## Abstract

This paper gives and proves correct a simulation interval for any schedule generated by a deterministic and memoryless scheduler (i.e., one where the scheduling decision is the same and unique for any two identical system states) for identical multiprocessor platforms. We first consider independent periodic tasks, then generalize the simulation interval to tasks sharing critical resources, and subject to precedence constraints or self-suspension. The simulation interval is based only on the periods, release times and deadlines, and is independent from any other parameters. It is proved large enough to cover any feasible schedule produced by any deterministic and memoryless scheduler on multiprocessor platforms, including non conservative schedulers. To the best of our knowledge, this simulation interval covers the largest class of task systems and scheduling algorithms on identical multiprocessor platforms ever studied. This simulation interval is used to derive a simulation algorithm using a linear space complexity. Finally, a generic exact schedulability test based on simulation is presented. This test can be applied only when sustainability is consistent with online variability of the tasks’ parameters.

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## Notes

- 1.
Duration between a job release and its completion.

- 2.
Period of continuous processor occupation starting at the critical instant, ignoring tasks of lower priority.

- 3.
Necessary and sufficient.

- 4.
Where \(\doteq \) means “equals by definition”.

- 5.
See Definition 4.

- 6.
Here exact means that a shorter interval would not be a simulation interval.

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## Acknowledgments

The authors sincerely thank the reviewers, whose remarks really helped to improve the paper. They also thank Pascal Richard, Gerard Fohler and Yves Sorel for very inspiring discussions. This research has been partially funded by the projects PIA CORAC Panda, CS ISAE-ENSMA, FUI Waruna, FP7 IP PROXIMA, IP LEOC Capacites and IP BGLE Departs.

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Goossens, J., Grolleau, E. & Cucu-Grosjean, L. Periodicity of real-time schedules for dependent periodic tasks on identical multiprocessor platforms.
*Real-Time Syst* **52, **808–832 (2016). https://doi.org/10.1007/s11241-016-9256-1

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### Keywords

- Real-time scheduling
- Simulation intervals
- Multiprocessor