1. CPE555A: Real-Time Embedded Systems
Lecture 6
Ali Zaringhalam
Stevens Institute of Technology
Spring 2015, arz
CS555A – Real-Time Embedded Systems Stevens Institute of Technology 1

2. Outline Scheduling Earliest-Due-Date (EDD)
Earliest-Deadline-First (EDF)
Latest Deadline First (LDF)
Rate Monotonic Scheduling (RMS)
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3. The Scheduling Problem
Input
A set of n tasks Γ = {τ1, τ2, , τn},
A set of m processors P = {P1, P2, , Pm}
A set of s types of resources R = {R1,R2, , Rs}
Constraints: timing (deadline), precedence
Output
Assigning processors from P and resources from R to tasks from Γ in order to complete all tasks under the specified constraints
This problem, in its general form, has been shown to be computationally intractable
To make the problem of constructing feasible schedules tractable, certain assumptions are made
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4. Scheduling Classification - 1
Preemptive
The running task can be interrupted at any time to assign the processor to another active task, according to a predefined scheduling policy
Non-preemptive
A task, once started, is executed by the processor until completion.
All scheduling decisions are taken when the current task terminates its execution
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5. Scheduling Classification - 2
Static
Scheduling decisions are based on fixed parameters (e.g. priority), assigned to tasks before their activation
Dynamic
Scheduling decisions are based on dynamic parameters (e.g. priority) that may change during task execution
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6. Scheduling Classification - 3
Off-line
All tasks are known at start & never change
The scheduling algorithm is executed on the entire task set before task activation
The schedule is stored in a table and later executed by a dispatcher
On-line
Tasks can arrive at run time
The scheduling decisions are taken at runtime every time a new task enters the system or when a running task terminates
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7. Scheduling Classification - 4
Optimal
An optimal schedule minimizes some given cost function defined over the task set
If there is no cost function, then the only concern is to achieve a feasible schedule meeting deadlines
An algorithm is said to be optimal with respect to feasibility if it is able to find a feasible schedule, if one exists
Heuristic
Uses a heuristic approach in taking its scheduling decisions
A heuristic algorithm tends toward the optimal schedule, but does not guarantee finding it
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8. Guarantee-Based Algorithms
A dynamic scheduler with real-time deadline and admission control
A new task arrives
The task is accepted only if the deadlines of current tasks are unaffected
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9. Example Scheduler Metrics
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10. Earliest Due Date (EDD)
Jackson’s algorithm
A set of n aperiodic tasks
A single processor
Optimization metric
Minimizing maximum lateness
Lmax=Maxi(fi-di)
Designation: 1 | sync | Lmax
sync means that tasks arrive “synchronously”, at the same time
Each task can have a different computation time and deadline
But they all arrive at the same time
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11. EDD Algorithm
No random task arrival. All tasks arrive at the same time.
Given a set of n independent tasks, any algorithm that executes the tasks in order of non-decreasing deadlines is optimal with respect to minimizing the maximum lateness
Because all tasks arrive at the same time, preemption is not an issue
The static priority assigned to each task is the deadline: the shorter the deadline, the higher the priority
We can assume that all tasks arrive at t=0
Each task is characterized by computation time and deadline (C,d)
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12. Proof of EDD
Suppose there was a non-EDD schedule S. There must be two tasks a and b where a immediately precedes b but da > db. Why?
The schedule can be improved to reduce maximum lateness by interchanging a and b
No non-EDD schedule can do better in minimizing maximum lateness
The complexity of the EDD algorithm is O(nxlogn): the complexity of the sorting algorithm
Because if there were no such instance, then the schedule would be EDD.
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13. EDD & Non-EDD Schedules
Note that fj=f’i because tasks execute consecutively
Non-EDD
Non-EDD: i finishes earlier but has a later deadline than j. In this example both fi-di and fj-dj are negative.
EDD
Two cases for possible value of L’max:
L’max = f’i – di = fj-di <= fj – dj
L’max = f’j – dj <= fj – dj
The non-EDD Lmax = fj-dj
So in both cases L’max <= Lmax
So EDD is no worst than non-EDD
Tasks are scheduled as soon as the CPU is available.
There is no preemption. A task runs to completion.
If the schedule is feasible, all deadlines are met and fi-di is negative for all i.
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14. Example 1 J1 J2 J3 J4 J5 fi 1 8 4 7 3 di 10 5 Li -2 -3 -1
J4 has maximum lateness = -1
Queue J2 J4 J3 J5 J1
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Dequeue

15. EDD & Feasibility
EDD produces a schedule which minimizes maximum lateness
But it may not be a feasible schedule in the sense of meeting all deadlines
So EDD cannot be relied upon to obtain a feasible schedule meeting all deadlines
But if there does exist a feasible schedule, then EDD can find it
From any feasible schedule you can use EDD to obtain another feasible schedule which minimizes maximum lateness
Complexity of EDD
O(nlogn) – A heap sorting algorithm
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16. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
Example 2 J1 J2 J3 J4 J5 fi 1 4 2 10 6 di 5 8 Li -1 -2
Deadline missed
Queue J4 J5 J2 J3 J1
Dequeue
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17. Dynamic Run-Time Scheduler
The scheduler lets a task to continue to completion before scheduling another task
There is no timer interrupt
The scheduler is invoked by the ISR or the OS service call invoked by the user
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18. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
Typically a lower numerical value represents a higher priority
By user
Or by the scheduler to prevent starvation of lower-priority tasks
Tasks are switched in and out when the timer interrupt kicks in and invokes the scheduler .
A non-preemptive priority-based scheduler allows a task to complete. The next task to run is determined by priority.
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19. Earliest Deadline First (EDF)
Horn’s algorithm
N independent tasks with arbitrary arrival time. Tasks are not synchronous
Preemption is allowed
1 | preemp | Lmax
Optimization metric is the same as in EDD
Minimizing maximum lateness
Lmax=Maxi(fi-di)
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20. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
EDF Algorithm
Given a set of n independent tasks with arbitrary arrival times, any algorithm that at any instant executes the task with the earliest absolute deadline among all the ready tasks is optimal with respect to minimizing the maximum lateness.
Complexity
O(n2) if ready queue is a linear list
O(nlogn) if the ready queue is implemented as a heap
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21. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
EDF Proof
At time t=4, the executing task J4 has a deadline of td=15.
s(t)=arrival time; s(tE)=execution time.
Task J2 arrives at t=4. The scheduler rethinks the existing schedule and makes changes.
This is a non-EDF feasible schedule
Maximum lateness is Lmax=0 associated with J1
Preemption is allowed: so break up time in slices. Tasks can execute in disjointed intervals
Interchanging the execution of J4 and J2 at t=4 does not increase maximum lateness
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22. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
EDF Proof - Continued
After the interchange the resulting schedule is also feasible because the deadline of J4 is after the deadline associated with J2.
Interchange the execution of J2 and J4 at t=4.
Maximum lateness remains the same at L=0 for J1
Continue this in all time slices
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23. Example Completed tasks None CS555A – Real-Time Embedded Systems
Both J1 and J2 arrive at t=0. J1 is scheduled first because its deadline is (d1=2) < (d2=5).
Completed tasks
None
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24. Example - Continued Completed tasks J1
J3 arrives at t=2. J3 has a deadline of d3=4 whereas the currently executing task J2 has a deadline of d2=5. Because d3 < d2, J2 is preempted and J3 is dispatched for execution.
Completed tasks J1
preemption
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25. Example - Continued Completed tasks J1
J4 arrives at t=3. Its deadline is d4=10. The earliest deadline is still d3=4. So J3 remains in execution. No preemption at t=3.
Completed tasks J1
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26. Example - Continued Lmax = 0 Completed tasks J1 J2 J3
J5 arrives at t=6. Its deadline is d5=9. which is earlier than the d4=10 of J4 which is currently executing. So J4 is preempted and J5 is dispatched.
Lmax = 0
Completed tasks J1 J2 J3
preemption
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27. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
Some Observations
A schedule is computed at t=0 based on the ready tasks
The schedule is re-examined only when a new task arrives
You can view the task with the earliest deadline as having the highest priority
If a feasible schedule (one that meets all deadlines) exists, EDF will find it. It also has the minimum lateness
However an EDF schedule is not necessarily feasible (may not meet deadlines)
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28. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
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29. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
After task 1, task 4 has the earliest deadline but it cant’ be scheduled because it has a dependency on task 2 which must be scheduled first. So task 3 is the first EDF task meeting dependency requirements that can be scheduled.
NO.
Task 1: meets deadline d=2
Task 2: meets deadline d=5
Task 3: meets deadline d=4
Task 4: does NOT meet deadline d=3
Task 5: meets deadline d=5
Task 6: meets deadline d=6
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30. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
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31. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
LDF assumes a fixed, finite set of tasks with deadlines. No arrival of new tasks. No preemption. 6
Stack
Push
What’s nice about a leaf node is that no other task is dependent on it.
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32. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
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33. CS555A – Real-Time Embedded Systems Stevens Institute of Technology6
Stack
Push
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34. CS555A – Real-Time Embedded Systems Stevens Institute of Technology6
Stack
Push
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35. CS555A – Real-Time Embedded Systems Stevens Institute of Technology3 5 6
Stack
Push
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36. CS555A – Real-Time Embedded Systems Stevens Institute of Technology 4 3 5 6
Stack
Push
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37. CS555A – Real-Time Embedded Systems Stevens Institute of Technology 2 4 3 5 6
Stack
Push
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38. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
Stack
Pop 1 2 4 3 5 6
Stack
Push
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39. The complexity of LDF is O(n2).
So it is also optimal with respect to feasibility. Why?
The complexity of LDF is O(n2).
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40. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
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41. EDF With Precedence Assume all tasks arrive at ai=ri=0
All tasks require execution time ci=1
r4=0
c4=1
d4=3
r2=0
c2=1
d2=5 4 2
r5=0
c5=1
d5=5 5
r1=0
c1=1
d1=1 1
Rule of thumb for this strategy: the deadline of a task cannot exceed its own deadline or the minimum slack time of its dependent tasks. 3
r6=0
c6=1
d6=6 6
r3=0
c3=1
d3=4
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ej is extecution time of task j 41 41

42. Timing Definitions - Continued
/ri
Start time si is the time at which a task starts its execution
Finishing time fi is the time at which a task finishes its execution
Response time Ri is the difference between the finishing time and the request time: Ri = fi − ri
Lateness: Li = fi −di represents the delay of a task completion with respect to its deadline; note that if a task completes before the deadline, its lateness is negative
Laxity or Slack time: Xi = di − ai − Ci is the maximum time a task can be delayed on its activation to complete within its deadline
Xi = di − Ci when ai=0
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43. EDF With Precedence Start with i=1 Tasks immediately depending on 1
d2-c2=4
d3-c3=3
d1’=min(1, min(4,3))=1
r2=0
c2=1
d2=5
r4=0
c4=1
d4=3 4 2
r5=0
c5=1
d5=5 5
r1=0
c1=1
d1’’=1 1
r3=0
c3=1
d3=4 3
Slack time: Xi = di − ai − Ci = di – Ci
because ai = ri = 0
Rule of Thumb:The deadline of a task cannot exceed its own deadline or the minimum slack time of its dependent tasks.
r6=0
c6=1
d6=6 6
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44. EDF With Precedence Next select i=2 with d2=5
Tasks immediately depending on 2
D(2) = {4, 5}
d4-c4=2
d5-c5=4
d2’=min(5, min(2,4))=2
r2=0
c2=1
d2’’=2
r4=0
c4=1
d4=3 4 2
r5=0
c5=1
d5=5 5
r1=0
c1=1
d1’’=1 1
r3=0
c3=1
d3=4 3
Slack time: Xi = di − ai − Ci = di – Ci
because ai = ri = 0
Rule of Thumb:The deadline of a task cannot exceed its own deadline or the minimum slack time of its dependent tasks.
r6=0
c6=1
d6=6 6
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45. EDF With Precedence Next select i=3 with d3=4
Tasks immediately depending on 3
D(3) = {6}
d6-c6=5
d3’=min(4, min(5))=4
r2=0
c2=1
d2’’=2
r4=0
c4=1
d4=3 4 2
r5=0
c5=1
d5=5 5
r1=0
c1=1
d1’’=1 1
r3=0
c3=1
d3’=4 3
Slack time: Xi = di − ai − Ci = di – Ci
because ai = ri = 0
The deadline of a task cannot exceed its own deadline or the minimum slack time of its dependent tasks.
r6=0
c6=1
d6=6 6
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46. EDF With Precedence
The remaining nodes are leaf nodes. No other task depends on them
So their deadlines remain unchanged
r2=0
c2=1
d2’’=2
r4=0
c4=1
d4’=3 4 2
r5=0
c5=1
d5’=5 5
r1=0
c1=1
d1’’=1 1
r3=0
c3=1
d3’=4 3
Slack time: Xi = di − ai − Ci = di – Ci
because ai = ri = 0
The deadline of a task cannot exceed its own deadline or the minimum slack time of its dependent tasks.
r6=0
c6=1
d6’=6 6
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47. EDF With Precedence Now apply normal EDF: Task 1 Task 2 Task 4 Task 35
r1=0
c1=1
d1’’=1 1
r3=0
c3=1
d3’=4 3
r6=0
c6=1
d6’=6 6
EDF with precedence rationalizes deadlines. Instead of accepting the given deadline, the algorithm ensures that the assigned “virtual” deadlines take into account deadlines of successor. A task cannot have a later deadline than the minimum slack of its successors.
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48. Optimality
EDF with precedence is optimal in the sense of minimizing maximum lateness
A schedule obtained may not be feasible
Tasks may not meet their deadline
But if a feasible schedule exists, EDF with precedence will find it.
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49. Periodic Tasks T
A periodic task is a sequence of jobs that are released at times 0+fi, Ti +fi, 2 Ti +fi, …
A periodic task ti is specified using the parameters
Ti – period
Ci – maximum execution time
fi– phase or offset (almost always zero)
Di – relative deadline (almost always Ti)
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50. Sporadic Tasks
Sporadic tasks are characterized by a minimum inter-arrival time, rather than a fixed period (the earliest, the task arrives again)
Minimum spacing is necessary to ensure execution within C (maximum execution time)
Many of the analysis for periodic tasks apply to sporadic tasks if Ti is interpreted as the minimum time between arrivals
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51. Scheduling Points
Scheduling point is where the scheduler must examine the schedule and make scheduling decisions
This includes preemption of the currently running task
For periodic tasks, the scheduler kicks in at the fixed period
For sporadic tasks, the scheduler kicks in when tasks arrive
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52. Scheduling Assumptions - 1
The instances of a periodic task ti are regularly activated at a constant rate. The interval Ti between two consecutive activations is the period of the task
All instances of a periodic task ti have the same worst-case execution time Ci
All instances of a periodic task ti have the same relative deadline Di which is equal to the period Ti
What’s relative deadline?
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53. Scheduling Assumptions - 2
All tasks in the task set are independent; that is, there are no precedence relations and no resource constraints
No task can suspend itself, for example on I/O operations
All tasks are released as soon as they arrive (ai=ri)
All context-switching overheads are assumed to be zero
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54. Hyperperiod Or Major Cycle
A periodic system schedule can be defined by giving a finite-length schedule that is repeated over and over.
The length of the smallest such schedule is called the hyperperiod for that task set
For a purely periodic task set, the hyperperiod is given by the Least Common Multiple of the periods
Hyperperiod = LCM(Ti) for all i
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55. Harmonic Task Set A periodic task set is harmonic if
Every period evenly divides all larger periods
Every period is evenly divisible by all smaller periods
The hyperperiod of a harmonic task set is equal to the largest task period (because it is the LCM!)
Example: three tasks with periods 25, 50 and 100
100/50=2; 100/25=4; 50/25=2
100:50:25 = 4:2:1
Hyperperiod: 100
Scheduling algorithms may be simpler and more efficient for harmonic task sets
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56. Timeline Scheduling Time Line Scheduling
It is an off-line static scheduling strategy
Divide the time line into time slices for scheduling purposes
Use the Greatest Common Divisor of the Task Periods as the time slice
This is the same as the smallest period in a harmonic task set
It is referred to as the Minor Cycle
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57. Example Three tasks with periods 25, 50, 100
Hyperperiod
Must be scheduled
every 25 units.
Must be scheduled
every 50 units.
Must be scheduled
every 100 units.
Three tasks with periods 25, 50, 100
Hyperperiod/Major cycle (LCM): 100
Time slice/minor cycle (GCD) : 25
A feasible schedule exists if
CA+CB < 25
CA+CC < 25
These are sufficiency conditions. They are not necessary conditions (i.e., there may be feasible solutions even if these conditions are not met.
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58. Timeline Scheduling Notes
Advantage: design simplicity
A timer interrupts every “minor cycle”
A table-driven scheduler executes programs in the order computed in the major cycle
Disadvantage:
Could cause instability during CPU overload if a task does not terminate at the minor-cycle boundary
Difficult to maintain a static schedule when task periods or execution times change
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59. Rate-Monotonic Scheduling
Assume n tasks running periodically
Periods: T1, T2,….Tn
Worst-case execution time: C1, C2,….Cn
Relative deadline (relative to release time) Di = Ti
No precedence constraints
Preemptive with fixed/static priorities
Theorem: if any static priority assignment yields a feasible schedule, then priorities ordered by period (smallest period has the highest priority) also yields a feasible schedule
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60. Optimal Feasibility
Optimality with respect to feasibility means: if there is a feasible schedule S, then RMS can also schedule it
If RMS executes every task to completion once within its designated period, then it satisfies the feasibility objective
There may be other feasible schedules that are not RMS
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63. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
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64. finish The scheduler kicks in whenever a task arrives
Note that if we make the red task longer, at some point the blue task may not be able to complete within C1 and misses its deadline. In this case there is no feasible RMS solution.
finish
The scheduler kicks in whenever a task arrives
at its periodic interval.
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65. The worst-case processor demand occurs when all tasks are released simultaneously. This is referred to as the critical instant. The response time is longest.
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66. RMS solution: T2 has a higher priority
We have two tasks
RMS solution: T2 has a higher priority
Non-RMS solution: which task has a higher priority?
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67. CS555A – Real-Time Embedded Systems Stevens Institute of Technology
With only two tasks, the non-RMS schedule is the one where the blue task with the longer period has a higher priority. So we have to only consider feasible non-RMS scheduling scenarios.
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69. Comments The proof can be extended to an arbitrary number of tasks
The proof is for optimality with respect to feasibility.
Other optimality criteria such as processor utilization are not considered. In fact RMS schedulers cannot always achieve 100% utilization.
RMS is essentially a preemptive scheduling algorithm with fixed priority given by the task period
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70. What Have We Shown?
We have shown that in cases where there exists a feasible scheduling solution under conditions stated in the theorem, the RMS scheduling algorithm also works
RMS is optimal in the sense that it yields a feasible solution when there exists a feasible solution under any strategy
This is called: optimality with respect to feasibility
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71. What Have We Shown?
If C1+C2>T2, a non-RMS solution does not exist in the two-task example
RMS does not have to satisfy this constraint
Therefore it can cover more than non-RMS solutions
But there may be cases where there is no feasible solution. In these cases RMS can’t help either. Examples:
Suppose in the two-task example, the period of the longer task is shorter
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72. Example: Infeasible RMS Schedule
time
processing
period ) , ( = i C T P
Deadline Miss
1st
run
2nd
run
3rd
run ) 1 , 4 ( = P
1st
run
2nd
run ) 2 , 5 ( = P
1st
run
1st
run ) 2 , 7 ( 3 = P 5 10 15
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