X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/12334c0cc5e73de7a7a3c186b7bbbcd814f922af..64ae1ea15de81e7d26e31750c558df497b58ab44:/Heter_paper.tex?ds=inline

diff --git a/Heter_paper.tex b/Heter_paper.tex
index c77212a..690c6a1 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -53,7 +53,7 @@
 \newcommand{\Told}{\Xsub{T}{Old}} 
 \begin{document} 
 
-\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
+\title{Energy Consumption Reduction In a Heterogeneous Architecture Using DVFS}
 
 \author{%
   \IEEEauthorblockN{%
@@ -61,7 +61,7 @@
     RaphaÃ«l Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch
-  }
+  } 
   \IEEEauthorblockA{%
     FEMTO-ST Institute\\
     University of Franche-ComtÃ©\\
@@ -81,13 +81,30 @@
 
 \section{Introduction}
 \label{sec.intro}
-
+Modern processors continue to increased in a performance, achieved  maximum number of floating point operations per second (FLOPS), thus the energy consumption and the heat dissipation are increased drastically  according to this increase. The number of FLOPS is linearly related to power consumption of a CPU~\cite{51}.  
+As an example of more power hungry cluster, according to the Top500 list in June 2014 \cite{43}, Tianhe-2 has more than 3 millions of cores and consumed more  than 17.8 megawatt per second. Moreover, according to the U.S. annual energy outlook 2014 \cite{60}, the price of energy for 1 megawatt per hour is approximately equal to 70\$ (1.16\$ for megawatt per second). Therefore, we can consider the price of the energy consumption for the Tianhe-2 platform is approximately more than 390 millions dollars of megawatt per year. For this reason, the heterogeneous clusters must be offer more energy efficiency due to the increase in the energy cost and the environment influences. Therefore, a green computing clusters are require nowadays.  For example, the GSIC center of Tokyo  heterogeneous cluster became the top of the Green500 list in June 2014 \cite{59}. This platform  has more than four thousand of  MFLOPS per watt. Dynamic voltage and frequency scaling (DVFS) is a process used widely to reduce the energy consumption of the processor. In a heterogeneous clusters enabled DVFS, many researchers used DVFS  in a different ways. DVFS can be minimized the energy consumption but it lead to a disadvantage due to the performance degradation increase. Therefore,  researchers used different optimization strategies to overcame this problem. The best tradeoff relation between the energy reduction and performance degradation ratio is become a key challenges in a heterogeneous platforms. In this paper we are propose a heterogeneous scaling algorithm  that selects the optimal vector of the frequency scaling factors for distributed iterative application, producing minimum energy saving against minimum performance degradation ratio simultaneously. The algorithm has very small overhead, works online and not needs for any training or profiling.  
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents some
+related works from other authors.  Section~\ref{sec.exe} describes how the
+execution time of MPI programs can be predicted.  It also presents an energy
+model for heterogeneous platforms. Section~\ref{sec.compet} presents
+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 heterogeneous scaling algorithm.
+Section~\ref{sec.expe} presents the results of running  the NAS benchmarks on 
+the proposed heterogeneous platform. It also shows the comparison of three different power
+scenarios and it verifies the precision of the proposed algorithm.  Finally, we conclude 
+in Section~\ref{sec.concl} with a summary and some future works.
 
 \section{Related works}
 \label{sec.relwork}
-
-
-
+Energy reduction process for a high performance clusters recently performed using dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique  enabled in a modern processors to scaled down both of the  voltage and the frequency of the CPU while it is in the computing mode to reduce the energy consumption. DVFS is also  allowed in the graphical processors GPUs, to achieved the same goal. Applying DVFS has a dramatical side effect if it is applied to minimum levels to gain more energy reduction, producing a high percentage of performance degradations for the parallel applications.  Many researchers used different strategies to solve this nonlinear problem for example in~\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
+set of frequency scaling factors for a heterogeneous cluster to simultaneously optimize both the energy and the execution time  without adding a big overhead.
+This work is developed from our previous work of a homogeneous cluster~\cite{45}. Therefore we are interested to present some works that concerned the heterogeneous clusters enabled DVFS. In general, the heterogeneous cluster works fall into two categorizes: GPUs-CPUs heterogeneous clusters and CPUs-CPUs heterogeneous clusters. In GPUs-CPUs heterogeneous clusters some parallel tasks executed on a GPUs and the others executed on a CPUs. As an example of this works, Luley et al.~\cite{51}, proposed  a heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal is to determined the energy efficiency as a function of performance per watt, the best tradeoff is done when the performance per watt function is maximized. In the work of Kia Ma et al.~\cite{49}, They developed a scheduling algorithm to distributed different workloads proportional to the computing power of the node to be executed on a CPU or a GPU, emphasize all tasks must be finished in the same time. 
+Recently, Rong et al.~\cite{50}, Their study explain that a heterogeneous clusters enabled DVFS using GPUs and CPUs gave better energy and performance efficiency 
+than other clusters composed of only CPUs. The CPUs-CPUs heterogeneous clusters consist of number of computing nodes  all of the type CPU. Our work in this paper can be classified to this type of the clusters. As an example of this works see  Naveen et al.~\cite{52} work, They developed a policy to dynamically assigned the frequency to a heterogeneous cluster. The goal is to minimizing a fixed metric of $energy*delay^2$. Where our proposed method is automatically optimized  the relation between the energy and the delay of the iterative applications. Other works such as Lizhe et al.~\cite{53}, their algorithm divided the executed tasks into two types: the critical and non critical tasks. The algorithm scaled down the frequency of the non critical tasks as function to the  amount of the slack and communication times that have with maximum of performance degradation percentage of 10\%. In our method there is no fixed bounds for performance degradation percentage and the bound is dynamically computed according to the energy and the performance tradeoff relation of the executed application. 
+There are some approaches used a heterogeneous cluster composed from two different types of Intel and AMD processors such as~\cite{54} and \cite{55}, they predicated  both the energy and the performance for each frequency gear, then the algorithm selected the best gear that gave the best tradeoff. In contrast our algorithm works over a heterogeneous  platform composed of four different types of processors. Others approaches such as \cite{56} and \cite{57}, they are selected the best frequencies for a specified heterogeneous clusters offline using some heuristic methods. While our proposed algorithm works online during the execution time of iterative application. Greedy dynamic approach used by Chen et al.~\cite{58},  minimized the power consumption of a heterogeneous severs  with time/space complexity, this approach had considerable overhead. In our proposed scaling algorithm has very small overhead and it is works without any previous analysis for the application time complexity. 
 
 \section{The performance and energy consumption measurements on heterogeneous architecture}
 \label{sec.exe}
@@ -107,18 +124,19 @@ power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
 have the same network bandwidth and latency.
 
 
-\begin{figure}[t]
+
+The  overall execution time  of a distributed iterative synchronous application over a heterogeneous platform  consists of the sum of the computation time and the communication time for every iteration on a node. However, due to the heterogeneous computation power of the computing nodes, slack times might occur when fast nodes have to
+ wait, during synchronous communications, for  the slower nodes to finish  their computations (see Figure~(\ref{fig:heter}). 
+ Therefore,  the overall execution time  of the program is the execution time of the slowest
+ task which have the highest computation time and no slack time.
+  
+ \begin{figure}[t]
   \centering
     \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 
- The  overall execution time  of a distributed iterative synchronous application over a heterogeneous platform  consists of the sum of the computation time and the communication time for every iteration on a node. However, due to the heterogeneous computation power of the computing nodes, slack times might occur when fast nodes have to
- wait, during synchronous communications, for  the slower nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
- Therefore,  the overall execution time  of the program is the execution time of the slowest
- task which have the highest computation time and no slack time.
- 
 Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
 of a CPU by scaling down its voltage and frequency.  Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound.  The frequency reduction process can be  expressed by the scaling factor S which is the ratio between  the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
 \begin{equation}
@@ -151,7 +169,6 @@ This prediction model is based on our model for predicting the execution time of
 
 
 \subsection{Energy model for heterogeneous platform}
-
 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
 consumed as long as the computing unit is turned on, the latter is only consumed during
@@ -211,8 +228,7 @@ reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is
 The static power is related to the power leakage of the CPU and is consumed during computation and even when idle. As in~\cite{3,46}, we assume that the static power of a processor is constant during idle and computation periods, and for all its available frequencies. 
 The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}), 
 the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related  
-to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy 
-of a processor after scaling its frequency is computed as follows: 
+to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy of a processor after scaling its frequency is computed as follows: 
 \begin{equation}
   \label{eq:Estatic}
  E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)
@@ -262,7 +278,7 @@ In the same way, we normalize the energy by computing the ratio between the cons
   E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
   {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
  \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i@+eYd162 \cdot T_{Old})}}
+ \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
 \end{multline} 
 Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
 
@@ -273,11 +289,6 @@ 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.  
 
-Many researchers used different strategies to solve this nonlinear problem for example
-in~\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
-set of frequency scaling factors to simultaneously optimize both energy and execution time
- without adding a big overhead. \textbf{put the last two phrases in the related work section}
  
   
 Our solution for this problem is to make the optimization process for energy and execution time follow the same
@@ -340,7 +351,10 @@ maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as fol
  F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
 \end{equation}
 If the computed initial frequency for a node is not available in the gears of that node, the computed initial frequency is replaced by the nearest available frequency.
-In  figure (\ref{fig:st_freq}), the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are coloured in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal vector of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the frequency selecting factors algorithm starts its search method from these initial frequencies and takes a downward search direction. If the algorithm starts to search from the first frequencies of all nodes, regardless the higher bound frequencies, at each step the predicted performance and energy  are degreased  together, then the best distance be unreachable. This case is similar to homogeneous scaling algorithm when all nodes in the cluster has the same computing power, therefore there is a smaller distance between the performance and the energy curves, while in a heterogeneous cluster the distance is bigger and the energy saving against smaller execution time is higher, as an example see figure~(\ref{fig:r1} and \ref{fig:r2}). 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 EQ(\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 function EQ(\ref{eq:max}).
+In  figure (\ref{fig:st_freq}), the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are colored in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal vector of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the algorithm that selects the frequency scaling factors starts the search method from these initial frequencies and takes a downward search direction 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 EQ(\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 function EQ(\ref{eq:max}).
+
+The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an application running on a homogeneous platform and a heterogeneous platform respectively while increasing the scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor should be started from the maximum frequency because the performance and the consumed energy is decreased since  the beginning of the plot. On the other hand, in  the heterogeneous platform the performance is  maintained at the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes have computing powers lower than the slowest node. In other words, until they reach the higher bound. It can also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger the maximum distance between the energy curve and the performance curve is while varying the scaling factors which results in bigger energy savings. 
 \begin{figure}[t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}
@@ -419,23 +433,12 @@ In  figure (\ref{fig:st_freq}), the nodes are  sorted by their computing powers
 
 \section{Experimental results}
 \label{sec.expe}
-To evaluate the efficiency and the overall energy consumption reduction of algorithm~\ref{HSA}), it was applied to the NAS parallel benchmarks NPB v3.3 
+To evaluate the efficiency and the overall energy consumption reduction of algorithm~(\ref{HSA}), it was applied to the NAS parallel benchmarks NPB v3.3 
 \cite{44}. The experiments were executed on the simulator SimGrid/SMPI
 v3.10~\cite{casanova+giersch+legrand+al.2014.versatile} which offers easy tools to create a heterogeneous platform and run message passing applications over it. The heterogeneous platform that was used in the experiments, had one core per node because just one process was executed per node. The heterogeneous platform  was composed of four types of nodes. Each type of nodes had different characteristics such as the maximum CPU frequency, the number of
 available frequencies and the computational power, see table
-(\ref{table:platform}). The characteristics of these different types of  nodes are inspired   from the specifications of real Intel processors. The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, for example if  a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were chosen proportionally to  its computing power (FLOPS).  In the initial heterogeneous platform,  while computing with highest frequency, each node  consumed power proportional to its computing power which 80\% of it was dynamic power and the rest was 20\% was static power, the same assumption  was made in \cite{45,3}. Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
-
+(\ref{table:platform}). The characteristics of these different types of  nodes are inspired   from the specifications of real Intel processors. The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, for example if  a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were chosen proportionally to  its computing power (FLOPS).  In the initial heterogeneous platform,  while computing with highest frequency, each node  consumed power proportional to its computing power which 80\% of it was dynamic power and the rest was 20\% for the static power, the same assumption  was made in \cite{45,3}. Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
 
-\textbf{modify the characteristics table by replacing the similar column with the computing power of the different types of nodes in flops}
-
-
- The proposed scaling algorithm 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 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 section.
 
 \begin{table}[htb]
   \caption{Heterogeneous nodes characteristics}
@@ -443,20 +446,21 @@ needs  from 12 to 20 iterations to select the best vector of frequency scaling f
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\
-    type     & to          & Freq. GHz  & Freq. GHz    & Freq. GHz       & power        & power \\
+    Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
+    type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
+                  &           & GHz      & GHz          &GHz             &              &       \\
     \hline
-    1       & core-i3       & 2.5         & 1.2          & 0.1           & 20~w         &4~w    \\
-            &  2100T        &             &              &               &              &  \\
+    1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
+                  &           &          &              &                &              &  \\
     \hline
-    2       & Xeon          & 2.66        & 1.6          & 0.133         & 25~w         &5~w    \\
-            & 7542          &             &              &               &              &  \\
+    2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
+                  &           &          &              &                &              &  \\
     \hline
-    3       & core-i5       & 2.9         & 1.2          & 0.1           & 30~w         &6~w    \\
-            & 3470s         &             &              &               &              &  \\
+    3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
+                  &           &          &              &                &              &  \\
     \hline
-    4       & core-i7       & 3.4         & 1.6          & 0.133         & 35~w         &7~w    \\
-            & 2600s         &             &              &               &              &  \\
+    4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
+                  &           &          &              &                &              &  \\
     \hline
   \end{tabular}
   \label{table:platform}
@@ -470,8 +474,8 @@ needs  from 12 to 20 iterations to select the best vector of frequency scaling f
 \label{sec.res}
 
 
-The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in this paper, only the results of the biggest class, C, are presented while being run on different number of nodes, ranging  from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the benchmarks CG, MG, LU and FT should be executed on $2^1, 2^2, 2^4 or 2^8$ nodes. The other benchmarks such as BT and SP should be executed on $2^1, 2^2, 2^4 or 2^9$ nodes.
-\textbf{there must be an error in the number of nodes }
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in this paper, only the results of the biggest class, C, are presented while being run on different number of nodes, ranging  from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes. The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
+
  
  
 \begin{table}[htb]
@@ -639,36 +643,27 @@ The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
 The overall energy consumption was computed for each instance according to the energy consumption  model EQ(\ref{eq:energy}), with and without applying the algorithm. The execution time was also measured for all these experiments. Then, the energy saving and performance degradation percentages were computed for each instance.  
 The results are presented in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}). All these results are the average values from many experiments for  energy savings and performance degradation.
 
-The tables  show the experimental results for running the NAS parallel benchmarks on different number of nodes. The experiments show that the algorithm reduce significantly the energy consumption (up to 35\%) and tries to limit the performance degradation. They also show that the  energy saving percentage is decreased  when the number of the computing nodes is increased. This reduction is due to the increase of the communication times compared to the execution times when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C, are executed on different number of nodes, so the computation required for each iteration is divided by the number of computing nodes.   On the other hand, more communications are required when increasing the number of nodes so the static energy is increased linearly according to the communication time and the dynamic power is less relevant in the overall energy consumption. Therefore, reducing the frequency with algorithm~\ref{HSA}) have less effect in reducing the overall energy savings. It can also be noticed that for the benchmarks EP and SP that contain little or no communications,  the energy savings are not significantly affected with the high number of nodes. No experiments were conducted using bigger classes such as D, because they require a lot of memory(more than 64GB) when being executed by the simulator on one machine.
+The tables  show the experimental results for running the NAS parallel benchmarks on different number of nodes. The experiments show that the algorithm reduce significantly the energy consumption (up to 35\%) and tries to limit the performance degradation. They also show that the  energy saving percentage is decreased  when the number of the computing nodes is increased. This reduction is due to the increase of the communication times compared to the execution times when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C, are executed on different number of nodes, so the computation required for each iteration is divided by the number of computing nodes.   On the other hand, more communications are required when increasing the number of nodes so the static energy is increased linearly according to the communication time and the dynamic power is less relevant in the overall energy consumption. Therefore, reducing the frequency with algorithm~(\ref{HSA}) have less effect in reducing the overall energy savings. It can also be noticed that for the benchmarks EP and SP that contain little or no communications,  the energy savings are not significantly affected with the high number of nodes. No experiments were conducted using bigger classes such as D, because they require a lot of memory(more than 64GB) when being executed by the simulator on one machine.
 The maximum distance between the normalized energy curve and the normalized performance for each instance is also shown in the result tables. It is decreased in the same way as the energy saving percentage. The tables also show that the performance degradation percentage is not significantly increased when the number of computing nodes is increased because the computation times are small when compared to the communication times.  
 
 
  
 \begin{figure}
   \centering
-  \subfloat[CG, MG, LU and FT benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
+  \subfloat[Energy saving]{%
+    \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%
   \quad%
-  \subfloat[BT and SP benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
+  \subfloat[Performance degradation ]{%
+    \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}
-  \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
+  \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
 \end{figure}
 
  
-   The average of values of these three objectives are plotted to the number of
-nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}).  In CG, MG, LU, and
-FT benchmarks the average of energy saving is decreased when the number of nodes
-is increased because the communication times is increased as mentioned
-before. Thus, the average of distances (our objective function) is decreased
-linearly with energy saving while keeping the average of performance degradation approximately is 
-the same. In BT and SP benchmarks, the average of the  energy saving is not decreased
-significantly compare to other benchmarks when the number of nodes is
-increased. Nevertheless, the average of performance degradation approximately
-still the same ratio. This difference is depends on the characteristics of the
-benchmark such as the computations to communications ratio that has.
-
-\textbf{All the previous paragraph should be deleted, we need to talk about it}
+  \textbf{ The energy saving and performance degradation of all benchmarks are plotted to the number of
+nodes as in plots (\ref{fig:energy} and \ref{fig:per_deg}). As shown in the plots, the energy saving percentage of the benchmarks MG, LU, BT and FT is decreased linearly  when the the number of nodes increased. While in EP benchmark the energy saving percentage is approximately the same percentage when the number of computing nodes is increased, because in this benchmark there is no communications. In the SP benchmark the energy saving percentage is decreased when it runs on a small number of nodes, while this percentage is increased when it runs on a big number of nodes. The energy saving of the GC benchmarks  is significantly decreased when the number of nodes is increased, because  this benchmark has more communications compared to other benchmarks. The performance degradation percentage of the benchmarks CG, EP, LU and BT is decreased when they run on a big number of nodes. While in MG benchmark has a higher percentage of performance degradation when it runs on a big number of nodes. The inverse happen in SP benchmark has smaller performance degradation percentage when it runs on a big number of nodes.} 
+
+
 \subsection{The results for different power consumption scenarios}
 
 The results of the previous section were obtained while using processors that consume during computation an overall power which is 80\% composed of  dynamic power and 20\% of static power. In this
@@ -677,9 +672,9 @@ section, these ratios are changed and two new power scenarios are considered in
 \item 70\% dynamic power  and 30\% static power
 \item 90\% dynamic power  and 10\% static power
 \end{itemize}
-The NAS parallel benchmarks were executed again over processors that follow the the new power scenarios. The class C of each benchmark was run over 8 or 9 nodes and the results are presented in  tables (\ref{table:res_s1} and \ref{table:res_s2}).\textbf{should explain the tables more}
+The NAS parallel benchmarks were executed again over processors that follow the the new power scenarios. The class C of each benchmark was run over 8 or 9 nodes and the results are presented in  tables (\ref{table:res_s1} and \ref{table:res_s2}). \textbf{These tables show that the energy saving percentage of the 70\%-30\% scenario is less for all benchmarks compared to the energy saving of the 90\%-10\% scenario, because this scenario uses higher percentage of dynamic dynamic power that is quadratically related to scaling factors. While the performance degradation percentage is less in 70\%-30\% scenario  compared to 90\%-10\%  scenario, because the first scenario used higher percentage for static power consumption that is linearly related to scaling factors and thus the execution time. }
 
-The two new power scenarios are compared to the old one  in figure (\ref{fig:sen_comp}). It shows the average of the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. The comparison shows that  the energy saving ratio is proportional to the dynamic power ratio: it is increased when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the the most relevant in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in the overall consumed energy and lowering the frequency do not returns big energy savings.
+The two new power scenarios are compared to the old one in figure (\ref{fig:sen_comp}). It shows the average of the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. The comparison shows that  the energy saving ratio is proportional to the dynamic power ratio: it is increased when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the the most relevant in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in the overall consumed energy and lowering the frequency do not returns big energy savings.
 Moreover, the average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed algorithm optimizes the energy consumption when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens when using a higher ratio for  static  power, the algorithm proportionally  selects  smaller scaling values which results in less energy saving but less performance degradation. 
 
 
@@ -745,7 +740,7 @@ Moreover, the average of the performance degradation is decreased when using a h
   \subfloat[Comparison the average of the results on 8 nodes]{%
     \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
   \quad%
-  \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
+  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
     \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
@@ -758,8 +753,11 @@ Moreover, the average of the performance degradation is decreased when using a h
 The precision of the proposed algorithm mainly depends on the execution time prediction model defined in EQ(\ref{eq:perf}) and the energy model computed by EQ(\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 the work 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 for all  the NAS parallel benchmarks running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise, the maximum normalized difference between  the predicted execution time  and the real execution time is equal to 0.03 for all the NAS benchmarks.
 
-Since  the proposed algorithm is not an exact method and do not test all the possible solutions (vectors of scaling factors) in the search space and to prove its efficiency, it was compared on small instances to a brute force search algorithm that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with 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. 
-\textbf{should put the paragraph about the overhead here}
+Since  the proposed algorithm is not an exact method and do not test all the possible solutions (vectors of scaling factors) in the search space and to prove its efficiency, it was compared on small instances to a brute force search algorithm that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with 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 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 section (\ref{sec.res}).
+
 \section{Conclusion}
 \label{sec.concl}
 
