Date: Thu Apr 09 2015 - 05:39:25 EST

On Wed, Apr 08, 2015 at 01:59:40PM +0200, Luca Abeni wrote:
> schedulability tests for global EDF, and references to real-time literature,
> ---
> 1 file changed, 71 insertions(+), 10 deletions(-)
>
> index ffaf95f..da5a8d7 100644
> @@ -160,7 +160,8 @@ CONTENTS
> maximum tardiness of each task is smaller or equal than
> ((M â 1) Â WCET_max â WCET_min)/(M â (M â 2) Â U_max) + WCET_max
> where WCET_max = max_i{WCET_i} is the maximum WCET, WCET_min=min_i{WCET_i}
> - is the minimum WCET, and U_max = max_i{WCET_i/P_i} is the maximum utilisation.
> + is the minimum WCET, and U_max = max_i{WCET_i/P_i} is the maximum
> + utilisation[12].
>
> If M=1 (uniprocessor system), or in case of partitioned scheduling (each
> real-time task is statically assigned to one and only one CPU), it is
> @@ -202,15 +203,52 @@ CONTENTS
>
> On multiprocessor systems with global EDF scheduling (non partitioned
> systems), a sufficient test for schedulability can not be based on the
> - utilisations (it can be shown that task sets with utilisations slightly
> - larger than 1 can miss deadlines regardless of the number of CPUs M).
> - However, as previously stated, enforcing that the total utilisation is smaller
> - than M is enough to guarantee that non real-time tasks are not starved and
> - that the tardiness of real-time tasks has an upper bound.
> -
> - SCHED_DEADLINE can be used to schedule real-time tasks guaranteeing that
> - the jobs' deadlines of a task are respected. In order to do this, a task
> - must be scheduled by setting:
> + utilisations or densities: it can be shown that even if D_i = P_i task
> + sets with utilisations slightly larger than 1 can miss deadlines regardless
> + of the number of CPUs.

+ \newline (add som breathing space)

"Consinder a set of M+1 tasks on a system with M CPUs [...]"

As 'M' is normally used to denote the number of cores available and it is
much easier to grasp the context of "<some number> + 1" rather than "<some
number - 1"-CPUs.

> + CPUs, with the first M - 1 tasks having a small worst case execution time
> + WCET_i=e and period equal to relative deadline P_i=D_i=P-1. The last task

Normally, 'e' is used to denote an _arbitrarily_ small value, and I suspect
that this is indeed the case here as well (you're going to describe
Dhall's effect, right?). Perhaps make that point explicit?

T_i = {P_i, e, P_i}

> + (Task_M) has period, relative deadline and worst case execution time
> + equal to P: P_M=D_M=WCET_M=P.

T_M = {P, P, P}

> + If all the tasks activate at the
> + same time t, global EDF schedules the first M - 1 tasks first (because
> + their absolute deadlines are equal to t + P - 1, hence they are smaller
> + than the absolute deadline of Task_M, which is t + P). As a result, Task_M
> + can be scheduled only at time t + e, and will finish at time t + e + P,
> + after its absolute deadline t + P. The total utilisation of the task set
^^^^^^
Drop this, the text is full of equations as it is.

> + is (M - 1) Â e / (P - 1) + P / P = (M - 1) Â e / (P - 1) + 1, and for
> + small values of e this can become very close to 1. This is known as "Dhall's
> + effect"[7].

This gives the impression that 'e' must be constant, but all it really
means is that e is an 'arbitrarily small value which can be *almost* 0' and
that they will be picked _before_ the heavy task by EDF.

> + More complex schedulability tests for global EDF have been developed in
> + real-time literature[8,9], but they are not based on a simple comparison
> + between total utilisation (or density) and a fixed constant. If all tasks
> + have D_i = P_i, a sufficient schedulability condition can be expressed in
> + a simple way:
> + sum_i WCET_i / P_i <= M - (M - 1) Â U_max

sum_i; as stated in another comment, just juse 'sum' (IMHO)

> + where U_max = max_i {WCET_i / P_i}[10]. Notice that for U_max = 1,
> + M - (M - 1) Â U_max becomes M - M + 1 = 1 and this schedulability condition
> + just confirms the Dhall's effect. A more complete survey of the literature
> + about schedulability tests for multi-processor real-time scheduling can be
> + found in [11].
> +
> + As seen, enforcing that the total utilisation is smaller than M does not
> + guarantee that global EDF schedules the tasks without missing any deadline
> + (in other words, global EDF is not an optimal scheduling algorithm). However,
> + a total utilisation smaller than M is enough to guarantee that non real-time
> + tasks are not starved and that the tardiness of real-time tasks has an upper
> + bound[12] (as previously noticed). Different bounds on the maximum tardiness
> + experienced by real-time tasks have been developed in various papers[13,14],
> + but the theoretical result that is important for SCHED_DEADLINE is that if
> + the total utilisation is smaller or equal than M then the response times of
> + the tasks are limited.
> +
> + Finally, it is important to understand the relationship between the
> + described above (which represent the real temporal constraints of the task).

"
Finally, it is important to understand the relationship between the
described above.

The task itself supplies a _relative_ deadline, i.e. an offset after the
release of the task at which point it must be complete whereas
the form

D_i = r_i + d_i
"
Or somesuch? I may be overdoing this.

> + If an admission test is used to guarantee that the scheduling deadlines are
> + respected, then SCHED_DEADLINE can be used to schedule real-time tasks
> + guaranteeing that the jobs' deadlines of a task are respected.
> + In order to do this, a task must be scheduled by setting:
>
> - runtime >= WCET
> @@ -242,6 +280,29 @@ CONTENTS
> Concerning the Preemptive Scheduling of Periodic Real-Time tasks on
> One Processor. Real-Time Systems Journal, vol. 4, no. 2, pp 301-324,
> 1990.
> + 7 - S. J. Dhall and C. L. Liu. On a real-time scheduling problem. Operations
> + research, vol. 26, no. 1, pp 127-140, 1978.
> + 8 - T. Baker. Multiprocessor EDF and Deadline Monotonic Schedulability
> + Analysis. Proceedings of the 24th IEEE Real-Time Systems Symposium, 2003.
> + 9 - T. Baker. An Analysis of EDF Schedulability on a Multiprocessor.
> + IEEE Transactions on Parallel and Distributed Systems, vol. 16, no. 8,
> + pp 760-768, 2005.
> + 10 - J. Goossens, S. Funk and S. Baruah, Priority-Driven Scheduling of
> + Periodic Task Systems on Multiprocessors. Real-Time Systems Journal,
> + vol. 25, no. 2â3, pp. 187â205, 2003.
> + 11 - R. Davis and A. Burns. A Survey of Hard Real-Time Scheduling for
> + Multiprocessor Systems. ACM Computing Surveys, vol. 43, no. 4, 2011.
> + http://www-users.cs.york.ac.uk/~robdavis/papers/MPSurveyv5.0.pdf
> + 12 - U. C. Devi and J. H. Anderson. Tardiness Bounds under Global EDF
> + Scheduling on a Multiprocessor. Real-Time Systems Journal, vol. 32,
> + no. 2, pp 133-189, 2008.
> + 13 - P. Valente and G. Lipari. An Upper Bound to the Lateness of Soft
> + Real-Time Tasks Scheduled by EDF on Multiprocessors. Proceedings of
> + the 26th IEEE Real-Time Systems Symposium, 2005.
> + 14 - J. Erickson, U. Devi and S. Baruah. Improved tardiness bounds for
> + Global EDF. Proceedings of the 22nd Euromicro Conference on
> + Real-Time Systems, 2010.
> +
>
> 4. Bandwidth management
> =======================
> --
> 1.7.9.5
>

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