis called.
Whenconsideringthe next candidateto move into the sortedlist, all tasksin the unsortedlist are
examinedin turn. To makethesearchmoreefficient,we selectthe“most promising”candidatefirst, using
a heuristicfunction,describedbelow. If thealgorithmfails to find a solutionwith thatpartialassignment,
it will backtrackandthenselectthenext task. Additionally, we pruneinfeasiblepathsby not considering
tasksthatcannotbemadeschedulableat thecurrentpriority level.
Figure 3 givesthepseudocodeof thesearchalgorithm,which is presentedasa recursive algorithm.
It takestwo parameters:7 , which is theunsortedpart(containingall thetaskswaiting for priority assign-
ment),and � , which is the next priority to assign. The tasksthat have beenassignedpriorities arekept
separately(andnot explicitly shown in thepseudo-code)for preemptionthresholdassignmentat theendof
thealgorithm.Thelist of candidatesto searchis createdin Ú . Thecomputationof worst-caseresponsetimes
assumesthatall taskswith unassignedprioritieshavethehighestpriority, andthatall unassignedpreemption
thresholdsareequalto thetaskpriority.
Pruning Infeasible Paths. First, we tentatively assigna task the currentpriority andcomputeits
responsetimewith its preemptionthresholdsetto thehighestpriority in thesystem.If its computedresponse
timeexceedsits deadline,thenthetaskcannotbemadeschedulableat thispriority level (Theorem4.2).This
is because,we assumeat this stagethatall othertaskshave preemptionthresholdsequalto their priorities.
Therefore,weprunesuchabranchto makethesearchmoreefficient.
Heuristic Function. To computetheheuristicfunction,we computetheresponsetime for the task
by tentativelyassigningit thecurrentpriority andassumingthatthepreemptionthresholdequalsits priority.
Let Q ¹ bethecomputedresponsefor a task �¹ in thismanner. Then,theheuristicfunctionis givenby:
Û ¹ " ÜÝ Þ g Ú ¹ ß^à Q ¹ c2� ¹Q ¹ 0À� ¹âá�ãAä�á (11)
whereg Ú ¹ denotesthe blocking limit for �¹ . The blocking limit representsthe maximumblocking that
the taskcanget while still meetingits responsetime. Note that at this stagesincewe have assumedthat
prioritiesequalpreemptionthresholds,thereis noblocking.However, oncetheprioritiesarefully assigned,
it is possiblethat in the preemptionthresholdassignmentstagea lower priority taskmay be assigneda
thresholdthat is higherthanthis task,andcancauseblocking. Theblocking limit capturesthe maximum
blocking that a task can toleratewhile still meetingits deadline. This can be computedby assigninga
blockingtermto thetask,repeatingtheworst-caseresponsetimecomputation,andcheckingif it still meets
thedeadline.
Theblocking limit is meaningfulif Q ¹ c¢� ¹ . Otherwise,it is still possiblethat ¹ maybeschedu-
lableat this priority with anappropriatepreemptionthreshold.Thustasksthatneeda smallerreductionin
interferencefrom higherpriority tasksarebettercandidatesfor selection.Accordingly, weassignaheuristic
valueof � ¹ 0åQ ¹ for eachtask.Notethatthesevaluesarenegative,whileg Ú ¹ is positive.Thus,suchtasks
have a lowerheuristicvalue,which is asdesired.
4.4 A GreedyAlgorithm
Theefficiency of theoptimalalgorithmshown abovedependsheavily onthecharacteristicsof thetask
set.In theworstcase,it hasexponentialsearchspacein termof thenumberof tasks.Clearly, thisalgorithm
becomesinfeasibleto use,even with a modestnumberof tasks. Therefore,we have developeda greedy-
heuristicalgorithm,which we usein our simulationsfor schedulabilitycomparison.Thebasicideaof this
greedyalgorithmis thesameastheoptimalalgorithm.Theonly differencelies in thebranchingpart. The
optimalalgorithmwill try all possiblebranchesbeforeit find a solution. However, theheuristicalgorithm
will only try theonethatis mostpromising,or in anotherword,theoneat theheadof thecandidatelist.
This heuristicalgorithmdominatesthe preemptive schedulingalgorithm,i.e., if a tasksetis schedu-
lable with preemptive scheduling,then the algorithm will be able to find a feasibleassignmentaswell.
This is not surprisingsincethealgorithmextendsAudsley’soptimalalgorithmfor priority assignment.On
the otherhand,therearecaseswhenthe algorithmis not ableto find a feasibleassignment,whena non-
preemptive priority assignmentalgorithm is able to find a feasibleassignment.Sincea non-preemptive
priority assignmentis alsoa feasiblesolutionfor our model,thealgorithmcanbetrivially extendedto use
thenon-preemptivepriority assignmentalgorithmfirst, andthenusethis algorithm.Without actuallydoing
so,we assumethatthis is thecase,andthis extendedalgorithmis usedin our simulations.In this way, our
extendedalgorithmdominatesbothpreemptiveandnon-preemptiveschedulingalgorithms.
5 Schedulability Impr ovement
Sinceour schedulingmodelsubsumesbothpreemptive andnon-preemptive scheduling,any taskset
that is schedulablewith eithera purepreemptive or a non-preemptive scheduleris alsoschedulablein our
model.In section1.1,we gavea simpleexampletaskset,which is not schedulableundereitherpreemptive
or non-preemptive scheduling,but canbescheduledwith our model. Nonetheless,it maybethecasethat
any suchschedulabilityimprovementcomesonly in exceptionalcases,andany suchimprovementmayonly
bemarginal. Accordingly, in this sectionwe wantto quantifytheschedulabilityimprovementthatmaybe
achieved by usingthis new model. Our strategy is to userandomlygeneratedtasksetsandtest for their
schedulabilityunderthedifferentschedulingmodels.
To comparetheschedulabilityof a tasksetunderdifferentschedulingpolicies,we needa quantitative
measurement.Onepossibilityis to usethebinarymeasureof schedulability, i.e.,eithera tasksetis schedu-
lableor not. A bettermeasure,andonethat is commonlyusedis thenotionof breakdownutilization [11].
Thebreakdown utilizationof a tasksetis definedastheassociatedutilizationof a tasksetat which a dead-
line is first missedwhenthecomputationtimeof eachtaskin thetasksetis scaledby a factor. In thispaper
weusebreakdown utilizationsto measureschedulability.
5.1 Simulation Design
We userandomlygeneratedperiodictasksetsfor our simulations.Eachtaskis characterizedby its
computationtime � � andits period � � . To keepthenumberof variablessmall,we assumethat � ��" � � for
all tasks.We vary two parametersin our simulations:(1) numberof tasks (W�£æ S/ç�S , from 5 to 50; and(2)
maximumperiodfor tasks� æ�èeé V�ê _�ë X , from 10 to 1000.For any givenpair of (W�£æ S�ç�S and � æ�èWé V/ê _°ë X ,werandomlygenerate100tasksets.
A tasksetis generatedby randomlyselectinga periodandcomputationtime for eachof the (W�£æ S�ç�S .First,a period �C� is assignedrandomlyin therangeF - : � æ�èeé V�ê _°ë X G with auniformprobabilitydistribution
function.Then,weassignautilization ìI� in therangeF 3 ; 34. : 3 ; .�G ,againwith uniformprobabilitydistribution
function.Thecomputationtimeof thetaskis thenassignedas ��� " �C�Wí�ìI� .For eachrandomlygeneratedtaskset,we measurethebreakdown utilization for eachof (1) purepre-
emptive scheduling,(2) non-preemptive scheduling,and(3) schedulingwith preemptionthreshold.We do
thisby scalingthecomputationtimesto getdifferenttasksetutilization,andthentestingfor schedulability.
Sincetherandomlygeneratedtasksetsmayhave utilization greaterthan1 to begin with, we initially scale
theutilization to 100%.We thenusebinarysearchto find themaximumutilization at which thetasksetis
schedulableunderaparticularschedulingalgorithm.
5.2 Attrib ute Assignment
Beforewe cando theresponsetime analysiswith a particularschedulingalgorithm,we mustderive
the schedulingattributes. In the caseof preemptive andnon-preemptive schedulingalgorithms,the only
schedulingattribute is taskpriority, while for the preemptionthresholdscheduling,both taskpriority and
preemptionthresholdmustbe derived. We derive optimal priority assignmentsfor both preemptive and
non-preemptive schedulingalgorithms. For the preemptive case,the optimal priority assignmentis sim-
ply therate-monotonicassignment[14]. For thegeneralcase,with arbitrarydeadlines,Audsley’s optimal
priority assignmentalgorithm[1] canbeused.For non-preemptivescheduling,we useoptimalpriority as-
signmentalgorithmpresentedin [5] which is basicallythe sameasAudsley’s algorithm,but adaptedfor
non-preemptive scheduling.For schedulingwith preemptionthresholds,we usethegreedy-heuristicalgo-
rithm presentedin lastsectionasanefficient approximationto assignprioritiesandpreemptionthresholds.
Thetime complexity of this algorithmis thesameasAudsley’salgorithm. This providesa fair groundfor
comparisonbetweenthealgorithms.
5.3 Simulation Results
In this section,we describethe resultsfor schedulabilityimprovement(asmeasuredby breakdown
utilization) usingpreemptionthresholds.As mentionedearlier, we controlledtwo parameters:(1) num-
ber of tasks( (W�£æ S/ç�S ), and (2) maximumperiod( � æ�èeé V/ê _�ë X ). We did the simulationsfor (W�£æ S/ç�S D8 . : -/3 : -/. : 143 : 1�. : # 3 : # . :<î 3 : .43 > and � æ�èWé V/ê _°ë X Dï8 -/3 : 143 : .43 : - 343 : .4363 : - 34343 > . For eachtaskset, we
measuredthebreakdownutilizationfor eachof thethreecases:purepreemptivescheduling,non-preemptive
scheduling,andpreemptiveschedulingwith preemptionthreshold.
In Figures4 and5weshow theschedulabilityimprovementasthenumberof tasksisvaried.Theresults
areshown for � æ�èWé V/ê _°ë X " 10 and100;theresultsaresimilar for othervalues.In eachcase,we plot the
averageandmaximumincreasein breakdown utilizationwhenusingpreemptionthresholds.Figure4 shows
schedulabilityimprovementascomparedto purepreemptive priority scheduling.As theplot shows, when
looking at averageimprovement,thereis a modestimprovementin schedulability(3%-6%), depending
on the numberof tasks. As the numberof tasksincreases,the improvementtendsto decrease.Perhaps,
moreinterestingis the plot for maximumincrease,which shows that the schedulabilityimprovementcan
be ashigh as18% improvementin breakdown utilization for selectedtasksets,althoughonceagainthe
improvementdecreasesasthenumberof tasksis increased.
Theresultsshowing theschedulabilityimprovementwith non-preemptiveschedulingaremorevaried.
First, for mostrangesof theparameters,theschedulabilityimprovementis muchmorethanthepreemptive
case(whichalsomeansthatpreemptiveschedulinggiveshigherbreakdown utilization,ascomparedto non-
preemptivescheduling).Theresultshouldnotbesurprisingsincenon-preemptiveschedulingperformsvery
badlyeven if onetaskhasa tight deadline,andany othertaskhasa largecomputationtime. In suchcases,
thebreakdown utilizationcanbearbitrarily low, ascanbeseenin Figure5(b).
Whilenon-preemptiveschedulingperformspoorlyin general,thereareselectedcaseswhenit performs
betterthanpreemptive scheduling,andbetterthanour heuristicalgorithmfor preemptionthreshold.Note,
however, thatweassumethatourheuristicalgorithmis augmentedwith theschedulabilitycheckusingnon-
preemptive scheduleraswell, andsoin theplotsthosecasessimply show upaszeropercentimprovement.
Theseresultscanbe seenin Figure5(a),wherenon-preemptive schedulingperformsthe bestof all three
when (W�£æ S/ç�S¬§ 143 ; similarresultsareobtainedfor othersmallervaluesof � æ�èeé V�ê _�ë X , but thiseffectgoes
awaywhen � æ�èWé V/ê _°ë X " -/363 or more.Thereasonfor this is thatwith largenumberof tasks,anda small
valueof � æ�èWé V/ê _°ë X , thecomputationtimesfor all tasksaresmall. This meansthatany blockingcaused
by non-preemptionhaslittle effect onschedulability, whichgivesriseto higherbreakdown utilization.
6 PreemptionOverheads
The applicationof our fixed priority schedulingmodelwith preemptionthresholdnot only provides
betterschedulability, but also may reducethe numberof preemptionsas comparedto pure preemptive
scheduling.Indeed,this wasoneof theprimarymotivationsbehindthedevelopmentof preemptionthresh-
old in theThreadXoperatingsystem[10]. Certainly, in theextremecaseif a tasksetis schedulableusing
non-preemptive scheduling,thenthereareno preemptions.Evenotherwise,we expectthat theuseof pre-
emptionthresholdsshouldpreventsomeunnecessarypreemptions.
Onepossiblebenefitbroughtby reducedpreemptionsis further improvementin schedulabilitywhen
theschedulingoverheadcausedbypreemptionsis takeninto account.Typically, suchanalysisassociatestwo
context switchespertask,andthatoverheadis addedto thecomputationtimeof a task[18]. Unfortunately,
it doesnot seemthatthis canbereduced(in theworst-case)with theuseof preemptionthresholds.On the
otherhand,if we canshow that therearereducednumberof preemptions(on anaverage),thenany such
timesavingsimpliesthattheprocessoris availablefor otherbackground/soft-real-timejobs,whichcanonly
bebeneficial.
In this section,we first show how thenumberof preemptionsin a given(schedulable)tasksetcanbe
reducedin a systematicmanner, while maintainingschedulability. We thenusesimulationresultsto seeif
thereis any substantialreductionin thenumberof preemptions.
6.1 PreemptionThresholdAssignmentto ReducePreemptions
Clearly, any feasibleassignmentof priorities andpreemptionthresholdswill be no worsethanpure
preemptive scheduling.However, for a given taskset(andassumingthat priorities aregiven), theremay
0
5
10
15
20
25
5 10 15 20 25 30 35 40 45 50
Incr
ease
in B
reak
dow
n U
tiliz
atio
n
ðNumber of Tasks
AverageMaximum
(a)MaximumPeriod= 10
0
5
10
15
20
25
5 10 15 20 25 30 35 40 45 50
Incr
ease
in B
reak
dow
n U
tiliz
atio
n
ðNumber of Tasks
AverageMaximum
(a)MaximumPeriod= 100
Figure 4. Schedulability Improvement with Preemption Threshold as compared to Preemptive
Scheduling.
beseveral feasiblethresholdassignments,andsomewill reducepreemptionsmorethanothers.Earlier in
this paper, we presentedanoptimalalgorithmto assignpreemptionthresholds(Figure 2). Thealgorithm
attemptsto assignthesmallestfeasiblepreemptionthresholdvalues,which wasnecessaryto ensureits op-
timality. However, it is possiblethatthereareotherfeasibleassignmentswith largerpreemptionthresholds
thatwill incur fewer preemptions.
Our approachto reducepreemptionsis to usethe assignmentgeneratedby our optimal threshold
assignmentasa startingpoint. Then,we iterateover thesolutionandattemptto assignthelargestfeasible
preemptionthresholdvaluesto taskssuchthatthetasksetremainsschedulable.Clearly, largerpreemption
0
20
40
60
80
100
5 10 15 20 25 30 35 40 45 50
Incr
ease
in B
reak
dow
n U
tiliz
atio
n
ðNumber of Tasks
AverageMaximum
(a)MaximumPeriod= 10
0
20
40
60
80
100
5 10 15 20 25 30 35 40 45 50
Incr
ease
in B
reak
dow
n U
tiliz
atio
n
ðNumber of Tasks
AverageMaximum
(a)MaximumPeriod= 100
Figure 5. Schedulability Improvement with Preemption Threshold as compared to Non-
Preemptive Scheduling.
thresholdvaluesreducethechancesof preemptions,andtherefore,shouldresultin lesspreemptions.
Figure6 givesthealgorithmthatattemptsto assignlargerpreemptionthresholdvaluesto tasks.The
algorithmconsidersonetaskatatime,startingfrom thehighestpriority task,andtriesto assignit thelargest
thresholdvaluethat will maintainthe feasibility of the system.We do this onestepat a time, andcheck
the responsetime of the affectedtaskto ensurethat the systemstaysschedulable.By goingfrom highest
priority taskto the lowestpriority task,we ensurethatany changein thepreemptionthresholdassignment
in latter(lowerpriority) taskscannotincreasetheassignmentof a former(higherpriority) task,andthuswe
only needto go throughthelist of tasksonce.
Algorithm: AssignMaximum PreemptionThresholds
/* Assumesthat taskpriorities arefixed,anda */
/* setof feasiblepreemptionthresholdsareassigned*/
(1) for (i := n down to 1)
(2) while (schedulable== TRUE) && ( � � ¡�( )
(3) � � += 1; /* try a largervalue*/
(4) Let LK bethetasksuchthat � K " � � ./* Calculatetheworst-caseresponsetimeof taskj
andcompare it with deadline*/
(5) Q K := WCRT( LK );(6) if ( Q K£M � K ) then // Taskj is not schedulable.
(7) schedulable:= FALSE
(8) ��� -= 1;
(9) endif
(10) end
(11) schedulable:= TRUE
(12) end
(13) return
Figure 6. Algorithm for Finding Maxim um Preemption Threshold
6.2 Simulation Designand Results
We usethesamerandomlygeneratedtasksetsthatwe usedin theprevioussection.We simulatethe
executionof a tasksetfor 100000time units,andtrack thenumberof preemptions.With � æ�èeé V�ê _�ë X "-/34363 , this givesat least1000 instancesof eachtask in a simulationrun. We want to seethe savings in
preemptionswhenusingpreemptionthresholdsascomparedto purepreemptive scheduling.Accordingly,
weusepercentagereductionin thenumberof preemptionsasthemetric,which is definedas:ñ�ò � é ê�V�V ��ó�ô _°ë�( S<õ 0 ñ�ò � é êöV�V ��ó�ô _°ë�( S<õ ÷ñ�ò � é ê�V/V ��ó�ô _�ëö( SLõ ÷ ík-/343
0
5
10
15
20
25
30
35
40
45
50
5 10 15 20 25 30 35 40 45 50
Per
cent
age
Red
uctio
n in
Pre
empt
ions
ø
Number of Tasks
MaxPeriod = 10MaxPeriod = 20MaxPeriod = 50
MaxPeriod = 100MaxPeriod = 500
MaxPeriod = 1000
Figure 7. Average Percentage Reduction in Number of Preemptions for Preemption Threshold
Scheduling as compared to Pure Preemptive Scheduling
whereñ�ò � é ê�V�V ��ó�ô _°ë�( S<õ and
ñ�ò � é ê�V/V ��ó�ô _�ë�( SLõ ÷ are the numberof preemptionsencounteredin a
particularsimulationrunwith preemptiveschedulingandpreemptionthresholdschedulingrespectively.
Wedid onesimulationrunfor eachtasksetgeneratedin thesimulationsof Section3. Thecomputation
timesfor the taskswerechosenby scalingthemto the largestvalueat which the tasksetwasschedulable
underpreemptivescheduling(i.e. thebreakdown utilization). We assignedprioritiesto thesetasksetsusing
therate-monotonic(optimal)algorithm. We usethesamepriorities for thepreemptionthresholdcase,but
additionallyassignedpreemptionthresholdsto the tasksasdescribedabove. That is, we first generateda
feasibleassignmentof preemptionthresholds.Then,we usedthealgorithmin Figure6 to assigna new set
of preemptionthresholds.During thesimulation,we randomlyassignedtheinitial arrival time of eachtask � in therangeF 3 : � � G .We plot the averagepercentagereductionin the numberof preemptionsfor preemptionthreshold
schedulingascomparedto preemptivescheduling.Theresultsareshown in Figure7. As canbeseenin the
figure, thereis a significantreductionin preemptionsfor smallnumberof tasks,but it tapersdown to less
than5%asthenumberof tasksis increased.Also, for any givennumberof tasks,thenumberof reductions
is largerfor largervaluesof � æ�èeé V/ê _�ë X , i.e.,whentheperiodrangeis larger.
7 RelatedWork
Like many papersin real-timeliterature,ourwork hasrootsin real-timescheduling.In thelasttwenty
years,schedulingtheoryhasbeenwidely studiedto provide guidelinesfor developinghardreal-timesys-
tems. Many resultshave beenproducedfor non-idling schedulingover a singleprocessor. Theseresults
fall into two categories: fixed-priority schedulersanddynamicpriority schedulers.Alternatively, onecan
categorizetheschedulersaspreemptive or non-preemptive. While dynamicpriority schedulerscanachieve
higherschedulabilityfor tasksets,fixed priority schedulersarecommonlypreferreddue to lower over-
headsin implementation.Moreover, mostcontemporaryreal-timeoperatingsystemsprovidedirectsupport
for fixedpriority preemptive scheduling,andalmostnoneprovide supportfor dynamicpriority scheduling
algorithmssuchasearliestdeadlinefirst (although,thepriority canbechangedfrom theapplication).
Schedulabilityanalysisin fixedpriority preemptiveschedulingis oftenbasedoncomputingtheworst-
caseresponsetimesandby comparingthemwith deadlines.Thenotionof level-i busyperiodwasintroduced
by Lehoczky [12] andcomputationof theworst-caseresponsetime for generaltasksetsis shown in [9, 8,
12, 18].
With fixed-priority preemptive scheduling,the deadlinemonotonicorderinghasbeenshown to be
optimal[13] for tasksetswith relativedeadlineslessthanor equalto periods.A specialcaseof thedeadline
monotonicorderingis the rate monotonicorderingwhen deadlinesequalperiods[14]. However, for a
generaltaskset,wheredeadlinesarenot relatedto the periods,Lehoczky [12] pointedout that deadline
monotonicorderingis not optimal. Audsley [1, 18] givesan optimal priority assignmentprocedurewith
complexity of ù'ú)ûCü ý .Deadlinemonotonicpriority orderingis no longeroptimal in the context of non-preemptive fixed-
priority schedulingof generaltaskset. However, it is optimal for a tasksetwith deadlineslessthanor
equalto periodsif smallerdeadlineimpliessmalleror equalcomputationtime[5]. Furthermore,theoptimal
priority assignmentalgorithmproposedby Audsley [1] is shown to bestill valid [5]. Thefeasibility is still
closelyrelatedwith computationof worst-caseresponsetime. Moreover, thelevel-i busyperiodanalysisis
alsovalid in thecontext of non-preemptivescheduling[5].
The schedulingwork that comesclosestto ours is the schedulingof taskswith varying execution
priorities [8, 4]. In [8], schedulabilityanalysisfor taskswith varyingexecutionpriorities is given,andin
fact it is recognizedthat the schedulabilityof a tasksetcanbe improved by increasinga task’s priority
sometimeduring its execution. The samegeneralideais usedin [4] with dual priority schedulingexcept
that the taskpriority is raisedaftera fixedamountof (real) time. The notionof preemptionthresholdcan
beviewedasaspecialcaseof schedulingwith varyingexecutionpriorities.However, ourwork differsfrom
thesein thatwegobeyondresponsetimeanalysis,andfocusonschedulingattributeassignment.
Our work on priority andpreemptionthresholdassignmentfor tasksis relatedto thegeneralproblem
of assigningfeasibletaskattributes. Besidesthe priority assignmentalgorithmsmentionedabove, several
researchershaveaddressedthisproblemin differentcontexts. In [6] feasibletaskattributessuchasperiods,
deadlines,andoffsetswerederived from end-to-endtiming requirementsspecifiedon systeminputsand
outputs.Thework hasbeenfollowedupby anumberof otherresearchers,addressingdifferentvariationsof
feasibletaskattributeassignmentproblem[3, 15, 16, 7].
Thenotionof preemptionthresholdis usedin variousothercontexts. Mostnotably, it is usedto avoid
unboundedpriority inversionsin priority ceilingprotocols[17, 2] by raisingthepriority of a taskwhile in a
critical section.
8 Concluding Remarks
We presenta generalizedfixed-priority schedulingmodelwith the notion of preemptionthreshold.
This modelbridgesthegapbetweenpreemptive andnon-preemptiveschedulingandincludesbothof them
as specialcases. It capturesthe bestof both models,and as a result it can feasibly scheduletask sets
whicharenot feasiblewith eitherpurepreemptivescheduling,or non-preemptivescheduling.In this paper,
we presentsimulationresultsover randomlygeneratedtasksetsshowing our modelachievessubstantial
improvementof the schedulableutilization of tasksetsover both preemptive andnon-preemptive cases.
Furthermore,usingtheconceptof level-i busyperiod,we derive theequationfor computingtheworst-case
responsetime of periodicor sporadictasksin thisgeneralmodel.
Usingthis analysis,we addressedtheproblemof findinganoptimalpriority orderingandpreemption
thresholdassignmentthatensuresschedulability. Basedon the theoreticalresultswe developedunderthe
model,we proposedanefficient algorithmfor finding optimalpreemptionthresholdassignmentfor a task
set with predefinedpriority. Finally using this thresholdassignmentalgorithm, we develop an optimal
searchalgorithmto find theoptimalprioritiesandpreemptionthresholds.A heuristicalgorithm,whichwill
dominatebothpreemptiveandnon-preemptiveschedulingis alsoproposedto achieve betterefficiency.
Oneinterestingaspectof this new schedulingmodelis that it achievesbetterschedulabilitywithout
increasingthe overheads.This is in sharpcontrastto dynamicpriority schemes,suchasearliestdeadline
first, which achievehigherschedulableutilizationat thecostof significantlyhigheroverheadsascompared
to fixed priority scheduling.The only additionalcostof this new schedulingmodelis anextra field to be
associatedwith tasks(for keepingtheirpreemptionthresholdvalues).With aminormodification,thekernel
canswitchbetweenthe normalpriority andthe preemptionthresholdof a task,asdictatedby the model.
Contrastthis with the dual-priorityschemeof Burns [4], wherethe implementationcostsarenon-trivial.
We alsonotethat the modelcanbeeasilyemulatedat userlevel, whenit is not supportedby a real-time
kernel.Of course,thereis anadditionalcostin thiscaseto changepriorities.
Not only doesthisnew schedulingmodelnotaddany additionaloverheads,it canreduceoverheadsby
incorporatingthebestaspectsof non-preemptive schedulinginto thepreemptive schedulingmodel. Thus,
themodelcanbeusedto introduceonly enoughpreemptabilityasis necessaryto achieve feasibility. In this
way, it canbeusedto reduceschedulingoverheadsby reducingthenumberof preemptions.Oursimulation
resultssupportthisconjecture,andshow thatin many casestherecanbeamarkedreductionin preemptions,
makingtheCPUbandwidthavailablefor backgroundandsoft real-timepriority tasks.
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