"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
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.
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.
On multiprocessor systems with global EDF scheduling (non partitioned
systems), a sufficient test for schedulability can not be based on the
- 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.
4. Bandwidth management
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projects / firefly-linux-kernel-4.4.55.git / blobdiff