Real-Time Systems,
COSC-4301-01,
Lecture 4
Stefan Andrei
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COSC-4301-01, Lecture 4
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Reminder of the last lecture
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Sporadic tasks
Scheduling nonpreemptable tasks
Scheduling nonpreemptable sporadic tasks
Scheduling nonpreemptable tasks with
precedence constraints
Communicating periodic tasks: deterministic
rendezvous model
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Overview of This Lecture
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Multiprocessor scheduling
Available scheduling tools
Available real-time operating systems
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Multiprocessor scheduling
Generalizing the scheduling problem from a
uniprocessor to a multiprocessor system increases
the problem complexity since we now have to
tackle the problem of assigning tasks to specific
processors.
In fact, for two or more processors, no scheduling
algorithm can be optimal without a priori knowledge
of the:
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Deadlines
Computation times
Start times of the tasks.
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Scheduling single-instance tasks
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Given n identical processors and m tasks at
time i, m > n, our objective is to ensure that
all tasks complete their execution by their
respective deadlines.
If m ≤ n (i.e., the number of tasks does not
exceed the number of processors), the
problem is trivial since each task has its own
processor.
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Schedule representation
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Static schedule representations:
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Dynamic schedule representations:
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Gantt charts
Timing diagrams
Scheduling game boards [Dertouzos, Mok; 1989]
Example of two-processor system (n=2) for
three single-instance tasks:
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J1: S1 = 0, c1 = 1, D1 = 2
J2: S2 = 0, c2 = 2, D2 = 3
J3: S3 = 0, c3 = 4, D3 = 4
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A Gantt chart for Example of slide 5
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The below figure shows the Gantt chart of a
feasible schedule for this task set.
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Figure 3.23 from [Cheng; 2005], page 66
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A timing diagram for Example of slide 5
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The below figure shows the timing diagram of
a feasible schedule for this task set:
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Figure 3.24 from [Cheng; 2005], page 67
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A game board for Example of slide 5
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The below figure shows the scheduling game
board representation of this task set at time i=0.
The x-axis shows the laxity of a task and the yaxis shows its remaining computation time.
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Figure 3.25 from [Cheng; 2005], page 66
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Scheduling single-instance tasks with
game board
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Let C(i) denote the remaining computation time of a
task at time i, and let L(i) denote the laxity (slack) of a
task at time i (i.e., L(i)=D(i)-C(i)-S(i)).
On the L-C plane of the scheduling board, executing
any n of the m tasks in parallel corresponds to moving
at most n of the m tokens one division (time unit)
downward and parallel to the C-axis.
Thus, for tasks executed:
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L(i+1) = L(i), C(i+1)=C(i)-1
Tokens corresponding to the remaining tasks that are
not executed move to the left toward the C-axis.
Thus, for tasks not executed:
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L(i+1) = L(i)-1, C(i+1)=C(i)
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Rules for the Scheduling Game Board
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Each configuration of tokens on the L-C plane
represents the scheduling problem at a point in time.
The rules for the scheduling game are:
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Initially, the starting L-C plane configuration with tokens
representing the tasks to be scheduled is given.
At each step of the game, the scheduler can move at most n
tokens one division downward toward the horizontal axis.
The rest of the tokens move leftward toward the vertical axis.
Any token reaching the horizontal axis can be ignored (it has
completed its execution).
The scheduler fails if any token crosses the vertical axis into
the second quadrant before reaching the horizontal axis.
The scheduler wins if no failure occurs.
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EDF scheduler fails (ex. From slide 5)
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Example of two-processor system (n=2) for three
single-instance tasks:
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J1: S1 = 0, c1 = 1, D1 = 2
J2: S2 = 0, c2 = 2, D2 = 3
J3: S3 = 0, c3 = 4, D3 = 4
J1 and J2 have earlier absolute deadline, so they are
assigned to start.
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Figure 3.26 from [Cheng; 2005], page 68
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LL scheduler wins (ex. From slide 5)
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At time 0, J3 has the lowest
laxity, so it is assigned to start.
The other one can be J1 (since
it has same laxity as J2).
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Figure 3.27 from [Cheng; 2005], page
69
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Conflict-Free Task Sets
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For two or more processors, no deadline
scheduling algorithm can be optimal without a
priori knowledge of the deadlines, computations
times, and start times of the tasks.
If no such a priori knowledge is available, optimal
scheduling is possible if the set of tasks does not
have subsets of conflict with each another
[Dertouzos, Mok; 1989].
A special case is that in which all tasks have unit
computation times (then EDF algorithm is optimal
even for the multiprocessor case).
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Conflict-Free Task Sets
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We divide the scheduling game board in 3
regions:
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R1(k)={Jj: Dj ≤ k}
R2(k)={Jj: Lj ≤ k and Dj > k}
R3(k)={Jj: Lj > k}
where k is the number time units.
Surplus computing power function:
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F(k,i)=kn-ΣR1Cj-ΣR2(k-Lj), for every positive integer k.
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A Necessary Condition for ConflictFree Task Sets
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F(k,i) provides a measure of the surplus
computing power of the multiprocessor
system in terms of available processor time
units between a given time instant i and k
time units into the future.
A necessary condition for scheduling to meet
the deadlines of a set of tasks whose start
times are the same (at time i=0) is:
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For all k>0, F(k,0) ≥ 0.
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A Sufficient Condition for ConflictFree Task Sets
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Schedulability test 9:
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For a multiprocessor system, if a schedule exists
that meets the deadlines of a set of singleinstance tasks whose start times are the same,
then the same set of tasks can be scheduled at
run-time even if their start times are different and
not known a priori.
Knowledge of pre-assigned deadlines and
computation times alone is enough to schedule
using LL-algorithm.
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Scheduling single-instance tasks.
Example
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Let us consider the following preemptable single-instance
task set:
J1: S1 = 0, c1 = 1, D1 = 2
J2: S2 = 0, c2 = 2, D2 = 4
J3: S3 = 0, c3 = 4, D3 = 5
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how many processors do the above tasks set need to execute?
check the applicability of EDF-scheduling method.
draw a game board and check the applicability LL-scheduling
method.
check the applicability of schedulability test 9.
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Scheduling single-instance tasks.
Example
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One processor does not suffice as at least
one job will miss its deadline.
Hence, there is a need of 2 processors to
execute the given task set.
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Scheduling game board. EDF strategy.
Initial state
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Scheduling game board. Time t=1(EDF)
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By EDF algorithm, J1 and J2 execute first since
their absolute deadlines are earlier than J3.
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Scheduling game board. Time t=2(EDF)
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Since J2 reaches horizontal axis, it is completed.
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Scheduling game board. Time t=5(EDF)
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J2 completes its execution at time t=5.
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Scheduling game board. LL strategy.
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To check whether LL strategy works, we
consider the scheduling board at the initial
state (slide 20).
At time t=0, both J1 and J3 have laxity 1.
Hence J1 and J3 move downward (vertically)
and J2 moves to the left (horizontally).
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Scheduling game board. Time t=1 (LL)
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time t=1, J1 completes execution so the remaining
tasks are J2 and J3.
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Scheduling game board. Time t=2 (LL)
 At time t=2, J2 and J3 again move downward.
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Scheduling game board. Time t=3 (LL)
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Scheduling game board. Time t=4 (LL)
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Schedulability test 9 applied to our
example
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Since there is an LL schedule for the above
task set, it means the task set can be
scheduled at run-time even if their start times
are different and not known a priori (using the
LL-algorithm).
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Scheduling Periodic Tasks
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LL scheduler is optimal for a set of singleinstance tasks satisfying a sufficient
condition.
This makes it possible to schedule tasks
without knowing their release times in
advance.
This LL scheduler is no longer optimal for
periodic tasks.
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A Sufficient Condition for Scheduling
Periodic Tasks
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Schedulability test 10:
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Given a set of k independent, preemptable (at
discrete time instants), and periodic tasks on a
multiprocessor system with n processors with
U=c1/p1+…+ck/pk≤n, let
T=GCD(p1, …, pk), and t=GCD(T,T(c1/p1), …,
T(ck/pk)).
A sufficient condition for feasible scheduling of
this task set is t is integral.
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Scheduling Periodic Tasks. Example 1
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Consider two processors (n=2) and four tasks (period
= deadline):
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J1: c1=32, p1=40
J2: c2=3, p2=10
J3: c3=4, p3=20
J4: c4=7, p4=10.
U=32/40+3/10+4/20+7/10=2 ≤ n
T=GCD(40, 10, 20, 10) = 10
t=GCD(10, 10(32/40), 10(3/10), 10(4/20), 10(7/10)) =
GCD(10, 8, 3, 2, 7)=1.
Since 1 is integral, a feasible schedule exists for this
tasks set (according to Schedulability test 10).
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A Feasible Schedule for Example 1
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The tasks are assigned to processor 1 and “fill it up”
until we encounter a task that cannot be scheduled
on this processor.
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Figure 3.28 from [Cheng; 2005], page 71
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Scheduling Periodic Tasks. Example 2
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Let us consider the following preemptable, periodic,
and independent task set:
J1: c1=40, p1=50
J2: c2=4, p2=10
J3: c3=5, p3=25
J4: c4=6, p4=10.
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compute the utilization rate.
check the applicability of schedulability test 10.
find a feasible schedule for the above task set.
if the above tasks are non-preemptable, is there any
feasible schedule for it?
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Scheduling Periodic Tasks. Example 2
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U = 2, hence two processors are needed to
schedule the above task set.
T = GCD(50, 10, 25, 10) = 5
t = GCD(T, T(c1/p1), T(c2/p2), T(c3/p3), T(c4/p4))
= GCD(5, 5*40/50, 5*4/10, 5*5/25, 5*6/10) = 1.
Since the value of t is integral, a feasible
schedule exists for this task.
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Scheduling Periodic Tasks. Example 2
 Scheduling for preemptable task set using two
processors and RM Scheduling algorithm:
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Scheduling Periodic Tasks. Example 2
 Scheduling for non-preemptable task set using two
processors:
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Available scheduling tools
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A variety of tools are available for scheduling
and schedulability analysis of real-time tasks.
Three of these are:
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PERTS (also called RAPID RMA) is downloadable
from http://www.tripac.com/
PerfoRMAx is downloadable from
http://www.aonix.com/
TimeWiz is downloadable from (n.t., obsolete)
http://www.timesys.com/
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Available real-time operating systems
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The goals of conventional, non-real-time operating
systems are to provide a convenient interface
between the user and the computer hardware while
attempting to maximize average throughput, to
minimize average time for tasks, and to ensure the
fair and correct sharing of resources.
However, meeting task deadlines is not an
essential objective in non-real-time operating
systems since its scheduler usually does not
consider the deadlines of individual tasks when
making scheduling decisions.
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Available real-time operating systems
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For real-time applications in which task deadlines
must be satisfied, a real-time operating system
(RTOS) with an appropriate scheduler for
scheduling tasks with timing constraints must be
used.
Since the late 1980s, several experiments as well
as commercial RTOSs have been developed,
most of which are extensions and modifications of
existing operating systems such as UNIX.
Most current RTOSs conform to the IEEE POSIX
standard and its real-time extensions.
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Commercial real-time operating systems
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LynxOS (http://www.lynx.com/) is Lynux’s hard
RTOS based on the LINUX. It is scalable,
Linux-compatible, and highly deterministic.
RTMX O/S (http://www.rtmx.com/) has
support for X11 and motif on M68K, MIPS,
SPARC and PowerPC processors.
VxWorks and pSOSystem
(http://www.winddriver.com/products/html/os.ht
ml) are Wind River’s RTOS’s with a flexible,
scalable, and reliable architecture for most
CPU platforms.
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Summary
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Multiprocessor scheduling
Available scheduling tools
Available real-time operating systems
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Reading suggestions
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From [Cheng; 2002]
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Chapter 3, sections 3.3, 3.4, 3.5
Chapters 3, 10 and 11 of [Kopetz; 1997]
Chapter 2 of [Stankovic, Spuri, Ramamritham,
Buttazzo; 1998]
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Coming up next
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SAT-based Scheduling
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Thank you for your attention!
Questions?
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