1. Overview
2. Scheduling algorithm
3. Scheduling Real-Time Tasks
+ 3.1 Definitions
+ 3.2 Schedulability Analysis for Uniprocessor Systems
+ 3.3 Schedulability Analysis for Multiprocessor Systems
+ 3.4 Relationship with SCHED_DEADLINE Parameters
4. Bandwidth management
4.1 System-wide settings
4.2 Task interface
"admission control" strategy (see Section "4. Bandwidth management") is used
(clearly, if the system is overloaded this guarantee cannot be respected).
- Summing up, the CBS[2,3] algorithms assigns scheduling deadlines to tasks so
+ Summing up, the CBS[2,3] algorithm assigns scheduling deadlines to tasks so
that each task runs for at most its runtime every period, avoiding any
interference between different tasks (bandwidth isolation), while the EDF[1]
algorithm selects the task with the earliest scheduling deadline as the one
suited for periodic or sporadic real-time tasks that need guarantees on their
timing behavior, e.g., multimedia, streaming, control applications, etc.
+3.1 Definitions
+------------------------
+
A typical real-time task is composed of a repetition of computation phases
(task instances, or jobs) which are activated on a periodic or sporadic
fashion.
arrival time r_j (the time when the job starts), an amount of computation
time c_j needed to finish the job, and a job absolute deadline d_j, which
is the time within which the job should be finished. The maximum execution
- time max_j{c_j} is called "Worst Case Execution Time" (WCET) for the task.
+ time max{c_j} is called "Worst Case Execution Time" (WCET) for the task.
A real-time task can be periodic with period P if r_{j+1} = r_j + P, or
sporadic with minimum inter-arrival time P is r_{j+1} >= r_j + P. Finally,
d_j = r_j + D, where D is the task's relative deadline.
+ Summing up, a real-time task can be described as
+ Task = (WCET, D, P)
+
The utilization of a real-time task is defined as the ratio between its
WCET and its period (or minimum inter-arrival time), and represents
the fraction of CPU time needed to execute the task.
- If the total utilization sum_i(WCET_i/P_i) is larger than M (with M equal
+ If the total utilization U=sum(WCET_i/P_i) is larger than M (with M equal
to the number of CPUs), then the scheduler is unable to respect all the
deadlines.
Note that total utilization is defined as the sum of the utilizations
More precisely, it can be proven that using a global EDF scheduler the
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_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 utilization.
+ 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
+ utilization[12].
+
+3.2 Schedulability Analysis for Uniprocessor Systems
+------------------------
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
of all the tasks executing on a CPU if and only if the total utilization
of the tasks running on such a CPU is smaller or equal than 1.
If D_i != P_i for some task, then it is possible to define the density of
- a task as C_i/min{D_i,P_i}, and EDF is able to respect all the deadlines
- of all the tasks running on a CPU if the sum sum_i C_i/min{D_i,P_i} of the
- densities of the tasks running on such a CPU is smaller or equal than 1
- (notice that this condition is only sufficient, and not necessary).
+ a task as WCET_i/min{D_i,P_i}, and EDF is able to respect all the deadlines
+ of all the tasks running on a CPU if the sum of the densities of the tasks
+ running on such a CPU is smaller or equal than 1:
+ sum(WCET_i / min{D_i, P_i}) <= 1
+ It is important to notice that this condition is only sufficient, and not
+ necessary: there are task sets that are schedulable, but do not respect the
+ condition. For example, consider the task set {Task_1,Task_2} composed by
+ Task_1=(50ms,50ms,100ms) and Task_2=(10ms,100ms,100ms).
+ EDF is clearly able to schedule the two tasks without missing any deadline
+ (Task_1 is scheduled as soon as it is released, and finishes just in time
+ to respect its deadline; Task_2 is scheduled immediately after Task_1, hence
+ its response time cannot be larger than 50ms + 10ms = 60ms) even if
+ 50 / min{50,100} + 10 / min{100, 100} = 50 / 50 + 10 / 100 = 1.1
+ Of course it is possible to test the exact schedulability of tasks with
+ D_i != P_i (checking a condition that is both sufficient and necessary),
+ but this cannot be done by comparing the total utilization or density with
+ a constant. Instead, the so called "processor demand" approach can be used,
+ computing the total amount of CPU time h(t) needed by all the tasks to
+ respect all of their deadlines in a time interval of size t, and comparing
+ such a time with the interval size t. If h(t) is smaller than t (that is,
+ the amount of time needed by the tasks in a time interval of size t is
+ smaller than the size of the interval) for all the possible values of t, then
+ EDF is able to schedule the tasks respecting all of their deadlines. Since
+ performing this check for all possible values of t is impossible, it has been
+ proven[4,5,6] that it is sufficient to perform the test for values of t
+ between 0 and a maximum value L. The cited papers contain all of the
+ mathematical details and explain how to compute h(t) and L.
+ In any case, this kind of analysis is too complex as well as too
+ time-consuming to be performed on-line. Hence, as explained in Section
+ 4 Linux uses an admission test based on the tasks' utilizations.
+
+3.3 Schedulability Analysis for Multiprocessor Systems
+------------------------
On multiprocessor systems with global EDF scheduling (non partitioned
systems), a sufficient test for schedulability can not be based on the
- utilizations (it can be shown that task sets with utilizations slightly
- larger than 1 can miss deadlines regardless of the number of CPUs M).
- However, as previously stated, enforcing that the total utilization 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.
+ utilizations or densities: it can be shown that even if D_i = P_i task
+ sets with utilizations 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 utilization 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 utilization (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 utilization 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 utilization 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 utilization is smaller or equal than M then the response times of
+ the tasks are limited.
+
+3.4 Relationship with SCHED_DEADLINE Parameters
+------------------------
- 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:
+ Finally, it is important to understand the relationship between the
+ SCHED_DEADLINE scheduling parameters described in Section 2 (runtime,
+ deadline and period) and the real-time task parameters (WCET, D, P)
+ described in this section. Note that the tasks' temporal constraints are
+ represented by its absolute deadlines d_j = r_j + D described above, while
+ SCHED_DEADLINE schedules the tasks according to scheduling deadlines (see
+ Section 2).
+ 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 all the jobs' deadlines of a task are respected.
+ In order to do this, a task must be scheduled by setting:
- runtime >= WCET
- deadline = D
- period <= P
- IOW, if runtime >= WCET and if period is >= P, then the scheduling deadlines
+ IOW, if runtime >= WCET and if period is <= P, then the scheduling deadlines
and the absolute deadlines (d_j) coincide, so a proper admission control
allows to respect the jobs' absolute deadlines for this task (this is what is
called "hard schedulability property" and is an extension of Lemma 1 of [2]).
Symposium, 1998. http://retis.sssup.it/~giorgio/paps/1998/rtss98-cbs.pdf
3 - L. Abeni. Server Mechanisms for Multimedia Applications. ReTiS Lab
Technical Report. http://disi.unitn.it/~abeni/tr-98-01.pdf
+ 4 - J. Y. Leung and M.L. Merril. A Note on Preemptive Scheduling of
+ Periodic, Real-Time Tasks. Information Processing Letters, vol. 11,
+ no. 3, pp. 115-118, 1980.
+ 5 - S. K. Baruah, A. K. Mok and L. E. Rosier. Preemptively Scheduling
+ Hard-Real-Time Sporadic Tasks on One Processor. Proceedings of the
+ 11th IEEE Real-time Systems Symposium, 1990.
+ 6 - S. K. Baruah, L. E. Rosier and R. R. Howell. Algorithms and Complexity
+ 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
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