Return-Path: Received: (majordomo@vger.kernel.org) by vger.kernel.org via listexpand id S932843AbbDJKUZ (ORCPT ); Fri, 10 Apr 2015 06:20:25 -0400 Received: from mail0.unitn.it ([193.205.194.10]:53094 "EHLO mail0.unitn.it" rhost-flags-OK-OK-OK-OK) by vger.kernel.org with ESMTP id S932348AbbDJKUO (ORCPT ); Fri, 10 Apr 2015 06:20:14 -0400 From: Luca Abeni To: peterz@infradead.org Cc: henrik@austad.us, juri.lelli@gmail.com, raistlin@linux.it, mingo@kernel.org, linux-kernel@vger.kernel.org, linux-doc@vger.kernel.org, Luca Abeni Subject: =?UTF-8?q?=5BPATCH=206/8=5D=20Documentation/scheduler/sched-deadline=2Etxt=3A=20add=20some=20references?= Date: Fri, 10 Apr 2015 12:19:53 +0200 Message-Id: <1428661195-9609-7-git-send-email-luca.abeni@unitn.it> X-Mailer: git-send-email 1.7.9.5 In-Reply-To: <1428661195-9609-1-git-send-email-luca.abeni@unitn.it> References: <1428661195-9609-1-git-send-email-luca.abeni@unitn.it> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Sender: linux-kernel-owner@vger.kernel.org List-ID: X-Mailing-List: linux-kernel@vger.kernel.org Content-Length: 6335 Lines: 112 Add a description of the Dhall's effect, some discussion about schedulability tests for global EDF, and references to real-time literature, --- Documentation/scheduler/sched-deadline.txt | 73 +++++++++++++++++++++++++--- 1 file changed, 67 insertions(+), 6 deletions(-) diff --git a/Documentation/scheduler/sched-deadline.txt b/Documentation/scheduler/sched-deadline.txt index 9663d53..8bec2f5 100644 --- a/Documentation/scheduler/sched-deadline.txt +++ b/Documentation/scheduler/sched-deadline.txt @@ -162,7 +162,8 @@ CONTENTS maximum tardiness of each task is smaller or equal than ((M − 1) · WCET_max − WCET_min)/(M − (M − 2) · U_max) + WCET_max where WCET_max = max{WCET_i} is the maximum WCET, WCET_min=min{WCET_i} - is the minimum WCET, and U_max = max{WCET_i/P_i} is the maximum utilisation. + is the minimum WCET, and U_max = max{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 @@ -204,11 +205,48 @@ 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. + 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. + + Consider a set {Task_1,...Task_{M+1}} of M+1 tasks on a system with M + CPUs, with the first task Task_1=(P,P,P) having period, relative deadline + and WCET equal to P. The remaining M tasks Task_i=(e,P-1,P-1) have an + arbitrarily small worst case execution time (indicated as "e" here) and a + period smaller than the one of the first task. Hence, if all the tasks + activate at the same time t, global EDF schedules these M tasks first + (because their absolute deadlines are equal to t + P - 1, hence they are + smaller than the absolute deadline of Task_1, which is t + P). As a + result, Task_1 can be scheduled only at time t + e, and will finish at + time t + e + P, after its absolute deadline. The total utilisation of the + task set is U = M · e / (P - 1) + P / P = M · 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]. Note: the example in the original paper by Dhall has been + slightly simplified here (for example, Dhall more correctly computed + lim_{e->0}U). + + 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(WCET_i / P_i) <= M - (M - 1) · U_max + where U_max = max{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 noted). 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. 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 @@ -244,6 +282,29 @@ 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. 2–3, pp. 187–205, 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 -- To unsubscribe from this list: send the line "unsubscribe linux-kernel" in the body of a message to majordomo@vger.kernel.org More majordomo info at http://vger.kernel.org/majordomo-info.html Please read the FAQ at http://www.tux.org/lkml/