Real-Time Scheduling
Scheduling of Reactive Systems
Priority-Driven Scheduling
1
Current Assumptions
Single processor
Fixed number, n, of independent periodic tasks
i.e. there is no dependency relation among jobs
Jobs can be preempted at any time and never suspend
themselves
No aperiodic and sporadic jobs
No resource contentions
Moreover, unless otherwise stated, we assume that
Scheduling decisions take place precisely at
release of a job
completion of a job
(and nowhere else)
Context switch overhead is negligibly small
i.e. assumed to be zero
There is an unlimited number of priority levels
2
Fixed-Priority vs Dynamic-Priority Algorithms
A priority-driven scheduler is on-line
i.e. it does not precompute a schedule of the tasks
It assigns priorities to jobs after they are released and places the
jobs in a ready job queue in the priority order
with the highest priority jobs at the head of the queue
At each scheduling decision time, the scheduler updates the
ready job queue and then schedules and executes the job at the
head of the queue
i.e. one of the jobs with the highest priority
Fixed-priority = all jobs in a task are assigned the same priority
Dynamic-priority = jobs in a task may be assigned different priorities
3
Fixed-priority Algorithms – Rate Monotonic
Best known fixed-priority algorithm is rate monotonic (RM) scheduling
that assigns priorities to tasks based on their periods
The shorter the period, the higher the priority
The rate is the inverse of the period, so jobs with higher rate
have higher priority
RM is very widely studied and used
The deadline monotonic (DM) algorithm assigns priorities to tasks
based on their relative deadlines
the shorter the deadline, the higher the priority
4
Rate Monotonic vs Deadline Monotonic
T1 = (50, 50, 25, 100), T2 = (0, 62.5, 10, 20), T3 = (0, 125, 25, 50)
Priorities: T2 T3 T1
DM is optimal:
50 100 150 200 250
0 62.5 125 187.5 250
0 125 250
20 82.5 145 207.5
50 175
T3
T2
T1
RM is not optimal:
50 100 150 200 250
0 62.5 125 187.5 250
0 125 250
20 82.5 145 207.5
50 175
T3
T2
T1
5
Dynamic-priority Algorithms
Best known is earliest deadline first (EDF) that assigns
priorities based on current deadlines
At the time of a scheduling decision, the job queue is
ordered by earliest deadline
Example 1
T1 = (2, 1) and T2 = (5, 2.5)
0 1 2 3 4 5 6 7 8 9 10
T2
T1
6
Summary of Algorithms
In what follows we consider
Dynamic-priority algorithms: EDF
Fixed-priority algorithms: RM and DM
We consider the following questions:
Are the algorithms optimal?
How to efficiently (or even online) test for schedulability?
To measure abilities of scheduling algorithms and to get fast
online tests of schedulability we use a notion of utilization
7
Utilization
Utilization ui of a periodic task Ti with period pi and
execution time ei is defined by ui := ei/pi
The fraction of time a periodic task with period pi and
execution time ei keeps a processor busy
Total utilization UT of a set of tasks T = {T1, . . . , Tn} is
defined as the sum of utilizations of all tasks of T, i.e. by
UT
:=
n
i=1
ui
U is a schedulable utilization of an algorithm ALG if all sets
of tasks T satisfying UT ≤ U are schedulable by ALG.
Maximum schedulable utilization UALG of an algorithm ALG
is the supremum of schedulable utilizations of ALG.
That is
If UT
< UALG, then T is schedulable by ALG.
If U > UALG, then there is T with UT
≤ U that is not
schedulable by ALG.
8
Real-Time Scheduling
Priority-Driven Scheduling
Dynamic-Priority
9
Optimality of EDF
Theorem 2
Let T be a set of independent, preemptable periodic tasks with
Di ≥ pi. The following statements are equivalent:
1. T can be feasibly scheduled on one processor
2. total utilization UT of T is equal to or less than one
3. T is schedulable using EDF
(i.e. UEDF = 1)
Proof.
1.⇒2. We prove that UT
> 1 implies that T is not schedulable (whiteb.)
2.⇒3. Whiteboard ...
3.⇒1. Trivial
10
Optimality of EDF – proof – 2.⇒3.
11
Optimality of EDF – proof – 2.⇒3.
12
Density and EDF
What about tasks with Di < pi ?
Density of a task Ti with period pi, execution time ei and relative
deadline Di is defined by
ei/ min(Di, pi)
Total density ∆T of a set of tasks T is the sum of densities of
tasks in T
Note that if Di < pi for some i, then ∆T
> UT
Theorem 3
A set T of independent, preemptable, periodic tasks can be
feasibly scheduled on one processor if ∆T ≤ 1.
Note that this is NOT a necessary condition!
13
Schedulability Test For EDF
The problem: Given a set of independent, preemptable, periodic
tasks T = {T1, . . . , Tn} where each Ti has a period pi, execution time
ei, and relative deadline Di, decide whether T is schedulable by EDF.
Solution using utilization and density:
If pi ≤ Di for each i, then it suffices to decide whether UT ≤ 1.
Otherwise, decide whether ∆T ≤ 1:
If yes, then T is schedulable with EDF
If not, then T does not have to be schedulable
Note that
Phases of tasks do not have to be specified
Parameters may vary: increasing periods or deadlines, or
decreasing execution times does not prevent schedulability
14
Schedulability Test for EDF – Example
Consider a digital robot controller
A control-law computation
takes no more than 8 ms
the sampling rate: 100 Hz, i.e. computes every 10 ms
Feasible? Trivially yes ....
Add Built-In Self-Test (BIST)
maximum execution time 50 ms
want a minimal period that is feasible
With 250 ms still feasible ....
Add a telemetry task
maximum execution time 15 ms
want to minimize the deadline on telemetry
period may be large
Reducing BIST to once a second, deadline on telemetry
may be set to 100 ms ....
15