The Energy/Frequency Convexity Rule of Energy Consumption ... · Presentation’s Outline 1...

The Energy/Frequency Convexity Ruleof Energy Consumption for Programs:

Modeling, Thermosensitivity, and Applications

Karel De Vogeleer

Ph.D. defense

September 4th, 2015

Special thanks to Fondation TELECOM for funding this research

Introduction Motivation

Karel De Vogeleer (ParisTech) The Energy/Frequency Convexity Rule September 4th, 2015 1 / 27

Introduction Motivation


Introduction Overview

A Green IT Thinking

Off-line, includingI transistor design,I circuit design,I architecture,I software design,I software coding,I compiler optimization;

on-line, includingI system reconfiguration,I compiler optimization,I context placement.

Image source jiji.ng and wisegeek.com


Introduction Thesis’ Contributions

Contributions

Energy consumption analysis for computer systems:I analytical model,I Energy/Frequency Convexity Rule,I supportive measurement data;

temperature/power relationship demystified:I supportive measurement data,I guidelines for power measurement;

transient thermal model for microprocessors:I analytical model including radiation,I approximations,I applicability analysis.



Contributions




dat

a.2[

ind

ex,

”PA

15”]

time (s)2075 2175 2275 2375 2475 2575 2675

pow

er(W

)1.

252

1.25

61.

261.

264

1.26

81.

272

1.27

6

datalinear transformquadratic transform



Contributions






Contributions




0

222630

36

45

60

83

surface (m2)0 0.01 0.02 0.03 0.04 0.05

r cr

-0.4

-0.2

00.

20.

40.

60.

81



Contributions

energy consumption analysis for computer systems:I analytical model,I Energy/Frequency Convexity Rule,I supportive measurement data;


transient thermal model for microprocessors:I analytical model,I approximations,I applicability analysis.


Introduction Outline

Presentation’s Outline

1 Introduction

2 Energy Model

3 Practical Example

4 Parameter Sensitivity

5 Case Studies

6 Conclusion


Energy Model

1 Introduction

2 Energy Model

3 Practical Example


5 Case Studies

6 Conclusion


Energy Model

Preliminary Evidence of Energy/Frequency Curves

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 200 400 600 800 1000

Nor

mal

ized

Tot

al E

nerg

y

CPU Frequency (MHz)

2% Miss Ratio9% Miss Ratio

16% Miss Ratio

(a) Fan et al. [1] (b) Le Sueur and Heiser [3]

1.5 2 2.5Frequency [GHz]

0

400

800

1200

1600

2000

2400

Ener

gy to

solu

tion

[J]

DGEMM 8CDGEMM 4CRAY 8CRAY SMT 8C

(a)

(c) Hager et al. [2]

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

50 100 150 200 250 300 350 400 450

CPU Energy

CPU Frequency (MHz)

Model predicted energybasicmath

bitcntscelpgzipmpg

qsortsusan.corners

susan.edgessusan.smoothing

visionworstfft

inv_fftpatriciatypeset

(d) Snowdon et al. [4]


Energy Model General Framework

System Energy Consumption Model (Esys)

System’s energy consumption Esys definition

Esys =

∫ ∆t

0Psys dt

=

∫ ∆t

0( Pcpu + Pback ) dt;

Examples of Pback include:I LCD screen,I radio interface,I sensors (e.g. GPS);

If Pcpu and Pback are constant over ∆t:

Esys = (Pcpu + Pback) ·∆t.

everythingelse

CPU

syst

emKarel De Vogeleer (ParisTech) The Energy/Frequency Convexity Rule September 4th, 2015 7 / 27

Energy Model Power and Time Model

Microprocessor Power Model Execution Time Model

CPU power Pcpu consists of:

dynamic power Pdyn,

leakage current Pleak,

short-circuit current Psc,

Pcpu = Pdyn + Pleak + Psc

= ( 1 + γV ) · η αCV 2f

= (1 + γV ) · ξV 2f .

Execution time ∆t depends on:

ccb code size in clock cycles,

f CPU clock frequency,

fk frequency thieves,

β slack time per clock cycle,

∆t = ccb

(1

f − fk+ β

).


Energy Model Optimal Clock Frequency

Optimal Clock Frequency (fopt)

System’s energy consumption model

Esys(f ) = (Pcpu + Pback) ·∆t

= ([1 + γV ]ξV 2f + Pback) · ccb

(1

f − fk+ β

),

where {γ, ξ,Pback, ccb, fk, β} ∈ R+;

a single minimum for Esys(f ) exists at fopt when(∂Esys

∂f

)f =fopt

= 0, and∂2Esys

∂f 2> 0 holds;

V is approximately an affine map of f : V → m2f + m1.


Energy Model Optimal Clock Frequency

Supply Voltage/Frequency Relationship

A linear trend between V and f is observed: V = m2f + m1.

0

Exynos 4210Exynos 4x12Exynos 5250Intel MS3C6410PXA320

linear approximations

m1 = 13 , m2 = 4

5m1 = 1

3 , m2 = 45

frequency (GHz)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

sup

ply

volt

age

(V)

0.85

0.95

11.

051.

151.

251.

351.

45


Practical Example

1 Introduction

2 Energy Model

3 Practical Example


5 Case Studies

6 Conclusion


Practical Example Energy Measurement

Benchmark and Testbed

Benchmark: bit-reverse algorithm,part of the DFT algorithm:

void bitreverse_gold_rader

(int N, complex *data) {

int n = N, nm1 = n-1;

int i = 0, j = 0;

for (; i < nm1; i++) {

int k = n >> 1;

if (i < j) {

complex temp = data[i];

data[i] = data[j];

data[j] = temp;}

while (k <= j) {

j -= k; k >>= 1;}

j += k ;

}

}

testbed: Samsung Galaxy SII;

power Measurement: Monsoon.







int n = N, nm1 = n-1;

int i = 0, j = 0;

for (; i < nm1; i++) {

int k = n >> 1;

if (i < j) {


data[i] = data[j];

data[j] = temp;}

while (k <= j) {

j -= k; k >>= 1;}

j += k ;

}

}









int n = N, nm1 = n-1;

int i = 0, j = 0;

for (; i < nm1; i++) {

int k = n >> 1;

if (i < j) {


data[i] = data[j];

data[j] = temp;}

while (k <= j) {

j -= k; k >>= 1;}

j += k ;

}

}





The Energy/Frequency Convexity Rule

Energy consumption versus CPU clock frequency shows convex properties.

1

frequency (GHz)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

ener

gyp

erar

ray

elem

ent

(nJ)

3040

5060

70

Input size (2N)

N= 6N= 8N= 10

N= 12N= 14N= 16

measuredmodel


Parameter Sensitivity

1 Introduction

2 Energy Model

3 Practical Example


5 Case Studies

6 Conclusion


Parameter Sensitivity

Energy Model’s Parameter Sensitivity Analysis

Energy consumption model under analysis:

Esys = ([1 + γV ] · ξV 2f + Pback) · ccb

(1

f − fk+ β

),

(∂Esys

∂f

)f =fopt

= 0;

The aim is to find the conditions under which fopt is exploitable;

The following parameters will be looked at in more detail:I frequency thieves (overhead) fk,I background power Pback,I power gain ξ,I temperature γ(T );

Analysis based on energy profile of the Exynos 4210.


Parameter Sensitivity Frequency Thieves

Influence of frequency thieves fk on fopt


(1

f−fk+ β

)1

ξ (V−1)

0.1550.1620.1680.1740.181

Pback=0.5 (W)

fk (GHz)0 0.2 0.5 0.8 1 1.2 1.5 1.8 2

f opt

(GH

z)0.

20.

61

1.4

1.8

2.2

2.6

33.

4

≈ 30 MHz

≈ 50 MHz

(e) fopt(fk,ξ)

1Pback (W)

00.511.52

2.533.544.5

ξ=0.168 (V−1)

fk (GHz)0 0.2 0.5 0.8 1 1.2 1.5 1.8 2

f opt

(GH

z)0.

20.

61

1.4

1.8

2.2

2.6

33.

4

(f) fopt(fk,Pback)


Parameter Sensitivity Background Power Demands

Influence of Background Power Pback on fopt


(1

f−fk+ β

)1

ξ (V−1)

0.1550.1620.1680.1740.181

Pback (W)0 1 2 3 4 5 6

f opt

(GH

z)0

0.2

0.5

0.8

11.

21.

51.

82

2.2

≈ 100 MHz

≈ 0.5 W

(g) fopt(Pback, ξ)

1ξ (V−1)

0.1550.1620.1680.1740.181

Pback (W)0 1 2 3 4 5 6

Pback/P

cpu

00.

20.

40.

60.

81

1.2

1.4

≈ 0.05

(h) Pback/Pcpu ratio at fopt


Parameter Sensitivity Power Gain

Influence of Power Gain ξ(s) on fopt


(1

f−fk+ β

)

Cooperative microprocessorson the same die:

1 power-efficient: Cortex A7,2 high-performance: Cortex A15;

ξ is scaled by s between its lowerand upper bound: s ∈ {1, 2};Exynos 5410 power model.

1

0

0.25

0.5

0.75

1

1.5

2

2.5

0

0.0750.15

0.3

A15A7

power scaling (s)1 1.2 1.4 1.6 1.8 2

f opt

(GH

z)0.

20.

40.

60.

81

1.2

1.4

1.6

Numbers on the lines represent the background power for that line.


Parameter Sensitivity Temperature

Influence of Temperature γ(T ) on fopt


(1

f−fk+ β

)

γ is a function of temperature;

temperature/power model ofExynos 5410 is used;

temperature/power showsexponential behavior;

0

temperature (◦C)30 40 50 60 70 80

pow

er(W

)2.

42.

52.

62.

72.

8

dataexponential fitquadratic fitlinear fit

∆fopt ≈ 200 MHz when 25◦C < T < 85◦C.


Case Studies

1 Introduction

2 Energy Model

3 Practical Example


5 Case Studies

6 Conclusion


Case Studies Optimization Techniques Classification

Case Study 1: fopt Classification

fopt•

max(fmin, fk)•

frequency

ener

gy

fopt•

max(fmin, fk)•

fmax•

frequency

ener

gy

fopt•

fmax•

frequency

ener

gy

1 fopt < max(fmin, fk) the slower, the better2 max(fmin, fk) ≤ fopt ≤ fmax chase fopt3 fmax < fopt race-to-halt


Case Studies Multi-core Code Execution

Case Study 2: fopt and Multi-core Code Execution

Clock frequency scheduling schemes:

1 on-demand: binary (high/low) as work arrives;

2 selfish: each core is individually energy optimized;

3 thread-cooperation: all cores are collectively energy optimized.

t

thread

0 ta tb tc

A

B

C

(a) default clock scaling

t

thread

0 tatbtc

A

B

C

(b) cooperative clock scaling


Case Studies Multi-core Code Execution

Case Study 2: fopt and Multi-core Code Execution contd.

Problem statement:

n threads executed in parallel with common deadline tmax;

threads individually clock frequency fi scalable;

Etot(fi ) : Rm → R to be minimized over fi :

Etot(fi ) = Pbacktmax +n∑

i=0

[ccb,i

fiP+ +

(tmax −

ccb,ifi

)P◦],

subject to ∀i ,ccb,i

fi≤ tmax and fmin ≤ fi ≤ fmax;

{ccb, fi , tmax,Pback,P◦,P+} ∈ R+;

active power (P+) and idle power (P◦) are generated by the Exynos5410 power model.


Case Studies Performance Evaluations

Case Study 2: fopt and Multi-core Code Execution contd.

Performance evaluation of 4 clock frequency scalable parallel threads.

1

thread cooperationselfishon-demand

background power (W)0 1 2 3 4 5 6 7 8 9 10

ener

gyra

tio

(%)

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

(a) energy

1

thread cooperationselfishon-demand

background power (W)0 1 2 3

tim

era

tio

(%)

11.

11.

31.

51.

71.

92

(b) time


Case Studies Performance Evaluations

Case Study 3: big-LITTLE Heterogeneous Computing

Optimal clock frequency forcooperative microprocessors:

1 power-efficient: Cortex A7,2 high-performance: Cortex A15;

fopt is chosen on the core yieldingbest efficiency;

Exynos 5410 power model used.

1

0.5

0.75

1

1.5

2

2.5

0

0.0750.15

0.30

0.25

power scaling (s)1 1.2 1.4 1.6 1.8 2

f opt

(GH

z)0.

20.

40.

60.

81

1.2

1.4

1.6

Numbers on the lines represent the

background power for that line.


Conclusion Summary

1 Introduction

2 Energy Model

3 Practical Example


5 Case Studies

6 Conclusion


Conclusion Summary

Conclusion

System’s energy consumption shows convex properties over f ;

rules of thumb for an exploitable fopt:I Pback should be smaller than Pcpu,I overhead (fk) should be limited,I slack time β should be limited,I power profile (ξ) has minimal effect,I code size (ccb) has no effect;

energy gains could be from 10% up to 50% at fixed temperature;

temperature/Power relationship shows exponential behavior;

radiation can be omitted for small devices.


Conclusion What’s Next

Future Work

Including:

apply results to other domains:I multi-core,I HPC,I clock modulation,I interactive/performance;

exploit the thermal behavior;

better understanding of how muchenergy can practically be gained.


The Energy/Frequency Convexity Ruleof Energy Consumption for Programs:

Modeling, Thermosensitivity, and Applications

Karel De Vogeleer

Ph.D. defense

September 4th, 2015


Publications

K. DeVogeleer, G. Memmi, P. Jouvelot, and F. Coelho, “The Energy/FrequencyConvexity Rule: modeling and experimental validation on mobile devices,” inProceedings of the 10th Conference on Parallel Processing and AppliedMathematics. Springer Verlag, Sep. 2013.

K. DeVogeleer, G. Memmi, P. Jouvelot, and F. Coelho, “Modeling thetemperature bias of power consumption for nanometer-scale cpus in applicationprocessors,” in 14th International Conference on Embedded Computer Systems:Architectures, Modeling, and Simulation, Jul. 2014, pp. 172-180.

K. DeVogeleer, P. Jouvelot, and G. Memmi, “The impact of surface size on theradiative thermal behavior of embedded systems,” CoRR, vol. abs/1410.0628,2014, (submitted to IEEE TMC in 2014).

K. DeVogeleer, G. Memmi, and P. Jouvelot, “Parameter Sensitivity Analysis of theEnergy/Frequency Convexity Rule for Nanometer-scale Application Processors,”CoRR, vol. abs/1508.07740, 2015, (in submission to The Elsevier Journal ofParallel and Distributed Computing, 2015).



References I

Fan, X., Ellis, C. S., and Lebeck, A. R. The synergy between power-aware memory systems and processor voltage

scaling. In Proceedings of the Third international conference on Power - Aware Computer Systems (Berlin, Heidelberg,2004), Springer-Verlag, pp. 164–179.

Hager, G., Treibig, J., Habich, J., and Wellein, G. Exploring performance and power properties of modern

multi-core chips via simple machine models. Concurrency and Computation: Practice and Experience (2013), n/a–n/a.

Le Sueur, E., and Heiser, G. Dynamic voltage and frequency scaling: the laws of diminishing returns. In Proceedings

of the 2010 international conference on Power aware computing and systems (Berkeley, CA, USA, 2010), HotPower’10,pp. 1–8.

Snowdon, D. C., Ruocco, S., and Heiser, G. Power management and dynamic voltage scaling: Myths and facts. In

2005 WS Power Aware Real-time Comput. (New Jersey, USA, Sept. 2005).


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