Global EDF Schedulability Analysis for Synchronous Parallel Tasks on Multicore Platforms

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1 Global EDF Schedulablty Analyss for Synchronous Parallel Tasks on Multcore Platfors Hoon Sung Chwa, Jnkyu Lee, Keu-My Phan, Arvnd Easwaran, Insk Shn Dept. of Coputer Scence, KAIST, South Korea Dept. of Electrcal Engneerng and Coputer Scence, The Unversty of Mchgan, U.S.A. School of Coputer Engneerng, NTU, Sngapore Abstract The trend towards ult-core/any-core archtectures s well underway. It s therefore becong very portant to develop software n ways that take advantage of such parallel archtectures. Ths partcularly entals a shft n prograng paradgs towards fne-graned, thread-parallel coputng. Many parallel prograng odels have been ntroduced targetng such ntra-task thread-level parallels. However, ost successful results on tradtonal ult-core real-te schedulng are focused on sequental prograng odels. For exaple, thread-level parallels s not properly captured nto the concept of nterference, whch s key to any schedulablty analyss technques. Thereby, ost nterference-based analyss technques are not drectly applcable to parallel prograng odels. Motvated by ths, we extend the noton of nterference to capture thread-level parallels ore accurately. We then leverage the proposed noton of parallels-aware nterference to derve effcent EDF schedulablty tests that are drectly applcable to synchronous parallel task odels on ult-core platfors. Our evaluaton results ndcate that the proposed analyss sgnfcantly advances the state-of-the-art n EDF schedulablty analyss for synchronous parallel tasks. I. INTRODUCTION An rreversble shft towards ult-core processors s underway. Recent developent trends suggest that the chp ndustry s ovng to ult-/any-core archtectures to better anage trade-offs between perforance, power effcency, and relablty n deep subcron technology. For exaple, Intel desgned a research chp wth 80 cores [1], and Tlera ntroduced a range of processors wth up to 100 cores [2]. Gven the ncreasng ephass on ult-/any-core chp desgn, software parallels s lkely to be one of the greatest constrants on coputer perforance. Ths nherently entals a shft n prograng paradgs towards fne-graned thread-parallel coputng, rather than relatvely coarse-graned applcatonlevel parallels. A popular technque to acheve fne-graned, thread-level parallels operates on the prncple of dvde-and-conquer. It breaks down a larger task nto any saller subtasks, runs those subtasks n parallel, and erges the results once each subtask copletes coputaton. The synchronous parallel prograng odel ebodes ths knd of parallel decoposton. A synchronous parallel task conssts of a sequence of parallel regon, called segents, and each segent ncludes one or ore threads. Two portant aspects are assocated wth the synchronous parallel task odel: segent-level synchronzaton and thread-level parallels. All the threads belongng to the sae segent are released at the sae te and are not restrcted to run sultaneously wth at ost degree of parallels, where s the nuber of avalable cores. Segents are subject to synchronzaton rules between two consecutve segents. All the threads belongng to one segent ust coplete ther own executon n order to ove onward to the next segent. A good exaple s the fork-jon prograng odel, whch has been ncreasngly eployed n any prograng envronents, such as Java [3], OpenMP [4], and Clk [5]. A shft fro un-core to ult-core processors allows nter-task parallels, where several applcatons (tasks) can execute sultaneously on ult-core processors. In addton, a shft fro sngle-thread to ult-thread tasks allows ntratask parallels, where even a sngle task can have ultple threads runnng sultaneously to take full advantage of ultcore processng. Despte the growng portance of ntra-task parallels, ost real-te ultprocessor schedulng studes are focused on nter-task parallels of sequental tasks, and relatvely uch less attenton has been pad to understandng ntra-task parallels towards the schedulablty analyss of synchronous parallel tasks. For exaple, a large nuber of studes extensvely nvestgated the schedulablty analyss of sequental tasks under EDF (earlest-deadlne-frst) [6] and fxed-prorty global schedulng [6], producng any nfluental results. Such results nclude the concept of proble wndow and nterference [7], [8], nterference boundng technques [9], [10], response te coputaton ethods [8], [11], and optal prorty assgnent [12]. Many other schedulng algorths have been proposed for sequental odels n order to take advantage of ultprocessors ore effectvely, ncludng optal algorths such as pfar [13], DP-Far [14], and RUN [15]. In addton, soe approaches [16], [17] have been also proposed for schedulng tasks wth ppelne precedence constrants n dstrbuted real-te systes. However, the nsghts behnd those successful results are not drectly applcable to synchronous parallel tasks, due to the unque characterstcs of thread-level parallels. For exaple, the noton of nterference has been well defned n the sequental task case and serves as the bass for any schedulablty analyss ethods [7] [9]. However, the current noton of nterference does not capture thread-level parallels because t assues that each task has only a sngle thread to run at any te nstant. Hence, those analyss ethods are not drectly applcable or easly extensble to synchronous parallel tasks. Recently, a few analyss ethods have been proposed for schedulablty of synchronous parallel tasks [18] [22].

2 Those ethods can broadly fall nto two categores: drect and ndrect. Indrect analyss ethods [19] [21] share a prncple of task decoposton. Each sngle synchronous parallel task s transfored nto ultple ndependent sequental subtasks such that each ndvdual subtask s assgned ts own nteredate deadlne. Schedulablty analyss s then perfored over nteredate deadlnes after task decoposton at the expense of potentally ncurrng soe non-trval decoposton overheads. On the other hand, drect analyss ethods [18], [22] perfor analyss wthout task decoposton. A recent study [18] consders only a certan type of ntra-task threadlevel parallels (.e., gang schedulng), whch allows a fxed degree of parallels all threads run or none does. A ore recent study [22] consders a ore general parallel task odel, drected acyclc graph (DAG) odel, for the sngle DAG task case, yet leavng unresolved the ultple synchronous parallel task case. Motvated by ths, the goal of ths paper s drect schedulablty analyss for synchronous parallel tasks. The ratonale for ths goal s that ndrect EDF schedulablty analyss through task decoposton nvolves the addtonal overhead of nteredate deadlne assgnent and a substantal aount of pesss (dscussed n Secton VI). Contrbuton. The an contrbuton of ths paper s to ntroduce, to the authors best knowledge, the frst EDF schedulablty condton that s drectly applcable to synchronous (alleable) parallel task systes n whch parallel tasks are allowed to execute wth any arbtrary degree of thread-level parallels up to the nuber of avalable processors. To ths end, ths paper extends the concept of nterference capturng thread-level parallels ore accurately wth novel notons of crtcal nterference and p-depth crtcal nterference. Ths paper provdes evaluaton results, showng that the proposed schedulablty analyss sgnfcantly outperfors the state-ofthe-art approaches avalable for synchronous parallel tasks. II. RELATED WORK There has been an ncreasng attenton to parallel task odels n the context of real-te schedulng [18] [28]. The work n [23], [24] consders soft real-te schedulng focusng on boundng tardness upon deadlne ss, whle hard real-te systes a at ensurng all deadlnes are et. In ths paper, we consder hard real-te schedulng. In general, a parallel task s sad to be (1) oldable f the task executes on exactly a certan nuber of processors, whch s deterned before executon and reans unchanged, or (2) alleable f the task can execute on any varable nuber of processors, whch can be dynacally changng at runte. A recent study [18] consders gang schedulng [29] of oldable parallel tasks, n whch a group of threads (e.g., all threads of the sae task) run wth a predeterned degree of parallels or none do. Ths work ntroduces a new noton of nterference (nterference rectangle) to characterze nter-task nterference accordng to such behavor of all-or-none thread executon. It then derves a schedulablty analyss for global EDF gang schedulng. Snce such a new nterference noton does not capture the stuaton where any arbtrary nuber of threads execute n parallel, however, t s dffcult to apply the noton to the alleable parallel task case. More recent studes [19] [22] consder alleable parallel task odels, n whch threads can execute any arbtrary degree of parallels up to the nuber of avalable processors. One of the wdely used alleable parallel task odels s the forkjon odel [19]. A fork-jon task conssts of a sequence of segents such that every odd-nubered segent contans a sngle (aster) thread and every even-nubered segent conssts of ultple (worker) threads. The aster thread that runs sequentally forks off a nuber of worker threads whch execute blocks of code n parallel. After the executon of the parallelzed code, the worker threads jon back nto the aster thread, whch contnues onward to another parallel regon or the end of the progra. Relaxng the requreent of such parallel/non-parallel alternaton of the fork-jon odel, a ore general synchronous parallel task odel [20], [21] s consdered such that each segent can have any nuber of threads. In ths paper, we consder ths task odel. A few studes [19] [21] share a coon prncple of task decoposton for schedulablty analyss. They decopose a sngle synchronous parallel task nto ultple ndependent sequental sub-tasks through nteredate deadlne assgnent. Ths approach s safe satsfyng the nteredate deadlnes of all sub-tasks leads to eetng the deadlnes of ther aggregate synchronous parallel tasks. They then eploy exstng schedulablty analyss for those sequental sub-tasks. More specfcally, Lakshanan, et al. [19] propose a parttoned preeptve fxed-prorty schedulng algorth wth a provable perforance for fork-jon tasks, under the assupton that all parallel segents have the sae nuber of parallel threads. Safullah, et al. [20] decopose a parallel task nto a set of sequental sub-tasks such that the densty of each segent s upper bounded by soe value, and ths bound s used to derve a resource augentaton bound. Nelssen, et al. [21] decopose a parallel task such that the axu densty aong all segents n a parallel task s nzed. However, such an ndrect analyss va task decoposton can be pessstc, because task decoposton can ncur nontrval overheads (dscussed n Secton VI). Furtherore, t requres odfcatons to exstng operatng systes to support task decoposton. Thereby, ths paper seeks to derve drect schedulablty analyss for synchronous parallel tasks. Another recent study [22] consders the sporadc DAG odel that s a general for of the synchronous parallel task odel. In the DAG odel, each vertex corresponds to a sngle thread and each edge represents a precedence constrant. A thread can execute only after all of ts predecessors have been executed. Ths work presents a new noton of deand and load for the sporadc DAG task odel and apples t to derve an effcent EDF schedulablty test for the sngle sporadc DAG task wth a focus on the arbtrary deadlne case. However, ths work does not present schedulablty analyss for a case n whch ultple DAG tasks share processors, leavng t as future work. On the other hand, our work consders schedulng of ultple synchronous parallel tasks and presents EDF schedulablty tests for the case. III. SYSTEM MODEL We consder a ult-core platfor, where sporadc, synchronous parallel tasks run over dentcal processors under global EDF schedulng. A synchronous parallel task conssts

3 σ,1 1,1 LC,1 σ,2 σ,3 σ,s Θ 1,2,1 1 2 Θ,2,2 3,3 2 Θ,2,3 Θ,2,,2,2 D T,s ts own prorty under EDF schedulng. Thereby, dfferent jobs have dfferent prortes, but all threads wthn a sngle job have the sae prorty, breakng tes arbtrarly. In ths paper, we assue quantu-based te and wthout loss of generalty, let one te unt denote the quantu length. All task paraeters are assued to be specfed as ultples of ths quantu length. Fg. 1. A synchronous parallel task τ of a seres of segents, where each segent contans one or ore synchronous threads. A synchronous parallel task has two ajor propertes n ters of executon echans: segentlevel synchronzaton and thread-level parallel executon. All threads wthn a sngle segent are released sultaneously, are able to execute n parallel, and ust fnsh ther executon pror to proceedng to the next segent. Thereby, no segents wthn a sngle task can overlap each other. A set of tasks s denoted by τ. For a sporadc, synchronous parallel task τ, T s the nu separaton, D s the relatve deadlne, s s the nuber of segents, and σ,j represents the j-th segent (see Fgure 1). Each segent σ,j has ts own nternal threads θ,j,k, and,j represents the nuber of threads n the segent. The axu parallels of a task τ (denoted wth ) s then defned as the axu value of,j aong all segents,.e., = ax j,j. In ths paper, there s no restrcton on,j ; t can be larger than, the nuber of processors. For an ndvdual thread θ,j,k, C,j,k s ts worst-case executon te requreent (WCET). A segent σ,j s assocated wth two WCET paraeters: C,j and LC,j. C,j s defned as the axu aount of te to coplete the executon of all the threads wthn the segent σ,j on a sngle core. On the other hand, LC,j s defned as the nu aount of te needed to execute all threads n σ,j assung that t can use as any processors as possble for ts executon. That s,,j C,j = k=1 C,j,k and LC,j = ax 1 k,j {C,j,k }. (1) Slarly, C and LC represent the axu and nu executon tes of a task τ, respectvely. We then defne the as s s C = C,j and LC = LC,j. (2) j=1 A sporadc, synchronous parallel task τ nvokes a seres of jobs, each separated fro ts predecessor by a nu of T te unts. We consder a constraned deadlne D such that D T. It should be LC D but not necessarly C D. Let U denote the utlzaton of τ and be defned as U = C /T. We denote the k-th job of a task τ wth J k. We wll ot the superscrpt n the notaton for splcty when no abguty arses. For a job J k, rk and d k are ts release te and deadlne. The executon wndow of a job J k s then defned as nterval [r k, dk ). In ths paper, every sngle job J s assgned j=1 IV. INTERFERENCE-BASED SCHEDULABILITY ANALYSIS FOR SYNCHRONOUS PARALLEL TASKS In ths secton, we derve schedulablty analyss of global EDF schedulng for sporadc synchronous parallel task systes wth constraned deadlnes. In the real-te schedulng lterature, the noton of nterference has been eployed n any schedulablty analyss ethods [8], [9], [30] [32], usng the followng defntons: Interference I k (a, b): the su of all ntervals n whch τ k s ready for executon but cannot execute due to other hgher-prorty tasks n [a, b). Interference I,k (a, b): the su of all ntervals n whch τ s executng and τ k s ready to execute but not executng n [a, b). Wth the above defntons, the relaton between I k (a, b) and I,k (a, b) serves as an portant bass for dervng schedulablty analyss. In the sngle-thread task case, t s ntutve to construct such a relaton on processors as follows [8]: I k (a, b) = 1 τ τ I,k (a, b). (3) However, t s not straghtforward to buld such a relaton n the ult-thread task case, as llustrated n the followng exaple. Exaple 4.1: As an exaple, suppose that two threads of hgher-prorty τ and one thread of lower-prorty τ k are ready for executon on two processors at te t. Then, the two threads of τ wll run on two processors n [t, t + 1), delayng the executon of τ k. Accordng to the above defntons, τ poses nterference on τ k n [t, t + 1), yeldng I k (t, t + 1) = 1 and I,k (t, t + 1) = 1. However, Eq. (3) no longer supports such defntons. The above exaple suggests a need for extendng the concept of nterference for the parallel task odel, and ths rases three probles: (1) how to calculate the nterference on τ k when only soe (but not all) threads of τ k are nterfered, (2) how to calculate the nterference of τ on τ k when only soe (but not all) threads of τ nterfere wth τ k, and (3) how to calculate the nterference of threads of task τ k on other threads of the sae task τ k. To address proble (1), we ntroduce a new concept called crtcal nterference. A thread s sad to be a crtcal thread f t fnshes last aong all the threads belongng to the sae segent. Wth the noton of crtcal threads, we can now extend the tradtonal defnton of nterference towards the synchronous parallel task odel as follows: Crtcal nterference I k (a, b): the su of all ntervals n whch a crtcal thread of τ k s ready for executon

4 but cannot execute due to other hgher-prorty threads n [a, b). Crtcal nterference I,k (a, b): the su of all ntervals n whch at least one thread of τ s executng and the crtcal thread of τ k s ready to execute but not executng n [a, b). Note that when all tasks have a sngle thread, then the sngle thread s equal to the crtcal thread and our defnton s the sae as the tradtonal defnton of nterference. To address proble (2), we ntroduce a new concept called p-depth crtcal nterference. The p-depth crtcal nterference of a task τ on τ k characterzes not only the length of the delay τ causes to τ k but also the nuber of threads of τ that cause the delay. To address proble (3), we ncorporate the noton of ntratask nterference nto both the crtcal nterference and the p- depth crtcal nterference such that they nclude nterference on a crtcal thread by other non-crtcal threads of the sae task. A. Interference-based Schedulablty Analyss We frst seek to dentfy a necessary condton for any synchronous parallel task to ss a deadlne on processors. In synchronous parallel tasks, all the threads belongng to the sae segent σ k,u are released at the sae te, but they can coplete executon at dfferent te nstants dependng on ther executon behavor. A thread of a segent σ k,u s sad to be a crtcal thread (denoted as θ k,u,v ) f t fnshes last aong all the threads of the segent. A segent s then consdered as coplete as soon as ts crtcal thread copletes executon. We defne nterference on a crtcal thread θ k,u,v over nterval [a, b) (denoted as I k,u,v (a, b)) as the cuulatve length of all ntervals n whch the crtcal thread s ready to execute but not executng due to the executon of hgher-prorty threads belongng to other tasks as well as belongng to the sae task. To avod any confuson, t s worth notng that I k,u,v (a, b) ncludes ntra-task nterference that a crtcal thread θ k,u,v receves fro other threads θ k,u,x of the sae task τ k. Accordng to our defnton, I k (a, b) s a total nterference posed collectvely on all the crtcal threads of τ k,.e., I k (a, b) = σ k,u τ k I k,u,v (a, b). (4) We let Jk denote the job nstance of a synchronous parallel task τ k that receves the axu crtcal nterference aong all the jobs of τ k. For Jk, r k and d k are ts release te and deadlne. Suppose Jk ssed a deadlne. Then, one can see that at least one crtcal thread of Jk ust not execute for C k,u,v te unts, and all crtcal threads do not execute for LC k te unts n total. On the other hand, when Jk receves the aount of nterference saller than or equal to D k LC k, every job of τ k has suffcent te to coplete the executon of all crtcal threads pror to a deadlne under any workconservng schedulng. Ths observaton yelds the followng lea. LC=6,2,2,3,1 ( T =9, D =8 ),1,1,2,3,3,2,4,1 σ,1 σ,2 σ,3 σ,4 (a) Thread structure of processors,1,1,2,1,2,2,2,1,2,3 synchronzaton nstant,2,2,3,1,3,2 0 8 I (0,8) (b) Crtcal threads and deadlne ss Crtcal threads of Non-crtcal threads of Threads of other tasks,4,1 deadlne ss copleton te Fg. 2. An exaple of synchronous parallel task τ on 3 processors. (a) Task τ has 4 segents wth T = 9, D = 8, and LC = 6. (b) Task τ sses a deadlne at 8. Here, crtcal threads are θ,1,1, θ,2,2, θ,3,2, and θ,4,1, because they fnsh last aong all threads n the sae segent. Two crtcal threads, θ,2,2 and θ,3,2, were blocked for 3 te unts (.e., [1,2), [3,4), and [5,6)), yeldng I (0, 8) > D LC. Lea 1: A set of synchronous parallel tasks (denoted by τ) s schedulable on processors f te τ k τ, I k (r k, d k) D k LC k. (5) Fgure 2 shows an exaple that we wll consder throughout ths paper. Fgure 2(a) shows the thread structure of task τ and ts paraeters. In Fgure 2(b), a job of τ s released at 0 wth a deadlne of 8. The fgure shows that the executon of two crtcal threads, θ,2,2 and θ,3,2, were delayed for 3 te unts collectvely, resultng n I (0, 8) > D LC. Ths akes t nfeasble for the last thread θ,4,1 to fully execute for C,4,1 te unts before the deadlne of 8. Ths leads to the deadlne ss of task τ. As shown n Exaple 4.1, t s not as straghtforward as Eq. (3) to buld the relaton between I k (a, b) and I,k (a, b). Ths s anly because I,k (a, b) does not capture how any threads of τ nterfere wth the crtcal threads of τ k. We thereby ntroduce a new concept of p-depth crtcal nterference that characterzes the nuber of nterferng threads, and ths new noton wll brdge I k (a, b) and I,k (a, b) effectvely for synchronous parallel tasks. Let us defne the p-depth crtcal nterference I,k (p, a, b) of task τ on task τ k durng nterval [a, b) as the cuulatve length of all ntervals n whch (1) a crtcal thread of τ k s ready to execute but does not execute and (2) exactly p nuber of threads of τ are executng (see Fgure 3). It s worth notng that when t coes to the ntra-task nterference case, I k,k (p, a, b) corresponds to a case where a crtcal thread of τ k s not executng whle exactly p nuber of other non-crtcal threads of τ k are executng. The p- depth crtcal nterference enables to represent the behavor of parallel executon n ore detal, allowng to fgure out exactly how any threads of a task τ are executng sultaneously when τ delays the executon of another task τ k. A total crtcal nterference I,k (a, b) can be decoposed nto ndvdual p- depth crtcal nterferences as follows:

5 processors (=3) I,k (a,b) = 5 I +,k (3,a,b) = 1 I +,k (2,a,b) = 3 I +,k (1,a,b) = 5 a k Other tasks b te condton. Note that t s possble to upper bound the value of I +,k (p, r k, d k ) by D k LC k for all 1 p n the schedulablty analyss of τ k. Ths serves as a bass for workload bound technques, whch wll be shown n the next secton. Theore 1: A task set τ s schedulable under any workconservng algorth on dentcal processors f for each task τ k τ, Fg. 3. The noton of p-depth crtcal nterference and at least p-depth crtcal nterference. Suppose that task τ k has a lower prorty than task τ. A job of τ k s released at te nstant a, but t cannot execute n [a, a + 5) due to the executon of other hgher prorty tasks. In ths exaple, task τ executes a sngle thread n ntervals [a, a + 1) and [a + 3, a + 4), whch corresponds to 1-depth crtcal nterference on τ k. Ths yelds I,k (1, a, b) = 2. Task τ executes two threads n ntervals [a + 2, a + 3) and [a + 4, a + 5), leadng to I,k (2, a, b) = 2. And I,k (3, a, b) = 1. I,k (a, b) = I,k (p, a, b). (6) The p-depth crtcal nterference also akes t easy to consttute a total nterference I k (a, b) out of ndvdual nterferences of each task on task τ k on processors as follows. Lea 2: For any work-conservng algorth, the total crtcal nterference I k (a, b) posed on task τ k n nterval [a, b) s equal to the total aount of contrbuton of ndvdual threads to the nterference on θ k,u,v dvded by the nuber of processors,.e., I k (a, b) = 1 τ τ I,k (p, a, b) p. (7) Proof: Snce the schedulng algorth s work-conservng, n the te nstants n each of whch a crtcal thread of a task s ready but not executng, each processor ust be occuped by all the other threads of another task and ncludng tself. The total aount of the contrbuton to the crtcal nterference on τ k s τ τ I,k(p, a, b) p. If t s dvded by the nuber of processors, we can get the length of cuulatve ntervals n whch a crtcal thread of τ k s ready to execute but cannot n an nterval [a, b). For notatonal convenence, we defne at least p-depth crtcal nterference I +,k (p, a, b) as the su of ntervals n whch at least p nuber of threads of τ execute sultaneously delayng the executon of the crtcal thread of τ k n [a, b). By defnton, we have I +,k(p, a, b) = I,k (q, a, b). (8) q=p In the exaple shown n Fgure 3, at least 2-depth crtcal nterference of τ on τ k s calculated as the su of all ntervals n whch two or ore threads of τ are executng posng nterference on τ k. Then, I +,k (2, a, b) = I,k(2, a, b) + I,k (3, a, b) = 3. We note that I,k (a, b) s equal to I +,k (1, a, b) by defnton. Buldng upon the noton of at least p-depth crtcal nterference, the followng theore derves a schedulablty τ τ n(i,k(p, + rk, d k), D k LC k ) < (D k LC k ). (9) Proof: The proof s by contraposton. We wsh to prove that f a task set τ s not schedulable under any work-conservng algorth, τ k τ, τ τ n(i+,k (p, r k, d k ), D k LC k ) (D k LC k ). Suppose task τ k sses a deadlne. Then, by Lea 1, I k (r k, d k) > D k LC k. (10) We note that, by Lea 2 and Eq. (8), I k (r k, d k) = 1 = 1 τ τ τ τ I,k (p, rk, d k) p I,k(p, + rk, d k). (11) Let α denote {(, p) τ τ, 1 p, I +,k (p, r k, d k ) D k LC k }, and α denotes the nuber of eleents n α. If α, τ τ n(i,k(p, + rk, d k), D k LC k ) = α (D k LC k ) + (,p) / α I +,k(p, r k, d k) = α (D k LC k ) + I k (r k, d k) (,p) α I +,k(p, r k, d k) ( by Eq.(11)) α (D k LC k ) + I k (r k, d k) α I k (r k, d k) ( by I +,k(p, r k, d k) I k (r k, d k)) = α (D k LC k ) + ( α ) I k (r k, d k) > α (D k LC k ) + ( α ) (D k LC k ) ( by Eq.(10)) = (D k LC k ). Otherwse, α >, τ τ n(i,k(p, + rk, d k), D k LC k ) α (D k LC k ) > (D k LC k ).

6 V. WORKLOAD-BASED SCHEDULABILITY TEST Note that Theore 1 ncludes the nterference ters of I +,k (p, r k, d k ), but t s generally dffcult to calculate those values precsely. Most exstng approaches [8], [10], [31] [33] nstead seek to derve upper bounds on nterference based on workload: the workload W (a, b) of τ s the su of all ntervals n whch τ s executng n nterval [a, b). Ths secton derves a schedulablty test usng a workload-based nterference bound. To ths end, we seek to derve a bound on the whole su of such ndvdual bounded nterferences,.e., n(i+,k (p, r k, d k ), D k LC k ). Let us frst restrct our dscusson to a case where the nuber of threads n each segent σ,j s saller than or equal to the nuber of processors ( ), and we wll relax ths restrcton later. A. The nuber of threads n any segent s not larger than the nuber of processors ( ) for all. Along wth the noton of p-depth crtcal nterference, we ntroduce the noton of p-depth workload. The p-depth workload W,k (p, a, b) of task τ s the su of all ntervals n whch exactly p nuber of threads of τ are executng n a way that those p nuber of threads are all of hgher prorty than that of a crtcal thread of τ k n nterval [a, b). Slarly to I +,k (p, a, b), we defne the noton of at least p-depth workload (p, a, b) as follows: W +,k W +,k(p, a, b) = W,k (q, a, b). (12) q=p By defnton, we have that I,k (p, a, b) W,k (p, a, b) and that I +,k (p, a, b) W +,k (p, a, b). Thus, the followng nequalty hold for any J k of τ k : n(i,k(p, + r k, d k ), D k LC k ) n(w +,k(p, r k, d k ), D k LC k ) def. = Ŵ +,k. (13) Accordng to Inequalty (13), an upper-bound on the nterference ter can be obtaned by fndng the axu possble value of W +,k n any schedulng wndow of a job of τ k. For the tradtonal sngle-thread task odel, t requres us to dentfy the worst-case release pattern of τ. For ths ultthread task odel, n addton to the worst-case release pattern, we also need to dentfy the worst-case executon pattern n a segent, whch characterzes the executon of parallel threads n the sae segent. Note that the equaton of Ŵ +,k contans the n operaton (see Inequalty (13)), whch does not allow W +,k (p, r k, d k ) to contrbute to Ŵ +,k any greater than D k LC k. The worst-case release and executon patterns thus should axze Ŵ +,k n the presence of the n operaton. We next consder two cases to dscuss worst-case release and executon patterns for axzng Ŵ +,k : nter-task ( k) and ntra-task cases ( = k). r k Fg. 4. Fg. 5. W +,k(1, r k, d k ) =16 W +,k(2, r k, d k ) = 8 W +,k(3, r k, d k ) = 5 Dk J k The worst-case release pattern n whch Ŵ +,k s axzed. r LC r L σ,1 σ,2 σ,3 σ,4 The worst-case stuaton n whch the carry-n s axzed 1) Inter-Task Workload: To splfy the presentaton, we use the followng ters. A job s sad to be a carry-n job of an nterval [a, b) f t s released before a but has a deadlne wthn [a, b), or a body job f ts release te and deadlne are both wthn [a, b). Worst-case release pattern. Under a gven executon pattern, we can now deterne the worst-case release pattern. Fgure 4 shows the worst-case release pattern, n whch task τ has the axu aount of Ŵ +,k that nterferes wth job J k over nterval [r k, d k ) under EDF schedulng. As shown n the fgure, all the jobs of τ are released perodcally, and ts last body job (J ) of the nterval [r k, d k ) has a deadlne equal to that of J k (.e., d = d k). For the carry-n job, we consder the worst-case stuaton n whch all threads of the carry-n job are executed as late as possble (see Fgure 5). Wth ths release pattern we can nclude the largest nuber of segents of τ havng hgher prorty than J k n the nterval [r k, d k ), thus axzng the value of Ŵ +,k. We recaptulate the above result n the followng lea: Lea 3: Under a gven executon pattern, the release pattern of task τ that axzes Ŵ +,k s: (1) jobs are nally separated, (2) a deadlne of a job of τ algns wth the deadlne of the job of τ k, and (2) all threads of the carry-n jobs are executed as late as possble (rght before the deadlne). Worst-case executon pattern. If at all te nstants, task τ executes all of ts avalable threads (that are released and do not fnsh), then τ s sad to execute wth the axu degree of parallels. Note that snce the nuber of processors s larger than or equal to the nuber of threads n a segent, there are enough processors to execute all released threads of τ sultaneously. Lea 4: For a gven release pattern, when task τ executes wth the axu degree of parallels, W +,k s axzed. Proof: Note that by the assupton for all task J d d d k

7 τ, t s possble for τ to execute wth the axu degree of parallels. The proof s constructed by nducton on the axu degree of parallels of task τ : Base case: =1, the hypothess s trvally true. Now suppose the hypothess s true for all tasks τ j wth the axu degree of parallels j = x. We wll prove that the hypothess s true for all tasks τ wth the axu degree of parallels = x + 1. Frst we construct a task τ by deletng one thread fro every segent of τ. Then τ has = x. Thus the hypothess s true for τ. Now we add one thread to every segent of τ. The extra workload caused by the addtonal threads can contrbute to soe W +,k (p, r k, d k ). In the presence of the n operaton, the contrbuton of ths extra workload to Ŵ +,k can be axzed when t s added to as any sallest W +,k (p, r k, d k )s as possble (). By the defnton of W +,k (p, r k, d k ), W +,k (p, r k, d k ) W +,k (q, r k, d k ) when p > q (). Fro (), () can be satsfed when all x + 1 threads are executed wth the axu degree of parallels. Thus the hypothess s satsfed for = x+1. The lea s proven. Note that n the worst-case executon pattern defned above, all segent σ,j execute only for LC,j. Calculatng worst-case workload. By usng the worstcase executon pattern and the worst-case release pattern, we can now deterne the nterval of length D k whch axzes W +,k. Defne W,k (p, D k) as the value of W +,k (p, r k, d k ) n such an nterval. Fgure 5 shows how to copute the axu aount of carry-n workload. Let us denote by L the length of a carry-n nterval. In the fgure, when L = 4, the last three segents of σ,2, σ,3, and σ,4 are consdered as carry-n workloads, snce ther executon can fully/partally ft nto the carry-n nterval. If σ,3 executes only for a duraton saller than C,3, then n order to guarantee the schedulablty for task τ n the worst case, σ,2 stll needs to fnsh before d (LC,3 + LC,4 ), leavng enough roo for σ,3 to execute for C,3. Thus we can consder all threads to execute for ther WCET to calculate the carry-n jobs. Let us consder a bound BD +,k (p, D k) on the at least p- depth body-job workload n any nterval of length D k and another bound CI +,k (p, L ) on the at least p-depth carry-n workload n any carry-n nterval of length L. The axu nuber of body jobs of τ over an nterval of length D k s Dk T. Then, BD +,k (p, D k) and CI +,k (p, L ) are calculated as follows: CI +,k (p, L ) def. = BD +,k(p, D k ) = Dk T j:,j p LC,j, (14) 0, f L 0, s j=h:,j p LC,j, else f 0 < L LC and s j=h:,j p LC,j+,h 1 < p, else f 0 < L LC and (L s j=h LC,j),,h 1 p, j:,j p LC,j, otherwse, where h ndcates the earlest segent that s fully ncluded n the carry-n nterval. Then, the (h 1)-th segent can execute partally wthn the carry-n nterval. In the exaple n Fgure 5, h ndcates the 3rd segent, and the 2nd segent (the (h 1)-th segent) partally contrbutes to CI +,k (p, L ), for each p, by 1. Then, the at least p-depth workload W +,k (p, r k, d k ) that wll contrbute to the worst case, for k, s expressed as follows: W,k(p, D k ) = BD +,k(p, D k ) + CI +,k(p, D k %T ). (15) We can then copute bounds on the aount of nter-task nterference as follows: W +,k n(w,k(p, D k ), D k LC k ). (16) 2) Intra-Task Workload: The crtcal threads of task τ k can get nterference fro the other threads belongng to the sae task. For the ntra-task nterference, the worst-case release pattern s already deterned as the executon wndow of a job of τ k. Then, threads wthn a sngle job can nterfere wth each other f they belong to the sae segent, but not otherwse. For each segent, all threads except the crtcal thread can nterfere on the crtcal thread, and t s clear that the nterference of a sngle thread θ k,u,v on the crtcal thread θ k,u,v s upper bounded by C k,u,v. Wth the sae reasonng for the ntertask nterference case, Ŵ + k,k s axzed under the worstexecuton pattern where the axu possble nuber of threads of task τ k execute n parallel as uch as possble. Then, W + k,k (p, r k, d k ) that wll contrbute to the worst case s calculated as follows: Wk,k(p, D k ) def. = LC k,j. (17) j: k,j p+1 We can copute bounds on the aount of ntra-task nterference as k Ŵ + k,k n(wk,k(p, D k ), D k LC k ). (18) 3) Total Workload: Lea 5: When for all task τ, a task set τ s schedulable under global EDF schedulng on dentcal processors f for each task τ k τ, n(w,k(p, D k ), D k LC k ) k k + n(wk,k(p, D k ), D k LC k ) (D k LC k ). (19) Proof: Fro Leas 3 and 4, and Inequalty (13), the left-hand sde s an upper bound of all the nter-task nterferences of all tasks τ on task τ k, where k, and the ntra-task nterference of task τ k on tself. Then, by Theore 1, f Eq. (19) s satsfed for all tasks n a task set τ, the task set s schedulable under global EDF schedulng on dentcal processors.

8 B. The nuber of processors s saller than the nuber of threads n soe segents ( < ) for soe We now reove the restrcton of,j for each segent σ,j such that can be larger than. Lea 6: When < for soe tasks τ, Inequalty (19) stll holds. Proof: Let M = ax{ }. Suppose we have M processors. Then, fro Leas 3 and 4, the followng holds for any job J k of τ k : M n(w +,k (p, r k, d k ), D k LC k ) n(w,k (p, D k), D k LC k ) k k + n(wk,k (p, D k), D k LC k ). (20) When < M, the value of W +,k (p, r k, d k ) ay change because the executon pattern changes. However, we wll prove that: n(w +,k (p, r k, d k ), D k LC k ) M n(w +,k (p, r k, d k ), D k LC k ). (21) When we have M processors, the workload n the p-th processor can be consdered as the at least p-depth workload W +,k (p, a, b). Snce we decrease the nuber of processors fro M to < M, we have to re-dstrbute the workload n M processors to the reanng processors. Choose M sallest value of W +,k (p, a, b). Let W (M ) be the largest value aong the and W (M ) be the su of the. There are two cases. Case 1. (W (M ) < D k LC k ). Then n(w +,k (p, a, b), D k LC k ) = W +,k (p, a, b) for all the (M ) chosen values. Hence the aount subtracted fro the left-hand sde of Inequalty (20) s exactly W (M ). The aount then added to the reanng processors can be at ost W (M ). Hence Inequalty (21) holds. Case 2. (W (M ) D k LC k ). Then n(w +,k (p, a, b), D k LC k ) = D k LC k for all the non-chosen values. Hence when we add the M chosen workloads to the reanng workloads, all wll be dropped because the orgnal workload already exceed D k LC k. Thus soe aount ay be subtracted fro the left-hand sde of the Inequalty (20) and none s added. Hence Inequalty (21) holds. Fro Inequalty (20) and (21), Lea 6 holds. C. Schedulablty Test Fro Leas 5 and 6, we have the followng theore. Theore 2: A task set τ s schedulable under global EDF schedulng on dentcal processors f for each task τ k τ, n(w,k(p, D k ), D k LC k ) k k + n(wk,k(p, D k ), D k LC k ) (D k LC k ). (22) Coplexty. We denote the nuber of tasks n a task set by n. Note that t requres O(n) to calculate Eq. (22) for a gven τ k. Therefore, the schedulablty test n Theore 1 requres O(n 2 ). It s worth notng that W,k (p, D k) n Eq. (15) s a generalzaton of a workload-based nterference bound for the sngle-thread task case [30]. They are equvalent when task τ has a sngle segent wth a sngle thread. VI. EVALUATION The goal of ths paper s to develop global EDF schedulablty analyss that s drectly applcable to a set of synchronous parallel tasks, and ths secton presents sulaton results for the evaluaton of our proposed analyss. Sulaton Envronent. In order to understand how the proposed analyss behaves to synchronous parallel tasks, we eploy a sulaton paraeter n task set generaton. Ths paraeter controls the rato of the nuber of synchronous parallel tasks to the nuber of entre tasks n each task set fro 0% to 100%. We generate task sets by adaptng the technque proposed for the sequental task odel n [33]. For both sequental and parallel tasks τ, ther task paraeters are deterned as follows: perod and deadlnes (T = D ) 1 are unforly chosen n [100, 1000]. For the parallel task case, the nuber of segents (s ) and the nuber of threads wthn each segent σ,j (,j ) are unforly dstrbuted n [1, 5] and [1, 3/2], respectvely, where s the nuber of processors. All threads wthn the sae segent share the sae WCET, and the WCET s unforly chosen n [1, T /s ]. For the sequental task case, C = LC are unforly chosen n [1, T ]. We generate 40,000 task sets for = 4 and = 8 wth the parallel task rato fro 0% to 100%. Accordng to the paraeters deterned as descrbed above, we frst generate a set of tasks and then keep creatng an addtonal new task set by addng a new task nto the old set untl the syste utlzaton (.e., U sys = τ τ U ) becoes greater than. Table I characterzes the task sets generated. In the table, the parallels ndex of a task τ (denoted by PF ) s defned as the rato of ts WCET upper-bound C to ts WCET lower-bound LC (.e., PF = C /LC ), and t ndcates how uch a gven task can be allowed to explot ntra-task parallels. It s ntutve that PF grows as the rato of parallel tasks becoes larger. For the generated task sets, we perfor sulatons for our schedulablty test n Theore 2 (denoted by OUR). We 1 In ths secton, we only show the results of plct deadlne tasks due to space ltaton, but the behavors of constraned deadlne tasks are slar to those of plct ones.

9 The nuber of dedcated sets OUR NBG SAL Usys ( = 4 ) The nuber of dedcated sets OUR NBG SAL Usys ( = 8 ) The nuber of dedcated sets OUR NBG SAL Fg Rato of parallel tasks ( = 4 ) Schedulablty under dfferent syste utlzatons The nuber of dedcated sets OUR NBG SAL Rato of parallel tasks ( = 8 ) Fg. 7. TABLE I CHARACTERISTICS OF GENERATED TASK SETS FOR SIMULATION Parallel task rato Avg Nuber of Tasks U sys P F copare our tests wth two other related approaches [20], [21] (referred to as SAL and NBG, respectvely). 2 In those approaches, a sngle parallel task s decoposed nto ultple sequental sub-tasks such that each sub-task corresponds to a thread and s assgned ts own offset and deadlne. Then, sub-tasks belongng to dfferent segents wthn the sae task are separated by ther offsets and deadlnes. Ths way, sub-tasks are subject to experencng nterference fro sub-tasks belongng to other parallel tasks and the subtasks belongng to the sae segent. Whle those approaches are not desgned for deadlne-based analyss, we eploy an exstng deadlne-based schedulablty test [8], upon whch our 2 Other related analyss technques are not ncluded n our evaluaton, because those n [18], [19] are applcable to ore restrctve parallel task odels and the one n [22] s applcable only to the sngle DAG task case, whle the ultple parallel task case s of our nterest. Schedulablty under dfferent parallel task ratos proposed schedulablty tests are bult, for the decoposed sequental sub-tasks. 3 We note that the schedulablty tests used for SAL and NBG have the sae te coplexty of O(n 2 ) as our proposed schedulablty test, but SAL and NBG requre addtonal coputatons for assgnng the deadlnes of subtasks. Sulaton Results. In Fgure 6, we plot the nuber of task sets deeed schedulable by each schedulablty test, wth dfferent syste utlzatons for = 4 and = 8. The fgure shows that OUR sgnfcantly outperfors other schedulablty tests. In both cases of = 4 and = 8, OUR fnds 81% and 134% ore schedulable task sets whch are deeed schedulable by nether NBG nor SAL, respectvely. We can nterpret such a consstent gap as the beneft of usng drect schedulablty analyss for synchronous parallel tasks, copared to ndrect approaches based on task decoposton. Fgure 7 plots the sae sulaton results presented n Fgure 6 fro a dfferent angle, showng the over dfferent parallel task ratos fro 0% to 100%. Note that when the rato s 0% (.e., there are only sequental tasks n task sets), all the 3 Those decoposton-based approaches can work wth other schedulablty analyss, such as response te analyss (RTA) [9]. As deonstrated n the lterature [8], [9], t s possble to extend the concept and technque behnd deadlne-based analyss toward response te analyss. Thereby, our proposed noton of parallels-aware nterference can be extended toward response te analyss and t wll be possble to copare our approach wth others accordng to response te analyss. However, ths s beyond the scope of ths paper due to the lt of space.

10 analyses (OUR, SAL and NBG) yeld the sae results. Ths s because task decoposton s no longer necessarly appled to sequental tasks and our schedulablty test s reduced to the exstng one [8] for the sequental task case. One nterestng observaton s that task decopostonbased approaches fnd less task sets deeed schedulable as the parallel task rato ncreases, whle our approach fnds a larger nuber of such tasks on average. Ths can be nterpreted that the overheads of task decoposton are accuulated wth a growng nuber of parallel tasks. On the other hand, OUR s relatvely uch nsenstve to the parallel task rato, plyng that t s effectvely dealng wth the thread-level parallels and segent-level synchronzaton of the synchronous parallel task odel. More techncally, Table I shows that when the parallel task rato grows, the parallels ndex ncreases and thereby the WCET lower-bound LC k of a task τ k generally decreases. Ths gves task τ k ore roo to accoodate larger nterference fro other tasks, leadng to better schedulablty. However, t can be nterpreted that such a potental for schedulablty proveent s not well exploted n NBG and SAL snce they do not consder ntra-task parallels drectly. In those approaches, the entre executon wndow of a synchronous parallel task s dvded nto saller ntervals of ts sub-tasks through nteredate deadlnes, and ths sees to severely lt the flexblty n executng subtasks; they now need to execute only wthn ther own artfcal executon wndows whle correspondng threads have flexblty n runnng even outsde such artfcal wndows n our approach. Such a dfference leads to a sgnfcant gap between schedulablty for synchronous parallel tasks. VII. CONCLUSION The otvaton for our work was the desre to understand the thread-level parallels and segent-level synchronzaton of synchronous parallel tasks n the context of hard real-te ult-core schedulng. In ths paper, we extended the noton of nterference foralzng t at a fner-graned thread level and buldng a connecton to the noton at a task level. We then generalzed nterference-based analyss ethods accordng to the new proposed noton of nterference, ntroducng the frst global EDF schedulablty condtons that are drectly applcable to a set of synchronous (alleable) parallel tasks. Our evaluaton results showed that t sgnfcantly proves the state-of-the-art analyss technques avalable for synchronous parallel tasks. Ths paper ncorporated thread-level parallels drectly nto schedulablty analyss focusng on the EDF algorth. However, we beleve the schedulablty of synchronous parallel tasks can be advanced uch ore sgnfcantly f thread-level parallels s drectly reflected nto schedulng algorths as well. Hence, a drecton of our future work ncludes developng new real-te schedulng algorths that support ntra-task parallels and synchronzaton drectly. 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