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A parallel branch-and-bound algorithm to compute a tighter tardiness bound for preemptive global EDF

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Abstract

In this paper we present a parallel exact algorithm to compute an upper bound to tardiness of preemptive global EDF (G-EDF) schedulers, named harmonic bound, which has been proved to be up to 30% tighter than previously proposed bounds. Tightness is a crucial property of tardiness bounds: a too loose bound may cause a feasible soft real-time application to be mistakenly deemed unfeasible. Unfortunately, no polynomial-time algorithm is known to date to compute the harmonic bound. Although there is no proof of hardness of any sort either, the complex formula of the bound apparently provides no hints to devise algorithms with sub-exponential worst-case cost. In this paper we address this issue by proposing a parallel, exact, branch-and-bound algorithm to compute the harmonic bound, called harm-BB, which proves to be extremely fast in a large number of experiments. More specifically, we compare its execution times with those of existing polynomial-time algorithms for other known tardiness bounds on 630,000 random task sets. harm-BB outperforms, or is comparable to, the competitor algorithms in all scenarios but the ones with the highest number of processors (7 and 8) and tasks (\(\sim \) 50). In the latter scenarios harm-BB is indeed slower than the other algorithms; yet, it was still feasible, as it takes only about 2.8 s to compute the bound on a commodity dual-core CPU. Even better, we show that harm-BB has a high parallel efficiency, thus its execution time may be largely cut down on highly-parallel platforms.

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Notes

  1. These are the scenarios for which wrong execution times, in the order of just milliseconds, were reported for the sequential version of harm-BB in Leoncini et al. (2017).

  2. The way this partition is made, as well as the number of generated subspaces, is clearly problem dependent.

  3. All experiments can be repeated by applying two patches to SchedCAT (2014) and running an ad hoc script.

  4. We name cva2 the oldest one between the last two algorithms, just to preserve the same naming as the one used for the bounds in Valente (2016).

  5. according to Fig. 3 and to the fact that the dual-core CPU exposes four logical CPUs, this means that harm-BB generated 4 threads.

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Acknowledgements

Authors wish to thank Prof. Daniele Funaro for providing a compact proof of Lemma 3.

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Authors and Affiliations

  1. Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via Campi 213/b, Modena, Italy

    Mauro Leoncini, Manuela Montangero & Paolo Valente

  2. Istituto di Informatica e Telematica, CNR Via Moruzzi 1, Pisa, Italy

    Mauro Leoncini

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  1. Mauro Leoncini
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  2. Manuela Montangero
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  3. Paolo Valente
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Correspondence to Manuela Montangero.

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Leoncini, M., Montangero, M. & Valente, P. A parallel branch-and-bound algorithm to compute a tighter tardiness bound for preemptive global EDF. Real-Time Syst 55, 349–386 (2019). https://doi.org/10.1007/s11241-018-9319-6

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