corrections

[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 c09b3df3f409bc5f337264904a61e67e0f148d10..329d52665d9659a659f1ff1ccc978bbe67aabd6e 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -107,9 +107,9 @@
 
 
 
 
 
 

-\title{Energy Consumption Reduction with DVFS for Message \\
-         Passing Iterative Applications on \\
-                    Grid Architecture} 
+\title{Optimizing the Energy Consumption \\ 
+of Message Passing Applications with Iterations \\ 
+Executed over Grids} 

   
 
 
   
 
 


@@ -143,10 +143,10 @@ scaling (DVFS) is one of them. It can be used to reduce the power consumption of
   In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.  
   It selects the frequencies and tries to give the best
   trade-off between energy saving and performance degradation, for each node
   In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.  
   It selects the frequencies and tries to give the best
   trade-off between energy saving and performance degradation, for each node

-  computing the message passing iterative application. 
+  computing the message passing  application with iterations. 

   The algorithm has a small
   overhead and works without training or profiling. It uses a new energy model
   The algorithm has a small
   overhead and works without training or profiling. It uses a new energy model

-  for message passing iterative applications running on a  grid. 
+  for message passing  applications with iterations 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 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
   The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
   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


@@ -192,33 +192,40 @@ This heterogeneous platform executes more than 7 GFlops per watt while consuming
 50.32 kilowatts.
 
 Besides platform improvements, there are many software and hardware techniques
 50.32 kilowatts.
 
 Besides platform improvements, there are many software and hardware techniques

-to lower the energy consumption of these platforms, such as scheduling, DVFS,
-\dots{} DVFS is a widely used process to reduce the energy consumption of a
+to lower the energy consumption of these platforms, such as DVFS, scheduling and other techniques.
+ DVFS is a widely used process to reduce the energy consumption of a

 processor by lowering its frequency
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
 the number of FLOPS executed by the processor which 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
 processor by lowering its frequency
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
 the number of FLOPS executed by the processor which 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

-\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.
+\cite{Our_first_paper} and \cite{pdsec2015}, a frequency selecting algorithm
+was proposed to reduce the energy consumption of message passing 
+applications with iterations running over homogeneous and heterogeneous clusters respectively.

 The results of the experiments showed significant energy consumption
 reductions. All the experimental results were conducted over the SimGrid
 The results of the experiments showed significant energy 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. %
-\AG{[\dots], which offers easy tools to describe homogeneous and heterogeneous
-  platforms, and to simulate the execution of 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
+simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous  platforms, and to simulate the execution of message passing parallel
+applications over them. 
+
+
+This paper presents the following contributions :
+\begin{enumerate}
+\item two new energy and performance models for message passing 
+  synchronous applications with iterations running over a heterogeneous grid platform. Both models
+  take into account communications and slack times. The models can predict the
+  required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous grid
+  platforms. The algorithm has a very small overhead and does not need any
+  training nor profiling. It uses a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing  synchronous application with iterations.  The algorithm  was applied to the NAS

 parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
 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 frequencies that simultaneously tries to
-offer the maximum energy reduction and minimum performance degradation
-ratios. The algorithm has a very small overhead, works online and does not need
-any training or profiling.
+\cite{grid5000}.
+
+\end{enumerate}
+

 
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some
 
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some


@@ -231,9 +238,10 @@ 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 also evaluates the algorithm over multi-cores per node architectures and over three different power scenarios. Moreover, it shows the
+It also evaluates the algorithm over multi-core 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}
 


@@ -255,10 +263,10 @@ The methods could be heuristics, exact or brute force methods that satisfy
 varied objectives such as energy reduction or performance. They also could be
 adapted to the execution's environment and the type of the application such as
 sequential, parallel or distributed architecture, homogeneous or heterogeneous
 varied objectives such as energy reduction or performance. They also could be
 adapted to the execution's environment and the type of the application such as
 sequential, parallel or distributed architecture, homogeneous or heterogeneous

-platform, synchronous or asynchronous application, \dots{}
+platform, synchronous or asynchronous application.

 
 

-In this paper, we are interested in reducing energy for message passing
-iterative synchronous applications running over heterogeneous grid platforms.  Some
+In this paper, we are interested in reducing the energy consumption of message passing
+ synchronous applications with iterations running over heterogeneous grid platforms.  Some

 works have already been done for such platforms and they can be classified into
 two types of heterogeneous platforms:
 \begin{itemize}
 works have already been done for such platforms and they can be classified into
 two types of heterogeneous platforms:
 \begin{itemize}


@@ -302,44 +310,33 @@ some heuristic.  Chen et
 al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
 programming approach to minimize the power consumption of heterogeneous servers
 while respecting given time constraints.  This approach had considerable
 al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
 programming approach to minimize the power consumption of heterogeneous servers
 while respecting given time constraints.  This approach had considerable

-overhead.  In contrast to the above described papers, this paper presents the
-following contributions :
-\begin{enumerate}
-\item two new energy and performance models for message passing iterative
-  synchronous applications running over a heterogeneous grid platform. Both models
-  take into account communication and slack times. The models can predict the
-  required energy and the execution time of the application.
-
-\item a new online frequency selecting algorithm for heterogeneous grid
-  platforms. The algorithm has a very small overhead and does not need any
-  training nor profiling. It uses a new optimization function which
-  simultaneously maximizes the performance and minimizes the energy consumption
-  of a message passing iterative synchronous application.
-
-\end{enumerate}
+overhead.

 
 
 
 \section{The performance and energy consumption measurements on heterogeneous grid architecture}
 \label{sec.exe}
 
 
 
 
 \section{The performance and energy consumption measurements on heterogeneous grid architecture}
 \label{sec.exe}
 

-\subsection{The execution time of message passing distributed iterative
-  applications on a heterogeneous platform}
+\subsection{The execution time of message passing distributed 
+  applications with iterations on a heterogeneous platform}

 
 In this paper, we are interested in reducing the energy consumption of message
 
 In this paper, we are interested in reducing the energy consumption of message

-passing distributed iterative synchronous applications running over
+passing distributed  synchronous applications with iterations running over

 heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
 heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
 and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
 
 heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
 heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
 and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
 

-The overall execution time of a distributed iterative synchronous application 
+The overall execution time of a distributed  synchronous application  with iterations running 

 over a heterogeneous grid consists of the sum of the computation time and 
 over a heterogeneous grid consists of the sum of the computation time and 

-the communication time for every iteration on a node. However, due to the
-heterogeneous computation power of the computing clusters, slack times may occur
-when fast nodes have to wait, during synchronous communications, for the slower
-nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
+the communication time for every iteration on a node. 
+However, nodes from distinct clusters in a grid have different computing powers, thus
+while the application, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
+periods are called idle or slack times. 
+Therefore, the

 overall execution time of the program is the execution time of the slowest task 
 overall execution time of the program is the execution time of the slowest task 

-which has the highest computation time and no slack time.
+which has the highest computation time and almost no slack time. For example, in Figure \ref{fig:heter}, task 1 is the slower task and it does not have to wait for the other nodes to communicate with them because they all finish their computations before it.  

 
 \begin{figure}[!t]
   \centering
 
 \begin{figure}[!t]
   \centering


@@ -360,6 +357,8 @@ as in (\ref{eq:s}).
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}

+where $\Fmax$ is the maximum frequency before applying any DVFS and $\Fnew$  is the new frequency after applying DVFS.
+

 The execution time of a compute bound sequential program is linearly
 proportional to the frequency scaling factor $S$.  On the other hand, message
 passing distributed applications consist of two parts: computation and
 The execution time of a compute bound sequential program is linearly
 proportional to the frequency scaling factor $S$.  On the other hand, message
 passing distributed applications consist of two parts: computation and


@@ -374,36 +373,38 @@ synchronously sent or received.
 Since in a heterogeneous grid each cluster has different characteristics,
 especially different frequency gears, when applying DVFS operations on the nodes 
 of these clusters, they may get different scaling factors represented by a scaling vector:
 Since in a heterogeneous grid each cluster has different characteristics,
 especially different frequency gears, when applying DVFS operations on the nodes 
 of these clusters, they may get different scaling factors represented by a scaling vector:

-$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
-be able to predict the execution time of message passing synchronous iterative
-applications running over a heterogeneous grid, for different vectors of
+$(S_{11}, S_{12},\dots, S_{NM_i})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
+be able to predict the execution time of message passing synchronous 
+applications with iterations running over a heterogeneous grid, for different vectors of

 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any

-vector of scaling factors can be predicted using (\ref{eq:perf}).
+vector of scaling factors can be predicted using Equation (\ref{eq:perf}).

 %
 \begin{equation}
   \label{eq:perf}
 %
 \begin{equation}
   \label{eq:perf}

-  \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij}) 
-  +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
+  \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots,M_h}}  (\Tcm[hj])

 \end{equation}
 %
 \end{equation}
 %

-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 $i$ 
+where $N$ is the number of  clusters in the grid, $M_i$ is the number of  nodes in
+ cluster $i$, $\TcpOld[ij]$ 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 execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
 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\AG[]{It means that\dots} 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
+and the  communication time of the slowest node without slack time during one iteration.
+ The slowest node $h$ is the node which takes the  maximum execution time to execute an iteration  before scaling down its  frequency.
+It means that only the communication time without any slack time is taken into account.
+Therefore, the execution time of the  application is equal to
+the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the

 number of iterations of that application.
 
 number of iterations of that application.
 

-This prediction model is developed from the model to predict the execution time
-of message passing distributed applications for homogeneous and heterogeneous clusters
-~\cite{Our_first_paper,pdsec2015}.  The execution time prediction model is
-used in the method to optimize both the energy consumption and the performance
-of iterative methods, which is presented in the following sections.
+This model is an adaptation of the one developed in ~\cite{Our_first_paper}  which predicts the execution time
+of message passing  applications with iterations running on homogeneous clusters. 
+In a  homogeneous cluster only one scaling factor denoted as $S$ was used because all the nodes in the cluster have the same computing power.
+In a heterogeneous cluster, each node may have a different scaling factor denoted as  $(S_i)$ where $i$ is the index of the node. In a grid, each node in each cluster may have a scaling factor. The whole set of scaling factors of all the computing nodes in the grid  is denoted by a two dimensional array of scales  
+$(S_{11}, S_{12},\dots, S_{NM_i})$ where $N$ is the number of used clusters and $M_i$ is the number of nodes in cluster $i$.  
+
+The execution time model, Equation \ref{eq:perf}, is used in the algorithm presented in section \ref{sec.optim}. The latter selects the scaling factors that optimize both the energy consumption and the performance of message passing applications with iterations running on grids.

 
 
 \subsection{Energy model for heterogeneous grid platform}
 
 
 \subsection{Energy model for heterogeneous grid platform}


@@ -474,7 +475,7 @@ by the new time of computation and is given by the following equation:
 \end{equation}
 The static power is related to the power leakage of the CPU and is consumed
 during computation and even when idle. As
 \end{equation}
 The static power is related to the power leakage of the CPU and is consumed
 during computation and even when idle. As

-in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
+in~\cite{Rauber_Analytical.Modeling.for.Energy, Zhuo_Energy.efficient.Dynamic.Task.Scheduling},

 the static power of a processor is considered as constant during idle and
 computation periods, and for all its available frequencies.  The static energy
 is the static power multiplied by the execution time of the program.  According
 the static power of a processor is considered as constant during idle and
 computation periods, and for all its available frequencies.  The static energy
 is the static power multiplied by the execution time of the program.  According


@@ -490,34 +491,36 @@ processor after scaling its frequency is computed as follows:
 
 In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
 different dynamic and static powers from the nodes of the other clusters, 
 
 In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
 different dynamic and static powers from the nodes of the other clusters, 

-noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Therefore, even if the distributed 
-message passing iterative application is load balanced, the computation time of each CPU $j$ 
-in cluster $i$ noted $\Tcp[ij]$ may be different and different frequency scaling factors may be
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Moreover, even if the distributed 
+message passing  application with iterations is load balanced, the computation time of each CPU $j$ in cluster $i$
+ noted $\Tcp[ij]$ may be slightly different due to the delay caused by the scheduler of the operating system. Therefore, different frequency scaling factors may be

 computed in order to decrease the overall energy consumption of the application
 computed in order to decrease the overall energy consumption of the application

-and reduce slack times.  The communication time of a processor $j$ in cluster $i$ is noted as
+and reduce the slack times.  The communication time of a processor $j$ in cluster $i$ is noted as

 $\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
 see Figure~\ref{fig:heter}.  Therefore, all nodes do not have equal
 communication times. While the dynamic energy is computed according to the
 frequency scaling factor and the dynamic power of each node as in
 (\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
 $\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
 see Figure~\ref{fig:heter}.  Therefore, all nodes do not have equal
 communication times. While the dynamic energy is computed according to the
 frequency scaling factor and the dynamic power of each node as in
 (\ref{eq:Edyn}), the static energy is computed as the sum of the execution time

-of one iteration multiplied by the static power of each processor.  The overall
-energy consumption of a message passing distributed application executed over a
+of one iteration multiplied by the static power of each processor. 
+ The CPU during the communication times consumes only the static power. While 
+in the computation times, it consumes both the dynamic and the static powers, for more information refer to \cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.
+The overall energy consumption of a message passing distributed application executed over a

 heterogeneous grid platform during one iteration is the summation of all dynamic and
 heterogeneous grid platform during one iteration is the summation of all dynamic and

-static energies for $M$ processors in $N$ clusters.  It is computed as follows:
+static energies for $M_i$ processors in $N$ clusters.  It is computed as follows:

 \begin{multline}
   \label{eq:energy}
 \begin{multline}
   \label{eq:energy}

- E = \sum_{i=1}^{N} \sum_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
- \sum_{i=1}^{N} \sum_{j=1}^{M} (\Ps[ij] \cdot {} \\
-  (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij}) 
-  +\mathop{\min_{j=1,\dots M}} (\Tcm[hj]) ))
+ E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
+ \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot {} \\
+  (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))

 \end{multline}
 
 
 Reducing the frequencies of the processors according to the vector of scaling
 \end{multline}
 
 
 Reducing the frequencies of the processors according to the vector of scaling

-factors $(S_{11}, S_{12},\dots, S_{NM})$ may degrade the performance of the application
+factors $(S_{11}, S_{12},\dots, S_{NM_i})$ may degrade the performance of the application

 and thus, increase the static energy because the execution time is
 increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
 and thus, increase the static energy because the execution time is
 increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption

-for the iterative application can be measured by measuring the energy
+for a synchronous  application with iterations can be measured by measuring the energy

 consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
 of iterations of that application.
 
 consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
 of iterations of that application.
 


@@ -542,10 +545,11 @@ works, \cite{Our_first_paper} and \cite{pdsec2015}, two methods that select the
 frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed. 
 Both methods selects the frequencies that gives the best trade-off between 
 energy consumption reduction and performance for  message passing
 frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed. 
 Both methods selects the frequencies that gives the best trade-off between 
 energy consumption reduction and performance for  message passing

-iterative synchronous applications.   In this work we
-are interested in grids that are composed of heterogeneous clusters were the nodes have different characteristics such  as  dynamic power, static power, computation power, frequencies range, network latency and bandwidth. 
-Due to the
-heterogeneity of the processors, a vector of scaling factors should be selected
+ synchronous applications with iterations.   In this work we
+are interested in grids that are composed of heterogeneous clusters. The nodes from distinct clusters may have 
+ different characteristics such  as  dynamic power, static power, computation power, 
+frequencies range, network latency and bandwidth. 
+Due to the heterogeneity of the processors, a vector of scaling factors should be selected

 and it must give the best trade-off between energy consumption and performance.
 
 The relation between the energy consumption and the execution time for an
 and it must give the best trade-off between energy consumption and performance.
 
 The relation between the energy consumption and the execution time for an


@@ -564,12 +568,13 @@ maximum frequency for all nodes) as follows:
 \end{equation}
 %
 where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).
 \end{equation}
 %
 where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).

-%
+

 \begin{equation}
   \label{eq:told}
 \begin{equation}
   \label{eq:told}

-   \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])             
+   \Told = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} ) 
+  +\mathop{\min_{j=1,\dots,M_h}}  (\Tcm[hj])    

 \end{equation}
 \end{equation}

-%
+

 In the same way, the energy is normalized by computing the ratio between the
 consumed energy while scaling down the frequency and the consumed energy with
 maximum frequency for all  nodes:
 In the same way, the energy is normalized by computing the ratio between the
 consumed energy while scaling down the frequency and the consumed energy with
 maximum frequency for all  nodes:


@@ -584,8 +589,8 @@ computed as in (\ref{eq:eorginal}).
 %
 \begin{equation}
   \label{eq:eorginal}
 %
 \begin{equation}
   \label{eq:eorginal}

-    \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot  \Tcp[ij])  + 
-     \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M} (\Ps[ij] \cdot \Told)       
+    \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M_i} ( \Pd[ij] \cdot  \Tcp[ij])  + 
+     \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M_i} (\Ps[ij] \cdot \Told)       

 \end{equation}
 
 While the main goal is to optimize the energy and execution time at the same
 \end{equation}
 
 While the main goal is to optimize the energy and execution time at the same


@@ -624,12 +629,12 @@ Figure~\ref{fig:r2}. Then the objective function has the following form:
 \begin{equation}
   \label{eq:max}
   \MaxDist =
 \begin{equation}
   \label{eq:max}
   \MaxDist =

-\mathop{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
+\mathop{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}}_{k=1,\dots,F_j}

       (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
        \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
 \end{equation}
       (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
        \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
 \end{equation}

-where $N$ is the number of clusters, $M$ is the number of nodes in each cluster and
-$F$ is the number of available frequencies for each node.  Then, the optimal set 
+where $N$ is the number of clusters, $M_i$ is the number of nodes in the cluster $i$ and
+$F_j$ is the number of available frequencies in the node $j$.  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 important 
 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 important 


@@ -647,7 +652,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \Require ~
     \begin{description}
     \item [{$N$}] number of clusters in the grid. 
     \Require ~
     \begin{description}
     \item [{$N$}] number of clusters in the grid. 

-    \item [{$M$}] number of nodes in each cluster.
+    \item [{$M_i$}] number of nodes in each cluster.

     \item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
     \item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
     \item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
     \item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
     \item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
     \item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.


@@ -707,11 +712,11 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 
 
 In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 
 
 
 In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 

-is presented. It selects the vector of the frequency
+is presented. It selects the vector of  frequency

 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing
 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing

-synchronous iterative application executed on a  grid. It works
-online during the execution time of the iterative message passing program.  It
+synchronous  application with iterations executed on a  grid. It works
+online during the execution time of the  application.  It

 uses information gathered during the first iteration such as the computation
 time and the communication time in one iteration for each node. The algorithm is
 executed after the first iteration and returns a vector of optimal frequency
 uses information gathered during the first iteration such as the computation
 time and the communication time in one iteration for each node. The algorithm is
 executed after the first iteration and returns a vector of optimal frequency


@@ -719,20 +724,18 @@ scaling factors that satisfies the objective function (\ref{eq:max}). The
 program applies DVFS operations to change the frequencies of the CPUs according
 to the computed scaling factors.  This algorithm is called just once during the
 execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
 program applies DVFS operations to change the frequencies of the CPUs according
 to the computed scaling factors.  This algorithm is called just once during the
 execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed

-scaling algorithm is called in the iterative MPI program.
+scaling algorithm is called in the application.

 
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/init_freq}
 
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/init_freq}

-  \caption{Selecting the initial frequencies}
+  \caption{Selecting the initial frequencies in a grid platform}

   \label{fig:st_freq}
 \end{figure}
 
   \label{fig:st_freq}
 \end{figure}
 

-Nodes from distinct clusters in a grid have different computing powers, thus
-while executing message passing iterative synchronous applications, fast nodes
-have to wait for the slower ones to finish their computations before being able
-to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
-periods are called idle or slack times.  The algorithm takes into account this
+
+
+The algorithm takes into account this

 problem and tries to reduce these slack times when selecting the vector of the frequency
 scaling factors. At first, it selects initial frequency scaling factors
 that increase the execution times of fast nodes and minimize the differences
 problem and tries to reduce these slack times when selecting the vector of the frequency
 scaling factors. At first, it selects initial frequency scaling factors
 that increase the execution times of fast nodes and minimize the differences


@@ -743,15 +746,15 @@ frequency scaling factors are computed as a ratio between the computation time
 of the slowest node and the computation time of the node $i$ as follows:
 \begin{equation}
   \label{eq:Scp}
 of the slowest node and the computation time of the node $i$ as follows:
 \begin{equation}
   \label{eq:Scp}

-  \Scp[ij] =  \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]}
+  \Scp[ij] =  \frac{ \mathop{\max\limits_{i=1,\dots N}}\limits_{j=1,\dots,M_i}(\Tcp[ij])} {\Tcp[ij]}

 \end{equation}
 Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
 algorithm computes the initial frequencies for all nodes as a ratio between the
 \end{equation}
 Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
 algorithm computes the initial frequencies for all nodes as a ratio between the

-maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+maximum frequency of node and its computed scaling factor as

 follows:
 \begin{equation}
   \label{eq:Fint}
 follows:
 \begin{equation}
   \label{eq:Fint}

-  F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M}
+  F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M_i}

 \end{equation}
 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
 \end{equation}
 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


@@ -773,9 +776,7 @@ Therefore, the algorithm iterates on all remaining frequencies, from the higher
 bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
 energy consumption and performance and selects the optimal vector of the frequency scaling
 factors. At each iteration the algorithm determines the slowest node
 bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
 energy consumption and performance and selects the optimal vector of the frequency scaling
 factors. At each iteration the algorithm determines the slowest node

-according to Equation~\ref{eq:perf}
-%\AG[]{Be consistent: remove word ``Equation'' and add parentheses around equation number, here and all along the rest of the text.}
-and keeps its frequency unchanged,
+according to 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
 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


@@ -789,11 +790,7 @@ 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  grid platform the performance is maintained at the beginning of the
 plot even if the frequencies of the faster nodes decrease until the computing
 consumed energy decrease from the beginning of the plot. On the other hand, in
 the  grid 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,
-\AG[]{That's not a sentence.}
-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, which results in bigger energy savings. 
+power of scaled down nodes are lower than the slowest node. 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, which results in bigger energy savings. 

 
 
 \section{Experimental results}
 
 
 \section{Experimental results}


@@ -803,7 +800,7 @@ in this paper real experiments were conducted over the Grid'5000 platform.
 
 \subsection{Grid'5000 architecture and power consumption}
 \label{sec.grid5000}
 
 \subsection{Grid'5000 architecture and power consumption}
 \label{sec.grid5000}

-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,
+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.
 Each site of the grid is composed of a few heterogeneous 
 computing clusters and each cluster contains many homogeneous nodes. In total,
 which is the French National Telecommunication Network for Technology.
 Each site of the grid is composed of a few heterogeneous 
 computing clusters and each cluster contains many homogeneous nodes. In total,


@@ -812,7 +809,7 @@ 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 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 
 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 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, \dots{} For more details refer to
+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. 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}. 
 \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}. 


@@ -851,21 +848,20 @@ selected clusters and are presented in Table~\ref{table:grid5000}.
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/power_consumption.pdf}
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/power_consumption.pdf}

-  \AG{I don't understand the labels on the horizontal axis: 10:30:37, 10:30:38,
-    etc.}

   \caption{The power consumption by one core from the Taurus cluster}
   \label{fig:power_cons}
 \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.
   \caption{The power consumption by one core from the Taurus cluster}
   \label{fig:power_cons}
 \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 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,  class D was used for all benchmarks in all the experiments presented in the next sections. 
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These benchmarks are considered as message passing  applications with iterations because the same block of operations is executed many times.
+ 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 the next sections, the  class D was used for all the benchmarks in all the experiments. 

 
 
   
 \begin{table}[!t]
 
 
   
 \begin{table}[!t]

-  \caption{CPUs characteristics of the selected clusters}
+  \caption{The characteristics of the CPUs in the selected clusters}

   % title of Table
   \centering
   \begin{tabular}{|*{7}{c|}}
   % title of Table
   \centering
   \begin{tabular}{|*{7}{c|}}


@@ -899,9 +895,7 @@ The benchmarks have seven different classes, S, W, A, B, C, D and E, that repres
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 

-to the NAS parallel benchmarks are presented. 
-
-As mentioned previously, the experiments 
+to the NAS parallel benchmarks are presented. Each experiment  has been executed many times and the results presented in the figures are the average values of many executions. As mentioned previously, the experiments 

 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}
 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}


@@ -913,7 +907,7 @@ Two scenarios were considered while selecting the clusters from these two sites
 The main reason 
 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 
 The main reason 
 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 

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

 
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
 
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 


@@ -922,7 +916,7 @@ Table~\ref{tab:sc} shows the number of nodes used from each cluster for each sce
 
 \begin{table}[h]
 
 
 \begin{table}[h]
 

-\caption{The different clusters scenarios}
+\caption{The different grid scenarios}

 \centering
 \begin{tabular}{|*{4}{c|}}
 \hline
 \centering
 \begin{tabular}{|*{4}{c|}}
 \hline


@@ -933,7 +927,7 @@ Table~\ref{tab:sc} shows the number of nodes used from each cluster for each sce
                                       & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                       & Griffon       & Nancy        & 6                      \\ 
 \hline
                                       & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                       & Griffon       & Nancy        & 6                      \\ 
 \hline

-\multirow{3}{*}{Tow sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
+\multirow{3}{*}{Two sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 

                                       & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                       & Griffon     &Nancy           & 12                     \\ 
 \hline
                                       & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                       & Griffon     &Nancy           & 12                     \\ 
 \hline


@@ -949,8 +943,16 @@ Table~\ref{tab:sc} shows the number of nodes used from each cluster for each sce
  \label{tab:sc}
 \end{table}
 
  \label{tab:sc}
 \end{table}
 

-
-
+\begin{figure}
+  \centering
+  \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios    
+           ]{%
+    \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]{%
+    \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}

 
 The NAS parallel benchmarks are executed over these two platforms
  with different number of nodes, as in Table~\ref{tab:sc}. 
 
 The NAS parallel benchmarks are executed over these two platforms
  with different number of nodes, as in Table~\ref{tab:sc}. 


@@ -958,7 +960,7 @@ The overall energy consumption of all the benchmarks solving the class D instanc
 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 power showed in Table~\ref{table:grid5000}
 and the static 
 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 power showed in Table~\ref{table:grid5000}
 and the static 

-power is assumed to be equal to 20\% of the dynamic power. The execution
+power is assumed to be equal to 20\% of the dynamic power as in \cite{Rauber_Analytical.Modeling.for.Energy}. 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 


@@ -969,18 +971,17 @@ for  16 and 32 nodes is lower than the energy consumed while using two sites.
 The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption. 
 
 The execution times of these benchmarks 
 The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption. 
 
 The execution times of these benchmarks 

-over one site with 16 and 32 nodes are also lower when  compared to those of the  two sites 
-scenario. Moreover, most of the benchmarks running over the one site scenario have their execution times  approximately divided by two  when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).
+over one site with 16 and 32 nodes are also lower than those of the  two sites 
+scenario. Moreover, most of the benchmarks running over the one site scenario have their execution times  approximately halved  when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).

 
 

-However, the execution times and the energy consumptions of EP and MG
+However, the execution times and the energy consumptions of the EP and MG

 benchmarks, which have no or small communications, are not significantly
 affected in both scenarios, even when the number of nodes is doubled.  On the
 benchmarks, which have no or small communications, are not significantly
 affected in both scenarios, even when the number of nodes is doubled.  On the

-other hand, the communications\AG[]{the communication time?} of the rest of the benchmarks increases when
-using long distance communications between two sites or increasing the number of
+other hand, the communication times of the rest of the benchmarks increase when
+using long distance communications between two sites or when increasing the number of

 computing nodes.
 
 
 computing nodes.
 
 

-

 The energy saving percentage is computed as the ratio between the reduced 
 energy consumption, Equation~\ref{eq:energy}, and the original energy consumption,
 Equation~\ref{eq:eorginal}, for all benchmarks as in Figure~\ref{fig:eng_s}. 
 The energy saving percentage is computed as the ratio between the reduced 
 energy consumption, Equation~\ref{eq:energy}, and the original energy consumption,
 Equation~\ref{eq:eorginal}, for all benchmarks as in Figure~\ref{fig:eng_s}. 


@@ -992,18 +993,20 @@ results in  a lower energy consumption. Indeed, the dynamic  consumed power
 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 
 benchmarks being executed. The results of benchmarks CG, MG, BT and FT show more 
 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 
 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    
-           ]{%
-    \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]{%
-    \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}
+energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes.  LU and SP consume more energy with 16 nodes than 32 nodes on one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 

 
 

+\begin{figure*}[!h]
+  \centering
+  \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%
+    \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]{%
+    \includegraphics[width=.48\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.4cm}%
+    \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks  
+      over different scenarios]{%
+    \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}
+  \label{fig:exp-res}
+  \caption{The experimental results of different scenarios}
+\end{figure*}

 
 
 
 
 
 


@@ -1020,30 +1023,19 @@ 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*}[t]
-  \centering
-  \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%
-    \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]{%
-    \includegraphics[width=.48\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.4cm}%
-    \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks  
-      over different scenarios]{%
-    \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}
-  \label{fig:exp-res}
-  \caption{The experimental results of different scenarios}
-\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. 
 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 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 
+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 high 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\% and 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. 
 
 The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to 
 the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its 
 performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
 nodes when the communications occur in high speed network does not decrease the computations to 
 communication ratio. 
 
 The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to 
 the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its 
 performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.

-The rest of the benchmarks showed different performance degradation percentages, which decrease
+The rest of the benchmarks showed different performance degradation percentages which decrease

 when the communication times increase and vice versa.
 
 Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The trade-off distance percentage can be 
 when the communication times increase and vice versa.
 
 Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The trade-off distance percentage can be 


@@ -1058,25 +1050,25 @@ which increase the computations to communications ratio  and the latter uses lon
 
 
 
 
 
 

-\subsection{The experimental results over multi-cores clusters}
+\subsection{The experimental results over multi-core clusters}

 \label{sec.res-mc}
 
 The  clusters of Grid'5000 have different number of cores embedded in their nodes
 as shown in Table~\ref{table:grid5000}. In 
 this section, the proposed scaling algorithm is evaluated over the  Grid'5000 platform  while using multi-cores nodes selected according to the one site scenario described in  Section~\ref{sec.res}.
 \label{sec.res-mc}
 
 The  clusters of Grid'5000 have different number of cores embedded in their nodes
 as shown in Table~\ref{table:grid5000}. In 
 this section, the proposed scaling algorithm is evaluated over the  Grid'5000 platform  while using multi-cores nodes selected according to the one site scenario described in  Section~\ref{sec.res}.

-The one site scenario uses  32 cores from multi-cores nodes instead of 32 distinct nodes. For example if 
+The one site scenario uses  32 cores from multi-core nodes instead of 32 distinct nodes. For example if 

 the participating number of cores from a certain cluster is equal to 14, 
 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}. 
-The energy consumptions and execution times of running  class D of the NAS parallel 
-benchmarks over these two different scenarios are presented 
+in the multi-core scenario  4 nodes are selected and
+3 or 4 cores from each node are used. The platforms with one  
+core per node and  multi-core 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 two different platforms are presented 

 in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
 
 
 in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
 
 

-\begin{table}[]
+\begin{table}[!h]

 \centering
 \centering

-\caption{The multicores scenarios}
+\caption{The multi-core scenarios}

 \begin{tabular}{|*{4}{c|}}
 \hline
 Scenario name                          & Cluster name & Nodes per cluster & 
 \begin{tabular}{|*{4}{c|}}
 \hline
 Scenario name                          & Cluster name & Nodes per cluster & 


@@ -1084,7 +1076,7 @@ Scenario name                          & Cluster name & Nodes per cluster &
 \multirow{3}{*}{One core per node}    & Graphite     & 4               & 1                   \\  \cline{2-4}
                                        & Graphene     & 14              & 1                   \\  \cline{2-4}
                                        & Griffon      & 14              & 1                   \\ \hline
 \multirow{3}{*}{One core per node}    & Graphite     & 4               & 1                   \\  \cline{2-4}
                                        & Graphene     & 14              & 1                   \\  \cline{2-4}
                                        & Griffon      & 14              & 1                   \\ \hline

-\multirow{3}{*}{Multi-cores per node}  & Graphite     & 1               &  4              \\  \cline{2-4}
+\multirow{3}{*}{Multi-core per node}  & Graphite     & 1               &  4              \\  \cline{2-4}

                                        & Graphene     & 4               & 3 or 4              \\  \cline{2-4}
                                        & Griffon      & 4               & 3 or 4                   \\ \hline
 \end{tabular}
                                        & Graphene     & 4               & 3 or 4              \\  \cline{2-4}
                                        & Griffon      & 4               & 3 or 4                   \\ \hline
 \end{tabular}


@@ -1092,54 +1084,54 @@ Scenario name                          & Cluster name & Nodes per cluster &
 \end{table}
 
 
 \end{table}
 
 

-\begin{figure}
+\begin{figure}[!h]

   \centering
   \centering

-  \subfloat[Comparing the  execution times of running NAS benchmarks over one core and multicores scenarios]{%
+  \subfloat[Comparing the  execution times of running the NAS benchmarks over one core and multi-core scenarios]{%

     \includegraphics[width=.48\textwidth]{fig/time.eps}\label{fig:time-mc}} \hspace{0.4cm}%
     \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 the NAS benchmarks over one core and multi-core scenarios]{%

     \includegraphics[width=.48\textwidth]{fig/eng_con.eps}\label{fig:eng-cons-mc}}
     \label{fig:eng-cons}
     \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}
+  \caption{The energy consumptions and execution times of the  NAS benchmarks running over one core and multi-core per node architectures}

 \end{figure}
 
 
 
 \end{figure}
 
 
 

-The execution times for most of  the NAS  benchmarks are higher over the multi-cores per node scenario 
-than over single core per node  scenario. Indeed,  
- the communication times  are higher in the one site multi-cores scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.    
+The execution times for most of  the NAS  benchmarks are higher over the multi-core per node scenario 
+than over the single core per node  scenario. Indeed,  
+ the communication times  are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.    

 Moreover, the energy consumptions of the NAS benchmarks are lower over the 
 Moreover, the energy consumptions of the NAS benchmarks are lower over the 

- one core scenario  than over the multi-cores scenario because 
+ one core scenario  than over the multi-core scenario because 

 the first scenario had less execution time than the latter which results in less static energy being consumed.
 The computations to communications ratios of the NAS benchmarks are higher over 
 the first scenario had less execution time than the latter which results in less static energy being consumed.
 The computations to communications ratios of the NAS benchmarks are higher over 

-the one site one core scenario  when compared to the ratio of the multi-cores scenario. 
+the one site one core scenario  when compared to the ratio of the multi-core scenario. 

 More energy reduction can be gained when this ratio is big because it pushes the proposed scaling algorithm to select smaller frequencies that decrease the dynamic power consumption. These experiments also showed that the energy 
 consumption and the execution times of the EP and MG benchmarks do not change significantly over these two
 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.
 \begin{figure*}[t]
   \centering
 More energy reduction can be gained when this ratio is big because it pushes the proposed scaling algorithm to select smaller frequencies that decrease the dynamic power consumption. These experiments also showed that the energy 
 consumption and the execution times of the EP and MG benchmarks do not change significantly over these two
 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.
 \begin{figure*}[t]
   \centering

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

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

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

-    \subfloat[The trade-off distance of running NAS benchmarks over one core and multicores scenarios]{%
+    \subfloat[The trade-off distance of running NAS benchmarks over one core and multicore scenarios]{%

     \includegraphics[width=.48\textwidth]{fig/dist_mc.eps}\label{fig:dist-mc}}
   \label{fig:exp-res2}
     \includegraphics[width=.48\textwidth]{fig/dist_mc.eps}\label{fig:dist-mc}}
   \label{fig:exp-res2}

-  \caption{The experimental results of one core and multi-cores scenarios}
+  \caption{The experimental results of one core and multi-core scenarios}

 \end{figure*}  
   
 \end{figure*}  
   

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

-core and the multi-cores scenarios
+core and the multi-core scenarios

 are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.
 The energy consumption is reduced at the same rate in the two scenarios when compared to the energy consumption of the executions without DVFS. 
 
 
 The performance degradation percentages of the NAS benchmarks are presented in
 are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.
 The energy consumption is reduced at the same rate in the two scenarios when compared to the energy consumption of the executions without DVFS. 
 
 
 The performance degradation percentages of the NAS benchmarks are presented in

-Figure~\ref{fig:per-d-mc}. It shows that the performance degradation percentages are 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 are lower because  the computations to communications ratios are smaller than the ratios of the other scenario. 
+Figure~\ref{fig:per-d-mc}. It shows that the performance degradation percentages are higher for the NAS benchmarks executed over the  one core per node scenario  (on average equal to 10.6\%)  than over the  multi-core scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-core scenario are lower because  the computations to communications ratios are smaller than the ratios of the other scenario. 

 
 

-The trade-off distances percentages of the NAS benchmarks over the two scenarios are presented 
-in ~Figure~\ref{fig:dist-mc}. These  trade-off distances 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  trade-off distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).
+The trade-off distances percentages of the NAS benchmarks over both scenarios are presented 
+in ~Figure~\ref{fig:dist-mc}. These  trade-off distances 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  trade-off distance percentages are on average   bigger over the multi-core scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).

 
 
 
 
 
 


@@ -1155,7 +1147,7 @@ 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, 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. 
+In these experiments, the class D of the NAS parallel benchmarks were executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, were used in this experiment. 

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


@@ -1181,18 +1173,18 @@ In these experiments, class D of the NAS parallel benchmarks are executed over t
 \end{figure}
 
 The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 
 \end{figure}
 
 The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 

-in Figure~\ref{fig:eng_sen}. This figure shows that the  10\% of static power scenario 
+in Figure~\ref{fig:eng-pow}. This figure shows that the  10\% of static power scenario 

 gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
 scenarios. The small value of the static power consumption makes the proposed 
 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. 
 gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
 scenarios. The small value of the static power consumption makes the proposed 
 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.
+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 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 trade-off distance percentage for the NAS benchmarks with these three static power scenarios 
 
 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 trade-off distance percentage for the NAS benchmarks with these three static power scenarios 

-are presented in Figure~\ref{fig:dist}. 
+are presented in Figure~\ref{fig:dist-pow}. 

 It shows that the best  trade-off
 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.
 It shows that the best  trade-off
 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.


@@ -1208,21 +1200,8 @@ distance percentages for these static power scenarios are not significantly diff
 Finding the frequencies that give the best trade-off 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}. 
 Finding the frequencies that give the best trade-off 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 
+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-core 

 architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
 architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.

-
-To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  Equation~\ref{eq:energy} and
-execution time model, Equation~\ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
-Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.
-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 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 trade-off distance percentages, are 
-presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
-
-

 \begin{figure*}[t]
   \centering
   \subfloat[The energy reduction induced by the Maxdist method and the EDP method]{%
 \begin{figure*}[t]
   \centering
   \subfloat[The energy reduction induced by the Maxdist method and the EDP method]{%


@@ -1234,17 +1213,27 @@ presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dis
   \label{fig:edp-comparison}
   \caption{The comparison results}
 \end{figure*}
   \label{fig:edp-comparison}
   \caption{The comparison results}
 \end{figure*}

+To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  Equation~\ref{eq:energy} and
+execution time model, Equation~\ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
+Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.
+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 running 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 trade-off distance percentages, are 
+presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.

 
 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. 
 
 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. 
+The proposed algorithm gives better results than the EDP method because the former selects the set of frequencies that  
+gives the best tradeoff between energy saving and performance. 

 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
 These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
 The high positive values of the trade-off distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
 These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
 The high positive values of the trade-off 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 
-maximum number of available frequencies. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average  $0.01 ms$  to compute the best frequencies while EDP is on average ten times slower over the same architecture.  
+The  complexity of both algorithms, Maxdist and EDP,  are of order $O(N \cdot M_i \cdot F_j)$ and 
+$O(N \cdot M_i \cdot F_j^2)$ respectively, where $N$ is the number of the clusters, $M_i$ is the number of nodes and $F_j$ is the 
+maximum number of available frequencies. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average  $0.01$  $ms$  to compute the best frequencies while the EDP method is on average ten times slower over the same architecture.  

 
 
 \section{Conclusion}
 
 
 \section{Conclusion}


@@ -1253,24 +1242,24 @@ This paper presents a new online frequencies selection algorithm.
  The algorithm selects the best vector of 
 frequencies that maximizes  the trade-off 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 trade-off distance 
 between the predicted energy consumption and the predicted execution time of the distributed 

-iterative applications running over a heterogeneous grid. A new energy model 
+ applications with iterations running over a heterogeneous grid. A new energy model 

 is used by the proposed algorithm to predict the energy consumption 
 is used by the proposed algorithm to predict the energy consumption 

-of the distributed iterative message passing application running over a grid architecture.
+of the application.

 To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
 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 experiments on 16 nodes, distributed over three clusters, showed that the algorithm   on average reduces by 30\% the energy consumption
+NAS parallel benchmarks   and the  class D instance was executed over the  Grid'5000 testbed platform. 
+The experiments executed 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. 
 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 
+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-core 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 
 that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better trade-off  between energy consumption reduction and performance. 
 
 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 
 that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better trade-off  between energy consumption reduction and performance. 
 

-In the near future, we would like to develop a similar method that is adapted to
-asynchronous iterative applications where iterations are not synchronized and communications are overlapped with computations. 
+In the near future, we will adapt the proposed algorithm to take into consideration the variability between some iterations.  For example, the proposed algorithm can be executed twice: after the first iteration the frequencies are scaled down according to the execution times measured in the first iteration, then after a fixed number of iterations, the frequencies are adjusted according to the execution times measured during the fixed number of iterations. If the  computing power of the system is constantly changing, it would be interesting to implement a mechanism that detects this change and adjusts the frequencies according to the variability of the system. We would like also to develop a similar method that is adapted to
+asynchronous  applications with iterations where iterations are not synchronized and communications are overlapped with computations. 

 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 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.
+the global convergence of the iterative system. Finally, it would be interesting to evaluate the scalability of the proposed algorithm by running it on large platforms composed of many thousands of cores. The scalability of the algorithm can be improved by distributing it  in a hierarchical manner where a leader is chosen for each cluster or a group of nodes  to compute their scaled frequencies and by using asynchronous messages to exchange the the data measured at the first iteration.