DTIC AD1014923: Reducing Response Time Bounds for DAG-Based Task Systems on Heterogeneous Multicore Platforms

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Reducing Response-Time Bounds for DAG-Based Task Systems on
Heterogeneous Multicore Platforms ∗
KechengYang,MingYang,andJamesH.Anderson
DepartmentofComputerScience,UniversityofNorthCarolinaatChapelHill
Abstract donebycellularbasestationsinwirelessnetworks.Dueto
spaceconstraints,werefrainfromdelvingintospecificsre-
Thispaperconsidersforthefirsttimeend-to-endresponse-
garding this particular application domain, opting instead
time analysis for DAG-based real-time task systems im-
foramoreabstracttreatmentoftheproblemathand.
plemented on heterogeneous multicore platforms. The spe-
This problem involves the scheduling of real-time data-
cific analysis problem that is considered was motivated by
flowsonheterogeneouscomputationalelements(CEs),such
anindustrialcollaborationinvolvingwirelesscellularbase
as CPUs, digital signal processors (DSPs), or one of many
stations. The DAG-based systems considered herein allow
typesofhardwareaccelerators.Eachdataflowisrepresented
intra-taskparallelism:whileeachinvocationofatask(i.e.,
byaDAG,thenodes(resp.,edges)ofwhichrepresenttasks
DAG node) is sequential, successive invocations of a task
(resp.,producer/consumerrelationships).Agiventaskisre-
mayexecuteinparallel.Intheproposedanalysis,thischar-
strictedtorunonaspecificCEtype.Taskpreemptionmay
acteristicisexploitedtoreduceresponse-timebounds.Ad-
beimpossibleforsomeCEsandshouldinanycasebedis-
ditionally,thereissomeleewayinchoosinghowtosettasks’
couraged. Each DAG has a single source task that is in-
relativedeadlines.Itisshownthatbyresolvingsuchchoices
voked periodically. Intra-task parallelism is allowed in the
holistically via linear programming, response-time bounds
sense that consecutive jobs (i.e., invocations) of the same
canbefurtherreduced.Finally,intheconsideredusecase,
task can execute in parallel (but each job executes sequen-
DAGsaredefinedbaseduponjustafewtemplatesandindi-
tially).Infact,alaterjobcanfinishearlierduetovariations
viduallyoftenhavequitelowutilizations.Itisshownthat,by
in running times.1 The DAGs to be supported are defined
combining many such DAGs into one of higher utilization,
using a relatively small number of “templates,” i.e., many
response-timeboundscanoftenbedrasticallylowered.The
DAGs may exist that are structurally identical. The chal-
effectiveness of these techniques is demonstrated via both
lenge is to devise a multi-resource, real-time scheduler for
case-studyandschedulabilityexperiments.
supportingdataflowsasdescribedherewithaccompanying
per-dataflow end-to-end response-time analysis. Note that,
1 Introduction
althoughresponse-timeboundsarerequired(andshouldnot
The multicore revolution is currently undergoing a second betoolarge),strictdeadlinesarenotenforced.
waveofinnovationintheformofheterogeneoushardware Relatedwork. Theliteratureonreal-timesystemsincludes
platforms. In the domain of real-time embedded systems, muchworkpertainingtotheschedulingofDAG-basedtask
suchplatformsmaybedesirabletouseforavarietyofrea- systemsonidenticalmultiprocessorplatforms[1,3,4,5,6,
sons.Forexample,ARM’sbig.LITTLEmulticorearchitec- 8,9,11,13,15,16,17,18,19,20,21,22].However,weare
ture[7]enablesperformanceandenergyconcernstobebal- notawareofanycorrespondingworkdirectedatheteroge-
anced by providing a mix of relatively slower, low-power neousplatforms.Moreover,theproblemabovehasfacets—
coresandfaster,high-powerones.Unfortunately,themove such as allowing intra-task parallelism and defining poten-
towards greater heterogeneity is further complicating soft- tially many DAGs based upon relatively few templates—
ware design processes that were already being challenged that have not been fully explored. (Intra-task parallelism
onaccountofthesignificantparallelismthatexistsin“con- has been considered in a limited context in DAG-related
ventional” multicore platforms with identical processors. workpertainingtoidenticalmultiprocessors[6,8,21].)Ad-
Suchcomplicationsareimpedingadvancementsintheem- ditionally,inpriorworkonthereal-timeschedulingoftask
beddedcomputingindustrytoday. systems—whicharenotDAG-based—uponheterogeneous
Problem considered herein. In this paper, we report on platforms, task-to-CE assignments have been a paramount
our efforts towards solving a particular real-time analysis concern. In our setting, this is a non-issue, as this assign-
problem concerning heterogeneity motivated by an indus- mentispre-determinedbasedonCEfunctionalities.
trialcollaboration.Thisproblempertainstotheprocessing Beyond the real-time systems community, DAG-based
systems implemented on heterogeneous platforms have
∗WorksupportedbyNSFgrantsCPS1239135,CNS1409175,andCPS
1446631, AFOSR grant FA9550-14-1-0161, ARO grant W911NF-14-1- 1Intheconsideredapplicationdomain,thedataproducedbysubsequent
0499,andfundingfrombothGeneralMotorsandFutureWeiCorp. jobscanbebuffereduntilallpriorjobsofthesametaskhavecompleted.
1
beenconsideredbefore(e.g.,[2,14,24]).However,allsuch 2 SystemModel
work known to us focuses on one-shot, aperiodic DAG-
Inthissection,weformalizethedataflow-schedulingprob-
basedjobs,ratherthanperiodicorsporadicDAG-basedtask
lemdescribedinSec.1andintroducerelevantterminology.
systems. Moreover, real-time issues are considered only
EachdataflowisrepresentedbyaDAG,asdiscussedearlier.
obliquelyfromtheperspectivesofjobadmissioncontrolor
We specifically consider a system G = {G ,G ,
jobmakespanminimization. 1 2
...,G } comprised of N DAGs. The DAG G con-
N i
Contributions. Inthispaper,weformalizetheproblemde-
sists of n nodes, which correspond to n tasks, de-
i i
scribedaboveandthenaddressitbyproposingascheduling noted τ1,τ2,...,τni. Each task τv releases a (po-
approachandassociatedend-to-endresponse-timeanalysis. tentiallyiinfiinite) siequence of jobs iJv , Jv ,.... The
Inthefirstpartofthepaper,weattacktheproblembypre- i,1 i,2
edges in G reflect producer/consumer relationships. A
i
senting a transformation process whereby successive task particular task τv’s producers are those tasks with
models are introduced such that: (i) the first task model outgoing edges idirected to τv, and its consumers
directly formalizes the problem above; (ii) prior analysis are those with incoming edgies directed from τv.
can be applied to the last model to obtain response-time Thejth joboftaskτv,Jv ,can- i
boundsunderearliest-deadline-first(EDF)scheduling;and not commence execiutioi,nj until 1
(iii)eachsuccessivemodelisarefinementofthepriorone the jth jobs of all of its produc- 1
inthesensethatallDAG-basedprecedenceconstraintsare
ershavecompleted;thisensures
preserved. Such a transformation approach was previously that its necessary input data is 2 3
usedbyLiuandAnderson[18]inworkonDAG-basedsys- 1 1
available. Such job dependen-
tems, but that work focused on identical multiprocessors.
cies only exist with respect to
Moreover, our work differs from theirs in that we allow the same invocation of a DAG, 4
intra-task parallelism. This enables much smaller end-to- 1
and not across different invoca-
endresponse-timeboundstobederived.
tions. That is, while jobs must
Afterpresentingthistransformationprocess,wediscuss
execute sequentially, intra-task Figure1:ADAGG .
two techniques that can reduce the response-time bounds 1
parallelismisallowed.
enabledbythisprocess.Thefirsttechniqueexploitsthefact
Example1. Fig.1showsanexampleDAG,G .Taskτ4’s
thatsomeleewayexistsinsettingtasks’relativedeadlines. 1 1
producers are tasks τ2 and τ3, thus for any j, J4 needs
Bysettingmoreaggressivedeadlinesfortasksalong“long” 1 1 1,j
pathsinaDAG,theoverallend-to-endresponse-timebound inputdatafromeachofJ12,j andJ13,j,soitmustwaituntil
of that DAG can be reduced. We show that such deadline those jobs complete. Because intra-task parallelism is al-
adjustmentscanbemadebysolvingalinearprogram. lowed,J14,j andJ14,j+1couldpotentiallyexecuteinparallel.
The second technique exploits the fact that, in the con- To simplify analysis, we assume that each DAG G has
i
sideredcontext,DAGsaredefinedusingrelativelyfewtem- exactlyonesourcetaskτ1,whichhasonlyoutgoingedges,
i
platesandtypicallyhavequitelowutilizations.Thesefacts and one sink task τni, which has only incoming edges.
i
enable us to reduce response-time bounds by combining Multi-source/multi-sink DAGs can be supported with the
manyDAGsintooneoflargerutilization.Asaverysimple additionofsingular“virtual”sourcesandsinksthatconnect
example,twoDAGswithaperiodof10timeunitsmightbe multiplesourcesandsinks,respectively.Virtualsourcesand
combinedintoonewithaperiodof5timeunits.Aresponse- sinkshaveaworst-caseexecutiontime(WCET)ofzero.
time-boundreductionisenabledbecausetheseboundstend We consider the scheduling of DAGs as just described
tobeproportionaltoperiods.Intheconsideredapplication onaheterogeneoushardwareplatformconsistingofdiffer-
domain,theextentofcombiningcanbemuchmoreexten- ent types of CEs. A given CE might be a CPU, DSP, or
sive:upwardsof40DAGsmaybecombinable. somespecializedhardwareaccelerator(HAC).TheCEsare
As a final contribution, we evaluate our proposed organizedinM CEpools,whereeachCEpoolπ consists
k
techniques via case-study and schedulability experiments. ofm identicalCEs.Eachtaskτv hasaparameterPv that
k i i
These experiments show that our techniques can signifi- denotes the particular CE pool on which it must run, i.e.,
cantlyreduceresponse-timebounds.Furthermore,ouranal- Pv =π meansthateachjobofτv mustbescheduledona
i k i
ysis supports “early releasing” [10] (see Sec. 7) to im- CEintheCEpoolπ .TheWCEToftaskτvisdenotedCv.
k i i
prove observed end-to-end response times. We experimen-
AlthoughtheproblemdescriptioninSec.1indicatedthat
tallydemonstratetheefficacyofthisaswell.
sourcetasksarereleasedperiodically,wegeneralizethisto
Organization. In the rest of this paper, we formalize the allow sporadic releases, i.e., for the DAG G , the job re-
i
considered problem (Sec. 2), present the refinements men- leases of τ1 have a minimum separation time, denoted T .
i i
tionedabovethatenabletheuseofprioranalysis(Secs.3– Anon-sourcetaskτv (v > 1)releasesitsjth jobJv when
i i,j
4), show that the bounds arising from this analysis can be thejth jobsofallitsproducertasksinG havecompleted.
i
improved via linear programming (Sec. 5) and DAG com- That is, letting av and fv denote the release (or arrival)
i,j i,j
bining(Sec.6),discussearlyreleasing(Sec.7),presentour andfinishtimesofJv ,respectively,
i,j
case-study(Sec.8)andschedulability(Sec.9)experiments,
andconclude(Sec.10). av =max{fw |τw isaproducerofτv}. (1)
i,j i,j i i
2
J11,1 J11,2
WealsouseΓ todenotethesetoftasksthatarerequiredto
τ1 k
1 executeontheCEpoolπ ,i.e.,
k
J12,1 J12,2 Γ ={τv |Pv =π }. (4)
τ2 k i i k
1
The overutilization of a CE pool could cause unbounded
J13,1 J13,2 responsetimes,sowerequireforeachk,
τ3
1 (cid:88)
uv ≤m . (5)
J14,2 i k
τiv∈Γk
τ4 J14,1
1
3 Offset-BasedIndependentTasks
end-to-end response time
end-to-end response time In this section, we present a second task model, which as
Time shown in Sec. 4 can be viewed as a refinement of that just
0 5 10 15 20
presented. The prior model is somewhat problematic be-
Job Release Job Deadline Job Completion CPU Execution DSP Execution cause of difficult-to-analyze dependencies among jobs. In
(Assume depicted jobs are scheduled alongside other jobs, which are not shown.) particular, by (1), the release times of jobs of non-source
tasksdependonthefinishtimesofotherjobs,andhenceon
Figure2:ExampleschedulefortheDAGinG inFig.1. their execution times. By (2), deadlines (and hence priori-
1
ties)ofjobsareaffectedbysimilardependencies.
Inordertoeaseanalysisdifficultiesassociatedwithsuch
The response time of job Jv is definedas fv −av , and
i,j i,j i,j jobdependencies,weintroduceheretheoffset-basedinde-
theend-to-endresponsetimeoftheDAGG asfni −a1 .
i i,j i,j pendent task (obi-task) model. Under this model, tasks are
Example 2. Fig. 2 depicts an example schedule for the partitionedintogroups.Theithsuchgroupconsistsoftasks
DAGG1 inFig.1,assumingtaskτ12 isrequiredtoexecute denoted τi1,τi2,...,τini, where τi1 is a designated source
on a DSP and the other tasks are required to execute on a task that releases jobs sporadically with a minimum sepa-
CPU.Thefirst(resp.,second)jobofeachtaskhasalighter rationofTi.Thatis,foranypositiveintegerj,
(resp., darker) shading to make them easier to distinguish.
a1 −a1 ≥T . (6)
Tasksτ2 andτ3 haveonlyoneproducer,τ1,sowhentask i,j+1 i,j i
1 1 1
τ1finishesajobattimes3and8,τ2andτ3releaseajobim-
1 1 1 Job releases of each non-source task τv are governed by a
mediately.Incontrast,taskτ4hastwoproducers,τ2andτ3. i
1 1 1 new parameter Φv, called the offset of τv. Specifically, τv
Therefore,τ4cannotreleaseajobattime5whenτ3finishes i i i
1 1 releases its jth job exactly Φv time units after the release
ajob,butrathermustwaituntiltime6whenτ2alsofinishes i
1 timeofthejthjobofthesourcetaskτ1ofitsgroup.Thatis,
ajob.Notethatconsecutivejobsofthesametaskmightexe- i
cuteinparallel(e.g.,J4 andJ4 executeinparallelduring av =a1 +Φv. (7)
1,1 1,2 i,j i,j i
[11,12)).Furthermore,foragiventask,alater-releasedjob
(e.g., J4 ) may even finish earlier than an earlier-released Forconsistency,wedefine
1,2
one(e.g.,J4 )duetoexecution-timevariations. Φ1 =0. (8)
1,1 i
Scheduling. SincemanyCEsarenon-preemptible,weuse Undertheobi-taskmodel,ajobofataskτv canbesched-
i
thenon-preemptiveglobalEDF(G-EDF)schedulingalgo- uledatanytimeafteritsreleaseindependentlyoftheexecu-
rithmwithineachCEpool.ThedeadlineofjobJv isgiven tionofanyotherjobs,evenjobsofthesametaskτv.
i,j i
by Thedefinitionssofarhavedealtwithjobreleases.Addi-
dv =av +Dv, (2) tionally,thetwoper-taskparametersCvandPvfromSec.2
i,j i,j i i i
areretainedwiththesamedefinitions.
whereDvistherelativedeadlineoftaskτv.Forexample,in
i i Thefollowingpropertyshowsthateveryobi-taskτv has
theexamplescheduleinFig.2,relativedeadlinesofD1 = i
1 aminimumjob-releaseseparationofT .
7,D2 =4,D3 =6,andD4 =8areassumed. i
1 1 1
Property1. Foranyobi-taskτv,av −av ≥T .
In the context of this paper, deadlines mainly serve the i i,j+1 i,j i
purpose of determining jobs’ priorities, rather than strict Proof.
timing constraints for individual jobs. Therefore, deadline av −av = {by(7)}
i,j+1 i,j
misses are acceptable as long as the end-to-end response
(a1 +Φv)−(a1 +Φv)
timeofeachDAGcanbereasonablybounded. i,j+1 i i,j i
≥ {by(6)}
Utilization. Wedenotetheutilizationoftaskτv by
i T
i
Cv
uv = i . (3)
i T
i
3
4 Response-TimeBounds instantiation of the npc-sporadic task τv = (Cv,T ,Dv).
i i i i
Hence,anyinstantiationofanobi-taskset{τv |Pv = π }
In this section, we establish two results that enable prior i i k
is an instantiation of the npc-sporadic task set {τv | Pv =
worktobeleveragedtoestablishresponse-timeboundsfor i i
π }.Also,sinceobi-tasksexecuteindependentlyofonean-
DAG-basedtasksystems.First,weshowthat,undertheobi- k
other, obi-tasks executing in different CE pools cannot af-
taskmodelwitharbitraryoffsetsettings,per-taskresponse-
fecteachother.SinceeachCEpoolπ hasm identicalpro-
time bounds can be derived by exploiting prior work per- k k
cessors,theproblemwemustconsideristhatofscheduling
taining to a task model called the npc-sporadic task model
aninstantiationofthenpc-sporadictaskset{τv|Pv =π }
(“npc” stands for “no precedence constraints”—this refers i i k
on m identical processors. Since Theorem 1 applies to a
tothelackofprecedenceconstraintsamongjobsofthesame k
non-concretenpc-sporadictaskset,itappliestoeverycon-
task) [12, 25]. Second, we show that, by properly setting
creteinstantiationofsuchataskset.Thus,wehavethefol-
offsets,any DAG-basedtask systemcanbe transformedto
lowingresponse-timeboundforeachobi-taskτv:
acorrespondingobi-tasksystem. i
 
4.1 Response-TimeBoundsforObi-Tasks 1 (cid:88) (cid:88)
Riv =m Div· uwl + (uwl ·max{0,Tl−Dlw})
Annpc-sporadictaskτi isspecifiedby(Ci,Ti,Di),where k τlw∈Γk τlw∈Γk
C isitsWCET,T istheminimumseparationtimebetween
i i m −1
consecutivejobreleasesofτ ,andD isitsrelativedeadline. + max {Cw}+ k Cv, (9)
Asbefore,τ ’sutilizationisiu =Ci/T . τlw∈Γk l mk i
i i i i
The main difference between the conventional sporadic whereΓ ={τw |Pw =π }.
taskmodelandthenpc-sporadictaskmodelisthatthefor- k l l k
Notethat(9)isapplicableaslongasallrelativedeadlines
mer requires successive jobs of each task to execute in se-
arenon-negative,andappliestoanyarbitraryoffsetsetting.
quence while the latter allows them to execute in paral-
lel. That is, under the conventional sporadic task model,
4.2 FromDAG-BasedTaskSetstoObi-TaskSets
jobJ cannotcommenceexecutionuntilitspredecessor
i,j+1
J completes, even if a , the release time of J , We now show that, by properly setting offsets, any DAG-
i,j i,j+1 i,j+1
haselapsed.Incontrast,underthenpc-sporadictaskmodel, based task set can be transformed to an obi-task set, and
anyjobcanexecuteassoonasitisreleased.Notethat,al- per-DAG end-to-end response-time bounds can be derived
thoughweallowintra-taskparallelism,eachindividualjob byleveragingtheobi-taskresponse-timeboundsjuststated.
stillmustexecutesequentially. AnyDAG-basedtasksetcanbeimplementedbyanobi-
Yang and Anderson [25] investigated the G-EDF tasksetinanobviousway:eachDAGbecomesanobi-task
schedulingofnpc-sporadictasksonuniformheterogeneous groupwiththesametaskdesignatedasitssource,andallTi,
multiprocessor platforms where different processors may Civ, and Piv parameters are retained without modification.
havedifferentspeeds.Bysettingeachprocessor’sspeedto Whatislessobviousishowtodefinetaskoffsetsunderthe
be1.0,thefollowingtheoremfollowsfromtheirwork. obi-task model. This is done by setting each Φvi (v (cid:54)= 1)
parametertobeaconstantsuchthat
Theorem 1. (Follows from Theorem 4 in [25]) Consider
the scheduling of a set of npc-sporadic tasks τ on m iden- Φv ≥ max {Φk+Rk}, (10)
i i i
ticalmultiprocessors.Undernon-preemptiveG-EDF,each τk∈prod(τv)
i i
npc-taskτ ∈τ hasthefollowingresponse-timebound,pro-
(cid:80) i where prod(τv) denotes the set of obi-tasks corresponding
vided u ≤m. i
τl∈τ l totheDAG-basedtasksthataretheproducersoftheDAG-
basedtaskτv inG ,andRk denotesaresponse-timebound
(cid:32) (cid:18) (cid:19)(cid:33) i i i
1 (cid:88) (cid:88) fortheobi-taskτk.Fornow,weassumethatRk isknown,
D · u + u ·max{0,T −D } i i
m i τl∈τ l τl∈τ l τl∈τ l l butlater,wewillshowhowtocomputeit.
m−1 Example3. ConsideragaintheDAGG1inFig.1.Assume
+mτl∈aτx{Cl}+ m Ci. thhaavte, arfetseproanpspel-ytiimngetbhoeuanbdosvoeftrRa1ns=for9m,aRtio2n=, th5e,oRb3i-t=ask7s,
1 1 1
We now show that Theorem 1 can be applied to obtain andR1 =8,respectively.Then,wecansetΦ1 =0,Φ2 =9,
1 1 1
per-taskresponse-timeboundsforanyobi-taskset. Φ3 = 9,andΦ4 = 16,respectively,andsatisfy(10).With
1 1
these response-time bounds, the end-to-end response-time
Concretevs.non-concrete. Aconcretesequenceofjobre-
boundthatcanbeguaranteedisdeterminedbyR1,R3,and
leases that satisfies a task’s specification (under either the 1 1
R1andisgivenbyR =24.Fig.3depictsapossiblesched-
obi-ornpc-sporadictaskmodel)iscalledaninstantiationof 1 1
ule for these obi-tasks and illustrates the transformation.
thattask.Aninstantiationofataskset isdefinedsimilarly.
LikeinFig.2,thefirst(resp.,second)jobofeachtaskhasa
Incontrast,ataskoratasksetthatcanhavemultiple(poten-
lighter(resp.,darker)shading,andintra-taskparallelismis
tiallyinfinite)instantiationssatisfyingitsspecification(e.g.,
possible(e.g.,J4 andJ4 intimeinterval[23,24)).
minimumreleaseseparation)iscallednon-concrete. 1,1 1,2
By Property 1, any instantiation of an obi-task τv is an Thefollowingpropertiesfollowfromthistransformation
i
4
Thus, a DAG-based task set can be transformed to an
R1
1 obi-tasksetwiththesameper-taskparameters.Giventhese
τ11 per-task parameters, a response-time bound for each obi-
Φ11= 0 R21 taskcanbecomputedby(9)foranyarbitraryoffsetsetting.
τ2 Then, we can properly set the offsets for each obi-task ac-
1
Φ12 R13 ceoacrdhinDgAtGo (i1n0t)opboylocgoincsaildeorridnegr,thstearctoinrrgeswpiothndΦin1g=task0sfoinr
i
τ3 eachsourcetaskτ1,by(8).ByProperty2,theresultingobi-
1 i
Φ3 tasksetsatisfiesallrequirementsoftheoriginalDAG-based
1 R4
1 tasksystem,andbyProperty3,anend-to-endresponse-time
τ4
1 boundR canbecomputedforeachDAGG .
Φ4 i i
1 (Note that the response-time bound for a virtual
end-to-end response time
source/sinkisnotcomputedby(9),butiszerobydefinition,
end-to-end response time
sinceitsWCETiszero.Anyjobofsuchataskcompletesin
R1 : the end-to-end response-time bound for G1 Time zerotimeassoonasitisreleased.)
0 5 10 15 20 25 30
Job Release Job Deadline Job Completion CPU Execution DSP Execution 5 SettingRelativeDeadlines
(Assume depicted jobs are scheduled alongside other jobs, which are not shown.)
Inthepriorsections,weshowedthat,byapplyingourpro-
posed transformation techniques, an end-to-end response-
Figure3:Examplescheduleoftheobi-taskscorrespondingtothe time bound for each DAG can be established, given ar-
DAG-basedtasksinG1inFig.1. bitrary but fixed relative-deadline settings. That is, given
Dv ≥ 0 for any i,v, we can compute correspond-
i
process.AccordingtoProperty2,aDAG-basedtasksetcan ing end-to-end response-time bounds (i.e., Ri for each i)
beimplementedbyacorrespondingsetofobi-tasks,andall by(9),(8),(10),and(13).
producer/consumerconstraintsintheDAG-basedspecifica- SimilarDAGtransformationapproacheshavebeenpre-
tion will be implicitly guaranteed, provided the offsets of sented previously [11, 18], but under the assumption that
theobi-tasksareproperlyset(i.e.,satisfy(10)). intra-taskprecedenceconstraintsexist(i.e.,jobsofthesame
task must execute in sequence). Moreover, in this prior
Property 2. If τk is a producer of τv in the DAG-based work, per-task relative deadlines have been defined in a
i i
tasksystem, then forthe jth jobsofthe correspondingtwo DAG-obliviousway.Byconsideringtheactualstructureof
obi-tasks,fk ≤av . such a DAG, it may be possible to reduce its end-to-end
i,j i,j
response-timeboundbysettingitstasks’relativedeadlines
Proof.By(7),ak =a1 +Φk,andbythedefinitionofRk,
i,j i,j i i soastofavorcertaincriticalpaths.
fk ≤ak +Rk.Thus,
i,j i,j i Consider, for example, the DAG 10
(8)
illustrated in Fig. 4. Suppose that
fk ≤a1 +Φk+Rk. (11)
i,j i,j i i theprioranalysisyieldsaresponse-
10
time bound of 10 for each task, (8)
By(7),av =a1 +Φv,andby(10),Φv ≥Φk+Rk.Thus,
i,j i,j i i i i as depicted within each node. The 10
(15)
correspondingend-to-endresponse-
av ≥a1 +Φk+Rk. (12) 10
i,j i,j i i time bound would then be 40 and (8)
isobtainedbyconsideringtheright-
By(11)and(12),fk ≤av .
i,j i,j sidepath.Now,supposethatweal- 10
Property 3 shows how to compute an end-to-end ter the tasks’ relative-deadline set- (8)
response-timeboundR . tings to favor the tasks along this
i
path at the possible expense of the Figure 4: More
Property 3. In the obi-task system, for each j, all jobs remaining task on the left. Further, highly prioritizing
Ji1,j,Ji2,j,··· ,Jin,ji finish their execution within Ri time suppose this modification changes the right-side path
unitsaftera1i,where the per-task response-time bounds in this DAG de-
creases its end-to-
to be as depicted in parentheses.
R =Φni +Rni. (13) end response-time
i i i Then,this modificationwouldhave
bound.
the impact of reducing the end-to-
Proof.By(7)andthedefinitionofRv,Jv finishesbytime
i i,j endboundto32.
a1 +Φv+Rv.Thus,Jni inparticularfinisheswithinΦni+
i,j i i i,j i Inthissection,weshowthattheproblemofdetermining
Rini = Ri time units after a1i. Also, by (10), Φvi +Riv ≤ the“best”relative-deadlinesettingscanbecastasalinear-
Φnii,sinceτini isthesinglesinkinGi.BecauseΦnii ≤ Ri, programmingproblem,whichcanbesolvedinpolynomial
thisimpliesthat,foranyv,Jiv,j finisheswithinRitimeunits time.Theproposedlinearprogram(LP)isdevelopedinthe
aftera1. nexttwosubsections.
i
5
5.1 LinearProgram Thereare|E|distinctconstraintsinthisset,where|E|isthe
totalnumberofedgesinallDAGsinthissystem.
InourLP,therearethreevariablespertaskτv:Dv,Φv,and
i i i Finally,wehaveasetofconstraintsthatarelinearequal-
Rv.TheparametersT ,Cv,andm areviewedasconstants.
i i i k ityconstraints.
Thus,thereare3|V|variablesintotal,where|V|isthetotal
numberoftasks(i.e.,nodes)acrossallDAGsinthesystem. ConstraintSet(iii): WithConstraintsSet(i),itisclearthat
Beforestatingtherequiredconstraints,wefirstestablishthe wecanre-write(9)asfollows,foreachtaskτv,
i
followingtheorem,whichshowsthatarelative-deadlineset-
 
tingofDy >T ispointlesstoconsider.
Tthheeroersepmonxs2e.-tIifmxDexbyo>undTxof, ttahseknτbyy,wseitltlindgecDrexyase=,aTnxd,eRacxyh, Riv =m1k Div·τlw(cid:88)∈Γkuwl +τlw(cid:88)∈Γk(cid:0)uwl ·(Tl−Dlw)(cid:1)
x m −1
othertask’sresponse-timeboundwillremainthesame. + max {Cw}+ k Cv, (14)
Proof. To begin, note that, by (9), τy does not impact τlw∈Γk l mk i
x
the response-time bounds of those tasks executing on CE where Γ = {τw|Pw = π }. Moreover, by (8), for each
pools other than Pxy. Therefore, the response-time bounds DAGGik, l l k
{Riv | Piv (cid:54)= Pxy} for such tasks are not altered by any Φ1i =0.
changetoDy.
x
Intheremainderoftheproof,weconsiderataskτv such Constraint Set (iii) yields |V| + |G| linear equations,
i
that Pv = Py. Let Rv and R(cid:48)v denote the response-time where|G|denotesthenumberofDAGs.
i x i i
boundsforτv beforeandafter,respectively,reducingDy to ConstraintSets(i),(ii),and(ii)fullyspecifyourLP,with
i x
T .Ifi=xandv =y,thenby(9), theexceptionoftheobjectivefunction.InthisLP,thereare
x
3|V|variables,2|V|+|E|inequalityconstraints,and|V|+
Ry −R(cid:48)y =Dxy−Tx · (cid:88) uw+ |G|linearequalityconstraints.
x x m l
k
τw∈τk
l 5.2 ObjectiveFunction
uy
x (max{0,T −Dy}−max{0,T −T })
m x x x x Different objective functions can be specified for our LP
k
thatoptimizeend-to-endresponse-timeboundsindifferent
>{sinceDy >T }
x x senses.Here,weconsiderafewexamples.
0.
Single-DAG systems. For systems where only a single
Alternatively,ifi(cid:54)=xorv (cid:54)=y,thenby(9), DAG exists, the optimization criterion is rather clear. In
order to optimize the end-to-end response-time bound of
Rv−R(cid:48)v =uyx (max{0,T −Dy}−max{0,T −T }) thesingleDAG,theobjectivefunctionshouldminimizethe
i i m x x x x end-to-endresponse-timeboundoftheonlyDAG,G .That
k 1
={sinceDy >T } is,thedesiredLPisasfollows.
x x
0. minimize Φn1 +Rn1
1 1
subjectto ConstraintSets(i),(ii),and(iii)
Thus,thetheoremfollows.
By Theorem 2, the reduction of Dy mentioned in the
x Multiple-DAG systems. For systems containing multiple
theoremdoesnotincreasetheresponse-timeboundforany
DAGs, choices exist as to the optimization criteria to con-
task. By (10), this implies that none of the offsets, {Φv},
i sider.Welisttwohere.
needstobeincreased.Therefore,byProperty3,noend-to-
endresponse-timeboundincreases.Thesepropertiesmoti-
Minimizingtheaverageend-to-endresponse-time
vateourfirstsetoflinearconstraints.
bound:
ConstraintSet(i): Foreachtaskτv,
i (cid:88)
minimize (Φni +Rni)
0≤Dv ≤T . i i
i i i
subjectto ConstraintSets(i),(ii),and(iii)
2|V|individuallinearinequalitiesarisefromthisconstraint
set,where|V|isthetotalnumberoftasks.
Another issue we must address is that of ensuring that Minimizing the maximum end-to-end response-
the offset settings, given by (10), are encoded in our LP. timebound:
Thisgivesrisetothenextconstraintset.
minimize Y
ConstraintSet(ii): ForeachedgefromτiwtoτivinaDAG subjectto ∀i: Φni +Rni ≤Y
G , i i
i
Φv ≥Φw+Rw. ConstraintSets(i),(ii),and(iii)
i i i
6
T if all of its producers have already finished. For example,
in Fig. 3, J3 cannot execute until time 9, even though its
1,1
G1 correspondingproducerjob,J1 ,hasfinishedbytime3.
1,1
Observed response times can be improved under
G2 deadline-based scheduling without altering analytical
response-timeboundsbyusingatechniquecalledearlyre-
T/2
leasing[10].Whenearlyreleasingisallowed,ajobiseligi-
G bleforexecutionassoonasallofitscorrespondingproducer
[1,2]
Corresponding jobshavefinished,evenifthisconditionissatisfiedbefore
to Invocations of: G1 G2 G1 G2 G1 G2 itsactualreleasetime.Earlyreleasingdoesnotimpactana-
DAG Releases Identical DAGs lyticalresponse-timebounds.Inanappendix,whyprovide
a brief explanation as to why this is the case and consider
Figure5:IllustrationofDAGcombining. anexampleschedulewhereearlyreleasingisallowed.
6 DAGCombining 8 CaseStudy
In the application domain that motivates our work, the To illustrate the computational details of our analysis, we
DAGs to be scheduled are typically of quite low utiliza- considerhereacase-studysystemconsistingofthreeDAGs,
tionsandaredefinedbasedonarelativelysmallnumberof G , G , and G , which are specified in Fig. 6. π is a CE
1 2 3 1
templates that define various computational patterns. Two poolconsistingoftwoidenticalCPUs,andπ isaCEpool
2
DAGsdefinedusingthesametemplatearestructurallyiden- consisting of two identical DSPs. Thus, m = m = 2.
1 2
tical:theyaredefinedbygraphsthatareisomorphic,corre- ThesethreeDAGshavefewernodesandhigherutilizations
spondingnodesfromthetwographsperformidenticalcom- thantypicallyfoundinourconsideredapplicationdomain.
putations, the source nodes are released at the same time, However,onecanimaginethatthesegraphswereobtained
etc.Suchstructurallyidenticalgraphscanbecombinedinto fromcombiningmanyidenticalgraphsoflowerutilization.
one graph with a reduced period and larger utilization, as Whileitwouldhavebeendesirabletoconsiderlargergraphs
long any overutilization of the underlying hardware plat- withmorenodes,graphsfromourchosendomaintypically
form is avoided. Such combining can be a very effective have tens of nodes, and this makes them rather unwieldy
technique,becauseastheexperimentspresentedlatershow, to discuss. Still, the general conclusions we draw here are
ourresponse-timeboundstendtobeproportionaltoperiods. applicabletolargergraphs.
We illustrate this idea with a simple example. Consider
Utilization check. First, we must calculate the total uti-
twoDAGsG andG withacommonperiodofT thatare
1 2 lization of all tasks assigned to each CE pool to make
structurally identical. A schedule of these two DAGs is il- sure that neither is overutilized. We have (cid:80) uv =
lustratedabstractlyatthetopofFig.5.Asillustratedatthe τiv∈Γ1 i
bottomofthefigure,ifthesetwoDAGsarecombined,then 200+100+300 + 133+78+197+73+5 = 1.686 < 2, and
500 1000
theyarereplacedbyastructurallyidenticalgraph,denoted (cid:80) uv = 380 + 16+83+242 =1.101<2.
τiv∈Γ2 i 500 1000
here as G , with a period of T/2. With this change, the
[1,2] Virtual source/sink. Note that all DAGs have a single
providedresponse-timeboundshavetobeslightlyadjusted.
sourceandsink,exceptforG ,whichhastwosinks.Forit,
2
Forexample,ifG hasaresponse-timeboundofR ,
[1,2] [1,2] we connect its two sinks to a single virtual sink τ6, which
then this would also be a response-time bound for G , but 2
1 hasaWCETof0andaresponse-timeboundof0.Wecall
thatforG wouldbeR + T,becauseincombiningthe
2 [1,2] 2 the resulting DAG G(cid:48). For convenience, we include a de-
two graphs, the releases of G are effectively shifted for- 2
2 pictionofthisgraphinFig.10inanappendix.
ward by T time units. While this graph combining idea is
2 Implicit deadlines. We now show how to compute
reallyquitesimple,theexperimentspresentedlatersuggest
response-time bounds assuming implicit deadines, i.e.,
thatitcanhaveaprofoundimpactintheconsideredappli-
Dv = 500 for 1 ≤ v ≤ 4, Dv = 1000 for 1 ≤ v ≤ 5
cation domain. In particular, in that domain, per-DAG uti- 1 2
(the relative deadline of the virtual sink is irrelevant), and
lizations are low enough that upwards of 40 DAGs can be
Dv =1000for1≤v ≤3.Inordertoderiveanend-to-end
combinedintoone.Thus,theactualperiodreductionisnot 3
merelybyafactorof 1 butbyafactorashighas 1 . response-timebound,wefirsttransformtheoriginalDAG-
2 40 based tasks into obi-tasks as described in Sec. 3. Next, we
calculate a response-time bound Rv for each obi-task τv
7 EarlyReleasing i i
by(9).Theresultingtaskresponse-timebounds,{Rv},are
i
Transforming a DAG-based task system to a correspond- listedinTable2,whichisgivenintheappendix.Notethat,
ing obi-task system enabled us to derive an end-to-end asavirtualsink,theresponse-timeboundforthevirtualsink
response-timeboundforeachDAG.However,suchatrans- τ6 doesnotneedtobecomputedby(9),butis0bydefini-
2
formationmayactuallycauseobservedend-to-endresponse tion.By(8)and(10),theoffsetsoftheobi-taskscannowbe
timesatruntimetoincrease,becausetheoffsetsintroduced computed in topological order with respect to each DAG.
inthetransformationmaypreventajobfromexecutingeven The resulting offsets, {Φv}, are also shown in Table 2.
i
7
T = 500
1 τ2 T2= 1000 τ3 τ4
1 2 2 T = 1000
τ11 CP1221== 3π820 τ14 τ21 τ22 CP2233== π823 CP2244== π1197 3 τ31 τ32 τ33
CP1111== π2100 C13τ=13 100 CP1441== π3100 CP1221== π1133 CP2222== π126 C25τ=25 78 CP1133== π713 CP3322== π2242 CP3333== π51
P31= π1 P5= π
2 1
(a)G1 (b)G2 (c)G3
Figure6:DAGsinthecase-studysystem.G hastwosinks,sotoanalyzeit,avirtualsinkτ6mustbeaddedthathasaWCETof0anda
2 2
response-timeboundof0.WeshowtheresultinggraphinFig.10inanappendix.
Finally, by Property 3, we have an end-to-end response- 9 SchedulabilityStudies
time bound for each DAG: R = Φ4 + R4 = 2538.25,
1 1 1 Inthissection,weexpanduponthespecificcasestudyjust
R =Φ6+R6 =4361.5,andR =Φ3+R3 =3376.5.
2 2 2 3 3 3 describedbyconsideringgeneralschedulabilitytrendsseen
LP-based deadline settings. If we use LP techniques to forrandomlygeneratedtasksystems.
optimizeend-to-endresponse-timebounds,thenchoicesex-
istregardingtheobjectivefunction,becauseoursystemhas 9.1 ImprovementsEnabledbyBasicTechniques
multipleDAGs.Weconsiderthreechoiceshere.
We first consider the improvements enabled by the ba-
Minimizing the average end-to-end response-time sic techniques covered in Secs. 4 and 5 that underlie our
bound. For this choice, relative-deadline settings, obi-task work: allowing intra-task parallelism as provided by the
response-timebounds,andobi-taskoffsetsareasshownin npc-sporadictaskmodel,anddeterminingrelative-deadline
Table3(a),foundintheappendix.Theresultingend-to-end settingsbysolvinganLP.
response-time bounds are R = Φ4 + R4 = 3134.5,
1 1 1 Random system generation. We considered a heteroge-
R =Φ6+R6 =2341.2,andR =Φ3+R3 =1736.2.
2 2 2 3 3 3 neousplatformcomprisedofthreeCEpools,eachconsist-
Minimizing the maximum end-to-end response-time ingofeightidenticalCEs.Eachpoolwasassumedtohave
bound. For this choice, relative-deadline settings, obi-task thesametotalutilization.Weconsideredallchoicesoftotal
response-timebounds,andobi-taskoffsetsareasshownin per-poolutilizationsintherange[1,8]inincrementsof0.5.
Table3(b).Theresultingend-to-endresponse-timebounds We generated DAG-based task systems using a method
are R1 = Φ41 +R14 = 2650.4, R2 = Φ62 +R26 = 2650.4, similar to that used by others [3, 15]. These systems were
andR3 =Φ33+R33 =2650.4. generated by first specifying the number of DAGs in the
Minimizing the maximum proportional end-to-end system, N, and the number of tasks per DAG, n. For
response-timebound.Forthischoice,relative-deadlineset- each considered pair N and n, we randomly generated 50
tings, obi-task response-time bounds, and obi-task off- task-systemstructures,eachcomprisedofN DAGswithn
sets are as shown in Table 3(c). The resulting end-to-end nodes.Eachnodeinsuchastructurewasrandomlyassigned
response-time bounds are R = Φ4 + R4 = 2208.9, tooneoftheCEpools,andforeachDAGinthestructure,
1 1 1
R =Φ6+R6 =4417.8,andR =Φ3+R3 =4261.0. one node was designated as its source, and one as its sink.
2 2 2 3 3 3
Further,eachpairofinternalnodes(notasourceorasink)
Early releasing. As discussed in Sec. 7, early releasing was connected by an edge with probability edgeProb,
can improve observed response times without compromis- a settable arameter. Such an edge was directed from the
ing response-time bounds. The value of allowing early re- lower-indexednodetothehigher-indexednode,topreclude
leasing can be seen in the results reported in Table 1. This cycles.Finally,anedgewasaddedfromthesourcetoeach
tablegivesthelargestobservedend-to-endresponsetimeof internalnodewithnoincomingedges,andtothesinkfrom
each DAG, assuming implicit deadlines with and without eachinternalnodewithnooutgoingedges.
earlyreleasing,inaschedulethatwassimulatedfor50,000 Foreachconsideredper-pooluntilizationandeachgen-
timeunits.Analyticalboundsareshownaswell. erated task-system structure, we randomly generated 50
actual task systems by generating task utilizations using
theMATLABfunctionrandfixedsum()[23].Accord-
G1 G2 G3 ing to the application domain that motivates this work,2
Earlyreleasing 1006 897 453 we defined each DAG’s period to be 1 ms. (A task’s
Noearlyreleasing 1966.75 3536.25 2586.0 WCET is determined by its utilization and period.) For
Bounds 2538.25 4361.5 3376.5
2Inapplicationsusuallyconsideredinthereal-time-systemscommu-
Table1:Observedend-to-endresponsetimeswith/withoutearly
nity,muchlargerperiodsarethenorm.Theconsidereddomainisquite
releasingandanalyticalend-to-endresponse-timeboundsforthe different.
implicit-deadlinesetting.
8
identical DAGs will exist. We proposed the technique of
ms)40
Bounds (35 cnnoppnccv--sseppnootirroaanddaiiccl sttaapssokkrssa,,d iLimcP pt-abliscakists ed, deim addpellaiincdietl isndeesadlines DduAcGerceospmobnisnei-ntgiminebSoeuc.nd6st.oWeexpnloowitdthisiscufassctsctohefduurtlhaebrilritey-
Time 30 experimentsthatweconductedtoevaluatethistechnique.
End-to-End Response-122505 RrcssayturanssundtsceeodtmdumormilesnystrsScugyeocescmtntu.eeprm9rera.si1tsig,encedegonxmoecsfrepyaprsNitttseioetmthdneasm.to,ptWfhilnaaeNsttteeesimas.DdApAsloidoGmfdysigie,lteadionrwnaeetaroapllgtryitoen,hncgaweetsrteaasdtsiiekonsd--f-
m
mu10 troduced a new parameter K that indicates the number of
Maxi 5 identical DAGs per template. A period of 1 ms was still
Average 01 2 3 4 5 6 7 8 aCsosomcpiaatreidsownitsheteuapch. DWAeGco.mparedtwostrategies:(i)dono
Total Utilization in Each CE Pool
combining, and compute end-to-end response-time bounds
Figure 7: AMERBsasafunctionoftotalutilizationineachCE assuming N · K independent DAGs; (ii) combine identi-
poolinthecasewhereeachtasksethasfiveDAGs,20tasksper cal DAGs, and compute end-to-end response-time bounds
DAG,andedgeProb=0.5. assuming N DAGs, making adjustments as discussed in
Sec.6toobtainactualresponse-timeboundsfortheDAGs
each considered value of N, n, and total per-pool utiliza- thatwerecombined.Underbothstrategies,thegeneraltech-
tion(onepointinoneofourgraphs),weconsidered50(task niquesevaluatedinSec.9.1wereapplied.
systemstructures)×50(utilizations)=2,500tasksets.
Results. Inallcasesthatweconsidered,theDAGcombin-
Comparison setup. We compared three strategies: (i) ing technique improved end-to-end response-time bounds
transforming to a conventional sporadic task system and significantly.Duetospaceconstraints,wepresenthereonly
using implicit relative deadlines, which is a strategy used the case where each system has five templates, each of
in prior work on identical platforms [18]; (ii) transform- which has 20 nodes, and edgeProb = 0.5. Other results
ing to an npc-sporadic task system and using implicit rel- can be found online. For the considered case, Fig. 8 plots
ative deadlines; and (iii) transforming to an npc-sporadic AMERBs as a function of total per-pool utilization, when
task system and using LP-based relative deadlines. When the number of identical DAGs per template is fixed to 40
applying our LP techniques, we chose the objective func- (thisnumberisclosetowhatwouldbeexpectedintheappli-
tionthatminimizesthemaximumend-to-endresponse-time cation domain that motivates this work). Also, Fig. 9 plots
bound.Althoughanidenticalplatformwasassumedin[18], AMERBs as a function of the number of identical DAGs
thetechniquesfromthatpapercanbeextendedtoheteroge- per template, when every CE pool is fully utilized (i.e.,
neousplatformsinasimilarwaytothispaper. the total utilization of each pool is eight). In this case, the
AMERB metric was calculated over all task sets that have
Results. Inallcasesthatweconsidered,thetwoevaluated
thesamenumberofidenticalDAGspertemplate.Notethat
techniquesimprovedend-to-endresponse-timebounds,of-
theAMERBsinFig.8aremuchlowerthanthoseinFig.7,
tensignificantly.Duetospaceconstraints,wepresenthere
only the case where N = 5, n = 20, and edgeProb = even before applying the DAG combining technique. This
is because the systems considered in Fig. 8 have far more
0.5.Foreachgeneratedtaskset,werecordedthemaximum
DAGsthanthoseinFig.7.Asaresult,foreachgiventotal
end-to-endresponse-timeboundamongitsfiveDAGs.For
per-poolutilization,thesystemsinFig.8havemuchlower
eachgiventotalper-poolutilizationpoint,wereportherethe
per-DAGandper-taskutilizations.
averageofthemaximumend-to-endresponse-timebounds
amongthe2,500tasksetsgeneratedforthatpoint.Wecall Accordingtoourindustrypartners,intheconsideredap-
thismetrictheaveragemaximumend-to-endresponse-time plicationdomain,aDAG’send-to-endresponse-timebound
bound(AMERB).Fig.7plotsAMERBsasafunctionofto- shouldtypicallybeatmost2.35ms.AsobservedinFig.8,
talper-CE-poolutilization.Asseen,theapplicationofboth intheabsenceofDAGcombining,AMERBsinthisexper-
techniquesreducedAMERBsby39.42%to81.65%. iment were as high as 8.2 ms. However, the introduction
ofDAGcombiningenabledadroptolessthan2.0ms,even
In addition to this plot, we also considered other cases
with different values for N, n, and edgeProb. These whentheplatformwasfullyutilized.Thisdemonstratesthat
DAG combining—as simple as it may seem—can have a
other results show similar trends and can be found in an
online appendix (available at http://cs.unc.edu/ powerfulimpactinthetargeteddomain.
˜anderson/papers.html).
10 Conclusion
9.2 ImprovementsEnabledbyDAGCombining
We presented task-transformation techniques to provide
AsmentionedinSec.6,intheapplicationdomainthatmo- end-to-endresponse-timeboundsforDAG-basedtasksim-
tivates our work, DAGs are usually defined using several plementedonheterogenousmultiprocessorplatformswhere
well-definedcomputationaltemplates,andasaresult,many intra-taskparallelismisallowed.WealsopresentedanLP-
9
msBounds ()10 DCoo mnboitn ceo imdebnintiec aidl eDnAtiGcasl DAGs [3] Sgwtre.iinbtBhueaotberuiudnsaSphdya.isscttkIermiimnbpgusr,ttoe1evd5ceh(rd2ne)aim:ql1-uu0teil7mtsi–.pe1IrE1os8cyEe,sEst2es0Tmo0rras4ng.bsloyabceatxilopsnlcoshioteindnuPglaasrbcaihllleietdyluaalnendahlDoylsiesiss-
me 8 ofsporadicDAGtasksystems.In26thECRTS,2014.
Ti
Response- 6 [[54]] SSra..dBBicaarDruuAaahhG..tFaTeshkdeesryfaesttededemrsasct.ehdIendsu1cl8hintehgdDuolAfinTsgpEo,or2fa0dc1ioc5n.DstAraGinetads-kdesaydsltienmess.pIon-
End 29thIPDPS,2015.
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m
Maximu 2 [7] bpirgo.cLeIsTseTsL.EInP3r3orcdeRssTinSgS.,20h1tt2p.://www.arm.com/products/processors
Average 01 2 3 4 5 6 7 8 [8] V/stie.bcBihliontnyoilfaoangcaiiel,ysAs/bi.siMgilnaitrttclhehepertsotpic-oSerspasadicniccga.DpmhAepGl.a,tSas.kStmilloedr,eal.ndInA.2W5thiesEeC.RFTeaS-,
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