Narrowing the speedup factor gap of partitioned EDF

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Abstract

Schedulability is a fundamental problem in analyzing real-time systems, but it often has to be approximated because of the intrinsic computational hardness. Partitioned earliest deadline first (EDF) is one of the most popular polynomial-time and practical scheduler on multiprocessor platforms, and it was shown to have a speedup factor of at most 2.63221/m. This paper further improves the factor to 2.55561/m for both the constrained-deadline case and the arbitrary-deadline case, and it is very close to the known (non-tight) lower bound of 2.51/m. The key ideas are that we develop a novel method to discretize and regularize sporadic task sets that are schedulable on uniprocessors, and we find that the ratio (ρ) of the approximate demand bound value to the machine capacity is upper-bounded by 1.5556 for the arbitrary-deadline case, which plays an important role in estimating the speed factor of partitioned EDF.

Introduction

Scheduling plays a fundamental role in real-time systems. Basically, given a finite set of tasks, each sequentially releasing infinitely many jobs, the mission of real-time scheduling is to allocate computing resources so that all the jobs are done in a timely manner. Formally, a schedule defines at each time instant which jobs receive the required computing resources (while others must wait). The fundamental question of schedulability naturally arises: is it possible at all to successfully schedule these tasks so as to meet all the deadlines?

Unfortunately, answering this question is often not “easy”; for example, the schedulability of a set of constrained-deadline1 sporadic tasks, which is the focus of this article, is co-NP-hard even on a uniprocessor platform [2]. For the multiprocessor case, it remains NP-hard for partitioned scheduling, even for implicit-deadline task sets, where the relative deadline of each task equals its period [3]. Here partitioned scheduling means that once a task is assigned to a processor, all the jobs released by the task will be scheduled on the dedicated processor. These hardness results imply that it is impossible to exactly decide schedulability in polynomial time, unless P=NP.

Because of the hardness, real-time schedulability problems are usually solved approximately by pessimistic algorithms that always answer “no” unless some sufficient conditions for schedulability are met. To evaluate the performance of such an approximate algorithm (say, A), the concept of the speedup factor, also known as the resource augmentation bound, has been proposed. Specifically, algorithm A has a speedup factor of s1 if whenever a set of tasks is schedulable (by an optimal approach) on a platform with speed 1, A will return “yes” when the speed of the platform is increased to s. Despite some recent discussion on potential pitfalls [4], [5], [6], the speedup factor has been a major metric and standard theoretical tool for assessing scheduling algorithms since the seminal work of Kalyanasundaram and Pruhs [7] in 2000.

Recent years have witnessed impressive progress in finding scheduling algorithms with low speedup factors. For preemptive scheduling (i.e., running jobs might be interrupted by emergent ones), global EDF has a speedup factor of 21/m [8] for scheduling tasks on m identical processors, and there is a polynomial-time algorithm for uniprocessors whose speedup factor is 1+ϵ [9], where ϵ>0 is arbitrarily small. For nonpreemptive scheduling, there are also a variety of results [10], [11]. In addition to the speedup factor, there are several articles concerning the utilization bound [12], [13], [14].

Although the speedup factor on uniprocessors is already known to be tight, the multiprocessor case remains open. Among all schedulers, partitioned scheduling is of particular interest because of its implementation friendliness, simplicity, and capability of extending most uniprocessor results to the multiprocessor scenario directly under naive “partition” heuristics; i.e., once the task-to-core mapping is fixed, the scheduling in the multiprocessor case is reduced to multiple uniprocessor scheduling problems, where classical solutions exist. Since EDF is an optimal preemptive scheduler on a uniprocessor, this article focuses on partitioned EDF.2 Note that partitioned-deadline-monotonic [15] is also commonly implemented, with a best known speedup factor of 2.8431, while global EDF is not a partitioned paradigm.

A breakthrough in partitioned EDF was made in 2005, when Baruah and Fisher [16] established an upper bound of 31/m for the speedup factor on constrained-deadline task sets and an upper bound of 42/m for the speedup factor on arbitrary-deadline task sets, where m is the number of identical processors. In 2011, Chen and Chakraborty [1] further improved the speedup factor to 2.63221/m for the constrained-deadline case and to 31/m for the arbitrary-deadline case. Also, they established an asymptotical lower bound of 2.5 for the speedup factor for the constrained-deadline case. Since then, the speedup factor bounds have never been improved.

Deriving the upper bound of the speedup factor of partitioned EDF relies heavily on a quantity ρ concerning scheduling on uniprocessors. The quantity ρ, called the relaxation factor in this article and formally defined in formula (1) in Section 2, roughly indicates how much the approximate demand bound function (defined in Section 2) deviates from the machine capacity. Baruah and Fisher [16] bridged the relaxation factor and the speedup factor of partitioned EDF by showing that in the case of constrained deadlines, the speedup factor is at most 1+ρ1/m. As a result, upper-bounding the speedup factor is reduced to upper-bounding of ρ, and it is in this manner that both Baruah and Fisher [16] and Chen and Chakraborty [1] obtained their estimates of the speedup factor. Hence, the relaxation factor itself deserves deep investigation. Baruah and Fisher [16] upper-bounded it by 2, and Chen and Chakraborty [1] narrowed its range to [1.5,1.6322].

On this ground, we explore a better upper bound of the relaxation factor, and on this basis provide a better estimate of the speedup factor of partitioned EDF for sets of constrained-deadline sporadic tasks. The contributions are summarized as follows:

  • 1.

    We improve the best existing upper bound of the relaxation factor from 1.6322 to 1.5556 (Theorem 1), which is very close to the lower bound of 1.5 for the uniprocessor case. The result holds for both constrained-deadline tasks and arbitrary-deadline tasks. Accordingly, the speedup factor of partitioned EDF for constrained-deadline tasks decreases from 2.6321/m to 2.55561/m (Theorem 2) for the multiprocessor case.

  • 2.

    We identify a lossless way to discretize and regularize the tasks. As a result, the execution times of the tasks of interest can be fixed to be 1 and the deadlines can be fixed to be 1,2,,n, where n is number of tasks to be scheduled (Lemma 3, Lemma 7, Lemma 8). The only parameter that varies is the period. The transformation is lossless in the sense that the relaxation factor does not change although the parameters are extremely simplified.

  • 3.

    We invent a method to further transform the tasks so that the period of each task ranges over integers between 1 and 2n (Lemma 9). Although this transformation is not guaranteed to be lossless, the loss, if any, is negligible since we prove that the relaxation factor increases by at most 0.0556 (for both constrained-deadline task sets and arbitrary-deadline task sets). These transformation techniques may be further applied to real-time scheduling analysis or other problems.

The rest of this article is organized as follows. Section 2 presents the model and preliminaries. Section 3 focuses on the uniprocessor case, and in it we derive a new upper bound (14/9) of the relaxation factor. Section 4 provides a new upper bound (23/91/m) of the speedup factor for partitioned EDF. Finally, Section 5 concludes the article and provides some potential future directions.

Section snippets

System model and preliminaries

We consider a finite set τ of sporadic tasks. Each task τi can be represented by a triple τi=(ei,di,pi), where ei is the worst-case execution time, di is its relative deadline, and pi is the minimum interarrival separation length (also known as the period). Such a task releases infinitely many jobs, each of which has an execution time of at most ei and has to be finished within time di after arrival, while the interarrival time of consecutive jobs is at least pi. The task τi is said to be a

Improved upper bound of the relaxation factor

To estimate the speedup factor for multiprocessor partitioned scheduling, we first analyze the relaxation factor and hence focus on uniprocessors. The main result of this section is Theorem 1, which establishes 14/9 as an upper bound of the relaxation factor for sporadic tasks.

The basic idea of our proof is to discretize any given task set into a regular form, thus reducing the problem to an optimization one on bounded integers with several constraints (MP4). Roughly speaking, Lemma 3 ensures

Partitioned scheduling on multiprocessors

This section is devoted to partitioning sporadic tasks on multiprocessors, where the tasks are assumed to have constrained deadlines. Note that although Theorem 1 holds for arbitrary deadlines, the extension to multiprocessors applies only for the constrained-deadline case. We focus on the algorithm of deadline-monotonic partitioned EDF; namely, the algorithm PARTITION in [16].

Basically, the algorithm PARTITION assigns tasks sequentially in nondecreasing order of relative deadlines to

Conclusion and future work

In this article, we improved the upper bound of the speedup factor of (polynomial-time) partitioned EDF from 2.63221/m to 2.55561/m for constrained-deadline sporadic tasks on m identical processors, narrowing the gap between the upper and lower bounds from 0.1322 to 0.0556. This is an immediate corollary of our improvement of the upper bound of the relaxation factor from 1.6322 to 1.5556, which holds for both the constrained-deadline scenario and the arbitrary-deadline scenario.

Technically,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors thank Prof. Sanjoy Baruah from Washington University at St. Louis and Prof. Yungang Bao from the Institute of Computing Technology, Chinese Academy of Sciences, for fruitful discussions. This work was partially supported by the National Key Research and Development Program of China (Grant No. 2016YFB1000201), the Key-Area Research and Development Program of Guangdong Province (2020B010164003), the National Natural Science Foundation of China (11971091, 62072433, and 62090020),

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