end of english correction

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 50fc783f09869fee9ea5478943a807a4f0593bc8..9d743db707e2ffa874517ab225d0d211b722b9f3 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -514,7 +514,7 @@ function has the following form:
 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 
 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 
+(static and dynamic powers). However, the most important 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}.
 
 \section{The scaling factors selection algorithm for heterogeneous platforms }
 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}.
 
 \section{The scaling factors selection algorithm for heterogeneous platforms }


@@ -555,34 +555,41 @@ and the computation scaling factor $Scp_i$ as follows:
   \label{eq:Fint}
  F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
 \end{equation}
   \label{eq:Fint}
  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 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 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 
-function (\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. 
+If the computed  initial frequency for a  node is not available in  the gears of
+that  node,  it  is replaced  by  the  nearest  available frequency.  In  figure
+(\ref{fig:st_freq}), the nodes are sorted by their computing power 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
+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 function (\ref{eq:max}).
+
+Figures~\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 start  from the maximum frequency because  the performance and the
+consumed energy decrease from 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 decrease  until the
+computing power of scaled down  nodes are 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
+ the scaling factors are varying which results in bigger energy savings.

 \begin{figure}[t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}
 \begin{figure}[t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}


@@ -660,18 +667,21 @@ which results in bigger energy savings.
 
 \subsection{The evaluation of the proposed algorithm}
 \label{sec.verif.algo}
 
 \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 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 
-\cite{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 does not test all the possible solutions (vectors of scaling factors) 
+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 to  the
+execution time  and the dynamic energy  is related to the  computation time. So,
+all the works presented  in this paper are 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  \cite{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 it does not test all the possible solutions (vectors of scaling factors) 

 in the search space. 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 
 in the search space. 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 


@@ -684,22 +694,27 @@ vector of frequency scaling factors that gives the results of the next sections.
 
 \section{Experimental results}
 \label{sec.expe}
 
 \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. The experiments were executed 
-on the simulator SimGrid/SMPI 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\% for the static power, the same assumption  was made in \cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. 
-Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
+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. The
+experiments were executed on the  simulator SimGrid/SMPI 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 an amount  of power proportional  to its
+computing  power  (which  corresponds to  80\%  of  its  dynamic power  and  the
+remaining  20\%  to  the  static   power),  the  same  assumption  was  made  in
+\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.    Finally,  These
+nodes were connected via an ethernet network with 1 Gbit/s bandwidth.

 
 
 \begin{table}[htb]
 
 
 \begin{table}[htb]


@@ -736,12 +751,14 @@ Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwi
 \label{sec.res}
 
 
 \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, 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.
+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 had to be executed on $1,
+2, 4, 8, 16, 32, 64, 128$ nodes.   The other benchmarks such as BT and SP had to
+be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.

 
  
  
 
  
  


@@ -907,32 +924,38 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \end{tabular}
   \label{table:res_128n}
 \end{table}
   \end{tabular}
   \label{table:res_128n}
 \end{table}

-The overall energy consumption was computed for each instance according to the energy 
-consumption  model (\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 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.  
+The overall energy  consumption was computed for each  instance according to the
+energy  consumption  model  (\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
+significantly reduces the energy consumption (up to 35\%) and tries to limit the
+performance  degradation.  They  also  show that  the  energy saving  percentage
+decreases when the  number of computing nodes increases.   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  numbers  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  increases  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}) is
+less effective  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 by the high  number of nodes.  No
+experiments were conducted  using bigger classes than 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
+decrease 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.

 
 
  
 
 
  


@@ -944,65 +967,81 @@ compared to the communication times.
   \subfloat[Performance degradation ]{%
     \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}
   \subfloat[Performance degradation ]{%
     \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}

-  \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
+  \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}

 \end{figure}
 
 \end{figure}
 

-Plots (\ref{fig:energy} and \ref{fig:per_deg}) present the energy saving and performance degradation 
-respectively for all the benchmarks according to the number of used nodes. As shown in the first plot, 
-the energy saving percentages of the benchmarks MG, LU, BT and FT are decreased linearly  when the  
-number of nodes is increased. While for the  EP and SP benchmarks, the energy saving percentage is not 
-affected by the increase of the number of computing nodes, because in these benchmarks there are little or 
-no communications. Finally, the energy saving of the GC benchmark  is significantly decreased when the number 
-of nodes is increased because  this benchmark has more communications than the others. The second plot 
-shows that the performance degradation percentages of most of the benchmarks are decreased when they 
-run on a big number of nodes because they spend more time communicating than computing, thus, scaling 
-down the frequencies of some nodes have less effect on the performance. 
+Figures  \ref{fig:energy} and  \ref{fig:per_deg} present  the energy  saving and
+performance  degradation respectively for  all the  benchmarks according  to the
+number of used nodes. As shown  in the first plot, the energy saving percentages
+of the benchmarks MG,  LU, BT and FT decrease linearly when  the number of nodes
+increase. While  for the EP and  SP benchmarks, the energy  saving percentage is
+not affected by the increase of  the number of computing nodes, because in these
+benchmarks there are little or  no communications. Finally, the energy saving of
+the  GC benchmark  significantly  decrease  when the  number  of nodes  increase
+because this benchmark has more  communications than the others. The second plot
+shows that  the performance  degradation percentages of  most of  the benchmarks
+decrease when  they run on a  big number of  nodes because they spend  more time
+communicating than computing,  thus, scaling down the frequencies  of some nodes
+has less effect on the performance.

 
 
 
 
 \subsection{The results for different power consumption scenarios}
 \label{sec.compare}
 
 
 
 
 \subsection{The results for different power consumption scenarios}
 \label{sec.compare}

-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 section, 
-these ratios are changed and two new power scenarios are considered in order to evaluate how the proposed  
-algorithm adapts itself according to the static and dynamic power values.  The two new power scenarios 
-are the following: 
+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 of 20\% of static power. In this section, these ratios are changed and
+two new  power scenarios are  considered in order  to evaluate how  the proposed
+algorithm adapts itself  according to the static and  dynamic power values.  The
+two new power scenarios are the following:

 
 \begin{itemize}
 
 \begin{itemize}

-\item 70\% dynamic power  and 30\% static power
-\item 90\% dynamic power  and 10\% static power
+\item 70\% of dynamic power  and 30\% of static power
+\item 90\% of dynamic power  and 10\% of static power

 \end{itemize}
 
 \end{itemize}
 

-The NAS parallel benchmarks were executed again over processors that follow 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}. 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. Indeed, in the latter 
-more dynamic power is consumed when nodes are running on their maximum frequencies, thus, scaling down the frequency 
-of the nodes results in higher energy savings than in the 70\%-30\% scenario. On the other hand,  the performance 
-degradation percentage is less in the 70\%-30\% scenario  compared to the 90\%-10\%  scenario. This is due to the 
-higher static power percentage in the first scenario which makes it more relevant in the overall consumed energy. 
-Indeed, the static energy is related to the execution time and if the performance is  degraded the total consumed 
-static energy is directly increased. Therefore, the proposed algorithm do not scales down much the frequencies of the 
-nodes  in order to limit the increase of the execution time and thus limiting the effect of the consumed static energy.
-
-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  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. 
+The NAS parallel benchmarks were  executed again over processors that follow 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}. These tables  show that the energy saving  percentage of the
+70\%-30\% scenario is  smaller for all benchmarks compared  to the energy saving
+of the 90\%-10\% scenario. Indeed, in  the latter more dynamic power is consumed
+when  nodes are running  on their  maximum frequencies,  thus, scaling  down the
+frequency of  the nodes results in  higher energy savings than  in the 70\%-30\%
+scenario. On the  other hand, the performance degradation  percentage is smaller
+in the 70\%-30\% scenario compared to the 90\%-10\% scenario. This is due to the
+higher  static  power percentage  in  the first  scenario  which  makes it  more
+relevant in the  overall consumed energy.  Indeed, the  static energy is related
+to the execution time and if  the performance is degraded the amount of consumed
+static  energy directly  increas.  Therefore,  the proposed  algorithm  does not
+really significantly  scale down much the  frequencies of the nodes  in order to
+limit the  increase of the  execution time and  thus limiting the effect  of the
+consumed static energy.
+
+Both   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  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 decreases  when the 70\%-30\%
+scenario is  used because  the dynamic  energy is less  relevant in  the overall
+consumed energy and  lowering the frequency does not  return 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  Figure (\ref{fig:scales_comp}). The  opposite happens when  using a
+higher  ratio for  static power,  the algorithm  proportionally  selects smaller
+scaling  values which result  in less  energy saving  but also  less performance
+degradation.

 
 
  \begin{table}[htb]
 
 
  \begin{table}[htb]

-  \caption{The results of 70\%-30\% powers scenario}
+  \caption{The results of the 70\%-30\% power scenario}

   % title of Table
   \centering
   \begin{tabular}{|*{6}{l|}}
   % title of Table
   \centering
   \begin{tabular}{|*{6}{l|}}


@@ -1031,7 +1070,7 @@ results in less energy saving but less performance degradation.
 
 
 \begin{table}[htb]
 
 
 \begin{table}[htb]

-  \caption{The results of 90\%-10\% powers scenario}
+  \caption{The results of the 90\%-10\% power scenario}

   % title of Table
   \centering
   \begin{tabular}{|*{6}{l|}}
   % title of Table
   \centering
   \begin{tabular}{|*{6}{l|}}


@@ -1060,7 +1099,7 @@ results in less energy saving but less performance degradation.
 
 \begin{figure}
   \centering
 
 \begin{figure}
   \centering

-  \subfloat[Comparison  of the results on 8 nodes]{%
+  \subfloat[Comparison  between 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]{%
     \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]{%


@@ -1080,12 +1119,14 @@ They developed a green governor that regularly applies an online frequency selec
 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.
 
 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, 
+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 provides 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\%. 
 
 
 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. 
+For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
+terms 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.

 
 
 
 
 
 


@@ -1123,16 +1164,31 @@ Spiliopoulos et al. algorithm in term of energy and performance tradeoff, see fi
 
 \section{Conclusion}
 \label{sec.concl} 
 
 \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. 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 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.
+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  platforms. 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 terms 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 platforms 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  other tasks  to finish  their  works.  The
+development of such a 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}
 
 
 \section*{Acknowledgment}