1. Elastic Task Model For Adaptive Rate Control
Hehe Li
12/5/2018

2. Outline Introduction The Elastic Model
Equivalence with a Spring System
Compressing Tasks’ Utilizations
Theoretical Validation of the Model
Experimental Results
Conclusions
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Introduction

3. Abstract
Greater flexibility requested by multimedia and adaptive control applications
Tasks’ periods are treated as springs with given elastic coefficients in ETM
Controlling QoS as a function of the current load.
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Introduction

4. Classic Framework Stability Feasibility Schedulability
Constant rate. (e.g. digital control system)
Feasibility
No change of task period (e.g. critical control system)
Schedulability
Rigid assumptions (e.g. RM, EDF)
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Introduction

5. Assumptions in EDF & RM
A task has two parameters: computation time Ci and period Ti
Remain constant for all task instances
Reasonable for most real-time control systems
Too restrictive for applications (e.g. multimedia)
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Introduction

6. Multimedia Systems
Missing deadline decreases QoS but doesn’t cause critical system faults. (e.g. MPEG video)
Waste of CPU resource if treated as hard real-time tasks
Codeing/decoding vary significantly
WCET >> mean execution time
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Introduction

7. Advantages of ETM
Allowing tasks to intentionally change their execution rate to provide different QoS
Handling overload situations in a more flexible way
Providing a simple and efficient method for controlling the QoS of the system as a function of the current load
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Introduction

8. Outline Introduction The Elastic Model
Equivalence with a Spring System
Compressing Tasks’ Utilizations
Theoretical Validation of the Model
Experimental Results
Conclusions
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The Elastic Model

9. Task Parameters Computation time Ci (fixed) Nominal period Ti0
Minimum period Timin
Maximum period Timax
Elastic coefficient ei >= 0
A task = i(Ci, Ti0, Timin, Timax, ei)
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The Elastic Model

10. Parameter Constraints
Ti is the actual period of task I.
Ti  [Timin, Timax]
Any variation is subject to an elastic guarantee and is accepted only if there exists a feasible schedule in which all the other periods are within their range.
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The Elastic Model

11. I reduces its period to 50
Schedulable by EDF.
Up = 10/ / /70 = < 1
I reduces its period to 50
Up = 10/ / /50 = 1.05 > 1
Not schedulable
Change T1 to 22 and T1 to 45
Up = 0.977
Schedulable again
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The Elastic Model

12. Example Cont. I reduces its period to 40
Schedulable (left as excise )
But T1 = 35 must be rejected, since there’s no feasible schedule with T1 and T1 within their range
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The Elastic Model

13. Configuration Policy
Implicitly encoded in the elastic coefficients provided by the user
Each task is varied based on its current elastic status and a feasible configuration is found, if there exists one.
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The Elastic Model

14. More Features
The other direction (decompressoin): task terminates or decreases its rate.
Other tasks gain back bandwidth
Hard tasks
Ti0 = Timin = Timax
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The Elastic Model

15. Equivalence with a Spring System
Outline
Introduction
The Elastic Model
Equivalence with a Spring System
Compressing Tasks’ Utilizations
Theoretical Validation of the Model
Experimental Results
Conclusions
12/5/2018
Equivalence with a Spring System

16. Equivalence with a Spring System
Spring Parameters
A spring Si
Actual length xi
Nominal length xi0
Minimum period ximin
Maximum period ximax
Rigidity coefficient ki
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Equivalence with a Spring System

17. Comparison (recall slide 9)
xi = Ui = Ci / Ti (utilization factor)
ki = 1 / ei
L = i[1,n]xi is equvalent to Up = i[1,n]Ui
In the spring system, the problem is stated:
Given a set of n springs w/ known rigidity and length constraints, if L > Lmax, find a set of new lengths x’i s.t. x’i  [ximin, ximax] and L’ = Ld, where Ld is any length s.t. Ld < Lmax
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Equivalence with a Spring System

18. Springs w/o Length Constraints
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Equivalence with a Spring System

19. Equivalence with a Spring System
Solve the Problem
L0 = i[1,n]xi0
Ld = i[1,n]xi (1)
0 = Ld = L0 and for all i 0 < xi < xi0
i F = ki(xi0 – xi) (2)
Solving (1) and (2)
i xi = xi0 – (L0 – Ld) K// / ki
where K// = 1 / (i[1,n] 1/ki)
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Equivalence with a Spring System

20. Introducing Length Constraints
xi  [ximin, ximax]
If during compression one or more springs reach their minimum length
Let task set  = f + v
f: Fixed springs having minimum length
v:Variable springs that can still be compressed
Siv xi = xi0 – (L0 – Ld + Lf) Kv / ki (5)
Lf = Sif ximin (6)
Kv = 1 / (Siv 1/ki) (7)
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Equivalence with a Spring System

21. Equivalence with a Spring System
Iterative Solution
Whenever there exists some spring for which equation (5) gives xi < ximin , the length of that spring has to be fixed at its minimum value, sets f and v must be updated, and equations (5), (6) and (7) recomputed for the new set v
Algorithm for compressing a set of springs with length constraints on next page
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Equivalence with a Spring System

22. Equivalence with a Spring System
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Equivalence with a Spring System

23. Compressing Tasks’ Utilizations
Outline
Introduction
The Elastic Model
Equivalence with a Spring System
Compressing Tasks’ Utilizations
Theoretical Validation of the Model
Experimental Results
Conclusions
12/5/2018
Compressing Tasks’ Utilizations

24. Compressing Tasks’ Utilizations
Rewrite (5), (6) and (7) by substituting all length parameters with all length parameters with the corresponding utilization factors, and rigidity coefficients ki and Kv with the corresponding elastic coeffients ei and Ev
Algorithm on next page
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Compressing Tasks’ Utilizations

25. Compressing Tasks’ Utilizations
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Compressing Tasks’ Utilizations

26. Compressing Tasks’ Utilizations
Decompression
A task decreases its rate or returns to its nominal period.
Compressed tasks expand their utilizations according to their elastic coefficients
If total utilization is less than least upper bound, then all tasks can return to their nominal periods
Otherwise, view it as a compression
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Compressing Tasks’ Utilizations

27. Theoretical Validation of the Model
Outline
Introduction
The Elastic Model
Equivalence with a Spring System
Compressing Tasks’ Utilizations
Theoretical Validation of the Model
Experimental Results
Conclusions
12/5/2018
Theoretical Validation of the Model

28. Theoretical Validation of the Model
If tasks’ periods are changed at opportune instants the task set remains schedulable and no deadline is missed.
I will show proofs to the theorems on blackboard
The following lemmas states two properties of the EDF algorithm that are useful for proving the main theorem.
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Theoretical Validation of the Model

29. Theoretical Validation of the Model
Lemma 1
In any feasible EDF schedule , the following condition holds:
t >0  i[1,n] ri (t)/t >= Up
Where Up =  i[1,n] Ci / Ti and ri (t) is the cumulative time executed by all the instances of task i up to t.
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Theoretical Validation of the Model

30. Theoretical Validation of the Model
Lemma 2
In any feasible EDF schedule , the following condition holds:
t >0  i[1,n] ci(t) <=  i[1,n] [ci(t) – t]Ui
Where Ui = Ci / Ti, ci (t) is the remaining execution time of the current instance of task i at time t, and vi (t) is the next release time of i greater than or equal to t
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Theoretical Validation of the Model

31. Theoretical Validation of the Model
Theorem 1
Given a feasible task set , with total utilization factor Up =  i[1,n] Ci/Ti <= 1, if at time t all the periods are increased from Ti to T’i >= Ti, then for all L > 0,
D(t, t+L) <= LU’ p
Where D(t1, t2) is the total processor demand of  in [t1, t2], and U’p =  i[1,n] Ci/T’i
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Theoretical Validation of the Model

32. Theoretical Validation of the Model
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Theoretical Validation of the Model

33. Theoretical Validation of the Model
Note on Compression
Thm 1 doesn’t hold in case of compression
The period of compressed task can be decreased only at its next release time.
Example on next slide
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Theoretical Validation of the Model

34. Theoretical Validation of the Model
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Theoretical Validation of the Model

35. Outline Introduction The Elastic Model
Equivalence with a Spring System
Compressing Tasks’ Utilizations
Theoretical Validation of the Model
Experimental Results
Conclusions
12/5/2018
Experimental Results

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Experimental Results

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Experimental Results

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Experimental Results

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Experimental Results

40. Outline Introduction The Elastic Model
Equivalence with a Spring System
Compressing Tasks’ Utilizations
Theoretical Validation of the Model
Experimental Results
Conclusions
12/5/2018
Conclusions

41. Conclusions
Periodic tasks can intentionally change their execution rate to provide different QoS, and other tasks canautomatically adapt their peroids to keep the system underloaded.
Policy is encoded in the elastic coefficient
Useful for supporting both multimedia and control applications, in which the execution rates of some computational activities have to be dynamically tuned as a function of the current system.
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Conclusions