From: Luca Abeni
Date: Thu Apr 09 2015 - 06:06:29 EST

Hi Henrik,

On 04/09/2015 11:39 AM, Henrik Austad wrote:
[...]
- 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)
Ok

"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.
Yes, this is what I originally wrote (and is the example I teach to students:
http://disi.unitn.it/~abeni/RTOS/multiprocessor.pdf, slide 7). But then I
re-read the original paper, and I see Dhall used m tasks (and n CPUs, just to
confuse people :). So I rewrote the example in this way... Also because in this
more complex (because of the _{M+1}). But I can rewrite using M+1 tasks and M 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
Right. This was a \epsilon in the original paper (actually, Dhall used 2\epsilon
and I decided to simplify things a little bit).

(you're going to describe
Dhall's effect, right?). Perhaps make that point explicit?

T_i = {P_i, e, P_i}

+ equal to P: P_M=D_M=WCET_M=P.

T_M = {P, P, P}
Ok

+ 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
+ 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.
Ok

+ 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'
Right. The original paper uses "\lim_{\epsilon -> 0} ...", but I decided to
simplify the description (maybe I oversimplified?). A constant and small e
should be ok to give an intuition of Dhall's effect: if e becomes very small,
the utilisation becomes very near to 1. But if you think this confuses the reader,

and that they will be picked _before_ the heavy task by EDF.
Right. This is because these tasks have period (and relative deadline) equal to P-1.

+ 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)
Ok; if other people agree, I'll add a patch to the patchset to convert all the
"sum_" into "sum".

+ 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
+
+ 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.
This is not the point I wanted to make... Absolute deadlines (equal to r + D)
have been previously defined in the document for real-time tasks too.
"absolute vs relative". The difference is that SCHED_DEADLINE cannot know the
according to the CBS rules (described in Section 2).
Now, if a task is developed according to the Liu&Layland model (does not block
before the end of the job) and the SCHED_DEADLINE parameters are properly assigned
(runtime >= WCET, period <= P) then the task's absolute deadlines and the scheduling
deadlines coincides, so it is possible to guarantee the respect of the task's temporal
constraints.