1. Jian-Jia Chen and Tei-Wei Kuo
Energy-Efficient Scheduling of Periodic Real-Time Tasks over Homogeneous Multiprocessors
Jian-Jia Chen and Tei-Wei Kuo
Department of Computer Science and Information Engineering, Graduate Institute of Networking and Multimedia, National Taiwan University

2. Agenda Introduction Problem Definition Non-linear Programming
An Approximation Algorithm
Performance Evaluation
Conclusions and Future Work
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

3. Dynamic Voltage Scaling
A Dynamic Voltage Scaling (DVS) system is a system that can execute tasks at different speeds.
A higher supply voltage results in a higher frequency (or higher execution speed).
s = k * (Vdd-Vt)2/(Vdd), where
s is the corresponding speed of the supply voltage Vdd and
Vt is the threshold voltage
The power consumption function P() of the execution speeds of a processor is a convex function:
P(s) = Cef Vdd2 s, in which Cef is the switch capacitance related to tasks under executions
P(s) = Cef s3/k2 , when Vt = 0
P(s) / s, where  is between 2 and 3
In general, we could formulate the power consumption function P(s) as a proportional function to the \alpha-th exponent of s
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

4. Related Work on Energy-Efficient Scheduling
Uniprocessor
[Aydin ECRTS’01, Mejia-Alvarez RTAS’03, Chen EMSOFT’05]
Independent periodic real-time tasks
[Yao FOCS’95, Bansal FOCS’04, Irani SODA’03, Ishihara ISLPED’98, Kwon DAC’03]
Independent aperiodic real-time tasks
[Fang RTSS’02]
Real-time tasks with resource competitions
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

5. Related Work on Energy-Efficient Scheduling
Multiprocessor
[Gruian ASP-DAC’01, Zhang DAC’02]:
Heuristic algorithms based on the well-known list-scheduling
[Mishra IPDPS’03]:
Heuristic algorithms based on the well-known list-scheduling for tasks with precedence constraints with communication costs
[Anderson ICDCS’04]:
Heuristic partition algorithm to trade the number of processors with the energy consumption
[Aydin IPDPS’03, AlEnawy RTAS’05]:
Heuristic algorithms based on traditional bin packing strategies
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

6. Related Work on Energy-Efficient Scheduling
Multiprocessor
[Chen ECRTS’04, Yang DATE’05, Chen ICPP’05]:
Frame-based real-time tasks
First known approximation algorithms for multiprocessor energy-efficient scheduling
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

7. Agenda Introduction Problem Definition Non-linear Programming
An Approximation Algorithm
Performance Evaluation
Conclusions and Future Work
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

8. Task Model i = (ci, pi, Pi())
ci: computational requirement for i, in CPU cycles
pi: period of task i, where the relative deadline of i is equal to pi
Pi(): the power consumption function of i
Pi(s) = his , where hi 2 R+ and  is a hardware constant between 2 and 3
The relative deadline of a task is equal to its period
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

9. S
power
DVS Model
Pi() is identical on each of the available identical processors.
DVS constraints:
Each processor can adjust its processor speed independently of each another.
Available speeds are continuous in [0, 1].
The number of CPU cycles executed in a time interval is linearly proportional to the processor speed.
Executing of task i at the processor speed s for t time units consumes Pi(s)£t energy.
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

10. Problem Definition
The Minimization Problem of the Energy Consumption for Multiprocessor Scheduling:
Input:
A set T of independent tasks over M identical processors.
Each task i  T is characterized by (pi, ci, Pi()).
No task migration is allowed
Output:
A schedule for all of the tasks in T so that each task completes in time, and the energy consumption in the hyper-period L of T is minimized.
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

11. Power and Energy Consumption Functions( s ) = h e i ( t ) = P c h 1
Pi(s) is a convex and strictly increasing function of speeds.
ei(t) is a convex and strictly decreasing function of execution times.
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

12. A Trivial Case: |T|  M
Executing each task i on the i-th processor at the speed ci/pi is an optimal schedule, for i=1, …, |T|.
We focus on other cases for the rest of this talk.
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

13. Agenda Introduction Problem Definitions Non-Linear Programming
An Approximation Algorithm
Performance Evaluation
Conclusions and Future Work
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

14. An Optimal Solution to Multiprocessor Energy-Efficiency
Applying a proof similar to that by Aydin et al.
All the jobs of a task execute at a common speed
The partial schedule on each processor is with 100% utilization
H. Aydin, R. Melhem, D. Moss´e, and P. Mej´ıa-Alvarez. Determining optimal processor speeds for periodic real-time tasks with different power characteristics. In Proceedings of the IEEE EuroMicro Conference on Real-Time Systems, pages 225–232, 2001.
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

15. A Convex Programming Formulation: a b n r y v l e t o d c w h s g u - p f k . E ( ) j L , = P 1 W z 2 T ; M
> 8
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

16. Relaxation I f w e r l a x t h - 1 c o n s i u d b y g v , p : z P E (m i n z e P 2 T E ( t ) s u b j c o x = p 1 ; f r : M
> 8 a d
Relaxation I f w e r l a x t h - 1 c o n s i m u d b y g v , p : z P 2 T E ( ) ; j = M
< K L . L e m a : W h n t i
< p d j , E ( ) = w r v s o f c l y .
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

17. An Optimal Solution to the Relaxed Problemf o l w i n g a r t m d v s x c u y p : F - E ( ) ; ` 1
< M I j + 2 L , = b O .
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

18. Agenda Introduction Problem Definitions Non-linear Programming
An Approximation Algorithm
Performance Evaluation
Conclusions and Future Work
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

19. An Approximation Algorithm
Terminologies:
Estimated execution time ti* is the execution time of task i in the optimal execution time assignment for the relaxed problem.
Estimated utilization ui* is ti*/pi.
Estimated energy consumption ei* is Ei(ti*)=L/pi(ci/(ti*)-1).
Estimated execution speed si* is ci/ti*.
By the Lagrange multiplier method, ei*/ui* = ej*/uj* if ui* < 1 and uj* < 1.
Since E’i(ti*)pi = E’j(tj*)pj, we know
Our proposed algorithm is to turn the optimal solution of the MEESM problem to a feasible schedule of the MEES problem. Because the optimal energy consumption of the MEESM problem is a lower bound of that of the MEES problem, we could prove the approximation ratio by comparing the energy consumption of our derived schedule to the optimal energy consumption of the MEESM problem. h i L p c ( t ) = j , w m l e s u .
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

20. Algorithm Largest-Estimated-Utilization First (LEUF)3
speed 5 1 2 4
Estimated Utilizations
Sort tasks in a non-increasing order of their estimated utilizations
Assign tasks in a greedy manner to the processor with the smallest total estimated utilizations
Adjust execution speeds
si à si* (Um) p1 1 p1 1 p2 5 p2 5 2 2 3 3 p3 p3 4 4 1 1
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

21. Property
The maximum total estimated utilization (Uz) is at most twice of the minimum total estimated utilization (Un)
Case 1: Uz = 1, Un must be 1
Case 2: Uz > 1
Let k be the last task inserted onto processor z
uk* · Uz - uk* · Un
Uz · 2Un p1 1 p2 5 2 3 p3 4 D k
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

22. Algorithm LEUF is with 1.412-Approximationd b - c g y k w = 1 v A L E U . S + B C
< , z ( )
Speed
Estimated utilization p1 p1 p2 p2
First of all, we know that for any task with equal estimated execution time to D, the energy consumption for this task is the same in the optimal solution of the MEESM problem and the derived schedule of Algorithm LEET. p3 p3 1 1
er*/ur*
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

23. F L e t M b h s o f p r c a i g n d w k u < 1 . T y m l v A E U P (
Um
Umin Um
2Umin F L e t M b h s o f p r c a i g n d w k u
< 1 . T y m l v A E U P 2 ( ) = x j : S Z , + X ; 2x x x x
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

24. Algorithm LEUF is with 1.412-Approximation( x ) = k 2 + ^ M r a i t v n m b d - g , w h
< 1 : A l W 3 4 .
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

25. Agenda Introduction Problem Definition Non-linear Programming
An Approximation Algorithm
Performance Evaluation
Conclusions and Future Work
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

26. Performance Evaluation
Replot figures
ci is uniform at random in (0, pi], hi is uniform at random in [2,10], =3
For each ratio v of number of tasks to number of processors, we roll an integral dice in [10, 30] to obtain M and generate the floor function of vM tasks independently
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

27. Agenda Introduction Problem Definition Non-linear Programming
An Approximation Algorithm
Performance Evaluation
Conclusions and Future Work
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

28. Conclusion
A approximation algorithm is proposed when task migration is not allowed
The proposed approximation algorithm provides near-optimal solutions in the performance evaluation.
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

29. Future Work
We will explore how to apply our algorithm to processors with speed constraints.
Copyright: All rights reserved, Jian-Jia Chen, Embedded Systems and Wireless Networkin Lab., Department of CSIE, National Taiwan University.

30. Questions and Suggestions?