sched/dl/Documentation: Add some notes on EDF schedulability

 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.

 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 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 sum(WCET_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).

 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

      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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