X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/bf4693302ab924b37253fdb2079ad75a8808a987..HEAD:/mpi-energy2-extension/Heter_paper.tex

diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index 88731ec..9108cb1 100644
--- 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} 
   
 
 
@@ -143,11 +143,11 @@ 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
-  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
-  for message passing iterative applications running on a  grid. 
-  The proposed algorithm is evaluated on a real grid, the grid'5000 platform, while
+  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
    by \np[\%]{3.2}. Finally, the algorithm is 
@@ -188,30 +188,44 @@ the Tianhe-2 platform is approximately more than \$10 million each year.  The
 computing platforms must be more energy efficient and offer the highest number
 of FLOPS per watt possible, such as the Shoubu-ExaScaler from RIKEN
 which became the top of the Green500 list in June 2015 \cite{Green500_List}.
-This heterogeneous platform executes more than 7 GFLOPS per watt while consuming
+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
-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
-\cite{Our_first_paper} and \cite{pdsec2015} , a frequency selecting algorithm was proposed to reduce
-the energy consumption of message passing iterative applications running over
-homogeneous  and heterogeneous clusters respectively.  
-The results of the experiments showed significant energy
-consumption reductions. All the experimental results were conducted over the
-Simgrid simulator \cite{SimGrid}, which offers easy tools to create homogeneous and heterogeneous platforms and runs message passing parallel applications over them. In this paper, a new frequency selecting algorithm,
-adapted to  grid platforms composed of heterogeneous clusters, is presented. It is applied to the NAS parallel benchmarks and evaluated over a real testbed, 
-the grid'5000 platform \cite{grid5000}. It selects  for a grid platform running a message passing iterative
-application the vector of
-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{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
+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
+\cite{grid5000}.
+
+\end{enumerate}
+
 
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some
@@ -223,10 +237,11 @@ energy-performance objective function that maximizes the reduction of energy
 consumption while minimizing the degradation of the program's performance.
 Section~\ref{sec.optim} details the proposed 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
+NAS parallel benchmarks and executing them on the Grid'5000 testbed. 
+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.
+
 \section{Related works}
 \label{sec.relwork}
 
@@ -248,10 +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
-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}
@@ -266,7 +281,7 @@ heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
 goal was to maximize the energy efficiency of the platform during computation by
 maximizing the number of FLOPS per watt generated.
 In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
-al. developed a scheduling algorithm that distributes workloads proportional to
+al. developed a scheduling algorithm that distributes workload proportional to
 the computing power of the nodes which could be a GPU or a CPU. All the tasks
 must be completed at the same time.  In~\cite{Rong_Effects.of.DVFS.on.K20.GPU},
 Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
@@ -295,36 +310,34 @@ 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
-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 or 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}
 
-\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
-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.
 
+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 
+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 
+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
   \includegraphics[scale=0.6]{fig/commtasks}
@@ -332,15 +345,6 @@ and higher latency than the local networks of the clusters. Each computing clust
   \label{fig:heter}
 \end{figure}
 
-The overall execution time of a distributed iterative synchronous application 
-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
-overall execution time of the program is the execution time of the slowest task 
-which has the highest computation time and no slack time.
-
 Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
 modern processors, that reduces the energy consumption of a CPU by scaling
 down its voltage and frequency.  Since DVFS lowers the frequency of a CPU
@@ -353,6 +357,8 @@ as in (\ref{eq:s}).
   \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
@@ -367,35 +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:
-$(S_{11}, S_{12},\dots, S_{NM})$ 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
-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}
-  \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}
-
-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 the slowest communication time without slack time during one iteration. 
-The latter is equal to the  communication time of the slowest node in the slowest cluster $h$.
-It means only the communication time without any slack time is taken into account.  
-Therefore, the execution time of the iterative application is equal to
-the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
+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  communication time of the slowest node without slack time during one iteration.
+The slowest node in cluster $h$ 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.
 
-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}
@@ -432,7 +441,7 @@ communication and no slack time.
 The main objective of DVFS operation is to reduce the overall energy
 consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  The operational
 frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot
-F$ with some constant $\beta$.~This equation is used to study the change of the
+F$ with some constant $\beta$. This equation is used to study the change of the
 dynamic voltage with respect to various frequency values
 in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction process of the
 frequency can be expressed by the scaling factor $S$ which is the ratio between
@@ -466,7 +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
-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
@@ -482,34 +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, 
-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
-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
-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
-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}
- 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_h}} (\Tcm[hj]) ))
 \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
-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.
 
@@ -532,12 +543,13 @@ increasing significantly the execution time.
 In our previous
 works, \cite{Our_first_paper} and \cite{pdsec2015}, two methods that select the optimal
 frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed. 
-Both methods selects the frequencies that gives the best tradeoff between 
+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
@@ -549,45 +561,48 @@ are not measured using the same metric.  To solve this problem, the execution
 time is normalized by computing the ratio between the new execution time (after
 scaling down the frequencies of some processors) and the initial one (with
 maximum frequency for all nodes) as follows:
+%
 \begin{equation}
   \label{eq:pnorm}
   \Pnorm = \frac{\Tnew}{\Told}                 
 \end{equation}
+%
+where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).
 
-
-Where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{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}
+
 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:
+%
 \begin{equation}
   \label{eq:enorm}
   \Enorm = \frac{\Ereduced}{\Eoriginal} 
 \end{equation}
-
-Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
+%
+where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
 computed as in (\ref{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
 time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. 
-According to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
-vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy
-and the execution time simultaneously.  But the main objective is to produce
+According to (\ref{eq:pnorm}) and (\ref{eq:enorm}), the
+vector of frequency scaling factors $S_{11},S_{12},\dots,S_{NM_i}$ reduces both the energy
+and the execution time,  but the main objective is to produce
 maximum energy reduction with minimum execution time reduction.
 
 This problem can be solved by making the optimization process for energy and
 execution time follow the same evolution according to the vector of scaling factors
-$(S_{11}, S_{12},\dots, S_{NM})$. Therefore, the equation of the
+$(S_{11}, S_{12},\dots, S_{NM_i})$. Therefore, the equation of the
 normalized execution time is inverted which gives the normalized performance
 equation, as follows:
 \begin{equation}
@@ -609,17 +624,17 @@ Then, the objective function can be modeled in order to find the maximum
 distance between the energy curve (\ref{eq:enorm}) and the performance curve
 (\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
 represents the minimum energy consumption with minimum execution time (maximum
-performance) at the same time, see Figure~\ref{fig:r1} or
+performance) at the same time, see Figure~\ref{fig:r1} and
 Figure~\ref{fig:r2}. Then the objective function has the following form:
 \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}
-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 
@@ -637,7 +652,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \Require ~
     \begin{description}
     \item [{$N$}] number of clusters in the grid. 
-    \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.
@@ -645,7 +660,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \item[{$\Ps[ij]$}] array of the static powers for all nodes.
     \item[{$\Fdiff[ij]$}] array of the differences between two successive frequencies for all nodes.
     \end{description}
-    \Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$,  a vector of scaling factors that gives the optimal tradeoff between energy consumption and execution time
+    \Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$,  a vector of scaling factors that gives the optimal trade-off between energy consumption and execution time
 
     \State $\Scp[ij] \gets \frac{\max_{i=1,2,\dots,N}(\max_{j=1,2,\dots,M_i}(\Tcp[ij]))}{\Tcp[ij]} $
     \State $F_{ij} \gets  \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
@@ -653,8 +668,8 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \If{(not the first frequency)}
           \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
     \EndIf
-    \State $\Told \gets $ computed as in equations (\ref{eq:told}).
-    \State $\Eoriginal \gets $ computed as in equations (\ref{eq:eorginal}) .
+    \State $\Told \gets $ computed as in Equation \ref{eq:told}.
+    \State $\Eoriginal \gets $ computed as in Equation \ref{eq:eorginal}.
     \State $\Sopt[ij] \gets 1,~i=1,\dots,N,~{j=1,\dots,M_i}. $
     \State $\Dist \gets 0 $
     \While {(all nodes have not reached their  minimum   \newline\hspace*{2.5em} frequency \textbf{or}  $\Pnorm - \Enorm < 0 $)}
@@ -662,8 +677,8 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
         \State $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
         \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
         \EndIf
-       \State $\Tnew \gets $ computed as  in equations (\ref{eq:perf}). 
-       \State $\Ereduced \gets $ computed as  in equations (\ref{eq:energy}). 
+       \State $\Tnew \gets $ computed as  in Equation \ref{eq:perf}. 
+       \State $\Ereduced \gets $ computed as  in Equation \ref{eq:energy}. 
        \State $\Pnorm \gets \frac{\Told}{\Tnew}$,  $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
         \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
@@ -696,12 +711,12 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 \end{algorithm}
 
 
-In this section, the scaling factors selection algorithm for  grids, algorithm~\ref{HSA}, 
-is presented. It selects the vector of the frequency
+In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 
+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
-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
@@ -709,20 +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
-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}
-  \caption{Selecting the initial frequencies}
+  \caption{Selecting the initial frequencies in a grid platform}
   \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
@@ -733,15 +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}
-  \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
-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}
-  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
@@ -763,11 +776,11 @@ 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
-according to the equation (\ref{eq:perf}) 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
-highest distance according to the objective function (\ref{eq:max}).
+highest distance according to the objective function~\ref{eq:max}.
 
 Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
 consumed energy for an application running on a homogeneous cluster and a
@@ -777,35 +790,32 @@ 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
-power of scaled down nodes are lower than the slowest node. In other words,
-until they reach the higher bound. It can also be noticed that the higher the
-difference between the faster nodes and the slower nodes is, the bigger the
-maximum distance between the energy curve and the performance curve is, 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}
 \label{sec.expe}
 While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid}, 
-in this paper real experiments were conducted over the grid'5000 platform. 
+in this paper real experiments were conducted over the Grid'5000 platform. 
 
 \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,
-grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
+Grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
 the clusters and their nodes are connected via  high speed local area networks. 
 Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
 
-Since grid'5000 is dedicated to  testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture 
-the power consumption  for each node in those sites. The measured power is the overall consumed power  by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ...  For more details refer to
+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. For more details refer to
 \cite{Energy_measurement}. In order to correctly measure the CPU power of one core in a node $j$, 
  firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumptions represents the 
-dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 
+dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 
 
 
-The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
+The dynamic power $\Pd[j]$ is computed as in Equation~\ref{eq:pdyn}
 \begin{equation}
   \label{eq:pdyn}
     \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
@@ -819,20 +829,20 @@ measured value in maximum powers vector and the minimum measured value in the id
 
 On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in Grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.
 
-In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
+In the experiments presented in the following sections, two sites of Grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as shown on Figure~\ref{fig:grid5000}.
 
 Four clusters from the two sites were selected in the experiments: one cluster from 
 Lyon's site, Taurus, and three clusters from Nancy's site, Graphene, 
 Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
-frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 
-the detailed characteristics of these four clusters. Moreover, the dynamic powers were computed  using equation (\ref{eq:pdyn}) for all the nodes in the 
-selected clusters and are presented in table  \ref{table:grid5000}.
+frequency ranges and local network features: the bandwidth and the latency.  Table~\ref{table:grid5000} shows 
+the detailed characteristics of these four clusters. Moreover, the dynamic powers were computed  using Equation~\ref{eq:pdyn} for all the nodes in the 
+selected clusters and are presented in Table~\ref{table:grid5000}.
 
 
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=1]{fig/grid5000}
-  \caption{The selected two sites of grid'5000}
+  \caption{The selected two sites of Grid'5000}
   \label{fig:grid5000}
 \end{figure}
 \begin{figure}[!t]
@@ -844,19 +854,20 @@ selected clusters and are presented in table  \ref{table:grid5000}.
 
 
 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]
-  \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|}}
     \hline
                 &             & Max   & Min   & Diff. &                 &               \\
-    Cluster     & CPU         & Freq. & Freq. & Freq. & No. of cores    & Dynamic power \\
+    Cluster     & CPU         & Freq. & Freq. & Freq. & Cores           & Dynamic power \\
     Name        & model       & GHz   & GHz   & GHz   & per CPU         & of one core   \\
     \hline
                 & Intel       &       &       &         &           &              \\
@@ -884,9 +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} 
-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}
@@ -898,27 +907,27 @@ 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 
-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 
 is different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
-Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
+Table~\ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
 
 \begin{table}[h]
 
-\caption{The different clusters scenarios}
+\caption{The different grid scenarios}
 \centering
 \begin{tabular}{|*{4}{c|}}
 \hline
 \multirow{2}{*}{Scenario name}        & \multicolumn{3}{c|} {The participating clusters} \\ \cline{2-4} 
-                                      & Cluster & Site           & No. of  nodes     \\ 
+                                      & Cluster & Site           & Nodes per cluster     \\ 
 \hline
 \multirow{3}{*}{Two sites / 16 nodes} & Taurus & Lyon                & 5                      \\ \cline{2-4} 
                                       & 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
@@ -934,57 +943,70 @@ Table \ref{tab:sc} shows the number of nodes used from each cluster for each sce
  \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}. 
+ with different number of nodes, as in Table~\ref{tab:sc}. 
 The overall energy consumption of all the benchmarks solving the class D instance and
 using the proposed frequency selection algorithm is measured 
-using the equation of the reduced energy consumption, equation 
-(\ref{eq:energy}). This model uses the measured dynamic power showed in Table \ref{table:grid5000} and the static 
-power is assumed to be equal to 20\% of the dynamic power. The execution
+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 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 
-presented in  plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 
+presented in Figures~\ref{fig:eng_sen} and \ref{fig:time_sen} respectively.
 
 For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
 for  16 and 32 nodes is lower than the energy consumed while using two sites. 
 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 their execution times  are 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).  
-
-However, the  execution times and the energy consumptions of 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 other hand, the communications of the rest of the benchmarks increases when using long distance communications between two sites or increasing the number of computing 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 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
+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.
 
 
 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}. 
+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}. 
 This figure shows that the energy saving percentages of one site scenario for
 16 and 32 nodes are bigger than those of the two sites scenario which is due
 to the higher  computations to communications ratio in the first scenario   
-than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which 
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computation times are bigger than the communication times which
 results in  a lower energy consumption. Indeed, the dynamic  consumed power
 is exponentially related to the CPU's frequency value. On the other 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*}
 
 
 
@@ -1001,70 +1023,60 @@ 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\%.
 
-\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 tradeoff 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. 
-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 
+the other benchmarks. Indeed, in the EP benchmark, there are no communication and no 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 tradeoff distance percentage can be 
-computed as in equation \ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
-tradeoff, on average it is equal to  26.8\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
-tradeoff comparing to the two sites scenario  because the former has high speed local communications 
+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 
+computed as in Equation~\ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
+trade-off, on average it is equal to  26.8\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
+trade-off comparing to the two sites scenario  because the former has high speed local communications 
 which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 
 
- Finally, the best energy and performance tradeoff depends on all of the following:
+ Finally, the best energy and performance trade-off depends on all of the following:
 1) the computations to communications ratio when there are  communications and slack times, 2) the heterogeneity of the computing powers of the nodes and 3) the heterogeneity of the consumed  static and dynamic powers of the nodes.
 
 
 
 
-\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 the section \ref{sec.res}.
-The one site scenario uses  32 cores from multi-cores nodes instead of 32 distinct nodes. For example if 
+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-core nodes instead of 32 distinct nodes. For example if 
 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  figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
-
-\begin{table}[]
+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.
+
+
+\begin{table}[!h]
 \centering
-\caption{The multicores scenarios}
+\caption{The multi-core scenarios}
 \begin{tabular}{|*{4}{c|}}
 \hline
-Scenario name                          & Cluster name & \begin{tabular}[c]{@{}c@{}}No. of  nodes\\ in each cluster\end{tabular} & 
-                                       \begin{tabular}[c]{@{}c@{}}No. of  cores\\ for each node\end{tabular}  \\ \hline
+Scenario name                          & Cluster name & Nodes per cluster & 
+                                       Cores per node  \\ \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}
@@ -1072,54 +1084,54 @@ Scenario name                          & Cluster name & \begin{tabular}[c]{@{}c@
 \end{table}
 
 
-\begin{figure}
+\begin{figure}[!h]
   \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}%
-  \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}
-  \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}
 
 
 
-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  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 
- 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 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
-    \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}%
-    \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}%
-    \subfloat[The tradeoff 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-res}
-  \caption{The experimental results of one core and multi-cores scenarios}
+  \label{fig:exp-res2}
+  \caption{The experimental results of one core and multi-core scenarios}
 \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 
-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
-figure \ref{fig:per-d-mc}. It shows that the performance degradation percentages is higher for the NAS benchmarks over the  one core per node scenario  (on average equal to 10.6\%)  than over the  multi-cores scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-cores scenario is lower because  the computations to communications ratio is smaller than the ratio of the other scenario. 
+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 tradeoff distance percentages of the NAS benchmarks over the two scenarios are presented 
-in figure \ref{fig:dist-mc}. These  tradeoff distance between energy consumption reduction and performance  are used to verify which scenario is the best in both terms  at the same time. The figure shows that  the  tradeoff distance percentages are on average   bigger over the multi-cores scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).
+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\%).
 
 
 
@@ -1130,12 +1142,12 @@ in figure \ref{fig:dist-mc}. These  tradeoff distance between energy consumption
 \subsection{Experiments with different static power scenarios}
 \label{sec.pow_sen}
 
-In section \ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 
+In Section~\ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 
 
 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]
@@ -1144,7 +1156,7 @@ In these experiments, class D of the NAS parallel benchmarks are executed over t
     \includegraphics[width=.48\textwidth]{fig/eng_pow.eps}\label{fig:eng-pow}} \hspace{0.4cm}%
   \subfloat[The performance degradation percentages for the NAS benchmarks over the three power scenarios]{%
     \includegraphics[width=.48\textwidth]{fig/per_pow.eps}\label{fig:per-pow}}\hspace{0.4cm}%
-    \subfloat[The tradeoff distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios]{%
+    \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios]{%
       
     \includegraphics[width=.48\textwidth]{fig/dist_pow.eps}\label{fig:dist-pow}}
   \label{fig:exp-pow}
@@ -1161,23 +1173,23 @@ 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 
-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. 
-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 performance degradation percentages are presented in Figure~\ref{fig:per-pow}.
 The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
 had  selected big frequencies for the CPUs. While, 
-the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The tradeoff distance percentage for the NAS benchmarks with these three static power scenarios 
-are presented in figure \ref{fig:dist}. 
-It shows that the best  tradeoff
+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-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.
 
-In the EP benchmark, the energy saving, performance degradation and tradeoff 
+In the EP benchmark, the energy saving, performance degradation and trade-off 
 distance percentages for these static power scenarios are not significantly different because there is no communication in this benchmark. Therefore, the static power is only consumed during computation and   the proposed scaling algorithm selects similar frequencies for the three scenarios.  On the other hand,  for the rest of the benchmarks,  the scaling algorithm  selects  the values of the frequencies according to the communication times of each benchmark because the static energy consumption increases  proportionally to the  communication times.
 
 
@@ -1185,79 +1197,77 @@ distance percentages for these static power scenarios are not significantly diff
 \subsection{Comparison of the proposed frequencies selecting algorithm }
 \label{sec.compare_EDP}
 
-Finding the frequencies that give the best tradeoff between the energy consumption and the performance for a parallel 
+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.
-
-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 tradeoff distance percentages, are 
-presented in the 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]{%
     \includegraphics[width=.48\textwidth]{fig/edp_eng}\label{fig:edp-eng}} \hspace{0.4cm}%
     \subfloat[The performance degradation induced by  the Maxdist method and the EDP method]{%
     \includegraphics[width=.48\textwidth]{fig/edp_per}\label{fig:edp-perf}}\hspace{0.4cm}%
-    \subfloat[The tradeoff distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method]{%
+    \subfloat[The trade-off distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method]{%
     \includegraphics[width=.48\textwidth]{fig/edp_dist}\label{fig:edp-dist}}
   \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. 
-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 tradeoff values for some benchmarks in the two sites scenarios.
-These negative tradeoff values mean that the performance degradation percentage is higher than the energy saving percentage.
-The high positive values of the tradeoff distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
-The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and 
-$O(N \cdot M \cdot F^2)$ respectively, where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the 
-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.  
+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  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}
 \label{sec.concl}
 This paper presents a new online frequencies selection algorithm.
  The algorithm selects the best vector of 
-frequencies that maximizes  the tradeoff distance 
+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 
-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  
- 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. 
-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 tradeoff  between energy consumption reduction and performance. 
+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 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. 
 
 
 
 \section*{Acknowledgment}
 
 This work  has been  partially supported by  the Labex ACTION  project (contract
-``ANR-11-LABX-01-01'').  Computations  have been performed  on the Grid'5000 platform. As  a  PhD student,
+``ANR-11-LABX-01-01'').  Computations  have been performed  on the Grid'5000
+platform and on the mÃ©socentre of Franche-ComtÃ©. As  a  PhD student,
 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.
 
@@ -1266,4 +1276,15 @@ supporting his work.
 
 \end{document}
 
-
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% fill-column: 80
+%%% ispell-local-dictionary: "american"
+%%% End:
+
+%  LocalWords:  DVFS Fanfakh Charr Franche ComtÃ© IUT MarÃ©chal Juin cedex NAS et
+%  LocalWords:  supercomputing Tianhe Shoubu ExaScaler RIKEN GFlops CPUs GPUs
+%  LocalWords:  Luley Xeon NVIDIA GPU Rong Naveen Lizhe al AMD ij hj RENATER
+%  LocalWords:  Infiniband Graphene consumptions versa multi Spiliopoulos Labex
+%  LocalWords:  Maxdist ANR LABX
