Analysis of Federated and Global Scheduling for Parallel Real-Time Tasks

Transcription

1 Analysis of Federaed and Global Scheduling for Parallel Real-Time Tasks Jing Li, Jian-Jia Chen, Kunal Agrawal, Chenyang Lu, Chris Gill, Abusayeed Saifullah Washingon Universiy in S. Louis, U.S.A. TU Dormund Universiy, Germany {kunal, lu, Absrac This paper considers he scheduling of parallel realime asks wih implici deadlines. Each parallel ask is characerized as a general direced acyclic graph (DAG). We analyze hree differen real-ime scheduling sraegies: wo well known algorihms, namely global earliesdeadline-firs and global rae-monoonic, and one new algorihm, namely federaed scheduling. The federaed scheduling algorihm proposed in his paper is a generalizaion of pariioned scheduling o parallel asks. In his sraegy, each high-uilizaion ask (uilizaion 1) is assigned a se of dedicaed cores and he remaining low-uilizaion asks share he remaining cores. We prove capaciy augmenaion bounds for all hree schedulers. In paricular, we show ha if on uni-speed cores, a ask se has oal uilizaion of a mos m and he criicalpah lengh of each ask is smaller han is deadline, hen federaed scheduling can schedule ha ask se on m cores of speed ; G-EDF can schedule i wih speed ; and G-RM can schedule i wih speed We also provide lower bounds on he speedup and show ha he bounds are igh for federaed scheduling and G-EDF when m is sufficienly large. I. Inroducion In he las decade, mulicore processors have become ubiquious and here has been exensive work on how o exploi hese parallel machines for real-ime asks. In he real-ime sysems communiy, here has been significan research on scheduling ask ses wih iner-ask parallelism where each ask in he ask se is a sequenial program. In his case, increasing he number of cores allows us o increase he number of asks in he ask se. However, since each ask can only use one core a a ime, he compuaional requiremen of a single ask is sill limied by he capaciy of a single core. Recenly, here has been some ineres in he design and analysis of scheduling sraegies for ask ses wih inra-ask parallelism (in addiion o iner-ask parallelism), where individual asks are parallel programs and can poenially uilize more han one core in parallel. These models enable asks wih higher execuion demands and igher deadlines, such as hose used in auonomous vehicles [31], video surveillance, compuer vision, radar racking and real-ime hybrid esing [8] In his paper, we consider he general direced acyclic graph (DAG) model. We analyze hree differen scheduling sraegies: a new sraegy, namely federaed scheduling, and wo classic sraegies, namely global EDF and global rae-monoonic scheduling. We prove ha all hree sraegies provide srong performance guaranees, in he form of capaciy augmenaion bounds, for scheduling parallel DAG asks wih implici deadlines. One can generally derive wo ypes of performance bounds for real-ime schedulers. The radiional bound is called a resource augmenaion bound (also called a processor speed-up facor). A scheduler S provides a resource augmenaion bound of b 1 if i can successfully schedule any ask se τ on m cores of speed b as long as he ideal scheduler can schedule τ on m cores of speed 1. A resource augmenaion bound provides a good noion of how close a scheduler is o he opimal schedule, bu i has a drawback. Noe ha he ideal scheduler is only a hypoheical scheduler, meaning ha i always finds a feasible schedule if one exiss. Unforunaely, since we ofen canno ell wheher he ideal scheduler can schedule a given ask se on uni-speed cores, a resource augmenaion bound may no provide a schedulabiliy es. Anoher bound ha is commonly used for sequenial asks is a uilizaion bound. A scheduler S provides a uilizaion bound of b if i can successfully schedule any ask se which has oal uilizaion a mos m/b on m cores. 1 A uilizaion bound provides more informaion han a resource augmenaion bound; any scheduler ha guaranees a uilizaion bound of b auomaically guaranees a 1 A uilizaion bound is ofen saed in erms of 1/b; we adop his noaion in order o be consisen wih he oher bounds saed here.

2 resource augmenaion bound of b as well. In addiion, i acs as a very simple schedulabiliy es in iself, since he oal uilizaion of he ask se can be calculaed in linear ime and compared o m/b. Finally, a uilizaion bound gives an indicaion of how much load a sysem can handle; allowing us o esimae how much over-provisioning may be necessary when designing a plaform. Unforunaely, i is ofen impossible o prove a uilizaion bound for parallel sysems due o Dhall s effec; ofen, we can consruc pahological ask ses wih uilizaion arbirarily close o 1, bu which canno be scheduled on m cores. Li e al. [35] defined a concep of capaciy augmenaion bound which is similar o he uilizaion bound, bu adds a new condiion. A scheduler S provides a capaciy augmenaion bound of b if i can schedule any ask se τ which saisfies he following wo condiions: (1) he oal uilizaion of τ is a mos m/b, and () he worscase criical-pah lengh of each ask L i (execuion ime of he ask on an infinie number of cores) is a mos 1/b fracion of is deadline. A capaciy augmenaion bound is quie similar o a uilizaion bound: i also provides more informaion han a resource augmenaion bound does; any scheduler ha guaranees a capaciy augmenaion bound of b auomaically guaranees a resource augmenaion bound of b as well. I also acs as a very simple schedulabiliy es. Finally, i can also provide an esimaion of he load a sysem is expeced o handle. There has been some recen research on proving boh resource augmenaion bounds and capaciy augmenaion bounds for various scheduling sraegies for parallel asks. This work falls in wo caegories. In decomposiion-based sraegies, he parallel ask is decomposed ino a se of sequenial asks and hey are scheduled using exising sraegies for scheduling sequenial asks on muliprocessors. In general, decomposiion-based sraegies require explici knowledge of he srucure of he DAG off-line in order o apply decomposiion. In non-decomposiion based sraegies, he program can unfold dynamically since no off-line knowledge is required. For a decomposed sraegy, mos prior work considers synchronous asks (subcaegory of general DAGs) wih implici deadlines. Lakshmanan e al. [3] proved a capaciy augmenaion bound of 3.4 for pariioned fixed-prioriy scheduling for a resriced caegory of synchronous asks 3 under decomposed deadline monoonic scheduling. Saifullah e al. [45] provide a differen decomposiion sraegy for general parallel synchronous asks and prove a capaciy augmenaion bound of 4 when he decomposed asks are scheduled using global EDF and 5 when scheduled using pariioned DM. Kim e al. [31] provide anoher decomposiion sraegy and prove a capaciy augmenaion bound of Criical-pah lengh of a sequenial ask is equal o is execuion ime 3 Fork-join ask model, in heir erminology using global deadline monoonic scheduling. Nelissen e al. [40] proved a resource augmenaion bound of for general synchronous asks. More recenly, Saifullah e al. [44] provide a decomposiion sraegy for general DAG asks ha provides a capaciy augmenaion bound of 4. For non-decomposiion sraegies, researchers have sudied primarily global earlies deadline firs (G-EDF) and global rae-monoonic (G-RM). Andersson and Niz [4] show ha G-EDF provides resource augmenaion bound of for synchronous asks wih consrained deadlines. Boh Li e al. [35] and Bonifaci e al. [15] concurrenly showed ha G-EDF provides a resource augmenaion bound of for general DAG asks wih arbirary deadlines. In heir paper, Bonifaci e al. also proved ha G-RM provides a resource augmenaion bound of 3 for parallel DAG asks wih arbirary deadlines; Li e al. also provide a capaciy augmenaion bound of 4 for G-EDF for ask ses wih implici deadlines. In summary, he bes known capaciy augmenaion bound for implici deadlines asks are 4 for DAG asks using G-EDF, and 3.73 for parallel synchronous asks using decomposiion combined wih G-DM. The conribuions of his paper are as follows: 1 We propose a novel federaed scheduling sraegy. Here, each high-uilizaion ask (uilizaion 1) is allocaed a dedicaed cluser (se) of cores. A muliprocessor scheduling algorihm is used o schedule all low-uilizaion asks, each of which is run sequenially, on a shared cluser composed of he remaining cores. Federaed scheduling can be seen as a pariioned sraegy generalized o parallel asks. This is he bes known capaciy augmenaion bound for any scheduler for parallel DAGs. We prove ha he capaciy augmenaion bound for his federaed scheduler is. In addiion, we also show no scheduler can provide a beer capaciy augmenaion bound of 1/m for parallel asks. Therefore, a bound of for federaed scheduling is igh when m is large enough. 3 We improve he capaciy augmenaion bound of G- EDF o for DAGs. When m is large, here is a maching lower bound for G-EDF [35]; hence, his resul closes he gap for large m. This is he bes known capaciy augmenaion bound for any global scheduler for parallel DAGs. 4 We show ha G-RM has a capaciy augmenaion bound of This is he bes known capaciy augmenaion bound for any fixed-prioriy scheduler for DAG asks. Even if resriced o synchronous asks, his is sill he bes bound for global fixed prioriy scheduling wihou decomposiion. The paper is organized as follows. Secion II defines he DAG model for parallel asks and provides some definiions. Secion III presens our federaed scheduling

3 algorihm and proves he augmenaion bound. Secion IV proves a lower bound for any scheduler for parallel asks. Secion V presens a canonical form o give an upper bound of he work of a DAG ha should be done in a specified inerval lengh. Secion VI proves ha G-EDF provides a capaciy augmenaion bound of.618. Secion VII shows ha G-RM provides a capaciy augmenaion bound of We discuss some pracical consideraions of he hree schedulers in Secion VIII. Secion IX discusses relaed work and Secion X concludes his paper. II. Sysem Model We now presen he deails of he DAG ask model for parallel asks and some addiional definiions. We consider a se τ of n independen sporadic realime asks {τ 1, τ,..., τ n }. A ask τ i represens an infinie sequence of arrivals and execuions of ask insances (also called jobs). We consider he sporadic ask model [9, 9] where, for a ask τ i, he minimum iner-arrival ime (or period) T i represens he ime beween consecuive arrivals of ask insances, and he relaive deadline represens he emporal consrain for execuing he job. If a ask insance of τ i arrives a ime, he execuion of his insance mus be finished no laer han he absolue deadline + and he release of he nex insance of ask τ i mus be no earlier han plus he minimum iner-arrival ime, i.e. +T i. In his paper, we consider implici deadline asks where each ask τ i s relaive deadline is equal o is minimum iner-arrival ime T i ; ha is, T i =. We consider he schedulabiliy of his ask se on a uniform mulicore sysem consising of m idenical cores. Each ask τ i τ is a parallel ask and is characerized as a direced acyclic graph (DAG). Each node (subask) in he DAG represens a sequence of insrucions (a hread) and each edge represens a dependency beween nodes. A node (subask) is ready o be execued when all is predecessors have been execued. Throughou his paper, as i is no necessary o build he analysis based on specific srucure of he DAG, only wo parameers relaed o he execuion paern of ask τ i are defined: oal execuion ime (or work) C i of ask τ i : This is he summaion of he wors-case execuion imes of all he subasks of ask τ i. criical-pah lengh L i of ask τ i : This is he lengh of he criical-pah in he given DAG, in which each node is characerized by he wors-case execuion ime of he corresponding subask of ask τ i. Criical-pah lengh is he wors-case execuion ime of he ask on an infinie number of cores. Given a DAG, obaining work C i and criical-pah lengh L i [46, pages ] can boh be done in linear ime. For breviy, he uilizaion Ci T i = Ci of ask τ i is denoed by u i for implici deadlines. The oal uilizaion of he ask se is U = τ u i τ i. Uilizaion-Based Schedulabiliy Tes. In his paper, we analyze schedulers in erms of heir capaciy augmenaion bounds. The formal definiion is presened here: Definiion 1. Given a ask se τ wih oal uilizaion of U, a scheduling algorihm S wih capaciy augmenaion bound b can always schedule his ask se on m cores of speed b as long as τ saisfies he following condiions on uni speed cores. Uilizaion does no exceed oal cores, u i m (1) τ i τ For each ask τ i τ, he criical pah L i () Since no scheduler can schedule a ask se τ on m uni speed cores unless Condiions (1) and () are me, a capaciy augmenaion bound auomaically leads o a resource augmenaion bound. This definiion can be equivalenly saed (wihou reference o he speedup facor) as follows: Condiion (1) says ha he oal uilizaion U is a mos m/b and Condiion () says ha he criical-pah lengh of each ask is a mos 1/b of is relaive deadline, ha is, L i /b. Therefore, in order o check if a ask se is schedulable we only need o know he oal ask se uilizaion, and he maximum criical-pah uilizaion. Noe ha a scheduler wih a smaller b is beer han anoher wih a larger b, since when b = 1 S is an opimal scheduler. III. Federaed Scheduling This secion presens he federaed scheduling sraegy for parallel asks wih implici deadlines. We prove ha i provides a capaciy augmenaion bound of on m-core machine parallel real-ime asks. A. Federaed Scheduling Algorihm Given a ask se τ, he federaed scheduling algorihm works as follows: Firs, asks are divided ino wo disjoin ses: τ high conains all high-uilizaion asks asks wih wors-case uilizaion a leas one (u i 1), and τ low conains all he remaining low-uilizaion asks. Consider a high-uilizaion ask τ i wih wors-case execuion ime C i, wors-case criical-pah lengh L i, and deadline (which is equal o is period T i ). We assign n i dedicaed cores o τ i, where n i is Ci L i n i = (3) L i We use n high = τ i τ high n i o denoe he oal number of cores assigned o high-uilizaion asks τ high. We assign he remaining cores o all low-uilizaion asks τ low, denoed as n low = m n high. The federaed scheduling algorihm admis he ask se τ, if n low is non-negaive and n low τ i τ low u i. Afer a valid core allocaion, runime scheduling proceeds as follows: (1) Any greedy (work-conserving) parallel scheduler can be used o schedule a high-uilizaion ask 3

4 τ i on is assigned n i cores. Informally, a greedy scheduler is one ha never keeps a core idle if some node is ready o execue. () Low-uilizaion asks are reaed and execued as hough hey are sequenial asks and any muliprocessor scheduling algorihm (such as pariioned EDF [37], or various rae-monoonic schedulers [3]) wih a uilizaion bound of a mos 1/ can be used o schedule all he lowuilizaion asks on he allocaed n low cores. The imporan observaion is ha we can safely rea low-uilizaion asks as sequenial asks since C i and parallel execuion is no required o mee heir deadlines. 4 B. Capaciy Augmenaion Bound of for Federaed Scheduling Theorem 1. The federaed scheduling algorihm has a capaciy augmenaion bound of. To prove Theorem 1, we consider a ask se τ ha saisfies Condiions (1) and () from Definiion 1 for b =. Then, we (1) sae he relaively obvious Lemma 1; () prove ha a high uilizaion ask τ i mees is deadline when assigned n i cores; and (3) show ha n low is nonnegaive and saisfies n low b τ i τ low u i and herefore all low uilizaion asks in τ will mee deadlines when scheduled using any muliprocessor scheduling sraegy wih uilizaion bound no less han b (i.e. can afford oal ask se uilizaion of m/b = 50%m). These hree seps complee he proof. Lemma 1. A ask se τ is classified ino disjoin subses s 1, s,..., s k, and each subse is assigned a dedicaed cluser of cores wih size n 1, n,..., n k respecively, such ha i n i m. If each subse s j is schedulable on is n j cores using some scheduling algorihm S j (possibly differen for each subse), hen he whole ask se is guaraneed o be schedulable on m cores. High-Uilizaion Tasks Are Schedulable. Assume ha a machine s execuion ime is divided ino discree quana called seps. During each sep a core can be eiher idle or performing one uni of work. We say a sep is complee if no core is idle during ha sep, and oherwise we say i is incomplee. A greedy scheduler never keeps a cores idle if here is ready work available. Then, for a greedy scheduler on n i cores, we can sae wo sraighforward lemmas [35]. Lemma. [Li13] Consider a greedy scheduler running on n i cores for ime seps. If he oal number of incomplee seps during his period is, he oal work F done during hese ime seps is a leas F n i (n i 1). 4 Even if hese asks are expressed as parallel programs, i is easy o enforce correc sequenial execuion of parallel asks any opological ordered execuion of he nodes of he DAG is a valid sequenial execuion. Lemma 3. [Li13] If a job of ask τ i is execued by a greedy scheduler, hen every incomplee sep reduces he remaining criical-pah lengh of he job by 1. From Lemmas and 3, we can esablish Theorem. Theorem. If an implici-deadline deerminisic parallel C ask τ i is assigned n i = i L i L i dedicaed cores, hen all is jobs can mee heir deadlines, when using a greedy scheduler. Proof: For conradicion, assume ha some job of a highuilizaion ask τ i misses is deadline when scheduled on n i cores by a greedy scheduler. Therefore, during he ime seps beween he release of his job and is deadline, here are fewer han L i incomplee seps; oherwise, by Lemma 3, he job would have compleed. Therefore, by Lemma, he scheduler mus have finished a leas n i (n i 1)L i work. = n i (n i 1)L i = n i ( L i ) + L i Ci L i ( L i ) + L i L i C i L i ( L i ) + L i = C i L i Since he job has work of a mos C i, i mus have finished in seps, leading o a conradicion. Low-Uilizaion Tasks are Schedulable. We firs calculae a lower bound on n low, he number of oal cores assigned o low-uilizaion asks, when a ask se τ ha saisfies Condiions (1) and () of Definiion 1 for b = is scheduled using a federaed scheduling sraegy. Lemma 4. The number of cores assigned o low-uilizaion asks is a leas n low low u i. Proof: Here, for breviy of he proof, we denoe σ i = Di L i. I is obvious ha = σ i L i and hence C i = u i = σ i u i L i. Therefore, Ci L i σi u i L i L i σi u i 1 n i = = = L i σ i L i L i Since each ask τ i in ask se τ saisfies Condiion () of Definiion 1 for b = ; herefore, he criical-pah lengh of each ask is a mos 1/b of is relaive deadline, ha is, L i /b = σ i b =. By he definiion of high-uilizaion ask τ i, we have 1 u i. Togeher wih σ i, we know ha: 0 (u i 1)(σ i ) 4

5 From he definiion of ceiling, we can derive σi u i 1 n i = < σ iu i = σ iu i + σ i σ iu i + σ i + (u i 1)(σ i ) = σ iu i + σ i + σ i u i u i σ i + = σ iu i u i = bu i = u i() = u i In summary, for each high-uilizaion ask, n i < bu i. So, heir sum τ high saisfies n high = high n i < b high u i. Since he ask se also saisfies Condiion (1), we have n low = m n high > b all u i b high u i = b low Thus, he number of remaining cores allocaed o lowuilizaion asks is a leas n low > low u i. Corollary 1. For ask ses saisfying Condiions (1) and (), a muliprocessor scheduler wih uilizaion bound of a leas 50% can schedule all he low-uilizaion asks sequenially on he remaining n low cores. Proof: Low-uilizaion asks are allocaed n low cores, and from Lemma 4 we know ha he oal uilizaion of he low uilizaion asks is less han n low /b = 50%n low. Therefore, any muliprocessor scheduling algorihm ha provides a uilizaion bound of (i.e., can schedule any ask se wih oal wors-case uilizaion raio no more han 50%) can schedule i. Many muliprocessor scheduling algorihms (such as pariioned EDF or pariioned RM) provide a uilizaion bound of 1/ (i.e., 50%) o sequenial asks. Tha is, given n low cores, hey can schedule any ask se wih a oal wors-case uilizaion up o n low /. Using any of hese algorihms for low-uilizaion asks will guaranee ha he federaed algorihm mees all deadlines wih capaciy augmenaion of. Therefore, since we can successfully schedule boh high and low-uilizaion asks ha saisfy Condiions (1) and (), we have proven Theorem 1 (using Lemma 1). As menioned before, a capaciy augmenaion bound acs as a simple schedulabiliy es. However, for federaed scheduling, his es can be pessimisic, especially for asks wih high parallelism. Noe, however, ha he federaed scheduling algorihm described in Secion III-A can also be direcly used as a (polynomial-ime) schedulabiliy es: given a ask se, afer assigning cores o each highuilizaion ask using his algorihm, if he remaining cores are sufficien for all low-uilizaion asks, hen he ask se u i is schedulable and we can admi i wihou deadline misses. This schedulabiliy es admis a sric superse of asks admied by he capaciy augmenaion bound es, and in pracice, i may admis ask ses wih uilizaion greaer han m/. IV. Lower Bound on Capaciy Augmenaion of Any Scheduler for Parallel Tasks On a sysem wih m cores, consider a ask se τ wih a single ask, τ 1, which sars wih sequenial execuion for 1 ime and hen forks m subasks wih execuion ime. Here, we assume is an arbirarily small posiive number. Therefore, he oal work of ask τ 1 is C 1 = m and is criical-pah lengh L i = 1. The deadline (and also minimum iner-arrival ime) of τ 1 is 1. Theorem 3. The capaciy augmenaion bound for any scheduler for parallel asks on m cores is a leas 1 m, when + 0. Proof: Consider he sysem defined above. The finishing ime of τ 1 by running a speed is no earlier han 1 m 1 + m = 1 m. If > 1 m and + 0, hen 1 m > 1, hen ask τ 1 misses is deadline. Therefore, we reach he conclusion. Since Lemma 3 works for any scheduler for parallel asks, we can conclude ha he lower bound on capaciy augmenaion of federaed scheduling is a leas, when m is sufficienly large. Since we have shown ha he upper bound on capaciy augmenaion of federaed scheduling is also, herefore, we have closed he gap beween he lower and upper bound of federaed scheduling for large m. Moreover, for sufficienly large m, federaed scheduling has he bes capaciy augmenaion bound, among all schedulers for parallel asks. V. Canonical Form of a DAG Task In his secion, we inroduce he concep of a DAG s canonical form. Noe each ask can have an arbirarily complex DAG srucure which may be difficul o analyze and may no even be known before runime. However, given he known ask se parameers (work, criical pah lengh, uilizaion, criical-pah uilizaion, ec.) we represen each ask using a canonical DAG ha allows us o upper bound he demand of he ask in any given inerval lengh. These resuls will play an imporan role when we analyze he capaciy augmenaion bounds for G-EDF in Secion VI and G-RM in Secion VII. Recall ha in his paper, we analyze asks wih implici deadline, so period equals o deadline (T i = ). Recall ha we classify each ask τ i as a low-uilizaion if u i = C i / < 1 (and hence C i < ); or high-uilizaion ask, if τ i s uilizaion u i 1. 5

6 For analyical purposes, insead of considering he complex DAG srucure of individual asks τ i, we consider a canonical form τi of ask τ i. The canonical form of a ask is represened by a simpler DAG. In paricular, each subask (node) of ask τi has execuion ime, which is posiive and arbirarily small. Noe ha is a hypoheical uni-node execuion ime. Therefore, i is safe o assume ha Di and Ci are boh inegers. Low and high-uilizaion asks have differen canonical forms described below. The canonical form τi of a low-uilizaion ask τ i is simply a chain of C i / nodes, each wih execuion ime. Noe ha ask τi is a sequenial ask. The canonical form τi of a high-uilizaion ask τ i sars wih a chain of / 1 nodes each wih execuion ime. The oal work of his chain is. The las node of he chain forks all he remaining nodes. Hence, all he remaining (C i + )/ nodes have an edge from he las node of his chain. Therefore, all hese nodes can execue enirely in parallel. Figure 1 provides an example for such a ransformaion for a high-uilizaion ask. I is imporan o noe ha he canonical form τi does no depend on he DAG srucure of τ i a all. I depends only on he ask parameers of τ i (a) original DAG nodes.. (b) canonical form: heavy ask Fig. 1: A high-uilizaion DAG ask τ i wih L i = 1, C i = 0, T i = = 16, and u i = 1.5 and is canonical form, where he number in each node is is execuion ime. As an addiional analysis ool, we define a hypoheical scheduling sraegy S ha mus schedule a ask se τ on an infinie number of cores, ha is, m =. Wih infinie number of cores, he prioriizaion of he sub-jobs becomes unnecessary and S can obain an opimal schedule by simply assigning a sub-job o a core as soon as ha subjob becomes ready for execuion. Using his schedule, all he asks finish wihin heir criical-pah lengh; herefore, if L i for all asks τ i in τ, he ask se always mees he deadlines. We denoe his schedule as S. Similarly, S, is he resuling schedule when A schedules asks on cores of speed 1. Noe ha S, finishes a job of ask τ i exacly L i / ime unis afer i is released. We now define some noaions based on S,. Le q i (, ) be he oal work finished by S, beween he arrival ime r i of ask τ i and ime r i +. Therefore, in he inerval from r i + o r i + (inerval of lengh ) he remaining C i q i (, ) workload has o be finished. We define maximum load, denoed by work i (, ), for 4+ nodes ask i as he maximum amoun of work (compuaion) ha S, mus do on he sub-jobs of τ i in any inerval of lengh. We can derive work i (, ) as follows: work i (, ) = { Ci q i (, ) C i + work i (, ) >. Clearly, boh q i (, ) and work i (, ) for a ask depend on he srucure of he DAG. We similarly define qi (, ) for he canonical form τ i. As he canonical form in ask τi is well-defined, we can derive qi (, ) direcly. Noe ha can be arbirarily small, and, hence, is impac is ignored when calculaing qi (, ). We can now define he canonical maximum load worki (, ) as he maximum workload of he canonical ask τi in any inerval in schedule S,. For a lowuilizaion ask τ i, where C i / < 1, and τi is a chain, i is easy o see ha he canonical workload is worki (, ) = (5) 0 < Ci ( ( Ci )) Ci C i + worki (, ) >. Similarly, for high-uilizaion asks, where C i / 1, when is arbirarily small, we have worki (, ) = (6) 0 < Di C i + ( ( Di )) Di C i + worki (, ) >. Figure shows he qi (, ), q i(, ), worki (, ), and work i (, ) of he high-uilizaion ask τ i in Figure 1 when = 16, = 1, and =. Noe ha worki (, ) work i (, ). In fac, he following lemma proves ha worki (, ) work i(, ) for any > 0 and 1. Lemma 5. For any > 0 and 1, worki (, ) work i (, ). Proof: For low-uilizaion asks, he enire work C i is sequenial. When < Ci, q i (, ) is, so q i(, ) = qi Ci (, ). When <, q i (, ) = C i = qi (, ). Similarly, for high-uilizaion asks, he firs unis of work is sequenial, so when < Di, q i (, ) =. In addiion, S, finishes τ i exacly Li ime unis afer i is released, while i finishes he τi a Di. Since he criicalpah lengh L i for all τ i and τi a uni-speed sysem, when < Li, q i(, ) = qi Li (, ). When <, q i(, ) = qi ( Di, ) > q i (, ), When L i <, Lasly, when Di <, q i (, ) = C i = qi (, ) We can conclude ha qi () q i() for any 0 (4) 6

7 q i (, ) q i (, ) q i (, 1) qi (, 1) (a) q i (, ) and q i(, ) worki (, 1) work i (, 1) work i (, ) worki (, ) (b) work i (, ) and work i(, ) Fig. : q i (, ), q i(, ), work i (, ) and work i(, ) for he high-uilizaion ask τ i wih = 0 in Figure 1. <. Combining wih he definiion of work(, ) (Equaion (4)), we complee he proof. We classify ask τ i as a ligh or heavy ask. A ask is a ligh ask if u i = C i / <. Oherwise, we say ha τ i is heavy (u i ). The following lemmas provide an upper bound on he densiy (he raio of workload ha has o be finished o he inerval lengh) for heavy and ligh asks. Lemma 6. For any ask τ i, > 0 and 1 <, we have { work i (, ) work i (, ) ui (0 u i < ) (7) ( u i ) u i Proof: The firs inequaliy in Inequaliy (7) comes from Lemma 5. We now show ha he second inequaliy also holds for any ask. Noe ha he righ hand side is posiive, since 1 > u i > 0. There are wo cases: Case 1: 0 <. If τ i is a low-uilizaion ask, where worki (, ) is defined in Equaion (5). For any 0 <, we have work i (, ) C i ( + C i ) C i = ( )( C i ) 0 where we rely on assumpions: (a) ; (b) since τ i is a low-uilizaion ask, C i < 1; and (c) > 1. Then, work i (,) Ci = u i. If τ i is a high-uilizaion ask, where worki (, ) is defined in Equaion (6). Inequaliy (7) holds rivially when 0 < < Di, since he lef side is 0 and righ side is posiive. For Di, we have worki (, ) = C i + = + ( u i ), Therefore, work i (,) is maximized eiher (a) when = Di if u i 0 or (b) = if u i < 0. 7 If τ i is a ligh ask wih 1 u i < and hence (b) is rue, hen we have work i (Di,) = u i. If heavy ask τ i wih u i and hence (a) is rue, hen work i ( Di, ) Di = C i Di = u i Therefore, Inequaliy (7) holds for 0 <. Case : > Suppose ha is k +, where k is and 0 <. When u i <, by Equaion (5) and Equaion (6), we have worki (, ) = kc i + worki (, ) k + ku i + u i k + = u i When u i, we can derive ha u i Equaion (6), we have worki (, ) = kc i + worki (, ) k + k ui 1 D 1 1 i + ui k + u i 1 ui 1. By 1 1 ku i + ui k Hence, Inequaliy (7) holds for any ask and any > 0. We denoe τ L and τ H as he se of ligh and heavy asks in a ask se, respecively; τ H as he number of heavy asks in he ask se; and oal uilizaion of ligh and heavy asks as U L = τ L u i and U H = τ H u i, respecively. Lemma 7. For any ask se, he following inequaliy holds: ( ) U U L τ H (8) W = ( n i=1 work i (, )) ( m 1 1 ) (9) Proof: By Lemma 6, for any > 1, i is clear ha W = work τ L+τ H sup i (,) >0 τ L u i + τ H (u i 1) 1 1 τ u = L i τ u L i+ τ u H i τ 1 H 1 = U U L τ H 1

8 where sup is he supremum of a se of numbers, τ L and τ H are he ses of heavy asks (u i ) and ligh asks (u i < ), respecively. Noe ha τ L u i + τ H u i = U L + U H = U m. Since for τ i τ H, u i, U H = τ H u i τ H, we can derive he following upper bound: U U L τ H sup(u U L τ H ) = U τ This is because for any ask se, here are wo cases: If τ H = 0 and hence U = U L, hen U U L τ H = 0. If τ H 1, hen U U L = U H U and τ H. Therefore, U U L τ H U Togeher wih he definiion of U and τ H 1 = τ H, W ( U U L τ H 1 ) ( U 1 ) ( m 1 which proves he Inequaliies 8 and 9 of Lemma 7. We use Lemma 7 in Secions VI and VII o derive bounds on G-EDF and G-RM scheduling. VI. Capaciy Augmenaion of Global EDF In his secion, we use he resuls from Secion V o prove he capaciy augmenaion bound of (3 + 5)/ for G-EDF scheduling of parallel DAG asks. In addiion, we also show a maching lower bound when m 3. A. Upper Bound on Capaciy Augmenaion of G-EDF Our analysis builds on he analysis used o prove he resource augmenaion bounds by Bonifaci e al. [15]. We firs review he paricular lemma from he paper ha we will use o achieve our bound. Lemma 8. If > 0, (m m+1) n i=1 work i(, ), he ask se is schedulable by G-EDF on speed- cores. Proof: This is based on a reformulaion of Lemma 3 and Definiion 10 in [15] considering cores wih speed. Theorem 4. The capaciy augmenaion bound for G-EDF is 3 m m + 4 m ( 3+ 5, when m is large). ) Proof: From Lemma 7 Inequaliy (9), we have > 0, n i=1 work i(, ) ( m m 1 ). If 1 (m m + 1), by Lemma 8 he schedulabiliy es for G-EDF holds. To calculae, we solve he derived equivalen inequaliy m (3m ) + (m 1) 0. ) which solves o = (3 5 m + 8m + 4m /. We now sae a more general corollary relaing he more precise bound using more informaion of a ask se. Corollary. If a ask se has oal uilizaion U, he oal heavy ask uilizaion U H and he number of capaciy augmenaion bound U / H =0.94 U / H = U / H = U / H = U / m 1 Fig. 3: The required speedup of G-EDF when m is sufficienly large and U L = 0 (i.e. U = U H ). capaciy augmenaion bound m=3 1.4 m=6 1. m=1 m= U / m Fig. 4: The required speedup of G-EDF when τ H = 1 and U L = 0 (i.e. U = U H ). heavy ask τ H, hen his ask se will be schedulable under G-EDF on a m-core machine wih speed = + U τ H 1 m + 4(U H τ H ) m + (U τ H 1) m. Proof: The proof is he same as in he proof of Theorem 4, bu wihou using he Inequaliy (9). Insead, we direcly use Inequaliy (8). If U U L τ H 1 (m m + 1), by Lemma 8 he schedulabiliy es for G-EDF holds for his ask se. Solving his, we can ge he required speedup for he schedulabiliy of he ask se. However, noe ha heavy asks are defined as he se of all asks τ i wih uilizaion u i. Therefore, given a ask se, o accuraely calculae, we sar wih he upper ) bound on, which is ˆ = (3 5 m + 8m + 4m /; hen for each ieraion i, we can calculae he required speedup i by using he U i 1 H and τ H i 1 from he (i 1)-h ieraion; we ieraively classify more asks ino he se of heavy asks and we sop when no more asks can be added o his se, i.e., τ H i 1 = τ H i. Through hese ieraive seps, we can calculae an accurae speedup. 8

9 Figure 3 illusraes he required speedup of G-EDF provided in Corollary when m is sufficienly large (i.e., m = and 1/m = 0) and U L = 0 (i.e. U = U H ). We vary U U m and τ. Noe ha U H τ = UH H τ H is he average uilizaion of all heavy asks, which should be no less han (3 + 5)/.618. I can be also be seen ha he bound is geing closer o (3+ 5)/, when U τ H is larger, which resuls from τ H = 1 and m =. Figure 4 illusraes he required speedup of G-EDF provided in Corollary when τ H 1 and U L = 0 (i.e. U = U H ) wih varying m. Noe ha > 1 is required by he proof of Corollary. And τ H = 1 can only be rue, if U m m (i.e. U m 1/3 for m = 3). B. Lower Bound on Capaciy Augmenaion of G-EDF As menioned above, Li e al. s lower bound [35] demonsraes he ighness of he above bound for large m. We now provide a lower bound for he capaciy augmenaion bound for small m. Consider a ask se τ wih wo asks, τ 1 and τ. Task τ 1 sars wih sequenial execuion for 1 ime and hen forks m + 1 subasks wih execuion ime. Here, we assume is an arbirarily small posiive number and hence 1 i is safe o assume ha m is a posiive ineger. Therefore, he oal work of ask τ 1 is C 1 = m 1 and is criical-pah lengh is L i = 1. The minimum iner-arrival ime of τ 1 is 1. Task τ is simply a sequenial ask wih work (execuion ime) of 1 1 and minimum iner-arrival ime also 1 1, where > 1 will be defined laer. Clearly, he oal uilizaion is m and he criical-pah lengh of each ask is a mos he relaive deadline (minimum iner-arrival ime). Lemma 9. When < 3 m δ+ g() and m 3, hen 1 Proof: By solving 1 he equaliy holds when + m m < 3 m + + m m 5 1 m + 4 m and δ = + > holds. = 1 1 1, we know ha 5 1 m + 4 m g() where g() is a posiive funcion of 1, which approaches o 0 when approaches 0. Now, by seing δ o + g(), we reach he conclusion. Theorem 5. The capaciy augmenaion bound for G-EDF is a leas 3 m m + 4 m, when + 0. Proof: Consider he sysem wih wo asks τ 1 and τ defined in he beginning of Secion VI-B. Suppose ha he arrival of ask τ 1 is a ime 0, and he arrival of ask τ is a ime 1 +. By definiion, he firs jobs of τ 1 and τ have absolue deadlines a 1 and 1 +. Hence, G-EDF capaciy augmenaion bound upper bound lower bound m Fig. 5: The upper bound of G-EDF provided in Theorem 4 and he lower bound in Theorem 5 wih respec o he capaciy augmenaion bound. will execue he sequenial execuion of ask τ 1 and he sub-jobs of ask τ 1 firs, and hen execue τ. The finishing ime of τ 1 a speed is no earlier han 1 + m m = 1 ask τ is no earlier han 1 If 1 + m m m m. Hence, he finishing ime of + m m > 1 +, hen ask τ misses is deadline. By Lemma 9, we reach he conclusion. Figure 5 illusraes he upper bound of G-EDF provided in Theorem 4 and he lower bound in Theorem 5 wih respec o he capaciy augmenaion bound. I can be easily seen ha he upper and lower bounds are geing closer when m is larger. When m is 100, he gap beween he upper and he lower bounds is roughly abou I is imporan o noe ha he more precise speedup in Corollary is igh even for small m. This is because in he above example ask se, U = m, oal high ask uilizaion U H = m 1 and number of heavy ask τ H = 1, hen according o he Corollary, he capaciy augmenaion bound for his ask se under G-EDF is = + U τ H 1 m + m m + 4(U H τ H ) m + (U τ H 1) m 4(m 1 1) m + (m 1 1) m = 3 m m + 4 m which is exacly he lower bound in Theorem 5 for + 0. VII. G-RM Scheduling This secion proves ha G-RM provides a capaciy augmenaion bound of + 3 for large m. The srucure of he proof is very similar o he analysis in Secion VI. Again, we use a lemma from [15], resaed below. Lemma 10. If > 0, 0.5( m m + 1) i work i(, ) he ask se is schedulable by G-RM on speed- cores. 9

10 capaciy augmenaion bound U / H =9.86 U / H =14.93 U / H = U / H = U / m Fig. 6: The required speedup of G-RM when m is sufficienly large and U L = 0 (i.e. U = U H ). Proof: This is based on a reformulaion of Lemma 6 and Definiion 10 in [15]. Noe ha he analysis in [15] is for deadline-monoonic scheduling, by giving a sub-job of a ask higher prioriy if is relaive deadline is shorer. As we consider asks wih implici deadlines, deadline-monoonic scheduling is he same as rae-monoonic scheduling. Proofs like hose in Secion VI-A give us he bounds below for G-RM scheduling. Theorem 6. The capaciy augmenaion bound for G-RM is 4 3 m m + 9 m ( + 3, when m is large). Proof: (Similar o he proof of Theorem 4:) Firs, we know from Lemma 7 ha > 0, n i=1 work i(, ) ( m m 1 ); Second, if 1 0.5(m m + 1). we can also conclude ha he schedulabiliy es for G-RM in Lemma 10 holds. By solving he inequaliy above, we have 4 3 m m + 9 m, and prove Theorem 6. The resul in Theorem 6 is he bes known resul for he capaciy augmenaion bound for global fixed-prioriy scheduling for general DAG asks wih arbirary srucures. Ineresingly, Kim e al. [31] ge he same bound of + 3 for global fixed-prioriy scheduling of parallel synchronous asks (a subse of DAG asks). The sraegy used in [31] is quie differen. In heir algorihm, he asks undergo a srech ransformaion which generaes a se of sequenial subask (each wih is release ime and deadline) for each parallel ask in he original ask se. These subasks are hen scheduled using a G-DM scheduling algorihm [11]. Noe ha even hough he parallel asks in he original ask se have implici deadlines, he ransformed sequenial asks have only consrained deadlines hence he need for deadline monoonic scheduling insead of rae monoonic scheduling. Corollary 3. If a ask se has oal uilizaion U, he oal high ask uilizaion U H and he number of heavy ask τ H, hen his ask se will be schedulable under G-RM on a m-core machine wih speed = + U τ H 1 m + 8(U H τ H ) m + (U τ H 1) m. Proof: The proof is he same as in he proof of Corollary, excep ha insead of using Lemma 8, i uses Lemma 10. Figure 6 illusraes he required speedup of G-RM provided in Corollary 3 when m is sufficienly large (i.e., m = and 1/m = 0) and U L = 0 (i.e. U = U H ). We vary U U m and τ. Again, U H τ H is he average uilizaion of all heavy asks. I can be also be seen ha he bound is geing closer o + 3, when U τ H is larger, which resuls from τ H = 1 and m =. VIII. Pracical Consideraions As have shown in he previous secions, he capaciy augmenaion bound for federaed scheduling, G-EDF and G-RM are,.618 and 3.73, respecively. In his secion, we consider heir run-ime efficiency and efficacy from a pracical perspecive. We consider four dimensions saic vs. dynamic prioriies, and global vs. pariioned scheduling, overheads due o scheduling and synchronizaion overheads, and work-conserving vs. no. Praciioners have generally found i easier o implemen fixed (ask) prioriy schedulers (such as RM) han dynamic prioriy schedulers (such as EDF). Fixed prioriy schedulers have always been well-suppored in almos all real-ime operaing sysems. Recenly, here has been effors on efficien implemenaions of job-level dynamic prioriy (EDF and G-EDF) schedulers for sequenial asks [16, 34]. Federaed scheduling does no require any prioriy assignmen for high-uilizaion asks (since hey own heir cores exclusively) and can use eiher fixed or dynamic prioriy for low-uilizaion asks. Thus, i is relaively easier o implemen G-RM and federaed scheduling. For sequenial asks, in general, global scheduling may incur more overhead due o hread migraion and he associaed cache penaly, he exen of which depends on he cache archiecure and he ask ses. In paricular, for parallel asks, he overheads for global scheduling could be worse. For sequenial asks, preempions and migraions only occur when a new job wih higher prioriy is released. In conras, for parallel asks, a preempion and possibly migraion could occur whenever a node in he DAG of a job wih higher prioriy is enabled. Since nodes in a DAG ofen represen a fine-grained unis of compuaion, he number of nodes in he ask se can be larger han he number of asks. Hence, we can expec a larger number of such evens. Since federaed scheduling is a generalizaion of pariioned scheduling o parallel asks, i has advanages similar o pariioning. In fac, if we use a pariioned RM or pariioned EDF sraegy for low-uilizaion asks, 10

11 here are only preempions bu no migraion for lowuilizaion asks. Meanwhile, federaed scheduling only allocaes he minimum number of dedicaed cores o ensure he schedulabiliy of each high-uilizaion ask, so here is no preempions for high-uilizaion asks and he number of migraions is minimized. Hence, we expec ha federaed scheduling will have less overhead han global schedulers. In addiion, parallel runime sysems have addiional parallel overheads, such as synchronizaion and scheduling overheads. These overheads (per ask) usually are approximaely linear in he number of cores allocaed o each ask. Under federaed scheduling, a minimum number of cores is assigned. However, depending on he paricular implemenaion, global scheduling may execue a ask on all he cores in he sysem and may have higher overheads. Finally, noe ha federaed scheduling is no a greedy (work conserving) sraegy for he enire ask se, alhough i uses a greedy schedule for each individual ask. In many real sysems, he wors-case execuion imes are normally over-esimaed. Under federaed scheduling cores allocaed o asks wih overesimaed execuion imes may idle due o resource over-provisioning. In conras, work-conserving sraegies (such as G-EDF and G-RM) and can uilize available cores hrough hread migraion dynamically. IX. Relaed Work In his secion, we review closely relaed work on realime scheduling, concenraing primarily on parallel asks. Real-ime muliprocessor scheduling considers scheduling sequenial asks on compuers wih muliple processors or cores and has been sudied exensively (see [10, 3] for a survey). In addiion, plaforms such as Limus RT [17, 19] have been designed o suppor hese ask ses. Here, we review a few relevan heoreical resuls. Researchers have proven resource augmenaion bounds, uilizaion bounds and capaciy augmenaion bounds. The bes known resource bound for G-EDF for sequenial asks on a muliprocessor is [7]; a capaciy augmenaion bound of 1 m + for small [14]. Pariioned EDF and versions pariioned saic prioriy schedulers also provide a uilizaion bound of [3, 37]. G-RM provides a capaciy augmenaion bound of 3 [] o implici deadline asks. For parallel real-ime asks, mos early work considered inra-ask parallelism of limied ask models such as malleable asks [, 30, 33] and moldable asks [39]. Kao e al. [30] sudied he Gang EDF scheduling of moldable parallel ask sysems. Researchers have since considered more realisic ask models ha represen programs generaed by commonly used parallel programming languages such as Cilk family [13, 1], OpenMP [41], and Inel s Thread Building Blocks [43]. These languages and libraries suppor primiives such as parallel for-loops and fork/join or spawn/sync in order o expose parallelism wihin he programs. Using hese consrucs generaes asks whose srucure can be represened wih differen ypes of DAGs. Tasks wih parallel synchronous asks have been sudied more han ohers in he real-ime communiy. These asks are generaed if we use only parallel-for loops o generae parallelism. Lakshmanan e al. [3] proved a (capaciy) augmenaion bound of 3.4 for a resriced synchronous ask model which is generaed when we resric each parallel-for loop in a ask o have he same number of ieraions. General synchronous asks (wih no resricion on he number of ieraions in he parallel-for loops), have also been sudied [4, 31, 40, 45]. (More deails on hese resuls were presened in Secion I) Chwa e al. [0] provide a response ime analysis. If we do no resric he primiives used o parallel-for loops, we ge a more general ask model mos easily represened by a general direced acyclic graph. A resource augmenaion bound of 1 m for G-EDF was proved for a single DAG wih arbirary deadlines [8] and for muliple DAGs [15, 35]. A capaciy augmenaion bound of 4 m was proved in [35] for asks wih for implici deadlines. Liu e al. [36] provide a response ime analysis for G-EDF. There has been significan work on scheduling non-realime parallel sysems [5, 6, 4 6, 4]. In his conex, he goal is generally o maximize hroughpu. Various provably good scheduling sraegies, such as lis scheduling [18, 7] and work-sealing [1] have been designed. In addiion, many parallel languages and runime sysems have been buil based on hese resuls. While muliple asks on a single plaform have been considered in he conex of fairness in resource allocaion [1], none of his work considers real-ime consrains. X. Conclusions In his paper, we consider parallel asks in he DAG model and prove ha for parallel asks wih implici deadlines he capaciy augmenaion bounds of federaed scheduling, G-EDF and G-RM are,.618 and 3.73 respecively. In addiion, he bound for federaed scheduling and he bound of.618 for he G-EDF are boh igh for large m, since here exis maching lower bounds. Moreover, he hree bounds are he bes known bounds for hese schedulers for DAG asks. There are several direcions of fuure work. The G-RM capaciy augmenaion bound is no known o be igh. The curren lower bound of G-RM is.668, inheried from he sequenial sporadic real-ime asks wihou DAG srucures [38]. Therefore, i is worh invesigaing a maching lower bound or lowering he upper bound. In addiion, since he lower bound of any scheduler is 1 m, i would be ineresing o invesigae if i is possible o design schedulers ha reach his bound. Finally, all he known 11

12 capaciy augmenaion bound resuls are resriced o implici deadline asks; we would like o generalize hem o consrained and arbirary deadline asks. Acknowledgmen This research was suppored in par by he prioriy program Dependable Embedded Sysems (SPP spp1500.iec.ki.edu), by DFG, as par of he Collaboraive Research Cener SFB876 (hp://sfb876.u-dormund.de/), by NSF grans CCF (CPS) and CCF (XPS). The auhors hank anonymous reviewers for heir suggesions on improving his paper. References [1] K. Agrawal, C. E. Leiserson, Y. He, and W. J. Hsu. Adapive work-sealing wih parallelism feedback. In: ACM Trans. Compu. Sys. 6 (008), pp [] B. Andersson, S. Baruah, and J. Jonsson. Saic-prioriy scheduling on muliprocessors. In: RTSS [3] B. Andersson and J. Jonsson. The uilizaion bounds of pariioned and pfair saic-prioriy scheduling on muliprocessors are 50%. In: ECRTS [4] B. Andersson and D. de Niz. Analyzing Global-EDF for Muliprocessor Scheduling of Parallel Tasks. 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