X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/80fa286273f1324fb10bd3a0e5d4c482a4032096..7cdfe38eb1150188a1f43cb954d2ddb5103aaab6:/Heter_paper.tex?ds=inline

diff --git a/Heter_paper.tex b/Heter_paper.tex
index d2180d9..c29b2a4 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -8,7 +8,7 @@
 \usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
-
+\usepackage{multirow}
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
@@ -53,7 +53,7 @@
 \newcommand{\Told}{\Xsub{T}{Old}} 
 \begin{document} 
 
-\title{Energy Consumption Reduction in a Heterogeneous Architecture Using DVFS}
+\title{Energy Consumption Reduction for Message Passing Iterative  Applications in Heterogeneous Architecture Using DVFS}
  
 \author{% 
   \IEEEauthorblockN{%
@@ -84,12 +84,13 @@ Therefore, the frequency that gives the best  tradeoff between the energy consum
 application must be selected. 
 
 In this paper, a new online frequencies selecting algorithm for heterogeneous platforms is presented. 
-It selects the frequency that give the best tradeoff between energy saving and performance degradation, 
+It selects the frequency that try to give the best tradeoff between energy saving and performance degradation, 
 for each node computing the message passing iterative application. The algorithm has a small overhead and 
 works without training or profiling. It uses a new energy model for message passing iterative applications 
-running on a heterogeneous platform. The proposed algorithm evaluated  on the Simgrid simulator while 
+running on a heterogeneous platform. The proposed algorithm is evaluated  on the Simgrid simulator while 
 running the NAS parallel benchmarks. The experiments demonstrated that it reduces the energy consumption 
-up to 35\% while limiting the performance degradation as much as possible.
+up to 35\% while limiting the performance degradation as much as possible. Finally, the algorithm is compared to an existing method and the comparison results show that it outperforms the latter.
+
 \end{abstract}
 
 \section{Introduction}
@@ -107,9 +108,9 @@ Tianhe-2 platform is approximately more than \$10 millions each year.
 The computing platforms must be more energy efficient and offer the highest number of FLOPS per watt possible, 
 such as the L-CSC from the GSI Helmholtz Center which  
 became the top of the Green500 list in November 2014 \cite{Green500_List}. 
-This heterogeneous platform executes more than 5  GFLOPS per watt while consuming 57.15 kilowatts.
+This heterogeneous platform executes more than 5  GFLOPS per watt while consumed 57.15 kilowatts.
 
-Besides hardware improvements, there are many software techniques to lower the energy consumption of these platforms, 
+Besides platform improvements, there are many software and hardware techniques to lower the energy consumption of these platforms, 
 such as scheduling, DVFS, ... DVFS is a widely  used process to reduce the energy consumption of a processor by lowering 
 its frequency \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also  reduces the number of FLOPS 
 executed by the processor which might increase  the execution time of the application running over that processor.
@@ -127,8 +128,9 @@ the energy-performance objective function that maximizes the reduction of energy
 consumption while minimizing the degradation of the program's performance.
 Section~\ref{sec.optim} details the proposed frequency selecting algorithm then the precision of the proposed algorithm is verified. 
 Section~\ref{sec.expe} presents the results of applying the algorithm on  the NAS parallel benchmarks and executing them 
-on a heterogeneous platform. It also shows the results of running three 
-different power scenarios and comparing them. 
+on a heterogeneous platform. It shows the results of running three 
+different power scenarios and comparing them. Moreover, it also shows the comparison results
+between the proposed method and an existing method.
 Finally, in Section~\ref{sec.concl} the paper is ended with a summary and some future works.
 
 \section{Related works}
@@ -136,7 +138,7 @@ Finally, in Section~\ref{sec.concl} the paper is ended with a summary and some f
 DVFS is a technique enabled 
 in modern processors to scale down both the voltage and the frequency of 
 the CPU while computing, in order to reduce the energy consumption of the processor. DVFS is 
-also  allowed in the GPUs to achieve the same goal. Reducing the frequency of a processor lowers its number of FLOPS and might degrade the performance of the application running on that processor, especially if it is compute bound. Therefore selecting the appropriate frequency for a processor to satisfy some objectives and while taking into account all the constraints, is not a trivial operation.  Many researchers used different strategies to tackle this problem. Some of them developed online methods that compute the new frequency while executing the application, such as ~\cite{Hao_Learning.based.DVFS,Dhiman_Online.Learning.Power.Management}. Others used offline methods that might need to run the application and profile it before selecting the new frequency, such as ~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The methods could be heuristics, exact  or brute force methods that satisfy varied objectives such as energy reduction or performance. They also could be adapted to the execution's environment and the type of the application such as sequential, parallel or distributed architecture, homogeneous or heterogeneous platform,  synchronous or asynchronous application, ... 
+also  allowed in the GPUs to achieve the same goal. Reducing the frequency of a processor lowers its number of FLOPS and might degrade the performance of the application running on that processor, especially if it is compute bound. Therefore selecting the appropriate frequency for a processor to satisfy some objectives and while taking into account all the constraints, is not a trivial operation.  Many researchers used different strategies to tackle this problem. Some of them developed online methods that compute the new frequency while executing the application, such as ~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}. Others used offline methods that might need to run the application and profile it before selecting the new frequency, such as ~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The methods could be heuristics, exact  or brute force methods that satisfy varied objectives such as energy reduction or performance. They also could be adapted to the execution's environment and the type of the application such as sequential, parallel or distributed architecture, homogeneous or heterogeneous platform,  synchronous or asynchronous application, ... 
 
 In this paper, we are interested in reducing energy for message passing iterative synchronous applications running over heterogeneous platforms.
 Some works have already been done for such platforms and they can be classified into two types of heterogeneous platforms: 
@@ -252,7 +254,7 @@ vector of scaling factors can be predicted using (\ref{eq:perf}).
 \end{equation}
 Where:\\
 \begin{equation}
-\label{eq:perf}
+\label{eq:perf2}
  MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
 \end{equation}
 where $TcpOld_i$ is the computation time  of processor $i$ during the first 
@@ -356,7 +358,7 @@ The communication time of a processor $i$ is noted as $Tcm_{i}$ and could contai
 if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do 
 not have equal communication times. While the dynamic energy is computed according to the frequency 
 scaling factor and the dynamic power of each node as in (\ref{eq:Edyn}), the static energy is 
-computed as the sum of the execution time of each processor multiplied by its static power. 
+computed as the sum of the execution time of one iteration multiplied by static power of each processor. 
 The overall energy consumption of a message passing  distributed application executed over a 
 heterogeneous platform during one iteration is the summation of all dynamic and static energies 
 for each  processor.  It is computed as follows:
@@ -403,7 +405,7 @@ factors is nonlinear, for more details refer to~\cite{Freeh_Exploring.the.Energy
 Moreover, they are not measured using the same metric.  To solve this problem,  the
 execution time is normalized by computing the ratio between the new execution time (after 
 scaling down the frequencies of some processors) and the initial one (with maximum 
-frequency for all nodes,) as follows:
+frequency for all nodes) as follows:
 \begin{multline}
   \label{eq:pnorm}
   P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
@@ -427,13 +429,13 @@ Where $E_\textit{Reduced}$ and $E_\textit{Original}$ are computed using (\ref{eq
 While the main 
 goal is to optimize the energy and execution time at the same time, the normalized 
 energy and execution time curves are not in the same direction. According 
-to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the vector  of frequency
+to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the vector  of frequency
 scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
 time simultaneously.  But the main objective is to produce maximum energy
 reduction with minimum execution time reduction.  
   
 This problem can be solved by making the optimization process for energy and 
-execution time follow the same direction.  Therefore, the equation of the 
+execution time following the same direction.  Therefore, the equation of the 
 normalized execution time is inverted which gives the normalized performance equation, as follows:
 \begin{multline}
   \label{eq:pnorm_inv}
@@ -456,8 +458,8 @@ normalized execution time is inverted which gives the normalized performance equ
 \end{figure}
 
 Then, the objective function can be modeled   as finding the maximum distance
-between the energy curve EQ~(\ref{eq:enorm}) and the  performance
-curve EQ~(\ref{eq:pnor@+eYd162m_inv}) over all available sets of scaling factors.  This
+between the energy curve (\ref{eq:enorm}) and the  performance
+curve (\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
 represents the minimum energy consumption with minimum execution time (maximum 
 performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then the objective
 function has the following form:
@@ -468,8 +470,8 @@ function has the following form:
       (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
        \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
 \end{equation}
-where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
-Then, the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}) can be selected.  
+where $N$ is the number of nodes and $F$ is the  number of available frequencies for each node. 
+Then, the optimal set of scaling factors that satisfies (\ref{eq:max}) can be selected.  
 The objective function can work with any energy model or any power values for each node 
 (static and dynamic powers). However, the most energy reduction gain can be achieved when 
 the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
@@ -478,14 +480,14 @@ the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynam
 \label{sec.optim}
 
 \subsection{The algorithm details}
-In this section algorithm~(\ref{HSA}) is presented. It selects the frequency scaling factors 
+In this section, algorithm \ref{HSA} is presented. It selects the frequency scaling factors 
 vector that gives the best trade-off between minimizing the energy consumption  and maximizing 
 the performance of a message passing synchronous iterative application executed on a heterogeneous 
 platform. It works online during the execution time of the iterative message passing program.  
 It uses information gathered during the first iteration such as the computation time and the 
 communication time in one iteration for each node. The algorithm is executed  after the first 
 iteration and returns a vector of optimal frequency scaling factors   that satisfies the objective 
-function (\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs 
+function (\ref{eq:max}). The program applies DVFS operations to change the frequencies of the CPUs 
 according to the computed scaling factors.  This algorithm is called just once during the execution 
 of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called 
 in the iterative MPI program.
@@ -524,7 +526,7 @@ scaling factors starts the search method from these initial frequencies and take
 toward lower frequencies. The algorithm iterates on all left frequencies, from the higher bound until all 
 nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select 
 the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node 
-according to (\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of  
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of  
 all other nodes by one gear.
 The new overall energy consumption and execution time are computed according to the new scaling factors. 
 The optimal set of frequency scaling factors is the set that gives the highest distance according to  the objective 
@@ -569,12 +571,11 @@ which results in bigger energy savings.
     \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
     \If{(not the first frequency)}
           \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
-          \State where $i=1,\dots,N$ means for loop.
     \EndIf 
     \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
     \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
-    \State  $Sopt_{i} \gets \frac{Fmax_i}{F_i},~i=1,\dots,N. $
-    \State Computing the initial distance  $Dist \gets\Pnorm(Sopt_i) - \Enorm(Sopt_i) $
+    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
+    \State $Dist \gets 0 $
     \While {(all nodes not reach their  minimum  frequency)}
         \If{(not the last freq. \textbf{and} not the slowest node)}
         \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
@@ -616,12 +617,12 @@ which results in bigger energy savings.
   \label{dvfs}
 \end{algorithm}
 
-\subsection{The verifications of the proposed algorithm}
+\subsection{The evaluation of the proposed algorithm}
 \label{sec.verif.algo}
 The precision of the proposed algorithm mainly depends on the execution time prediction model defined in 
 (\ref{eq:perf}) and the energy model computed by (\ref{eq:energy}). 
 The energy model is also significantly dependent  on the execution time model because the static energy is 
-linearly related the execution time and the dynamic energy is related to the computation time. So, all of 
+linearly related to the execution time and the dynamic energy is related to the computation time. So, all of 
 the works presented in this paper is based on the execution time model. To verify this model, the predicted 
 execution time was compared to  the real execution time over SimGrid/SMPI simulator, v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}, 
 for all  the NAS parallel benchmarks NPB v3.3 
@@ -635,7 +636,7 @@ that tests all the possible solutions. The brute force algorithm was applied to
 different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical 
 and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time: 
 for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in 
-table~(\ref{table:platform}), it takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes 
+table~\ref{table:platform}, it takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes 
 to compute the best scaling factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number 
 of iterations and $N$ is the number of computing nodes. The algorithm needs  from 12 to 20 iterations to select the best 
 vector of frequency scaling factors that gives the results of the next sections.
@@ -671,16 +672,16 @@ Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwi
                   &           & GHz      & GHz          &GHz             &              &       \\
     \hline
     1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
-                  &           &          &              &                &              &  \\
+         
     \hline
     2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
-                  &           &          &              &                &              &  \\
+                  
     \hline
     3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
-                  &           &          &              &                &              &  \\
+                  
     \hline
     4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
-                  &           &          &              &                &              &  \\
+                  
     \hline
   \end{tabular}
   \label{table:platform}
@@ -709,7 +710,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &  64.64        & 3560.39        &34.16        &6.72               &27.44       \\
@@ -736,7 +737,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &36.11    	   &3263.49	        &31.25	      &7.12	               &24.13     \\
@@ -763,7 +764,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &31.74	       &4373.90	        &26.29	      &9.57	               &16.72          \\
@@ -790,7 +791,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &32.35	       &6704.21	        &16.15	      &5.30	               &10.85           \\
@@ -817,7 +818,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &46.65	       &17521.83	    &8.13	      &1.68	               &6.45           \\
@@ -845,7 +846,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance     \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
     \hline
     CG         &56.92	       &41163.36        &4.00	      &1.10	               &2.90          \\
@@ -965,7 +966,7 @@ results in less energy saving but less performance degradation.
   \centering
   \begin{tabular}{|*{6}{l|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
+    Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
     \hline
     CG         &4144.21          &22.42        &7.72                &14.70         \\
@@ -994,7 +995,7 @@ results in less energy saving but less performance degradation.
   \centering
   \begin{tabular}{|*{6}{l|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
+    Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
     \hline
     CG         &2812.38	         &36.36	       &6.80                &29.56         \\
@@ -1018,7 +1019,7 @@ results in less energy saving but less performance degradation.
 
 \begin{figure}
   \centering
-  \subfloat[Comparison the average of the results on 8 nodes]{%
+  \subfloat[Comparison  of the results on 8 nodes]{%
     \includegraphics[width=.33\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
 
   \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
@@ -1030,21 +1031,74 @@ results in less energy saving but less performance degradation.
 
 
 
+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
+In this section, the scaling  factors selection algorithm, called MaxDist,
+is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP. 
+They developed a green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Enegry*Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
+To fairly compare both algorithms, the same energy and execution time models, equations (\ref{eq:energy}) and  (\ref{eq:fnew}), were used for both algorithms to predict the energy consumption and the execution times. Also Spiliopoulos et al. algorithm was adapted to  start the search from the 
+initial frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm is an exhaustive search algorithm that minimizes the EDP and has the initial frequencies values as an upper bound.
+
+Both algorithms were applied to the parallel NAS benchmarks to compare their efficiency. Table \ref{table:compare_EDP}  presents the results of comparing the execution times and the energy consumptions for both versions of the NAS benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous nodes. The results show that our algorithm gives better energy savings than Spiliopoulos et al. algorithm, 
+on average it results in 29.76\% energy saving while their algorithm returns just 25.75\%. The average of performance degradation percentage is approximately the same for both algorithms, about 4\%. 
+
+
+For all benchmarks, our algorithm outperforms 
+Spiliopoulos et al. algorithm in term of energy and performance tradeoff, see figure (\ref{fig:compare_EDP}), because it maximizes the distance between the energy saving and the performance degradation values while giving the same weight for both metrics. 
+
+
+
+
+\begin{table}[h]
+ \caption{Comparing the proposed algorithm}
+ \centering
+\begin{tabular}{|l|l|l|l|l|l|l|l|}
+\hline
+\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Program \\ name\end{tabular}}} & \multicolumn{2}{l|}{Energy saving \%} & \multicolumn{2}{l|}{Perf.  degradation \%} & \multicolumn{2}{l|}{Distance} \\ \cline{3-8} 
+\multicolumn{2}{|l|}{}                                                                         & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
+\multicolumn{2}{|l|}{CG}                                                                       & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
+\multicolumn{2}{|l|}{MG}                                                                       & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
+\multicolumn{2}{|l|}{LU}                                                                       & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
+\multicolumn{2}{|l|}{EP}                                                                       & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
+\multicolumn{2}{|l|}{BT}                                                                       & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
+\multicolumn{2}{|l|}{SP}                                                                       & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
+\multicolumn{2}{|l|}{FT}                                                                       & 27.03           & 31.02            & 2.75           & 2.54              & 24.28        & 28.48           \\ \hline
+
+\end{tabular}
+\label{table:compare_EDP}
+\end{table}
+
+
+
+
+
+\begin{figure}[t]
+  \centering
+   \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
+  \caption{Tradeoff comparison for NAS benchmarks class C}
+  \label{fig:compare_EDP}
+\end{figure}
+
 
 \section{Conclusion}
 \label{sec.concl} 
 In this paper, a new online frequency selecting algorithm has been presented. It selects the best possible vector of frequency scaling factors that gives the maximum distance (optimal tradeoff) between the predicted energy and 
 the predicted performance curves for a heterogeneous platform. This algorithm uses a new energy model for measuring  
 and predicting the energy of distributed iterative applications running over heterogeneous 
-platform. To evaluate the proposed method, it  was  applied on the NAS parallel benchmarks and executed over a heterogeneous platform simulated by  Simgrid. The results of the experiments showed that the algorithm reduces up to 35\% the energy consumption of a message passing iterative method while limiting the degradation of the performance. The algorithm also  selects different scaling factors   according to the percentage of the computing and communication times, and according to the values of  the static and  dynamic powers of the CPUs. 
+platform. To evaluate the proposed method, it  was  applied on the NAS parallel benchmarks and executed over a heterogeneous platform simulated by  Simgrid. The results of the experiments showed that the algorithm reduces up to 35\% the energy consumption of a message passing iterative method while limiting the degradation of the performance. The algorithm also  selects different scaling factors   according to the percentage of the computing and communication times, and according to the values of  the static and  dynamic powers of the CPUs. Finally, the algorithm was compared to Spiliopoulos et al. algorithm and the results showed that it 
+ outperforms their algorithm in term of energy-time tradeoff.
 
 In the near future, this method will be applied to real heterogeneous platforms to evaluate its performance in a real study case. It would also be interesting to evaluate its scalability over large scale heterogeneous platform and measure the energy consumption reduction it can produce. Afterward, we would like  to develop a similar method that is adapted to asynchronous  iterative applications 
-where each task does not wait for others tasks to finish there works. The development of such method might require a new 
+where each task does not wait for others tasks to finish their works. The development of such method might require a new 
 energy model because the number of iterations is not 
 known in advance and depends on the global convergence of the iterative system.
 
 \section*{Acknowledgment}
 
+This work has been partially supported by the Labex
+ACTION project (contract âANR-11-LABX-01-01â). As a PhD student, 
+Mr. Ahmed Fanfakh, would like to thank the University of
+Babylon (Iraq) for supporting his work. 
 
 
 % trigger a \newpage just before the given reference
@@ -1056,7 +1110,7 @@ known in advance and depends on the global convergence of the iterative system.
 \bibliographystyle{IEEEtran}
 \bibliography{IEEEabrv,my_reference}
 \end{document}
-
+   
 %%% Local Variables:
 %%% mode: latex
 %%% TeX-master: t
