We first embark on analyzing **non**-idling **scheduling**. The **optimality** of the **non**- idling **non**-**preemptive** Earliest Deadline First **scheduling** policy is **revisited**. Then, we provide feasibility conditions in the presence of aperiodic or periodic traffic. Second, we examine the concept of idling **scheduling**, whereby a processor can remain idle in the presence of pending tasks. The **non**-idling **non**-**preemptive** Earli- est Deadline First **scheduling** policy is not optimal since it is possible to find feasi- ble task sets for which this policy fails to produce a valid schedule. An optimal algorithm to find a valid schedule (if any) is presented **and** its complexity analyzed. This paper shows that **preemptive** **and** **non**-**preemptive** **scheduling** are closely related. However, **non**-**preemptive** **scheduling** leads to more complex problems when combined with idling **scheduling**.

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Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des V[r]

[12] George, L., Rivierre, N., Spuri, M., “**Preemptive** **and** **Non**- **Preemptive** **Real**-**Time** UniProcessor **Scheduling**”, INRIA Research Report, No. 2966, September 1996.
[13] George, L., Muhlethaler, P., Rivierre, N., “**Optimality** **and** **Non**-**Preemptive** **Real**-**Time** **Scheduling** **Revisited**,” Rapport de Recherche RR-2516, INRIA, Le Chesnay Cedex, France, 1995. [14] Howell, R.R., Venkatrao, M.K., “On **non**-**preemptive** **scheduling** of recurring tasks using inserted idle **time**”, Information **and** computation Journal, Vol. 117, Number 1, Feb. 15, 1995. [15] K. Jeffay, D. F. Stanat, C. U. Martel, “On **Non**-**Preemptive** **Scheduling** of Periodic **and** Sporadic Tasks”, In Proc. RTSS, pages 129-139, 1991.

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We present simulation results of preemptive GDM scheduling algorithm and we compare the performance of DAG-Str algorithm and the Direct Scheduling approach at DAG-Level, by varying the n[r]

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To that end, our purpose is first to provide a general framework based, on the one hand, a representation of **preemptive**, **real**-**time** **scheduling** in an algebraic structure that enables us to evaluate the distance of the **optimality** of any **scheduling** algorithm ; **and** on the other hand, a consistent representation of the associated feasibility conditions that enables us to evaluate the number of basic operations. As a second step, considering several kinds of traf- fics, we initiate the comparison by a straight, but limited, application of our general frame- work. Our preliminary results will notably highlight, in the cases where deadlines are all greater than periods, that fixed priority schedulers (like deadline monotonic) behave as well as EDF while the worst-case response **time** analysis is less complex. The same observation is valid when the task sets are almost homogeneous but is in favor of EDF in the general case or when a simple feasibility analysis is needed.

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Although, some works allow the computation of the exact cost of preemptions in the **scheduling** analysis [4],
usually this cost is approximated as stated by Liu **and** Lay- land [1]. This approximation may lead to incorrect be- haviour during the **real**-**time** execution of the tasks or at least a waste of resources due to the WCET **and** memory margins the designer must take. In the same vein the over- head of **preemptive** **scheduling** algorithms is more difficult to characterize **and** predict than the one of **non**-**preemptive** **scheduling** algorithms. Since **scheduling** overhead is of- ten ignored in **scheduling** models, an implementation of a **non**-**preemptive** scheduler will be closer to the formal model than an implementation of a **preemptive** sched- uler. In this case, the cost of the scheduler itself could be taken into account in schedulability conditions. **Non**- **preemptive** **scheduling** on a uniprocessor naturally guar- antees exclusive access to shared resources **and** data, thus eliminating both the need for synchronization **and** its asso- ciated overhead. In control applications, the input-output delay **and** jitter are minimized for all tasks when using a **non**-**preemptive** **scheduling** discipline, since the inter- val between the start **and** end times is always equal to the task computation **time** [5]. This simplifies the tech- niques for delay compensation in the control design. In many practical **real**-**time** **scheduling** problems involving I/O **scheduling**, properties of device hardware **and** soft- ware either make preemption impossible of prohibitively expensive [6]. For these reasons, designers often use **non**- **preemptive** approaches even if the theoretical results do not extend easily to them [7].

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In [ZKG + 08], the authors propose a framework for the schedulability analysis of **real**-
**time** systems, where they dene a generalized model for sporadic tasks to more precisely characterizes the task arrival times. Each task is characterized by two constraints: higher instantaneous arrival rate, which bounds the maximum number of task arrivals during some small **time** interval; lower average arrival rate, which is used to specify the maxi- mum number of arrivals over some longer **time** interval. The work of [MCG13] considers systems with probabilistic execution times **and** probabilistic inter-arrival times. However it does not handle dynamic **scheduling** policies. Moreover, the method is a numerical analysis technique whose complexity is exponential in proportion to the number of sam- ples **and** tasks. In [TDP12], the authors propose a method to control the **preemptive** behavior of **real**-**time** sporadic task systems by the use of CPU frequency scaling. They introduced a new sporadic task model in which the task arrival may deviate, according to a discrete **time** probability distribution, from the minimum inter-arrival **time**. Based on the probability of arrivals, the authors propose an on-line algorithm computing CPU frequencies that guarantee **non**-preemptiveness of task behavior while preserving system schedulability.

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Abstract—Modern GPUs allow concurrent kernel execu- tion **and** preemption to improve hardware utilization **and** responsiveness. Currently, the decision on the simultaneous execution of kernels is performed by the hardware, which can lead to unreasonable use of resources. In this work, we tackle the problem of co-**scheduling** for GPUs in high competition scenarios. We propose a novel graph- based **preemptive** co-**scheduling** algorithm, with the focus on reducing the number of preemptions. We show that the optimal **preemptive** makespan can be computed by solving a Linear Program in polynomial **time**. Based on this solution we propose graph theoretical model **and** an algorithm to build **preemptive** schedules which minimizes the number of preemptions. We show, however, that finding the minimal amount of preemptions among all **preemptive** solutions of optimal makespan is a NP-hard problem. We performed experiments on **real**-world GPU applications **and** our approach can achieve optimal makespan by preempting 6 to 9% of the tasks.

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Fig 9 Task execution with DVFS
7.5 Comparison of results
We have proposed an approach which is to describe **and** compare the energy efficiency of various power management techniques **and** this combined with an EDF **preemptive** scheduler. Also, we could raise the energy profile of each processor in addition to the execution of the the tasks in order to give more visiblity concerning the performances of STORM to highlight the energy savings. In addition, we determined, via the simulator, the power consumed by each processor calculated for a period of 1 second. The CPU power consumption diagram of a processor shows over **time** its electrical power (in watts) computed according to the physical characteristics of its chip **and** its functioning states. Insofar, as the processors, generally, function with less possible downtime, the energy energy consumption **and** costs obtained for each technique of power management differ considerably. Energy saving can mostly reach 95% of the costs of the consumption compared to the use of EDF. At first glance, it should be noted that the techniques of energy management contribute contribute significantly to the reduction in power consumption, although the use of DVFS seems more efficient in terms of gain. It is about 0.05J only whereas without energy consideration it reaches 0.93J **and** this without negative incidence on the performance of the systems.

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52 State of art **Scheduling** independent **non**-**preemptive** strictly periodic tasks
A more particular **scheduling** problem is the one with **non**-**preemptive** strictly periodic tasks. The **non**-**preemptive** **scheduling** problem is known to be NP-hard computational complexity [65]. Adding the strict periodicity constraint increases the problem com- plexity. Korst et al. [80] were the first to study the problem of **scheduling** a set of **non**-**preemptive** strictly periodic tasks. Their work was motivated by **real**-**time** video signal processing. The authors considered this problem on a minimum number of processors [79]. They showed that the problem is NP-complete in the strong sense, even in the case of a single processor, but that it is solvable in polynomial **time** if the periods **and** execution times are divisible. Thus, they proposed an approximation algorithm based on assigning tasks to processors according to some priority rule. Besides, they proposed a necessary **and** sufficient condition for the schedulability of two strictly periodic tasks. Later, Kermia et al [70] proposed a sufficient schedulability condition that generalizes the previous condition for a set of tasks. They imposed that the sum of the tasks execution times is less or equal to the greatest common divisors (GCDs) of task periods. In [92], Marouf **and** Sorel proved that this sufficient condition is very restrictive (pessimistic). They also gave a schedulability condition for implicit-deadline strictly periodic tasks **and** proposed an heuristic based on this condition. In contrast, our approach is not restricted to implicit-deadline tasks.

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† University of Rennes 1, UEB, IRISA, Rennes, France
Email: {jmsm,nelis,smp}@isep.ipp.pt , Isabelle.Puaut@irisa.fr
Abstract—In **real**-**time** systems, there are two distinct trends for **scheduling** task sets on unicore systems: **non**-**preemptive** **and** preemp- tive **scheduling**. **Non**-**preemptive** **scheduling** is obviously not subject to any preemption delay but its schedulability may be quite poor, whereas fully **preemptive** **scheduling** is subject to preemption delay, but benefits from a higher flexibility in the **scheduling** decisions. The **time**-delay involved by task preemptions is a major source of pessimism in the analysis of the task Worst-Case Execution **Time** (WCET) in **real**-**time** systems. **Preemptive** **scheduling** policies including **non**-**preemptive** regions are a hybrid solution between **non**-**preemptive** **and** fully **preemptive** **scheduling** paradigms, which enables to conjugate both world’s benefits. In this paper, we exploit the connection between the progression of a task in its operations, **and** the knowledge of the preemption delays as a function of its progression. The pessimism in the preemption delay estimation is then reduced in comparison to state of the art methods, due to the increase in information available in the analysis.

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There are some works in the case of **non**-**preemptive** tasks with strict periods. Al Sheikh **and** al. study the par- tition **scheduling** on an IMA (Integrated Modular Avion- ics) platform where the avionic functions are strictly peri- odic. They gave an exact algorithm with excessive com- putation **time**, based on a linear programming formula- tion, to solve the problem. Korst **and** al. proved in [11] a necessary **and** sufficient schedulability condition for two tasks, which becomes a sufficient condition for more than two tasks as proved by Kermia in [12]. However, as men- tioned in [13], this later condition is very restrictive. In [14] Eisenbrand **and** al. proposed **scheduling** algorithms in the case of harmonic **and** **non**-harmonic tasks.

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For **Scheduling**-TPN, we tackle the problem of the state-space explosion by a two-stage analysis. First, we pre-compute the state space of the **Scheduling**- TPN as a linear hybrid automaton. This first step is performed by a fast DBM- based algorithm. Although this algorithm is over-approximating, the produced linear hybrid automaton is proved to be **time**-bisimilar to the initial **Scheduling**- TPN i.e. the additional locations generated by the approximation are actually not reachable. As a consequence, the cost of the translation is fairly low. The second step consists of an exact analysis of that LHA with the HyTech model- checker. For this second step to be efficient, the number of variables (clocks) must be kept as low as possible. To this effect, the translation algorithm offers a number of reduction mechanisms **and** thus produces a LHA that has, in general, a fairly lower number of variables than what is required for a direct modeling as a product of linear hybrid automata.

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2. **Scheduling** Extended **Time** Petri Nets
We propose an extension for the TPN 1 that enables to take into account the way
the **real**-**time** tasks of an application distributed over different processors are scheduled. At first this extension consists in adding two parameters to the TPN’s places ; we call them “p r o c e s s o r ” **and** “priority”, **and** they correspond respectively to the allocation **and** the priority of the task which is associated with the place. However all places of a TPN do not require such parameters. Actually when a place does not represent a true activity for a processor (for example a register or memory state), neither a processor nor a priority have to be attached to it. In this specific case, the semantics remains unchanged with respect to a standard TPN. One can notice that it is equivalent to attach to this place a processor for its exclusive use **and** any priority (it does not matter in this case).

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Rapport de recherche n° 8234 — February 2013 — 17 pages
Résumé : Les techniques d’analyse de pire temps d’exécution (WCET) ont atteint une très
bonne précision dans l’analyse de programmes séquentiels s’exécutant sur architectures monopro- cesseurs. Dans cet article, nous étendons une technique récente d’analyse WCET et l’outil associé pour permettre de calculer des estimations précises du temps de réponse (WCRT) d’applications parallèles **non**-préemptives s’exécutant sur des plates-formes multi-coeurs. La technique proposée est intégrée dans le sens où elle calcule en même temps les estimations WCET des fragments de code séquentiel et le WCRT global. L’utilisation de cette methode produit des estimations plus précises que les approches découplées plus classiques où le calcul du WCRT est réalisé à partir de valeurs WCET précalculées séparément pour chacun des fragments de code séquentiel. Sur 2 exemples d’applications de contrôle embarqué notre technique d’analyse améliore les estimations WCRT de 21% en moyenne.

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that have been completed while receiving service for a total duration of **time** t when being in state n.
In order to derive stability **and** asymptotic **optimality** results, we will be interested in limits of the fluid-scaled process. In Section IV-A we will characterize a generic description of weak fluid limits (usually not unique) of Equation (3), following the same reasoning as in [15]. In Section IV-B we focus instead on a special class of policies for which we can prove convergence in probability towards a unique limit, which will be referred to as the strong fluid limit. (In [20] similar is done but only for a subset of the state space.) We discuss the differences between weak **and** strong fluid limits in more detail in Remark 5, after having introduced formally both concepts.

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IV. C ONCLUSION **AND** F UTURE W ORK
Our goal of our research is to propose **real**-**time** energy-efficient **scheduling** for embedded many-core architecture with more general recurring task model which can be more appropriate for industrial purpose **and** to take into account the new trends, for example using processor/core groups. The algorithm presented here aims to minimize preemptions in shared memory systems. We are also working on dynamic **scheduling** with hierarchical memory **and** allowing both local **and** global reconfiguration. We would like to define how to evaluate costs for both of them. Another way is to work on thread parallelism: it is a part of a job J i executed simultaneously on several cores/processors

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chooseAction **and** (3) its timing by the function chooseDelay (see Algorithm 1).
The rswtioco relation does not allow either of “standard” outputs **and** ‘’indicators’’ to be emitted in advance or on late, by the system. Also, this relation allows having more information about the **non**-conformance of a system. So, when the system emits an indicator or an output that was not expected at that **time**, then we can know if that indicator (resp. output) must be an active output (resp. an indicator) or nothing (see algorithm 1). The proposed rswtioco relation makes it possible to answer another question: “does some action a resume at the expected date? i.e. rswtioco does not allow a suspended action to be resumed in advance or on late. Under assumptions of input enabledness, the rswtioco relation coincides with relativized timed trace inclusion. Timed Traces of the SUT operating under an environment must be included in those of the specification under the cover of the same environment.

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