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 b2c2c21e7a3a5dea5693aac7e66121ceb2fe2195..329d52665d9659a659f1ff1ccc978bbe67aabd6e 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -1,4 +1,41 @@
-\documentclass[conference]{IEEEtran}
+\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[T1]{fontenc}
 \usepackage[utf8]{inputenc}


@@ -6,6 +43,7 @@
 \usepackage{algpseudocode}
 \usepackage{graphicx}
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 \usepackage{graphicx}
 \usepackage{algorithm}

+\usepackage{setspace}

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


@@ -60,57 +98,246 @@
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Told}{\Xsub{T}{Old}}
 
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Told}{\Xsub{T}{Old}}
 

+
+
+

 \begin{document}
 
 \begin{document}
 

-\title{Energy Consumption Reduction with DVFS for \\
-  Message Passing Iterative Applications on \\
-  Heterogeneous Architectures}
-
-\author{%
-  \IEEEauthorblockN{%
-    Jean-Claude Charr,
-    Raphaël Couturier,
-    Ahmed Fanfakh and
-    Arnaud Giersch
-  }
-  \IEEEauthorblockA{%
-    FEMTO-ST Institute, University of Franche-Comté\\
+\begin{frontmatter}
+
+
+
+\title{Optimizing the Energy Consumption \\ 
+of Message Passing Applications with Iterations \\ 
+Executed over Grids} 
+  
+
+
+
+\author{Ahmed Fanfakh,
+        Jean-Claude Charr,
+        Raphaël Couturier,
+        and Arnaud Giersch}
+
+\address{FEMTO-ST Institute, University of Franche-Comté\\

     IUT de Belfort-Montbéliard,
     19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël
     % Fax: \mbox{+33 3 84 58 77 81}\\      % Dept Info
     IUT de Belfort-Montbéliard,
     19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël
     % Fax: \mbox{+33 3 84 58 77 81}\\      % Dept Info

-    Email: \email{{jean-claude.charr,raphael.couturier,ahmed.fanfakh_badri_muslim,arnaud.giersch}@univ-fcomte.fr}
+    Email: \email{{ahmed.fanfakh_badri_muslim,jean-claude.charr,raphael.couturier,arnaud.giersch}@univ-fcomte.fr}

    }
    }

-  }
-
-\maketitle

 
 \begin{abstract}
 
 \begin{abstract}

-  

 
 

+  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
+them. To minimize the operating costs of these platforms many
+techniques have been used. Dynamic voltage and frequency
+scaling (DVFS) is one of them. It can be used to reduce the power consumption of the CPU 
+  while computing, by lowering its frequency. However, lowering the frequency of 
+  a CPU may increase the execution time of an application running on that 
+  processor. Therefore, the frequency that gives the best trade-off between 
+  the energy consumption and the performance of an application must be selected. 
+  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  application with iterations. 
+  The algorithm has a small
+  overhead and works without training or profiling. It uses a new energy model
+  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}
 
 \end{abstract}
 

-\section{Introduction}
-\label{sec.intro}

 
 

+\begin{keyword}
+
+Dynamic voltage and frequency scaling \sep Grid computing\sep Green computing and  frequency scaling online algorithm.

 
 

+%% keywords here, in the form: keyword \sep keyword
+
+%% MSC codes here, in the form: \MSC code \sep code
+%% or \MSC[2008] code \sep code (2000 is the default)
+
+\end{keyword}
+
+\end{frontmatter}
+
+
+
+\section{Introduction}
+\label{sec.intro}
+The need for more computing power is continually increasing. To partially
+satisfy this need, most supercomputers constructors just put more computing
+nodes in their platform. The resulting platforms may achieve higher floating
+point operations per second (FLOPS), but the energy consumption and the heat
+dissipation are also increased.  As an example, the Chinese supercomputer
+Tianhe-2 had the highest FLOPS in June 2015  according to the Top500 list
+\cite{TOP500_Supercomputers_Sites}.  However, it was also the most power hungry
+platform with its over 3 million cores consuming around 17.8 megawatts.
+Moreover, according to the U.S.  annual energy outlook 2015
+\cite{U.S_Annual.Energy.Outlook.2015}, the price of energy for 1 megawatt-hour
+was approximately equal to \$70.  Therefore, the price of the energy consumed by
+the Tianhe-2 platform is approximately more than \$10 million each year.  The
+computing platforms must be more energy efficient and offer the highest number
+of FLOPS per watt possible, such as the Shoubu-ExaScaler from RIKEN
+which became the top of the Green500 list in June 2015 \cite{Green500_List}.
+This heterogeneous platform executes more than 7 GFlops per watt while consuming
+50.32 kilowatts.
+
+Besides platform improvements, there are many software and hardware techniques
+to lower the energy consumption of these platforms, such as DVFS, scheduling and other techniques.
+ DVFS is a widely used process to reduce the energy consumption of a
+processor by lowering its frequency
+\cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
+the number of FLOPS executed by the processor which may increase the execution
+time of the application running over that processor.  Therefore, researchers use
+different optimization strategies to select the frequency that gives the best
+trade-off between the energy reduction and performance degradation ratio. In
+\cite{Our_first_paper} and \cite{pdsec2015}, a frequency selecting algorithm
+was proposed to reduce the energy consumption of message passing 
+applications with iterations running over homogeneous and heterogeneous clusters respectively.
+The results of the experiments showed significant energy consumption
+reductions. All the experimental results were conducted over the SimGrid
+simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous  platforms, and to simulate the execution of message passing parallel
+applications over them. 
+
+
+This paper presents the following contributions :
+\begin{enumerate}
+\item two new energy and performance models for message passing 
+  synchronous applications with iterations running over a heterogeneous grid platform. Both models
+  take into account communications and slack times. The models can predict the
+  required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous grid
+  platforms. The algorithm has a very small overhead and does not need any
+  training nor profiling. It uses a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing  synchronous application with iterations.  The algorithm  was applied to the NAS
+parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
+\cite{grid5000}.
+
+\end{enumerate}
+
+
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents some
+related works from other authors.  Section~\ref{sec.exe} describes how the
+execution time of message passing programs can be predicted.  It also presents
+an energy model that predicts the energy consumption of an application running
+over a grid platform. Section~\ref{sec.compet} presents the
+energy-performance objective function that maximizes the reduction of energy
+consumption while minimizing the degradation of the program's performance.
+Section~\ref{sec.optim} details the proposed frequencies selecting algorithm.
+Section~\ref{sec.expe} presents the results of applying the algorithm on the 
+NAS parallel benchmarks and executing them on the Grid'5000 testbed. 
+It also evaluates the algorithm over multi-core per node architectures and over three different power scenarios. Moreover, it shows the
+comparison results between the proposed method and an existing method.  Finally,
+in Section~\ref{sec.concl} the paper ends with a summary and some future works.

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

+DVFS is a technique used in modern processors to scale down both the voltage and
+the frequency of the CPU while computing, in order to reduce the energy
+consumption of the processor. DVFS is also allowed in GPUs to achieve the same
+goal. Reducing the frequency of a processor lowers its number of FLOPS and may
+degrade the performance of the application running on that processor, especially
+if it is compute bound. Therefore selecting the appropriate frequency for a
+processor to satisfy some objectives, while taking into account all the
+constraints, is not a trivial operation.  Many researchers used different
+strategies to tackle this problem. Some of them developed online methods that
+compute the new frequency while executing the application, such
+as~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}.
+Others used offline methods that may need to run the application and profile
+it before selecting the new frequency, such
+as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
+The methods could be heuristics, exact or brute force methods that satisfy
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+sequential, parallel or distributed architecture, homogeneous or heterogeneous
+platform, synchronous or asynchronous application.
+
+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}
+\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
+\item the platform is only composed of heterogeneous CPUs.
+\end{itemize}

 
 

-\section{The performance and energy consumption measurements on heterogeneous architecture}
+For the first type of platform, the computing intensive parallel tasks are
+executed on the GPUs and the rest are executed on the CPUs.  Luley et
+al.~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a
+heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
+goal was to maximize the energy efficiency of the platform during computation by
+maximizing the number of FLOPS per watt generated.
+In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
+al. developed a scheduling algorithm that distributes 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
+DVFS gave better energy and performance efficiency than other clusters only
+composed of CPUs.
+
+The work presented in this paper concerns the second type of platform, with
+heterogeneous CPUs.  Many methods were conceived to reduce the energy
+consumption of this type of platform.  Naveen et
+al.~\cite{Naveen_Power.Efficient.Resource.Scaling} developed a method that
+minimizes the value of $\mathit{energy}\times \mathit{delay}^2$ (the delay is
+the sum of slack times that happen during synchronous communications) by
+dynamically assigning new frequencies to the CPUs of the heterogeneous cluster.
+Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed an
+algorithm that divides the executed tasks into two types: the critical and non
+critical tasks. The algorithm scales down the frequency of non critical tasks
+proportionally to their slack and communication times while limiting the
+performance degradation percentage to less than \np[\%]{10}.
+In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
+heterogeneous cluster composed of two types of Intel and AMD processors. They
+use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
+\cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
+frequencies for a specified heterogeneous cluster are selected offline using
+some heuristic.  Chen et
+al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
+programming approach to minimize the power consumption of heterogeneous servers
+while respecting given time constraints.  This approach had considerable
+overhead.
+
+
+
+\section{The performance and energy consumption measurements on heterogeneous grid architecture}

 \label{sec.exe}
 
 \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}


@@ -118,15 +345,6 @@ 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


@@ -139,6 +357,8 @@ as in (\ref{eq:s}).
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}

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

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


@@ -153,34 +373,38 @@ synchronously sent or received.
 Since in a heterogeneous grid each cluster has different characteristics,
 especially different frequency gears, when applying DVFS operations on the nodes 
 of these clusters, they may get different scaling factors represented by a scaling vector:
 Since in a heterogeneous grid each cluster has different characteristics,
 especially different frequency gears, when applying DVFS operations on the nodes 
 of these clusters, they may get different scaling factors represented by a scaling vector:

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

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

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

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

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

 \end{equation}
 \end{equation}

-
-where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
-each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
+%
+where $N$ is the number of  clusters in the grid, $M_i$ is the number of  nodes in
+ cluster $i$, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 

 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
 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.
 
 number of iterations of that application.
 

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

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


@@ -217,7 +441,7 @@ communication and no slack time.
 The main objective of DVFS operation is to reduce the overall energy
 consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  The operational
 frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot
 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


@@ -251,7 +475,7 @@ by the new time of computation and is given by the following equation:
 \end{equation}
 The static power is related to the power leakage of the CPU and is consumed
 during computation and even when idle. As
 \end{equation}
 The static power is related to the power leakage of the CPU and is consumed
 during computation and even when idle. As

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

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


@@ -267,33 +491,36 @@ processor after scaling its frequency is computed as follows:
 
 In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
 different dynamic and static powers from the nodes of the other clusters, 
 
 In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
 different dynamic and static powers from the nodes of the other clusters, 

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

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

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

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

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

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

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

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

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

 \end{multline}
 
 \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 a synchronous  application with iterations can be measured by measuring the energy

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


@@ -312,13 +539,17 @@ appropriate frequency scaling factor for each processor while considering the
 characteristics of each processor (computation power, range of frequencies,
 dynamic and static powers) and the task executed (computation/communication
 ratio). The aim being to reduce the overall energy consumption and to avoid
 characteristics of each processor (computation power, range of frequencies,
 dynamic and static powers) and the task executed (computation/communication
 ratio). The aim being to reduce the overall energy consumption and to avoid

-increasing significantly the execution time.  In our previous
-work~\cite{Our_first_paper,pdsec2015}, we proposed a method that selects the optimal
-frequency scaling factor for a homogeneous and heterogeneous clusters executing a message passing
-iterative synchronous application while giving the best trade-off between the
-energy consumption and the performance for such applications.  In this work we
-are interested in heterogeneous grid as described above.  Due to the
-heterogeneity of the processors, a vector of scaling factors should be selected
+increasing significantly the execution time.
+In our previous
+works, \cite{Our_first_paper} and \cite{pdsec2015}, two methods that select the optimal
+frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed. 
+Both methods selects the frequencies that gives the best trade-off between 
+energy consumption reduction and performance for  message passing
+ synchronous applications with iterations.   In this work we
+are interested in grids that are composed of heterogeneous clusters. The nodes from distinct clusters may have 
+ different characteristics such  as  dynamic power, static power, computation power, 
+frequencies range, network latency and bandwidth. 
+Due to the heterogeneity of the processors, a vector of scaling factors should be selected

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


@@ -330,40 +561,43 @@ 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})

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

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

 \end{equation}
 \end{equation}

+

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

+%

 \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


@@ -376,13 +610,12 @@ equation, as follows:
   \Pnorm = \frac{\Told}{\Tnew}          
 \end{equation}
 
   \Pnorm = \frac{\Told}{\Tnew}          
 \end{equation}
 

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

   \centering
   \subfloat[Homogeneous cluster]{%
   \centering
   \subfloat[Homogeneous cluster]{%

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


@@ -391,17 +624,17 @@ Then, the objective function can be modeled in order to find the maximum
 distance between the energy curve (\ref{eq:enorm}) and the performance curve
 (\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
 represents the minimum energy consumption with minimum execution time (maximum
 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 


@@ -412,12 +645,14 @@ 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.


@@ -425,7 +660,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \item[{$\Ps[ij]$}] array of the static powers for all nodes.
     \item[{$\Fdiff[ij]$}] array of the differences between two successive frequencies for all nodes.
     \end{description}
     \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 $F_{ij} \gets  \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$


@@ -433,8 +668,8 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \If{(not the first frequency)}
           \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
     \EndIf
     \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 $)}


@@ -442,10 +677,9 @@ 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 $\Dist \gets \Pnorm - \Enorm$


@@ -464,8 +698,7 @@ 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.


@@ -478,12 +711,12 @@ 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}, is presented. It selects the vector of the frequency
+In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 
+is presented. It selects the vector of  frequency

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

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

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


@@ -491,20 +724,18 @@ scaling factors that satisfies the objective function (\ref{eq:max}). The
 program applies DVFS operations to change the frequencies of the CPUs according
 to the computed scaling factors.  This algorithm is called just once during the
 execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
 program applies DVFS operations to change the frequencies of the CPUs according
 to the computed scaling factors.  This algorithm is called just once during the
 execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed

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

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

-  \includegraphics[scale=0.45]{fig/init_freq}
-  \caption{Selecting the initial frequencies}
+  \includegraphics[scale=0.6]{fig/init_freq}
+  \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


@@ -515,15 +746,15 @@ frequency scaling factors are computed as a ratio between the computation time
 of the slowest node and the computation time of the node $i$ as follows:
 \begin{equation}
   \label{eq:Scp}
 of the slowest node and the computation time of the node $i$ as follows:
 \begin{equation}
   \label{eq:Scp}

-  \Scp[ij] =  \frac{ \mathop{\max_{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


@@ -545,11 +776,11 @@ Therefore, the algorithm iterates on all remaining frequencies, from the higher
 bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
 energy consumption and performance and selects the optimal vector of the frequency scaling
 factors. At each iteration the algorithm determines the slowest node
 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


@@ -559,35 +790,32 @@ factor should start from the maximum frequency because the performance and the
 consumed energy decrease from the beginning of the plot. On the other hand, in
 the  grid platform the performance is maintained at the beginning of the
 plot even if the frequencies of the faster nodes decrease until the computing
 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])


@@ -599,70 +827,64 @@ $\lbrace\Theta_1,\Theta_2\rbrace$ 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}
   \label{fig:grid5000}
 \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. 
-
-
-
-

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

-  \caption{The power consumption by one core from Taurus cluster}
+  \caption{The power consumption by one core from the Taurus cluster}

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

+The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over Grid'5000.
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These benchmarks are considered as message passing  applications with iterations because the same block of operations is executed many times.
+ These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In the next sections, the  class D was used for all the benchmarks in all the experiments. 
+

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

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

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


@@ -672,43 +894,40 @@ 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}
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}

-In this section, the results of the the application of the scaling factors selection algorithm \ref{HSA} 
-to the NAS parallel benchmarks are presented. 
-
-As mentioned previously, the experiments 
-were conducted over two sites of grid'5000,   Lyon and Nancy sites. 
+In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 
+to the NAS parallel benchmarks are presented. Each experiment  has been executed many times and the results presented in the figures are the average values of many executions. As mentioned previously, the experiments 
+were conducted over two sites of Grid'5000,  Lyon and Nancy sites. 

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

-are connected via a long distance network.
-\item In the second scenario nodes from three clusters that are 
-located in one site, Nancy site.  
+ via a long distance network.
+\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 
-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.
+for using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
+scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 
+is very low due to the higher communication times which 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


@@ -717,8 +936,8 @@ Table \ref{tab:sc} shows the number of nodes used from each cluster for each sce
                                       & Griffon         & Nancy        & 6                      \\ 
 \hline
 \multirow{3}{*}{One site / 32 nodes}  & Graphite   & Nancy             & 4                      \\ \cline{2-4} 
                                       & Griffon         & Nancy        & 6                      \\ 
 \hline
 \multirow{3}{*}{One site / 32 nodes}  & Graphite   & Nancy             & 4                      \\ \cline{2-4} 

-                                      & Graphene      & Nancy          & 12                     \\ \cline{2-4} 
-                                      & Griffon          & Nancy       & 12                       \\ 
+                                      & Graphene      & Nancy          & 14                     \\ \cline{2-4} 
+                                      & Griffon          & Nancy       & 14                       \\ 

 \hline
 \end{tabular}
  \label{tab:sc}
 \hline
 \end{tabular}
  \label{tab:sc}


@@ -726,399 +945,334 @@ Table \ref{tab:sc} shows the number of nodes used from each cluster for each sce
 
 \begin{figure}
   \centering
 
 \begin{figure}
   \centering

-  \includegraphics[scale=0.5]{fig/eng_con_scenarios.eps}
-  \caption{The energy consumptions of NAS benchmarks over different scenarios }
-  \label{fig:eng_sen}
-\end{figure}
-
-
-
-\begin{figure}
-  \centering
-  \includegraphics[scale=0.5]{fig/time_scenarios.eps}
-  \caption{The execution times of NAS benchmarks over different scenarios }
-  \label{fig:time_sen}
+  \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios    
+           ]{%
+    \includegraphics[width=.48\textwidth]{fig/eng_con_scenarios.eps}\label{fig:eng_sen}} \hspace{0.4cm}%
+  \subfloat[The execution times of the NAS benchmarks over different scenarios]{%
+    \includegraphics[width=.48\textwidth]{fig/time_scenarios.eps}\label{fig:time_sen}}
+  \label{fig:exp-time-energy}
+  \caption{The  energy consumption and execution time of NAS  Benchmarks over different scenarios}

 \end{figure}
 
 The NAS parallel benchmarks are executed over these two platforms
 \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 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. 
 
 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.
-
-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.
-
-\begin{figure}
-  \centering
-  \includegraphics[scale=0.5]{fig/eng_s.eps}
-  \caption{The energy saving of NAS benchmarks over different scenarios }
-  \label{fig:eng_s}
-\end{figure}
+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 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).

 
 

-\begin{figure}
-  \centering
-  \includegraphics[scale=0.5]{fig/per_d.eps}
-  \caption{The performance degradation of NAS benchmarks over different scenarios }
-  \label{fig:per_d}
-\end{figure}
+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}
-  \centering
-  \includegraphics[scale=0.5]{fig/dist.eps}
-  \caption{The tradeoff distance of NAS benchmarks over different scenarios }
-  \label{fig:dist}
-\end{figure}
-

 The energy saving percentage is computed as the ratio between the reduced 
 The energy saving percentage is computed as the ratio between the reduced 

-energy consumption, equation (\ref{eq:energy}), and the original energy consumption,
-equation (\ref{eq:eorginal}), for all benchmarks as in figure \ref{fig:eng_s}. 
+energy consumption, Equation~\ref{eq:energy}, and the original energy consumption,
+Equation~\ref{eq:eorginal}, for all benchmarks as in Figure~\ref{fig:eng_s}. 

 This figure shows that the energy saving percentages of one site scenario for
 16 and 32 nodes are bigger than those of the two sites scenario which is due
 to the higher  computations to communications ratio in the first scenario   
 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 higher than the communication times which 
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computation times are bigger than the communication times which

 results in  a lower energy consumption. Indeed, the dynamic  consumed power
 results in  a lower energy consumption. Indeed, the dynamic  consumed power

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

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

-benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more 
-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 there computations to 
-communications ratio is not affected by the increase of the number of local communications. 
+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 ]{%
+    \includegraphics[width=.48\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{0.4cm}%
+  \subfloat[The performance degradation of the NAS benchmarks over different scenarios]{%
+    \includegraphics[width=.48\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.4cm}%
+    \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks  
+      over different scenarios]{%
+    \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}
+  \label{fig:exp-res}
+  \caption{The experimental results of different scenarios}
+\end{figure*}
+

 
 
 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 site 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. 
 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  
+
+In fact, high differences between the nodes' computing powers make the proposed frequencies selecting  

 algorithm  select smaller frequencies for the powerful nodes which 
 produces less energy consumption and thus more energy saving.
 algorithm  select smaller frequencies for the powerful nodes which 
 produces less energy consumption and thus more energy saving.

-The best energy saving percentage was obtained in the one site scenario with 16 nodes, The energy consumption was on average reduced up to 30\%.
-
-
-Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks.
-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. 
-
- \textcolor{red}{ 
-The proposed scaling algorithm selecting smaller frequencies in two sites scenario,
-due to decreasing in the computations to communications ratio when the number of nodes is increased and
-leads to less performance degradation percentage.  
-In contrast, 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. 
-The inverse is happens in this scenario when the number of computing nodes is increased 
-the performance degradation percentage is decreased. So, using double number of computing 
-nodes when the communications occur in high speed network not decreased the computations to 
-communication ratio. Moreover, as shown in the figure \ref{fig:time_sen}, the execution time of one site scenario with 32 nodes 
-are less by approximately double, linear speed-up, for most of the benchmarks comparing to the one site with 16 nodes scenario. 
-This leads to increased the number of the critical nodes which any one of them may increased the overall the execution time of the benchmarks.
-The EP benchmarks is gives the bigger performance degradation ratio, because there is no 
-communications and no slack times in this benchmarks which their performance controlled by 
-the computing powers of the nodes.
-The tradeoff between these scenarios can be computed as in the tradeoff function \ref{eq:max}.
-Figure \ref{fig:dist}, presents the tradeoff distance for all benchmarks  over all 
-platform scenarios.  The one site scenario with 16 and 32 nodes had the best tradeoff distance 
-compared to the two sites scenarios, due to the increase or decreased in the communications as mentioned before.
-The one site scenario with 16 nodes is the best scenario in term of energy and performance tradeoff,
-which on average is up 26\%.  Therefore, the tradeoff distance is related linearly to the  energy saving  
-percentage. Finally, the best energy and performance tradeoff depends on the all of the following:
-1) the computations to communications ratio when there is a communications and slack times, 2) the differences in computing powers 
-between the computing nodes and 3) the differences in static and the dynamic powers of the nodes.}
-
-
-
-\subsection{The experimental results of multicores clusters}
+The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
+
+
+Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
+The performance degradation percentage for the benchmarks running on two sites  with
+16 or 32  nodes is on average equal to 8.3\% or 4.7\% respectively. 
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are 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.
+The rest of the benchmarks showed different performance degradation percentages which decrease
+when the communication times increase and vice versa.
+
+Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The trade-off distance percentage can be 
+computed as in Equation~\ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
+trade-off, on average it is equal to  26.8\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
+trade-off comparing to the two sites scenario  because the former has high speed local communications 
+which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 
+
+ Finally, the best energy and performance trade-off depends on all of the following:
+1) the computations to communications ratio when there are  communications and slack times, 2) the heterogeneity of the computing powers of the nodes and 3) the heterogeneity of the consumed  static and dynamic powers of the nodes.
+
+
+
+
+\subsection{The experimental results over multi-core clusters}

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

-The grid'5000 clusters have different number of cores embedded in their nodes
-as in the Table \ref{table:grid5000}. Moreover, the cores of each node are 
-connected via shared memory model, the data transfer between cores' local 
-memories achieved via the global memory \cite{rauber_book}. Therefore, in 
-this section the proposed scaling algorithm is implemented over the  grid'5000 
-clusters which are included multicores in the selected nodes as same as the 
-two previous platform scenarios that mentioned in the section \ref{sec.res}.
-The two platform scenarios, the two sites and one site scenarios,  with 32 
-nodes are reconfigured to used multicores for each node. For example if 
-the participating number of nodes from a certain cluster is equal to 12 nodes, 
-in the multicores scenario the selected nodes is equal to 3 nodes with using 
-4 cores for each of them to produced 12 cores. These scenarios with one  
-core and  multicores are  demonstrated 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 represented 
-in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
-The execution times of NAS  benchmarks over the one site multicores scenario 
-is higher than the execution time of those running over one site multicores scenario. 
-The reason in the one site multicores scenario the communication is increased significantly,
-and all node's cores  share  the same node network link which increased 
-the communication times. Whereas, the execution times of the NAS benchmarks  over 
-the two site  multicores scenario is less than those executed over the two 
-sites one core scenario. This goes back when using multicores is decreasing the communications. 
-As explained previously, the cores shared same nodes' linkbut  the communications between the cores 
-are still less than the communication times between the nodes over the long distance 
-networks, and thus the over all execution time decreased. Generally, executing 
-the NAS benchmarks over the one site one core scenario gives smaller execution times 
-comparing to other scenarios. This due to each node in this scenario has it's 
-dedicated network link that used independently by one core, while in the other 
-scenarios the communication times are higher when using long distance communications 
-link or using the  shared link communications between cores of each node. 
-On the other hand, the energy consumptions of the NAS benchmarks over the 
-one site one cores is less than the one site multicores scenario because 
-this scenario had less execution time as mentioned before. Also, in the 
-one site one core scenario the computations to communications ratio is 
-higher, then the new scaled frequencies are decreased the dynamic energy 
-consumption which is decreased exponentially 
-with the new frequency scaling factors. These experiments also showed, the energy 
-consumption and the execution times of EP and MG benchmarks over these four 
-scenarios are not change a lot, because there are no or small communications 
-which are increase or decrease the static power consumptions. 
-The other benchmarks were showed that their energy consumptions and execution times 
-are changed according to the decreasing or increasing in the communication 
-times that are different from scenario to other or due to the amount of 
-communications in each of them.
-
-The energy saving percentages of all NAS benchmarks, as in figure 
-\ref{fig:eng-s-mc}, running over these four scenarios are presented. The figure 
-showed the energy saving percentages of NAS benchmarks over two sites multicores scenario is higher 
-than two sites once core scenario, because  the computation 
-times in this scenario is higher than the other one, then the more reduction in the 
-dynamic energy can be obtained as mentioned previously. In contrast, in the one site one 
-core and one site multicores scenarios the energy saving percentages 
-are approximately equivalent, on average they are up to 25\%. In these  both scenarios there are a small difference  in the 
-computations to communications ratio, leading the proposed scaling algorithm 
-to selects the frequencies proportionally to these ratios and keeping 
-as much as possible the energy saving percentages the same. The 
-performance degradation percentages of NAS benchmarks are presented in
-figure \ref{fig:per-d-mc}. This figure indicates that performance 
-degradation percentages of running NAS benchmarks over two sites 
-multocores scenario, on average is equal to 7\%,  gives more performance degradation percentage
-than two sites  one core scenario, which on average is equal to 4\%. 
-Moreover, using the two sites multicores scenario increased 
-the computations to communications ratio, which may be increased 
-the overall execution time  when the proposed scaling algorithm is applied and scaling down the frequencies.  
-The inverse was happened when the benchmarks are executed  over one 
-site one core scenario their performance degradation percentages, on average
-is equal to 10\%, are higher than those executed over one sit one core, 
-which on average is equal to 7\%. So, in one site 
-multicores scenario the computations to communications ratio is decreased
-as mentioned before, thus selecting new frequencies are not increased 
-the overall execution time. The tradeoff distances of all NAS 
-benchmarks over all scenarios are presented in the figure \ref{fig:dist-mc}.
-These  tradeoff distances are used to verified which scenario is the best in term of 
-energy and performance ratio. The one sites multicores scenario is the best  scenario in term of
-energy and performance tradeoff, on average is equal to 17.6\%, when comparing to the one site one core 
-scenario, one average is equal to 15.3\%.  The one site multicores  scenario 
-has the same  energy saving percentages of the one site one core scenario but
-with less performance degradation. The two sites multicores scenario is gives better 
-energy and performance tradeoff, one average is equal to 14.7\%, than the two sites
-one core, on average is equal to 13.3\%.
-Finally, using multicore in both scenarios increased the energy and performance tradeoff 
-distance. This generally due to using multicores was increased the computations to communications 
-ratio in two sites scenario and thus the energy saving percentage increased over the performance degradation percentage, whereas  this ratio was decreased  
-in one site  scenario causing the performance degradation percentage decreased over the energy saving percentage.
-
-
-
-
-
-\begin{table}[]
-\centering
-\caption{The multicores scenarios}

 
 

+The  clusters of Grid'5000 have different number of cores embedded in their nodes
+as shown in Table~\ref{table:grid5000}. In 
+this section, the proposed scaling algorithm is evaluated over the  Grid'5000 platform  while using multi-cores nodes selected according to the one site scenario described in  Section~\ref{sec.res}.
+The one site scenario uses  32 cores from multi-core nodes instead of 32 distinct nodes. For example if 
+the participating number of cores from a certain cluster is equal to 14, 
+in the multi-core scenario  4 nodes are selected and
+3 or 4 cores from each node are used. The platforms with one  
+core per node and  multi-core nodes are  shown in Table~\ref{table:sen-mc}. 
+The energy consumptions and execution times of running the class D of the NAS parallel 
+benchmarks over these two different platforms are presented 
+in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+
+
+\begin{table}[!h]
+\centering
+\caption{The 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}{*}{Two sites/ one core}   & Taurus       & 10              & 1                   \\ \cline{2-4}
-                                       & Graphene     & 10              & 1                   \\ \cline{2-4}
-                                       & Griffon      & 12              & 1                   \\ \hline
-\multirow{3}{*}{Two sites/ multicores} & Taurus       & 3               & 3 or 4              \\ \cline{2-4}
-                                       & Graphene     & 3               & 3 or 4              \\  \cline{2-4}
-                                       & Griffon      & 3               & 4                   \\ \hline
-\multirow{3}{*}{One site/ one core}    & Graphite     & 4               & 1                   \\  \cline{2-4}
-                                       & Graphene     & 12              & 1                   \\  \cline{2-4}
-                                       & Griffon      & 12              & 1                   \\ \hline
-\multirow{3}{*}{One site/ multicores}  & Graphite     & 3               & 3 or 4              \\  \cline{2-4}
-                                       & Graphene     & 3               & 3 or 4              \\  \cline{2-4}
-                                       & Griffon      & 3               & 4                   \\ \hline
+Scenario name                          & Cluster name & Nodes per cluster & 
+                                       Cores per node  \\ \hline
+\multirow{3}{*}{One core per node}    & Graphite     & 4               & 1                   \\  \cline{2-4}
+                                       & Graphene     & 14              & 1                   \\  \cline{2-4}
+                                       & Griffon      & 14              & 1                   \\ \hline
+\multirow{3}{*}{Multi-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}
 \label{table:sen-mc}
 \end{table}
 
 \end{tabular}
 \label{table:sen-mc}
 \end{table}
 

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

   \centering
   \centering

-  \includegraphics[scale=0.5]{fig/eng_con.eps}
-  \caption{Comparing the  energy consumptions of running NAS benchmarks over one core and multicores scenarios }
-  \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}
+  \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}
 
 

-  \begin{figure}
-  \centering
-  \includegraphics[scale=0.5]{fig/time.eps}
-  \caption{Comparing the  execution times of running NAS benchmarks over one core and multicores scenarios }
-  \label{fig:time-mc}
-\end{figure}

 
 

- \begin{figure}
+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 
+ one core scenario  than over the multi-core scenario because 
+the first scenario had less execution time than the latter which results in less static energy being consumed.
+The computations to communications ratios of the NAS benchmarks are higher over 
+the one site one core scenario  when compared to the ratio of the multi-core scenario. 
+More energy reduction can be gained when this ratio is big because it pushes the proposed scaling algorithm to select smaller frequencies that decrease the dynamic power consumption. These experiments also showed that the energy 
+consumption and the execution times of the EP and MG benchmarks do not change significantly over these two
+scenarios  because there are no or small communications. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
+\begin{figure*}[t]

   \centering
   \centering

-  \includegraphics[scale=0.5]{fig/eng_s_mc.eps}
-  \caption{The energy saving of running NAS benchmarks over one core and multicores scenarios }
-  \label{fig:eng-s-mc}
-\end{figure}
+    \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
+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
-  \includegraphics[scale=0.5]{fig/per_d_mc.eps}
-  \caption{The performance degradation of running NAS benchmarks over one core and multicores scenarios }
-  \label{fig:per-d-mc}
-\end{figure}

 
 

-\begin{figure}
-  \centering
-  \includegraphics[scale=0.5]{fig/dist_mc.eps}
-  \caption{The tradeoff distance of running NAS benchmarks over one core and multicores scenarios }
-  \label{fig:dist-mc}
-\end{figure}

 
 

-\subsection{The results of using different static power consumption scenarios}
+\subsection{Experiments with different static power scenarios}

 \label{sec.pow_sen}
 \label{sec.pow_sen}

-The static power consumption for one core of the computing node is the leakage power
-consumption when this core is in the idle state. The node's idle state power value that measured 
-as in section \ref{sec.grid5000} had many power consumptions embedded  such as 
-all cores static powers in addition to the power consumption of the other devices. So, the static power for one core 
-can't measured precisely. On the other hand, while the static power consumption of
-one core representing the core's power when there is no any computation,  thus 
-the majority of ratio of the total power consumption is depends on the dynamic power consumption. 
-Despite that, the static power consumption is becomes more important  when the execution time 
-increased using DVFS. Therefore, the objective of this section is to verify the ability of the proposed 
-frequencies selecting algorithm when the static power consumption is changed. 
-
-All the results obtained in the previous sections depend on the measured dynamic power 
-consumptions as in table \ref{table:grid5000}. Moreover, the static power consumption is assumed for
-one core represents  20\% of the measured dynamic power of that core. 
-This assumption is extended in this section to taking into account others ratios for the static power consumption.
-In addition to the previous ratio of the static power consumption, two other scenarios are used which 
-all of these scenarios can be denoted as follow: 
-\begin{itemize}
-\item 10\% of static power scenario 
-\item 20\% of static power scenario 
-\item 30\% of static power scenario 
-\end{itemize}

 
 

-These three scenarios represented the ratio of the static power consumption that can be computed from 
-the dynamic power consumption of the core. The NAS benchmarks of class D are executed over 16 nodes
-in the Nancy site using three clusters: Graphite, Graphene and Griffon. As same as used before, the one site 16 nodes
-platform scenario explained in the last experiments, as in table \ref{tab:sc}, is uses to run 
-the NAS benchmarks with these static power scenarios. 
+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. 

 
 

- \begin{figure}
-  \centering
-  \includegraphics[scale=0.5]{fig/eng_pow.eps}
-  \caption{The energy saving percentages for NAS benchmarks of the three power scenario}
-  \label{fig:eng-pow}
-\end{figure}
+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 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
   \centering

-  \includegraphics[scale=0.5]{fig/per_pow.eps}
-  \caption{The performance degradation percentages for NAS benchmarks of the three power scenario}
-  \label{fig:per-pow}
-\end{figure}
+  \subfloat[The energy saving percentages for the nodes executing the NAS benchmarks over the three power scenarios]{%
+    \includegraphics[width=.48\textwidth]{fig/eng_pow.eps}\label{fig:eng-pow}} \hspace{0.4cm}%
+  \subfloat[The performance degradation percentages for the NAS benchmarks over the three power scenarios]{%
+    \includegraphics[width=.48\textwidth]{fig/per_pow.eps}\label{fig:per-pow}}\hspace{0.4cm}%
+    \subfloat[The 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}
+\end{figure*}

 
 
 
 

-\begin{figure}
-  \centering
-  \includegraphics[scale=0.5]{fig/dist_pow.eps}
-  \caption{The tradeoff distance for NAS benchmarks of the three power scenario}
-  \label{fig:dist-pow}
-\end{figure}

 
 \begin{figure}
   \centering
 
 \begin{figure}
   \centering

-  \includegraphics[scale=0.47]{fig/three_scenarios.pdf}
-  \caption{Comparing the selected frequencies of MG benchmarks for three static power scenarios}
+  \includegraphics[scale=0.5]{fig/three_scenarios.pdf}
+  \caption{Comparing the selected frequency scaling factors for the MG benchmark over the three static power scenarios}

   \label{fig:fre-pow}
 \end{figure}
 
   \label{fig:fre-pow}
 \end{figure}
 

-The energy saving percentages of NAS benchmarks with these three static power scenarios are presented 
-in figure \ref{fig:eng_sen}. This figure showed the 10\% of static power scenario 
-gives the biggest energy saving percentage comparing to 20\% and 30\% static power 
-scenario. When using smaller ratio of static power consumption, the proposed 
-frequencies selecting algorithm selects smaller frequencies, bigger scaling factors, 
-because the static energy consumption not increased significantly the overall energy 
-consumption. Therefore, more energy reduction can be achieved  when the frequencies are scaled down.
-For example figure \ref{fig:fre-pow}, illustrated that the proposed algorithm  
-proportionally scaled down the new computed frequencies with the overall predicted energy 
-consumption. The results of 30\% static power scenario gives the smallest energy saving percentages 
-because the new selected frequencies produced smaller ratio in the reduced energy consumption. 
-Furthermore, The proposed algorithm tries to limit selecting smaller frequencies that increased 
-the static energy consumption if the static power consumption is increased.
-The performance degradation percentages are presented in the figure \ref{fig:per-pow},
-the 30\% of static power scenario had less performance degradation percentage, because
-bigger frequencies was selected due to the big ratio in the static power consumption.
-The inverse was happens in the 20\% and 30\% scenario, the algorithm was selected 
-biggest frequencies, smaller scaling factors, according to this increased in the static power ratios.
-The tradoff distance for the NAS benchmarks with these three static powers scenarios 
-are presented in the figure \ref{fig:dist}. The results showed that the tradeoff
-distance is the best when the  10\% of static power scenario is used, and this percentage 
-is decreased for the other two scenarios propositionally to their static power ratios.
-In EP benchmarks, the results of energy saving, performance degradation and tradeoff 
-distance are showed small differences when the these static power scenarios were used, 
-because this benchmark not has communications. The proposed algorithm is selected 
-same frequencies in this benchmark when all these static power scenarios are used. 
-The small differences in the results are due to the static power is consumed during the computation 
-times side by side to the dynamic power consumption, knowing that the dynamic power consumption 
-representing the highest ratio in the total power consumption of the core, then any change in 
-the static power during these times have less affect on the overall energy consumption. While the 
-inverse was happens for the rest of the benchmarks which have the communications 
-that increased the static energy consumption linearly to the mount of communications 
-in these benchmarks.
+The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 
+in Figure~\ref{fig:eng-pow}. This figure shows that the  10\% of static power scenario 
+gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
+scenarios. The small value of the static power consumption makes the proposed 
+scaling algorithm  select smaller frequencies for the CPUs. 
+These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives            less overall energy consumption. 
+The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors  according to the static power consumption ratio being used.
+
+The performance degradation percentages are presented in Figure~\ref{fig:per-pow}.
+The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
+had  selected big frequencies for the CPUs. While, 
+the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The trade-off distance percentage for the NAS benchmarks with these three static power scenarios 
+are presented in Figure~\ref{fig:dist-pow}. 
+It shows that the best  trade-off
+distance percentage is obtained with  the  10\% static power scenario  and this percentage 
+is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
+
+In the EP benchmark, the energy saving, performance degradation and 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 give the best trade-off between the energy consumption and the performance for a parallel 
+application is not a trivial task.  Many algorithms have been proposed to tackle this problem.  
+In this section, the proposed frequencies selecting algorithm is compared to a method that uses the well known  energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers  \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs}. 
+This objective function  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-core 
+architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
+\begin{figure*}[t]
+  \centering
+  \subfloat[The energy reduction induced by the Maxdist method and the EDP method]{%
+    \includegraphics[width=.48\textwidth]{fig/edp_eng}\label{fig:edp-eng}} \hspace{0.4cm}%
+    \subfloat[The performance degradation induced by  the Maxdist method and the EDP method]{%
+    \includegraphics[width=.48\textwidth]{fig/edp_per}\label{fig:edp-perf}}\hspace{0.4cm}%
+    \subfloat[The trade-off distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method]{%
+    \includegraphics[width=.48\textwidth]{fig/edp_dist}\label{fig:edp-dist}}
+  \label{fig:edp-comparison}
+  \caption{The comparison results}
+\end{figure*}
+To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  Equation~\ref{eq:energy} and
+execution time model, Equation~\ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
+Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.
+Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
+and selects the vector of frequencies that minimize the EDP product.
+
+Both algorithms were applied to the class D of the NAS benchmarks running over 16 nodes.
+The participating computing nodes are distributed  according to the two scenarios described in  Section~\ref{sec.res}. 
+The experimental results, the energy saving, performance degradation and trade-off distance percentages, are 
+presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
+
+As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
+The proposed algorithm gives better results than the EDP method because the former selects the set of frequencies that  
+gives the best tradeoff between energy saving and performance. 
+Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
+Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
+These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
+The high positive values of the trade-off distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
+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 presents a new online frequencies selection algorithm.
+ The algorithm selects the best vector of 
+frequencies that maximizes  the trade-off distance 
+between the predicted energy consumption and the predicted execution time of the distributed 
+ applications with iterations running over a heterogeneous grid. A new energy model 
+is used by the proposed algorithm to predict the energy consumption 
+of the application.
+To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
+NAS parallel benchmarks   and the  class D instance was executed over the  Grid'5000 testbed platform. 
+The experiments 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 
+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 will adapt the proposed algorithm to take into consideration the variability between some iterations.  For example, the proposed algorithm can be executed twice: after the first iteration the frequencies are scaled down according to the execution times measured in the first iteration, then after a fixed number of iterations, the frequencies are adjusted according to the execution times measured during the fixed number of iterations. If the  computing power of the system is constantly changing, it would be interesting to implement a mechanism that detects this change and adjusts the frequencies according to the variability of the system. We would like also to develop a similar method that is adapted to
+asynchronous  applications with iterations where iterations are not synchronized and communications are overlapped with computations. 
+The development of such a method might require a new energy model because the
+number of iterations is not known in advance and depends on
+the global convergence of the iterative system. Finally, it would be interesting to evaluate the scalability of the proposed algorithm by running it on large platforms composed of many thousands of cores. The scalability of the algorithm can be improved by distributing it  in a hierarchical manner where a leader is chosen for each cluster or a group of nodes  to compute their scaled frequencies and by using asynchronous messages to exchange the the data measured at the first iteration. 

 
 
 
 \section*{Acknowledgment}
 
 This work  has been  partially supported by  the Labex ACTION  project (contract
 
 
 
 \section*{Acknowledgment}
 
 This work  has been  partially supported by  the Labex ACTION  project (contract

-``ANR-11-LABX-01-01'').  Computations  have been performed  on the supercomputer
-facilities  of the  Mésocentre de  calcul de  Franche-Comté. As  a  PhD student,
+``ANR-11-LABX-01-01'').  Computations  have been performed  on the Grid'5000 platform. As  a  PhD student,

 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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