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Statically optimal dynamic soft real-time semi-partitioned scheduling

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Abstract

Semi-partitioned scheduling is an approach to multiprocessor real-time scheduling where most tasks are fixed to processors, while a small subset of tasks is allowed to migrate. This approach offers reduced overhead compared to global scheduling, and can reduce processor capacity loss compared to partitioned scheduling. Prior work has resulted in a number of semi-partitioned scheduling algorithms, but their correctness typically hinges on a complex intertwining of offline task assignment and online execution. This brittleness has resulted in few proposed semi-partitioned scheduling algorithms that support dynamic task systems, where tasks may join or leave the system at runtime, and few that are optimal in any sense. This paper introduces EDF-sc, the first semi-partitioned scheduling algorithm that is optimal for scheduling (static) soft real-time (SRT) sporadic task systems and allows tasks to dynamically join and leave. The SRT notion of optimality provided by EDF-sc requires deadline tardiness to be bounded for any task system that does not cause over-utilization. In the event that all tasks can be assigned as fixed, EDF-sc behaves exactly as partitioned EDF. Heuristics are provided that give EDF-sc the novel ability to stabilize the workload to approach the partitioned case as tasks join and leave the system.

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Notes

  1. Mode change protocols (Real and Crespo 2004; Nélis et al. 2011) have been extensively studied in the real-time literature for both uniprocessor and multiprocessor systems. While EDF-sc could certainly be made to support various types of mode change protocols, they are mentioned here mainly for illustrative purposes, and adding such support is outside the scope of this work.

  2. Because the set \(\tau\) (and several other sets defined in Sects. 3 and 5) changes over time, it may be more technically precise to use the notation \(\tau (t)\), but we omit the time parameter where it is obvious to avoid clutter.

  3. In prior work on dynamic task systems, the term weight is often used to refer to task utilizations. Changing a task’s utilization is referred to as reweighting the task (Block et al. 2005, 2008).

  4. Alternative reweighting rules could free system utilization more aggressively than the ones presented here. In particular, a removed migrating task’s utilization could be freed at the deadline of its last job. This would allow dynamic workload changes to be made more quickly, but would also create a blocking term in the tardiness analysis for fixed tasks to account for tasks that are being changed from migrating to fixed. To aid in understanding, we opt for more conservative reweighting rules in this work.

  5. New tardiness analysis techniques for GEDF (Erickson et al. 2010; Leoncini et al. 2018) have been proposed since Devi’s work, and could likely be applied to obtain reduced bounds for the extended sporadic task model. However, deriving new bounds for existing scheduling algorithms is beyond the scope of this work.

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Authors and Affiliations

  1. Department of Computer Science, UNC-Chapel Hill, 201 S. Columbia St., Chapel Hill, NC, 27599, USA

    Clara Hobbs, Zelin Tong, Joshua Bakita & James H. Anderson

Authors
  1. Clara Hobbs
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  2. Zelin Tong
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  3. Joshua Bakita
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  4. James H. Anderson
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Correspondence to Zelin Tong.

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Work supported by NSF grants CNS 1409175, CNS 1563845, CNS 1717589, and CPS 1837337, ARO grant W911NF-17-1-0294, ONR grant N00014-20-1-2698, and funding from General Motors.

Additional Figures

Additional Figures

In this appendix, we present additional experimental results that were omitted from Sect. 6 for brevity. We first show the full results of our schedulability study from Sect. 6.1, followed by the corresponding plots of minimum schedulable container periods. Finally, we show the mean analytical tardiness bounds that were computed for the dynamic task systems generated in Sect. 6.2.

1.1 Schedulability Experiments

In this section, we provide full results of the schedulability study from Sect. 6.1. The same general trend observed there holds throughout, with EDF-sc giving high weighted schedulability when periods are short, and both schedulers giving high weighted schedulability for task systems with longer periods (Figs. 13, 14, 15, 16, 17, and 18).

Fig. 13
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Schedulability results for medium task sets generated with a uniform distribution

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Fig. 14
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Schedulability results for medium task sets generated with a bimodal distribution

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Fig. 15
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Schedulability results for medium task sets generated with an exponential distribution

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Fig. 16
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Schedulability results for heavy task sets generated with a uniform distribution

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Fig. 17
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Schedulability results for heavy task sets generated with a bimodal distribution

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Fig. 18
figure 18

Schedulability results for heavy task sets generated with an exponential distribution

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1.2 Container Period Experiments

In this section, we show the full set of plots of the minimum container periods for which the task sets from the experiments in Sect. 6.1 were schedulable with EDF-sc. As mentioned in the main text, these curves all show a sharp upward trend once the container period reaches some value. This period at which this occurs varies between the different task set parameters, but is not more than 50 ms in any case, so this was used as a conservative container period in the subsequent experiments.(Figs. 19, 20, 21, 22, 23, and 24).

Fig. 19
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Minimum container periods required to schedule medium task sets generated with a uniform distribution and medium periods at different WSS

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Fig. 20
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Minimum container periods required to schedule medium task sets generated with a uniform distribution and differing periods at WSS \(=\) 512

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Fig. 21
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Minimum container periods required to schedule medium task sets generated with different distributions and medium periods at WSS \(=\) 512

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Fig. 22
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Minimum container periods required to schedule heavy task sets generated with a uniform distribution and medium periods at different WSS

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Fig. 23
figure 23

Minimum container periods required to schedule heavy task sets generated with a uniform distribution and differing periods at WSS \(=\) 512

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Fig. 24
figure 24

Minimum container periods required to schedule heavy task sets generated with different distributions and medium periods at WSS \(=\) 512

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1.3 Tardiness Bounds from Heuristic Comparison

Finally, in this section, we show the mean tardiness bounds for the dynamic task systems generated in Sect. 6.2 (Fig. 25).

Fig. 25
figure 25

Mean tardiness bounds for easy- (a), moderate- (b), and hard-to-partition (c) dynamic workloads, as computed using Corollaries 1 and 2. Note the different scales on the Y axis

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Hobbs, C., Tong, Z., Bakita, J. et al. Statically optimal dynamic soft real-time semi-partitioned scheduling. Real-Time Syst 57, 97–140 (2021). https://doi.org/10.1007/s11241-020-09359-8

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