some changes

[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 8c6061c8aea45247fd3f4bdff1afaf6521d580e6..0973f102920fffb8b120050e6579f17d31a5ffc5 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -1,11 +1,49 @@
-\documentclass[3p,times]{elsarticle-1}
-\usepackage{ecrc}
+\documentclass[review]{elsarticle}
+
+\usepackage{lineno,hyperref}
+\modulolinenumbers[5]
+
+\journal{Journal of Computational Science}
+
+%%%%%%%%%%%%%%%%%%%%%%%
+%% Elsevier bibliography styles
+%%%%%%%%%%%%%%%%%%%%%%%
+%% To change the style, put a % in front of the second line of the current style and
+%% remove the % from the second line of the style you would like to use.
+%%%%%%%%%%%%%%%%%%%%%%%
+
+%% Numbered
+%\bibliographystyle{model1-num-names}
+
+%% Numbered without titles
+%\bibliographystyle{model1a-num-names}
+
+%% Harvard
+%\bibliographystyle{model2-names.bst}\biboptions{authoryear}
+
+%% Vancouver numbered
+%\usepackage{numcompress}\bibliographystyle{model3-num-names}
+
+%% Vancouver name/year
+%\usepackage{numcompress}\bibliographystyle{model4-names}\biboptions{authoryear}
+
+%% APA style
+%\bibliographystyle{model5-names}\biboptions{authoryear}
+
+%% AMA style
+%\usepackage{numcompress}\bibliographystyle{model6-num-names}
+
+%% `Elsevier LaTeX' style
+\bibliographystyle{elsarticle-num}
+%%%%%%%%%%%%%%%%%%%%%%%
+

 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 \usepackage[english]{babel}
 \usepackage{algpseudocode}
 \usepackage{graphicx}
 \usepackage{algorithm}
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 \usepackage[english]{babel}
 \usepackage{algpseudocode}
 \usepackage{graphicx}
 \usepackage{algorithm}

+\usepackage{setspace}

 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{url}
 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{url}


@@ -61,122 +99,20 @@
 \newcommand{\Told}{\Xsub{T}{Old}}
 
 
 \newcommand{\Told}{\Xsub{T}{Old}}
 
 

-%% The ecrc package defines commands needed for running heads and logos.
-%% For running heads, you can set the journal name, the volume, the starting page and the authors
-
-%% set the volume if you know. Otherwise `00'
-\volume{00}
-
-%% set the starting page if not 1
-\firstpage{1}
-
-%% Give the name of the journal
-\journalname{Procedia Computer Science}
-
-%% Give the author list to appear in the running head
-%% Example \runauth{C.V. Radhakrishnan et al.}
-\runauth{}
-
-%% The choice of journal logo is determined by the \jid and \jnltitlelogo commands.
-%% A user-supplied logo with the name <\jid>logo.pdf will be inserted if present.
-%% e.g. if \jid{yspmi} the system will look for a file yspmilogo.pdf
-%% Otherwise the content of \jnltitlelogo will be set between horizontal lines as a default logo
-
-%% Give the abbreviation of the Journal.
-\jid{procs}
-
-%% Give a short journal name for the dummy logo (if needed)
-\jnltitlelogo{Procedia Computer Science}
-
-%% Hereafter the template follows `elsarticle'.
-%% For more details see the existing template files elsarticle-template-harv.tex and elsarticle-template-num.tex.
-
-%% Elsevier CRC generally uses a numbered reference style
-%% For this, the conventions of elsarticle-template-num.tex should be followed (included below)
-%% If using BibTeX, use the style file elsarticle-num.bst
-
-%% End of ecrc-specific commands
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%% The amssymb package provides various useful mathematical symbols
-\usepackage{amssymb}
-%% The amsthm package provides extended theorem environments
-%% \usepackage{amsthm}
-
-%% The lineno packages adds line numbers. Start line numbering with
-%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
-%% for the whole article with \linenumbers after \end{frontmatter}.
-%% \usepackage{lineno}
-
-%% natbib.sty is loaded by default. However, natbib options can be
-%% provided with \biboptions{...} command. Following options are
-%% valid:
-
-%%   round  -  round parentheses are used (default)
-%%   square -  square brackets are used   [option]
-%%   curly  -  curly braces are used      {option}
-%%   angle  -  angle brackets are used    <option>
-%%   semicolon  -  multiple citations separated by semi-colon
-%%   colon  - same as semicolon, an earlier confusion
-%%   comma  -  separated by comma
-%%   numbers-  selects numerical citations
-%%   super  -  numerical citations as superscripts
-%%   sort   -  sorts multiple citations according to order in ref. list
-%%   sort&compress   -  like sort, but also compresses numerical citations
-%%   compress - compresses without sorting
-%%
-%% \biboptions{comma,round}
-
-% \biboptions{}
-
-% if you have landscape tables
-\usepackage[figuresright]{rotating}

 
 

-% put your own definitions here:
-%   \newcommand{\cZ}{\cal{Z}}
-%   \newtheorem{def}{Definition}[section]
-%   ...
-
-% add words to TeX's hyphenation exception list
-%\hyphenation{author another created financial paper re-commend-ed Post-Script}
-
-% declarations for front matter

 
 \begin{document}
 
 \begin{frontmatter}
 
 
 \begin{document}
 
 \begin{frontmatter}
 

-%% Title, authors and addresses
-
-%% use the tnoteref command within \title for footnotes;
-%% use the tnotetext command for the associated footnote;
-%% use the fnref command within \author or \address for footnotes;
-%% use the fntext command for the associated footnote;
-%% use the corref command within \author for corresponding author footnotes;
-%% use the cortext command for the associated footnote;
-%% use the ead command for the email address,
-%% and the form \ead[url] for the home page:
-%%
-%% \title{Title\tnoteref{label1}}
-%% \tnotetext[label1]{}
-%% \author{Name\corref{cor1}\fnref{label2}}
-%% \ead{email address}
-%% \ead[url]{home page}
-%% \fntext[label2]{}
-%% \cortext[cor1]{}
-%% \address{Address\fnref{label3}}
-%% \fntext[label3]{}
-
-\dochead{}
-%% Use \dochead if there is an article header, e.g. \dochead{Short communication}
-\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} 

   
 
   
 

-%% use optional labels to link authors explicitly to addresses:
-%% \author[label1,label2]{<author name>}
-%% \address[label1]{<address>}
-%% \address[label2]{<address>}
+

 
 \author{Ahmed Fanfakh,
         Jean-Claude Charr,
 
 \author{Ahmed Fanfakh,
         Jean-Claude Charr,


@@ -192,7 +128,8 @@
    }
 
 \begin{abstract}
    }
 
 \begin{abstract}

-  In recent years, green computing topic  has  become an important topic 
+
+  In recent years, green computing   has  become an important topic 

   in the supercomputing research domain. However, the 
   computing platforms are still  consuming more and
 more energy due to the increasing number of nodes composing
   in the supercomputing research domain. However, the 
   computing platforms are still  consuming more and
 more energy due to the increasing number of nodes composing


@@ -206,22 +143,22 @@ 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. 
-  The proposed algorithm is evaluated on a real grid, the grid'5000 platform, while
-  running the NAS parallel benchmarks.  The experiments show that it reduces the
-  energy consumption on average by \np[\%]{30} while  the performance  is only degraded
-  on average by \np[\%]{3}. Finally, the algorithm is 
+  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 

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

-\textcolor{blue}{
-DVFS \sep heterogeneous grid \sep energy consumption  \sep performance prediction  \sep energy and performance trade-off  \sep frequencies selecting algorithm }
+
+Dynamic voltage and frequency scaling \sep Grid computing\sep Green computing and  frequency scaling online algorithm.

 
 %% keywords here, in the form: keyword \sep keyword
 
 
 %% keywords here, in the form: keyword \sep keyword
 


@@ -251,30 +188,34 @@ 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}.
 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
 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 frequencies 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 
-Simgrid simulator \cite{SimGrid}, which offers easy tools to create a homogeneous and heterogeneous platforms and run message passing parallel applications over them. In this paper, a new frequencies selecting algorithm,
-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. 
+
+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
+ application with iterations 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.

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


@@ -286,9 +227,8 @@ 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 
 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 shows the results of running different scenarios using multi-cores and one core per node and comparing them. 
-It also evaluates the algorithm over three different power scenarios. Moreover, it shows the
+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.
 
 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.
 


@@ -313,10 +253,10 @@ 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}


@@ -331,7 +271,7 @@ 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
 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
 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


@@ -363,16 +303,16 @@ 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}
 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
+\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 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
   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
+  training nor profiling. It uses a new optimization function which

   simultaneously maximizes the performance and minimizes the energy consumption
   simultaneously maximizes the performance and minimizes the energy consumption

-  of a message passing iterative synchronous application.
+  of a message passing  synchronous application with iterations.

 
 \end{enumerate}
 
 
 \end{enumerate}
 


@@ -381,15 +321,27 @@ following contributions :
 \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  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}
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/commtasks}


@@ -397,15 +349,6 @@ and higher latency than the local networks of the clusters. Each computing clust
   \label{fig:heter}
 \end{figure}
 
   \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
 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


@@ -418,6 +361,8 @@ 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


@@ -432,34 +377,35 @@ 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_i}}  (\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 
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 

-first  iteration. The model computes the maximum computation time with scaling factor 
-from each node added to the communication time of the slowest node in the slowest cluster $h$.
-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 $h$ 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
 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.
+~\cite{Our_first_paper,pdsec2015}. \textcolor{blue}{where the homogeneous cluster predication model was used one scaling factor denoted as $S$, because all the nodes in the cluster have the same computing powers. Whereas, in heterogeneous cluster prediction model all the nodes have different scales and the scaling factors have denoted as one dimensional vector $(S_1, S_2, \dots, S_N)$. The execution time prediction model for a grid Equation \ref{eq:perf} defines a two dimensional array of scales  
+$(S_{11}, S_{12},\dots, S_{NM_i})$}. This 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.

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


@@ -496,7 +442,7 @@ 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
 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
 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


@@ -510,11 +456,11 @@ follows:
 \end{equation}
 Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
 equation for dynamic power consumption:
 \end{equation}
 Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
 equation for dynamic power consumption:

-\begin{equation}
+\begin{multline}

   \label{eq:pdnew}
   \label{eq:pdnew}

-   \PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 
-    = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}
-\end{equation}
+   \PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 \\
+   {} = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}
+\end{multline}

 where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
 new frequency and the maximum frequency respectively.
 
 where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
 new frequency and the maximum frequency respectively.
 


@@ -530,7 +476,7 @@ 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


@@ -546,33 +492,36 @@ 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.  \textcolor{blue}{Therefore, even if the distributed 
+message passing  application \textcolor{blue}{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. 
+\textcolor{blue}{ The CPU during the communication times consumes only the static power. While 
+in the computation times, it consumes both the dynamic and the static power 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:
-\begin{equation}
+static energies for $M_i$ processors in $N$ clusters.  It is computed as follows:
+\begin{multline}

   \label{eq:energy}
   \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]) ))
-\end{equation}
+ 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
 
 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 the  application \textcolor{blue}{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.
 


@@ -595,12 +544,13 @@ 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. 
 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
 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 \textcolor{blue}{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


@@ -612,40 +562,43 @@ 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:
 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}
 \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}).
+\textcolor{blue}{

 \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_i}}  (\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:

+%

 \begin{equation}
   \label{eq:enorm}
   \Enorm = \frac{\Ereduced}{\Eoriginal} 
 \end{equation}
 \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}).
 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
 time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. 
 \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 $S_1,S_2,\dots,S_N$ 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
 maximum energy reduction with minimum execution time reduction.
 
 This problem can be solved by making the optimization process for energy and


@@ -661,9 +614,9 @@ equation, as follows:
 \begin{figure}
   \centering
   \subfloat[Homogeneous cluster]{%
 \begin{figure}
   \centering
   \subfloat[Homogeneous cluster]{%

-    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}} \hspace{2cm}%
+    \includegraphics[width=.48\textwidth]{fig/homo}\label{fig:r1}} \hspace{0.4cm}%

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

-    \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
+    \includegraphics[width=.48\textwidth]{fig/heter}\label{fig:r2}}

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


@@ -672,17 +625,17 @@ 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
 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 =
 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}
       (\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 


@@ -693,12 +646,14 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 \label{sec.optim}
 
 \begin{algorithm}
 \label{sec.optim}
 
 \begin{algorithm}

+\setstretch{1}

   \begin{algorithmic}[1]
     % \footnotesize
   \begin{algorithmic}[1]
     % \footnotesize

+    

     \Require ~
     \begin{description}
     \Require ~
     \begin{description}

-    \item [{$N$}] number of clusters in the grid.
-    \item [{$M$}] number of nodes in each cluster.
+    \item [{$N$}] number of clusters in the grid. 
+    \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.


@@ -706,7 +661,7 @@ 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}
     \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 $F_{ij} \gets  \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
 
     \State $\Scp[ij] \gets \frac{\max_{i=1,2,\dots,N}(\max_{j=1,2,\dots,M_i}(\Tcp[ij]))}{\Tcp[ij]} $
     \State $F_{ij} \gets  \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$


@@ -714,8 +669,8 @@ 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
     \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 $)}
     \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 $)}


@@ -723,10 +678,9 @@ 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 $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
         \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 $\Pnorm \gets \frac{\Told}{\Tnew}$
-       \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
+       \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. $
         \State $\Dist \gets \Pnorm - \Enorm$
       \If{$(\Pnorm - \Enorm > \Dist)$}
         \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
         \State $\Dist \gets \Pnorm - \Enorm$


@@ -745,8 +699,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}
       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}

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

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


@@ -759,12 +712,12 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 \end{algorithm}
 
 
 \end{algorithm}
 
 

-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
 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing
 is presented. It selects the vector of the 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 \textcolor{blue}{with iterations} executed on a  grid. It works
+online during the execution time of the  message passing program \textcolor{blue}{with iterations}.  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


@@ -772,20 +725,18 @@ 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  MPI program \textcolor{blue}{with iterations}.

 
 \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


@@ -796,15 +747,15 @@ 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_{i=1,\dots N}}_{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


@@ -826,11 +777,11 @@ 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
 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
 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
 
 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


@@ -840,35 +791,32 @@ 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
 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}, 
 
 
 \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 architature and power consumption}
+\subsection{Grid'5000 architecture and power consumption}

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

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

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

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

 computing clusters and each cluster contains many homogeneous nodes. In total,
 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.  
 
 the clusters and their nodes are connected via  high speed local area networks. 
 Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
 

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

 
 
 
 

-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])
 \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])


@@ -880,22 +828,22 @@ $\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle po
 Therefore, the dynamic power of one core is computed as the difference between the maximum 
 measured value in maximum powers vector and the minimum measured value in the idle powers vector.
 
 Therefore, the dynamic power of one core is computed as the difference between the maximum 
 measured value in maximum powers vector and the minimum measured value in the idle powers vector.
 

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

 
 

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

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

 Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
 Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 

-frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 
-the details characteristics of these four clusters. Moreover, the dynamic powers were computed  using the equation (\ref{eq:pdyn}) for all the nodes in the 
-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}
 
 
 \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]
   \label{fig:grid5000}
 \end{figure}
 \begin{figure}[!t]


@@ -906,37 +854,40 @@ selected clusters and are presented in table  \ref{table:grid5000}.
 \end{figure}
 
 
 \end{figure}
 
 

-The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
-The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
-The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 
+The 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. \textcolor{blue}{These benchmarks are message passing  applications with iterations compute
+the same block of operations several times, starting from the initial solution until reaching
+the acceptable approximation of the exact solution.}
+ 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|}}
     \hline
   % title of Table
   \centering
   \begin{tabular}{|*{7}{c|}}
     \hline

-    Cluster     & CPU         & Max   & Min   & Diff. & no. of cores    & dynamic power   \\
-    Name        & model       & Freq. & Freq. & Freq. & per CPU         & of one core     \\
-                &             & GHz   & GHz   & GHz   &                 &           \\
+                &             & Max   & Min   & Diff. &                 &               \\
+    Cluster     & CPU         & Freq. & Freq. & Freq. & Cores           & Dynamic power \\
+    Name        & model       & GHz   & GHz   & GHz   & per CPU         & of one core   \\

     \hline
     \hline

-    Taurus      & Intel       & 2.3  & 1.2  & 0.1     & 6               & \np[W]{35} \\
-                & Xeon        &       &       &       &                 &            \\
-                & E5-2630     &       &       &       &                 &            \\         
+                & Intel       &       &       &         &           &              \\
+    Taurus      & Xeon        & 2.3   & 1.2   & 0.1     & 6         & \np[W]{35}    \\
+                & E5-2630     &       &       &         &           &            \\         

     \hline
     \hline

-    Graphene    & Intel       & 2.53  & 1.2   & 0.133 & 4               & \np[W]{23} \\
-                & Xeon        &       &       &       &                 &            \\
-                & X3440       &       &       &       &                 &            \\    
+                & Intel       &       &       &         &           &             \\
+    Graphene    & Xeon        & 2.53  & 1.2   & 0.133   & 4         & \np[W]{23}  \\
+                & X3440       &       &       &         &           &             \\    

     \hline
     \hline

-    Griffon     & Intel       & 2.5   & 2     & 0.5   & 4               & \np[W]{46} \\
-                & Xeon        &       &       &       &                 &            \\
-                & L5420       &       &       &       &                 &            \\  
+                & Intel       &       &       &         &           &            \\
+    Griffon     & Xeon        & 2.5   & 2     & 0.5     & 4         & \np[W]{46}  \\
+                & L5420       &       &       &         &           &            \\  

     \hline
     \hline

-    Graphite    & Intel       & 2     & 1.2   & 0.1   & 8               & \np[W]{35} \\
-                & Xeon        &       &       &       &                 &            \\
-                & E5-2650     &       &       &       &                 &            \\  
+                & Intel       &       &       &         &           &            \\
+     Graphite   & Xeon        & 2     & 1.2   & 0.1     & 8         & \np[W]{35} \\
+                & E5-2650     &       &       &         &           &            \\  

     \hline
   \end{tabular}
   \label{table:grid5000}
     \hline
   \end{tabular}
   \label{table:grid5000}


@@ -947,41 +898,41 @@ 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. 
+to the NAS parallel benchmarks are presented. \textcolor{blue}{Each experiment of this section and next sections has been executed many times and the results presented in the figures are the average values of many execution.}

 
 As mentioned previously, the experiments 
 
 As mentioned previously, the experiments 

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

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

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

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

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

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

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

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

-16 and 32 nodes for each scenario. The number of participating computing nodes form each cluster 
-are different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
-Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
+16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
+is different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
+Table~\ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 

 
 \begin{table}[h]
 
 
 \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} 
 \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
 \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
                                       & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                       & Griffon     &Nancy           & 12                     \\ 
 \hline


@@ -997,75 +948,79 @@ 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    
            ]{%
 \begin{figure}
   \centering
   \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios    
            ]{%

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

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

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

   \label{fig:exp-time-energy}
   \caption{The  energy consumption and execution time of NAS  Benchmarks over different scenarios}
 \end{figure}
 
 The NAS parallel benchmarks are executed over these two platforms
   \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 
 The overall energy consumption of all the benchmarks solving the class D instance and
 using the proposed frequency selection algorithm is measured 

-using the equation of the reduced energy consumption, equation 
-(\ref{eq:energy}). This model uses the measured dynamic and static 
-power values  showed in Table \ref{table:grid5000}. The execution
+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 \textcolor{blue}{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 

-presented in the 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 
 
 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).  
+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 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.
+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.

 
 
 
 

-\begin{figure}
+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}. 
+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 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 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 ]{%
   \centering
   \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%

-    \includegraphics[width=.33\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{0.08cm}%
+    \includegraphics[width=.48\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{0.4cm}%

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

-    \includegraphics[width=.33\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.08cm}%
-    \subfloat[The tradeoff distance between the energy reduction and the performance of the NAS benchmarks  
+    \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]{%
       over different scenarios]{%

-    \includegraphics[width=.33\textwidth]{fig/dist.eps}\label{fig:dist}}
+    \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}

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

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

 
 

-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}. 
-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 
-results in  a lower energy consumption. Indeed, the dynamic  consumed power
-is exponentially related to the CPU's frequency value. On the other side, the increase in the number of computing nodes can 
-increase the communication times and thus produces less energy saving depending on the 
-benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more 
-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. 

 
 
 The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 
 
 
 The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 

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

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

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

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


@@ -1076,56 +1031,57 @@ The best energy saving percentage was obtained in the one site scenario with 16
 
 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
 
 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\% or 4\% 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\% or 10\% respectively. In opposition to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
+16 or 32  nodes is on average equal to 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 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.
 
 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\%. 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. 
 
 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.
 
 
 
 
 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 of multi-cores clusters}
+\subsection{The experimental results over multi-core clusters}

 \label{sec.res-mc}
 \label{sec.res-mc}

-\textcolor{blue}{
-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 grid while using multi-core 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, 
 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 the NAS parallel 
-benchmarks, class D, over these four different scenarios are presented 
-in the 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
 \centering

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

 \begin{tabular}{|*{4}{c|}}
 \hline
 \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
-\multirow{3}{*}{One site/ one core}    & Graphite     & 4               & 1                   \\  \cline{2-4}
+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
                                        & Graphene     & 14              & 1                   \\  \cline{2-4}
                                        & Griffon      & 14              & 1                   \\ \hline

-\multirow{3}{*}{One site/ multicores}  & 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}


@@ -1133,86 +1089,84 @@ Scenario name                          & Cluster name & \begin{tabular}[c]{@{}c@
 \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]{%
-    \includegraphics[width=.4\textwidth]{fig/time.eps}\label{fig:time-mc}} \hspace{1cm}%
-  \subfloat[Comparing the  energy consumptions of running NAS benchmarks over one core and multi-cores scenarios]{%
-    \includegraphics[width=.4\textwidth]{fig/eng_con.eps}\label{fig:eng-cons-mc}}
+  \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 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}
     \label{fig:eng-cons}

-  \caption{The energy consumptions and execution times of NAS benchmarks over one core and multi-cores scenarios}
+  \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}
 
 

-\textcolor{blue}{
-The execution times for most of  the NAS  benchmarks are higher over the one site multi-cores per node scenario 
-than the execution time of those running over one site 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 experiments showed that for most of the NAS benchmarks,      
-the one site one core scenario gives the best execution times because the communication times are the lowest. 
-Indeed, in this scenario each core has a dedicated network link and memory bus.  
+
+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 site one core scenario  than over the one site 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. 
-More energy reduction was achieved when this ratio is increased because the proposed scaling algorithm selects smaller frequencies that decrease the dynamic power consumption. 
-These experiments also showed that the energy 
+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
 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,  
-which could increase or decrease the static power consumptions. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
-The energy saving percentages of all NAS benchmarks running over these two scenarios are presented in the figure \ref{fig:eng-s-mc}. It shows that  the energy saving percentages in the one site one 
-core and one site multi-cores scenarios
-are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively. In both scenarios there 
-are a small difference  in the computations to communications ratios, which leads 
-the proposed scaling algorithm to select similar frequencies for both scenarios.  
+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 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 multi-core scenarios
+      ]{%
+    \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 multicore scenarios]{%
+    \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-core scenarios}
+\end{figure*}  
+  
+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-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
 The performance degradation percentages of the NAS benchmarks are presented in

-figure \ref{fig:per-d-mc}. It shows that the performance degradation percentages for the NAS benchmarks over one site one core is on average equal to 10.6\% and is higher than these executed over the one site multi-cores scenario, which is on average equal to 7.5\%. 
-The performance degradation percentages over one site multi-cores is lower because  the computations to communications ratio is decreased. Therefore, selecting big
-frequencies by the scaling algorithm are proportional to this ratio, and thus the execution time do not increase significantly.
-The tradeoff distance percentages of the NAS benchmarks over the two scenarios are presented 
-in the figure \ref{fig:dist-mc}. 
-These  tradeoff distance percentages are used to verify which scenario is the best in terms of energy reduction and performance. The figure shows that using muti-cores  scenario gives bigger tradeoff distance percentages, on overage equal to 17.6\%  than using one core per node scenario,  on average  equal to 15.3\%.}
+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 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\%).

 
 
 
 
 
 

-\begin{figure}
-  \centering
-    \subfloat[The energy saving of running NAS benchmarks over one core and multicores scenarios]{%
-    \includegraphics[width=.33\textwidth]{fig/eng_s_mc.eps}\label{fig:eng-s-mc}} \hspace{0.08cm}%
-    \subfloat[The performance degradation of running NAS benchmarks over one core and multicores scenarios
-      ]{%
-    \includegraphics[width=.33\textwidth]{fig/per_d_mc.eps}\label{fig:per-d-mc}}\hspace{0.08cm}%
-    \subfloat[The tradeoff distance of running NAS benchmarks over one core and multicores scenarios]{%
-    \includegraphics[width=.33\textwidth]{fig/dist_mc.eps}\label{fig:dist-mc}}
-  \label{fig:exp-res}
-  \caption{The experimental results of one core and multi-cores scenarios}
-\end{figure}

 
 
 
 
 
 

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

 \label{sec.pow_sen}
 
 \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.
 
 The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
 In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.
 The experiments have been executed with these two new static power scenarios  over the one site one core per node scenario.

-In these experiments, the class D of the NAS parallel benchmarks are executed over Nancy's site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, where used in this experiment. 
+In these experiments, 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}
+\begin{figure*}[t]

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

-    \includegraphics[width=.33\textwidth]{fig/eng_pow.eps}\label{fig:eng-pow}} \hspace{0.08cm}%
+    \includegraphics[width=.48\textwidth]{fig/eng_pow.eps}\label{fig:eng-pow}} \hspace{0.4cm}%

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

-    \includegraphics[width=.33\textwidth]{fig/per_pow.eps}\label{fig:per-pow}}\hspace{0.08cm}%
-    \subfloat[The tradeoff distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios]{%
-    \includegraphics[width=.33\textwidth]{fig/dist_pow.eps}\label{fig:dist-pow}}
+    \includegraphics[width=.48\textwidth]{fig/per_pow.eps}\label{fig:per-pow}}\hspace{0.4cm}%
+    \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}
   \caption{The experimental results of different static power scenarios}
   \label{fig:exp-pow}
   \caption{The experimental results of different static power scenarios}

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

 
 
 
 
 
 


@@ -1224,95 +1178,92 @@ In these experiments, the class D of the NAS parallel benchmarks are executed ov
 \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 the figure \ref{fig:per-pow}.
+The performance degradation percentages are presented in Figure~\ref{fig:per-pow}.

 The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
 had  selected big frequencies for the CPUs. While, 
 The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
 had  selected big frequencies for the CPUs. While, 

-the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The tradeoff distance percentage for the NAS benchmarks with these three static power scenarios 
-are presented in the figure \ref{fig:dist}. 
-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.
 
 distance percentage is obtained with  the  10\% static power scenario  and this percentage 
 is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
 

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

 
 
  
 
 
  

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

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

-Finding the frequencies that gives the best tradeoff between the energy consumption and the performance for a parallel 
+Finding the frequencies that give the best 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}. 
 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 the 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}
+\begin{figure*}[t]

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

-    \includegraphics[width=.33\textwidth]{fig/edp_eng}\label{fig:edp-eng}} \hspace{0.08cm}%
+    \includegraphics[width=.48\textwidth]{fig/edp_eng}\label{fig:edp-eng}} \hspace{0.4cm}%

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

-    \includegraphics[width=.33\textwidth]{fig/edp_per}\label{fig:edp-perf}}\hspace{0.08cm}%
-    \subfloat[The tradeoff distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method]{%
-    \includegraphics[width=.33\textwidth]{fig/edp_dist}\label{fig:edp-dist}}
+    \includegraphics[width=.48\textwidth]{fig/edp_per}\label{fig:edp-perf}}\hspace{0.4cm}%
+    \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}
   \label{fig:edp-comparison}
   \caption{The comparison results}

-\end{figure}
+\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.
 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.

-Whereas, the EDP algorithm gives sometimes negative tradeoff values for some benchmarks in the two sites scenarios.
-These negative tradeoff values mean that the performance degradation percentage is higher than energy saving percentage.
-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}
 
 
 \section{Conclusion}
 \label{sec.concl}

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

  The algorithm selects the best vector of 
  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 
 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 \textcolor{blue}{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 distributed  message passing application \textcolor{blue}{with iterations} running over a grid architecture.

 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 experimental results showed that the algorithm reduces  on average 30\% of the energy consumption
-for all the NAS benchmarks   while  only degrading by 3\% on average  the performance. 
-The Maxdist algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or \textcolor{blue}{between using one core and multi-cores per node} or in the values of the consumed static power. The algorithm selects different vector of frequencies according to the 
+NAS parallel benchmarks   and the  class D instance was executed over the  Grid'5000 testbed platform. 
+The experiments 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-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 
 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. 
- The development of
-such a method might require a new energy model because the
+In the near future, \textcolor{blue}{we will adapt the proposed algorithm to take the variability between some iterations in two steps. In the first step,  the algorithm selects the best frequencies at the end of the first iterations and apply them to the system. In the second step, after some iterations (e.g. 5 iterations) the algorithm recomputes the frequencies depending on the average of the communication and computation times  for all  previous iterations. It will change the frequency of each node if the new frequency is different from the old one. Otherwise, it keeps the old frequency.} 
+Also, we would like to develop a similar method that is adapted to
+asynchronous  applications \textcolor{blue}{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.
 
 number of iterations is not known in advance and depends on
 the global convergence of the iterative system.
 


@@ -1325,10 +1276,20 @@ This work  has been  partially supported by  the Labex ACTION  project (contract
 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.
 
 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.
 

-
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+%\section*{References}

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