0 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT)
Energy Efficient Task Partitioning based on the SingleFrequency Approximation Scheme
Santiago Pagani and Jian-Jia Chen
2013 IEEE 34th Real-Time Systems Symposium (RTSS)December 3-6 2013, Vancouver, Canada
KIT – University of the State of Baden-Wuerttemberg andNational Research Center of the Helmholtz Association www.kit.edu
Outline
1 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Outline
2 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Introduction
3 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Importance of Energy Efficiency:
Slow increases of battery capacity.Less Energy Consumption⇒ Prolong Battery Lifetime of EmbeddedSystems.
Increasing costs of energy.Less Energy Consumption⇒ Lower Power Bills for Servers.
Outcome for Computing Systems:
Motivated to move from single-core to multi-core.
Techniques for power management.
Introduction
3 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Importance of Energy Efficiency:
Slow increases of battery capacity.Less Energy Consumption⇒ Prolong Battery Lifetime of EmbeddedSystems.
Increasing costs of energy.Less Energy Consumption⇒ Lower Power Bills for Servers.
Outcome for Computing Systems:
Motivated to move from single-core to multi-core.
Techniques for power management.
Introduction
4 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Dynamic Power Management (DPM):
Technique for putting cores in a low-power mode: idle, sleep, off, etc.
Dynamic Voltage and Frequency Scaling (DVFS):
Technique for scaling the voltage and frequency of cores.
Per-core DVFS:Individual voltage and frequency for cores.
Optimal, but too expensive to manufacture.
Global DVFS:All cores share the same voltage.
Energy inefficient.
Introduction
4 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Dynamic Power Management (DPM):
Technique for putting cores in a low-power mode: idle, sleep, off, etc.
Dynamic Voltage and Frequency Scaling (DVFS):
Technique for scaling the voltage and frequency of cores.
Per-core DVFS:Individual voltage and frequency for cores.
Optimal, but too expensive to manufacture.
Global DVFS:All cores share the same voltage.
Energy inefficient.
Introduction
4 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Dynamic Power Management (DPM):
Technique for putting cores in a low-power mode: idle, sleep, off, etc.
Dynamic Voltage and Frequency Scaling (DVFS):
Technique for scaling the voltage and frequency of cores.
Per-core DVFS:Individual voltage and frequency for cores.
Optimal, but too expensive to manufacture.
Global DVFS:All cores share the same voltage.
Energy inefficient.
Introduction
4 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Dynamic Power Management (DPM):
Technique for putting cores in a low-power mode: idle, sleep, off, etc.
Dynamic Voltage and Frequency Scaling (DVFS):
Technique for scaling the voltage and frequency of cores.
Per-core DVFS:Individual voltage and frequency for cores.
Optimal, but too expensive to manufacture.
Global DVFS:All cores share the same voltage.
Energy inefficient.
Introduction
5 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Dynamic Voltage and Frequency Scaling (DVFS):
Multiple Voltage Islands:Compromise between Per-core DVFS and Global DVFS.
Cores are grouped into Voltage Islands.
Islands can have different voltages.
Figure: Intel’s SCC snapshot
Intel Corporation. Single-chip Cloud Computer (SCC). URL:http://www.intel.com/content/www/us/en/research/intel-labs-single-chip-cloud-computer.html
Introduction
6 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
CMOS-core Power Model
P (s) = Pdynamic (s) + Pstatic
Considering that:
Pdynamic (s) = CeffV2dds
s ∝(Vdd − Vt )
2
Vdd
We can approximate to:
P (s) = αsγ + β
Introduction
6 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
CMOS-core Power Model
P (s) = Pdynamic (s) + Pstatic
Considering that:
Pdynamic (s) = CeffV2dds
s ∝(Vdd − Vt )
2
Vdd
We can approximate to:
P (s) = αsγ + β
Introduction
6 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
CMOS-core Power Model
P (s) = Pdynamic (s) + Pstatic
Considering that:
Pdynamic (s) = CeffV2dds
s ∝(Vdd − Vt )
2
Vdd
We can approximate to:
P (s) = αsγ + β
Introduction
6 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
CMOS-core Power Model
P (s) = Pdynamic (s) + Pstatic
Considering that:
Pdynamic (s) = CeffV2dds
s ∝(Vdd − Vt )
2
Vdd
We can approximate to:
P (s) = αsγ + β
0 0.5 1 1.5 20
5
10
15
1.40
7.71
Frequency [GHz]
Pow
er [W
atts
]
Figure: α = 1.76 WattsGHz3 , γ = 3 and β = 0.5 Watts
Introduction
7 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Energy Consumption
E (s) = (αsγ + β)∆cs
Critical Frequency:
scrit =γ
√β
(γ− 1) α
0 0.5 1 1.5 20
2
4
6
8
10
Frequency [GHz]
Ene
rgy
[Jou
le]
Figure: α = 1.76 WattsGHz3 , γ = 3, β = 0.5 Watts and
∆c = 109 cycles
Outline
8 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Motivation
9 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
In each voltage island (or Global DVFS), for energy minimization:
Task partitioning and DVFS schedule:Play a major role in energy minimization.
Task Partitioning:
Convexity of E (s): For the same workload⇒ 2 · E (s) ≤ E (2 · s).
In general, load balancing reduces the dynamic energy consumption.
Balanced solution: with high complexity and not always feasible.
Good option: polynomial time algorithm based on load balancing, e.g.,Largest-Task-First (LTF)1 strategy.
1Chuan-Yue Yang, Jian-Jia Chen, and Tei-Wei Kuo. “An Approximation Algorithm for Energy-Efficient Scheduling on A Chip
Multiprocessor”. In: Conference on Design, Automation, and Test in Europe (DATE). 2005, pp. 468–473
Motivation
9 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
In each voltage island (or Global DVFS), for energy minimization:
Task partitioning and DVFS schedule:Play a major role in energy minimization.
Task Partitioning:
Convexity of E (s): For the same workload⇒ 2 · E (s) ≤ E (2 · s).
In general, load balancing reduces the dynamic energy consumption.
Balanced solution: with high complexity and not always feasible.
Good option: polynomial time algorithm based on load balancing, e.g.,Largest-Task-First (LTF)1 strategy.
1Chuan-Yue Yang, Jian-Jia Chen, and Tei-Wei Kuo. “An Approximation Algorithm for Energy-Efficient Scheduling on A Chip
Multiprocessor”. In: Conference on Design, Automation, and Test in Europe (DATE). 2005, pp. 468–473
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4τ5
τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1
τ2 τ3
τ4τ5
τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2
τ3
τ4τ5
τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4τ5
τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4
τ5τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4τ5
τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4τ5
τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4τ5
τ6
τ7
τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4τ5
τ6
τ7τ8
Motivation
10 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Largest-Task-First (LTF) Strategy Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 3 task sets: {T1,T2,T3}
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8
T1 T2 T3
τ1 τ2 τ3
τ4τ5
τ6
τ7τ8
Motivation
11 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
DVFS Schedule→ Single Frequency Approximation (SFA)2 Scheme:
Use the lowest voltage/frequency, satisfying the timing constraints.
Linear time complexity. Is the simplest and most intuitive strategy.
Not optimal, but significantly reduces the management overhead.
No frequency alignment between cores =⇒ Any uni-core DPMtechnique can be adopted individually in each core.
s
tCore 1:
su
s
tCore 2:
su
s
tCore 3:
su
s
tCore 4:
su = max {scrit,w4}
2Santiago Pagani and Jian-Jia Chen. “Energy Efficiency Analysis for the Single Frequency Approximation (SFA) Scheme”. In: Proceedings
of the 19th IEEE International Conference on Embedded and Real-Time Computing Systems and Applications (RTCSA). 2013
Motivation
12 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Combining LTF and SFA for energy minimization:
Is a practical solution.
Very easy to implement.
What is the worst-case performancein terms of energy efficiency?
Problem Definition
13 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
For periodic real-time tasks, assigned on a voltage island.
Using Earliest-Deadline-First (EDF) on individual cores.
Contributions
Present a simple and practical solution for energy minimization.Task Partitioning: Double-Largest-Task-First (DLTF).
DVFS schedule: SFA.
Theoretically analyse such a solution:
AFDLTFSFA = max
EDLTFSFA
E∗OPT≤ max
EDLTFSFA
E∗↓
Problem Definition
13 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
For periodic real-time tasks, assigned on a voltage island.
Using Earliest-Deadline-First (EDF) on individual cores.
Contributions
Present a simple and practical solution for energy minimization.Task Partitioning: Double-Largest-Task-First (DLTF).
DVFS schedule: SFA.
Theoretically analyse such a solution:
AFDLTFSFA = max
EDLTFSFA
E∗OPT≤ max
EDLTFSFA
E∗↓
Problem Definition
13 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
For periodic real-time tasks, assigned on a voltage island.
Using Earliest-Deadline-First (EDF) on individual cores.
Contributions
Present a simple and practical solution for energy minimization.Task Partitioning: Double-Largest-Task-First (DLTF).
DVFS schedule: SFA.
Theoretically analyse such a solution:
AFDLTFSFA = max
EDLTFSFA
E∗OPT≤ max
EDLTFSFA
E∗↓
Outline
14 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Double-Largest-Task-First (DLTF)
15 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Initial solution: Task partitioning with LTF.
Tasks are regrouped, shutting down all possible cores.
This reduces the energy consumption for idling under SFA:
Example: LTF and SFAs
ts
ts
ts
ts
tbreak-even time
Core 1: τ1
Core 2: τ2
Core 3: τ3
Core 4: τ4
Core 5: τ5
Example: DLTF and SFAs
ts
ts
ts
ts
tbreak-even time
Core 1: τ1
Core 2: τ2
Core 3: τ3
Core 4:
τ4
Core 5:
τ5
Double-Largest-Task-First (DLTF)
15 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Initial solution: Task partitioning with LTF.
Tasks are regrouped, shutting down all possible cores.
This reduces the energy consumption for idling under SFA:
Example: LTF and SFAs
ts
ts
ts
ts
tbreak-even time
Core 1: τ1
Core 2: τ2
Core 3: τ3
Core 4: τ4
Core 5: τ5
Example: DLTF and SFAs
ts
ts
ts
ts
tbreak-even time
Core 1: τ1
Core 2: τ2
Core 3: τ3
Core 4:
τ4
Core 5:
τ5
Double-Largest-Task-First (DLTF)
15 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Initial solution: Task partitioning with LTF.
Tasks are regrouped, shutting down all possible cores.
This reduces the energy consumption for idling under SFA:
Example: LTF and SFAs
ts
ts
ts
ts
tbreak-even time
Core 1: τ1
Core 2: τ2
Core 3: τ3
Core 4: τ4
Core 5: τ5
Example: DLTF and SFAs
ts
ts
ts
ts
tbreak-even time
Core 1: τ1
Core 2: τ2
Core 3: τ3
Core 4:
τ4
Core 5:
τ5
Double-Largest-Task-First (DLTF)
16 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 4 task sets:{
TDLTF1 ,TDLTF
2 , . . . ,TDLTF4
}
TDLTF1 TDLTF
2 TDLTF3 TDLTF
4
τ1
τ2 τ3 τ4
τ5τ6τ7
τ8
τ4
τ5
τ5
τ6
τ5
Double-Largest-Task-First (DLTF)
16 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 4 task sets:{
TDLTF1 ,TDLTF
2 , . . . ,TDLTF4
}
TDLTF1 TDLTF
2 TDLTF3 TDLTF
4
τ1
τ2 τ3 τ4
τ5τ6τ7
τ8
τ4
τ5
τ5
τ6
τ5
Double-Largest-Task-First (DLTF)
16 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 4 task sets:{
TDLTF1 ,TDLTF
2 , . . . ,TDLTF4
}
TDLTF1 TDLTF
2 TDLTF3 TDLTF
4
τ1
τ2 τ3 τ4
τ5τ6τ7
τ8
τ4
τ5
τ5
τ6
τ5
Double-Largest-Task-First (DLTF)
16 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 4 task sets:{
TDLTF1 ,TDLTF
2 , . . . ,TDLTF4
}
TDLTF1 TDLTF
2 TDLTF3 TDLTF
4
τ1
τ2 τ3 τ4
τ5τ6τ7
τ8
τ4
τ5
τ5
τ6
τ5
Double-Largest-Task-First (DLTF)
16 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 4 task sets:{
TDLTF1 ,TDLTF
2 , . . . ,TDLTF4
}
TDLTF1 TDLTF
2 TDLTF3 TDLTF
4
τ1
τ2 τ3 τ4
τ5τ6τ7
τ8
τ4
τ5
τ5
τ6
τ5
Double-Largest-Task-First (DLTF)
16 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 4 task sets:{
TDLTF1 ,TDLTF
2 , . . . ,TDLTF4
}
TDLTF1 TDLTF
2 TDLTF3 TDLTF
4
τ1
τ2 τ3 τ4
τ5τ6τ7
τ8
τ4
τ5
τ5
τ6
τ5
Double-Largest-Task-First (DLTF)
16 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Example:
8 tasks: {τ1, τ2, . . . , τ8}
Partitioned into 4 task sets:{
TDLTF1 ,TDLTF
2 , . . . ,TDLTF4
}
TDLTF1 TDLTF
2 TDLTF3 TDLTF
4
τ1
τ2 τ3 τ4
τ5τ6τ7
τ8
τ4
τ5
τ5
τ6
τ5
Properties of Double-Largest-Task-First (DLTF)
17 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
If wLTFM < scrit, then:
wDLTF1 ≤ wDLTF
1 ≤ · · · ≤ wDLTFM < scrit
All cores run at the critical frequency: su = scrit
The energy consumption for execution is minimized.
EDLTFSFA = E∗OPT ⇒ AFDLTF
SFA = 1
If wLTFM ≥ scrit, then:
wDLTFM = wLTF
M
If there is only one task in TDLTFM ⇒ wDLTF
M ≤ w∗M
If there are at least two tasks in TDLTFM ⇒ wDLTF
Mw∗M≤ θLTF = 4
3 −1
3M
Properties of Double-Largest-Task-First (DLTF)
17 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
If wLTFM < scrit, then:
wDLTF1 ≤ wDLTF
1 ≤ · · · ≤ wDLTFM < scrit
All cores run at the critical frequency: su = scrit
The energy consumption for execution is minimized.
EDLTFSFA = E∗OPT ⇒ AFDLTF
SFA = 1
If wLTFM ≥ scrit, then:
wDLTFM = wLTF
M
If there is only one task in TDLTFM ⇒ wDLTF
M ≤ w∗M
If there are at least two tasks in TDLTFM ⇒ wDLTF
Mw∗M≤ θLTF = 4
3 −1
3M
Outline
18 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Negligible Leakage Power Consumption
19 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Energy Consumption for DLTF and SFA (when β = 0):
EDLTFSFA = P (su)
Lsu
M
∑i=1
wDLTFi ⇒ EDLTF
SFA = αL(
wDLTFM
γ−1) M
∑i=1
w∗i
Lower Bound Energy Consumption (when β = 0):
Unroll periodic tasks in a hyper-period⇒ frame-based tasks.
Use the results from the SFA analysis paper 3:
E∗↓(β=0) = αL
[M∑
i=1
(w∗i −w∗i−1
) γ√
M − i + 1]γ
3Santiago Pagani and Jian-Jia Chen. “Energy Efficiency Analysis for the Single Frequency Approximation (SFA) Scheme”. In: Proceedings
of the 19th IEEE International Conference on Embedded and Real-Time Computing Systems and Applications (RTCSA). 2013
Negligible Leakage Power Consumption
19 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Energy Consumption for DLTF and SFA (when β = 0):
EDLTFSFA = P (su)
Lsu
M
∑i=1
wDLTFi ⇒ EDLTF
SFA = αL(
wDLTFM
γ−1) M
∑i=1
w∗i
Lower Bound Energy Consumption (when β = 0):
Unroll periodic tasks in a hyper-period⇒ frame-based tasks.
Use the results from the SFA analysis paper 3:
E∗↓(β=0) = αL
[M∑
i=1
(w∗i −w∗i−1
) γ√
M − i + 1]γ
3Santiago Pagani and Jian-Jia Chen. “Energy Efficiency Analysis for the Single Frequency Approximation (SFA) Scheme”. In: Proceedings
of the 19th IEEE International Conference on Embedded and Real-Time Computing Systems and Applications (RTCSA). 2013
Negligible Leakage Power Consumption
20 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max
(wDLTF
M
)γ−1 M∑
i=1w∗i[
M∑
i=1
(w∗i −w∗i−1
)γ√
M − i + 1]γ
Critical Cycle Utilization Distribution: Minimizes E∗↓ , for a fixed
M∑
i=1w∗i
· · ·w∗1
w∗2w∗M-1
w∗M
· · ·w ′1 w ′2 w ′M-1
w ′M
w ′1 = w ′2 = · · · = w ′M−1 = Average(w∗1 ,w
∗2 , . . . ,w∗M−1
)Utilization Ratio: 0 ≤ δ =
Average(w∗1 ,w∗2 ,...,w
∗M−1)
w∗M≤ 1
Negligible Leakage Power Consumption
20 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max
(wDLTF
M
)γ−1 M∑
i=1w∗i[
M∑
i=1
(w∗i −w∗i−1
)γ√
M − i + 1]γ
Critical Cycle Utilization Distribution: Minimizes E∗↓ , for a fixed
M∑
i=1w∗i
· · ·w∗1
w∗2w∗M-1
w∗M
· · ·w ′1 w ′2 w ′M-1
w ′M
w ′1 = w ′2 = · · · = w ′M−1 = Average(w∗1 ,w
∗2 , . . . ,w∗M−1
)Utilization Ratio: 0 ≤ δ =
Average(w∗1 ,w∗2 ,...,w
∗M−1)
w∗M≤ 1
Negligible Leakage Power Consumption
21 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max
(
wDLTFMw∗M
)γ−1
· h (δ)
where
h (δ) =1− δ + δM(
1− δ + δγ√
M)γ ≤ h (δmax)
and
δmax =
1γ−1
(γ− 1 + M − γ
γ√
M)
(M γ√
M −M − γ√
M + 1)
Negligible Leakage Power Consumption
21 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max
(
wDLTFMw∗M
)γ−1
· h (δ)
where
h (δ) =1− δ + δM(
1− δ + δγ√
M)γ ≤ h (δmax)
and
δmax =
1γ−1
(γ− 1 + M − γ
γ√
M)
(M γ√
M −M − γ√
M + 1)
Negligible Leakage Power Consumption
21 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max
(
wDLTFMw∗M
)γ−1
· h (δ)
where
h (δ) =1− δ + δM(
1− δ + δγ√
M)γ ≤ h (δmax)
and
δmax =
1γ−1
(γ− 1 + M − γ
γ√
M)
(M γ√
M −M − γ√
M + 1)
h (δ) for γ = 3:
0 0.5 11
1.5
2
2.5 M=32
δ
h(δ)
Negligible Leakage Power Consumption
21 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max
(
wDLTFMw∗M
)γ−1
· h (δ)
where
h (δ) =1− δ + δM(
1− δ + δγ√
M)γ ≤ h (δmax)
and
δmax =
1γ−1
(γ− 1 + M − γ
γ√
M)
(M γ√
M −M − γ√
M + 1)
h (δ) for γ = 3:
0 0.5 11
1.5
2
2.5 M=32
δ
h(δ)
δmax
Negligible Leakage Power Consumption
21 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max
(
wDLTFMw∗M
)γ−1
· h (δ)
where
h (δ) =1− δ + δM(
1− δ + δγ√
M)γ ≤ h (δmax)
and
δmax =
1γ−1
(γ− 1 + M − γ
γ√
M)
(M γ√
M −M − γ√
M + 1)
h (δ) for γ = 3:
0 0.5 11
1.5
2
2.5
δmax
M=2M=4
M=8
M=16
M=32
δ
h(δ)
Negligible Leakage Power Consumption
22 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
If w∗M ≥ wDLTFM ⇒ w ′M = wDLTF
M → Same AF as for fixed task sets
0 8 16 24 321
2
3
M
AF
n.p.
SFA
(β=0)
1.81
1.41
1.17
AFSFAn.p. (β = 0) (γ=3)
AFSFAn.p. (β = 0) (γ=2)
Note: AFn.p.SFA
(β=0)only depends on the values of γ and M.
Negligible Leakage Power Consumption
23 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
If w∗M < wDLTFM ⇒
If δ > δmax ⇒ h (δ) < h (δmax):
0 0.5 11
1.5
2
2.5
M=2M=4
M=8
M=16
M=32
δ
h(δ)
We prove that:
δ ≥w ′1w ′M≥ 4M + 1
6M≥ 0.66
The AF for this case is ≤ θγ−1LTF · h
(4M+1
6M
)
Negligible Leakage Power Consumption
23 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
If w∗M < wDLTFM ⇒
If δ > δmax ⇒ h (δ) < h (δmax):
0 0.5 11
1.5
2
2.5
M=2M=4
M=8
M=16
M=32
δ
h(δ)
We prove that:
δ ≥w ′1w ′M≥ 4M + 1
6M≥ 0.66
The AF for this case is ≤ θγ−1LTF · h
(4M+1
6M
)
Negligible Leakage Power Consumption
23 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Approximation Factor for DLTF and SFA (when β = 0):
If w∗M < wDLTFM ⇒
If δ > δmax ⇒ h (δ) < h (δmax):
0 0.5 11
1.5
2
2.5
M=2M=4
M=8
M=16
M=32
δ
h(δ)
We prove that:
δ ≥w ′1w ′M≥ 4M + 1
6M≥ 0.66
The AF for this case is ≤ θγ−1LTF · h
(4M+1
6M
)
Negligible Leakage Power Consumption
24 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Worst-case Approximation Factor for DLTF and SFA (when β = 0):
AFDLTFSFA
(β=0) ≤ max{
h (δmax) , θγ−1LTF · h
(4M + 1
6M
)}
0 8 16 24 32 40 481
2
3
4
M
AFDLTF
SFA
(β=0)
AFDLTFSFA
(β=0 )whenγ = 3
θγ−1LTF · h(
4M +16M ) when γ = 3
h(δmax ) when γ = 3
AFDLTFSFA
(β=0 )whenγ = 2
θγ−1LTF · h(
4M +16M ) when γ = 2
h(δmax ) when γ = 2
Note: AFDLTFSFA
(β=0)only depends on the values of γ and M.
Outline
25 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Non-negligible Leakage Power Consumption
26 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
We approximate the Lower Bound Energy Consumption:
0 0.5 1 1.5 2 2.50
20
40
60
80 if wM
* > sdyn
:
P(s) = α sγ
if wM
* ≤ sdyn
:
P(s) = α scrit
γ + β
smin
scrit
sdyn
smax
w∗
M[GHz]
E∗ ↓(w
∗ M)[Joule]
Non-negligible Leakage Power Consumption
26 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
We approximate the Lower Bound Energy Consumption:
0 0.5 1 1.5 2 2.50
20
40
60
80 if wM
* > sdyn
:
P(s) = α sγ
if wM
* ≤ sdyn
:
P(s) = α scrit
γ + β
smin
scrit
sdyn
smax
w∗
M[GHz]
E∗ ↓(w
∗ M)[Joule]
Non-negligible Leakage Power Consumption
26 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
We approximate the Lower Bound Energy Consumption:
0 0.5 1 1.5 2 2.50
20
40
60
80 if wM
* > sdyn
:
P(s) = α sγ
if wM
* ≤ sdyn
:
P(s) = α scrit
γ + β
smin
scrit
sdyn
smax
w∗
M[GHz]
E∗ ↓(w
∗ M)[Joule]
Negligible Leakage Power Consumption
27 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Worst-case Approximation Factor for DLTF and SFA (when β 6= 0):
AFDLTFSFA ≤ max
γ− 1
[γγh (δmax)]1
γ−1
+ h (δmax) ,γ− 1
θLTF
[γγh
(4M+1
6M
)] 1γ−1
+ θγ−1LTF h
(4M + 1
6M
)
0 8 16 24 32 40 481
2
3
4
M
AFDLTF
SFA
AFDLTFSFA when γ = 3
γ−1
θL T F[γ γh( 4M +16M )]
1γ− 1
+ θγ−1LTFh(
4M +16M ) when γ = 3
γ−1
[γ γh(δmax)]1
γ− 1+ h(δmax) whenγ = 3
AFDLTFSFA when γ = 2
γ−1
θL T F[γ γh( 4M +16M )]
1γ− 1
+ θγ−1LTFh(
4M +16M ) when γ = 2
γ−1
[γ γh(δmax)]1
γ− 1+ h(δmax) whenγ = 2
Note: AFDLTFSFA only depends on the values of γ and M.
Outline
28 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Non-negligible Sleeping Overhead
29 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Our strategy can be combined with any DPM schemes.
Two cases:
∑Mi=1 w∗i < scrit:
DLTF assigns all tasks on one core and SFA executes at scrit.
Uniprocessor scheduling problem.
For example, using Left-To-Right (LTR) algorithm 4 → 2-approximation.
∑Mi=1 w∗i ≥ scrit:
Using any DPM scheme, against the optimal solution:
AFDLTFSFA-DPM ≤ AFDLTF
SFA +γ− 1
γ
4Sandy Irani, Sandeep Shukla, and Rajesh Gupta. “Algorithms for power savings”. In: Symposium on Discrete Algorithms (SODA).
Baltimore, Maryland, 2003, pp. 37–46
Non-negligible Sleeping Overhead
29 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Our strategy can be combined with any DPM schemes.
Two cases:
∑Mi=1 w∗i < scrit:
DLTF assigns all tasks on one core and SFA executes at scrit.
Uniprocessor scheduling problem.
For example, using Left-To-Right (LTR) algorithm 4 → 2-approximation.
∑Mi=1 w∗i ≥ scrit:
Using any DPM scheme, against the optimal solution:
AFDLTFSFA-DPM ≤ AFDLTF
SFA +γ− 1
γ
4Sandy Irani, Sandeep Shukla, and Rajesh Gupta. “Algorithms for power savings”. In: Symposium on Discrete Algorithms (SODA).
Baltimore, Maryland, 2003, pp. 37–46
Non-negligible Sleeping Overhead
29 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Our strategy can be combined with any DPM schemes.
Two cases:
∑Mi=1 w∗i < scrit:
DLTF assigns all tasks on one core and SFA executes at scrit.
Uniprocessor scheduling problem.
For example, using Left-To-Right (LTR) algorithm 4 → 2-approximation.
∑Mi=1 w∗i ≥ scrit:
Using any DPM scheme, against the optimal solution:
AFDLTFSFA-DPM ≤ AFDLTF
SFA +γ− 1
γ
4Sandy Irani, Sandeep Shukla, and Rajesh Gupta. “Algorithms for power savings”. In: Symposium on Discrete Algorithms (SODA).
Baltimore, Maryland, 2003, pp. 37–46
Outline
30 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Simulation Results
31 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
For negligible overhead for sleeping:
Power parametersmodelled from SCC.
150 cases of synthetictasks for every M, withdifferent:
Amount of tasks.
Cycle utilizations.
Resulting hyper-periods.
Task Partitioning:DLTF for SFA.
Lower bound for theoptimal solution.
Simulation Results
31 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
For negligible overhead for sleeping:
Power parametersmodelled from SCC.
150 cases of synthetictasks for every M, withdifferent:
Amount of tasks.
Cycle utilizations.
Resulting hyper-periods.
Task Partitioning:DLTF for SFA.
Lower bound for theoptimal solution.
0 8 16 24 32 40 481
2
3
4
M
EDLTF
SFA
(wDLTF
M)
E∗ ↓(w
∗ M)
pea
k
AFDLTFSFA
EDLTFSFA (wDLTF
M)
E∗
↓(w∗
M)
peak
Outline
32 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Introduction
Motivation and Problem Definition
Task Partitioning Scheme: Double-Largest-Task-First (DLTF)
Approximation Factor Analysis (energy consumption): DLTF and SFANegligible Leakage Power Consumption
Non-negligible Leakage Power Consumption
Non-negligible Sleeping Overhead
Simulations
Conclusions
Conclusions
33 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
For periodic tasks, combining DLTF and SFA is a practical solution forenergy minimization.
Approximation factor of DLTF and SFA for energy efficiency:Considered cases: negligible leakage, non-negligible leakage, andcombinations with DPM.
Bounded by γ and M (for all cases).
Simulations show a small gap compared with our analysis (for theworst-case).
Combining DLTF and SFA is an acceptable scheme based on theworst-case analysis.
34 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Thank you!
Questions?
34 Dec. 2013 Santiago Pagani and Jian-Jia ChenEnergy Efficient Task Partitioning based on the Single Frequency Approximation Scheme
KIT
Thank you!
Questions?