figures

[mpi-energy2.git] / mpi-energy2-extension / Heter_paper.tex

diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index 928c20b28f86376b3f85fed78d9521a7d3d0750b..3e1bb461175032175abd0d22b3fd90e0406061d8 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -148,9 +148,9 @@ scaling (DVFS) is one of them. It can be used to reduce the power consumption of
   overhead and works without training or profiling. It uses a new energy model
   for message passing iterative applications running on a  grid. 
   The proposed algorithm is evaluated on a real grid, the grid'5000 platform, while
   overhead and works without training or profiling. It uses a new energy model
   for message passing iterative applications running on a  grid. 
   The proposed algorithm is evaluated on a real grid, the grid'5000 platform, while

-  running the NAS parallel benchmarks.  The experiments show that it reduces the
-  energy consumption on average by \np[\%]{30} while  the performance  is only degraded
-  on average by \np[\%]{3.2}. Finally, the algorithm is 
+  running the NAS parallel benchmarks.  The experiments on 16 nodes, distributed on three clusters, show that it reduces  on average the
+  energy consumption  by \np[\%]{30} while  the performance  is on average only degraded
+   by \np[\%]{3.2}. Finally, the algorithm is 

   compared to an existing method. The comparison results show that it outperforms the
   latter in terms of energy consumption reduction and performance.
 \end{abstract}
   compared to an existing method. The comparison results show that it outperforms the
   latter in terms of energy consumption reduction and performance.
 \end{abstract}


@@ -200,12 +200,12 @@ the number of FLOPS executed by the processor which may increase the execution
 time of the application running over that processor.  Therefore, researchers use
 different optimization strategies to select the frequency that gives the best
 trade-off between the energy reduction and performance degradation ratio. In
 time of the application running over that processor.  Therefore, researchers use
 different optimization strategies to select the frequency that gives the best
 trade-off between the energy reduction and performance degradation ratio. In

-\cite{Our_first_paper} and \cite{pdsec2015} , a frequencies selecting algorithm was proposed to reduce
+\cite{Our_first_paper} and \cite{pdsec2015} , a frequency selecting algorithm was proposed to reduce

 the energy consumption of message passing iterative applications running over
 homogeneous  and heterogeneous clusters respectively.  
 The results of the experiments showed significant energy
 the energy consumption of message passing iterative applications running over
 homogeneous  and heterogeneous clusters respectively.  
 The results of the experiments showed significant energy

-consumption reductions. All the experimental results were conducted over 
-Simgrid simulator \cite{SimGrid}, which offers easy tools to create a homogeneous and heterogeneous platforms and run message passing parallel applications over them. In this paper, a new frequencies selecting algorithm,
+consumption reductions. All the experimental results were conducted over the
+Simgrid simulator \cite{SimGrid}, which offers easy tools to create homogeneous and heterogeneous platforms and runs message passing parallel applications over them. In this paper, a new frequency selecting algorithm,

 adapted to  grid platforms composed of heterogeneous clusters, is presented. It is applied to the NAS parallel benchmarks and evaluated over a real testbed, 
 the grid'5000 platform \cite{grid5000}. It selects  for a grid platform running a message passing iterative
 application the vector of
 adapted to  grid platforms composed of heterogeneous clusters, is presented. It is applied to the NAS parallel benchmarks and evaluated over a real testbed, 
 the grid'5000 platform \cite{grid5000}. It selects  for a grid platform running a message passing iterative
 application the vector of


@@ -224,11 +224,9 @@ consumption while minimizing the degradation of the program's performance.
 Section~\ref{sec.optim} details the proposed frequencies selecting algorithm.
 Section~\ref{sec.expe} presents the results of applying the algorithm on the 
 NAS parallel benchmarks and executing them on the grid'5000 testbed. 
 Section~\ref{sec.optim} details the proposed frequencies selecting algorithm.
 Section~\ref{sec.expe} presents the results of applying the algorithm on the 
 NAS parallel benchmarks and executing them on the grid'5000 testbed. 

-%It shows the results of running different scenarios using multi-cores and one core per node and comparing them. 
-It also evaluates the algorithm over three different power scenarios. Moreover, it shows the
+It also evaluates the algorithm over multi-cores per node architectures and over three different power scenarios. Moreover, it shows the

 comparison results between the proposed method and an existing method.  Finally,
 in Section~\ref{sec.concl} the paper ends with a summary and some future works.
 comparison results between the proposed method and an existing method.  Finally,
 in Section~\ref{sec.concl} the paper ends with a summary and some future works.

-

 \section{Related works}
 \label{sec.relwork}
 
 \section{Related works}
 \label{sec.relwork}
 


@@ -385,8 +383,9 @@ vector of scaling factors can be predicted using (\ref{eq:perf}).
 where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
 each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
 where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
 each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 

-first  iteration. The model computes the maximum computation time with scaling factor 
-from each node added to the communication time of the slowest node in the slowest cluster $h$.
+first  iteration. the execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors 
+ and the slowest communication time without slack time during one iteration. 
+The latter is equal to the  communication time of the slowest node in the slowest cluster $h$.

 It means only the communication time without any slack time is taken into account.  
 Therefore, the execution time of the iterative application is equal to
 the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
 It means only the communication time without any slack time is taken into account.  
 Therefore, the execution time of the iterative application is equal to
 the execution time of one iteration as in (\ref{eq:perf}) multiplied by the


@@ -599,9 +598,9 @@ equation, as follows:
 \begin{figure}
   \centering
   \subfloat[Homogeneous cluster]{%
 \begin{figure}
   \centering
   \subfloat[Homogeneous cluster]{%

-    \includegraphics[width=.4\textwidth]{fig/homo}\label{fig:r1}} \hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/homo}\label{fig:r1}} \hspace{0.4cm}%

   \subfloat[Heterogeneous grid]{%
   \subfloat[Heterogeneous grid]{%

-    \includegraphics[width=.4\textwidth]{fig/heter}\label{fig:r2}}
+    \includegraphics[width=.48\textwidth]{fig/heter}\label{fig:r2}}

   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}
   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}


@@ -684,8 +683,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}
       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}

-        \State Gather all times of computation and\newline\hspace*{3em}%
-               communication from each node.
+        \State Gather all times of computation and communication from each node.

         \State Call Algorithm \ref{HSA}.
         \State Compute the new frequencies from the\newline\hspace*{3em}%
                returned optimal scaling factors.
         \State Call Algorithm \ref{HSA}.
         \State Compute the new frequencies from the\newline\hspace*{3em}%
                returned optimal scaling factors.


@@ -790,20 +788,20 @@ maximum distance between the energy curve and the performance curve is, which re
 While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid}, 
 in this paper real experiments were conducted over the grid'5000 platform. 
 
 While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid}, 
 in this paper real experiments were conducted over the grid'5000 platform. 
 

-\subsection{Grid'5000 architature and power consumption}
+\subsection{Grid'5000 architecture and power consumption}

 \label{sec.grid5000}
 \label{sec.grid5000}

-Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed over all metropolitan France and Luxembourg. All the sites are connected together via        a special long distance network called RENATER,
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed all over  metropolitan France and Luxembourg. All the sites are connected together via       a special long distance network called RENATER,

 which is the French National Telecommunication Network for Technology.
 which is the French National Telecommunication Network for Technology.

-Each site of the grid is composed of few heterogeneous 
+Each site of the grid is composed of a few heterogeneous 

 computing clusters and each cluster contains many homogeneous nodes. In total,
 grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
 the clusters and their nodes are connected via  high speed local area networks. 
 Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
 
 computing clusters and each cluster contains many homogeneous nodes. In total,
 grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
 the clusters and their nodes are connected via  high speed local area networks. 
 Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
 

-Since grid'5000 is dedicated for testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture 
-the power consumption  for each node in those sites. The measured power is the overall consumed power by by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ...  For more details refer to
-\cite{Energy_measurement}. To just measure the CPU power of one core in a node $j$, 
- firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumption represents the 
+Since grid'5000 is dedicated to  testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture 
+the power consumption  for each node in those sites. The measured power is the overall consumed power  by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ...  For more details refer to
+\cite{Energy_measurement}. In order to correctly measure the CPU power of one core in a node $j$, 
+ firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumptions represents the 

 dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 
 
 
 dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 
 
 


@@ -819,15 +817,15 @@ $\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle po
 Therefore, the dynamic power of one core is computed as the difference between the maximum 
 measured value in maximum powers vector and the minimum measured value in the idle powers vector.
 
 Therefore, the dynamic power of one core is computed as the difference between the maximum 
 measured value in maximum powers vector and the minimum measured value in the idle powers vector.
 

-On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in Grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.

 
 In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
 
 Four clusters from the two sites were selected in the experiments: one cluster from 
 
 In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
 
 Four clusters from the two sites were selected in the experiments: one cluster from 

-Lyon's site, Taurus cluster, and three clusters from Nancy's site, Graphene, 
+Lyon's site, Taurus, and three clusters from Nancy's site, Graphene, 

 Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
 frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 
 Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
 frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 

-the details characteristics of these four clusters. Moreover, the dynamic powers were computed  using the equation (\ref{eq:pdyn}) for all the nodes in the 
+the detailed characteristics of these four clusters. Moreover, the dynamic powers were computed  using equation (\ref{eq:pdyn}) for all the nodes in the 

 selected clusters and are presented in table  \ref{table:grid5000}.
 
 
 selected clusters and are presented in table  \ref{table:grid5000}.
 
 


@@ -845,9 +843,9 @@ selected clusters and are presented in table  \ref{table:grid5000}.
 \end{figure}
 
 
 \end{figure}
 
 

-The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
+The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over Grid'5000.

 The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
 The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.

-The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work,  class D was used for all benchmarks in all the experiments presented in the next sections. 

 
 
   
 
 
   


@@ -861,20 +859,20 @@ The benchmarks have seven different classes, S, W, A, B, C, D and E, that repres
     Name        & model       & Freq. & Freq. & Freq. & per CPU         & of one core     \\
                 &             & GHz   & GHz   & GHz   &                 &           \\
     \hline
     Name        & model       & Freq. & Freq. & Freq. & per CPU         & of one core     \\
                 &             & GHz   & GHz   & GHz   &                 &           \\
     \hline

-    Taurus      & Intel       & 2.3  & 1.2  & 0.1     & 6               & \np[W]{35} \\
-                & Xeon        &       &       &       &                 &            \\
+                & Intel       & 2.3  & 1.2  & 0.1     & 6               & \np[W]{35} \\
+    Taurus      & Xeon        &       &       &       &                 &            \\

                 & E5-2630     &       &       &       &                 &            \\         
     \hline
                 & E5-2630     &       &       &       &                 &            \\         
     \hline

-    Graphene    & Intel       & 2.53  & 1.2   & 0.133 & 4               & \np[W]{23} \\
-                & Xeon        &       &       &       &                 &            \\
+                & Intel       & 2.53  & 1.2   & 0.133 & 4               & \np[W]{23} \\
+    Graphene    & Xeon        &       &       &       &                 &            \\

                 & X3440       &       &       &       &                 &            \\    
     \hline
                 & X3440       &       &       &       &                 &            \\    
     \hline

-    Griffon     & Intel       & 2.5   & 2     & 0.5   & 4               & \np[W]{46} \\
-                & Xeon        &       &       &       &                 &            \\
+                & Intel       & 2.5   & 2     & 0.5   & 4               & \np[W]{46} \\
+    Griffon     & Xeon        &       &       &       &                 &            \\

                 & L5420       &       &       &       &                 &            \\  
     \hline
                 & L5420       &       &       &       &                 &            \\  
     \hline

-    Graphite    & Intel       & 2     & 1.2   & 0.1   & 8               & \np[W]{35} \\
-                & Xeon        &       &       &       &                 &            \\
+                & Intel       & 2     & 1.2   & 0.1   & 8               & \np[W]{35} \\
+     Graphite   & Xeon        &       &       &       &                 &            \\

                 & E5-2650     &       &       &       &                 &            \\  
     \hline
   \end{tabular}
                 & E5-2650     &       &       &       &                 &            \\  
     \hline
   \end{tabular}


@@ -889,22 +887,22 @@ In this section, the results of the application of the scaling factors selection
 to the NAS parallel benchmarks are presented. 
 
 As mentioned previously, the experiments 
 to the NAS parallel benchmarks are presented. 
 
 As mentioned previously, the experiments 

-were conducted over two sites of grid'5000,  Lyon and Nancy sites. 
+were conducted over two sites of Grid'5000,  Lyon and Nancy sites. 

 Two scenarios were considered while selecting the clusters from these two sites :
 \begin{itemize}
 \item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected 
  via a long distance network.
 Two scenarios were considered while selecting the clusters from these two sites :
 \begin{itemize}
 \item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected 
  via a long distance network.

-\item In the second scenario nodes from three clusters that are located in one site, Nancy site.  
+\item In the second scenario nodes from three clusters located in one site, Nancy site, were selected.  

 \end{itemize}
 
 The main reason 
 \end{itemize}
 
 The main reason 

-behind using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
+for using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 

 scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 
 scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 

-is very low due to the higher communication times which reduces the effect of DVFS operations.
+is very low due to the higher communication times which reduce the effect of DVFS operations.

 
 The NAS parallel benchmarks are executed over 
 
 The NAS parallel benchmarks are executed over 

-16 and 32 nodes for each scenario. The number of participating computing nodes form each cluster 
-are different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
+16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
+is different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.

 Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
 
 \begin{table}[h]
 Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
 
 \begin{table}[h]


@@ -944,12 +942,14 @@ The NAS parallel benchmarks are executed over these two platforms
 The overall energy consumption of all the benchmarks solving the class D instance and
 using the proposed frequency selection algorithm is measured 
 using the equation of the reduced energy consumption, equation 
 The overall energy consumption of all the benchmarks solving the class D instance and
 using the proposed frequency selection algorithm is measured 
 using the equation of the reduced energy consumption, equation 

-(\ref{eq:energy}). This model uses the measured dynamic and static 
-power values  showed in Table \ref{table:grid5000}. The execution
+(\ref{eq:energy}). This model uses the measured dynamic power showed in Table \ref{table:grid5000}
+
+and the static 
+power is assumed to be equal to 20\% of the dynamic power. The execution

 time is measured for all the benchmarks over these different scenarios.  
 
 The energy consumptions  and the execution times for all the benchmarks are 
 time is measured for all the benchmarks over these different scenarios.  
 
 The energy consumptions  and the execution times for all the benchmarks are 

-presented in the plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 
+presented in  plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 

 
 For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
 for  16 and 32 nodes is lower than the energy consumed while using two sites. 
 
 For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
 for  16 and 32 nodes is lower than the energy consumed while using two sites. 


@@ -972,17 +972,17 @@ This figure shows that the energy saving percentages of one site scenario for
 to the higher  computations to communications ratio in the first scenario   
 than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which 
 results in  a lower energy consumption. Indeed, the dynamic  consumed power
 to the higher  computations to communications ratio in the first scenario   
 than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which 
 results in  a lower energy consumption. Indeed, the dynamic  consumed power

-is exponentially related to the CPU's frequency value. On the other side, the increase in the number of computing nodes can 
+is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can 

 increase the communication times and thus produces less energy saving depending on the 
 increase the communication times and thus produces less energy saving depending on the 

-benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more 
+benchmarks being executed. The results of benchmarks CG, MG, BT and FT show more 

 energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While, LU and SP consume more energy with 16 nodes than 32 in one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 
 \begin{figure}
   \centering
   \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios    
            ]{%
 energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While, LU and SP consume more energy with 16 nodes than 32 in one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 
 \begin{figure}
   \centering
   \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios    
            ]{%

-    \includegraphics[width=.4\textwidth]{fig/eng_con_scenarios.eps}\label{fig:eng_sen}} \hspace{1cm}%
+    \includegraphics[width=.48\textwidth]{fig/eng_con_scenarios.eps}\label{fig:eng_sen}} \hspace{0.4cm}%

   \subfloat[The execution times of the NAS benchmarks over different scenarios]{%
   \subfloat[The execution times of the NAS benchmarks over different scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/time_scenarios.eps}\label{fig:time_sen}}
+    \includegraphics[width=.48\textwidth]{fig/time_scenarios.eps}\label{fig:time_sen}}

   \label{fig:exp-time-energy}
   \caption{The  energy consumption and execution time of NAS  Benchmarks over different scenarios}
 \end{figure}
   \label{fig:exp-time-energy}
   \caption{The  energy consumption and execution time of NAS  Benchmarks over different scenarios}
 \end{figure}


@@ -991,11 +991,11 @@ energy saving percentage in one site scenario when executed over 16 nodes compar
 
 
 The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 
 
 
 The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 

-scenario, except for the   EP benchmark which has  no communications. Therefore, the energy saving percentage of this benchmark is 
+scenario, except for the   EP benchmark which has  no communication. Therefore, the energy saving percentage of this benchmark is 

 dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 
 dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 

-in the one site scenario, the graphite cluster is selected but in the two sits scenario 
-this cluster is replaced with Taurus cluster which is more powerful. 
-Therefore, the energy saving of EP benchmarks are bigger in the two sites scenario due 
+in the one site scenario, the graphite cluster is selected but in the two sites scenario 
+this cluster is replaced with the Taurus cluster which is more powerful. 
+Therefore, the energy savings of the EP benchmark are bigger in the two sites scenario due 

 to the higher maximum difference between the computing powers of the nodes. 
 
 In fact, high differences between the nodes' computing powers make the proposed frequencies selecting  
 to the higher maximum difference between the computing powers of the nodes. 
 
 In fact, high differences between the nodes' computing powers make the proposed frequencies selecting  


@@ -1003,23 +1003,23 @@ algorithm  select smaller frequencies for the powerful nodes which
 produces less energy consumption and thus more energy saving.
 The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
 
 produces less energy consumption and thus more energy saving.
 The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
 

-\begin{figure}
+\begin{figure*}[t]

   \centering
   \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%
   \centering
   \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%

-    \includegraphics[width=.4\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{0.4cm}%

   \subfloat[The performance degradation of the NAS benchmarks over different scenarios]{%
   \subfloat[The performance degradation of the NAS benchmarks over different scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.4cm}%

     \subfloat[The tradeoff distance between the energy reduction and the performance of the NAS benchmarks  
       over different scenarios]{%
     \subfloat[The tradeoff distance between the energy reduction and the performance of the NAS benchmarks  
       over different scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/dist.eps}\label{fig:dist}}
+    \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}

   \label{fig:exp-res}
   \caption{The experimental results of different scenarios}
   \label{fig:exp-res}
   \caption{The experimental results of different scenarios}

-\end{figure}
+\end{figure*}

 Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
 The performance degradation percentage for the benchmarks running on two sites  with
 16 or 32  nodes is on average equal to 8.3\% or 4.7\% respectively. 
 For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with
 Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
 The performance degradation percentage for the benchmarks running on two sites  with
 16 or 32  nodes is on average equal to 8.3\% or 4.7\% respectively. 
 For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with

-16 or 32  nodes is on average equal to 3.2\% or 10.6\% respectively. In opposition to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
+16 or 32  nodes is on average equal to 3.2\% or 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 

 nodes when the communications occur in high speed network does not decrease the computations to 
 communication ratio. 
 
 nodes when the communications occur in high speed network does not decrease the computations to 
 communication ratio. 
 


@@ -1052,9 +1052,9 @@ the participating number of cores from a certain cluster is equal to 14,
 in the multi-core scenario the selected nodes is equal to  4 nodes while using 
 3 or 4 cores from each node. The platforms with one  
 core per node and  multi-cores nodes are  shown in Table \ref{table:sen-mc}. 
 in the multi-core scenario the selected nodes is equal to  4 nodes while using 
 3 or 4 cores from each node. The platforms with one  
 core per node and  multi-cores nodes are  shown in Table \ref{table:sen-mc}. 

-The energy consumptions and execution times of running the class D of the NAS parallel 
-benchmarks over these four different scenarios are presented 
-in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+The energy consumptions and execution times of running  class D of the NAS parallel 
+benchmarks over these two different scenarios are presented 
+in  figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.

 
 \begin{table}[]
 \centering
 
 \begin{table}[]
 \centering


@@ -1077,9 +1077,9 @@ Scenario name                          & Cluster name & \begin{tabular}[c]{@{}c@
 \begin{figure}
   \centering
   \subfloat[Comparing the  execution times of running NAS benchmarks over one core and multicores scenarios]{%
 \begin{figure}
   \centering
   \subfloat[Comparing the  execution times of running NAS benchmarks over one core and multicores scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/time.eps}\label{fig:time-mc}} \hspace{1cm}%
+    \includegraphics[width=.48\textwidth]{fig/time.eps}\label{fig:time-mc}} \hspace{0.4cm}%

   \subfloat[Comparing the  energy consumptions of running NAS benchmarks over one core and multi-cores scenarios]{%
   \subfloat[Comparing the  energy consumptions of running NAS benchmarks over one core and multi-cores scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/eng_con.eps}\label{fig:eng-cons-mc}}
+    \includegraphics[width=.48\textwidth]{fig/eng_con.eps}\label{fig:eng-cons-mc}}

     \label{fig:eng-cons}
   \caption{The energy consumptions and execution times of NAS benchmarks over one core and multi-cores per node architectures}
 \end{figure}
     \label{fig:eng-cons}
   \caption{The energy consumptions and execution times of NAS benchmarks over one core and multi-cores per node architectures}
 \end{figure}


@@ -1099,7 +1099,7 @@ consumption and the execution times of the EP and MG benchmarks do not change si
 scenarios  because there are no or small communications. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
   
   
 scenarios  because there are no or small communications. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
   
   

-The energy saving percentages of all NAS benchmarks running over these two scenarios are presented in the figure \ref{fig:eng-s-mc}. 
+The energy saving percentages of all NAS benchmarks running over these two scenarios are presented in figure \ref{fig:eng-s-mc}. 

 The figure shows that  the energy saving percentages in the one 
 core and the multi-cores scenarios
 are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.
 The figure shows that  the energy saving percentages in the one 
 core and the multi-cores scenarios
 are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.


@@ -1110,26 +1110,26 @@ The performance degradation percentages of the NAS benchmarks are presented in
 figure \ref{fig:per-d-mc}. It shows that the performance degradation percentages is higher for the NAS benchmarks over the  one core per node scenario  (on average equal to 10.6\%)  than over the  multi-cores scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-cores scenario is lower because  the computations to communications ratio is smaller than the ratio of the other scenario. 
 
 The tradeoff distance percentages of the NAS benchmarks over the two scenarios are presented 
 figure \ref{fig:per-d-mc}. It shows that the performance degradation percentages is higher for the NAS benchmarks over the  one core per node scenario  (on average equal to 10.6\%)  than over the  multi-cores scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-cores scenario is lower because  the computations to communications ratio is smaller than the ratio of the other scenario. 
 
 The tradeoff distance percentages of the NAS benchmarks over the two scenarios are presented 

-in the figure \ref{fig:dist-mc}. These  tradeoff distance between energy consumption reduction and performance  are used to verify which scenario is the best in both terms  at the same time. The figure shows that  the  tradeoff distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).
+in figure \ref{fig:dist-mc}. These  tradeoff distance between energy consumption reduction and performance  are used to verify which scenario is the best in both terms  at the same time. The figure shows that  the  tradeoff distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).

 
 
 
 
 
 

-\begin{figure}
+\begin{figure*}[t]

   \centering
     \subfloat[The energy saving of running NAS benchmarks over one core and multicores scenarios]{%
   \centering
     \subfloat[The energy saving of running NAS benchmarks over one core and multicores scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/eng_s_mc.eps}\label{fig:eng-s-mc}} \hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/eng_s_mc.eps}\label{fig:eng-s-mc}} \hspace{0.4cm}%

     \subfloat[The performance degradation of running NAS benchmarks over one core and multicores scenarios
       ]{%
     \subfloat[The performance degradation of running NAS benchmarks over one core and multicores scenarios
       ]{%

-    \includegraphics[width=.4\textwidth]{fig/per_d_mc.eps}\label{fig:per-d-mc}}\hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/per_d_mc.eps}\label{fig:per-d-mc}}\hspace{0.4cm}%

     \subfloat[The tradeoff distance of running NAS benchmarks over one core and multicores scenarios]{%
     \subfloat[The tradeoff distance of running NAS benchmarks over one core and multicores scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/dist_mc.eps}\label{fig:dist-mc}}
+    \includegraphics[width=.48\textwidth]{fig/dist_mc.eps}\label{fig:dist-mc}}

   \label{fig:exp-res}
   \caption{The experimental results of one core and multi-cores scenarios}
   \label{fig:exp-res}
   \caption{The experimental results of one core and multi-cores scenarios}

-\end{figure}
+\end{figure*}

 
 
 
 
 
 

-\subsection{Experiments with different static and dynamic powers consumption scenarios}
+\subsection{Experiments with different static power scenarios}

 \label{sec.pow_sen}
 
 In section \ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 
 \label{sec.pow_sen}
 
 In section \ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 


@@ -1137,18 +1137,18 @@ In section \ref{sec.grid5000}, since it was not possible to measure the static p
 The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
 In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.
 The experiments have been executed with these two new static power scenarios  over the one site one core per node scenario.
 The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
 In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.
 The experiments have been executed with these two new static power scenarios  over the one site one core per node scenario.

-In these experiments, the class D of the NAS parallel benchmarks are executed over Nancy's site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, where used in this experiment. 
+In these experiments, class D of the NAS parallel benchmarks are executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, where used in this experiment. 

 
 
 \begin{figure}
   \centering
   \subfloat[The energy saving percentages for the nodes executing the NAS benchmarks over the three power scenarios]{%
 
 
 \begin{figure}
   \centering
   \subfloat[The energy saving percentages for the nodes executing the NAS benchmarks over the three power scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/eng_pow.eps}\label{fig:eng-pow}} \hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/eng_pow.eps}\label{fig:eng-pow}} \hspace{0.4cm}%

   \subfloat[The performance degradation percentages for the NAS benchmarks over the three power scenarios]{%
   \subfloat[The performance degradation percentages for the NAS benchmarks over the three power scenarios]{%

-    \includegraphics[width=.4\textwidth]{fig/per_pow.eps}\label{fig:per-pow}}\hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/per_pow.eps}\label{fig:per-pow}}\hspace{0.4cm}%

     \subfloat[The tradeoff distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios]{%
       
     \subfloat[The tradeoff distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios]{%
       

-    \includegraphics[width=.4\textwidth]{fig/dist_pow.eps}\label{fig:dist-pow}}
+    \includegraphics[width=.48\textwidth]{fig/dist_pow.eps}\label{fig:dist-pow}}

   \label{fig:exp-pow}
   \caption{The experimental results of different static power scenarios}
 \end{figure}
   \label{fig:exp-pow}
   \caption{The experimental results of different static power scenarios}
 \end{figure}


@@ -1170,24 +1170,24 @@ scaling algorithm  select smaller frequencies for the CPUs.
 These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives            less overall energy consumption. 
 The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors   according to the static power consumption ratio being used.
 
 These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives            less overall energy consumption. 
 The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors   according to the static power consumption ratio being used.
 

-The performance degradation percentages are presented in the figure \ref{fig:per-pow}.
+The performance degradation percentages are presented in figure \ref{fig:per-pow}.

 The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
 had  selected big frequencies for the CPUs. While, 
 the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The tradeoff distance percentage for the NAS benchmarks with these three static power scenarios 
 The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
 had  selected big frequencies for the CPUs. While, 
 the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The tradeoff distance percentage for the NAS benchmarks with these three static power scenarios 

-are presented in the figure \ref{fig:dist}. 
+are presented in figure \ref{fig:dist}. 

 It shows that the best  tradeoff
 distance percentage is obtained with  the  10\% static power scenario  and this percentage 
 is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
 
 In the EP benchmark, the energy saving, performance degradation and tradeoff 
 It shows that the best  tradeoff
 distance percentage is obtained with  the  10\% static power scenario  and this percentage 
 is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
 
 In the EP benchmark, the energy saving, performance degradation and tradeoff 

-distance percentages for the these static power scenarios are not significantly different because there is no communication in this benchmark. Therefore, the static power is only consumed during computation and   the proposed scaling algorithm selects similar frequencies for the three scenarios.  On the other hand,  for the rest of the benchmarks,  the scaling algorithm  selects  the values of the frequencies according to the communication times of each benchmark because the static energy consumption increases  proportionally to the  communication times.
+distance percentages for these static power scenarios are not significantly different because there is no communication in this benchmark. Therefore, the static power is only consumed during computation and   the proposed scaling algorithm selects similar frequencies for the three scenarios.  On the other hand,  for the rest of the benchmarks,  the scaling algorithm  selects  the values of the frequencies according to the communication times of each benchmark because the static energy consumption increases  proportionally to the  communication times.

 
 
  
 
 
  

-\subsection{The comparison of the proposed frequencies selecting algorithm }
+\subsection{Comparison of the proposed frequencies selecting algorithm }

 \label{sec.compare_EDP}
 
 \label{sec.compare_EDP}
 

-Finding the frequencies that gives the best tradeoff between the energy consumption and the performance for a parallel 
+Finding the frequencies that give the best tradeoff between the energy consumption and the performance for a parallel 

 application is not a trivial task.  Many algorithms have been proposed to tackle this problem.  
 In this section, the proposed frequencies selecting algorithm is compared to a method that uses the well known  energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers  \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs}. 
 This objective function  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-cores 
 application is not a trivial task.  Many algorithms have been proposed to tackle this problem.  
 In this section, the proposed frequencies selecting algorithm is compared to a method that uses the well known  energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers  \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs}. 
 This objective function  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-cores 


@@ -1199,30 +1199,30 @@ Moreover, both algorithms start the search space from the upper bound computed a
 Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
 and selects the vector of frequencies that minimize the EDP product.
 
 Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
 and selects the vector of frequencies that minimize the EDP product.
 

-Both algorithms were applied to the class D of the NAS benchmarks over 16 nodes.
+Both algorithms were applied to class D of the NAS benchmarks over 16 nodes.

 The participating computing nodes are distributed  according to the two scenarios described in  section \ref{sec.res}. 
 The experimental results, the energy saving, performance degradation and tradeoff distance percentages, are 
 presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
 
 
 The participating computing nodes are distributed  according to the two scenarios described in  section \ref{sec.res}. 
 The experimental results, the energy saving, performance degradation and tradeoff distance percentages, are 
 presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
 
 

-\begin{figure}
+\begin{figure*}[t]

   \centering
   \subfloat[The energy reduction induced by the Maxdist method and the EDP method]{%
   \centering
   \subfloat[The energy reduction induced by the Maxdist method and the EDP method]{%

-    \includegraphics[width=.4\textwidth]{fig/edp_eng}\label{fig:edp-eng}} \hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/edp_eng}\label{fig:edp-eng}} \hspace{0.4cm}%

     \subfloat[The performance degradation induced by  the Maxdist method and the EDP method]{%
     \subfloat[The performance degradation induced by  the Maxdist method and the EDP method]{%

-    \includegraphics[width=.4\textwidth]{fig/edp_per}\label{fig:edp-perf}}\hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/edp_per}\label{fig:edp-perf}}\hspace{0.4cm}%

     \subfloat[The tradeoff distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method]{%
     \subfloat[The tradeoff distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method]{%

-    \includegraphics[width=.4\textwidth]{fig/edp_dist}\label{fig:edp-dist}}
+    \includegraphics[width=.48\textwidth]{fig/edp_dist}\label{fig:edp-dist}}

   \label{fig:edp-comparison}
   \caption{The comparison results}
   \label{fig:edp-comparison}
   \caption{The comparison results}

-\end{figure}
+\end{figure*}

 
 As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
 The proposed algorithm gives better results than EDP  because it 
 maximizes the energy saving and the performance at the same time. 
 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative tradeoff values for some benchmarks in the two sites scenarios.
 
 As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
 The proposed algorithm gives better results than EDP  because it 
 maximizes the energy saving and the performance at the same time. 
 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative tradeoff values for some benchmarks in the two sites scenarios.

-These negative tradeoff values mean that the performance degradation percentage is higher than energy saving percentage.
+These negative tradeoff values mean that the performance degradation percentage is higher than the energy saving percentage.

 The high positive values of the tradeoff distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
 The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and 
 $O(N \cdot M \cdot F^2)$ respectively, where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the 
 The high positive values of the tradeoff distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
 The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and 
 $O(N \cdot M \cdot F^2)$ respectively, where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the 


@@ -1231,7 +1231,7 @@ maximum number of available frequencies. When Maxdist is applied to a benchmark
 
 \section{Conclusion}
 \label{sec.concl}
 
 \section{Conclusion}
 \label{sec.concl}

-This paper has presented a new online frequencies selection algorithm.
+This paper presents a new online frequencies selection algorithm.

  The algorithm selects the best vector of 
 frequencies that maximizes  the tradeoff distance 
 between the predicted energy consumption and the predicted execution time of the distributed 
  The algorithm selects the best vector of 
 frequencies that maximizes  the tradeoff distance 
 between the predicted energy consumption and the predicted execution time of the distributed 


@@ -1239,10 +1239,10 @@ iterative applications running over a heterogeneous grid. A new energy model
 is used by the proposed algorithm to predict the energy consumption 
 of the distributed iterative message passing application running over a grid architecture.
 To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
 is used by the proposed algorithm to predict the energy consumption 
 of the distributed iterative message passing application running over a grid architecture.
 To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  

- NAS parallel benchmarks   and the  class D instance was executed over the  grid'5000 testbed platform. 
- The experimental results showed that the algorithm reduces  on average 30\% of the energy consumption
-for all the NAS benchmarks   while  only degrading by 3.2\% on average  the performance. 
-The Maxdist algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or between using one core and multi-cores per node or in the values of the consumed static power. The algorithm selects different vector of frequencies according to the 
+ NAS parallel benchmarks   and the  class D instance was executed over the  Grid'5000 testbed platform. 
+ The experiments on 16 nodes, distributed over three clusters, showed that the algorithm   on average reduces by 30\% the energy consumption
+for all the NAS benchmarks   while  on average only degrading by 3.2\%   the performance. 
+The Maxdist algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or  use multi-cores per node architecture or consume different static power values. The algorithm selects different vectors of frequencies according to the 

 computations and communication times ratios, and  the values of the static and measured dynamic powers of the CPUs. 
 Finally, the proposed algorithm was compared to another method that uses
 the well known energy and delay product as an objective function. The comparison results showed 
 computations and communication times ratios, and  the values of the static and measured dynamic powers of the CPUs. 
 Finally, the proposed algorithm was compared to another method that uses
 the well known energy and delay product as an objective function. The comparison results showed 


@@ -1264,7 +1264,7 @@ This work  has been  partially supported by  the Labex ACTION  project (contract
 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.
 
 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.
 

-\section*{References}
+%\section*{References}

 \bibliography{my_reference}
 
 \end{document}
 \bibliography{my_reference}
 
 \end{document}