Response-Time Analysis of Synchronous Parallel Tasks in Multiprocessor Systems

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1 Response-Tie Analysis of Synchronous Parallel Tasks in Multiprocessor Systes Cláudio Maia CISTER/INESC TEC, ISEP Porto, Portugal Marko Bertogna University of Modena Modena, Italy Luis Miguel Pinho CISTER/INESC TEC, ISEP Porto, Portugal Luís Nogueira CISTER/INESC TEC, ISEP Porto, Portugal ABSTRACT Prograers resort to user-level parallel fraeworks in order to exploit the parallelis provided by ultiprocessor platfors. While such general fraeworks do not support the stringent tiing requireents of real-tie systes, they offer a useful odel of coputation based on the standard fork/join, for which the analysis of tiing properties akes sense. Very few works analyse the schedulability of synchronous parallel real-tie tasks, which is a generalisation of the standard fork/join odel. This paper proposes to narrow the gap by presenting a odel that analyses the response-tie of synchronous parallel real-tie tasks. The odel under consideration targets tasks with fixed priorities, coposed of several segents with an arbitrary nuber of parallel and independent units of execution. We contribute to the state-of-the-art by analysing the response-tie behaviour of synchronous parallel tasks. To accoplish this, we take into account concepts previously proposed in the literature and define new concepts such as carry-out decoposition and sliding window technique in order to copute the worst-case workload in a window of interest. Results show that the proposed approach is significantly better than current approaches, iproving the stateof-the-art analysis of parallel real-tie tasks. Categories and Subject Descriptors D.4.1 [Operating Systes]: Process Manageent Scheduling General Ters Design, Algoriths, Theory Perission to ake digital or hard copies of all or part of this work for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for coponents of this work owned by others than ACM ust be honored. Abstracting with credit is peritted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. Request perissions fro Perissions@ac.org. RTNS 2014, October , Versailles, France Copyright 2014 ACM /14/10...$ Keywords Parallel Task Model, Job-level Parallelis, Real-Tie 1. INTRODUCTION The real-tie systes doain is currently facing the challenge of how to exploit the iense coputing power offered by the next-generation of any-core systes for tiecritical applications [1]. To this extent, several techniques have already been proposed for scheduling real-tie tasks in ultiprocessor systes [2]. However, the latest arket requireents and technology advanceents require both predictability and perforance fro real-tie applications. In order to explore the inherently parallel coputing power available, new real-tie coputing odels have recently been proposed with a special focus on job-level parallelis or intra-task parallelis [3, 4]. For exaple, in the synchronous parallel task odel proposed in [4] (depicted in Figure 1, real-tie jobs are coposed of consecutive segents, each containing a different nuber of independent sub-jobs. Different naes are used in the literature to nae these independent sub-jobs, i.e, thread, sub-task, parallel job or, in short, p-job. Segents have precedence constraints aong theselves, eaning that p-jobs belonging to a certain segent can only start their execution after all the p-jobs in the previous segent have been copleted. Moreover, there is no restriction on the nuber of segents per job, neither on the nuber of p-jobs executing in a segent. Such a odel is a generalisation of the fork/join odel presented in [3] to odel Java Fork/Join [5] or OpenMP [6] constructs. This paper focuses on the schedulability analysis of synchronous parallel tasks in ultiprocessor systes coposed of identical processors, where tasks have a fixed priority and the structure of each task is known a priori. Our approach iproves over the work reported in [7], providing tighter schedulability conditions and extending the analysis to fixedpriority task systes. Tighter upper-bounds on the workload within a window of interest are derived by coputing the response-tie upper bounds of the interfering jobs, siilarly to the technique proposed in [8] for sequential task sets. However, there are subtle differences to consider when dealing with parallel tasks. The analysis in [8] cannot be applied as it is, but needs to be properly odified to take into account the different task structure in a predictable way.

2 Figure 1: Exaple of a synchronous parallel task τ i coposed of three segents, with one, four and two p-jobs, respectively. Contributions: Our contributions are the following: We present a schedulability analysis for synchronous parallel sporadic tasks scheduled with global Fixed Priority on identical ultiprocessors. Existing schedulability analyses are based on global Earliest Deadline First [7], or consider partitioned approaches [9]. We highlight the ain issues in the response-tie analysis of parallel tasks for deriving predictable upper bounds on the interfering contributions in a window of interest. We provide a pseudo-polynoial algorith to copute an upper bound on the response tie of each parallel task. We present an extensive set of experients showing that the proposed approach outperfors the state-ofthe-art schedulability techniques for parallel real-tie task systes. The reainder of this paper is organised as follows. Section II presents the state-of-the-art of parallel real-tie tasks. Section III describes the syste odel. Section IV introduces soe preliinary results based on the notion of critical interference. Using these results, Section V details the schedulability analysis for fixed-priority synchronous parallel tasks. Section VI presents the siulation results obtained fro experients conducted using synthetically generated task sets. Finally, section VII concludes the paper and presents the future work. 2. RELATED WORK Goossens and Berten [10] redefined a classification for different types of parallel tasks. In this classification, a job ay be classified as rigid, oldable or alleable. In a rigid job, the nuber of processors assigned to each segent is fixed and deterined a priori, so that either all p-jobs of a segent are executed, or none of the is scheduled for execution. In a oldable job, the nuber of processors assigned to each segent is deterined by the scheduler, but cannot change once the segent starts executing. Finally, a alleable job allows the nuber of processors assigned to each segent to change dynaically. The proble of ultiprocessor scheduling of parallel realtie tasks was covered by Han and Lin in [11] where the authors prove that the proble of scheduling fixed-priority parallel jobs is NP-Hard. Drozdowski [12] focuses on the proble of scheduling parallel tasks with the objective of iniizing the akespan. Concerning rigid tasks, Goossens and Berten [10] proposed a scheduling algorith for parallel rigid real-tie tasks based on gang scheduling. Moldable tasks were studied by Maniaran et al. [13] where they proposed a non-preeptive Earliest Deadline First (EDF approach that considers parallel real-tie tasks. Kato and Ishikawa [14] proposed the Gang-EDF algorith, which applies EDF to the traditional gang scheduling schee. Jansen [15], Collette et al. [16], and Korsgaard and Hendseth [17] studied alleable tasks. Jansen [15] focused on iniizing the akespan but without considering real-tie constraints. Collette et al. [16] studied the proble of global scheduling of sporadic task systes on ultiprocessors considering job-level parallelis. Korsgaard and Hendseth [17] proposed a schedulability test for alleable tasks scheduled with global EDF. Lakshanan et al. [3] study the scheduling of periodic fork/join real-tie tasks on ultiprocessor platfors. Each task is divided into sequential and parallel segents. Parallel segents ust be preceded and followed by a sequential segent. All parallel segents ust have the sae nuber of threads, and the nuber of threads cannot be greater than the nuber of processors in the platfor. In order to schedule such tasks in a ultiprocessor platfor, the authors propose the decoposition of fork/join tasks by applying a task stretch transfor algorith. Moreover, a resource augentation bound is derived for the decoposed task set when partitioned deadline onotonic scheduling is used. Saifullah et al. [4] generalise the fork/join odel presented in [3], naed synchronous parallel task odel. In this odel, parallel tasks ay have any nuber of segents, and the nuber of parallel threads within any segent can be greater than the nuber of cores in the platfor. In [4], the decoposition of periodic parallel tasks into constrained-deadline sequential tasks is proposed. For the decoposed task sets, a resource augentation bound is derived for global EDF and partitioned deadline onotonic scheduling policies. A schedulability condition for synchronous parallel tasks scheduled with global EDF has been presented in [7], extending the traditional interference-based analysis for serial task odels. The authors introduce the concept of critical interference to capture the interference of parallel threads. In this paper, we will borrow the concept of critical interference to propose tighter schedulability conditions based on the response-tie analysis of synchronous parallel tasks. In [9], a worst-case response-tie analysis is presented for fork/join tasks under partitioned fixed-priority scheduling. The analysis iterates over the segents of a fork/join task, selecting the worst-case response tie of each segent. The authors show that fork/join tasks ay present a larger worst-case response tie due to the interference of sequential tasks. In [18], the Fork/Join OS (FJOS is presented, an operating syste based on Coposite OS, coparing its behaviour with the GOMP ipleentation on Linux. Moreover, the schedulability analysis technique proposed in [9] is adapted to include overheads based on real easureents in FJOS. As in [9], this approach is also based on partitioned fixedpriority scheduling.

3 3. SYSTEM MODEL We consider the proble of scheduling synchronous parallel real-tie tasks with fixed priority on a syste coposed of identical processors with unifor eory access. In our odel, each task releases a sequence of jobs where each job instance is allowed to execute in ore than one core at the sae instant. Without loss of generality, all tie intervals and task paraeters are assued to be integer ultiples of the syste clock. Let τ = {τ 1,..., τ n} denote the set of n sporadic tasks. Assue tasks are indexed in priority order, task τ 1 being the highest priority one. Each task τ i in the task set τ releases an infinite sequence of jobs separated by at least T i tie units. Each task has a deadline D i T i (i.e., constrained deadline odel, eaning that each of its jobs needs to coplete its execution at ost D i tie units after its release. Moreover, each task τ i is characterised by a sequence of segents s i = {σ i,1,..., σ i,si }, where each segent σ i,j is coposed of a set of i,j parallel jobs (or in short p-jobs, {J i,j,1,..., J i,j,i,j }, each one having the sae priority of the task that spawns it. The parallel jobs are independent sequential threads that can be executed in parallel, i.e., in different processors at the sae tie instant (see Figure 1. Before a segent starts executing any of its p-jobs, all the p-jobs of the preceding segent (if any ust have been copleted. Besides these precedence constraints, parallel jobs are independent, and there are no other shared resources than processing units. To clarify, i,j denotes the nuber of p-jobs within the j-th segent of task τ i. In this work, we allow the nuber of p-jobs of a segent to be greater than the nuber of cores, so that i,j ay be greater than for soe segent σ i,j. The axiu degree of parallelis of a task is denoted as i and is defined as i = ax j{ i,j}. A fully preeptive syste is assued where any executing p-job ay be preepted and resued later without any cost. At any given instant, the ready p-jobs with the highest priority are the ones executing in the cores. Ties are broken arbitrarily. Each p-job instance J i,j,k is characterized by a worst-case execution tie C i,j,k. The worst-case execution tie of each segent σ i,j is given by: i,j C i,j = C i,j,k. (1 k=1 The overall worst-case execution tie of a task τ i is then defined as: s i C i = C i,j. (2 j=1 The above equations represent the tie it takes to execute a segent (Equation 1 or a task (Equation 2 in a dedicated single processor platfor, i.e., with no parallelis. The iniu worst-case execution tie P i of a task τ i is the tie task τ i requires to execute when the nuber of processing units is infinite, i.e., the critical path length. Forally, P i is defined as: where P i,j s i P i = P i,j, (3 j=1 represents the worst-case execution tie of the largest p-job(s of segent σ i,j. Forally, P i,j = i,j ax k=1 {C i,j,k}. (4 The utilisation U i of task τ i is the ratio between the task s overall worst-case execution tie and period, U i = C i T i. For the task set τ, the total utilisation is defined as U(τ = n i=1 Ui. For iplicit-deadline sequential task sets, a necessary and sufficient condition for feasibility is U(τ ([19]. Nevertheless, for fork/join tasks this condition is only necessary [3], as it is for the synchronous parallel task odel adopted in this paper. Moreover, another necessary condition for schedulability of synchronous parallel tasks is P i D i. It is iportant to note that with the synchronous task odel there ay be feasible task sets in which soe task has a utilisation larger than 100%. Serialisation techniques are not possible with such tasks, as the derived sequential task would be clearly unschedulable. The worst-case response-tie of τ i, denoted as R i, is given by the axiu aount of tie that elapses between the release tie (r i of any job of τ i and its copletion tie. For parallel tasks, R i clearly depends on several factors such as the inter-task and intra-task interferences; precedence constraints between the segents of the task itself; the degree of parallelis of each region; and the nuber of cores in the hardware platfor. As it ay be extreely difficult to derive the exact worst-case response tie of a task in the addressed setting, a typical approach in the real-tie literature is to copute an upper bound Ri ub on the response-tie of task τ i. Table 1 presents a suary of the iportant notation defined and used throughout the paper for quick reference and clarity. 4. CRITICAL INTERFERENCE OF PARAL- LEL TASKS We propose a global fixed-priority approach for synchronous parallel tasks. In this approach, all the ready p-jobs are inserted in a global queue, fro where processors pick the highest priority p-jobs. The proposed approach considers for each job its worst-case execution tie and the interference that it suffers. If the syste is schedulable, the interference is bounded and the job execution tie plus the iposed interference is always less than or equal to the job s deadline. Interference is an iportant concept widely used in realtie systes. For traditional sequential task sets, the interference a task τ k suffers over an interval of length L, denoted as I k (L, is defined as the su of all intervals of tie in which τ k is ready to execute but it cannot execute due to the execution of other higher priority tasks in the syste. In particular, the interference of a higher priority task τ i over task τ k in an interval of length L is denoted as I i,k (L, and is defined as the su of all tie-intervals in which τ i is executing but τ k is not, even though it is ready to execute. Intuitively, the interference that a task suffers cannot be greater than the total aount of workload of the higher priority jobs. When dealing with synchronous parallel tasks two types of interference ay occur: inter-task and intra-task inter-

4 Sybol n τ U i U(τ T i D i C i P i s i i C i,j P i,j i,j C i,j,k r i d i R i Ri ub L I k (L I i,k (L I p i,k (L W p i Table 1: Suary of notation Description Nuber of processors in the platfor Nuber of tasks in the task set Set of periodic or sporadic tasks Utilisation of task τ i, i.e., C i T i Total utilisation of the task set τ Period of task τ i Relative Deadline of task τ i Overall worst-case execution tie requireent of τ i Miniu worst-case execution tie of task τ i Nuber of segents in task τ i Maxiu degree of parallelis of task τ i Overall worst-case execution tie of segent σ i,j Miniu worst-case execution tie of segent σ i,j Nuber of p-jobs within segent σ i,j Worst-case execution tie of p-job J i,j,k Release tie of a job of task τ i Absolute deadline of a job of task τ i Worst-case response tie of task τ i Upper-bound of R i Generic interval [r k, r k + Rk ub ] Critical interference on task τ k in any interval of length L Critical interf. of τ i on τ k in any L Critical interf. of τi on τ k with depth at least p in any L (L Workload of task τi for at least p p-jobs in the interval L ference. Inter-task interference is the interference caused by other higher priority tasks executing in the syste. This is the standard notion of interference widely used in traditional sequential odels. Instead, intra-task interference is peculiar to parallel task systes, and is defined as the selfinterference due to parallel jobs of the sae task instance. In order to copute the interference of a parallel task, we adopt the concept of critical thread 1, as previously defined in [7]. Definition 1. A thread is critical if it is the last one to coplete aong the threads belonging to the sae segent. For deriving the worst-case response tie of a task, it is then sufficient to characterise the interference iposed to its critical threads, as they are the ones suffering the largest interference. Definition 2. The critical interference I k (L on task τ k, in any interval of length L, is defined as the cuulative tie in which a critical thread of task τ k is ready to execute but it cannot due to the execution of other parallel jobs. Given the above definitions, the following theore siply follows. 1 While we prefer using the ter parallel job instead of thread, we decided here to keep the nae thread for hoogeneity with the original definition. However, both ters are interchangeably used in this paper. Theore 1. Given a set of synchronous parallel tasks τ scheduled by any work-conserving 2 algorith on identical cores, the worst-case response-tie of each task τ k can be upper bounded by Rk ub if P k + I k (R ub k R ub k. (5 Proof. Consider the job of τ k that leads to the worstcase response tie R k. Let r k be its release tie. Within a scheduling window [r k, r k + Rk ub ], Equation (5 guarantees that all s k critical threads have sufficient tie to execute P k tie-units, while accoodating the interference suffered fro other threads, accounted for in I k (Rk ub. Since the execution requireent of each critical thread cannot exceed the iniu worst-case execution tie of the corresponding segent, Equation (3 guarantees that all critical threads coplete their execution within the considered interval, proving the theore. The proble of the above theore is that coputing the exact interference iposed on the considered task is difficult. To sidestep this proble, a coon approach is to express the total interference as a function of individual task interfering contributions, and upper bound such contributions with the worst-case workload executed by each task in the considered window. Definition 3. The critical interference I i,k (L iposed by task τ i on task τ k in any interval of length L is defined as the cuulative workload executed by p-jobs of task τ i while a critical thread of τ k is ready to execute but is not executing. Differently fro the sequential case, each task τ i ay contribute with different p-jobs at the sae tie to the individual interference on a task τ k. In the particular case when i = k, the critical interference I k,k (L ay include the interfering contributions of (non critical p-jobs of task τ k on itself, i.e., the intra-task interference. The next lea allows expressing the total interference as a function of single task interferences. Lea 1. For any work-conserving algorith, the following relation holds: I k (L = 1 I i,k (L. (6 τ i Proof. Fro the work-conserving property of the considered scheduler, it follows that whenever a critical thread of τ k is interfered, all cores are busy executing other p- jobs. Therefore, the total aount of workload executed by p-jobs interfering with critical threads of τ k within the considered window is I k (L: τ i I i,k (L = I k (L. The lea siply follows by rephrasing the ters. As previously entioned, the individual interference I i,k (L accounts for all p-jobs of τ i interfering with τ k, including p- jobs that are executing at the sae tie. In order to capture how any parallel jobs of τ i ay siultaneously interfere 2 A scheduling algorith is said to be work-conserving if it never idles a core when there is a ready task waiting to be executed.

5 Figure 2: Task τ i interfering on task τ k. with task τ k, we will borrow fro [7] the concept of at least p-depth critical interference 3. Definition 4. The at least p-depth critical interference of τ i on τ k in any interval of length L, denoted as I p i,k (L, is defined as the total aount of tie in which a critical thread of τ k is ready to execute but cannot execute while there are at least p threads of task τ i siultaneously executing in the syste. To better understand the eaning of I p i,k (L, consider the exaple in Figure 2, where a task τ i interferes τ k with two threads for five tie-units, one thread for seven tieunits, and three threads for three tie-units. In this case, Ii,k(L 1 = 15, Ii,k(L 2 = 8, and Ii,k(L 3 = 3. Note that by definition (Definitions 2 and 4 the following inequality holds: I p i,k (L I k(l. The following lea allows establishing a relation between the overall critical interference on a task τ k and the at least p-depth critical interference of each task τ i on τ k. Lea 2. For any work-conserving algorith, the following relation holds: I k (L = 1 I p i,k (L. (7 Proof. Considering each single interfering task τ i, the aount of execution by all p-jobs of τ i interfering with τ k within the considered window equals p=1 Ip i,k (L. The Lea follows fro Lea 1. We will now extend to the parallel task odel considered in this paper two results proved in [8] for sequential tasks. Lea 3. in ( I p i,k (L, x x I k (L x. Proof. If. We would like to prove that if I k (L x, then ( in I p i,k (L, x x. For a given length L, let ξ be the nuber of at least p- depth critical interferences I p i,k (L x, naely: ξ = { I p i,k (L x}. i,p 3 Note that we are siplifying the analysis and notations with respect to [7], without aking use of the exact p- depth interference, which, to our belief, is not needed for the purposes of this paper. Also the theores presented in this section have therefore subtle differences fro the corresponding ones in [7]. For instance, the case of Lea 2, which differs fro a siilar result proved in [7] in that the notion of at least p-depth critical interference is used instead of the exact p-depth critical interference. If ξ >, then τ i p=1 in ( I p i,k (L, x ξx > x. Otherwise, ( ξ 0, and, using Lea 2, in ( I p i,k (L, x = ξx + I p i,k (L τ i p:i p i,k <x = ξx + I k (L (L [Lea 2] τ i ξx + I k (L ξi k (L = ξx + ( ξi k (L p:i p i,k x I p i,k [ I p i,k (L I k(l ] ξx + ( ξx = x. [using I k (L x] Only if. Fro Lea 2, we have I k (L = 1 I p i,k (L 1 in ( I p i,k (L, x 1 x = x. Theore 2. Given a set of synchronous parallel tasks τ scheduled by any work-conserving algorith on identical cores, the worst-case response-tie of each task τ k can be upper bounded by Rk ub if ( in I p i,k (Rub k, Rk ub P k + 1 < (Rk ub P k + 1 Proof. If the inequality holds, Lea 3 gives I k (R ub k < R ub k P k + 1. Since an integral tie odel is used, I k (R ub k R ub k P k. The theore then follows fro Theore 1. In the following section, the above theore will be used to derive a sufficient schedulability test for synchronous parallel task systes scheduled with global fixed priority algorith. 5. RESPONSE-TIME ANALYSIS In order to exploit the theore proved in the previous section to analyse the schedulabilty of a parallel task syste, it is necessary to copute the critical interference ters. Since finding such ters is known to be a difficult proble for ultiprocessor systes, a coon approach is to use upper bounds that are easier to copute. An upper bound on the interference of a task τ i in a window of length L is given by the axiu workload that τ i can execute within the considered window. However, coputing the axiu workload that can be executed by τ i in a generic window is also a difficult task. To sidestep this proble, a typical technique is to consider pessiistic scenarios in which the workload in a given window cannot be saller than in the worst-case situation. We hereafter describe the pessiistic scenario considered in this paper. Consider a window of length L that spans the interval [r k, r k +L] of a given (interfered task τ k. We call this interval of tie the window of interest. Within this window, we provide an upper bound on the execution tie of an interfering task τ i. As coonly adopted in the literature, we

6 Figure 3: Densest possible packing of threads within a window of interest. will call carry-in job the first instance of τ i executing in the window of interest, having a release tie before and deadline inside the window of interest. By contrast, the carry-out job has its release tie within (or before and deadline after the window of interest. Note that, by convention, a job that has both release tie and deadline outside the window is considered to be a carry-out job. All τ i s instances whose release tie and deadline are entirely contained within the considered window will be naed body jobs. As shown in [8], the densest possible packing of sequential jobs of a task τ i is found when: 1. A job starts executing at the beginning of the window of interest, and copletes as close as possible to its response tie. In other words, the job starts executing R i P i tie-units after its release tie, in correspondence to the beginning of the window of interest. 2. All subsequent jobs of τ i are executed as soon as possible after being released, i.e., respecting the period T i. Such a situation is depicted in Figure 3 for a parallel task τ i in the window of interest. 5.1 Sliding window technique An iportant observation to ake is that the scenario described above ay not represent the worst-case workload in the synchronous parallel task odel considered in this paper. This occurs because the parallel task structure is characterized by precedence constraints that ay affect the densest possible packing of p-jobs. Consider the exaple in Figure 3, where a task coposed of three segents is considered in the above scenario. The carry-in job is fully contained inside the window of interest L, while the carry-out is only partially contained. Now, if the window of interest is shifted right by one segent (as in the window L, the carry-in contribution decreases by one p-job, while the carry-out contribution increases by three p-jobs, leading to a larger task workload within the considered window. In order to properly consider the worst-case workload contribution of each task in a window of interest, we check all different eaningful alignents of the window of interest with respect to the task structure. Note that shifting right the window of interest, the workload contribution has a discontinuity whenever one of the extree points of the window coincides with a segent boundary. Therefore, we can check all possible scenarios in which the window of interest is shifted to the right fro the original configuration, such that either (i the window starts at the beginning of a segent of the carry-in job, or (ii the window ends at the end of a segent of the carry-out job. Forally, we consider the worst-case workload of a task τ i in a window of length L, taking the axiu workload of the considered task, over all possible configurations in which the window is shifted right fro the original configuration by a Γ 1 Γ 2, where Γ 1 and Γ 2 are the sets of significant offsets to check corresponding to scenario (i and (ii, respectively (see Figure 6. Before deriving the foral offset values to check, let η i(a, L be the carry-out length for task τ i in a window of length L and offset a. Then, η i(a, L = in (L, (L + R i P i + a od T i. We note that the eaningful offsets to consider in scenario (i correspond to the best-case starting ties of each segent σ i,j of τ i, i.e., j x=1 Pi,x, j [1, si]. Moreover, all offsets greater than P i η i(0, L can be ignored, since they would cause the end of the window to fall beyond the end of the carry-out job, resulting in a saller workload. Therefore, } Γ 1. = { j x=1 P i,x P i η i(0, L, j [1, s i] The offsets to consider in scenario (ii correspond to the difference (when positive between the best-case starting ties of each segent σ i,j and the original carry-out length η i(0, L, i.e., ( }. j Γ 2 = {ax 0, P i,x η i(0, L, j [1, s i]. x=1 5.2 Decoposing the carry-out job One last observation concerns predictability, as defined in [20] 4. A schedulability test needs to be predictable, in that it should consider all possible execution ties of a task syste, as long as they do not exceed the given worst-case execution tie. In other words, we would like the responsetie provided by our analysis to be sufficiently robust to consider all possible execution requireents of the given tasks, including when soe segent σ i,j requires less than C i,j tie-units, or when a task ay skip soe of the segents. A schedulability test that does not properly consider situations when execution requireents are reduced is by no eans sufficiently robust for critical applications. The proble with the above approach is that a larger workload ay fit the considered window if the carry-out skips soe segent. Consider the exaple in Figure 4. In the upper scenario, the original situation is depicted, with the carry-out job contributing to the workload in the window of interest with its first two segents. However, when the second segent of the carry-out job is skipped, a worse situation is found, as shown in the lower part of the figure, since a segent with a higher parallelis ay enter the window, resulting in a larger workload. Considering all possible cobinations of execution ties appears overly coplicated as it requires a cobinatorial 4 In [21], a broader concept is defined, i.e., sustainability, which generalizes the notion of predictability..

7 Figure 4: Densest possible packing of threads when a task skips soe segent. Figure 5: Exaple of a decoposed job exploration of the possible segent instances of each task. To solve this proble, by allowing our analysis to be sufficiently robust, we will consider a pessiistic situation in which the carry-out job is decoposed, re-aligning the parallel segents such that the segents with higher parallelis are shifted to the beginning of the job s execution. Thus, segents are ordered by their nuber of p-jobs following a non-increasing pattern where segents with a higher nuber of p-jobs execute first, as depicted in Figure 5. Replacing the original carry-out job by a decoposed job results in placing the parallel segents with higher parallelis within the window of interest, which allows us to obtain a sound upper bound on the workload of the carry-out job. We are now ready to derive an upper bound of the workload that each task ay ipose on a window of length L. 5.3 Workload of a task within a window Before presenting the analytical derivation of the workload coponents, we introduce the notion of at least p-depth workload. Definition 5. The at least p-depth workload of a task τ i in a window of length L, denoted as W p i (L, is the su of all intervals in which at least p threads of τ i execute siultaneously in parallel. Note that the following relation holds by the definition of I p i,k (L: I p i,k (L W p i (L. The above relation, together with Theore 2, gives the following lea. Lea 4. Given a set of synchronous parallel tasks τ scheduled by any work-conserving algorith on identical cores, the worst-case response-tie of each task τ k can be upper bounded by Rk ub if ( in W p i (Rub k, Rk ub P k + 1 < (Rk ub P k + 1 It now only reains to derive an upper bound on W p i (L. We will copute such an upper bound by considering the at least p-depth contributions of carry-in, body and decoposed carry-out of each task τ i in the worst-case scenario suarized in Figure 6, for all significant offsets a Γ 1 Γ 2. To copute the at least p-depth workload of the decoposed carry-out job, it is necessary to consider the first η i(a, L units of the decoposed carry-out job. The following function coputes the at least p-depth workload executed within the first x units of a generic job of τ i. 0, if x 0 z j=1: i,j p Pi,j + (x z j=1 Pi,j, if 0 < x Pi and i,z+1 p g p i (x = z j=1: i,j p Pi,j, if 0 < x Pi and i,z+1 < p j: i,j p Pi,j, otherwise, (8 where z represents the index of the last segent that is fully included in the interval, so that (z + 1 is the index of the segent that ay execute partially within the carry-out interval. The nuber of body jobs of τ i executing in L is given by L + Ri P i β i(l = 1. (9 Note that β i(l does not depend on a because the range in which a is varied never influences the nuber of body jobs. The at least p-depth workload of the body jobs of τ i executing in L is then given by b p i (L = βi(l T i j: i,j p P i,j. (10 The carry-in length α i(a, L can be derived as 5 α i(a, L = L η i(a, L β i(lt i. The at least p-depth carry-in contribution can then be derived by coputing the workload executed within the last α i(a, L units of the carry-in job. The following function (fro [7] coputes the at least p-depth workload executed within the last x units of a job of τ i. 0, if x 0 f p i (x = si j=h: i,j p Pi,j + (x s i j=h Pi,j, si j=h: i,j p Pi,j, if 0 < x Pi and i,h 1 p if 0 < x Pi and i,h 1 < p j: i,j p Pi,j, otherwise, (11 where h represents the index of the earliest segent that is fully included in the interval, so that (h 1 is the index of 5 When R i = P i and L T i, the first job of τ i executing in the window of interest is accounted for in the carry-in and not in the body contribution despite it has both release tie and deadline within the window.

8 Figure 6: Response-tie analysis details the segent that ay execute partially within the carry-in interval. Considering Equation 8, Equation 10 and Equation 11, an upper bound on the at least p-workload of a task τ i in a window of length L and offset a is given by: Ŵ p i (L, a = f p i (αi(a, L + bp i (L + gp i (ηi(a, L, (12 where g denotes that the function g is applied to the decoposed job. An upper bound on the worst-case workload of τ i with depth at least p in a window of length L is then derived as Ŵ p i (L = ax {W p i (L, a}, (13 a Γ 1 Γ 2 where Γ 2 denotes that the offsets in this set are coputed, again, considering the decoposed job. Note that the above expression can be used to bound the inter-task workload fro interfering tasks. Before applying Lea 4, a bound should also be provided to the intra-task interference, accounting for the workload of p-jobs fro the sae task. An upper bound on the intra-task workload of task τ k with depth at least p can be given by: Ŵ p k = P k,j, (14 j: k,j p+1 where the su is extended over all segents with parallelis at least p + 1 instead of p since the p-jobs of the critical threads do not contribute to the critical interference. 5.4 Schedulability condition Given the worst-case inter-task and intra-task workloads presented in the previous sections, we are now in a position for deriving an upper bound on the worst-case response tie of a parallel task. Lea 5. Given a set of synchronous parallel tasks τ scheduled by any work-conserving algorith on identical cores, the worst-case response-tie of each task τ k can be upper bounded by Rk ub if i p in (Ŵ i (Rub k, Rk ub P k + 1 τ i k p=1 k p + in (Ŵ k, Rub k P k + 1 p=1 < (R ub k P k + 1. Proof. The proof siply follows fro Lea 4, using the derived upper bounds instead of the real p-depth workload, and extending the p-indexed su over the axiu nuber of p-jobs of each task 6. For the special case of global fixed-priority scheduling, the interfering workload ay be liited to the set of tasks having higher priority than τ k. The following theore can then be used to derive Rk ub in a fixed priority setting. Theore 3. Given a set of synchronous parallel tasks τ scheduled by global fixed-priority on identical cores, an upper bound Rk ub on the worst-case response-tie of a task τ k can be derived by the fixed-point iteration of the following expression, starting with Rk ub = P k : R ub k P k + ( 1 i i<k p=1 p=1 p in (Ŵ i (Rub k, Rk ub P k + 1 k p + in (Ŵ k, Rub k P k + 1. Proof. If the iteration ends before Rk ub reaches D k, it is easy to see that the condition of Lea 5 is satisfied, proving the theore. A siilar theore holds for general work-conserving scheduling algoriths, extending the outer su to all tasks τ i k. A schedulability test for systes scheduled with global fixed-priority is easily derived by coputing Rk ub for each task τ k in priority order, starting fro the highest priority 6 As in [7], we are not taking advantage of the fact that carryin and carry-out contributions ay be less dense than in the considered scenario when there is soe segent σ i,j with a parallelis i,j greater than the nuber of processors.

9 one, and checking whether Rk ub D k for all tasks. If not, the test is not able to guarantee the schedulability of the syste. Note that, updating response tie upper bounds in priority order allows optially exploiting Theore 3, since every task can use the ost updated response ties of the higher priority tasks, leading to saller inter-task interferences. 5.5 Coplexity The coplexity of the proposed response-tie analysis is pseudo-polynoial in the task paraeters, as is the original response-tie analysis for sequential task sets presented in [8]. However, with respect to the sequential analysis, an additional s i ter has to be considered to account for the sliding window technique that repeats the workload coputation for all segent starting ties of the carry-in and carry-out jobs. To obtain a faster analysis, a siple ethod is to consider the coplete execution of the carry-in and carry-out job instances. To do that, it is sufficient to replace Ŵ p i (L in Theore 3 with the following ter: ( Ŵ p L + i (L = Ri P i + 1 P i,j. (15 T i j: i,j p As we will show in the experiental section, this ethod allows obtaining a faster worst-case response tie coputation without significant schedulability losses. 6. EVALUATION This section presents the siulation results to evaluate the behaviour of our schedulability analysis, coparing it to other approaches existing in the literature. We only show the results for the iplicit deadline case, which are however representative of the general behaviour. Concerning the siulation environent, we use a siilar setting as in [7]. We start by generating a task set with tasks, creating new task sets by adding a new task to the previous one until the task set utilization exceeds the nuber of processors. The above procedure is repeated until 40,000 task sets are generated. The percentage of parallel tasks in the task set is controlled by a paraeter that generates a rando nuber fro 0% to 100%. Periods of sequential tasks are uniforly generated in [100, 1000], with C i uniforly chosen fro [1, T i]. For parallel tasks, the nuber of segents s i is uniforly generated in [1, 5]; the nuber of threads per segent i,j is uniforly generated in the interval [1, 3/2]; the worstcase execution ties of the threads in each of the segents is uniforly chosen in the interval [1, T i/s i]; periods are uniforly generated in [100, 10000]. For the generated task sets, we copare the nuber of schedulable task sets detected by our analysis (PAR-RTA with the approach proposed in [7], denoted as PAR-EDF. In the sae paper, the authors copare their test with other existing approaches that use a decoposition technique to schedule parallel tasks, and show that PAR-EDF outperfors all of the. We also show the perforance of the faster ethod (PAR-RTA-UP presented in Section 5.5 that uses the workload upper bound in Equation (15. Figure 7a shows the results for = 4. Both our approaches clearly outperfor PAR-EDF, detecting 230% ore schedulable task sets. Interestingly, the faster ethod using the siplified upper bound has a perforance very siilar to the coplete ethod (within 1% 7. Increasing the nuber of processors, the situation is siilar. Figure 7b shows the case with = 8. While the nuber of schedulable task sets detected by all tests decreases, the relative perforances reain the sae. 7. CONCLUSION Parallel task odels are becoing iportant in the realtie systes counity due to the recent shift to ulti and any-core architectures, as well as the increase in the ubiquity of parallel prograing odels and fraeworks. This paper contributes by filling the schedulability gap of synchronous parallel tasks by presenting an iproved schedulability analysis for globally scheduled fixed-priority systes. More specifically, a response-tie analysis has been proposed for work-conserving schedulers, detecting the worstcase scenarios leading to the largest possible interference. A first test based on a sliding window technique and carry-out decoposition has been proposed. Then, a siplified test has been presented with a saller coputational coplexity and coparable perforance. Both tests are shown to significantly iprove over the state of the art, in ters of nuber of schedulable task sets detected aong randoly generated workloads. Different future works are foreseen. We believe that the analysis could be refined by reducing the nuber of carry-in instances to consider, exploiting techniques presented in [22] for the sequential task odel. Also, we intend to extend the response-tie analysis fraework presented in this paper to other task odels (DAG-based, arbitrary deadlines, etc. and scheduling policies, including global EDF. 8. ACKNOWLEDGMENTS This research work was partially supported by National Funds through FCT (Portuguese Foundation for Science and Technology and by ERDF (European Regional Developent Fund through COMPETE (Operational Prograe Theatic Factors of Copetitiveness, within projects Ref. FCOMP FEDER (CISTER, ref. FCOMP FEDER (REGAIN; by the European Union, under the Seventh Fraework Prograe (FP7/ , grant agreeent n o (P-SOCRATES; also by FCT and by ESF (European Social Fund through POPH (Portuguese Huan Potential Operational Progra, under PhD grant SFRH / BD / / REFERENCES [1] K. Asanovic et al., The landscape of parallel coputing research: A view fro berkeley, EECS Departent, University of California, Berkeley, Tech. Rep. UCB/EECS , Dec [Online]. Available: TechRpts/2006/EECS htl [2] R. I. Davis and A. 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