Scheduling Parallel Real-Time Recurrent Tasks on Multicore Platforms

Transcription

1 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 Scheduling Parallel Real-Tie Recurrent Tasks on Multicore Platfors Risat Pathan, Petros Voudouris, and Per Stenströ Abstract We consider the scheduling of a real-tie application that is odeled as a collection of parallel and recurrent tasks on a ulticore platfor. Each task is a directed-acyclic graph (DAG) having a set of subtasks (i.e., nodes) with precedence constraints (i.e., directed edges) and ust coplete the execution of all its subtasks by soe specified deadline. Each task generates potentially infinite nuber of instances where the releases of consecutive instances are separated by soe iniu inter-arrival tie. Each DAG task and each subtask of that DAG task is assigned a fixed priority. A two-level preeptive global fixed-priority scheduling (GFP) policy is proposed: a task-level scheduler first deterines the highest-priority ready task and a subtask-level scheduler then selects its highest-priority subtask for execution. To our knowledge, no earlier work considers a two-level GFP scheduler to schedule recurrent DAG tasks on a ulticore platfor. We derive a schedulability test for our proposed two-level GFP scheduler. If this test is satisfied, then it is guaranteed that all the tasks will eet their deadlines under GFP. We show that our proposed test is not only theoretically better but also epirically perfors uch better than the state-of-the-art test in scheduling randoly generated parallel DAG task sets. Index Ters Real-Tie Systes, Parallel DAG Tasks, Global Fixed-priority Scheduling, Schedulability Analysis, Multicore Processors INTRODUCTION The increasing deand for ore advanced functions in todays prevailing tie-critical systes is, and will be, et using coputation power of ulticore processors. One of the ain challenges for such systes is to axiize the utilization of the parallel ulticore architecture while eeting the real-tie deadlines of the application tasks. Sequential prograing has been the priary paradig to ipleent real-tie tasks on uni- and ulticore platfors []. However, the sequential paradig does not allow intra-task parallelis: each task ust execute sequentially, which liits the extent to which processing capacity of a parallel architecture can be exploited (according to Adahl s law [2]). On the other hand, the high-perforance coputing counity has developed several parallel prograing odels task parallelis (e.g., Cilk [3]), data parallelis (e.g., OpenMp loops [4]) to better exploit the parallel ulticore architecture. Parallel prograing paradigs allow both inter- and intra-task parallelis: each classical sequential task is ipleented as a collection of parallel subtasks that can execute in parallel on ultiple cores. Many of the the parallel applications like Sort, Strassen, and FFn the Barcelona OpenMP Task Suites are widely used in any real-tie applications [5]. The work in [6] considers a 3D ultigrid solver which is ipleented as a parallel task for executing on MERASA ulticore platfor. During the last couple of years, researchers have proposed real-tie scheduling algoriths and schedulability analysis of several parallel task odels for ulticore architectures [7], [8], [9], [], [], [2], [3], [4], [5], [6]. The directed acyclic graph (DAG) based task odel is the ost general parallel task odel proposed in R. Pathan, P. Voudouris, P. Stenströ are with the Departent of Coputer Science and Engineering, Chalers University of Technology, SE-42 96, Sweden. E-ail: {risat, petrosv, pers}@chalers.se Manuscript received, ; revised. the literature. A DAG task consists of a set of nodes and directed edges where each node is a sequential subtask and each directed edge is a precedence constraint between two subtasks. Tasks on a ulticore platfor can be scheduled statically or dynaically. In static scheduling (also known as partitioned scheduling) of a parallel task, the subtasks are statically preassigned to fixed cores [7] which ay severely underutilizes hardware resources due to load ibalance or counication overheads. On the other hand, dynaic scheduling (also known as global scheduling) in which the subtasks are allowed to execute on any core can significantly iprove resource utilization [8]. However, the schedulability analysis to guarantee that all the parallel tasks eet their deadlines under dynaic scheduling paradig is ore coplex due to any possible interleavings of the threads across the cores. Therefore, overly pessiistic assuptions are ade to analyze parallel real-tie DAG tasks which does not allow to enjoy the full speedup on a ulticore architecture. This paper considers dynaic (i.e., global) scheduling of a collection of recurrent DAG tasks. Each recurrent DAG task generates potentially infinite DAG instances where two consecutive instances are separated by soe iniu inter-arrival tie (also called the period). Each DAG task and each subtask of that DAG task are assigned fixed priorities. The tasks are scheduled using a two-level preeptive global fixed-priority (GFP) scheduler. A task-level scheduler first deterines the highest-priority ready task G hp and a subtask-level scheduler then selects the highest-priority subtask of G hp for execution. If a lower-priority subtask is in execution while all the cores are busy, a newly released higherpriority subtask preepts the execution of the lower-priority subtask. The GFP scheduling allows a subtask to execute on any core of a ulticore processor even when it resues execution after being preepted by a higher-priority subtask. To apply a particular real-tie scheduling algorith in the

2 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 2 doain of real-tie safety-critical systes like autootive or aerospace, the syste designer needs to verify offline (i.e., before the syste is in ission) that all the tiing constraints are et. The ajor tool that is used to do such verification is offline analysis of the scheduling algorith to derive a schedulability test for the two-level GFP scheduler. If this test is satisfied for a given task set and a processing platfor, it is guaranteed that all the tiing constraints will also be et during actual runtie. The ain endeavor of this paper is to present the schedulability analysis of our proposed two-level GFP scheduling algorith to derive a schedulability test. One of the ajor sources of pessiis for the analysis of global GFP scheduling is the coputation of intra-task and intertask interference that a subtask under analysis suffers fro its higher-priority subtasks due to the coplex internal structures of the DAG tasks (e.g., nuber of parallel subtasks, precedence constraints, etc). No work on GFP scheduling of DAG tasks exploits the internal structures of individual DAG task to deterine both intra- and inter-task interference. Melani et al. [6] recently proposed a response-tie based schedulability analysis for GFP scheduling of recurrent DAG tasks. However, the work in [6] does not consider any particular subtask-level scheduler and hides the internal structures of the DAG tasks by ebracing pessiis in their analysis, which in turn requires a relatively larger nuber of cores to eet the deadlines of all the tasks (it will be deonstrated in our experiental Section 6). It is worth entioning at this point that the work in [6] considers a relative ore general DAG task odel (known as conditional DAG tasks) in which soe subtasks of a DAG task ay or ay not be generated depending on soe conditional (e.g., if-then-else) stateent. In other words, the DAG structures of different instances of a conditional DAG task ay be different while the sae DAG is generated for all the instances of a nonconditional DAG task. We learned that the degree of pessiis in the schedulability analysis in [6] is not due to the consideration of a ore general DAG task odel rather such pessiis equally applies for the analysis of non-conditional DAG tasks. To that end we consider non-conditional DAG tasks in this paper and show the effectiveness of our analysis in significantly reducing the pessiis of the analysis in [6] even for such non-conditional DAG tasks. Extending our analysis for conditional DAG tasks is an interesting future work. There are any works in non-real-tie systes doains that consider assigning priorities to the subtasks with the ain ai of reducing the average copletion tie [3], [9], [2], [2]. For exaple, the HEFT algorith [2] assigns higher priority to a subtask with a relatively higher upward rank. However, the HEFT algorith is designed for iproving the average-case perforance and the copletion tie of DAG ay be very large for soe not-so-frequent (i.e., worst) case and could iss the deadline of the task. This paper proposes scheduling algorith and its analysis considering the worst-case, i.e., a safe upper bound on the copletion tie of the DAG can be coputed where subtasks are assigned priorities, which is a ajor contribution of this paper. This paper has the following contributions. First, we consider a specific subtask-level GFP scheduler to schedule the subtasks to efficiently utilize the processing cores and to reduce the pessiis in schedulability analysis in coparison to earlier approaches. A. The response tie of a DAG task is also known as the akespan, which is the longest possible tie any instance of the task requires to coplete its execution. siple but very effective heuristic to assign the fixed priorities to the subtasks of each DAG task is proposed by unfolding the internal structures (i.e., topology) of the DAG tasks. Second, we propose new techniques to copute both intra- and inter-task interference. To this end, we propose a new response-tie-based schedulability test for GFP scheduling of recurrent DAG tasks. Third, we deonstrate the effectiveness of our proposed test based on epirical study. Experiental results using randoly generated DAG task sets show that our proposed test outperfors the stateof-the-art test for various (i) loads/utilizations of the application, (ii) nuber of cores, and (iii) nuber of tasks of the application. The rest of the paper is organized as follows. The syste odel and iportant definitions are presented in Section 2. The overview of our schedulability analysis fraework is presented in Section 3. The details of the schedulability analysis of the twolevel GFP scheduler is presented in Section 4. The pessiis of the state-of-the-art test is presented in Section 5 using an exaple. We present our results in Section 6. Related works are presented in Section 7 before we conclude in Section 8. 2 SYSTEM MODEL AND USEFUL DEFINITIONS We consider the preeptive GFP scheduling of n recurrent DAG tasks in set Γ={G,...G n } on a ulticore processor having cores. The (noralized) speed of each core is. Each DAG task G k Γ generates potentially infinite nuber of DAG instances, called the jobs of the task, such that the releases of two consecutive jobs of G k is separated by a iniu distance. Each DAG task G k is characterized using four paraeters G k =(T k,d k,v k,e k ) where T k is the iniu inter-arrival tie of consecutive jobs (also, called the period) of task G k ; D k is the relative deadline such that D k T k ; V k = {v k,,...v k,nk } is a set of n k subtasks 2 ; and E k (V k V k ) is a set of directed arcs (or edges). The paraeter D k specifies the real-tie constraint: ifthe source (i.e., the first) subtask of soe job of task G k becoes ready for execution at tie t, then the execution of all the subtasks of that job ust coplete by tie (t + D k ). Each subtask v k,j V k represents a sequential chunk of execution having worst-case execution tie (WCET) equal to C k,j.if(v k,p,v k,q ) E k, then subtask v k,q can start execution after subtask v k,p copletes its execution. An exaple of a DAG task G k is shown in Figure where T k = and D k =52. Let the task G k in Figure first arrives at tie x. In other words, the first job or instance of this DAG task G k becoes ready for execution at tie x. This job has a deadline (x+d k )= (x + 52) by which all its subtasks ust finish their execution. The second job of task G k is released no earlier than tie (x + T k )= (x + ) since the iniu inter-arrival tie is T k =. Although the second job of the task G k is released no earlier than tie (x + ), its actual release tie ay be later than (x + ) because we are considering sporadic tasks. Let the second job is released at tie (x + 2). Then the deadline of the second job is D k =52tie units after its release tie, i.e., at tie (x + 2) + 52 = (x + 72). The third job of this task is released no earlier than tie (x + 22), and so on. Note that the deadline is specified for the entire job of the task, i.e., the execution of all the subtasks ust be copleted before that job s deadline. 2. There are works where the ters workflow and task are respectively used to ean task and subtask as are used in this paper.

3 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 3 Figure : A DAG task G k with six subtasks. The WCET of each subtask is shown inside each shape. The priority ordering is v k, (highest) v k,3 v k,2 v k,5 v k,4 v k,6 (lowest). A subtask with no incoing (res. outgoing) edge is called a source (resp. sink) subtask. Without loss of generality we assue that there is exactly one source (denoted as vk src ) and one sink (denoted as vk sink )ofg k. If there is ore than one source (sink) subtask, a duy subtask with WCET equal to zero as a new source (sink) subtask is added with arcs to (fro) all actual source (sink) subtasks. We define a path in the DAG of task G k originating at subtask v k,a as a sequence of subtasks (v k,a,...v k,b ) such that (v k,j,v k,j+ ) E k, a j < b. The length of a path is the su of the WCETs of the subtasks in the path. The longest path of G k is denoted by L k which is the length of the longest path in G k. For exaple, L k =46, which corresponds to the path (v k,,v k,3,v k,4,v k,6 ) in Figure. We denote by W k the total work of task G k which is equal to the su of WCETs of all the subtasks of V k, i.e., W k = n k j= C k,j. For exaple, W k =64in Figure. For each subtask v k,j V k, the ancestors of v k,j, denoted by Ancst j k, is the set of subtasks v k,p V k such that there exists a path fro v k,p to v k,j. Subtask v k,j V k becoes ready for execution when each of the subtasks in Ancst j k copletes execution (i.e., all dependencies are released). The values of L k, W k, and the set Ancst j k can be coputed in polynoial tie in the representation of G k [22]. Task-level Priority Assignent. The fixed priorities of the n tasks {G,G 2,...G n } are assigned based on Deadline- Monotonic (DM) priority ordering (i.e., a task having shorter relative deadline has higher fixed priority). The set of tasks having higher fixed priorities than that of task G k is denoted by hpt k. Subtask-level Priority Assignent. The fixed priorities to the subtasks of G k are assigned based on a topological order of the subtasks of G k. A topological order is such that if there is an edge fro subtask v k,j to subtask v k,g, then subtask v k,j appears before subtask v k,g in the topological order. A topological order can be coputed in linear tie in the size of the DAG of G k [22]. A subtask v k,j V k is assigned a higher fixed priority than subtask v k,g V k if and only if v k,j appears before subtask v k,g in the topological order of G k. This subtask-level priority assignent policy exploits the internal structure of the DAG task G k based on its topology. The topological sort of DAG G k in Figure is given as follows (u w iplies u has higher priority than w): v k, v k,3 v k,2 v k,5 v k,4 v k,6 During the subtask-level priority assignent, a subtask at a lower level of the DAG is given higher priority in coparison to the one at a higher level. If two subtasks have the sae level, then the subtask having relatively higher (subtask) index is given higher priority. Given a subtask v k,j V k of particular task G k, the set of subtasks in V k having higher fixed priorities than that of v k,j is denoted by hpst j k. The two-level GFP Scheduler. The subtasks of the DAGs are executed based on a two-level scheduler. The task-level scheduler deterines the highest-priority ready task G hp and the subtasklevel scheduler selects the highest-priority subtask of G hp for execution. A newly released relatively higher-priority subtask is allowed to preept the execution of a lower-priority subtask when all the cores are busy. Under the subtask-level fixed-priority assignent policy, subtask v k,j has lower fixed priority than all the subtasks in set Ancst j k. All the ancestors of v k,j are also in the set of higher priority subtasks of v k,j, i.e., Ancst j k hpstj k since subtasks are assigned priorities based on topological sort. For exaple, for the subtask v k,4 in Figure we have Ancst 4 k = {v k,,v k,2,v k,3 } and hpst 4 k = {v k,,v k,2,v k,3,v k,5 }. Note that Ancst 4 k hpst4 k. In this paper, we deterine the response tie (also known as akespan) of each task G k Γ. The response tie of a task G k is the largest possible tie that all the subtasks of any job of task G k requires to finish its execution relative to the tie when the source subtask of the job becoes ready for execution. The response tie of task G k is denoted as R k. We copute R k based on coputing the response tie of each subtask v k,j V k of task G k. The response tie of subtask v k,j of task G k is denoted as R j k. 3 OVERVIEW OF OUR ANALYSIS FRAMEWORK This section presents an overview of our schedulability analysis fraework that is used to derive a schedulability test. If this test is when satisfied, then it is guaranteed that all the tasks eet their deadlines (i.e., all the real-tie constraints are et). The fraework is siilar to that of in [6]. To deterine whether each task G k Γ eets its deadlines or not, the response tie R j k of each subtask v k,j V k of an arbitrary job of task G k is coputed based on worst-case schedulability analysis. The schedulability analysis of each subtask v k,j is perfored in an interval [rdy j k, rdyj k + t), called the proble window of length t, such that the subtask v k,j is released (i.e., becoes ready for execution) at tie rdy j k relative to the release tie of the source subtask of the arbitrary job. A subtask is said to be released when all its ancestors have copleted their execution (i.e., there

4 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 4 is no dependency). The proble window of v k,j is depicted in Figure 2. it is guaranteed that all the jobs of G k eet their deadlines. To copute R k, the response tie R j k of each subtask v k,j V k of task G k is coputed. The response tie R k is equal to the response tie of the sink subtask vk sink and is given as follows: R k = ax v k,j V k {R j k } = Rsink k () Figure 2: Proble window [rdy j k, rdyj k + t) of subtask v k,j that becoes ready for execution at tie rdy j k relative to the release tie of the source subtask. Within the proble window [rdy j k, rdyj k + t), the execution of subtask v i,j ay be interfered by the higher priority subtasks in set hpst j k, and also by the subtasks of the higher-priority tasks in set hpt k. The response tie R j k of v k,j is derived by coputing the total interfering workload and interference. Before coputing these ters, their definitions are forally presented. Total Interfering Workload. The intra-task interfering workload of a higher priority subtask v k,h hpst j k in the proble window of v k,j is the cuulative length of intervals during which v k,h executes in that window. Siilarly, the inter-task interfering workload of a higher-priority task G i hpt k in the proble window of v k,j is the cuulative length of intervals during which the subtasks of the jobs of task G i execute in that window. Since coputing the exact interfering workload considering all possible release ties of the tasks is coputationally infeasible, we will deterine an upper bound on the intra- and inter-task interfering workloads. The total interfering workload is the su of intra- and inter-task interfering workloads. Interference. The interference on subtask v k,j within its proble window is the cuulative length of the intervals during which subtask v k,j is ready but not executing. Under GFP scheduling, the execution of subtask v k,j in an interval [a, b] is interfered only if all the processing cores are siultaneously busy executing the higher-priority subtasks (i.e., subtasks in hpst j k and/or subtasks of the tasks in hpt k )in[a, b]. The interference is calculated based on total interfering workload as follows. A note on the relationship between total interfering workload and interference. Let I k,j (t) is the total interfering workload due to the execution of the higher-priority subtasks in hpst j k and the subtasks of the higher priority tasks in hpt k within the proble window of length t of subtask v k,j. By considering the worst-case in which the total interfering workload I k,j (t) in the proble window keeps all the cores siultaneously busy (i.e., no core is idle), the axiu interference that the subtask v k,j suffers in its proble window of length t is I k,j (t)/. The difference between the length of the proble window and interference in the proble window (i.e., t I k,j (t)/) is the cuulative length of intervals in the proble window during which there is at least one core on which subtask v k,j can execute. If the su of interference and the WCET of subtask v k,j is not larger than the length of the proble window, then its is guaranteed that the subtask v k,j can coplete its execution in the proble window. 4 RESPONSE TIME COMPUTATION This section presents the schedulability analysis of a DAG task G k Γ to deterine its response tie R k.ifr k D k, then 4. Coputing R j k for subtask v k,j The response tie of the higher-priority tasks in hpt k is coputed before coputing the response tie of (the lower-priority) task G k. We also copute the response ties of the subtasks of G k in their decreasing priority order, i.e., the response tie of the source subtask (i.e., the highest priority subtask) is coputed first and the response tie of the sink subtask (i.e., the lowest priority subtask) of G k is coputed the last. Therefore, the response tie of v k,j is coputed only after the response ties of all its ancestor 3 subtasks in Ancst j k are coputed. To deterine the response tie R j k of subtask v k,j, we follow the following four steps: Step (Find the Ready Tie): We first deterine the latest tie when subtask v k,j becoes ready (i.e, value of rdy j k ) for execution relative to the release tie of the source subtask v k,src of any (arbitrary) job of G k. By considering that the subtask v k,j ust have a response tie not saller than t, the schedulability analysis of v k,j to copute R j k is perfored within its proble window [rdy j k, rdyj k + t) of length t. If subtask v k,j does not coplete within [rdy j k, rdyj k +t), then the length t of the proble window is increased until subtask v k,j is guaranteed to coplete within [rdy j k, rdyj k + t) or the deadline is issed (this is a well-known approach to copute response tie, for exaple, in [6]). Step 2 (Find the Intra-Task Interfering Workload). We deterine the axiu interfering workload of the higherpriority subtasks in set hpst j k on subtask v k,j in the proble window [rdy j k, rdyj k + t). Step 3 (Find the Inter-Task Interfering Workload). We deterine the axiu interfering workload of the subtasks of the jobs of the higher-priority tasks in set hpt k on subtask v k,j in the proble window [rdy j k, rdyj k + t). Step 4 (Find Interference and the Response Tie). Based on the intra- and inter-task interfering workloads, we deterine the axiu interference that subtask v i,j suffers in its proble window. Finally, based on the interference we deterine the response tie R j k. Step (Find the Ready tie): Consider that an arbitrary job of task G k is released at tie r k, i.e., the source subtask v k,src of the job becoes ready at tie r k. In other words, the latest tie by which the source subtask v k,src of the job of G k becoes ready for execution relative to tie r k is. Therefore, we have rdy src k =. A (non-source) subtask v k,j becoes ready for execution after all the ancestor subtasks in Ancst j k coplete their execution (i.e., when all of its dependencies are released). Therefore, the latest tie when v k,j becoes ready for execution is the axiu 3. Recall fro the subtask-level priority assignent policy in Section 2 that the ancestors of v k,j have higher priority than the priority of v k,j.

5 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 5 response ties of all its ancestors. The latest ready tie rdy j k of subtask v k,j is: rdy j k = ax v k,a Ancst j k R a k (2) Since the response ties of all the ancestor subtasks of v k,j are coputed before coputing the response tie of v k,j, the value of rdy j k in Eq. (2) can be coputed based on the (already available) response ties of all its ancestors. Step 2 (Find the Intra-Task Interfering Workload). In this step, we deterine the axiu interfering workload of the higher-priority subtasks in set hpst j k. None of the ancestors of v k,j can interfere the execution of v k,j within the proble window [rdy j k, rdyj k +t) since all the ancestors of v k,j ust have copleted by tie rdy j k (according to Eq. (2)). In other words, the higher-priority subtasks fro set hpst j k that ay interfere the execution of subtask v k,j are in set S j k which is given as follows: S j k = hpstj k Ancstj k (3) The intra-task interfering workload in the proble window is now coputed based on the interfering workload of each of the subtasks in S j k. Since rdyj k is the tie when subtask v k,j becoes ready for execution, a higher-priority subtask v k,h S j k executes within the proble window [rdy j k, rdyj k +t) only if v k,h has not copleted its execution by tie rdy j k. Recall that for each higherpriority subtask v k,h S j k, the response tie Rh k has already been coputed before coputing rdy j k. If Rk h > rdyj k, then the higher-priority subtask v k,h copletes its execution after tie rdy j k and contributes at ost (Rk h rdyj k ) as the intra-task interfering workload within the proble window. Otherwise, we have Rk h rdyj k which iplies that subtask v k,h has already copleted its execution by tie rdy j k and cannot contribute to the intra-task interfering workload. Therefore, subtask v k,h S j k can execute at ost ax{,rk h rdyj k } tie units inside the proble window of v k,j. However, since the WCET of subtask v k,h is C k,h, the quantity ax{,rk h rdyj k } can be upper bounded by C k,h. Consequently, the higher-priority subtask v k,h S j k can execute for at ost in{c k,h,ax{,rk h rdyj k }} tie units inside the proble window of v k,j. The total intra-task interfering workload, denoted by Wk intra, due to all the higher-priority subtasks in set S j k =(hpstj k Ancstj k ) is W intra k = v k,h S j k in{c k,h,ax{,rk h rdy j k }} (4) Step 3 (Find the Inter-Task Interfering Workload). We now deterine the axiu interfering workload of the higher-priority tasks in set hpt k within the proble window [rdy j k, rdyj k + t) of length t. Given a proble window of length t, we denote 4 by Wk,t inter the axiu inter-task interfering workload of the jobs of the higher priority tasks in hpt k.to deterine Wk,t inter, we copute the inter-task interfering workload 4. Note that the right-hand side of Eq. (4) is independent of the length of the proble window, i.e., value of t. Therefore, the notation Wk intra does not include the sybol t. On the other hand (as it will be evident shortly) the total inter-task interfering workload depends on the length of the proble window t and we include the sybol t inwk,t inter. of each individual higher-priority task G i hpt k within the proble window [rdy j k, rdyj k + t) of subtask v k,j. The execution of different jobs of task G i hpt k in the proble window [rdy j k, rdyj k + t) of subtask v k,j is divided in to three categories: carry-in job, body jobs, and carry-out job. The carry-in job is the first job of task G i that executes in [rdy j k, rdyj k + t) such that its release tie is before rdyj k and its deadline is in [rdy j k, rdyj k + t). The carry-out job is the last job of task G i that executes in [rdy j k, rdyj k + t) such that its release tie is before (rdy j k + t) and deadline is after (rdyj k + t). All other jobs of task G i executing in [rdy j k, rdyj k +t) are body jobs. To deterine the axiu inter-task interfering workload of the jobs of the higher priority task G i hpt k inside the proble window, we need to consider the worst-case release pattern of the carry-in, body, and carry-out jobs of task G i such that their contribution to the inter-task interfering workload is axiized. The worst-case release pattern refers to the scenario in which the releases and execution of the jobs of the higher priority task G i hpt k inflict axiu interference on a lower priority task inside its proble window. We consider the following worst-case release pattern that is depicted in Figure 3 where the carry-out job 5 starts its execution at tie (rdy j k + t W i/) and copletes W i aount of work by tie (rdy j k + t); all the earlier jobs arrive as late as possible (i.e., strictly periodically); and the carry-in job of G i finishes its execution at tie (rdy j k + t cin) such that (rdy j k + t cin R i ) is the release tie of the carry-in job of G i for soe t cin > where R i is the (already coputed) response tie of task G i. Siilar to the body and carry-out jobs, we ay also consider that the carry-in job executes W i aount of work in W i / tie units and copletes at tie (rdy j k + t cin), which is an upper bound on the work done by the carry-in job in the proble window. However, we will copute the workload of the carryin job ore precisely by exploring the internal structure of G i. Therefore, the work done by the carry-in job, unlike the body and carry-out job, is not depicted in Figure 3 to coplete W i aount of work in a duration equal to W i /. Before we present the atheatical expressions to copute the inter-task interfering workload of task G i hpt k in the proble window based on Figure 3, we first prove in Lea that the release pattern in Figure 3 is in fact the worst-case, i.e., it axiizes the inter-task interfering workload of task G i in the proble window [rdy j k, rdyj k + t). Lea. An upper bound on the interfering workload of task G i in a proble window of length t can be coputed based on the worst-case release pattern of the carry-in, body and carry-out jobs of G i that is given in Figure 3. Proof. We prove this lea by showing that shifting the proble window in Figure 3 either to the right or to the left cannot increase workload in the proble window. Consider shifting the window to the right where the arrival patter reains the sae and we ove [rdy j k and rdyj k + t] to the 5. Copleting W i aount of work in W i / tie units contributes to axiu work of the carry-out job in the least possible length of the proble window, which leaves axiu length for the execution of the carry-in and carry-out job. This leads us to copute the worst-case workload of G i.

6 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 6 Figure 3: Worst-case release pattern. The (larger) upward dotted arrow and (shorter) downward dotted arrow respectively represents the release tie and the deadline of a job. right. Note that when a body job becoes a carry-in job during such right shift, we consider its execution siilar to the carryin job shown in the non-shifted window in Figure 3. Shifting the window to the right by δ tie units is sae as shifting the window to the left by ( (δ od )) tie units. Therefore, we only consider left shifting the proble window. Consider shifting the window in Figure 3 to left by δ tie units where δ W i /. By shifting we ean that we keep the arrival pattern of the jobs unchanged but the proble window [rdy j k, rdyj k + t] is oved leftward. In such case, the workload is reduced by (δ ) fro right side of the window and at ost (δ ) aount of work ay be increased fro the left. Shifting the window to the left by δ tie units, where W i / < δ <, would reduce at least W i aount of work fro the right and the aount of work that can be increased fro the left is at ost W i. This is because at ost W i aount of additional work can be generated fro the left in an interval not larger than in Figure 3. Shifting the window to the left by δ tie units, where δ, produces scenario equivalent to shifting the window left by (δ od ) tie units, which has already been considered. Therefore, the workload cannot increase for any possible shift of the proble window. According to Figure 3, the carry-in and the body jobs execute in the interval [rdy j k, rdyj k + t W i/]. The length of the interval [rdy j k, rdyj k + t W i/] is given (denoted by X i (t)) as follows: X i (t) =ax{, rdy j k + t W i } (5) where rdy j k is already coputed using Eq. (2). Next we will deterine the value of t cin which is the length of the interval where the carry-in job of τ i executes in the proble window. The axiu nuber of body jobs that execute in the proble window is Xi(t). Therefore, the length of the interval inside the proble window where all the body jobs execute is ). The length of the interval inside the proble window where the carry-out job executes is W i /. ( Xi(t) According to Figure 3, the carry-in job copletes its execution at tie (rdy j k +t cin). Note that there is an interval of length ( R i ), just after the tie instant (rdy j k +t cin), during which no job of G i can be released since the releases of two consecutive jobs of G i is separated by at least the period of task G i. Therefore, the length of the interval inside the proble window where the carry-in job execute, (i.e., value of t cin ) is coputed as follows: Xi (t) t cin = t ( R i ) W i To avoid negative length of t cin in Eq. (6), we lower bound the value of t cin by zero and t cin is given as follows: { t cin = ax,t ( R i ) Xi (t) W } i Note that if t cin =, then there is no carry-in job. The only jobs of G i that execute inside the proble window are the body jobs and the carry-out job. Now we copute the inter-task interfering workload of the carry-in, body and carry-out jobs. We denote by CR i (t cin ) the axiu inter-task interfering workload of the carry-in job of G i in [rdy j k, rdyj k + t cin). The coputation of CR i (t cin ) is presented below (after Eq. (8)). The axiu total work of the body jobs in the proble window is ( Xi(t) W i ) where Xi(t) is the nuber of body jobs in the proble window. The axiu total work of the carryout job during the proble window is at ost W i. Therefore, the inter-task interfering workload of task G i hpt k is (CR i (t cin )+ Xi(t) W i + W i ). The inter-task interfering workload Wk,t inter of all the tasks in hpt k within the proble window of v k,j is (6) (7)

7 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 7 W inter k,t = G i hpt k Coputing CR i (t cin ) ( CR i (t cin )+ X i(t) W i + W i We deterine the value of CR i (t cin ) by considering the work done by the subtasks of the carry-in job of task G i hpt k in the interval [rdy j k, rdyj k + t cin). Consider a subtask v i,g V i of the carry-in job of G i. Note that R g i is the response tie of the subtask v i,g V i, which is already coputed before the response tie of v k,j is coputed (recall that the response tie of all the subtasks of the higher priority task G i are coputed before the response tie of v k,j is coputed). Subtask v i,g of task G i executes in [rdy j k, rdyj k + t cin) only if its response tie R g i is > rdyj k. Based on the siilar analysis in Step 2, the higher-priority subtask v i,g of the carry-in job of G i can execute at ost in{c i,g,ax{,r g i rdyj k }} tie units inside the proble window of v k,j. An upper bound on the total interfering workload of all the higher-priority subtasks in set V i of the carry-in job of task G i, denoted by A i,is A i = in{c i,g,ax{,r g i rdyj k }} (9) v i,g V i However, the inter-task interfering workload of the carry-in job of G i in [rdy j k, rdyj k + t cin) cannot be larger than ( t cin ) since at ost ( t cin ) aount of work can be copleted within an interval of length t cin on cores. Therefore, the value of CR i (t cin ) is given as follows: ) (8) CR i (t cin )=in{ t cin, A i } () Step 4 (Interference and the Response Tie). The total interfering workload, denoted by I k,j (t), within the proble window of length t for subtask v k,j is the su of intra-task and inter-task interfering workloads: I k,j (t) =Wk intra where the values of W intra + W inter k,t () k and Wk,t inter are coputed using Eq. (4) and Eq. (8), respectively. The total interfering workload I k,j (t) causes axiu interference on v k,j in its proble window if all the cores are siultaneously busy executing the higher-priority interfering workload I k,j (t). In other words, the execution of subtask v i,j is interfered, after it becoes ready for execution, by at ost I k,j(t) tie units within its proble window [rdy j k, rdyj k +t). The response tie of subtask v k,j is given using the following recurrence: t h+ rdy j k + I k,j(t h ) + C k,j (2) where the first ter on the right-hand side in Eq. (2) is the latest tie when subtask v k,j becoes ready for execution, the second ter represents the interference on subtask v k,j within a proble window of size t h, and finally the third ter C k,j is the WCET of subtask v k,j. The recurrence in Eq. (2) can be solved by searching iteratively the least fixed point that satisfies Eq. (2) starting with t = C k,j for the right-hand side of Eq. (2). This recursion stops when either (i) t h+ = t h (i.e., subtask v k,j copletes at or before the deadline of the task G k ) and we have R j k = th+ ; or (ii) t h+ >D k (i.e., task G k can not guaranteed to be schedulable). If t h+ >D k,wesetr j k = to specify that the subtask v k,j is not guaranteed to eet the deadline of the task G k. Given the response tie R j k of each subtask v k,j of task G k, the response tie of task G k can be be coputed based on Eq. (). Now we prove in Theore that the recurrence to copute the response tie of each task using Eq. (2) is correct. Theore. All the tasks in set Γ={G,G 2,...G n } eet their deadlines if R k D k for k =, 2,...n. Proof. If R k D k, then we also have that R j k D k for each subtask v k,j V k fro Eq. (). We prove this theore by showing that Eq. (2) correctly coputes an upper bound on the response tie of each subtask v k,j V k of task G k for k =, 2...n. Based on Eq. (), we have I k,j (t h )=Wk intra +W inter k,t and h the recurrence in Eq. (2) can be re-written as t h+ rdy j k + Wintra k + C k,j + Winter k,t h (3) The intra- and inter-task interfering workloads Wk intra and Wk,t inter in Eq. (4) and Eq. (8) are coputed based on the WCETs of the subtasks in set hpst j k and the subtasks of the tasks in set hpt k, respectively. Lea proves that the release pattern of the jobs of the higher priority task G i hpt k according to Figure 3 axiizes the inter-task interfering workload Wk,t inter. The axiu intertask interference is Wk,t inter / in a proble window of size t. The intra-task interfering workload Wk intra is coputed by considering the axiu work done by each higher-priority subtask in hpst j k that are eligible to execute after tie rdyj k, where rdy j k is the latest tie when the node v k,j becoes ready for execution. Node v k,j does not start its execution until it is ready. Apart fro the inter-task interference, to coplete C k,j aount of execution of node v k,j it ay also suffer intra-task interference in the proble window. According to Eq. (3), the execution of node v k,j ay be delayed apart fro the inter-task interference Wk,t inter / by at ost rdy j k + Wintra k tie units. Note that rdy j k is the latest tie by which node v k,j ust be ready for execution. If the ancestors of v k,j during actual execution coplete earlier than rdy j k, then the higher-priority non-ancestor subtasks in set hpst j k ay execute longer in the proble window in coparison to the case when v k,j becoes ready for execution at tie rdy j k. Therefore, the intra-task interference ay be lager than Wintra k. We will now show that such increase in intratask interference does not invalidate Eq. (3). This is because if node v k,j becoes ready for execution Δ tie units earlier than rdy j k during actual execution, then the intra-task interference within the window [rdy j k Δ, rdyj k ) during which all the processors are busy cannot be larger than Δ. And, the intra-task interference beyond tie rdy j k is at ost Wintra k /. Therefore, the axiu delay in copleting the execution of v k,j is at ost the su of (i) the ready tie (rdy j k Δ), (ii) the intra-task interference (Δ + Wk intra /), (iii) its own execution tie C k,j, and (iv) the inter-task interference W inter k,t / within a proble h

8 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 8 window of length t h. The recurrence in such case to deterine the response tie is (denoted as W inter k,t ) of all the higher-priority tasks in hpt k in a proble window of length t is given as follows [6]: t h+ (rdy j k Wintra k Δ) + Δ + + C k,j + Winter k,t h which is equivalent to the recurrence in Eq. (3), which in turn is equivalent to Eq. (2). In suary, the cuulative length of the intervals during which v k,j does not execute (either because it is not ready or suffers interference) cannot be larger than rdy j k + I k,j(t h ) where I k,j (t h ) is the total interfering workload in the proble window of v k,j of length t h in Eq. (2). Therefore, the recurrence derived in Eq. (2) correctly coputes the response tie of subtask v k,j by considering the axiu interference that the subtask v k,j in a proble window of length t h suffers. 5 COMPARISON WITH THE STATE-OF-THE-ART TEST AND AN EXAMPLE Melani et al. [6] recently proposed a response-tie analysis of recurrent DAG tasks scheduled using GFP algorith. In this section, we present their proposed schedulability test and show that our test is theoretically ore precise than the test in [6]. In Section 6, we show that our proposed test epirically perfors uch better than the test in [6] for scheduling randoly generated DAG task sets. In [6], each DAG task has a fixed priority but the subtasks of a DAG task G k are not assigned any priority: the subtask-level scheduler is assued to be an arbitrary greedy scheduler that never idles a core if there is a subtask awaiting execution in the ready queue. In contrast, we consider specific fixed priorities for the subtasks in addition to the fixed priority of each task G k Γ to deterine which subtask of the highest-priority ready DAG task to execute. In other words, we consider a particular (fixed-prioritybased) subtask-level scheduler which is also greedy but selects subtasks for execution in decreasing priority order. Siilar to the analysis proposed in this paper, the response tie analysis of each task G k Γ in [6] is also perfored in a proble window [r k,r k + t) of length t where an arbitrary job of task G k is released at tie r k. The response tie of task G k is coputed by deterining an upper bound on both intra- and intertask interference that the longest path (i.e., that has length L k )of G k suffers in [r k,r k + t). In contrast, we deterine the response tie of G k by deterining an upper bound on both intra- and inter-task interference that each subtask v k,j V k of G k suffers in [r k,r k + t). Next we present the response tie test that is proposed by Melani et al. in [6] and then we show that our proposed test is theoretically better than the state-of-the-art test. According to the work in [6], the intra-task interference on the longest path of an (arbitrary) job of G k is W k L k where W k and L k are the total work of all the nodes and the length of the longest path in G k. For coputing the inter-task interference, the worst-case release pattern that axiizes the workload in the proble window (given in Figure 4) is derived in [6] by considering that (i) the carry-in job of G i hpt k starts execution at tie r k and finishes at tie (r k +W i /) such that (r k +W i / R i ) is the release tie of the carry-in job, (ii) each of the later jobs executes W i aount of work in W i / tie units as soon as possible. An upper bound on the inter-task interfering workload W inter k,t = G i hpt k ( t + Ri W i / W i { + in W i, (t + R i W }) i ) od (4) The total inter-task interference on the longest path of G k in the proble window is ( G i hpt k W inter i,t )/ and the intra-task interference on the longest path of G k is (W k L k )/. The response-tie of task G k for GFP scheduling is given in [6] using the following recurrence: + Winter k,t h t h+ L k + W k L k (5) Siilar to the recurrence in Eq. (2), the recurrence in Eq. (5) is solved by searching the least fixed point starting with t = L k fro the right-hand side. Deterining the workload in the proble window plays one of the ost iportant roles in gauging the effectiveness of the state-of-the-art test in Eq. (5). The analysis in [6] hides the internal structures of the DAG task G i by using only two paraeters to represent this task: W i and L i. Such abstraction while being siple to analyze leads to coputing the workload in the proble window ore pessiistically, which ay lead task sets to be deeed unschedulable using the proposed test in Eq. (5) while the task set is in fact schedulable. Our proposed technique to copute the workload is less pessiistic (will be shown shortly in Theore 2) due to exploiting the precedence constraint inforation (i.e., subtask-level priority) of the tasks. It will shown in Theore 2 that our proposed response tie test in Eq. (2) doinates the test in Eq. (5) in the sense that: if a task set is schedulable using the test in Eq. (5), then that task set is also guaranteed to be schedulable using our proposed test in Eq. (2); and there are task sets (see Exaple ) for which our test guarantees schedulability but the test in Eq. (5) cannot guarantee schedulability. Theore 2. The response-tie test in Eq. (2) doinates the response-tie test in Eq. (5). Proof. The response tie tests in Eq. (2) and Eq. (5) for task G k Γ can be rewritten as the su of the following two factors F and F2: F. Response tie of task G k without considering the inter-task interference, and F2. Inter-task interference in a proble window of length t. Based on Eq. (), we re-write Eq. (2) as follows. where t h+ rdy j k + Wintra k + C k,j + Winter k,t h (6) F: rdy j k + Wintra k + C k,j is the response tie of task G k without considering the inter-task interference; and W inter k,t h F2: is the inter-task interference in a proble window of length t h. Siilarly, in Eq. (5), we have

9 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 9 Figure 4: Worst-case release pattern for workload coputation in [6]. Figure 5: Left Shift of the worst-case release pattern for workload coputation in [6]. F: the ter L k + W k L k is the response tie of task G k without considering the inter-task interference; and W inter k,t h F2: is the inter-task interference in a proble window of length t h. To prove this theore, we first show that each of the ters F and F2 is never ore pessiistic in Eq. (2) in coparison to the state-of-the-art test test in Eq. (5). Factor F. The quantity L k +(W k L k )/ in Eq. (5) is the response tie of DAG G k under any work-conserving (i.e., greedy) algorith without considering the inter-task interference. In contrast, we assign fixed priorities to the subtasks and use a fixed-priority based subtask level scheduler to schedule the tasks. Since fixed-priority scheduler is also a work-conserving (i.e., greedy) scheduler, the response tie of DAG task G k, i.e., value of rdy j k + Wintra k +C k,j, is not larger than L k +(W k L k )/. Therefore, factor F in our test is never ore pessiistic than the factor F in the state-of-the-art test. Factor F2. We will show that W inter k,t Wk,t inter for any proble window of size t, which iplies that the inter-task interfering workload coputed using our approach is less than or equal to that of coputed by Melani et al. in [6] for the sae length of the proble window. First, we show another worst-case release pattern (Figure 5) that is equivalent to the release pattern in Figure 4 considered by Melani et al [6]. The inter-task interfering workload W inter i,t in Eq. (5) is exactly equal to the inter-task interfering workload of this equivalent release pattern in Figure 5. Second, we show that the release pattern in Figure 5 coputes the inter-task interfering workload ore pessiistically in coparison to the release pattern that we consider in Figure 3. Since the release patterns in Figure 4 and Figure 5 are equivalent, we conclude that Figure 4 (the state-of-the-art) coputes the inter-task interfering workload ore pessiistically in coparison to the release pattern that we consider in Figure 3. Therefore, W inter k,t W inter k,t. Consider shifting the proble window in Figure 4 to the right such that the body job finishes its execution of W i tie units in an interval of length W i / exactly at the end of the window, i.e., at tie (r k + t). This shifted window is shown in Figure 5. It

10 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 is evident that the inter-task interfering workload in the proble window in Figure 5 is sae as that of in Figure 4. This is because the aount of work that is decreased fro the left-hand side due to the right shift is exactly equal to the aount of work that is increased fro the right-hand side of the proble window. In other words, the inter-task interfering workload for the release pattern in Figure 5 is also W inter k,t. The release pattern in Figure 5 is quite siilar to the release pattern in Figure 3 except that within the window of length [r k,r k + t cin ), the work done by the carry-in job of the higherpriority task G i is exactly ( t cin ) in Figure 5 while it ay be saller than ( t cin ) in Figure 3. In other words, while the work done by the body and the carry-out jobs of G i is sae in our release pattern (Figure 3) and the equivalent release pattern (Figure 5), our approach to copute the work of the carry-in job of a higher priority task G i is less pessiistic. This is due to the in function in Eq. (): if CR i (t cin )=( t cin ) in Eq. (), then the carry-in workload in [r k,r k +t cin ) coputed using Eq. () is sae as that of coputed in [6]. If CR i (t cin ) < ( t cin ), then the carry-in workload coputed using Eq. () in [r k,r k + t cin ) is strictly saller than that of coputed in [6]. Consequently, the proposed technique to copute the work of the carry-in job of G i in [r k,r k + t cin ) is saller than or equal to that of in Figure 5. All the body jobs and the carry-out job of the higher priority task G i execute exactly for W i tie units in both Figure 3 (our analysis) and Figure 5. Since the inter-task interfering workload in Figure 5 is W inter i,t, our proposed technique to copute the carry-in workload in [r k,r k + t) is saller than or equal to that of proposed in [6], i.e., W inter i,t Wi,t inter for any proble window of length t. Consequently, the inter-task interference using our approach is saller than or equal to that of coputed in [6]. Since each of the factors F and F2 in our approach is never worse than the state-of-the-art in [6], the coputed response tie in Eq. () is never larger than the response tie of the state-of-theart in [6]. Exaple shows that the response tie coputation in Eq. () is strictly saller than that of coputed using Eq. (5) for an exaple DAG task in Figure 6, which show the doinance of the proposed test over the state-of-the-art test. Next we show an exaple of a single DAG task where the response tie coputed by Eq. (5) is larger than that of coputed using our approach in Eq. (2). When applied to a single DAG task G k (i.e., there is no inter-task interference), the response tie of G k based on Eq. (5) is R k = L k + W k L k (7) The response tie of each subtask v k,j of G k based on Eq. (2) is given as follows: R j k = rdyj k + Wintra k + C k,j (8) Exaple. Consider GFP scheduling of a single DAG task G k in Figure 6 on =2cores where D k =52and T k =. The length of the longest path is L k =46and total work of G k is W k =64. The response tie using Eq. (7) is L k +(W k L k )/ =46+(64 46)/2 =55. Since D k =52< 55, the state-of-the-art test cannot guarantee schedulability of G k. Now we copute the response tie of G k using Eq. (8). The fixed priorities of the subtasks of G k are v k, v k,3 v k,2 v k,5 v k,4 v k,6. The response tie of each subtask based on Figure 6: An exaple DAG task. Eq. (8) is given in the last colun in Table. The sink subtask v k,6 has response tie Rk 6 = 5.5. Therefore, R k = Rk 6 = 5.5 based on Eq. (). Since D k =52 R k =5.5, we can guarantee that the task G k eets its deadline using the proposed test. 6 EMPIRICAL STUDY In this section, we copare the perforance of our proposed schedulability test with the state-of-the-art schedulability test using randoly-generated parallel DAG task sets. The tasks are assigned fixed priorities based on deadline-onotonic (DM) priority ordering. We denote our proposed schedulability test in Eq. (2) by and the state-of-the-art schedulability tests in Eq. (5) by. Note that unlike test, test considers subtask-level fixed priority (i.e., topological sort in Section 2). The utilization u k of a task G k is the ratio between its total work and the period, i.e., u k = W k /T k. The utilization u k of the DAG in Figure 6 is W k /T k =64/ = 4. For a task set Γ, its total utilization is defined as U = G k Γ u k. The utilization U of a task set specifies its coputation load. Before presenting the experiental results, the task sets generation algorith is presented. 6. Task Sets Generation Algorith We use a siilar DAG task set generation algorith that is used in [6] which generates a series of parallel graphs by recursively expanding the non-terinal vertices for a given recursion depth. The purpose of such expansion is to generate either a new terinal vertices (i.e., sink subtasks) or a new parallel subgraphs. The axiu recursion depth for all our experients is set to 2. The probabilities to generate a terinal vertex and a parallel subgraph by expanding a non-terinal vertex are p ter and ( p ter ), respectively. The nuber of branches of a parallel subgraph is uniforly selected fro [2,n par ] where n par is the axiu branches (degree) for any parallel subgraph. Rando edges are added between pairs of subtasks with probability p add in a way so that there is no cycle in the graph. Once the subtasks and the edges of a DAG task G k are generated, the paraeters of each DAG task G k are generated as follows: the WCET C k,i of a subtask v k,i V k of task G k is uniforly selected in the range [, ]; the length of the longest path L k and the workload W k are coputed; the period T k (i.e., the iniu inter-arrival tie) is uniforly selected in the interval [L k,w k /β), where β is used to control the iniu utilization of task G k. The utilization u k of G k is in [β,w k /L k ]; finally,

11 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 subtask C k,j Ancst j k rdy j k = ax R v k,a Ancst j k a S j k = hpstj k Ancstj k Wk intra k R j k = rdyj k + Wintra k v k, 4 4 v k,3 2 v k, =24 v k,2 2 v k, 4 {v k,,v k,3 } {v k, } = {v k,3 } =26 v k,5 6 v k,,v k,3 24 {v k,2 } 26-24= =3 v k,4 4 v k,,v k,2,v k,3 26 {v k,5 } 3-26= =42.5 v k,6 8 v k,,...,v k, =5.5 Table : Coputing response tie of the DAG G k in Figure 6 where the priorities of the subtasks are v k, v k,3 v k,2 v k,5 v k,4 v k,6. Fro Eq. (4), we have W intra k = v k,h S j k in{c k,h,ax{,r h k rdyj k }}. +C k,j Total Utilization (U) Total Utilization (U) Total Utilization (U) (a) =4 (b) =8 (c) =6 Figure 7: Acceptance ratio of and tests for =4, 8, 6. the relative deadline D k of G k is uniforly selected fro the range [L k,t k ]. 6.2 Experiental Results We set p ter =.5, n par =5, p add =.and β =.for our experients. For a target total utilization U of a task set, new DAG tasks are repeatedly added to the task set until the total utilization of the task set is equal to U. In order to generate a task set with total utilization exactly equal to U, the period of the last task is adjusted so that the total utilization of the task set is exactly equal to U. A total of 5 task sets are generated at each of the utilization levels U [5,.5,...] where is the nuber of cores. Each of the 5 task sets that are generated at that particular utilization level U [5,.5,...] has total utilization U. For a specific schedulability test, and, U values, we consider the etric called acceptance ratio that denotes the fraction of task sets out of 5 DAG task sets that are guaranteed to be schedulable by the schedulability test at that utilization level on cores. We ipleented our experients in MATLAB and coputed the acceptance ratios of both tests for various values of U and. Figure 7 presents the acceptance ratios of test and test for = 4, 8, 6 cores. The x-axis in each plot in Figure 7 represents the total utilization U of each task set and the y-axis is the acceptance ratio. Our proposed Our- DM schedulability test outperfors the state-of-the-art MBBMB- DM test. For exaple, the acceptance ratios in Figure 7a at U = 2. for = 4 of test and test are respectively around 5% and 7%. In other words, around 2% ore task sets are deeed to be schedulable using test in coparison to MBBMB test at U =2.and =4. The difference in acceptance ratios between and MBBMB- DM tests increases with the increase in total utilization of the task sets for all = 4, 8, 6 in Figure 7a 7c. In Figure 7b for = 8, around 2% and 3% ore task sets are deeed to be schedulable respectively at U = 3. and U = 5. using test in coparison to MBBMB test. It is generally ore difficult to schedule task sets with relative higher total utilization or higher load. test is very effective in coparison to test in guaranteeing schedulability of relatively high-utilization task sets. Unlike the response tie coputation using test, the schedulability analysis used to derive test uses the internal structure of each DAG task to deterine which subtasks ay execute in parallel with the execution of the subtasks in the longest path of the DAG. Consequently, the response tie coputation using test is ore precise than that of MBBMM-DM test. Variation with nuber of cores (). For this set of experients, we vary the nuber of cores while keeping the total utilization U of each task set fixed. Figure 8 shows the acceptance ratio of test and test for various values of using three different (fixed) total utilizations: U =2, U =4, and U = 8. When the nuber of cores becoes larger, the acceptance ratio also increases in all the three plots in Figure 8a 8c. This is expected because a relative higher nuber of cores has higher likelihood of eeting all the deadline of the task sets for both tests. However, for any resource-constrained systes where the size, weight, and power are liited and costly, we cannot afford a uch larger nuber of cores. In such case (i.e., when the nuber of cores is sall), test has uch larger acceptance ratio than that of the test. For exaple in Figure 8a where U =2., test has 8% acceptance ratio when 8 while the test reaches 8% acceptance when 2 cores. In Figure 8c where U =8., test has larger than 8% acceptance ratio when 64 while the

12 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV Nuber of Cores () Nuber of Processors () Nuber of Cores () (a) U =2 (b) U =4 (c) U =8 Figure 8: Acceptance ratios of and tests for varying nuber of cores. test has 8% acceptance when cores. These results show the effectiveness of our proposed schedulability test in reducing the deand for hardware resources (particularly, the nuber of cores), which in turn reduces the cost of developing the systes for ass production. Variation with nuber of tasks (n). We vary the nuber of tasks while keeping the total utilization (U) of each task set and nuber of cores () fixed. Since the nuber of tasks (n) of each task set is fixed for a fixed value of U, the total utilization of n tasks in each randoly-generated task set ust be equal to the target utilization U. The UUnifast algorith, proposed by Bini and Buttazzo [23], with paraeters n and U is used to generate the utilization values of the n tasks in a task set such that the total utilization of the task set is exactly U. Figure 9 shows the acceptance ratios for various values of n for two given configurations of (U, ): Figure 9a 9b shows results for (U = 2, =4)and (U =4, =8), respectively. When the nuber of tasks (n) increases while the total utilization U is unchanged, the nuber of low-utilization tasks in a task set increases because the total utilization U is distributed across a relatively larger nuber of tasks of the task set. A lowutilization task uses less coputing resource and provides ore opportunity for other tasks to execute on the cores (i.e., easier binpacking). Consequently, when the nuber of tasks in the rando tasks sets is relatively larger (e.g., n 6 in Figure 9a and n 4 in in Figure 9b), then the acceptance ratios of both tests are %. When the nuber of tasks is relatively sall (i.e., there are ore high-utilization tasks in a task set), then test has higher acceptance ratio than that of the test. For exaple, when U =4and =8in Figure 9b, the acceptance ratio for n =5 of test is 65% while it is 4% for test. Therefore, our test is ore effective in scheduling task sets having a relatively larger nuber of highutilization tasks in coparison to task sets with higher nuber of low-utilization tasks for the sae total utilization U. 6.3 Other tests: Iplicit Deadlines Soe recent works [24], [25] have proposed schedulability tests for DAG tasks where the relative deadline D i is equal to the period for each task G i (known as iplicit-deadline task sets) by considering scheduling policies, like federated scheduling, decoposition-based scheduling. This subsection presents the acceptance ratios of the following four schedulability tests (in addition to and ) for iplicit deadline tasks: MBBMB-EDF: This test in [6] considers global EDF scheduling of DAG tasks. DECOM-EDF: This test in [25] considers decoposing each DAG task by assigning artificial release tie and artificial deadline to each subtask where such subtasks are scheduled using global EDF policy. FED-LI: This test in [24] considers federated scheduling where a DAG task with high utilization is assigned a dedicated nuber of processors while all the light-utilization tasks are executed sequentially and scheduled on a shared set of processors. SIM:For each set of DAG tasks, their execution is siulated using our proposed two-level GFP scheduler where the release tie of each DAG task is zero and the jobs of each task are released strictly periodically. The siulation of the schedule is run for 6 tie units. The acceptance ratio of SIM is an upper bound on the acceptance ratio the rando task sets that are actually schedulable using the proposed scheduler. The acceptance ratio of SIM depicts the tightness of our proposed schedulability test MBBMB- DM with respect to a hypothetical exact test. The acceptance ratios for SIM,,, FED- LI, DECOM-EDF and MBBMB-EDF tests are shown in Figure for = 4 and = 8 processors. It can be observed that our proposed perfors better than all other tests. More iportantly, the difference between the plot of is not very far fro the plot for SIM. This signifies that the preciseness of test is close to the (hypothetical) exact test for our proposed two-level GFP scheduling policy. 7 RELATED WORKS Many of the earlier works on parallel task odels proposed resource-augentation bounds and schedulability tests for various scheduling algoriths. The resource augentation bound B of a scheduling algorith A indicates that if there is a way to schedule a task set on identical unit-speed cores, then algorith A is guaranteed to successfully schedule the sae task set on cores with each core being B ties as fast as the original. However, resource-augentation bound cannot be used as a schedulability test since it is derived based on a hypothetical optial scheduler. The works on real-tie scheduling of parallel tasks on ulticores can be categorized in three groups considering the task odel that each group considers: (i) fork-join odel [7], (ii)

13 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV Nuber of tasks in a task set (n) Nuber of tasks in a task set (n) (a) U =2and =4 (b) U =4and =8 Figure 9: Acceptance ratios by varying n. SIM MBBMB-EDF DECOM-EDF FED-LI SIM MBBMB-EDF DECOM-EDF FED-LI Total Utilization (U) Total Utilization (U) Figure : Acceptance ratios by varying U for iplicit-deadline tasks for =4(left-hand side graph) and for =8(right-hand side graph). synchronous parallel task odel [8], [9], [], (iii) the dag task odel [2], [3], [4], [5], [6]. Each task in fork-join odel has alternating sequential and parallel segents [7]. The fork-join task odel is generalized in synchronous parallel task odel by allowing ultiple parallel segents to execute without a sequential segent in between [8], [9], []. Different resource augentation bounds and schedulability tests are proposed for both fork-join and synchronous parallel task odels [7], [8], [9], []. The third group of works considers dag task odel which is ore general than the synchronous task odel [2], [3], [4], [5], [6]. The work in [2] proposed both a polynoial and a pseudo-polynoial schedulability test for scheduling a single sporadic (i.e., recurrent) dag task using global dynaic-prioritybased Earliest-Deadline-First (GEDF) scheduling. The works in [3], [4] derived resource augentation bound of (2 /) for GEDF scheduling of ultiple sporadic dag tasks, where is the nuber of unit-speed cores. The work in [3] also derived a resource augentation bound of (3 /) for global fixed-priority-based deadline-onotonic scheduling. The work by Melani et al. [6] derived a schedulability test for GFP scheduling. The analysis in [6] is the state-of-the-art result for response-tie coputation of ultiple recurrent DAG tasks. We have shown that the perforance of our proposed test is uch better than the test in [6]. In addition to global scheduling, researchers have analyzed at least two other echaniss to schedule parallel DAG tasks: Federated scheduling [24] and Decoposition-based scheduling [25]. The ain idea of federated scheduling is the following: each DAG task with utilization larger than (called, high-utilization task) is assigned a dedicated nuber of processors while each of the other tasks with utilization no larger than (called, low-utilization task) executes sequentially on the reaining processors. It is shown in [24] that federated scheduling has a capacity augentation bound of 2 for the case when the nuber of processor is large. In decoposition based scheduling, each DAG task is transferred into a set of independent sporadic task by inserting artificial release tie and artificial deadline for each of the subtasks of the DAG task. The artificial release ties and deadlines to each subtask are assigned such that the precedence constraints of the original DAG task are satisfied. The decoposed subtasks of all the DAG tasks are scheduled based on GEDF scheduling policy in [25]. We have shown that our proposed test epirically perfors better than both the federated scheduling [24] and the decoposition-based scheduling in [25]. There are any works on scheduling parallel tasks on ultiprocessors without concerning real-tie requireents [3], [2], [2]. The Heterogeneous Earliest Finish Tie (HEFT) scheduler in [2] prioritizes the subtasks of the DAG based on the structure of the DAG. The HEFT algorith sorts the tasks based on the decreasing order of their upward rank. The task with the highest upward rank is dispatched to the processor that the task will have the sallest execution tie. Siilarly, the Critical-Path-ona-Processor (CPOP) scheduler, also proposed in [2], in addition to the upward rank takes into account also the downward rank and the tasks are sorted based on this two characteristics of the DAG.

14 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL., NO., NOV 27 4 For both HEFT and CPOP schedulers, two subtasks can have the sae rank and ties are broken randoly. Such rando tie breaking ay lead the subtasks to execute differently in different executions. Moreover, if soe subtask takes less than its WCET, then the DAG task can generate a relatively longer schedule than the schedule that is generated when each subtask takes its WCET, which is known as execution-tie based tiing anoalies [9]. There is no built-in strategy in such algorith to avoid tiing anoaly and therefore cannot be applied to real-tie systes. In short, scheduling algoriths designed for iproving the average case perforance, like the HEFT or CPOP algoriths, cannot be directly applied to guarantee real-tie constraints. 8 CONCLUSION This paper presents a schedulability analysis of recurrent DAG tasks considering GFP scheduling at both subtask and task levels. A siple ethod to assign fixed priorities to the subtasks of a DAG task is proposed based on the structure (i.e., topology) of each DAG task. To the best of our knowledge, assigning fixed priorities to the subtasks of a DAG task by exploring the internal structure of each DAG task has not been addressed in any earlier work. Based on the priorities of the subtasks, a new technique to copute the response tie of each subtasks is presented. Siulation results using randoly generated tasks show that our proposed test perfors better than the state-of-the-art test, particularly, for task sets with relatively larger utilization and also for task set with relatively higher nuber of high-utilization tasks. Finding a ore effective priority assignent for the subtasks of each task is left as a future work. ACKNOWLEDGMENTS This research has been funded by the MECCA project under the ERC grant ERC-23-AdG MECCA. REFERENCES [] G. Buttazzo, Hard Real-Tie Coputing Systes: Predictable Scheduling Algoriths and Applications. Third Edition, Springer. Third Edition, Springer, 2. [2] G. M. Adahl, Validity of the single processor approach to achieving large scale coputing capabilities, in Proc. of AFIPS, 967. [3] R. D. Bluofe and C. E. Leiserson, Scheduling ultithreaded coputations by work stealing, Journal of the ACM, vol. 46, no. 5, pp , 999. [4] Openp application progra interface. version 4., jul 23. [5] A. Duran, X. Teruel, R. Ferrer, X. Martorell, and E. Ayguade, Barcelona openp tasks suite: A set of bencharks targeting the exploitation of task parallelis in openp, in Parallel Processing, 29. ICPP 9. International Conference on. IEEE, 29, pp [6] C. Rochange, A. Bonenfant, P. Sainrat, M. Gerdes, J. Wolf, T. Ungerer, Z. Petrov, and F. Mikulu, Wcet analysis of a parallel 3d ultigrid solver executed on the erasa ulti-core, in OASIcs-OpenAccess Series in Inforatics, vol. 5. Schloss Dagstuhl-Leibniz-Zentru fuer Inforatik, 2. [7] K. Lakshanan, S. Kato, and R. Rajkuar, Scheduling parallel realtie tasks on ulti-core processors, in Proc. of RTSS, 2. [8] A. Saifullah, K. Agrawal, C. Lu, and C. Gill, Multi-core real-tie scheduling for generalized parallel task odels, in Proc. of RTSS, 2. [9] G. Nelissen, V. Berten, J. Goossens, and D. Milojevic, Techniques optiizing the nuber of processors to schedule ulti-threaded tasks, in Proc. of ECRTS, July 22, pp [] H. S. Chwa, J. Lee, K.-M. Phan, A. Easwaran, and I. Shin, Global edf schedulability analysis for synchronous parallel tasks on ulticore platfors, in Proc. of ECRTS, 23. [] C. Maia, M. Bertogna, L. Nogueira, and L. M. Pinho, Response-tie analysis of synchronous parallel tasks in ultiprocessor systes, in Proc. of RTNS, 24. [2] S. Baruah, V. Bonifaci, A. Marchetti-Spaccaela, L. Stougie, and A. Wiese, A generalized parallel task odel for recurrent real-tie processes, in Proc. of RTSS, 22. [3] V. Bonifaci, A. Marchetti-Spaccaela, S. Stiller, and A. Wiese, Feasibility analysis in the sporadic dag task odel, in Proc. of ECRTS, 23. [4] J. Li, K. Agrawal, C. Lu, and C. Gill, Analysis of global edf for parallel tasks, in Proc. of ECRTS, 23. [5] S. Baruah, Iproved ultiprocessor global schedulability analysis of sporadic dag task systes, in Proc. of ECRTS, 24. [6] A. Melani, M. Bertogna, V. Bonifaci, A. Marchetti-Spaccaela, and G. Buttazzo, Response-tie analysis of conditional dag tasks in ultiprocessor systes, in Proc. of ECRTS, 25. [7] B. Fechner, U. Honig, J. Keller, and W. Schiffann, Fault-tolerant static scheduling for grids, in Proc. of IPDPS, 28. [8] E. Agullo, O. Beauont, L. Eyraud-Dubois, and S. Kuar, Are static schedules so bad? a case study on cholesky factorization, in Proc. of IPDPS, 26. [9] R. L. Graha, Bounds on ultiprocessing tiing anoalies, SIAM journal on Applied Matheatics, vol. 7, no. 2, pp , 969. [2] Multi-objective energy-efficient workflow scheduling using list-based heuristics, Future Generation Coputer Systes, vol. 36, pp , 24. [2] H. Topcuoglu, S. Hariri, and M.-y. Wu, Perforance-effective and low-coplexity task scheduling for heterogeneous coputing, IEEE transactions on parallel and distributed systes, 22. [22] T. H. Coren, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algoriths. MIT Press, 2. [23] E. Bini and G. C. Buttazzo, Measuring the Perforance of Schedulability Tests, Real-Tie Systes, vol. 3, pp , 25. [24] J. Li, J. J. Chen, K. Agrawal, C. Lu, C. Gill, and A. Saifullah, Analysis of federated and global scheduling for parallel real-tie tasks, in Proc. of ECRTS, 24. [25] X. Jiang, X. Long, N. Guan, and H. Wan, On the decoposition-based global edf scheduling of parallel real-tie tasks, in Proc. of RTSS, 26. Risat Pathan is an assistant professor in the Departent of Coputer Science and Engineering at Chalers University of Technology, Sweden. He received the M.S., Lic.-Tech., and Ph.D. degrees fro Chalers University of Technology in 26, 2, and 22, respectively. He visited the Real-Tie Systes Group at The University He visited the Real-Tie Systes Group at The University of North Carolina at Chapel Hill, USA during fall 2. His ain research interests are real-tie scheduling on uni- and ulti-core processors fro efficient resource utilization, fault-tolerance and ixedcriticality perspectives. Risat is a eber of the IEEE and the ACM. Petros Voudouris is a PhD student in the Departent of Coputer Science and Engineering at Chalers University of Technology, Sweden. Petros received his B.Sc degree in Coputer Science fro University of Crete, Heraklion, Greece in 2. He received his M.Sc. degree in Ebedded Systes fro departent of Coputer Science and Matheatics, Technical University of Eindhoven, Netherlands, in 24. His ain research interests are the design and analysis of tie-predictable scheduling algorith and worst-case execution tie analysis for parallel coputer systes. Petros is a student eber of IEEE. Per Stenströ is professor at Chalers University of Technology. His research interests are in parallel coputer architecture. He has authored or co-authored four textbooks, ore than 5 publications and ten patents in this area. He has been progra chairan of several top-tier IEEE and ACM conferences including IEEE/ACM Syposiu on Coputer Architecture and acts as Senior Associate Editor of ACM TACO and Associate Editor-in-Chief of JPDC. He is a Fellow of the ACM and the IEEE and a eber of Acadeia Europaea, the Royal Swedish Acadey of Engineering Sciences and the Royal Spanish Acadey of Engineering Science.