An Improved Speedup Factor for Sporadic Tasks with Constrained Deadlines under Dynamic Priority Scheduling

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1 A Improved Speedup Factor for Sporadc Tasks wth Costraed Deadles uder Dyamc Prorty Schedulg X Ha Lag Zhao School of Software Techology Dala Uversty of Techology Emal: hax@dlut.edu.c lghta@qq.com Zhsha Guo Departmet of Computer Scece Mssour Uversty of S&T Emal: guozh@mst.edu Xgwu Lu Isttute of Computg Techology CAS Uversty of Chese Academy of Sceces Emal: luxgwu@ct.a.c arxv: v1 [cs.ds] 20 Jul 2018 Abstract Schedulablty s a fudametal problem real-tme schedulg but t has to be approxmated due to the trsc computatoal hardess. As the most popular algorthm for decdg schedulablty o multprocess platforms the speedup factor of parttoed-edf s challegg to aalyze ad s far from bee determed. Parttoed-EDF was frst proposed 2005 by Barush ad Fsher [1] ad was show to have a speedup factor at most 3 1/m meag that f the put of sporadc tasks s feasble o m processors wth speed oe parttoed-edf wll always retur succeeded o m processors wth speed 3 1/m. I 2011 ths upper boud was mproved to /m by Che ad Chakraborty [2] ao more mprovemets have appeared ever sce the. I ths paper we develop a ovel method to dscretze ad regularze sporadc tasks whch eables us to mprove the case of costraed deadles the speedup factor of parttoed-edf to /m very close to the asymptotc lower boud 2.5 [2]. Idex Terms Sporadc tasks resource augmetato parttoed schedulg demad boud fucto I. INTRODUCTION Schedulg s a hot topc the real-tme systems commuty. Bascally gve a fte set of tasks each sequetally releasg ftely may jobs the msso of realtme schedulg s to allocate computg resources so that all the jobs are doe a tmely maer. The fudametal questo of schedulablty aturally arses: Is t possble at all to successfully schedule these tasks such that all of them receves eough executo before ther deadles? Ufortuately aswerg ths questo s ofte ot easy ; e.g. the schedulablty of a set for costraed-deadle sporadc tasks whch s the focus of ths paper s co-np-hard eve o a uprocessor platform [3]. For multprocessor case t remas NP-hard for parttoed paradgm eve f the relatve deadles of the tasks are requred to equal ther perods [4]. Here parttoed paradgm meas that oce a task s assged o a processor all the legal jobs released by the task Work ported by NSFC ( NSF (CNS Correspodg author wll be scheduled o the dedcated processor. These hardess results mply that t s almost mpossble to exactly decde schedulablty polyomal tme. Due to the hardesses real-tme schedulablty problems are usually solved approxmately by pessmstc algorthms whch always aswer No uless some suffcet-oly codtos for schedulablty are met. To evaluate the performace of such a approxmate algorthm (say A the cocept of speedup factor also kow as resource augmetato boud has bee proposed. Specfcally wheever a set of tasks s schedulable o a platform wth speed oe algorthm A wll retur Yes o the same platform wth speed r 1. The mmum such r s referred to as the speedup factor of A. Despte of some recet dscusso o potetal ptfalls [5] [6] [7] speedup factor has bee a major metrc ad stadard theoretcal tool for assessg schedulg algorthms sce the semal work 2000 [8]. Recet years has wtessed mpressve progress o fdg schedulablty decso algorthms wth low speedup factors. For the preemptve case (.e. rug jobs mght be terrupted by emerget oes Global-EDF has a speedup factor 2 1/m [9] for schedulg set of tasks o m detcal processors ad there s a polyomal-tme algorthm for uprocessors whose speedup factor s 1 + ɛ [10] where ɛ > 0 s suffcetly small for uform processors refer to [11]. For the o-preemptve case there are also a varety of results refer to [12] [13]. Except the speedup factor there are may papers cocerg about the utlzato upper boud refer to [14] [16]. Although the speedup factor o uprocessors s tght the multprocessor case remas ope. Due to ts smplcty parttoed schedulg s of partcular terest ad has bee attractg more ad more atteto from researchers ad practtoers. Parttoed-EDF s the most popular schedulablty decso algorthm of parttoed style whle the abovemetoed Global-EDF s ot of parttoed paradgm. Breakthrough was made the year of 2005 whe Baruah ad Fsher [1] establshed a 4 2/m (3 1/m respectvely upper boud for the speedup factor of parttoed-edf o arbtrary-deadle 1

2 (costraed-deadle respectvely task sets where m s the umber of detcal processors. A set of tasks s sad to be costraed-deadle f the relatve deadle of each task s at most ts perod otherwse s arbtrary-deadle. The 2011 Che ad Chakraborty [2] further mproved the speedup factor to /m (3 1/m respectvely for the costraeddeadle case (arbtrarly deadle case respectvely. Also the same paper a lower boud 2.5 of the speedup factor was establshed for the costraed-deadle case. Throughout the last seve years the bouds [2] were ever mproved. It s worth otg that dervg the upper boud of the speedup factor of parttoed-edf reles heavly o a quatty about schedulg o uprocessors deoted by ρ whch s formally defed (1 of Secto II. Roughly speakg ρ measures how far the approxmate demad boud fucto (defed Secto II devates from the actual deadle tmepots. Baruah ad Fsher [1] brdged ρ ad the speedup factor of parttoed-edf by showg that case of costraed deadles the speedup factor s at most 1 + ρ 1/m. As a result upper-boudg the speedup factor s reduced to upper-boudg ρ ad t s ths maer that both [1] ad [2] obtaed ther estmatos of the speedup factor. Hece the quatty ρ tself deserves a deevestgato. Actually Baruah ad Fsher [1] upper-bouded t by 2 Che ad Chakraborty [2] arrowed ts rage to [ ]. O ths groud ths paper wll explore a better upper boud of ρ ad o ths bass provde a better estmato of the speedup factor of parttoed-edf for sets of costraeddeadle sporadc tasks. The cotrbutos are summarzed to the followg three aspects. 1 We mprove the best exstg upper boud of ρ for costraed-deadle tasks from to (Theorem 1 whch s much close to the lower boud 1.5. The speedup factor of parttoed-edf for the costraeddeadle case accordgly decreases from /m to /m (Theorem 2 whch s almost tght sce there s a asymptotc lower boud We fd a way to losslessly dscretze ad regularze the costraed-deadle tasks so that essetally the executo tmes of the tasks are all 1 ad the deadles are 1 2 respectvely where s umber of tasks to be scheduled (Lemmas The oly parameter that vares s the perod. The trasformato s lossless the sese that the quatty ρ does ot chage though the parameters are extremely smplfed. 3 We vet a method to further trasform the tasks so that the perod of each task rages over tegers betwee 1 to 2 (Lemma 6. Ths trasformato mght be lossful but the loss s eglgble sce ρ chages at most These techcs may be further appled to realtme schedulg or other problems. The rest of the paper s orgazed as follows: Secto 2 presets the model ad prelmares; Secto 3 focuses o uprocessor case ad derves a ew upper boud (14/9 of ρ for feasble sporadc tasks; Secto 4 provdes a ew upper boud (23/9 1/m of the speedup factor for parttoed- EDF. Fally Secto 5 cocludes the paper ad metos some potetal future drectos. II. SYSTEM MODEL AND PRELIMINARIES We cosder a fte set τ of sporadc tasks. Each task τ ca be represeted by a trple τ (e d where e s the worst-case executo tme d s ts relatve deadle ad s the mmum ter-arrval separato legth (also kow as perod respectvely. The task τ s sad to be costraeddeadle f d. Ths paper focuses o costraed-deadle tasks. Hereuder every task s costraed-deadle by default uless otherwse metoed. Gve a task τ we ca calculate ts demad boud fucto dbf(τ t [17] ad ts approxmate demad boud fucto dbf (τ t [10] the followg maer: { 0 f t < d dbf(τ t ( t d + 1 e otherwse ad dbf (τ t { 0 f t < d ( t d + 1 e otherwse. Smlarly for ay set τ of tasks we defe dbf(τ t τ τ dbf(τ t dbf (τ t τ τ dbf (τ t. To aalyze the speedup factor of parttoed-edf o multprocessor platforms the followg quatty plays a crtcal role: ρ τ dbf (τ d (1 d where τ rages over sporadc task sets that are feasble o uprocessor platforms ad d s the largest relatve deadle τ. Here feasble meas that the set of tasks allows a successful schedulg. We wll see that actually ρ s the optmum value of the followg math programmg MP 0 : dbf (τ (MP 0 (2 subject to dbf(τ t t t > 0 (3 d + > 1 1 (4 d 1 d 2 (5 Z + e d R + 1. (6 where Z + s the set of postve tegers whle R + stads for the set of postve real umbers. 2

3 Fg. 1. A example for task trasformato ad dbf modfcatos wth task parameters of e 2 d 3 5 ad 9. Lemma 1: ρ s the optmum value of MP 0. Proof: Let τ {τ (e d : 1 } be a arbtrary set of sporadc tasks that s feasble o a uprocessor wth speed 1. Assume that d 1 d 2. Apply the trasformato proposed [2]: e p d ( d d + 1 ( d d + 1 p ( d d e (7 (8 + d. (9 Let τ {τ 1 τ 2 τ } wth τ (e d p for ay 1. Please refer to Fgure 1 for a llustrato of the above metoed trasformato. For ay 1 p d d. Hece τ s costraed-deadle sce so s τ. I [2] t was prove that the followg results hold smultaeously: dbf (τ dbf (τ ; dbf(τ t dbf(τ t for t > 0 ; d < d + p for 1 ; v d. Ths mmedately leads to our lemma. III. IMPROVED BOUND FOR UNIPROCESSOR CASE I order to estmate the speedup factor for multprocessor parttoed schedulg we frst focus o the uprocessor case. The ma result of ths secto s Theorem 1 whch establshes 14/9 as a upper boud of ρ for sporadc tasks. The basc dea of our proof s to dscretze the tasks to regular form thus reducg the problem to a optmzato oe o bouded tegers. Roughly speakg Lemma 2 makes sure that ρ does ot chage f the parameters of the tasks are restrcted to be ratoal umbers Lemma 4 clams that further requrg e d d 1 for all keeps ρ uchaged the tred cotues by Lemma 5 eve f all the tasks are requred to have the same worst-case executo tme ad fally Lemma 6 eables us to oly cosder tasks wth bouded dscrete perods. These trasformatos reduce estmatg ρ to a smpler optmzato problem whch s solved approxmately Lemma 8. These results mmedately lead to Theorem 1. Specfcally we frst observe that the optmum value of MP 0 remas uchaged eve f the doma R + s replaced by Q + the set of postve ratoal umbers. dbf (τ (MP 1 (10 subject to dbf(τ t t t > 0 (11 d + > 1 1 (12 d 1 d 2 (13 Z + e d Q + 1. (14 Lemma 2: MP 0 ad MP 1 has the same optmum value. Proof: The lemma mmedately holds f all of the followg clams are true: 1 The objectve fuctos of MP 0 ad MP 1 are the same ad cotuous. 2 The doma of MP 1 s cluded that of MP 0. 3 For ay ɛ > 0 ad ay feasble soluto τ {τ (e d : 1 } to MP 0 there s a feasble soluto τ {τ (e d p : 1 } to MP 1 such that for ay 1 e e < ɛ d d < ɛ p < ɛ. (15 It suffces to prove Clam 3 sce the others are obvous. Let τ {τ (e d : 1 } be a arbtrary set of tasks that s a feasble soluto to MP 0 ad ɛ be a arbtrary postve real umber. Wthout loss of geeralty assume that ɛ < m 1 e. For ay 1 arbtrarly choose p ( + ɛ 2 + ɛ Q + d ( 1ɛ (d + 2 d + ɛ 2 Q+ e (e ɛ e Q +. Let τ deote the set of tasks {τ (e d p : 1 }. Obvously τ meets Codtos (14 ad (15. To proceed arbtrarly fx a teger 1. Note that p d > + ɛ 2 (d + ɛ 2 d. Ths together wth the fact that τ s costraed-deadle meas τ s also costraed-deadle. 3

4 Observe that d > d + ( 1ɛ 2 d 1 + ( 1ɛ 2 > d 1. Hece τ satsfes Codto (13 of MP 1. Because d + p ( 1ɛ > d ɛ 2 2 d + + ɛ 2 > + ɛ (sce τ satsfes (4 2 > d the task set τ satsfes Codto (12. As to Codto (11 arbtrarly fx t > 0. Whe t < d dbf(τ t 0 dbf(τ t. Whe t d because p > d > d e < e we have t d dbf(τ t p + 1 e ( t d + 1 e dbf(τ t. As a result we always have dbf(τ t dbf(τ t. The τ satsfes Codto (11 sce τ satsfes (3. Altogether τ s a feasble soluto to MP 1. Now we preset a techcal lemma that wll be frequetly used. Lemma 3: Suppose d p d p R + are such that d + p d + p ad d > d. For ay real umber t t d p > t d p f ad oly f t < d + p. Proof: Let δ d d p p. The t d p > t d p p (t d > p (t d p (t d + δ > (p + δ (t d p δ > δ (t d p > t d. Hereuder let d 0 0. The t s tme to show that the optmum value of MP 1 remas uchaged eve f we further requre e d d 1 for all 1. We defe a ew math programmg dbf (τ (MP 2 (16 subject to dbf(τ t t t > 0 (17 d + > 1 1 (18 d e + d 1 1 (19 Z + e d Q + 1. (20 Lemma 4: MP 1 ad MP 2 have the same optmum value. Proof: For ay feasble soluto τ {τ (e d : 1 } to MP 1 defe M(τ { : 1 d e + d 1 }. Obvously τ to MP 1 s a feasble soluto to MP 2 f ad oly f M(τ 0. Cosder the followg proposto: for ay feasble soluto τ to MP 1 wth M(τ > 0 there s a feasble soluto τ to MP 1 such that M(τ < M(τ ad the objectve value of τ s at least that of τ. If t s true oe ca easly prove the lemma by teratvely applyg the proposto. Hece the rest of the proof s devoted to showg ths proposto. Arbtrarly fx a feasble soluto τ {τ (e d : 1 } to MP 1. Suppose M(τ > 0. Assume k s the smallest dex such that e k d k d k 1 meag that e d d 1 for all < k. The we have k 1 e d k 1. (21 Sce d + > d k d for ay < k oe has k e k dbf(τ d k dbf(τ d k d k where the last equalty holds because τ satsfes Codto (11. Ths together wth (21 leads to e k d k d k 1. By the assumpto that e k d k d k 1 we get e k < d k d k 1. (22 Costruct τ {τ (e d p : 1 } where for ay k ad d d p e e e k e k d k d k 1 + e k p k d k + p k d k. By (21 ad (22 k e d k 1 + e k d k < d k. Obvously M(τ M(τ 1 < M(τ ad τ s costraed-deadle sce so s τ. Now we prove that τ s a feasble soluto to MP 1. Sce τ satsfes Codtos (12-(14 so does τ. To show that Codto (11 s satsfed by τ we arbtrarly choose t > 0 ad proceed case by case. Case 1: f t < d k. The dbf(τ t dbf(τ t 1 <k 1 <k dbf(τ t 1 dbf(τ t (because t < d j for j k dbf(τ t (because τ τ for < k t (because τ satsfes Codto (11. 4

5 Case 2: f d k t < d k. dbf(τ t dbf(τ t 1 k 1 k 1 k 1 ( t d p + 1 e e (because d + p > t for ay e d k t. Case 3: f d k t < d k + p k (. The t d dbf(τ k t k p + 1 e k k e k (because d k < d k t < d k + p k t dk + 1 e k p k where the last equalty s due to d k t < d k + p k d k + p k. For ay k dbf(τ t dbf(τ t sce τ τ. As a result dbf(τ t dbf(τ t t because τ satsfes Codto (11. Case 4: f t d k + p k. Because by Lemma 3 we have The dbf(τ t d k < d k ad p k + d k d k + p k 1 t d k p k dbf(τ t k dbf(τ t + ( t d k p k p k t d k p k. + 1 e k t dbf(τ dk t e k k k dbf(τ t + dbf(τ k t (sce τ τ for k dbf(τ t t (sce τ satsfes Codto (11. Altogether τ satsfes Codto (11 so t s a feasble soluto to MP 1. Fally we show that dbf (τ dbf (τ d. Whe k < we have d so t suffces to show dbf (τ dbf (τ d. By defto of τ for ay k dbf (τ dbf (τ d. Furthermore ote three facts: 1 p k + d k d k + p k ; 2 d k < d k; d 3 < d k + p k due to Codtos (12. By Lemma 3 these facts mea d k p k d d k p k whch mples dbf (τ k dbf (τ k d. As a result dbf (τ dbf (τ d. Whe k we have d <. For ay < dbf (τ e ( 1 + d d e ( 1 + p d < e ( 1 + p d d e ( d 1 + d d e d ( 1 + d d p (because τ τ dbf (τ d d where the equalty s due to d < ad d 0 (sce τ s costraed-deadle. I addto dbf (τ e < e d dbf (τ d d. Therefore we also get dbf (τ d as desred. We wll mpose further costrat o MP 2 wthout chagg the optmum value. As preseted the math programmg MP 3 the costrat s that all the e s are equal. dbf (τ d dbf (τ (MP 3 (23 subject to dbf(τ t t t > 0 (24 d + > 1 1 (25 d e + d 1 1 (26 e / 1 (27 Z + e d Q + 1. (28 Lemma 5: MP 2 ad MP 3 have the same optmum value. Proof: Let τ {τ (e d : 1 } be a arbtrary feasble soluto to MP 2. Due to Codto (20 we ca choose δ Q + such that k( e δ s a teger for ay 1. Let k(. For ay 1 l defe task τ l (e l d l p l as below where 1 ad 1 j k( are such that l m( j j + 1 h< k(h: e l δ d l d 1 + p l + d d l. j k( (d d 1 d 1 + jδ 5

6 Let τ ( {τ m(j : 1 j k(} for ay 1 ad τ τ (. Let d 0 0. Next we wll prove that τ s a feasble soluto to MP 3. Frst of all for ay 1 ad 1 j k( let l m( j. We have d l d ad p l + d d l. Thus τ s costraed-deadle because so s τ. Sce τ satsfes Codtos (25-(28 by defto ow vestgate Codto (24. Arbtrarly fx t > 0 ad proceed case by case. Case 1: t < d. Let teger l 0 be such that d l t < d l+1. The dbf(τ t 1 r l 1 r l 1 r l dbf(τ r t ( t d r 1 r dbf(τ r t p r e r d l t (because t < d l e r where the fourth equalty holds due to the equalty p r > t d r whch tur follows from three facts: 1 For ay 1 ad 1 j k( we have p m(j + d d m(j by defto; 2 For ay 1 + d > sce τ satsfes Codto (18; 3 d > t. Case 2: t d. It suffces to prove that dbf(τ ( t dbf(τ t for ay 1. Arbtrarly fx 1. Suppose t < d +. We observe that k( dbf(τ ( t dbf(τ m(j t j1 k( t d m(j + 1 δ j1 p m(j k(δ (because t < d + d m(j + p m(j e (By defto of k( dbf(τ t (because d t < d + The cosder t d +. For ay 1 j k( sce d > d m(j ad d + d m(j +p m(j Lemma 3 mples t d m(j p m(j t d Whch further leads to k( t d dbf(τ m(j ( t + 1 δ j1 j1 k( t d + 1 δ t d + 1 dbf(τ t e p m(j Altogether Codto (24 s satsfed both cases so τ s a feasble soluto to MP 3. The rest of the proof s to show that Note that d dbf (τ d dbf (τ. < + d d m(j + p m(j ad d m(j d for ay 1 1 j k(. Lemma 3 mples that d d m(j p m(j d. The for ay 1 we have ( k( dbf (τ ( d d d m(j + 1 δ j1 p m(j ( d d + 1 dbf (τ. Therefore dbf (τ d dbf (τ. It s stll hard to fd a good upper boud of the optmum value of MP 3 partly because Codto (24 s too strog ad Codto (25 s too weak. It has to be modfed accordgly. O the oe had we relax (24 by replacg the fucto dbf( wth f( : for ay task τ (e d ad tme t > 0 { dbf(τ t f t < d f(τ t + 2e otherwse Note that f(τ t dbf(τ t always holds. The frst argumet of f ca be aturally exteded to ay set τ of tasks: f(τ t τ τ f(τ t. O the other had stead of (25 we requre that the set of tasks τ should be alged as defed below: Defto 1: Gve a task set τ {τ (e d : 1 } a permutato π over {1 2 } s called a algg permutato of τ f d π( + p π( + d e 6

7 for ay 1. τ s sad to be alged f t has a algg permutato. We wll show that the optmum value of MP 3 does ot decrease after the modfcato. Specfcally defe a ew math programmg where the tasks are ot requred to be costraed-deadle: dbf (τ (MP 4 (29 subject to f(τ t t t > 0 (30 τ s alged (31 d e + d 1 1 (32 e / 1 (33 Z + e d Q + 1. (34 Lemma 6: The optmum value of MP 3 s ot more tha that of MP 4. Proof: Arbtrarly choose a feasble soluto τ {τ (e d : 1 } to MP 3. Let π be a permutato over {1 2 } such that d π(1 + p π(1 d π(2 + p π(2... d π( + p π(. (35 For ay 1 costruct a task τ (e d p where e e d d p + d π 1 ( d. Let τ {τ : 1 }. We wll show that τ s a feasble soluto to MP 4. Sce Codtos (31-(34 are satsfed by defto t suffces to vestgate Codto (30. Let s frst derve a equalty as tool. For ay 1 let j π 1 ( ad we have d + dbf(τ d + (sce τ satsfes Codto (24 dbf(τ π(l d + 1 l j + j<l 1 l j dbf(τ π(l d + 2e π(l + j<l e π(l 2j + ( j + d j (due to Codtos (26 ad (27 where the secod equalty s because d + d π(j + p π(j d π(l + p π(l for ay l j ad d + > d π(l for ay l. Hece we have by defto of τ. d + + d π 1 ( d + p (36 Now we cotue to prove τ satsfes Codto (30. For a arbtrary t > 0 ths ca be doe case by case. Case 1: t < + d 1. The for ay 1 d + d + p + d π 1 ( by (36 + d 1 > t Ths together wth the defto of τ mples that f(τ t dbf(τ t dbf(τ t. Because τ satsfes Codto (24 we have f(τ t t. Case 2: t + d 1. Choose the bggest 1 such that + d t. The for ay j > Thus p π(j + d π(j + d j > t > d π(j. f(τ t f(τ π(j t + f(τ π(j t 1 j 1 j + d t 2e π(j + <j <j e π(j Altogether Codto (30 s also satsfed. Furthermore for ay 1 by (36 ad d d we have p. Ths together wth e e d d for ay 1 mples dbf (τ d dbf (τ. As a result dbf (τ dbf (τ d. The lemma thus holds. We preset a techcal lemma before gog o. Lemma 7: For ay x 1 x 2 x R + such that x 2 we have d x 4 9. Proof: By Cauchy s Iequalty ( ( x ( 2. x Note that 3 2 ( xdx xdx

8 Therefore ( x Lemma 8: The optmum value of MP 4 s at most Proof: Arbtrarly choose a feasble soluto τ {τ (e d : 1 } to MP 4. Let δ d. By Codtos (32 ad (33 e δ ad d δ for ay 1. Let π be a algg permutato of τ. The we have p π( ( + d d π( 2 δ whch mples p π( δ 2. By Lemma 7 δ 4 p π( 9. j1 Hece d j + p j p j As a result dbf (τ d π( + p π( p π( d p π( δ p π( 4 9. dbf (τ 2δ δ (sce τ s alged ( 2 p + d e ( 2 p + d δ + d 2δ 4 9 δ 14 9 The lemma holds. We are ready to preset oe of the ma results of ths paper whch clams that the resource augmetato boud o a sgle processor s at most for ay set τ of costrateddeadle tasks such that dbf(τ t t for all t > 0 where d s the maxmum relatve deadle of the tasks τ. Proof: It follows from Lemmas ad 8. Theorem 1: dbf (τd d 14 9 IV. PARTITIONED SCHEDULING ON MULTIPROCESSORS Ths secto s devoted to parttog sporadc tasks o multprocessors where the tasks are assumed to have Costraed Deadles. We adopted the algorthm of Deadle-Mootoc Parttog [1]. It s preseted Algorthm 1 to make ths paper self-cotaed where e ad d stad for worst-case executo tme ad relatve deadle of task τ respectvely. Bascally Algorthm 1 assgs tasks sequetally the order of o-decreasg relatve deadles. Suppose τ(k s the set of tasks at processor k after the frst 1 tasks has bee assged. The task τ s assged to the frst processor (say processor No. k that ca safely serve the task amely e + dbf (τ(k d d. Remember that we have upper-bouded ρ τ dbf (τ d (37 d where τ rages over costraed-deadle sporadc task sets that are feasble o uprocessors ad d s the largest relatve deadle τ. The followg lemma s from refereces [1] ad [2] so the proof s omtted. Lemma 9: The speedup factor of Algorthm 1 s 1+ρ 1/m where m s the umber of processors. It s tme to preset the other ma result of ths paper. Theorem 2: The speedup factor for Algorthm 1 s at most /m. Proof: The theorem mmedately follows from Theorem 1 ad Lemma 9. Algorthm 1 Deadle-Mootoc Parttog Iput: sporadc tasks τ {τ 1... τ } to be parttoed o m detcal ut-capacty processors; The tasks are dexeo-decreasgly accordg to ther relatve deadles. For ay 1 k m let τ(k deote the set of tasks assged to the kth processor. 1: τ(k for ay 1 k m; 2: for 1 to do 3: f there exsts k such that e + dbf (τ(k d d the 4: Choose the smallest such k; 5: τ(k τ(k {τ }; 6: else 7: retur FAIL 8: ed f 9: ed for 10: retur feasble assgmet τ(1 τ(2... τ(m V. CONCLUSION AND FUTURE WORK I ths paper we mprove the upper boud of the speedup factor of parttoed-edf from /m to

9 1/m for costraed-deadle sporadc tasks o m detcal processors arrowg the gap betwee the upper ad the lower bouds from to Ths s a mmedate corollary of our mprovemet of the upper boud of ρ τ dbf (τ d/d from to The ew upper bouds are very close to the correspodg lower bouds. Techcally our mprovemets root at a ovel dscretzato that trasform the tasks to regular forms wthout decreasg ρ. The dscretzato essetally makes all the tasks have fxed executo tmes ad deadles. The oly parameter that vares s the perod whch s hghly restrcted so as to rage over the set {1 2 2} where s the umber of tasks to be scheduled. By ths trasformato estmatg ρ s reduced to a much smpler optmzato problem. We beleve that ths kack may work other problems or scearos. However we have ot yet proved that our trasformato s equvalet. Ths meas that the dscretzato mght strctly elarge ρ. The gooews s that the curred loss f ot zero at all s guarateed to be o more tha As to future drectos we cojecture that Theorems 1 ad 2 rema true f the costraed-deadle codto s removed. We also cojecture that our method ca derve a 1.5 upper boud for ρ thus closg the gap betwee the upper ad the lower bouds. If ths s the case the speedup factor of parttoed-edf s also fully determed at least the case of costraed deadles. [9] C. A. Phllps C. Ste E. Torg ad J. We Optmal tme-crtcal schedulg va resource augmetato Algorthmca vol. 32 o. 2 pp [10] K. Albers ad F. Slomka A evet stream drve approxmato for the aalyss of real-tme systems Real-Tme Systems ECRTS Proceedgs. 16th Euromcro Coferece o. IEEE 2004 pp [11] S. K. Baruah ad J. Goosses The EDF schedulg of sporadc task systems o uform multprocessors Proceedgs of the 29th IEEE Real-Tme Systems Symposum RTSS 2008 Barceloa Spa 30 November - 3 December pp [12] R. R. Devllers ad J. Goosses Lu ad laylad s schedulablty test revsted If. Process. Lett. vol. 73 o. 5-6 pp [13] R. I. Davs A. Thekklakattl O. Gettgs R. Dobr S. Puekkat ad J. Che Exact speedup factors ad sub-optmalty for o-preemptve schedulg Real-Tme Systems vol. 54 o. 1 pp [14] E. B ad G. C. Buttazzo Measurg the performace of schedulablty tests Real-Tme Systems vol. 30 o. 1-2 pp [15] E. B The quadratc utlzato upper boud for arbtrary deadle real-tme tasks IEEE Tras. Computers vol. 64 o. 2 pp [16] J. Thes ad G. Fohler Trasformato of sporadc tasks for offle schedulg wth utlzato ad respose tme trade-offs 19th Iteratoal Coferece o Real-Tme ad Network Systems RTNS 11 Nates Frace September Proceedgs 2011 pp [17] S. K. Baruah A. K. Mok ad L. E. Roser Preemptvely schedulg hard-real-tme sporadc tasks o oe processor Real-Tme Systems Symposum Proceedgs. 11th. IEEE 1990 pp ACKNOWLEDGMENT The authors would lke to thak Prof. Sajoy Baruah from Washgto Uversty at St. Lous for the frutful dscussos. REFERENCES [1] S. Baruah ad N. Fsher The parttoed multprocessor schedulg of sporadc task systems Real-Tme Systems Symposum RTSS th IEEE Iteratoal. IEEE 2005 pp [2] J.-J. Che ad S. Chakraborty Resource augmetato bouds for approxmate demad boud fuctos Real-Tme Systems Symposum (RTSS 2011 IEEE 32d. IEEE 2011 pp [3] F. Esebrad ad T. Rothvoß Edf-schedulablty of sychroous perodc task systems s cop-hard Proceedgs of the twety-frst aual ACM-SIAM symposum o Dscrete Algorthms. SIAM 2010 pp [4] A. K.-L. Mok Fudametal desg problems of dstrbuted systems for the hard-real-tme evromet Ph.D. dssertato Massachusetts Isttute of Techology [5] J.-J. Che G. vo der Brügge W.-H. Huag ad R. I. Davs O the Ptfalls of Resource Augmetato Factors ad Utlzato Bouds Real-Tme Schedulg 29th Euromcro Coferece o Real- Tme Systems (ECRTS 2017 ser. Lebz Iteratoal Proceedgs Iformatcs (LIPIcs 2017 pp. 9:1 9:25. [6] Z. Guo Regardg the optmalty of speedup bouds of mxedcrtcalty schedulablty tests Mxed Crtcalty o Multcore/Maycore Platforms (Dagstuhl Semar Reports vol [7] K. Agrawal ad S. Baruah Itractablty ssues mxed-crtcalty schedulg the 30th Euromcro Coferece o Real-Tme Systems (ECRTS to appear. [8] B. Kalyaasudaram ad K. Pruhs Speed s as powerful as clarvoyace J. ACM vol. 47 o. 4 pp