From 820c656c774ebf4354b0b3c49580d030621f8a9e Mon Sep 17 00:00:00 2001 From: Arnaud Giersch Date: Fri, 23 May 2014 16:20:22 +0200 Subject: [PATCH 01/16] Revert "Correct formulae and algorithm." This reverts commit deaad9c91bd56f904fe3c5ff93664431a18d4f44. --- paper.tex | 19 +++++++++++-------- 1 file changed, 11 insertions(+), 8 deletions(-) diff --git a/paper.tex b/paper.tex index 9c374ba..618e7ad 100644 --- a/paper.tex +++ b/paper.tex @@ -345,18 +345,18 @@ the consumed energy with scaled frequency and the consumed energy without scaled frequency: \begin{multline} \label{eq:enorm} - \Enorm(S) = \frac{ \Ereduced}{\Eoriginal} \\ - {} = \frac{\Pdyn \cdot S^{-2} \cdot + \Enorm = \frac{ \Ereduced}{\Eoriginal} \\ + {} = \frac{\Pdyn \cdot S_1^{-2} \cdot \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + - \Pstatic \cdot T_1 \cdot S \cdot N}{ + \Pstatic \cdot T_1 \cdot S_1 \cdot N}{ \Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + \Pstatic \cdot T_1 \cdot N } \end{multline} In the same way we can normalize the performance as follows: \begin{equation} \label{eq:pnorm} - \Pnorm(S) = \frac{\Tnew}{\Told} - = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{ + \Pnorm = \frac{\Tnew}{\Told} + = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{ \TmaxCompOld + \TmaxCommOld} \end{equation} The second problem is that the optimization operation for both energy and @@ -377,7 +377,7 @@ direction. Therefore, we inverse the equation of the normalized performance as follows: \begin{equation} \label{eq:pnorm_en} - \Pnorm^{-1}(S) = \frac{ \Told}{ \Tnew} + \Pnorm^{-1} = \frac{ \Told}{ \Tnew} = \frac{\TmaxCompOld + \TmaxCommOld}{\TmaxCompOld \cdot S + \TmaxCommOld} @@ -433,10 +433,13 @@ the objective function described above. \For {$j = 2$ to $\Pstates$} \State $\Fnew \gets \Fnew - \Fdiff$ \State $S \gets \Fmax / \Fnew$ + \State $S_i \gets S \cdot \frac{T_1}{T_i} + = \frac{\Fmax}{\Fnew} \cdot \frac{T_1}{T_i}$ + for $i=1,\dots,N$ \State $\Enorm \gets - \frac{\Pdyn \cdot S^{-2} \cdot + \frac{\Pdyn \cdot S_1^{-2} \cdot \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + - \Pstatic \cdot T_1 \cdot S \cdot N }{ + \Pstatic \cdot T_1 \cdot S_1 \cdot N }{ \Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + \Pstatic \cdot T_1 \cdot N }$ -- 2.39.5 From f0308f6359a61fd5ac3468415f50349fbfc8af62 Mon Sep 17 00:00:00 2001 From: Arnaud Giersch Date: Fri, 23 May 2014 16:25:35 +0200 Subject: [PATCH 02/16] Update. --- paper.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/paper.tex b/paper.tex index 618e7ad..f87dd56 100644 --- a/paper.tex +++ b/paper.tex @@ -285,11 +285,11 @@ function of the scaling factor $S$, as in EQ~\eqref{eq:energy}. \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) + \Pstatic \cdot T_1 \cdot S_1 \cdot N \end{equation} -where $N$ is the number of parallel nodes, $T_i$ and $S_i$ for $i=1,\dots,N$ are -the execution times and scaling factors of the sorted tasks. Therefore, $T_1$ is +where $N$ is the number of parallel nodes, $T_i$ for $i=1,\dots,N$ are +the execution times of the sorted tasks. Therefore, $T_1$ is the time of the slowest task, and $S_1$ its scaling factor which should be the highest because they are proportional to the time values $T_i$. The scaling -factors are computed as in EQ~\eqref{eq:si}. +factors $S_i$ are computed as in EQ~\eqref{eq:si}. \begin{equation} \label{eq:si} S_i = S \cdot \frac{T_1}{T_i} -- 2.39.5 From ce10e8df1478836eacce6e21a47c71d666d84561 Mon Sep 17 00:00:00 2001 From: jccharr Date: Tue, 27 May 2014 11:25:56 +0200 Subject: [PATCH 03/16] malll changes --- paper.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/paper.tex b/paper.tex index 2838f2d..fb35820 100644 --- a/paper.tex +++ b/paper.tex @@ -203,7 +203,7 @@ our paper is to present a new online scaling factor selection method which has t % paper in homogeneous clusters} -\section{Energy model for homogeneous platform} +\section{Energy model for a homogeneous platform} \label{sec.exe} Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into two power metrics: the static and the dynamic power. While the first one is -- 2.39.5 From c25c567d8c7358554605d785e62623b9fc7cda8e Mon Sep 17 00:00:00 2001 From: Arnaud Giersch Date: Tue, 27 May 2014 11:42:11 +0200 Subject: [PATCH 04/16] *a* homogeneous --- paper.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/paper.tex b/paper.tex index eb3cd18..da3fb81 100644 --- a/paper.tex +++ b/paper.tex @@ -136,7 +136,7 @@ the MPI program to choose the scaling factor. This algorithm has the ability to predict both energy consumption and execution time over all available scaling factors. The prediction achieved depends on some computing time information, gathered at the beginning of the runtime. We apply this algorithm to the NAS parallel benchmarks (NPB v3.3)~\cite{44}. Our experiments are executed using the simulator -SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over an homogeneous +SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over a homogeneous distributed memory architecture. Furthermore, we compare the proposed algorithm with Rauber and Rünger methods~\cite{3}. The comparison's results show that our algorithm gives better energy-time trade-off. @@ -682,7 +682,7 @@ trade-offs such as in BT and EP. In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed -application running on an homogeneous cluster. It uses the computation and +application running on a homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy consumption and the performance of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best trade-off -- 2.39.5 From af0f5992053409b1f86985b7208a110cc3790759 Mon Sep 17 00:00:00 2001 From: jccharr Date: Tue, 27 May 2014 11:50:53 +0200 Subject: [PATCH 05/16] changes --- paper.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/paper.tex b/paper.tex index eb3cd18..5ad7af7 100644 --- a/paper.tex +++ b/paper.tex @@ -228,7 +228,7 @@ our paper is to present a new online scaling factor selection method which has t % paper in homogeneous clusters} -\section{Energy model for a homogeneous platform} +\section{Energy model for an homogeneous platform} \label{sec.exe} Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into two power metrics: the static and the dynamic power. While the first one is -- 2.39.5 From c87d673a519126b07462c1167491c44c014bfdf1 Mon Sep 17 00:00:00 2001 From: jccharr Date: Tue, 27 May 2014 12:03:13 +0200 Subject: [PATCH 06/16] changes to a --- paper.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/paper.tex b/paper.tex index b4dc51c..da3fb81 100644 --- a/paper.tex +++ b/paper.tex @@ -228,7 +228,7 @@ our paper is to present a new online scaling factor selection method which has t % paper in homogeneous clusters} -\section{Energy model for an homogeneous platform} +\section{Energy model for a homogeneous platform} \label{sec.exe} Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into two power metrics: the static and the dynamic power. While the first one is -- 2.39.5 From 0928259450f000e2b9462375dfc0a4b9f59cd5b1 Mon Sep 17 00:00:00 2001 From: afanfakh Date: Wed, 28 May 2014 12:37:17 +0200 Subject: [PATCH 07/16] change of performance to execution time in a paper and plots --- fig/bt.eps | 58 ++++++------ fig/bt.pdf | Bin 7587 -> 0 bytes fig/cg.eps | 58 ++++++------ fig/cg.pdf | Bin 7967 -> 0 bytes fig/ep.eps | 246 +++++++++++++++++--------------------------------- fig/ep.pdf | Bin 7772 -> 0 bytes fig/file.eps | 10 +- fig/file.pdf | Bin 7276 -> 0 bytes fig/file3.eps | 12 ++- fig/file3.pdf | Bin 6512 -> 0 bytes fig/ft.pdf | Bin 7460 -> 0 bytes paper.tex | 55 +++++------ 12 files changed, 186 insertions(+), 253 deletions(-) mode change 100644 => 100755 fig/bt.eps delete mode 100644 fig/bt.pdf mode change 100644 => 100755 fig/cg.eps delete mode 100644 fig/cg.pdf mode change 100644 => 100755 fig/ep.eps delete mode 100644 fig/ep.pdf mode change 100644 => 100755 fig/file.eps delete mode 100644 fig/file.pdf mode change 100644 => 100755 fig/file3.eps delete mode 100644 fig/file3.pdf delete mode 100644 fig/ft.pdf diff --git a/fig/bt.eps b/fig/bt.eps old mode 100644 new mode 100755 index 2f44b28..0487747 --- a/fig/bt.eps +++ b/fig/bt.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-2.0 %%Title: bt.eps %%Creator: gnuplot 4.6 patchlevel 0 -%%CreationDate: Thu May 15 12:08:40 2014 +%%CreationDate: Tue May 27 14:43:24 2014 %%DocumentFonts: (atend) %%BoundingBox: 50 50 320 239 %%EndComments @@ -432,7 +432,7 @@ SDict begin [ /Author (afanfakh) % /Producer (gnuplot) % /Keywords () - /CreationDate (Thu May 15 12:08:40 2014) + /CreationDate (Tue May 27 14:43:24 2014) /DOCINFO pdfmark end } ifelse @@ -454,7 +454,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 602 448 M ( 0.3) Rshow /Helvetica findfont 140 scalefont setfont @@ -464,7 +464,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 602 799 M ( 0.4) Rshow /Helvetica findfont 140 scalefont setfont @@ -474,7 +474,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 0.5) Rshow /Helvetica findfont 140 scalefont setfont @@ -484,7 +484,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 0.6) Rshow /Helvetica findfont 140 scalefont setfont @@ -494,7 +494,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 0.7) Rshow /Helvetica findfont 140 scalefont setfont @@ -504,7 +504,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 0.8) Rshow /Helvetica findfont 140 scalefont setfont @@ -514,7 +514,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 0.9) Rshow /Helvetica findfont 140 scalefont setfont @@ -524,7 +524,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 1) Rshow /Helvetica findfont 140 scalefont setfont @@ -534,7 +534,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 1.1) Rshow /Helvetica findfont 140 scalefont setfont @@ -544,7 +544,7 @@ LTb 63 0 V 4398 0 R -63 0 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont -4482 0 R ( 1.2) Rshow /Helvetica findfont 140 scalefont setfont @@ -554,7 +554,7 @@ LTb 0 63 V 0 3100 R 0 -63 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 686 308 M ( 1) Cshow /Helvetica findfont 140 scalefont setfont @@ -564,7 +564,7 @@ LTb 0 63 V 0 3100 R 0 -63 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 0 -3240 R ( 1.5) Cshow /Helvetica findfont 140 scalefont setfont @@ -574,7 +574,7 @@ LTb 0 63 V 0 3100 R 0 -63 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 0 -3240 R ( 2) Cshow /Helvetica findfont 140 scalefont setfont @@ -584,7 +584,7 @@ LTb 0 63 V 0 3100 R 0 -63 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 0 -3240 R ( 2.5) Cshow /Helvetica findfont 140 scalefont setfont @@ -594,7 +594,7 @@ LTb 0 63 V 0 3100 R 0 -63 V -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 0 -3240 R ( 3) Cshow /Helvetica findfont 140 scalefont setfont @@ -609,26 +609,28 @@ LTb -4461 0 V Z stroke LCb setrgbcolor -/Helvetica findfont 140 scalefont setfont +/Helvetica findfont 200 scalefont setfont 112 2029 M currentpoint gsave translate -270 rotate 0 0 M -(Normalized energy and performance inv.) 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zrla#~UBfIe9_0`D@V~Q&0tk~$1$+i9IGrEb-|=7}7xIPCOcYbmV4$y$(+7Nk zDH;Zghw*072VhUr3 \Dist)$} + \State $\Pnorm \gets \Told / \Tnew$ + \If{$(\Pnorm - \Enorm > \Dist)$} \State $\Sopt \gets S$ - \State $\Dist \gets \PnormInv - \Enorm$ + \State $\Dist \gets \Pnorm - \Enorm$ \EndIf \EndFor \State Return $\Sopt$ @@ -524,9 +525,9 @@ Each node in the cluster has 18 frequency values from \np[GHz]{2.5} to \np[MHz]{800} with \np[MHz]{100} difference between each two successive frequencies. The nodes are connected via an ethernet network with 1Gbit/s bandwidth. -\subsection{Performance prediction verification} +\subsection{Execution time prediction verification} -In this section we evaluate the precision of our performance prediction method +In this section we evaluate the precision of our execution time prediction method based on EQ~\eqref{eq:tnew} by applying it to the NAS benchmarks. The NAS programs are executed with the class B option to compare the real execution time with the predicted execution time. Each program runs offline with all available @@ -566,14 +567,14 @@ and Rünger in~\cite{3}. The results showed that the algorithm selected different scaling factors for each program depending on the communication features of the program as in the plots from Figure~\ref{fig:nas}. These plots illustrate that there are different distances between the normalized energy and -the normalized inverted performance curves, because there are different +the normalized inverted execution time curves, because there are different communication features for each benchmark. When there are little or no -communications, the inverted performance curve is very close to the energy +communications, the inverted execution time curve is very close to the energy curve. Then the distance between the two curves is very small. This leads to small energy savings. The opposite happens when there are a lot of communication, the distance between the two curves is big. This leads to more energy savings (e.g. CG and FT), see Table~\ref{table:compareC}. All discovered -frequency scaling factors optimize both the energy and the performance +frequency scaling factors optimize both the energy and the execution time simultaneously for all NAS benchmarks. In Table~\ref{table:compareC}, we record all optimal scaling factors results for each benchmark running class C. These scaling factors give the maximum energy saving percentage and the minimum @@ -604,7 +605,7 @@ savings). In this section, we compare our scaling factor selection method with Rauber and Rünger methods~\cite{3}. They had two scenarios, the first is to reduce energy -to the optimal level without considering the performance as in +to the optimal level without considering the execution time as in EQ~\eqref{eq:sopt}. We refer to this scenario as $R_{E}$. The second scenario is similar to the first except setting the slower task to the maximum frequency (when the scale $S=1$) to keep the performance from degradation as mush as @@ -684,10 +685,10 @@ In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed application running on a homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy -consumption and the performance of the parallel application at every available +consumption and the execution time of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best trade-off between energy reduction and performance which is the maximum distance between -the energy and the inverted performance curves. To evaluate this method, we +the energy and the inverted eexecution time curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber and Rünger methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber and Rünger's methods in terms of energy-performance -- 2.39.5 From 526afcbbbf6d95852d6586ff18f662f97c195a85 Mon Sep 17 00:00:00 2001 From: afanfakh Date: Wed, 28 May 2014 12:50:57 +0200 Subject: [PATCH 08/16] correct of eexecution to execution --- paper.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/paper.tex b/paper.tex index b272684..d6d57d0 100644 --- a/paper.tex +++ b/paper.tex @@ -688,7 +688,7 @@ communication times measured at the first iteration to predict energy consumption and the execution time of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best trade-off between energy reduction and performance which is the maximum distance between -the energy and the inverted eexecution time curves. To evaluate this method, we +the energy and the inverted execution time curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber and Rünger methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber and Rünger's methods in terms of energy-performance -- 2.39.5 From b1439364c2206a717f8692414753c9a1066856d8 Mon Sep 17 00:00:00 2001 From: afanfakh Date: Wed, 28 May 2014 14:42:05 +0200 Subject: [PATCH 09/16] changing text of ep plot --- fig/ep.eps | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/fig/ep.eps b/fig/ep.eps index 45d7287..e1d29f4 100755 --- a/fig/ep.eps +++ b/fig/ep.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-2.0 %%Title: ep.eps %%Creator: gnuplot 4.6 patchlevel 0 -%%CreationDate: Tue May 27 15:28:04 2014 +%%CreationDate: Wed May 28 14:37:24 2014 %%DocumentFonts: (atend) %%BoundingBox: 50 50 320 239 %%EndComments @@ -432,7 +432,7 @@ SDict begin [ /Author (afanfakh) % /Producer (gnuplot) % /Keywords () - /CreationDate (Tue May 27 15:28:04 2014) + /CreationDate (Wed May 28 14:37:24 2014) /DOCINFO pdfmark end } ifelse @@ -588,7 +588,7 @@ LTb (Optimal scaling factor=1.04) Lshow /Helvetica findfont 140 scalefont setfont /Helvetica findfont 180 scalefont setfont -898 3381 M +806 3438 M (EP Class C) Lshow /Helvetica findfont 140 scalefont setfont 1.000 UL @@ -682,5 +682,5 @@ stroke grestore end showpage -%%Trailer +%%Trailer %%DocumentFonts: Helvetica -- 2.39.5 From 1d87f72a3089a46078b9e8aebef8d9b899d60e4c Mon Sep 17 00:00:00 2001 From: jccharr Date: Fri, 6 Jun 2014 11:26:43 +0200 Subject: [PATCH 10/16] Some changes int the performance and energy reduction trade-off --- paper.tex | 31 ++++++------------------------- 1 file changed, 6 insertions(+), 25 deletions(-) diff --git a/paper.tex b/paper.tex index d6d57d0..d8d84a2 100644 --- a/paper.tex +++ b/paper.tex @@ -311,8 +311,8 @@ EQ~\eqref{eq:sopt}. \section{Performance evaluation of MPI programs} \label{sec.mpip} -The execution time of parallel synchronous MPI applications depends -on the time of the slowest task. If there is no +The execution time of a parallel synchronous iterative application is +equal to the execution time of the slowest task. If there is no communication and the application is not data bounded, the execution time of a parallel program is linearly proportional to the operational frequency and any DVFS operation for energy reduction increases the execution time of the parallel @@ -337,11 +337,7 @@ for each processor as presented in the next section. \section{Performance and energy reduction trade-off} \label{sec.compet} -This section presents our approach for choosing the optimal scaling factor. -This factor gives maximum energy reduction while taking into account the execution -times for both computation and communication. The relation between the execution time -and the energy is nonlinear and complex. Thus, unlike the relation between the execution time and the scaling factor, the relation of energy with the scaling factor is nonlinear, for more details refer to~\cite{17}. Moreover, they are not measured using the same metric. To -solve this problem, we normalize the energy by calculating the ratio between +This section presents our method for choosing the optimal scaling factor that gives the best tradeoff between energy reduction and performance. This method takes into account the execution times for both computation and communication to compute the scaling factor. Since, the energy consumption and the performance are not measured using the same metric, a normalized value of both measurements can be used to compare them. The normalized energy is the ratio between the consumed energy with scaled frequency and the consumed energy without scaled frequency: \begin{multline} @@ -353,29 +349,14 @@ frequency: \Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + \Pstatic \cdot T_1 \cdot N } \end{multline} -In the same way we can normalize the time as follows: +In the same way, the normalized execution time of a program is computed as follows: \begin{equation} \label{eq:pnorm} \Tnorm = \frac{\Tnew}{\Told} = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{ \TmaxCompOld + \TmaxCommOld} \end{equation} -The second problem is that the optimization operation for both energy and -execution time is not in the same direction. In other words, the normalized energy -and the execution time curves are not at the same direction see -Figure~\ref{fig:rel}\subref{fig:r2}. While the main goal is to optimize the -energy and execution time in the same time. According to the -equations~\eqref{eq:enorm} and~\eqref{eq:pnorm}, the scaling factor $S$ reduce -both the energy and the execution time simultaneously. But the main objective is -to produce maximum energy reduction with minimum execution time reduction. Many -researchers used different strategies to solve this nonlinear problem for -example see~\cite{19,42}, their methods add big overheads to the algorithm to -select the suitable frequency. In this paper we present a method to find the -optimal scaling factor $S$ to optimize both energy and execution time -simultaneously without adding a big overhead. Our solution for this problem is -to make the optimization process for energy and execution time follow the same -direction. Therefore, we inverse the equation of the normalized execution time as -follows: +The relation between the execution time and the consumed energy of a program is nonlinear and complex. In consequences, the relation between the consumed energy and the scaling factor is also nonlinear, for more details refer to~\cite{17}. Therefore, the resulting normalized energy consumption curve and execution time curve, for different scaling factors, do not have the same direction see Figure~\ref{fig:rel}\subref{fig:r2}. To tackle this problem and optimize both terms, we inverse the equation of the normalized execution time as follows: \begin{equation} \label{eq:pnorm_en} \Pnorm = \frac{ \Told}{ \Tnew} @@ -393,7 +374,7 @@ follows: \label{fig:rel} \end{figure} Then, we can model our objective function as finding the maximum distance -between the energy curve EQ~\eqref{eq:enorm} and the inverse of execution time (performance) +between the energy curve EQ~\eqref{eq:enorm} and the inverse of the execution time (performance) curve EQ~\eqref{eq:pnorm_en} over all available scaling factors. This represents the minimum energy consumption with minimum execution time (better performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}. Then -- 2.39.5 From 70760e222f4b5ce42648eaec2037aaf40629c2e8 Mon Sep 17 00:00:00 2001 From: Arnaud Giersch Date: Fri, 6 Jun 2014 14:54:43 +0200 Subject: [PATCH 11/16] Remove comma after "Since". --- paper.tex | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/paper.tex b/paper.tex index d8d84a2..18115f5 100644 --- a/paper.tex +++ b/paper.tex @@ -337,8 +337,13 @@ for each processor as presented in the next section. \section{Performance and energy reduction trade-off} \label{sec.compet} -This section presents our method for choosing the optimal scaling factor that gives the best tradeoff between energy reduction and performance. This method takes into account the execution times for both computation and communication to compute the scaling factor. Since, the energy consumption and the performance are not measured using the same metric, a normalized value of both measurements can be used to compare them. The normalized energy is the ratio between -the consumed energy with scaled frequency and the consumed energy without scaled +This section presents our method for choosing the optimal scaling factor that +gives the best tradeoff between energy reduction and performance. This method +takes into account the execution times for both computation and communication to +compute the scaling factor. Since the energy consumption and the performance +are not measured using the same metric, a normalized value of both measurements +can be used to compare them. The normalized energy is the ratio between the +consumed energy with scaled frequency and the consumed energy without scaled frequency: \begin{multline} \label{eq:enorm} -- 2.39.5 From 0a5c63daac4a17dc16888d3374c97a6548e5feb5 Mon Sep 17 00:00:00 2001 From: Arnaud Giersch Date: Fri, 6 Jun 2014 14:56:36 +0200 Subject: [PATCH 12/16] Insert a \newpage before the references. --- paper.tex | 1 + 1 file changed, 1 insertion(+) diff --git a/paper.tex b/paper.tex index 18115f5..91fa988 100644 --- a/paper.tex +++ b/paper.tex @@ -701,6 +701,7 @@ supporting his work. % the document is modified later %\IEEEtriggeratref{15} +\newpage \bibliographystyle{IEEEtran} \bibliography{IEEEabrv,my_reference} \end{document} -- 2.39.5 From 6ca9e59079fd465ed14bdf659d8c3d9b6e0c0ab9 Mon Sep 17 00:00:00 2001 From: Arnaud Giersch Date: Fri, 6 Jun 2014 15:00:38 +0200 Subject: [PATCH 13/16] Generate pdf figures. --- fig/bt.pdf | Bin 0 -> 7555 bytes fig/cg.pdf | Bin 0 -> 7931 bytes fig/ep.pdf | Bin 0 -> 7234 bytes 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