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sched/dl/Documentation: Add some references
Add a description of the Dhall's effect, some discussion about schedulability tests for global EDF, and references to real-time literature. Signed-off-by: Luca Abeni <luca.abeni@unitn.it> Signed-off-by: Peter Zijlstra (Intel) <peterz@infradead.org> Cc: Linus Torvalds <torvalds@linux-foundation.org> Cc: Peter Zijlstra <peterz@infradead.org> Cc: Thomas Gleixner <tglx@linutronix.de> Cc: henrik@austad.us Cc: juri.lelli@gmail.com Cc: raistlin@linux.it Link: http://lkml.kernel.org/r/1431954032-16473-8-git-send-email-luca.abeni@unitn.it Signed-off-by: Ingo Molnar <mingo@kernel.org>
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@ -163,7 +163,8 @@ CONTENTS
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maximum tardiness of each task is smaller or equal than
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((M − 1) · WCET_max − WCET_min)/(M − (M − 2) · U_max) + WCET_max
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where WCET_max = max{WCET_i} is the maximum WCET, WCET_min=min{WCET_i}
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is the minimum WCET, and U_max = max{WCET_i/P_i} is the maximum utilization.
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is the minimum WCET, and U_max = max{WCET_i/P_i} is the maximum
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utilization[12].
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If M=1 (uniprocessor system), or in case of partitioned scheduling (each
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real-time task is statically assigned to one and only one CPU), it is
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@ -205,11 +206,48 @@ CONTENTS
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On multiprocessor systems with global EDF scheduling (non partitioned
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systems), a sufficient test for schedulability can not be based on the
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utilizations (it can be shown that task sets with utilizations slightly
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larger than 1 can miss deadlines regardless of the number of CPUs M).
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However, as previously stated, enforcing that the total utilization is smaller
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than M is enough to guarantee that non real-time tasks are not starved and
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that the tardiness of real-time tasks has an upper bound.
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utilizations or densities: it can be shown that even if D_i = P_i task
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sets with utilizations slightly larger than 1 can miss deadlines regardless
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of the number of CPUs.
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Consider a set {Task_1,...Task_{M+1}} of M+1 tasks on a system with M
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CPUs, with the first task Task_1=(P,P,P) having period, relative deadline
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and WCET equal to P. The remaining M tasks Task_i=(e,P-1,P-1) have an
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arbitrarily small worst case execution time (indicated as "e" here) and a
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period smaller than the one of the first task. Hence, if all the tasks
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activate at the same time t, global EDF schedules these M tasks first
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(because their absolute deadlines are equal to t + P - 1, hence they are
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smaller than the absolute deadline of Task_1, which is t + P). As a
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result, Task_1 can be scheduled only at time t + e, and will finish at
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time t + e + P, after its absolute deadline. The total utilization of the
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task set is U = M · e / (P - 1) + P / P = M · e / (P - 1) + 1, and for small
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values of e this can become very close to 1. This is known as "Dhall's
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effect"[7]. Note: the example in the original paper by Dhall has been
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slightly simplified here (for example, Dhall more correctly computed
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lim_{e->0}U).
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More complex schedulability tests for global EDF have been developed in
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real-time literature[8,9], but they are not based on a simple comparison
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between total utilization (or density) and a fixed constant. If all tasks
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have D_i = P_i, a sufficient schedulability condition can be expressed in
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a simple way:
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sum(WCET_i / P_i) <= M - (M - 1) · U_max
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where U_max = max{WCET_i / P_i}[10]. Notice that for U_max = 1,
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M - (M - 1) · U_max becomes M - M + 1 = 1 and this schedulability condition
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just confirms the Dhall's effect. A more complete survey of the literature
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about schedulability tests for multi-processor real-time scheduling can be
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found in [11].
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As seen, enforcing that the total utilization is smaller than M does not
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guarantee that global EDF schedules the tasks without missing any deadline
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(in other words, global EDF is not an optimal scheduling algorithm). However,
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a total utilization smaller than M is enough to guarantee that non real-time
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tasks are not starved and that the tardiness of real-time tasks has an upper
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bound[12] (as previously noted). Different bounds on the maximum tardiness
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experienced by real-time tasks have been developed in various papers[13,14],
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but the theoretical result that is important for SCHED_DEADLINE is that if
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the total utilization is smaller or equal than M then the response times of
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the tasks are limited.
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SCHED_DEADLINE can be used to schedule real-time tasks guaranteeing that
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the jobs' deadlines of a task are respected. In order to do this, a task
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@ -245,6 +283,29 @@ CONTENTS
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Concerning the Preemptive Scheduling of Periodic Real-Time tasks on
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One Processor. Real-Time Systems Journal, vol. 4, no. 2, pp 301-324,
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1990.
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7 - S. J. Dhall and C. L. Liu. On a real-time scheduling problem. Operations
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research, vol. 26, no. 1, pp 127-140, 1978.
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8 - T. Baker. Multiprocessor EDF and Deadline Monotonic Schedulability
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Analysis. Proceedings of the 24th IEEE Real-Time Systems Symposium, 2003.
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9 - T. Baker. An Analysis of EDF Schedulability on a Multiprocessor.
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IEEE Transactions on Parallel and Distributed Systems, vol. 16, no. 8,
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pp 760-768, 2005.
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10 - J. Goossens, S. Funk and S. Baruah, Priority-Driven Scheduling of
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Periodic Task Systems on Multiprocessors. Real-Time Systems Journal,
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vol. 25, no. 2–3, pp. 187–205, 2003.
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11 - R. Davis and A. Burns. A Survey of Hard Real-Time Scheduling for
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Multiprocessor Systems. ACM Computing Surveys, vol. 43, no. 4, 2011.
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http://www-users.cs.york.ac.uk/~robdavis/papers/MPSurveyv5.0.pdf
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12 - U. C. Devi and J. H. Anderson. Tardiness Bounds under Global EDF
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Scheduling on a Multiprocessor. Real-Time Systems Journal, vol. 32,
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no. 2, pp 133-189, 2008.
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13 - P. Valente and G. Lipari. An Upper Bound to the Lateness of Soft
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Real-Time Tasks Scheduled by EDF on Multiprocessors. Proceedings of
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the 26th IEEE Real-Time Systems Symposium, 2005.
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14 - J. Erickson, U. Devi and S. Baruah. Improved tardiness bounds for
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Global EDF. Proceedings of the 22nd Euromicro Conference on
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Real-Time Systems, 2010.
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4. Bandwidth management
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=======================
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