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Multiprocessor schedulability of arbitrary-deadline sporadic tasks: complexity and antichain algorithm

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Abstract

Baker and Cirinei (Lecture Notes in Computer Science, vol. 4878, Springer, pp. 62–75, 2007) have introduced an exact but naive algorithm, that consists in solving a state reachability problem in a finite automaton, to check whether a set of sporadic hard real-time tasks is schedulable on an identical multiprocessor platform. However, this algorithm suffers from poor performance due to the exponential size of the automaton relative to the size of the task set. In this paper, we build on the work of Baker and Cirinei, and rely on their formalism to characterise the complexity of this problem. We prove that it is PSpace-complete. In order to obtain an algorithm that is applicable in practice to systems of realistic sizes, we successfully apply techniques developed by the formal verification community, specifically antichain techniques (Doyen and Raskin in Lecture Notes in Computer Science, vol. 6015, Springer, pp. 2–22, 2010) to this scheduling problem. For that purpose, we define and prove the correctness of a simulation relation on Baker and Cirinei’s automaton. We show that our improved algorithm yields dramatically improved performance for the schedulability test and opens for many further improvements. This work is an extended and revised version of a previous conference paper by the same authors (Lindström et al., Proceedings of the 19th International Conference on Real-Time and Network Systems (RTNS 2011), pp. 25–34, 2011).

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Notes

  1. Here, and in the rest of the paper, a ‘request’, means the moment at which the scheduler captures the request. In practice, the request could have been generated before, by an external event, but we abstract those low-level concerns away from our model.

  2. Remark that by modeling the scheduler as a function, we restrict ourselves to deterministic schedulers.

  3. Recall that this is actually the moment at which the scheduler captures the request.

  4. Remark that the order does not matter.

  5. Remark that the classical definition of the universality problem asks whether the automaton recognises Σ ∗. Our definition of the problem differs slightly, as in our setting, automata bear labels on the nodes, instead of the edges. Hence, no such labeled automaton can accept ε. However, it is straightforward to check that this does not change the complexity of the problem.

  6. Remark that, in general V∖A∖B∪A=V∖B iff A∩B=∅.

  7. In some sense, the status of ‘NP versus PSpace’ is similar to that of ‘P versus NP’, as it is known that P⊆NP, and it is believed that P⊆̷NP and that all NP-complete problems are not in P.

  8. Remark that this was already known by the result of Baker and Cirinei (2007).

  9. Such as the well-known U≤1 for the mono-processor version of EDF, which is computable in polynomial time.

  10. Recall that S τ denotes the set of τ-request successors of S, see Definition 12.

  11. Remark that this does not contradict the measured time of 12 hours on 444,000 states given above. Indeed, the running time of the algorithm depends on the number of states of the automaton, but also on the branching degree of the automaton’s graph.

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Authors and Affiliations

  1. Département d’Informatique, Faculté des Sciences, Université libre de Bruxelles, Avenue Franklin D. Roosevelt 50, CP 212, 1050, Bruxelles, Belgium

    Gilles Geeraerts, Joël Goossens & Markus Lindström

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  1. Gilles Geeraerts
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  2. Joël Goossens
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  3. Markus Lindström
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Correspondence to Markus Lindström.

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Geeraerts, G., Goossens, J. & Lindström, M. Multiprocessor schedulability of arbitrary-deadline sporadic tasks: complexity and antichain algorithm. Real-Time Syst 49, 171–218 (2013). https://doi.org/10.1007/s11241-012-9172-y

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