1. Efficient and Flexible Fair Scheduling of Real-time Tasks on Multiprocessors
Anand Srinivasan
Department of Computer Science
University of North Carolina at Chapel Hill

2. What is a Real-time System?
A system with a dual notion of correctness:
Logical correctness (“it does the right thing”).
Temporal correctness (“it does it on time”).
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Anand Srinivasan

3. What is a Real-time System?
A system with a dual notion of correctness:
Logical correctness (“it does the right thing”).
Temporal correctness (“it does it on time”).
Example: a robot picking up objects from a conveyor belt.
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Anand Srinivasan

4. Current Trends
Sophisticated real-time applications with high computational demands are emerging.
Multiprocessors are becoming extremely cost-effective.
Multiprocessor real-time designs will continue to be of growing importance.
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Anand Srinivasan

5. An Example System
The Whisper acoustic tracking system at UNC-Chapel Hill.
Tracking Computer
Microphones on ceiling
Speakers on hands and feet
User
Let me describe one such sophisticated application that we are currently developing at UNC-Chapel Hill.
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Anand Srinivasan

6. Whisper Proposed multi-threaded model
Tasks
Microphone
Sampler
Measurement
Kalman filter
Microphone
Sampler
Measurement
Microphone
Sampler
Measurement
Real-time fine-grained adaptivity is necessary.
A multiprocessor system is required to handle these computationally intensive tasks.
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Anand Srinivasan

7. Periodic Tasks
A periodic task consists of an infinite sequence of jobs that are released periodically. 10 20 30 5 15 25
execution requirement
period
e1 = 2
p1 = 5
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Anand Srinivasan

8. Periodic Tasks Relative deadline of a task is equal to its period. 5
Deadline of a job equals release-time of next job of that task.
e1 = 2
p1 = 5
e2 = 9
p2 = 15 5 10 15 20 25 30
Weight of a task with execution requirement e and period p is .
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9. Real-time Systems Need two algorithms:
(i) Scheduler (typically, run-time).
(ii) Schedulability test (typically, offline).
Example: Scheduling periodic tasks on a uniprocessor.
(i) Scheduler is earliest-deadline-first (EDF).
(ii) Schedulability test is “ ”.
Yes
Run the system with EDF
Set of tasks
Re-design the system No
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Anand Srinivasan

10. Overview Fair scheduling on multiprocessors Flexibility Summary
Efficiency: The PD2 scheduling algorithm
Experimental comparison with other approaches
Flexibility
Dynamic task systems
Rate-based scheduling on multiprocessors
Summary
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Anand Srinivasan

11. Overview Fair scheduling on multiprocessors Flexibility Summary
Efficiency: The PD2 scheduling algorithm
Experimental comparison with other approaches
Flexibility
Dynamic task systems
Rate-based scheduling on multiprocessors
Summary
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Anand Srinivasan

12. Scheduling Problem
Given a set of n periodic tasks to be scheduled on m (< n) processors.
Determine whether they can be correctly scheduled.
Which scheduling approach and algorithm should we use?
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13. Assumptions Periods and worst-case execution requirements are given.
All values are integral.
All processors are identical and main memory is shared.
Preemptions and inter-processor migrations are allowed, but parallelism is not.
No context switch and migration overhead.
These are accounted for in the worst-case execution times of the tasks.
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14. Feasibility and Optimality
[Baruah et al.] A periodic task set  is feasible on m processors iff
weight(T) ≤ m.
A scheduling algorithm is said to be optimal if it can correctly schedule any feasible task set.
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Anand Srinivasan

15. Proportionate Fair Scheduling [Baruah et al.]
Scheduling decisions are based on the behavior of an ideal fluid system.
TIME A
weight = 2/5
2/5
Ideal schedule B
3/5
weight = 3/5
Quantum-based schedule
A Pfair schedule is a quantum-based schedule in which tasks are allocated fairly in proportion to their weights.
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Anand Srinivasan

16. Quantum-based schedule
Pfair Scheduling
Lag of a task is the difference between its allocations in the ideal and Pfair schedule.
TIME
Ideal share /5 6/5 9/5 12/ /5 21/5 24/5 27/5 6 B
Ideal schedule
weight = 3/5
3/5
Actual share
Quantum-based schedule
lag –2/5 1/5 –1/5 –3/ –2/5 1/5 –1/5 –3/5 0
Definition: A schedule is Pfair if each task’s allocation is strictly within one quantum of the ideal allocation at all times, i.e., –1 < lag < 1.
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17. Lag of a Periodic Task
Lag(T,u) = ideal share of T in [0,u) – Pfair(T,u).
lag(T, u) < 0  T is over-allocated at time u.
lag(T, u) > 0  T is under-allocated at time u. 6 5 4 3 2 1
ideal
Pfair
total share of task with weight 3/5.
Over-allocated
Under-allocated
time
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18. Scheduling of Periodic Tasks
Share of T in ideal schedule in [0,u) = ( )  u.
lag(T, u) = – Pfair(T, u).
If p divides u, then lag is an integer.
–1< lag < 1  lag = 0.
Hence, all jobs meet their deadlines.
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19. Scheduling of Subtasks
Each task is effectively divided into a series of quantum-length subtasks.
TIME
Subtask 3
Subtask 2
Subtask 1
Consider
a task T of
weight 3/7.
lag = 9/7
lag(T, 3) = (3/7)  3 – 0 = 9/7 > 1
So the first subtask must finish by time 3.
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Anand Srinivasan

20. Scheduling of Subtasks
TIME
Subtask 3
Subtask 2
Subtask 1
lag = 8/7
lag(T, 5) = (3/7)  5 – 1 = 8/7 > 1
So the second subtask must finish by time 5.
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Anand Srinivasan

21. Scheduling of Subtasks
TIME
Subtask 3
Subtask 2
Subtask 1
lag = –8/7
lag(T, 2) = (3/7)  2 – 2 = –8/7 < –1
So the second subtask cannot be scheduled at time 1.
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Anand Srinivasan

22. Pfair Windows
The lag bounds define evenly-spaced windows in which each subtask must execute.
TIME
Window 3
Window 2
Window 1
Thus, each subtask has a release-time and a deadline.
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Anand Srinivasan

23. Optimal Pfair Schedulers
PF, PD, and PD2.
Invoked at the beginning of each quantum.
Select m highest-priority subtasks for execution.
Use deadlines of subtasks to determine priority.
Based on EDF.
Differ in the choice of tie-breaking rules.
Tie-breaking rules are necessary for optimality on more than three processors.
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24. The PF Scheduling Algorithm
PF [Baruah et al. 1993]
T has higher priority at time t T … U t
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25. PF Swapping Proof
Proved optimal by an inductive swapping proof: Transform a correct schedule into a PF schedule.
Consider a two-processor schedule. T … U
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26. The PD Scheduling Algorithm [Baruah et al. 1995]
Uses four tie-breaking rules (applied in sequence).
Any further ties can be broken arbitrarily.
Polynomial time: O (m log n).
m processors, n tasks.
Proved to be optimal by showing that PD “closely” tracks PF.
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27. The PD2 Scheduling Algorithm [Anderson and Srinivasan, 2000]
Uses only two tie-breaking rules.
Polynomial time: O (m log n).
m processors, n tasks.
Optimality proved by a swapping proof.
Much more complicated than the proof for PF.
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28. Results about PD2
Most efficient among all known Pfair scheduling algorithms that are optimal.
If any of the tie-breaking rules is eliminated, task sets exist that miss deadlines.
For systems consisting of tasks with weight at most ½, only one tie-breaking rule is necessary.
For two-processor systems, no tie-breaking rules are required.
James H. Anderson and Anand Srinivasan, Early-release Fair Scheduling, IEEE Euromicro Conference on Real-time Systems, 2000.
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29. Overview Fair scheduling on multiprocessors Flexibility Summary
Efficiency: The PD2 scheduling algorithm
Experimental comparison with other approaches
Flexibility
Dynamic task systems
Rate-based scheduling on multiprocessors
Summary
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Anand Srinivasan

30. Comparison with Partitioning Approaches
Partition the set of tasks so that each task always runs on the same processor.
Steps:
Choose a uniprocessor scheduling algorithm.
Use the corresponding uniprocessor schedulability test to determine the partitions.
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31. Partitioning Approaches
Disadvantages:
Inherently sub-optimal.
Consider a two-processor system with three tasks of weight 2/3 (= 0.66) each.
Bin-packing is NP-hard in the strong sense.
Advantages:
Uniprocessor algorithms can be used.
No migration overhead is incurred.
It performs well in practice.
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32. Experimental Setup [Srinivasan et al.]
Overheads:
Scheduling overheads.
Context-switch cost (= 5μs).
Cache-related preemption delay ( [0,100μs]).
Approach:
Generate random periodic task sets with a given total utilization.
Calculate new worst-case execution times after incorporating the above overheads.
Determine the minimum number of processors required by PD2 and EDF with First-Fit.
First-fit: Each task is assigned to the first processor that can accept it.
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33. System of 50 Tasks
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34. System of 100 Tasks
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35. Why use PD2?
PD2 gives almost same performance as EDF with First-Fit under all cases.
Efficiently supporting critical sections is difficult with partitioning.
Efficient schemes have been presented for PD2 by Holman and Anderson [2002, 2003].
Partitioning is not suitable for highly-dynamic task systems.
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Anand Srinivasan

36. Overview Fair scheduling on multiprocessors Flexibility Summary
Efficiency: The PD2 scheduling algorithm
Experimental comparison with other approaches
Flexibility
Dynamic task systems
Rate-based scheduling on multiprocessors
Summary
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Anand Srinivasan

37. Dynamic Task Systems Whisper re-visited
Features of Whisper:
Real-time guarantees are required to ensure high fidelity response.
Multiprocessor system is required to handle computational demands.
Highly-dynamic system: Fine-grained adaptive nature implies that computational demands vary over time.
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Anand Srinivasan

38. Dynamic Task Systems … …
Unrestricted leaves and joins can lead to missed deadlines
Consider a uniprocessor system with one task of weight ½ and k tasks of weight 1/2k.
Leaves and immediately rejoins. … …
k-2 2k-1 2k
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39. Dynamic Task Systems
Join condition: Check feasibility with the new task.
Leave condition (trickier): Allow task to leave only if it is not over-allocated, i.e., lag  0.
This is sufficient on uniprocessors [Stoica et al. 1996]
This is not sufficient, in general, on multiprocessors. It is sufficient if the final job executes for its worst-case execution time.
We have also provided tighter conditions under which tasks are allowed to leave earlier.
Anand Srinivasan and James H. Anderson, Fair scheduling of dynamic task systems on multiprocessors, WPDRTS, 2003.
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Anand Srinivasan

40. Overview Fair scheduling on multiprocessors Flexibility Summary
Efficiency: The PD2 scheduling algorithm
Experimental comparison with other approaches
Flexibility
Dynamic task systems
Rate-based scheduling on multiprocessors
Summary
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Anand Srinivasan

41. Rate-based Scheduling
Consider the following EDF-scheduled tasks, where each
job of task U sends a message.
T = (c1, p1)
U = (c2, p2) p1 p2
2p1
2p2
3p1
3p2
4p1
5p1
6p1
4p2
messages
sent by U
These messages are not periodic.
So, task consuming these messages is not periodic.
It’s even worse if there’s a network in between.
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42. Intra-sporadic (IS) Tasks
The IS model allows subtasks to be released late, i.e., separation between consecutive subtask releases may be more than that for a periodic task.
In addition to release and deadline, each subtask Ti has an eligibility time, which may be less than its release time.
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43. IS Task Example 1 2 3 4 5 6 7 8 9 10 11 12 13
6 windows of task with weight 3/5. All subtasks are eligible at their Pfair release times.
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44. IS Task Example 1 2 3 4 5 6 7 8 9 10 11 12 13 IS delay1 2 3 4 5 6 7 8 9 10 11 12 13
6 windows of task with weight 6/10.
2nd subtask is delayed by one time unit, i.e. the 2nd subtask is eligible only at time 2. The Pfair deadlines also get delayed.
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45. IS Task Example
e < r 1 2 3 4 5 6 7 8 9 10 11 12 13
6 windows of task with weight 6/10.
2nd window is delayed by one time unit.
The 3rd subtask is eligible at time 3, and the 2nd subtask is scheduled in slot 2.
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46. Other Example Applications
Multimedia application:
Consider an application receiving packets over a network.
Packets arrive in bursts.
Possible delay sometimes.
Tasks that self-suspend (for example, due to RPC, I/O).
Tasks that block (for example, due to locks).
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47. Results about IS tasks
PD2 is optimal on any number of processors.
No tie-breaks are necessary on two processors.
No tie-breaks are necessary on three processors if all tasks have weight at most ½.
Feasibility: An intra-sporadic task system  is feasible on m processors iff
weight(T) ≤ m.
Anand Srinivasan and James H. Anderson, Optimal Rate-based Scheduling on Multiprocessors, STOC, 2002.
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48. Optimality of PD2 for IS Tasks Why swapping does not work… U
Future windows may not align “properly” for needed swappings to work.
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49. Overview Fair scheduling Flexibility Summary
Efficiency: The PD2 scheduling algorithm
Experimental comparison with other approaches
Flexibility
Dynamic task systems
Rate-based scheduling on multiprocessors
Summary
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Anand Srinivasan

50. Summary of Contributions
This talk:
The PD2 Pfair scheduling algorithm.
Dynamic real-time multiprocessor systems.
Rate-based scheduling on multiprocessors.
Other work:
Soft real-time multiprocessor systems.
in submission.
Scheduling of packet processing activities on a network processor.
Workshop on Network Processors 2003.
Scheduling of aperiodic tasks.
Euromicro Conference in Real-time Systems 2002.
Papers are available at
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51. Future Vision
Highly-dynamic and computation-intensive real-time applications are becoming increasingly common and important.
Automatic tracking systems.
Telepresence systems.
Processing data in disaster recovery and battlefield situations.
Long term goal: Design and develop a general framework that can efficiently support such applications.
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52. Short-term Plans
Multiprocessor scheduling issues in general-purpose operating systems.
Support for soft real-time applications
Quality-of-service (QoS) Guarantees in Networking:
Different set of constraints.
Intelligent use of spare bandwidth.
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53. This slide is intentionally left blank!
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54. Necessity of tie-breaks
Deadline missed!
Weight = ¾
3 Processors,
fully utilized: 2 x ¾ + 3 x ½ = 3. Hence, no processor must be idle.
Weight = ¾
Weight = ½
Weight = ½
Weight = ½
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