An Upper Bound to the Lateness of Soft Real-time Tasks Scheduled by EDF on Multiprocessors

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1 An Upper Bound o he Laeness of Sof Real-me Tasks Scheduled by EDF on ulprocessors Paolo Valene Scuola Superore S. Anna, Ialy pv@gandalf.sssup. Guseppe Lpar Scuola Superore S. Anna, Ialy lpar@sssup. Absrac ulprocessors are now commonplace for effcenly achevng hgh compuaonal power, even n embedded sysems. A consderable research effor s beng addressed o schedulably analyss of global schedulng n Symmerc ulprocessor Plaforms (SP), where here s a global queue of ready asks, and preempon and mgraon are allowed. In many sof real-me applcaons (as e.g. mulmeda and elecommuncaon) a bounded laeness s ofen oleraed. Unforunaely, when consderng prory-drven schedulng of perodc/sporadc asks, prevous resuls only focused on guaraneeng all deadlnes, and provded worscase ulzaon bounds ha are lower han he maxmum avalable compuaonal power. In parcular, unl now, he exsence of an upper bound on he laeness of sof real-me asks for a fully ulzed SP was sll an open problem. In hs paper we do solve hs problem by provdng an upper bound o he laeness of perodc/sporadc asks wh relave deadlnes equal o perods/mnmum ner-arrval mes scheduled by EDF on a SP, under he only assumpon ha he oal ulzaon s no hgher han he oal sysem capacy.. Inroducon ulprocessors are now commonplace n generalpurpose as well as n embedded sysems. They provde a cos-effecve soluon o acheve hgh compuaonal power. Besdes, due o echnologcal and physcal consrans, ncreasng he speed of sngle processors s becomng more and more dffcul. Hence mulprocessor plaforms seem o be he only opon for he mos compuaonally demandng applcaons. In he las year a large number of mul-core chps as well as mulprocessor archecures have been launched n he marke. For example, o mee he requremens of demandng embedded real-me applcaons, AR proposes PCore, a Ths work has been suppored n par by he European Commsson under conrac IST (ARTIST projec). synheszable mulprocessor core, whle oorola proposes s PowerPC-80 SP plaform. In he hgh-end general purpose processor marke, boh Inel, wh s Penum D brand, and AD, wh e.g. he Operon dual-core processor, envson mul-core processors as he archecure of choce for hgh performance applcaons. In hs paper we consder sof real-me asks o be execued on a Symmerc ul Processor (SP) plaforms, comprsed of dencal processors wh consan speed. Unforunaely, mulprocessor plaforms pose greaer dffcules han sngle processor ones when applcaons have me requremens. any negave resuls are known on he schedulng of real-me applcaons on mulprocessors, ncludngsps,,,8,,7,6,]. The resuls presened n hs paper are relaed o he class of sof real-me applcaons ha can be modeled as a se of perodc/sporadc asks,.e. sequences of jobs o execue, where each job s assocaed wh a relave compleon deadlne equal o he perod/mnmum ner-arrval me. In sof real-me applcaons, deadlnes are no crcal, bu s mporan o respec some Qualy of Servce (QoS) requremens. Examples of such QoS consrans are: lmed number of deadlne msses, lmed deadlne mss percenage, and so on. In hs paper we are neresed n sof real-me applcaons ha can olerae a bounded laeness wh respec o he desred deadlne. Ths knd of consran maches a large class of applcaons, lke mulmeda, elecommuncaon, and fnancal ones. As an example, consder a vdeo player: a gven frame-rae mus be guaraneed, bu a jer of few mllseconds n he frame-me does no sgnfcanly affec he qualy of he vdeo. In conras, audo qualy s exremely sensve o slence gaps. However, audo samples are ypcally buffered and played back a he desred rae by he audo devce. A bounded laeness n provdng new samples o he devce can be easly compensaed usng a pre-bufferng sraegy... Relaed work Research on real-me mulprocessor schedulng has been manly focused on guaraneeng src deadlne observance. The wo man approaches are paronng and global Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

2 schedulng. In paronng he ask se s dvded paroned no groups. Each group of asks s assgned o one of he processors, and processors are scheduled ndependenly. The man advanage of such an approach s s smplcy, as a mulprocessor schedulng problem s reduced o unprocessor ones. Furhermore, snce here s no mgraon, hs approach presens a low overhead. Unforunaely, here are varous negave drawbacks. Frs, fndng an opmal assgnmen of asks o processors s a bn-packng problem, whch s NP-hard n he srong sense. Hence, sub-opmal heurscs are usually adoped,, 9]. Second, here are ask ses ha are schedulable only f asks are no paroned 6]. Also, when asks are allowed o dynamcally ener and leave he sysem, a global re-assgnmen of asks o processors may be necessary o balance he load, oherwse he overall ulzaon may decrease dramacally. In global schedulng, jobs are nsered n a global proryordered ready queue, and a each me nsan he avalable processors are allocaed o he hghes prory jobs n he ready queue. Tasks are n general subjec o mgraon,.e. durng he sysem lfeme hey may be execued on dfferen processors. An mporan classfcaon s wheher a schedulng algorhm s prory-drven 8],.e. each job s assgned a fxed prory, or he prory of a job can vary over me. An mporan class of global schedulers of he second ype s he class of PFar schedulers 5, ]. PFar schedulers break jobs no smaller unform peces, whch are hen scheduled. Unforunaely, n case of eher paronng or prorydrven schedulng, meeng all deadlnes s pad n erms of schedulable ulzaon: any possble prory-drven and/or paroned schedulng algorhm has a oal wors-case ulzaon upper bound no larger han + 6]. On he conrary, PFar algorhms are he only known schedulers able o mee all he deadlnes sll achevng full ulzaon. Unforunaely hey may suffer from hgh schedulng and mgraon overhead... ovaon Unl now, sof real-me applcaons could be scheduled on mulprocessor plaforms eher usng effcen prorydrven schedulers and obanng zero laeness, bu wasng up o half of he avalable compuaonal power; or usng PFar algorhms, whch do acheve full ulzaon wh zero laeness, bu may cause hgh overhead. Excep for PFar schedulng, o he bes of he auhors knowledge, no laeness bound s avalable for sof real-me asks ha fully ulze a mulprocessor. In parcular, was no even known f laeness was acually bounded. When consderng paronng, s mpossble o reach full ulzaon wh bounded laeness, as shown by he followng example. Consder processors and asks, each one wh ulzaon /. There s no way o assgn all asks o he processors and acheve bounded laeness. In fac, eher we overload one of he processors, or we dscard one of he asks, achevng a oal ulzaon of /. Task E, P,,, Proc. Speed P P 5 ob arrvals 6 Dual proc. servce Fgure. Example of unbounded laeness wh fxed prory schedulng. When consderng global schedulng, no all prorydrven schedulng algorhms can acheve bounded laeness. Consder a sysem wh processors and asks, scheduled by fxed prory wh prory assgned accordng o Rae onoonc. Task and have compuaon me and perod ; ask has compuaon me and perod. The oal ulzaon s 5/ <. ob arrvals are shown n he op par of Fg.. Each arrvng job s depced as a recangle: he projecon of he lef corner of each recangle represens he arrval me of he correspondng job, whle he lengh of he base s equal o he execuon me of he job. The number on each recangle refers o he ask ha ssued he job. The schedule of he frs 6 nsans of me s shown n he boom par of he fgure. Noce ha ask sars accumulang nsances, and he laeness of each nsance ndefnely ncreases. In he prevous example, s easy o see ha EDF would have no suffered from he problem of unbounded laeness, because jobs whose deadlne s n he pas have larger prory han newly arrvng jobs. Inuvely, hs propery of EDF prores apparenly guaranees a bounded laeness. However, unl now, provdng an upper bound o he laeness of sof real-me asks for a fully ulzed SP, and under prory-drven schedulng, was sll an open problem... Conrbuon In hs paper we consder a class of global prory-drven schedulers, he DPS Fnsh Tme Schedulers (see Secon for a defnon of hs class), whch EDF belongs o. We prove ha hese schedulers guaranee bounded laeness even when he sysem s fully ulzed. We acheve hs resul by acually compung an upper bound o he maxmum laeness n a smple closed form. Is hs bound gh? We performed a large number of smulaon expermens o see how he acual maxmum laeness experenced by he asks compares o our wors-case bound. The bound resuled vrually gh n case of processors: he rao beween he measured maxmum laeness and he bound s Ths rao decreases as he number of processors ncreases, unl sablzes a approxmaely / for a number of processors hgher han 0. All he resuls are dscussed Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

3 more exensvely n Secon 5. The paper s organzed as follows. In Secon we formally nroduce he sysem and he noaons. In Secon we presen he man resuls, whereas n Secon we presen he proofs. Fnally, we repor smulaon resuls n Secon 5.. Sysem descrpon and noaons We consder a sysem conssng of N perodc or sporadc asks o be execued on a mulprocessor plaform wh dencal processors. All he processors have he same speed (capacy) R, measured n number of execuon cycles per me uns. Each ask consss of an nfne sequence of jobs j j =,... o be execued. Each job j s characerzed by an acvaon (arrval) me a j, a lengh L(j ), equal o he number of execuon cycles for compleng he job, and a compleon deadlne d j.wesayhaajobj has an execuon me e j L(j ) R. The followng relaons hold: a j a j + T d j = a j + T where T s he ask perod (mnmum ner-arrval me). The compleon (fnsh) me of he job j s denoed as f j. We defne as ] laeness of a job j he quany la j max 0,f j dj. { } We denoe, respecvely, wh L max j L( j ) and E L R he wors-case job lengh and he wors-case job execuon me for ask. Fnally, we defne U E T as he ulzaon of ask. We assume ha N = U. In 0] he concep of predcable scheduler s defned. A scheduler s predcable f, gven wo ses of jobs wh he same cardnaly and such ha, for each job n he frs se, here s a correspondng job n he second se wh he same arrval me and prory, and wh execuon me no larger han he job n he frs se, hen he fnsh me of each job n he frs se s no lower han he fnsh me of he correspondng job n he second se. They also proved ha any prory drven scheduler s predcable. Hence, for smplcy, n he remander we wll assume ha each job j has a lengh L( j )=L. We assume ha a job canno sar execung before he prevous job of he same ask has compleed. We refer o hs consran as he precedence consran. We sress he fac ha a job can arrve also before he prevous jobs of he same ask have compleed. We say ha a job j s pendng a me f and only f a j <fj (hence a job under servce s sll pendng). Every ask has a FIFO queue where s pendng jobs are sored. We say ha a ask s acve f has pendng jobs. We defne as oal speed and maxmum oal speed of a mulprocessor a me, respecvely, busy () R and R, where busy () s he number of busy processors a me. Wedefneasunder-loadandfull-load perodsheme nervals durng whch busy () <and busy () =, respecvely. In he remander of he paper we wll refer o he above defned sysem as he ul Processor Sysem (PS). As saed n he nroducon, we consder global prorydrven schedulng. A each me nsan he avalable processors are allocaed o he hghes prory jobs n he ready queue. We assume ha es are arbrarly broken. We allow preempon and mgraon,.e. jobs can be suspended and laer resumed on he same or on a dfferen processor, due o he arrval of some hgher prory job. In parcular, we wll focus on a specal class of global prory-drven schedulng algorhms defned n he nex subsecon ha ncludes EDF. We call a job fracon any poron of a job connuously execued beween wo consecuve sar (or resume) and suspend (or compleon) evens. We defne as prory of a job fracon he prory of he job he fracon belongs o. We defne a chan of jobs of a ask any sequence of job fracons belongng o he same ask and served back-o-back, and head of he chan he frs job fracon n he sequence. We wll assume any generc funcon f() of he me o be rgh connuous. Furhermore, for compacness, we se f(x ) = lm x f(), and we assume ha he exponenaon a x, wh exponen x =0, s always equal o (even when he base a s nfne)... The Dedcaed Processor Sysem In hs subsecon we nroduce he Dedcaed Processor Sysem (DPS), a specal reference sysem ha we wll use o defne he class of global prory-drven schedulers for whch our resuls hold. Defnon Gven a PS, we defne as s reference Dedcaed Processor Sysem (DPS) he sysem conssng of he same ask se and a mul-processor plaform conanng a dedcaed processor for each of he N asks n he PS; each dedcaed processor has a speed R DPS U R =,,..., N. For any job j we defne as s vrual fnsh me he me nsan F j a whch s compleed n he DPS. Snce T = E U, and snce we assumed ha all he jobs of he -h ask have wors-case lengh L, we have ha he DPS complees each job exacly on s deadlne and, hence, no laer han he arrval of he nex job of he same ask,.e. j F j = d j a j+. Consequenly j laj = fj F j. Ths s he crucal propery ha we wll explo o compue an upper bound o he maxmum laeness. An example of he servce provded by a PS and by s reference DPS s shown n Fg..A. The ask se s comprsed of perodc asks, all wh perod and job lengh. Arrvng jobs are depced usng he same convenons as n Fg.. obs are scheduled by EDF n he PS. Especally, snce es can be arbrarly broken, n hs example we chose o break es n favor of lower ndex asks o draw one of he Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

4 Task E, P,,,, Proc. Speed DP DP DP DP / / / / Proc. Speed P P P α() {,,, } {} {,,, } {, } A B b=8 {,,, } s=5 Fgure. Comparng he PS and he DPS. ob arrvals DPS servce PS servce R W S () W S () L() j a j, sj, fj F j L E L max E max lag () Speed of any of he processors Number of processors n he sysem Toal amoun of servce delvered by he sysem S durng 0,] Amoun of servce receved by he -h ask durng 0, ] n a sysem S Lengh (num. of execuon cycles) of job The j-h job of he -h ask Arrval me, sar me, fnsh me of j (Vrual) fnsh me of j n he DPS (Wors-case) lengh of -h ask (Wors-case) execuon me of -h ask axmum job lengh over all he asks axmum execuon me over all he asks Lag of ask (W DPS () W PS ()). possble schedules. The fgure clearly shows ha, whereas he DPS correcly schedules all he jobs, he PS msses e.g. he deadlne of job a me. Upon compleon, ask has laeness. The suaon ges worse durng he second perod, and boh and mss her deadlne a me 8. Hereafer we wll consder he followng wo sysems: a generc PS and s reference DPS. We wll refer o hese sysems as he PS and he DPS, respecvely. We can now defne he class of schedulers we wll focus on. Defnon We say ha a prory-drven scheduler for he PS s a DPS Fnsh Tme (DPS-FT) scheduler, f, denoed wh P j he prory of he generc job j, we have ha { P j = P k l F j = F k l j,l k P j >Pl k F j <Fl k and, a each me nsan, he avalable processors are allocaed o he hghes prory jobs. Tes are arbrarly broken. In oher words, n a DPS-FT scheduler he orderng among job prores s he oppose of he orderng beween job fnsh mes n he DPS. Snce j F j = dj, EDF s a DPS-FT scheduler. Hereafer, we wll assume ha a DPS-FT scheduler s used o schedule jobs n he PS. Under he assumpons of consan speed processors and of asks wh consan job lengh, any DPS-FT scheduler s equvalen o EDF (.e. generaes he same schedules). However, all he followng lemmas and heorems wll be acually proved n he more general case where all he processors have he same me-varyng speed R(), and where each dedcaed processor has me-varyng speed R DPS ()=U R(). In hs case, he class of DPS-FT schedulers can also nclude schedulers dfferen from EDF. Whle hs generalzaon does no complcae he proofs, paves he way for fuure more general resuls. We defne as W PS () and W DPS () he amoun of servce provded by, respecvely, he PS and he DPS o he -h ask durng 0,]. We defne he oal amoun of servce provded by he PS and he DPS durng 0,] as, re- () Table. Noaons used n hs paper. specvely, W PS () W PS () and W DPS () W DPS (). We defne as lag of he -haskame he followng quany: lag () W DPS () W PS () For brevy, gven wo me nsans >,wedefne W PS (, ) W PS ( ) W PS ( ). We use he same shor noaon for W DPS, W PS, W DPS and lag. In he proofs we wll ofen use he followng propery: snce R DPS R, he lag of a ask can no ncrease durng he servce of one of s job chans. For example, n Fg..A he lag of ask ncreases durng 0, ], and s equal o 9 a me. Conversely, decreases durng, 6], and s e.g. equal o a me. Snce he lag of a ask may be a useful fgure of mer, n hs paper we repor an upper bound o he maxmum perask lag n addon o he one on he maxmum laeness. The noaons nroduced unl now are summarzed n Table... axmum lag and maxmum laeness In hs secon we enuncae and brefly dscuss he followng heorems, whch consue he man resuls of hs paper. Theorem If an PS comprsed of dencal processors s scheduled usng a DPS-FT scheduler, he followng guaranees on he lag experenced by any ask hold:, lag () ( U ) L +U ( ) L max () ] j lag (f j ) U L +( ) L max Theorem If an PS comprsed of dencal consan speed processors s scheduled usng a DPS-FT scheduler, he followng guaranees on he job laeness hold: () Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

5 j la j E +( ) E max. () The formal proofs of he heorems (and of he nex corollary) are repored n he nex secon. We can noe ha processors are no requred o have consan speed for Theorem o hold (bu hey mus be dencal,.e. hey mus all have he same speed a any me nsan). Wh regard o Theorem, we hghlgh ha, when =, Eq. () collapses o he EDF guaranee j laj =0,snce( ) becomes equal o zero. I s easy o prove ha he rgh erms n Inequales (), () and () are all non-decreasng funcons of, andha (lm + ) = e. I follows ha: Corollary If an PS comprsed of dencal processors s scheduled usng a DPS-FT scheduler, he followng nequales hold:, lag () ( U ) L + e L max (5) j lag (f j ) U L + e L max ] (6) Furhermore, f processors have consan speed, he followng nequaly holds: j la j E + e E max. (7) As can be seen, he corollary provdes smpler bu more conservave upper bounds. Fnally, s worh nong ha all he aboveresulsholdalso whenasksfully ulze he sysem.. Proofs In hs secon we wll formally prove Theorems and, and Corollary. Frs, we nroduce he noaons used n he proofs and he proof sraegy. The proofs are essenally based on compung a bound o lag (), from whch he bound on he laeness wll be hen derved. The followng Lemma resrcs he me nsans o be consdered when compung an upper bound o he lag. Lemma The maxmum lag experenced by a ask s no hgher han he maxmum lag ha he ask can experence a he sar me of some of s job fracons. Proof. When a ask s nacve, s lag s necessarly no hgher han 0. Consder nsead a generc maxmal acve perod, ] of ask. LeX k be he k-h job fracon of ask served by he PS. Any me nsan, ] necessarly falls no a sub-nerval f k,f k], ] rangng from he fnsh me f k of a job fracon X k, and he fnsh me of he nex job fracon X k served by he PS (f X k s he frs job fracon of ask execued durng, ],hen we assume f k = ). Snce he lag can no ncrease durng he servce of a job, we have ha max lag ()=lag (s k f k,f k] ) where s k s he sar me of he fracon Xk. Consequenly, n he nex subsecon we wll focus on compung he maxmum lag of he ask a he sar me s of a generc job fracon X belongng o a job j. Frs, f s = a j,henlag (s) =0because boh he PS and DPS have fnshed all he pendng jobs a s. Le us hen consder he case s>a j (noe ha, n general, s mgh be even larger han F j,.e. larger han he job deadlne n case of EDF). To handle hs case, we defne as α() he se of he asks ownng pendng jobs wh prory no lower han X a me. For example, n case of EDF, α() ncludes all he asks ownng jobs wh deadlne no hgher han j (.e. han he job X belongs o) a me. Noe ha a j,s) α(), because j s pendng durng a j,s) and, by defnon, s prory s equal o he prory of s fracon X. Fg..B shows he values assumed by α() and α() durng 0, s), assumng he fracon X o concde wh he whole job, whch n urn sars servce a me s =5nFg..A. There are only wo possble causes for X o sar a me s>a j :)Xs blocked by prory, hasaleas asks own pendng jobs wh prory no lower han j a me s ( α(s ) ); ) X s blocked by he precedence consran a me s. In he second case, X belongs o a chan. Snce he lag of a ask does no ncrease whle s jobs are beng served, he maxmum lag of he ask s rvally upper bounded by s maxmum lag a he sar me of he chan head. Ths s n s urn equal o 0, unless he chan head s blocked by prory. As a concluson, he problem ha remans o solve s compung he maxmum lag of he ask a he sar me of a fracon blocked by prory. To hs am, we wll use he followng wo defnons. Defnon Gven a job fracon X blocked by prory, we defne as las prory blockng perod for X he me nerval b,s), whereb s he smalles me nsan b such ha b,s) α(),.e. such ha a leas asks are connuously acve and have pendng jobs wh prory no lower han X durng b,s). Fg..B shows he las prory blockng perod of he job. We noe ha b mgh n general precede aj.furhermore, we wll explo he followng wo properes of he las prory blockng perod: α(b ) <, and he PS s n full load durng b, s). Defnon We defne Γ as he se of he jobs ha receve servce n he PS durng b, s). Noce ha, by defnon of las prory blockng perod, he jobs n Γ have prory no lower han X. As an example, assumng agan X = n Fg., we have ha Γ={,,,,,,,, }. The PS sars servng X only afer servng par of he jobs n Γ. However, he jobs n Γ have prory no lower han Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

6 j, whch means ha hey fnsh no laer han j n he DPS. Hence, he DPS mus complee all he jobs n Γ before can complee j. Furhermore, snce he PS works a maxmum oal speed durng b,s], consumes he jobs n Γ a a pace no lower han he one a whch he DPS could consume hem durng he same me nerval. For hese reasons, nuvely, he maxmum value of lag (s) depends on how ahead s he DPS wh respec o he PS n he servce of he jobs n Γ a me b. ore formally, we wll show ha he maxmum value of lag (s) depends on he followng quany: j α lag j(b), whereα pos α(b) s he subse of pos he asks wh posve lag a me b. Wecalloal lag relaed o he fracon X he above quany. We can now defne he proof sraegy: we wll frs express he maxmum lag of a ask as a funcon of he oal lag n Subsecon.. Ths general formula wll serve wo purposes. We wll frs use n Subsecon. o compue an upper bound o he oal lag self. Then, n he las subsecon, we wll subsue he jus compued bound n he general formula, hus geng an upper bound o he lag experenced by a ask. Fnally, he laer bound wll be used o compue an upper bound o he laeness experenced by a job. A more dealed verson of any of he followng proofs can be found n 5]... Basc lemmas Ths subsecon conans four lemmas: he frs wo lemmas allows us o provde an upper bound o he lag of a ask as a funcon of he oal lag, whereas he las wo lemmas are jus algebrac facles ha wll be used n nex subsecon o compue he maxmum oal lag. As an nermedae sep for compung an upper bound o he maxmum lag of a ask, he nex lemma provdes an upper bound o he dfference beween he oal amoun of servce W PS (b, s) ha he PS provdes durng b, s] and he oal amoun of servce W DPS (b, F j ) ha he DPS mus provde durng b, F j ] o fnsh j. In he lemma, a specal se of asksσ s defned. We wll dscuss he use of σ jus afer enuncang he lemma. Lemma Le X be a generc job fracon, belongng o a job j, ha sars servce a me s n he PS afer beng blocked by prory. Le σ be any subse of he asks whose jobs are under execuon a me s, excludng ask. We have: W PS (b, s) W DPS (b, F j ) h α lag h(b) pos h σ lag h(s) L res ( j ) (8) where L res ( j ) s he dfference beween he lengh of j and he poron of j already served by he PS a me s, b s he begnnng of he las prory blockng perod of X, and α pos { α(b) lag (b) > 0}. Before he proof, a quck commen on he se σ. Icanbe any subse of he asks ha are acve a me s, excluded. For example, assumng X = n Fg., he possble values of σ are {}, {} and {, }. The lemma holds for any choce of σ. In he followng, we wll use hs same lemma wh dfferen values of σ o acheve dfferen resuls. In fac, f we se σ =, Inequaly (8) provdes an upper bound o W PS (b, s) W DPS (b, F j ) as a funcon of he oal lag, whch wll be used for compung an upper bound o he lag of any ask. Conversely, he case σ wll be used when compung an upper bound o he oal lag. The proof of he lemma follows. Proof. To prove Inequaly (8), we wll frs compue an upper bound o W PS (b, s), hen a lower bound o W DPS (b, F j ), and fnally subrac hem. Le Γ(σ) Γ be he subse of he jobs n Γ ssued by he asks n σ, and Γ(σ) Γ\Γ(σ). For boh bounds, we wll separae he conrbuon due o he jobs n Γ(σ) from he conrbuon due o he jobs n Γ(σ). We sar by compung an upper bound o W PS (b, s). Durng b,s) he PS serves only some fracons (a mos all he fracons) of he jobs n he prevous wo ses. Hence, defned as L(Γ(σ)), WΓ(σ) PS (b,s) he sum of he lengh of he jobs n Γ(σ), he servce ha he PS gave o he jobs n Γ(σ) before me b and he amoun of servce provded o he asks n σ by he PS durng b,s], respecvely, we have ha: (b) and W PS Γ(σ) W PS (b, s) L(Γ(σ)) W PS Γ(σ) PS (b)+wγ(σ) (b,s) (9) Second, we compue a lower bound for W DPS (b, F j ). Observe ha: ) no all he jobs n Γ have arrved a me b, ) he DPS mus complee all he jobs n Γ no laer han F j. I follows ha F j b. As done before, we separae he conrbuon of he jobs n Γ(σ) from he one of he jobs n Γ(σ). Wh regard o he laer se, he DPS mus ceranly have served boh all he jobs n Γ(σ), and he poron L res ( j ) before fnshng j. However, par of hese jobs mgh have arrved before b, and he DPS mgh have already (parally) served hem. The servce provded o hese jobs can be wren as WΓ(σ) DPS DPS (b)+w (b). In he end, we can wre: j L(Γ(σ)) + W DPS Γ(σ) (b,f j (W DPS Γ(σ) W DPS (b,f j j ) )+Lres ( j )+ DPS (b)+w (b)). j (0) Now we fnd an upper bound o he las erm. Consder a generc job Γ(σ). If a me b he DPS has already served a poron of larger han he poron already served by he PS, hen s pendng n he PS a me b and has prory no lower han X. Leq be he ask ha generaed. We have ha lag q (b) > 0. LeW PS (b) and W DPS (b) be he servce ha he PS and DPS gve o before b, respecvely. We have ha W DPS (b) >W PS (b) 0. Fnally, le W PS (b) and W DPS (b) be, respecvely, he servce ha he PS and he DPS gve o he jobs of q, excludng, before b. By defnon, q α pos \σ. oreover, from he defnon of lag q (b) we ge W DPS (W DPS (b) W PS (b)=lag q (b)+w PS (b)). SnceW DPS (b) >W PS (b) (b), Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

7 W DPS (b) W PS (b) 0. In he end, W DPS (b) lag q (b)+w PS (b). Summng over all he jobs Γ(σ) and applyng he above argumens o W DPS (b) as well, we ge: j WΓ(σ) DPS DPS (b)+w (b) j q α pos \σ lag q(b)+wγ(σ) PS(b) () q α lag q(b)+w PS pos Γ(σ) (b) We can dvde σ no wo subse, a subse σ α(b) and a subse σ such ha σ α(b)=. Fnally, defned σ pos σ as he subse of he asks whose lag s posve a me b, we have q α pos \σ lag q(b) = q α lag q(b) pos q σ pos lag q (b) lag q(b) () q αpos q σ lag q (b) = q α lag q(b) pos q σ lag q(b). where he las equaly follows from he fac ha q σ lag q (b)=0. Subsung () n (), we ge WΓ(σ) DPS DPS (b)+w (b) lag j q (b) lag q (b). q α pos q σ () A hs pon, we can reurn back o Inequaly (0) and wre: W DPS (b,f j j ) L(Γ(σ)) + WΓ(σ) DPS(b,F j )+Lres ( j )+ q α lag q(b)+ pos q σ lag q(b) WΓ(σ) PS(b) Subracng he las nequaly from (9), we ge: W PS (b, s) W DPS (b, F j ) WΓ(σ) PS DPS (b) WΓ(σ) (b,f j )+ L res ( j )+ q α lag q(b) pos q σ lag q(b). () We can smplfy he above expresson by consderng ha W PS Γ(σ) W PS Γ(σ) (b, s) W DPS Γ(σ) (b, F j ) DPS (b, s) WΓ(σ) (b, s) = (5) j σ lag j (b, s) Subsung (5) n (), we ge he hess. Usng he bound compued n he prevous lemma, we can now prove he followng lemma, whch expresses he maxmum lag of a ask as a funcon of he oal lag. Lemma Le X be a generc job fracon, belongng o a job j, ha sars servce a me s n he PS afer beng blocked by prory. Le σ be any subse of he asks, excludng ask, whose jobs are under execuon a me s. We have: lag (s) L res ( j )+ + U j α lag pos j (b) ] j σ lag j(s) L res ( j ) (6) where L res ( j ) s he dfference beween he lengh of j and he poron of j already served by he PS a me s, b s he begnnng of he las prory blockng perod of X, and α pos { α(b) lag (b) > 0}. Proof. The proof sraegy s as follows: we wll frs express lag (F j ) n a convenen form, hen we wll fnd an upper bound o lag (s) lag (F j ), fnally we wll sum hs bound o lag (F j ).Wehaveha: W DPS (F j PS )=W (s)+l res ( j ) (7) We can do some algebrac manpulaons: W PS (s)=w PS (F j )+ (8) where W PS (s) W PS (F j ). Subsung successvely (7) and (8) n he defnon of lag (F j ),wege lag (F j )=Lres ( j )+ (9) We wll now compue an upper bound o lag (s) lag (F j ). From (8) we ge: lag (s) lag (F j )= W DPS (s) W ] ) (0) We wan now o explo Lemma o fnd an upper bound o he dfference W DPS (s) W ) n (0). To hs am, we can frs pu hs dfference n a more convenen form: W DPS (s) W DPS (F j )=W DPS (b, s) W DPS (b, F j ) () We need a las sep arrve o a form smlar o he lef member of Inequaly (8). To hs am, we wll work on W DPS (b, s). Snceb, s] falls nsde a full-load perod (a leas asks are acve durng b, s)), hen b, s] he oal speed of he PS a me s R PS ()= R(). In conras, he oal speed R DPS () of he DPS s: R DPS ()= j A DPS () U j R()= j ADPS () Uj R PS () where A DPS () s he se of he asks acve under he DPS a me. Furhermore,W DPS (b, s) s maxmum f he DPS connuously serves he -h ask durng b, s]. Defnedχ () as he fracon of he oal speed ha he DPS dedcaes o he h ask a me b, s] under hs hypohess, we have U χ ()=. In he end: j A DP S Uj () W DPS (b, s) s b χ (τ) R DPS (τ) dτ = s U b j A DP S (τ) Uj j A DP S (τ) Uj R PS (τ) = U W PS (b, s) () As a concluson, subsung he las nequaly n () and explong (8), we ge W DPS (s) W ] ) U W PS (b, s) W DPS (b, F j ) U j α pos lag j (b) j σ lag j(s) L res ( j ) ] Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

8 Subsung he las nequaly n (0), and summng o (9), we ge he hess. The followng wo purely algebrac Lemmas wll prove useful n he nex subsecon. For space consrans, we removed her proofs, whch can be found n 5]. Lemma Defned he followng wo funcons: { x = g (x) A + E B ] j= g j(x) =,..., K wh A 0, B 0 and 0 E =,..., K, and K ; and defned f(k, A,...,A K,B,..., B K,E,..., E K,x) K = g (x) f(k, A,..., A K,B,..., B K,E,...,E K,x) s a non-decreasng funcon of x. Lemma 5 We defne h C + P D j= h j =,,..., K () wh C>0, D>0,0 P =,...,K, K, and assumng m j=l a j 0fl<m. We have z(k, C, D, P,P,..., P K ) K = h K C + D ( )K].. Boundng he oal lag We need a las nermedae lemma. Lemma 6 Le A( ) be any subse of V asks under servce a me. We have ha: j A( ) lag j( ) V L max+ + ( ] ] )V max j A( ) α pos lag j (b j ) where b j s he begnnng of he las prory blockng perod of he head X j of he chan, under servce a me,ofhej h ask n A( ), α pos j s he se of he asks whose pendng jobs have prory no lower han X j, and whose lag s posve a me b j. Proof. We wll frs fnd an upper bound o j A( ) lag j( ) wh he same form of funcon z n Lemma 5, hen we wll apply hs lemma o prove he hess. We can assume, whou losng generaly, ha he V asks n A( ) are he asks,,..., V, and ha hey are ordered by he sar me s j of he chan heads X j.le(x j ) be he job he fracon X j belongs o. We defne Lag max lag (b j ) j A( ) α pos j From Lemma and posng σ = {,,..., V }, we can wre lag V ( ) lag V (s V ) U V Lres ((X V )) + UV U V L max + UV Lag ] V j= lag j(s V ) Lag ] V j= lag j ( ) because he lag of a ask can no ncrease durng he execuon of one of s chans (asks,,..., V are connuously served durng s V,]). Now we defne λ j ( ) L max + U j ] j Lag lag ( ) = Consderng ha lag V ( ) λ V ( ) and addng lag V ( ) we ge lag V ( )+lag V ( ) lag V ( )+λ V ( ) = f(,a = L max, B =Lag V = lag ( ), E = UV, lag V ( )) () where he funcon f s he one defned n Lemma, and, as such, s a non-decreasng funcon of lag V ( ). Hence, o fnd an upper bound o s value, we look for he maxmum value ha can be assumed by lag V ( ). From Lemma, posng σ = {,,..., V }, and repeang he same seps as above, we ge lag V ( ) λ V ( ). Hence, consderng (), he prevous bound on lag V ( ) and Lemma, we have lag V ( )+lag V ( ) λ V ( )+λ V ( ) By nducvely applyng hs argumen, afer V seps we ge: V V lag j ( ) λ j ( ) j= j= Hence, he hess follows from applyng Lemma 5 o V j= λ j( ) =z(v, C = L max, D =Lag,P = U,P = U,..., P V = UV ). We can now compue an upper bound o he oal lag. Theorem For any job fracon X whch sars servce n he PS afer beng blocked by prory, we have lag j (b) ( ) ( ) L max (5) j α pos where b s he begnnng of he las prory blockng perod of X, and α pos { α(b) lag (b) > 0}. Proof. We wll proceed by nducon. For he base case, le X be he frs job fracon blocked by prory from he begnnng of he lfeme of he sysem. In such a case b concdes wh he begnnng of he frs congeson perod for he PS, whch mples j α pos lag j(b) 0. Hence (5) rvally holds. For he nducve sep, suppose ha (5) holds for all he job fracons blocked by prory ha sared servce before X. From he defnon of las prory blockng perod, follows ha he asks n α pos are less han and canno be Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

9 blocked by prory a me b. Hence, hey are all under servce a me b. From Lemma 6 and Inequaly (5), we can wre + ( ) αpos ] j α lag pos j(b) α pos L max+ ] ( ) ( ) L max ] Consder ha, as α pos ncreases, ( ) αpos decreases, hence he rgh erm n he prevous nequaly ncreases. Hence, snce α pos, he hess follows from seng α pos =... axmum lag and maxmum laeness Frs we prove our upper bounds o he lag. Proof of Theorem Due o space lmaons and snce he proof s que nuve, we repor jus a skech here. The full proof can be found n 5]. Gven a fracon X blocked by prory, s easy o prove ha lag (s) s upper bounded by he he rgh erm of (), by jus assumng σ = n Lemma and subsung n (6) he upper bound o he oal lag provded by Theorem. I s easy o prove ha he same bound holds f X s blocked by precedence, by consderng ha he lag of a ask can no ncrease whle he ask s beng served, and assumng ha he chan head s blocked by prory. Fnally, hanks o Lemma, he Inequaly () holds a any me nsan. The upper bound () on he compleon me of a job can be compued by applyng he above argumens o he las fracon of he job, and subracng, o he upper bound o lag (s), he dfference beween he amoun of servce receved by he ask durng he servce of he fracon and he maxmum amoun servce ha he ask can receve n he DPS durng he same me nerval. We can now prove our upper bound o he laeness. Proof of Theorem Recall ha la j = fj F j.iffj F j, he hess rvally holds. Consder he case fj >Fj.We wll prove he hess by conradcon. The schedules of (he fracons of) j n he PS and n he DPS, and hence he dfference f j F j, do no depend on wheher ask ssues new jobs afer a j. Suppose ha ndeed an ndefne number of jobs has been ssued by ask a me a j.insuchacase,f he upper bound () does no hold, we have ha W,fj ] ) > U L, max +( ) L max Furhermore, snce W DPS W DPS W DPS (F j U L +( whch conradcs Inequaly (). All s lef o prove s Corollary. PS )=W (f j ) we have lag (f j ) = (f j ) W PS (f j ) = (f j ) W ) = W,fj ] ) > ) L max Proof of Corollary I s mmedae o noe ha (), () and () are non-decreasng funcons of f and only f ( ) s a non decreasng funcon of. Assumng, and defned x 0,wehave ( ) =(+ x )x Defned f(x) ( + x )x, we can compue he frs dervave Df(x)] = e x log+ x] log ] + x] x+.iseasy o prove ha Df(x)] 0 x 0. Hence x 0 f(x) lm x > ( + x )x = e. The hess follows from subsung e n place of ( ) n he rgh erms of (), (6) and (). 5. Smulaons We smulaed EDF global schedulng over 9 SP plaforms, comprsed of,,..., 0 un-speed processors, respecvely. For each SP, we consdered four dfferen ypes of ask ses, all wh oal ulzaon equal o he number of avalable processors. The frs ype of ask ses was made only of lgh asks,.e. asks wh ulzaon no hgher han 0.5. In he second ype, half of he oal capacy was devoed o lgh asks, whle he oher half was devoed o heavy asks,.e. asks wh ulzaon hgher han 0.5. The hrd ype of ask ses was made only of heavy asks. The fourh ype was made only of very heavy asks,.e. asks wh ulzaon hgher han 0.8. For each SP and for each ype of ask se, 50 ask ses were randomly generaed. Fnally, for each ask se, he correspondng EDF schedule was smulaed for cks, 0 5 cks beng he maxmum ask perod. For he frs ype of ask ses, Fg..(a) shows, for each SP, he maxmum rao recorded, over he 50 smulaon runs, beween he laeness experenced by a ask and he value of he upper bound () for he ask. We wll shorly refer o he above quany as he maxmum rao. For each SP, he mean rao,.e. he rao beween he mean laeness experenced by he ask wh he maxmum rao and he upper bound () s repored as well. As can be seen, he maxmum rao decreases as he number of processors ncreases. Especally, s equal o 0.7 for processors, and sablzes a abou 0.5 for a number of processors larger han 0. The mean rao soon becomes neglgble. ovng o he successve ypes of ask ses, boh he maxmum and he mean rao ncrease. The hghes values are acheved n case of only very heavy asks, as shown n Fg..(b). Fg..(c) shows he values of he upper bound () for he asks ha experenced he rao repored n nse (b). Especally, he values of he upper bound repored n Fg..(c) concde wh he ones we obaned for he oher hree ypes of ask ses (n fac, accordng o (), he bound does no depend on he ulzaon of he asks). Fg..(b) shows ha he bound s vrually gh for wo processors. Then he maxmum rao sablzes a approxmaely / for a number of processors larger han 0. In he end, accordng o he smulaons, he bound s gh Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.

10 (a) Only lgh asks Rao maxmum laeness/th. bound Rao mean laeness/th. bound (b) Only very heavy asks Rao maxmum laeness/th. bound Rao mean laeness/th. bound cks (c) Only very heavy asks Theorecal bound CPUs CPUs CPUs Fgure. Inse (a) and (b): rao beween experenced laeness and upper bound. Inse (c): value of he upper bound. only for very heavy asks on processors, whle s oo conservave n he oher cases. However, we could no generae all he possble lgh and heavy ask ses durng smulaons. Fnally, he acual bound may depend on he characerscs of he asks (compuaon me and perods). Indeed, deermnng a possble relaonshp beween he properes of a ask se and he resulng wors-case laeness s sll an open problem. 6. Conclusons In hs paper we propose an upper bound o he laeness of sof real-me asks scheduled by EDF on a SP. Frs we show ha no all schedulng algorhms are able o provde a bounded laeness n he case of full ulzaon. Then, we propose a bound and prove s correcness. The proposed bound s n a smple closed form, and has been shown o be vrually gh for heavy ask ses on processors. Accordng o he smulaons, he bound s no gh for more han processors and for lgh ask ses. Obvously, we could no generae all he possble lgh and heavy ask ses durng smulaons. Hence, provng wheher he bound s gh, for lgh ask ses, and for more han processors, s sll an open problem. We beleve ha possble relaonshps beween he properes of he asks and he acual bound should be nvesgaed. Acknowledgemens We wsh o hank Enrco Bn for hs nsghful suggesons o mprove he presenaon of hs paper. References ]. Anderson and A. Srnvasan. xed pfar/erfar schedulng of asynchronous perodc asks. ournal of Compuer and Sysem Scences, 68():57 0, 00. ] B. Andersson. Sac-prory schedulng on mulprocessors. PhD hess, Deparmen of Compuer Engneerng, Chalmer Unversy of Technology, Goeborg, Sweden, 00. ] B. Andersson, S. Baruah, and. onsson. Sac-prory schedulng on mulprocessors. In IEEE, edor, Proceedngs of he IEEE Real-Tme Sysems Symposum, Dec 00. ] T. Baker. ulprocessor EDF and deadlne monoonc schedulably analyss. In Proceedngs of he h IEEE Inernaonal Real-Tme Sysems Symposum, RTSS 0, 00. 5] S. Baruah, N. Cohen, C. Plaxon, and D. Varvel. Proporonae progress: A noon of farness n resource allocaon. Algorhmca, 6, ]. Carpener, S. Funk, P. Holman, A. Srnvasan,. Anderson, and S. Baruah. Handbook of Schedulng: Algorhms, odels, and Performance Analyss, chaper A Caegorzaon of Real-me ulprocessor Schedulng Problems and Algorhms. Chapman Hall/ CRC Press, 00. 7] S. Funk,. Goossens, and S. Baruah. On-lne schedulng on unform mulprocessors. In IEEE, edor, Proceedngs of he IEEE Real-Tme Sysems Symposum, pages 8 9, Dec 00. 8]. Goossens, S. Funk, and S. Baruah. Prory-drven schedulng of perodc ask sysems on mulprocessors. Real-Tme Sysems, 5(-):87 05, Sep-Oc 00. 9] R. Graham. Compuer and ob Schedulng Theory, chaper Bounds on he performance of schedulng algorhms. Wley, New York, ] R. Ha and. W. S. Lu. Valdang mng consrans n mulprocessor and dsrbued real-me sysems. In h IEEE Inernaonal Conference on Dsrbued Compung Sysems, Los Alamos, 99. ] A. Khemka and R. K. Shyamasunda. ulprocessor schedulng of perodc asks n a hard real-me envronmen. Techncal repor, Taa Insue of Fundamenal Research, 990. ] A. ok and. Derouzos. ulprocessor schedulng n a hard real-me envronmen. In Proceedngs of he Sevenh Texas Conference on Compung Sysems, 978. ] Y. Oh and S. H. Son. Allocang fxed-prory perodc asks on mulprocessor sysems. ournal on Real Tme Sysems, 9, 995. ] A. Srnvasan and. Anderson. Effcen schedulng of sof real-me applcaons on mulprocessors. In Proceedngs of he a he 5h Euromcro Conference on Real-me Sysems, pages 5 59, uly 00. 5] P. Valene and G. Lpar. An upper bound o he laeness of edf on mulprocessors. Techncal Repor RETIS TR05-0, Scuola Superore S. Anna, 005. hp://feanor.sssup./~pv/edf_r.pdf. Proceedngs of he 6h IEEE Inernaonal Real-Tme Sysems Symposum (RTSS 05) /05 $ IEEE Auhorzed lcensed use lmed o: UNIVERSITA PISA S ANNA. Downloaded on Ocober 7, 008 a 6: from IEEE Xplore. Resrcons apply.