Real-Time Scheduling
Scheduling of Reactive Systems
[Some parts of this lecture are based on a real-time systems course
of Colin Perkins
http://csperkins.org/teaching/rtes/index.html]
1
Reminder of Basic Notions
Jobs are executed on processors and need resources
Parameters of jobs
temporal:
release time – ri
execution time – ei
absolute deadline – di
derived params: relative deadline (Di), completion time,
response time, ...
functional:
laxity type: hard vs soft
preemptability
interconnection
precedence constraints (independence)
resource
what resources and when are used by the job
Tasks = sets of jobs
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Reminder of Basic Notions
Schedule assigns, in every time instant, processors and
resources to jobs
valid schedule = correct (common sense)
Feasible schedule = valid and all hard real-time tasks meet
deadlines
Set of jobs is schedulable if there is a feasible schedule for it
Scheduling algorithm computes a schedule for a set of jobs
Scheduling algorithm is optimal if it always produces a feasible
schedule whenever such a schedule exists, and if a cost function
is given, minimizes the cost
We have considered scheduling of individual jobs
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Scheduling Reactive Systems
From this point on we concentrate on reactive systems
i.e. systems that run for unlimited amount of time
Recall that a task is a set of related jobs that jointly provide
some system function.
We consider various types of tasks
Periodic
Aperiodic
Sporadic
Differ in execution time patterns for jobs in the tasks
Must be modeled differently
Differing scheduling algorithms
Differing impact on system performance
Differing constraints on scheduling
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Periodic Tasks
A set of jobs that are executed repeatedly at regular time
intervals can be modeled as a periodic task
Time
Ji,1
ri,1
Ji,2
ri,2
Ji,3
ri,3
Ji,4
ri,4
· · ·
ϕi
Each periodic task Ti is a sequence of jobs
Ji,1, Ji,2, . . . Ji,n, . . .
The phase ϕi of a task Ti is the release time ri,1 of the ﬁrst
job Ji,1 in the task Ti ;
tasks are in phase if their phases are equal
The period pi of a task Ti is the minimum length of all time
intervals between release times of consecutive jobs in Ti
The execution time ei of a task Ti is the maximum execution
time of all jobs in Ti
The relative deadline Di is relative deadline of all jobs in Ti
(The period and execution time of every periodic task in the system are
known with reasonable accuracy at all times)
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Periodic Tasks – Notation
The 4-tuple Ti = (ϕi, pi, ei, Di) refers to a periodic task Ti with phase
ϕi, period pi, execution time ei, and relative deadline Di
For example: jobs of T1 = (1, 10, 3, 6) are
released at times 1, 11, 21, . . .,
execute for 3 time units,
have to be ﬁnished in 6 time units (the ﬁrst by 7, the second by 17, ...)
Default phase of Ti is ϕi = 0 and default relative deadline is di = pi
T2 = (10, 3, 6) satisﬁes ϕ = 0, pi = 10, ei = 3, Di = 6, i.e. jobs of T2 are
released at times 0, 10, 20, . . .,
execute for 3 time units,
have to be ﬁnished in 6 time units (the ﬁrst by 6, the second by 16, ...)
T3 = (10, 3) satisﬁes ϕ = 0, pi = 10, ei = 3, Di = 10, i.e. jobs of T3 are
released at times 0, 10, 20, . . .,
execute for 3 time units,
have to be ﬁnished in 10 time units (the ﬁrst by 10, the second by 20, ...)
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Periodic Tasks – Hyperperiod
The hyper-period H of a set of periodic tasks is the least
common multiple of their periods
If tasks are in phase, then H is the time instant after which the pattern of job
release/execution times starts to repeat
0 5 10 15 20 25 30
H H
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Aperiodic and Sporadic Tasks
Many real-time systems are required to respond to
external events
The tasks resulting from such events are sporadic and
aperiodic tasks
Sporadic tasks – hard deadlines of jobs
e.g. autopilot on/off in aircraft
Aperiodic tasks – soft deadlines of jobs
e.g. sensitivity adjustment of radar surveilance system
Inter-arrival times between consecutive jobs are identically
and independently distributed according to a probability
distribution A(x)
Execution times of jobs are identically and independently
distributed according to a probability distribution B(x)
In the case of sporadic tasks, the usual goal is to decide,
whether a newly released job can be feasibly scheduled with
the remaining jobs in the system
In the case of aperiodic tasks, the usual goal is to minimize the
average response time 8
Scheduling – Classiﬁcation of Algorithms
Static vs Dynamic
Static – decisions based on ﬁxed parameters assigned to
tasks/jobs before their activation
Dynamic – decisions based on dynamic parameters that
may change during computation
Off-line vs Online
Off-line – sched. algorithm is executed on the whole task
set before activation
Online – schedule is updated at runtime every time a new
task enters the system
Optimal vs Heuristic
Optimal – algorithm computes a feasible schedule and
minimizes cost of soft real-time jobs
Heuristic – algorithm is guided by heuristic function; tends
towards optimal schedule, may not give one
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Scheduling – Clock-Driven
Decisions about what jobs execute when are made at speciﬁc
time instants
these instants are chosen before the system begins
execution
Usually regularly spaced, implemented using a periodic
timer interrupt
Scheduler awakes after each interrupt, schedules jobs to
execute for the next period, then blocks itself until the next
interrupt
E.g. the helicopter example with the interrupt every 1/180 th of a
second
Typically in clock-driven systems:
All parameters of the real-time jobs are ﬁxed and known
A schedule of the jobs is computed off-line and is stored for
use at runtime; thus scheduling overhead at run-time can
be minimized
Simple and straight-forward, not ﬂexible
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Scheduling – Priority-Driven
Assign priorities to jobs, based on some algorithm
Make scheduling decisions based on the priorities, when events
such as releases and job completions occur
Priority scheduling algorithms are event-driven
Jobs are placed in one or more queues; at each event, the
ready job with the highest priority is executed
(The assignment of jobs to priority queues, along with rules such as
whether preemption is allowed, completely deﬁnes a priority-driven alg.)
Priority-driven algs. make locally optimal scheduling decisions
Locally optimal scheduling is often not globally optimal
Priority-driven algorithms never intentionally leave idle
processors
Typically in priority-driven systems:
Some parameters do not have to be ﬁxed or known
A schedule is computed online; usually results in larger
scheduling overhead as opposed to clock-driven scheduling
Flexible – easy to add/remove tasks or modify parameters
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Clock-Driven & Priority-Driven Example
T1 T2 T3
pi 3 5 10
ei 1 2 1
Clock-Driven:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
· · ·
· · ·
· · ·
Priority-driven: T1 T2 T3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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Real-Time Scheduling
Scheduling of Reactive Systems
Clock-Driven Scheduling
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Current Assumptions
Fixed number, n, of periodic tasks T1, . . . , Tn
Parameters of periodic tasks are known a priori
Execution time ei,k of each job Ji,k in a task Ti is ﬁxed
For a job Ji,k in a task Ti we have
ri,1 = ϕi = 0 (i.e., synchronized)
ri,k = ri,k−1 + pi
We allow aperiodic jobs
assume that the system maintains a single queue for
aperiodic jobs
Whenever the processor is available for aperiodic jobs, the
job at the head of this queue is executed
We treat sporadic jobs later
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Static, Clock-Driven Scheduler
Construct a static schedule ofﬂine
The schedule speciﬁes exactly when each job executes
The amount of time allocated to every job is equal to its
execution time
The schedule repeats each hyperperiod
i.e. it sufﬁces to compute the schedule up to hyperperiod
Can use complex algorithms ofﬂine
Runtime of scheduling algorithm is not relevant
Can compute a schedule that optimizes some
characteristics of the system
e.g. a schedule where the idle periods are nearly periodic (useful
to accommodate aperiodic jobs)
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Example
T1 = (4, 1), T2 = (5, 1.8), T3 = (20, 1), T4 = (20, 2)
Hyperperiod H = 20
0 4 8 12 16 20 24
Aper.
T1
T2
T3
T4
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Implementation of Static Scheduler
Store pre-computed schedule as a table
Each entry (tk , T(tk )) gives
a decision time tk
scheduling decision T(tk ) which is either a task to be
executed, or idle (denoted by I)
The system creates all tasks that are to be executed:
Allocates memory for the code and data
Brings the code into memory
Scheduler sets the hardware timer to interrupt at the ﬁrst
decision time t0 = 0
On receipt of an interrupt at tk :
Scheduler sets the timer interrupt to tk+1
If previous task overrunning, handle failure
If T(tk ) = I and aperiodic job waiting, start executing it
Otherwise, start executing the next job in T(tk )
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Example
T1 = (4, 1), T2 = (5, 1.8), T3 = (20, 1), T4 = (20, 2)
Hyperperiod H = 20
0 4 8 12 16 20 24
Aper.
T1
T2
T3
T4
tk 0.0 1.0 2.0 3.8 4.0 5.0 6.0 · · ·
T(tk ) T1 T3 T2 I T1 I T4 · · ·
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Implementation of Static Scheduler
Input: the table (tk , T(tk )) for k = 0, 1, . . . , n − 1
Task SCHEDULER:
set the number of decisions i := 0 and table entry k := 0;
set the timer to expire at t0;
do forever:
accept timer interrupt;
if an aperiodic job is executing, preempt it;
current task T := T(tk );
increment i by 1;
compute the next table entry k = i mod n;
set the timer to expire at i/n ∗ H + tk ;
if the current task T = I,
execute the job at the head of the aperiodic queue;
else
let the task T execute;
sleep;
end do.
End SCHEDULER
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Frame Based Scheduling
Arbitrary table-driven cyclic schedules ﬂexible, but
inefﬁcient
Relies on accurate timer interrupts, based on execution
times of tasks
High scheduling overhead
Easier to implement if a structure is imposed
Make scheduling decisions at periodic intervals (frames) of
length f
Execute a ﬁxed list of jobs within each frame;
no preemption within frames
How to choose the size of frames? How to compute
a schedule?
To simplify further development, assume that all parameters are in N and
choose frame sizes in N
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Frame Based Scheduling – Frame Size
0. Necessary condition for avoiding preemption of jobs is
f ≥ max
i
ei
(i.e. we want each job to have a chance to ﬁnish within a frame)
1. To minimize the number of entries in the cyclic schedule, the
hyper-period should be an integer multiple of the frame size, i.e.
pi/f − pi/f = 0
for some task Ti.
2. To allow scheduler to check that jobs complete by their deadline,
at least one frame should lie between release time of a job and
its deadline, i.e.
2 ∗ f − gcd(pi, f) ≤ Di
for all tasks Ti
Example: T1 = (4, 1.0), T2 = (5, 1.8), T3 = (20, 1.0), T4 = (20, 2.0)
Then f ∈ N satisﬁes 0.–2. iff f = 2.
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Frame Based Scheduling – Frame Size – Example
Example 1
T1 = (4, 1.0), T2 = (5, 1.8), T3 = (20, 1.0), T4 = (20, 2.0)
Then f ∈ N satisﬁes 0.–2. iff f = 2.
With f = 2 is schedulable:
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Frame Based Scheduling – Job Slices
Sometimes a system cannot meet all three frame size
constraints simultaneously (and even if it meets the
constraints, no non-preemptive schedule is feasible)
Can be solved by partitioning a job with large execution
time into slices with shorter execution times
This, in effect, allows preemption of the large job
To construct a schedule, we have to make three kinds of design
decisions (that cannot be taken independently):
Choose a frame size based on constraints
Partition jobs into slices
Place slices into frames
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Frame Based Scheduling – Job Slices – Example
Consider T1 = (4, 1), T2 = (5, 2, 7), T3 = (20, 5)
Cannot satisfy constraints: 1. ⇒ f ≥ 5 but 3. ⇒ f ≤ 4
Solve by splitting T3 into T3,1 = (20, 1), T3,2 = (20, 3), and
T3,3 = (20, 1)
(Other splits exist)
Result can be scheduled with f = 4
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Frame Based Scheduling – Job Slices
Assuming that preemption is allowed in arbitrary places, jobs
are independent and there are no resource contentions, there
is a (pseudo)polynomial time algorithm which
chooses an appropriate frame size f
partitions jobs into slices
places slices into frames
(whiteb.)
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Frame Based Scheduling – Algorithm
Compute all frame sizes satisfying conditions 1. and 2.
(not necessarily 0.)
For every frame size f construct a network ﬂow graph:
(the number of frames in one hyperperiod is F)
Vertices:
a vertex for each job Ji
a vertex for each frame j where j = 1, . . . , F
source and sink
Edges:
from Ji to j of capacity f if Ji can be scheduled in the frame j
from source to Ji of capacity ei
from every frame to sink of capacity f
Theorem 2
Max ﬂow assigned to every edge from source to Ji is ei
iff
the jobs can be partitioned into slices and the slices placed into
frames so that the resulting schedule is feasible. The ﬂows
assigned to edges of the form (Ji, j) determine the schedule.
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Frame Based Scheduling – Network Flow Example
The example with
T1 = (4, 1), T2 = (5, 2, 7),
T3 = (20, 5) and f = 4
Note that
T2 has four jobs J2,1, J2,2,
J2,3, J2,4 in one hyperper.
The only job of T3 released
in one hyperper. can be
placed into any frame
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Frame Based Scheduling – Cyclic Executive
Modify previous table-driven scheduler to be frame based
Table that drives the scheduler has F entries, where
F = H/f
The k-th entry L(k) lists the names of the job slices that are
to be scheduled in frame k (L(k) is called scheduling block)
Each job slice is implemented by a procedure
Cyclic executive executed by the clock interrupt that
signals the start of a frame:
If an aperiodic job is executing, preempts it; if a periodic
overruns, handles the overrun
Determines the appropriate scheduling block for this frame
Executes the jobs in the scheduling block
Executes jobs from the head of the aperiodic job queue for
the remainder of the frame
Less overhead than pure table driven cyclic scheduler,
since only interrupted on frame boundaries, rather than on
each job
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Scheduling Aperiodic Jobs
So far, aperiodic jobs scheduled in the background after all jobs
with hard deadlines
This may unnecessarily delay aperiodic jobs
Note: There is no advantage in completing periodic jobs early
Ideally, ﬁnish periodic jobs by their respective deadlines.
Slack Stealing:
Slack time in a frame = the time left in the frame after all
(remaining) slices execute
Schedule aperiodic jobs ahead of periodic in the slack time of
periodic jobs
The cyclic executive keeps track of the slack time left in
each frame as the aperiodic jobs execute, preempts them
with periodic jobs when there is no more slack
As long as there is slack remaining in a frame and the
aperiodic jobs queue is non-empty, the executive executes
aperiodic jobs, otherwise executes periodic
Reduces resp. time for aper. jobs, but requires accurate timers
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Example
Assume that the aperiodic queue is never empty.
Aperiodic at the ends of frames:
0 4 8 12 16 20 24
Aper.
T1
T2
T3
T4
Slack stealing:
0 4 8 12 16 20 24
Aper.
T1
T2
T3
T4
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Slack Stealing – cont.
Sl. S.
Standard
Rel. aper.
Period.
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Frame Based Scheduling – Sporadic Jobs
Let us allow sporadic jobs
i.e. hard real-time jobs whose release and exec. times are not known a priori
The scheduler determines whether to accept a sporadic job when it
arrives (and its parameters become known)
Perform acceptance test to check whether the new sporadic job
can be feasibly scheduled with all the jobs (periodic and
sporadic) in the system at that time
Acceptance check done at the beginning of the next frame; has to keep
execution times of the parts of sporadic jobs that have already executed
If there is sufﬁcient slack time in the frames before the new job’s
deadline, the new sporadic job is accepted; otherwise, rejected
Among themselves, sporadic jobs scheduled according to EDF
This is optimal for sporadic jobs
Note: rejection is often better than missing deadline
e.g. a robotic arm taking defective parts off a conveyor belt: if the arm cannot
meet deadline, the belt may be slowed down or stopped
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S1(17, 4.5) released at 3 with abs. deadline 17 and execution time 4.5;
acceptance test at 4; must be scheduled in frames 2, 3, 4; total slack in
these frames is 4, i.e. rejected
S2(29, 4) released at 5 with abs. deadline 29 and exec. time 4; acc. test
at 8; total slack in frames 3-7 is 5.5, i.e. accepted
S3(22, 1.5) released at 11 with abs. deadline 22 and exec. time 1.5;
acc. test at 12;
2 units of slack in frames 4, 5 as S3 will be executed ahead of the
remaining parts of S2 by EDF – check whether there will be enough
slack for the remaining parts of S2, accepted
S4(44, 5.0) is rejected (only 4.5 slack left)
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Handling Overruns
Overruns may happen due to failures
e.g. unexpectedly large data over which the system operates, hardware
failures, etc.
Ways to handle overruns:
Abort the overrun job at the beginning of the next frame;
log the failure; recover later
e.g. control law computation of a robust digital controller
Preempt the overrun job and ﬁnish it as an aperiodic job
use this when aborting job would cause “costly” inconsistencies
Let the overrun job ﬁnish – start of the next frame and the
execution jobs scheduled for this frame are delayed
This may cause other jobs to be delayed
depends on application
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Clock-drive Scheduling: Conclusions
Advantages:
Conceptual simplicity
Complex dependencies, communication delays, and
resource contention among jobs can be considered when
constructing the static schedule
Entire schedule in a static table
No concurrency control or synchronization needed
Easy to validate, test and certify
Disadvantages:
Inﬂexible
If any parameter changes, the schedule must be usually
recomputed
Best suited for systems which are rarely modiﬁed (e.g. controllers)
Parameters of the jobs must be ﬁxed
As opposed to most priority-driven schedulers
35
Real-Time Scheduling
Scheduling of Reactive Systems
Priority-Driven Scheduling
36
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
37
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
Note: In our case, a priority assigned to a job does not change. There are
job-level dynamic priority algorithms that vary priorities of individual jobs – we
won’t consider such algorithms.
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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
Example 3
T1 = (4, 1), T2 = (5, 2), T3 = (20, 5)
with rates 1/4, 1/5, 1/20, respectively
The priorities: T1 T2 T3
0 4 8 12 16 20
T3
T2
T1
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Fixed-priority Algorithms – Deadline Monotonic
The deadline monotonic (DM) algorithm assigns priorities to
tasks based on their relative deadlines
the shorter the deadline, the higher the priority
Observation: When relative deadline of every task matches its
period, then RM and DM give the same results
Proposition 1
When the relative deadlines are arbitrary DM can sometimes
produce a feasible schedule in cases where RM cannot.
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Rate Monotonic vs Deadline Monotonic
T1 = (50, 50, 25, 100), T2 = (0, 62.5, 10, 20), T3 = (0, 125, 25, 50)
DM is optimal (with priorities T2 T3 T1):
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 (with priorities T1 T2 T3):
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
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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
Another one is the least slack time (LST)
The job queue is ordered by least slack time
Recall that the slack time of a job Ji at time t is equal to di − t − x where x is
the remaining computation time of Ji at time t
Comments:
There is also a strict LST which reassigns priorities to jobs whenever
their slacks change relative to each other – won’t consider
Standard “non real-time” algorithms such as FIFO and LIFO are also
dynamic-priority algorithms
We focus on EDF here, leave some LST for homework
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EDF – Example
T1 = (2, 1) and T2 = (5, 2.5)
0 1 2 3 4 5 6 7 8 9 10
T2
T1
Note that the processor is 100% “utilized”, not surprising :-)
43
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?
44