1. Periodic Task Scheduling
張軒彬助理教授
中興大學資訊科學系
In this chapter, we introduce the periodic task scheduling.
Periodic activities represent the major computational demand in most real-time systems. For example, sensory data acquisition, control loops, system monitoring are all periodic tasks.

2. Outline Processor Utilization Factor Rate Monotonic Scheduling
Earliest Deadline First
In this chapter, we first introduce the processor utilization factor. Then, three periodic tasks scheduling algorithms, Rate Monotonic and Earliest Deadline First, are described sequentially.

3. Processor Utilization Factor
Given a set T of n periodic tasks, processor utilization factor U is the fraction of processor time spent in the execution of a task set
There exists a maximum value of U below which T is schedulable and above with T is not schedulable
Depend on the task set and used scheduling algorithm
Let Uub(T, A) denote the upper bound of the processor utilization factor for a task set T under a give algorithm A
If U = Uub(T, A), the set T is said to fully utilize the processor
Definition: Processor Utilization Factor
Given a set T of n periodic tasks, processor utilization factor U is the fraction of processor time spent in the execution of a task set
Where Ci is task i’s computation time and Ti is its period.
From the definition, processor utilization factor provides a measure of the CPU’s computational load due to the periodic task set.
Depend on the task set and applied scheduling algorithm, there exists a maximum value of U which task set T is schedulable and above with T is not schedulable.

4. Processor Utilization Factor (Cont.)
For a given algorithm A, the least upper bound Ulub(A) of the processor utilization factor is the minimum of the utilization factors over all task sets that fully utilize the processor
Uub1 T1
Uub2
The definitions of upper bound and the lease upper bound of the processor utilization factor are shown in the slides.
Example: as shown in the figure, each task set Ti differ in the number of tasks and their periods. Each task set may have different upper bound and the minimum one of is the least upper bound. T2
Uub3 T3
Uub4 T4
Uubm Tm U 1
Ulub

5. Processor Utilization Factor (Cont.)
Ulub(A) defines an important characteristics of a real-time scheduling algorithm.
It allow to easily verify the schedulability of a task set.
If a task set whose processor utilization factor is below Ulub(A), then it is schedulable by the algorithm A.
In contrast, if a task set’s processor utilization factor is above Ulub(A), it may or may not be schedulable by the algorithm A

6. Rate Monotonic Scheduling
Task model:
Periodic tasks
Arbitrary arrival times
Different computation time and deadlines
Preemptive
This slide shows the task model of Rate Monotonic.

7. Rate Monotonic Scheduling (Cont.)
Rate Monotonic (RM)
Assign priorities to tasks according to their request rates
Tasks with higher request rates, i.e., shorter periods, have higher priorities
Fixed-priority scheduling algorithm
Priority are assigned to tasks before execution and do not change over time
Intrinsically preemptive
The currently executing task is preempted by a newly arrived task with shorter period

8. Rate Monotonic Scheduling (Cont.)
Optimality: RM is optimal among all fixed-priority scheduling algorithms
No other fixed-priority algorithm can schedule a task set that cannot be scheduled by RM
Schedulability condition: given n tasks, the least upper bound of the processor utilization factor under RM is
In 1973, Liu and Layland showed that RM is optimal among all fixed-priority scheduling algorithms since no other fixed-priority algorithm can schedule a task set that cannot be scheduled by RM.
The least upper bound of the processor utilization factor in RM is n 1 2 3 4 5 6 7 8 9 10
Ulub
1.000
0.828
0.780
0.757
0.743
0.735
0.729
0.724
0.721
0.718

9. Rate Monotonic Scheduling (Cont.)
Concluding remarks
The schedulability condition is sufficient to guarantee the feasibility of any task set, but it is not necessary
Thus, if a task set has an utilization factor greater than Ulub and less than one, nothing can be said on the feasibility of the task set

10. Earliest Deadline First
Earliest Deadline First (EDF)
Selects tasks according to their (absolute) deadlines
Tasks with earlier deadlines will be executed at higher priority
Dynamic scheduling algorithm
The absolute deadline of a periodic task depends on the current ith instances
Thus, EDF is a dynamic priority assignment
Intrinsically preemptive
The current executing task is preempted whenever another periodic instance with earlier deadline become active

11. Earliest Deadline First (Cont.)
Optimality:
EDF is optimal among all dynamic-priority scheduling algorithms
Schedulability analysis
A set of periodic tasks is schedulable with EDF if and only if

12. Example Two tasks: Processor utilization factor
Task 1: execution time: 2, period: 5
Task 2: execution time: 4, period: 7
Processor utilization factor
U = 2/5 + 4/7 = 0.97
U > Ulcb of RM
RM cannot guarantee schedulability
U < Ulcb(=1) of EDF
EDP thus guarantee schedulability

13. Example RM T1 5 10 15 20 25 30 35
time overflow T2 7 14 21 28 35
EDF
Example, a task set whose processor utilization factor is 2/5 + 4/7 = 34/35 ~ 0.97.
Thus, the CPU spends its 97% of time in executing periodic tasks with 3% of time idle.
Since U > ln2, thus, the task set’s schedulability is not guaranteed in RM. Indeed, RM results in a miss deadline at t = 7.
In contrast, U < 1, thus the task set is schedulable under EDF. T1 5 10 15 20 25 30 35 T2 7 14 21 28 35

14. Reference
Giorgio C. Buttazzo, “Hard Real-Time Computing Systems: Predictable Scheduling Algorithms and Applications,” Kluwer Academic Publishers, 1997
Jane W. S. Liu, “Real-Time Systems,” Prentice Hall, 2002