Hi Henrik,On 04/09/2015 11:39 AM, Henrik Austad wrote:[...]>> - 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)Ok>>> + For example, consider a M tasks {Task_1,...Task_M} scheduled on M - 1>> Please consider rewriting this to>> "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 Ire-read the original paper, and I see Dhall used m tasks (and n CPUs, just toconfuse people :). So I rewrote the example in this way... Also because in thisway the last task is Task_M, instead of Task_{M+1} which would make the notationmore 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 wellRight. This was a \epsilon in the original paper (actually, Dhall used  2\epsilonand 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}>>> + (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}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>> + 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.Ok>>> + is (M - 1) · e / (P - 1) + P / P = (M - 1) · 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 tosimplify the description (maybe I oversimplified?). A constant and small eshould 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,I can add a note about \lim_{e -> 0}> 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) · 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) · 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>> + scheduling deadlines assigned by SCHED_DEADLINE and the tasks' deadlines>> + described above (which represent the real temporal constraints of the task).>> What about simething like>> "> Finally, it is important to understand the relationship between the> scheduling deadlines assigned by SCHED_DEADLINE and the tasks' deadlines> 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> SCHED_DEADLINE assigns an _absolute_ deadline (a specific point in time) on> 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.The difference between SCHED_DEADLINE's deadlines and tasks' deadlines is not"absolute vs relative". The difference is that SCHED_DEADLINE cannot know the"real" tasks' deadlines, so it uses "scheduling deadlines" that are generatedaccording to the CBS rules (described in Section 2).Now, if a task is developed according to the Liu&Layland model (does not blockbefore 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 schedulingdeadlines coincides, so it is possible to guarantee the respect of the task's temporalconstraints.This is the tricky (and confusing :) thing about SCHED_DEADLINE.I'll see if I can reword this paragraph to make it more clear.			Thanks,				Luca