X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/603efe31a934432abf3bffc42407ce61f68895f4..7c3b1f8043c9badb515d44ca95d88a77c8c3953c:/Heter_paper.tex?ds=inline

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 66dab36..2f78db5 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -8,7 +8,6 @@
 \usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
-
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
@@ -24,36 +23,47 @@
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 
-\newcommand{\Xsub}[2]{\ensuremath{#1_\textit{#2}}}
+\newcommand{\Xsub}[2]{{\ensuremath{#1_\mathit{#2}}}}
+
+%% used to put some subscripts lower, and make them more legible
+\newcommand{\fxheight}[1]{\ifx#1\relax\relax\else\rule{0pt}{1.52ex}#1\fi}
 
-\newcommand{\Dist}{\textit{Dist}}
+\newcommand{\CL}{\Xsub{C}{L}}
+\newcommand{\Dist}{\mathit{Dist}}
+\newcommand{\EdNew}{\Xsub{E}{dNew}}
 \newcommand{\Eind}{\Xsub{E}{ind}}
 \newcommand{\Enorm}{\Xsub{E}{Norm}}
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
-\newcommand{\Fdiff}{\Xsub{F}{diff}}
-\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Es}{\Xsub{E}{S}}
+\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
+\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}
 \newcommand{\Fnew}{\Xsub{F}{new}}
 \newcommand{\Ileak}{\Xsub{I}{leak}}
 \newcommand{\Kdesign}{\Xsub{K}{design}}
-\newcommand{\MaxDist}{\textit{Max Dist}}
+\newcommand{\MaxDist}{\mathit{Max}\Dist}
+\newcommand{\MinTcm}{\mathit{Min}\Tcm}
 \newcommand{\Ntrans}{\Xsub{N}{trans}}
-\newcommand{\Pdyn}{\Xsub{P}{dyn}}
-\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
+\newcommand{\PdNew}{\Xsub{P}{dNew}}
+\newcommand{\PdOld}{\Xsub{P}{dOld}}
 \newcommand{\Pnorm}{\Xsub{P}{Norm}}
-\newcommand{\Tnorm}{\Xsub{T}{Norm}}
-\newcommand{\Pstates}{\Xsub{P}{states}}
-\newcommand{\Pstatic}{\Xsub{P}{static}}
-\newcommand{\Sopt}{\Xsub{S}{opt}}
-\newcommand{\Tcomp}{\Xsub{T}{comp}}
-\newcommand{\TmaxCommOld}{\Xsub{T}{Max Comm Old}}
-\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
-\newcommand{\Tmax}{\Xsub{T}{max}}
+\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
+\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
+\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
+\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
+\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
+\newcommand{\Ppeak}[1][]{\Xsub{P}{peak}_{#1}}
+\newcommand{\Pidle}[1][]{\Xsub{P}{idle}_{\fxheight{#1}}}
+\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
-\newcommand{\Told}{\Xsub{T}{Old}} 
-\begin{document} 
+\newcommand{\Told}{\Xsub{T}{Old}}
+
+\begin{document}
 
-\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
+\title{Energy Consumption Reduction with DVFS for \\
+  Message Passing Iterative Applications on \\
+  Heterogeneous Architectures}
 
 \author{%
   \IEEEauthorblockN{%
@@ -63,8 +73,7 @@
     Arnaud Giersch
   }
   \IEEEauthorblockA{%
-    FEMTO-ST Institute\\
-    University of Franche-ComtÃ©\\
+    FEMTO-ST Institute, University of Franche-ComtÃ©\\
     IUT de Belfort-MontbÃ©liard,
     19 avenue du MarÃ©chal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % RaphaÃ«l
@@ -77,333 +86,373 @@
 
 \begin{abstract}
   
+
 \end{abstract}
 
 \section{Introduction}
 \label{sec.intro}
 
 
+
 \section{Related works}
 \label{sec.relwork}
 
 
-
-
 \section{The performance and energy consumption measurements on heterogeneous architecture}
 \label{sec.exe}
 
-% \JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'',
-%   can be deleted if we need space, we can just say we are interested in this
-%   paper in homogeneous clusters}
-
-\subsection{The execution time of message passing distributed iterative applications on a heterogeneous platform}
+\subsection{The execution time of message passing distributed iterative
+  applications on a heterogeneous platform}
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over
-heterogeneous platforms. We define a heterogeneous platform as a collection of
-heterogeneous computing nodes interconnected via a high speed homogeneous
-network. Therefore, each node has different characteristics such as computing
-power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
-have the same 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.
 
-\begin{figure}[t]
+\begin{figure}[!t]
   \centering
-    \includegraphics[scale=0.6]{fig/commtasks}
+  \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 
- The  overall execution time  of a distributed iterative synchronous application over a heterogeneous platform  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 nodes, slack times might 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 have 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 and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound.  The frequency reduction process can be  expressed by the scaling factor S which is the ratio between  the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
+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
+and consequently its computing power, the execution time of a program running
+over that scaled down processor may increase, especially if the program is
+compute bound.  The frequency reduction process can be  expressed by the scaling
+factor S which is the ratio between  the maximum and the new frequency of a CPU
+as in (\ref{eq:s}).
 \begin{equation}
   \label{eq:s}
- S = \frac{F_\textit{max}}{F_\textit{new}}
+  S = \frac{\Fmax}{\Fnew}
 \end{equation}
- 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 communication. The execution time of the computation part is linearly proportional to the frequency scaling factor $S$ but  the communication time is not affected by the scaling factor because  the processors involved remain idle during the  communications~\cite{17}. The communication time for a task is the summation of periods of time that begin with an MPI call for sending or receiving   a message till the message is synchronously sent or received.
-
-Since in a heterogeneous platform, each node has different characteristics,
-especially different frequency gears, when applying DVFS operations on these
-nodes, they may get different scaling factors represented by a scaling vector:
-$(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
+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
+communication.  The execution time of the computation part is linearly
+proportional to the frequency scaling factor $S$ but the communication time is
+not affected by the scaling factor because the processors involved remain idle
+during the communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  The
+communication time for a task is the summation of periods of time that begin
+with an MPI call for sending or receiving a message until the message is
+synchronously sent or received.
+
+Since in a heterogeneous grid each cluster has different characteristics,
+especially different frequency gears, when applying DVFS operations on the nodes 
+of these clusters, they may get different scaling factors represented by a scaling vector:
+$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
 be able to predict the execution time of message passing synchronous iterative
-applications running over a heterogeneous platform, for different vectors of
+applications running over a heterogeneous grid, for different vectors of
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
-vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
+vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
- \textit  T_\textit{new} = 
- \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  MinTcm
+  \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
 \end{equation}
-where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $MinTcm$ is the communication time of the slowest processor from 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, it means  only the
- communication time without any slack time. Therefore, we can consider the execution time of the iterative application is equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied by the number of iterations of that application.
 
-This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. The execution time prediction model is used in our method for optimizing both energy consumption and performance of iterative methods, which is presented in the following sections.
+where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
+each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
+and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
+first  iteration. The model computes the maximum computation time with scaling factor 
+from each node added to the communication time of the slowest node in the slowest cluster $h$.
+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
+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.
 
 
 \subsection{Energy model for heterogeneous platform}
 
-Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
-two power metrics: the static and the dynamic power.  While the first one is
-consumed as long as the computing unit is turned on, the latter is only consumed during
-computation times.  The dynamic power $P_{d}$ is related to the switching
-activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
-operational frequency $F$, as shown in EQ(\ref{eq:pd}).
+Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
+  Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
+  Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
+a processor into two power metrics: the static and the dynamic power.  While the
+first one is consumed as long as the computing unit is turned on, the latter is
+only consumed during computation times.  The dynamic power $\Pd$ is related to
+the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
+and operational frequency $F$, as shown in (\ref{eq:pd}).
 \begin{equation}
   \label{eq:pd}
-  P_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F
+  \Pd = \alpha \cdot \CL \cdot V^2 \cdot F
 \end{equation}
-The static power $P_{s}$ captures the leakage power as follows:
+The static power $\Ps$ captures the leakage power as follows:
 \begin{equation}
   \label{eq:ps}
-   P_\textit{s}  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
+   \Ps  = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak
 \end{equation}
-where V is the supply voltage, $N_{trans}$ is the number of transistors,
-$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
-technology-dependent parameter.  The energy consumed by an individual processor
+where V is the supply voltage, $\Ntrans$ is the number of transistors,
+$\Kdesign$ is a design dependent parameter and $\Ileak$ is a
+technology dependent parameter.  The energy consumed by an individual processor
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}
-   E_\textit{ind} =  P_\textit{d} \cdot Tcp + P_\textit{s} \cdot T
+  \Eind =  \Pd \cdot \Tcp + \Ps \cdot T
 \end{equation}
-where $T$ is the execution time of the program, $T_{cp}$ is the computation
-time and $T_{cp} \leq T$.  $T_{cp}$ may be equal to $T$ if there is no
+where $T$ is the execution time of the program, $\Tcp$ is the computation
+time and $\Tcp \le T$.  $\Tcp$ may be equal to $T$ if there is no
 communication and no slack time.
 
-The main objective of DVFS operation is to
-reduce the overall energy consumption~\cite{37}.  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 dynamic
-voltage with respect to various frequency values in~\cite{3}.  The reduction
-process of the frequency can be expressed by the scaling factor $S$ which is the
-ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
-The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. we can calculate the new frequency 
-$F_{new}$ from EQ(\ref{eq:s}) as follow:
+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
+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
+the maximum and the new frequency as in (\ref{eq:s}).  The CPU governors are
+power schemes supplied by the operating system's kernel to lower a core's
+frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
+follows:
 \begin{equation}
   \label{eq:fnew}
-   F_\textit{new} = S^{-1} \cdot F_\textit{max}
+   \Fnew = S^{-1} \cdot \Fmax
 \end{equation}
-Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic 
-power consumption:
+Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
+equation for dynamic power consumption:
 \begin{multline}
   \label{eq:pdnew}
-   {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
-   {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}
+   \PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 \\
+   {} = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}
 \end{multline}
-where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the new frequency and the maximum frequency respectively.
-
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 
-reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$.  The new dynamic energy is the  dynamic power multiplied by the new time of computation and is given by the following equation:
+where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
+new frequency and the maximum frequency respectively.
+
+According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
+$S^{-3}$ when reducing the frequency by a factor of
+$S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is
+proportional to the frequency of a CPU, the computation time is increased
+proportionally to $S$.  The new dynamic energy is the dynamic power multiplied
+by the new time of computation and is given by the following equation:
 \begin{equation}
   \label{eq:Edyn}
-   E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{dOld} \cdot  Tcp 
+   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp
 \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{3,46}, we assume that the static power of a processor is 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 to the execution time model in EQ(\ref{eq:perf}), 
-the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related  
-to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy 
-of a processor after scaling its frequency is computed as follows: 
+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},
+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
+to the execution time model in (\ref{eq:perf}), the execution time of the
+program is the sum of the computation and the communication times. The
+computation time is linearly related to the frequency scaling factor, while this
+scaling factor does not affect the communication time.  The static energy of a
+processor after scaling its frequency is computed as follows:
 \begin{equation}
   \label{eq:Estatic}
- E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)
+  \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
 \end{equation}
 
-In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $Tcp_{i}$ might be different and different frequency  scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $Tcm_{i}$ and could contain slack times if it is 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 EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing  distributed application executed over a heterogeneous platform during one iteration is the summation of all dynamic and static energies for each  processor.  It is computed as follows:
+In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
+different dynamic and static powers from the nodes of the other clusters, 
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Therefore, even if the distributed 
+message passing iterative application is load balanced, the computation time of each CPU $j$ 
+in cluster $i$ noted $\Tcp[ij]$ may be different and different frequency scaling factors may be
+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
+$\Tcm[ij]$ 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
+heterogeneous grid platform during one iteration is the summation of all dynamic and
+static energies for $M$ processors in $N$ clusters.  It is computed as follows:
 \begin{multline}
   \label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot  Tcp_i)} + {} \\
- \sum_{i=1}^{N} (Ps_{i} \cdot (\max_{i=1,2,\dots,N} (Tcp_i \cdot S_{i}) +
-  {MinTcm))}
- \end{multline}
-
-Reducing the frequencies of the processors according to the vector of
-scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
-application and thus, increase the static energy because the execution time is
-increased~\cite{36}. We can measure the overall energy consumption for the iterative 
-application by measuring  the energy consumption for one iteration as in EQ(\ref{eq:energy}) multiplied by 
-the number of iterations of that application.
+ E = \sum_{i=1}^{N} \sum_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
+ \sum_{i=1}^{N} \sum_{j=1}^{M} (\Ps[ij] \cdot {} \\
+  (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots M}} (\Tcm[hj]) ))
+\end{multline}
 
+Reducing the frequencies of the processors according to the vector of scaling
+factors $(S_{11}, S_{12},\dots, S_{NM})$ may degrade the performance of the application
+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
+consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
+of iterations of that application.
 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 
-Using the lowest frequency for each processor does not necessarily gives the most energy efficient execution of an application. Indeed, even though the dynamic power is reduced while scaling down the frequency of a processor, its computation power is proportionally decreased and thus the execution time might be drastically increased during which dynamic and static powers are being consumed. Therefore,  it might cancel any gains achieved by scaling down the frequency of all nodes to the minimum  and the overall energy consumption of the application might not be the optimal one. It is not trivial to select the 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) in order to reduce the overall energy consumption and not significantly increase the execution time. In our previous work~\cite{45}, we  proposed a method that selects the optimal 
-frequency scaling factor for a homogeneous cluster 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 clusters as described above. Due to the heterogeneity of the processors, not one but 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 application is complex and nonlinear, Thus, unlike the relation between the execution time 
-and the scaling factor, the relation of the energy with the frequency scaling
-factors is nonlinear, for more details refer to~\cite{17}.  Moreover, they are
-not measured using the same metric.  To solve this problem, we normalize the
-execution time 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{multline}
+Using the lowest frequency for each processor does not necessarily give the most
+energy efficient execution of an application. Indeed, even though the dynamic
+power is reduced while scaling down the frequency of a processor, its
+computation power is proportionally decreased. Hence, the execution time might
+be drastically increased and during that time, dynamic and static powers are
+being consumed.  Therefore, it might cancel any gains achieved by scaling down
+the frequency of all nodes to the minimum and the overall energy consumption of
+the application might not be the optimal one.  It is not trivial to select the
+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
+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
+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
+application is complex and nonlinear, Thus, unlike the relation between the
+execution time and the scaling factor, the relation between the energy and the
+frequency scaling factors is nonlinear, for more details refer
+to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  Moreover, these relations
+are not measured using the same metric.  To solve this problem, the execution
+time is normalized by computing the ratio between the new execution time (after
+scaling down the frequencies of some processors) and the initial one (with
+maximum frequency for all nodes) as follows:
+\begin{equation}
   \label{eq:pnorm}
-  P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
-       {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +MinTcm}
-           {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
-\end{multline}
+  \Pnorm = \frac{\Tnew}{\Told}                 
+\end{equation}
 
 
-In the same way, we normalize the energy by computing the ratio between the consumed energy while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
-\begin{multline}
+Where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told})
+\begin{equation}
+  \label{eq:told}
+   \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])             
+\end{equation}
+In the same way, the energy is normalized by computing the ratio between the
+consumed energy while scaling down the frequency and the consumed energy with
+maximum frequency for all  nodes:
+\begin{equation}
   \label{eq:enorm}
-  E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
-  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
-\end{multline} 
-Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
-
- While the main 
-goal is to optimize the energy and execution time at the same time, the normalized energy and execution time curves are not in the same direction. According 
-to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), 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 maximum energy
-reduction with minimum execution time reduction.  
-
-Many researchers used different strategies to solve this nonlinear problem for example
-in~\cite{19,42}, their methods add big overheads to the algorithm to select the
-suitable frequency.  In this paper we  present a method to find the optimal
-set of frequency scaling factors to simultaneously optimize both energy and execution time
- without adding a big overhead. \textbf{put the last two phrases in the related work section}
- 
-  
-Our solution for this problem is to make the optimization process for energy and execution time follow the same
-direction.  Therefore, we inverse the equation of the normalized execution time which gives 
-the normalized performance equation, as follows:
-\begin{multline}
-  \label{eq:pnorm_inv}
-  P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
-          = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
-            { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm} 
-\end{multline}
+  \Enorm = \frac{\Ereduced}{\Eoriginal} 
+\end{equation}
 
+Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
+computed as in ().
 
-\begin{figure}
+\textcolor{red}{A reference is missing}
+\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)       
+\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
+maximum energy reduction with minimum execution time reduction.
+
+This problem can be solved by making the optimization process for energy and
+execution time follow the same evolution according to the vector of scaling factors
+$(S_{11}, S_{12},\dots, S_{NM})$. Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance
+equation, as follows:
+\begin{equation}
+  \label{eq:pnorm_inv}
+  \Pnorm = \frac{\Told}{\Tnew}          
+\end{equation}
+
+\begin{figure}[!t]
   \centering
-  \subfloat[Homogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
-  \qquad%
-  \subfloat[Heterogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
+  \subfloat[Homogeneous cluster]{%
+    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+
+  \subfloat[Heterogeneous grid]{%
+    \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}
 
-Then, we can model our objective function as finding the maximum distance
-between the energy curve EQ~(\ref{eq:enorm}) and the  performance
-curve EQ~(\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 figure~(\ref{fig:r2}) .  Then our objective
-function has the following form:
+Then, the objective function can be modeled in order to find the maximum
+distance between the energy curve (\ref{eq:enorm}) and the performance curve
+(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
+represents the minimum energy consumption with minimum execution time (maximum
+performance) at the same time, see Figure~\ref{fig:r1} or
+Figure~\ref{fig:r2}. Then the objective function has the following form:
 \begin{equation}
   \label{eq:max}
-  Max Dist = 
-  \max_{i=1,\dots F, j=1,\dots,N}
-      (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
-       \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
+  \MaxDist =
+\mathop{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
+      (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
 \end{equation}
-where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
-Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).  Our objective function can
-work with any energy model or any power values for each node (static and dynamic powers).
-However, the most energy reduction gain can be achieved when the energy curve has a convex form as shown in~\cite{15,3,19}.
-
-\section{The scaling factors selection algorithm for heterogeneous platforms }
+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 
+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 
+energy reduction gain can be achieved when the energy curve has a convex form as shown 
+in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
+
+\section{The scaling factors selection algorithm for  grids }
 \label{sec.optim}
 
-In this section we  propose algorithm~\ref{HSA}) which selects the frequency scaling factors vector 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 heterogeneous platform.  
-IT works online during the execution time of the iterative message passing program.  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 scaling factors   that satisfies the objective function EQ(\ref{eq:max}). The program apply 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.
-
-
-The nodes in a heterogeneous platform 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.
-Our algorithm takes into account this problem and tries to reduce these slack times when selecting the frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase the execution times of fast nodes and  minimize the  differences between  the  computation times of fast and slow nodes. The value of the initial frequency scaling factor  for each node is inversely proportional to its computation time that was gathered from the first iteration. These initial frequency scaling factors are computed as   a ratio between the computation time of the slowest node and the computation time of the node $i$ as follows:
-\begin{equation}
-  \label{eq:Scp}
- Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
-\end{equation}
-Using the initial  frequency scaling factors computed in EQ(\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 follows:
-\begin{equation}
-  \label{eq:Fint}
- F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
-\end{equation}
-If the computed initial frequency for a node is not available in the gears of that node, the computed initial frequency is replaced by the nearest available frequency.
-In  figure (\ref{fig:st_freq}), the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are coloured in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal set of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the frequency selecting factors algorithm starts its search method from these initial frequencies and takes a downward search direction. The algorithm iterates on all left frequencies, from the higher bound until all nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node according to EQ(\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of  all other nodes by one gear. The new overall energy consumption and execution time are computed according to the new scaling factors. The optimal set of frequency scaling factors is the set that gives the highest distance according to  the objective function EQ(\ref{eq:max}).
-\begin{figure}[t]
-  \centering
-    \includegraphics[scale=0.5]{fig/start_freq}
-  \caption{Selecting the initial frequencies}
-  \label{fig:st_freq}
-\end{figure}
-
-
-This algorithm has a small
-execution time: for a heterogeneous cluster composed of four different types of
-nodes having the characteristics presented in table~(\ref{table:platform}), it  
-takes \np[ms]{0.04} on average for 4 nodes and \np[ms]{0.15} on average for 144
-nodes.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the
-number of iterations and $N$ is the number of computing nodes. The algorithm
-needs on average from 12 to 20 iterations to selects the best vector of frequency scaling factors that give the results of the next section. \textbf{put the lst paragraph in experiments}
-
-
-
-
-
-
 \begin{algorithm}
   \begin{algorithmic}[1]
     % \footnotesize
     \Require ~
     \begin{description}
-    \item[$Tcp_i$] array of all computation times for all nodes during one iteration and with highest frequency.
-    \item[$Tcm_i$] array of all communication times for all nodes during one iteration and with highest frequency.
-    \item[$Fmax_i$] array of the maximum frequencies for all nodes.
-    \item[$Pd_i$] array of the dynamic powers for all nodes.
-    \item[$Ps_i$] array of the static powers for all nodes.
-    \item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
+    \item [{$N$}] number of clusters in the grid.
+    \item [{$M$}] 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[{$\Pd[ij]$}] array of the dynamic powers for all nodes.
+    \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 $Sopt_1,Sopt_2 \dots, Sopt_N$ is a vector of optimal scaling factors
+    \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
 
-    \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
-    \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
-    \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
+    \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 Round the computed initial frequencies $F_i$ to the closest  available frequency for each node.
     \If{(not the first frequency)}
-          \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
-    \EndIf 
-    \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
-    \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
-    \State $Dist \gets 0$
-    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
-    \While {(all nodes not reach their  minimum  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 $\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 $)}
         \If{(not the last freq. \textbf{and} not the slowest node)}
-        \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
-        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,\dots,N.$
+        \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 $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm $
-       \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} + $  \hspace*{43 mm} 
-               $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
-       \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
-       \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
+       \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 $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
-        \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $
+        \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile
-    \State  Return $Sopt_1,Sopt_2,\dots,Sopt_N$
+    \State  Return $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$
   \end{algorithmic}
-  \caption{Heterogeneous scaling algorithm}
+  \caption{Scaling factors selection algorithm}
   \label{HSA}
 \end{algorithm}
 
@@ -416,7 +465,7 @@ needs on average from 12 to 20 iterations to selects the best vector of frequenc
       \If {$(k=1)$}
         \State Gather all times of computation and\newline\hspace*{3em}%
                communication from each node.
-        \State Call algorithm from Figure~\ref{HSA} with these times.
+        \State Call Algorithm \ref{HSA}.
         \State Compute the new frequencies from the\newline\hspace*{3em}%
                returned optimal scaling factors.
         \State Set the new frequencies to nodes.
@@ -427,352 +476,232 @@ needs on average from 12 to 20 iterations to selects the best vector of frequenc
   \label{dvfs}
 \end{algorithm}
 
+\subsection{The algorithm details}
+
+\textcolor{red}{Delete the subsection if there's only one.}
+
+In this section, the scaling factors selection algorithm for  grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency
+scaling factors  that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+synchronous iterative application executed on a  grid. It works
+online during the execution time of the iterative message passing program.  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
+scaling factors that satisfies the objective function (\ref{eq:max}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors.  This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
+scaling algorithm is called in the iterative MPI program.
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.45]{fig/init_freq}
+  \caption{Selecting the initial frequencies}
+  \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
+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
+between the computation times of fast and slow nodes. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
+\begin{equation}
+  \label{eq:Scp}
+  \Scp[ij] =  \frac{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]}
+\end{equation}
+Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
+algorithm computes the initial frequencies for all nodes as a ratio between the
+maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+follows:
+\begin{equation}
+  \label{eq:Fint}
+  F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M}
+\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
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing powers in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors.  The resulting new
+frequencies are highlighted in Figure~\ref{fig:st_freq}.  This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting higher frequencies
+than the higher bound will not improve the performance of the application and it
+will increase its overall energy consumption.  Therefore the algorithm that
+selects the frequency scaling factors starts the search method from these
+initial frequencies and takes a downward search direction toward lower
+frequencies until reaching the nodes' minimum frequencies or lower bounds. A node's frequency is considered its lower bound if the computed distance between the energy and performance at this frequency is less than zero.
+A negative distance means that the performance degradation ratio is higher than the energy saving ratio.
+In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances. 
+
+Therefore, the algorithm iterates on all remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
+energy consumption and performance and selects the optimal vector of the frequency scaling
+factors. At each iteration the algorithm determines the slowest node
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+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}).
+
+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
+ grid platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous cluster the search for the optimal scaling
+factor should start from the maximum frequency because the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand, in
+the  grid platform the performance is maintained at the beginning of the
+plot even if the frequencies of the faster nodes decrease until the computing
+power of scaled down nodes are lower than the slowest node. In other words,
+until they reach the higher bound. It can also be noticed that the higher the
+difference between the faster nodes and the slower nodes is, the bigger the
+maximum distance between the energy curve and the performance curve is, which results in bigger energy savings. 
+
+
 \section{Experimental results}
 \label{sec.expe}
+While in~\cite{mpi-energy2} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid.org}, 
+in this paper real experiments were conducted over the grid'5000 platform. 
+
+\subsection{Grid'5000 architature and power consumption}
+\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,
+which is the French National Telecommunication Network for Technology.
+Each site of the grid is composed of few heterogeneous 
+computing clusters and each cluster contains many homogeneous nodes. In total,
+ grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
+the clusters and their nodes are connected via  high speed local area networks. 
+Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
+
+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 $P\max[jx]$. 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}). 
+
+\textcolor{red}{why maximum and minimum, change peak in the equation and the figure}
+
+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} (P\max[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
+\end{equation}
 
-The experiments of this work are executed on the simulator SimGrid/SMPI
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We are configure the
-simulator to use a heterogeneous cluster with one core per node. The proposed
-heterogeneous cluster has four different types of nodes. Each node in the cluster
-has different characteristics such as the maximum frequency speed, the number of
-available frequencies and dynamic and static powers values, see table
-(\ref{table:platform}). These different types of processing nodes are simulate some
-real Intel processors. The maximum number of nodes that supported by the cluster
-is 144 nodes according to characteristics of some MPI programs of the NAS
-benchmarks that used. We are use the same number from each type of nodes when we
-run the iterative MPI programs, for example if  we are execute the program on 8 node, there
-are 2 nodes from each type participating in the computation. The dynamic and
-static power values is different from one type to other. Each node has a dynamic
-and static power values proportional to their computing power (FLOPS), for more
-details see the Intel data sheets in \cite{47}.  Each node has a percentage of
-80\% for dynamic power and 20\% for static power of the total power
-consumption of a CPU, the same assumption is made in \cite{45,3}. These nodes are
-connected via an ethernet network with 1 Gbit/s bandwidth.
-\begin{table}[htb]
-  \caption{Heterogeneous nodes characteristics}
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\
-    type     & to          & Freq. GHz  & Freq. GHz    & Freq GHz       & power        & power \\
-    \hline
-    1       & core-i3       & 2.5         & 1.2          & 0.1           & 20~w         &4~w    \\
-            &  2100T        &             &              &               &              &  \\
-    \hline
-    2       & Xeon          & 2.66        & 1.6          & 0.133         & 25~w         &5~w    \\
-            & 7542          &             &              &               &              &  \\
-    \hline
-    3       & core-i5       & 2.9         & 1.2          & 0.1           & 30~w         &6~w    \\
-            & 3470s         &             &              &               &              &  \\
-    \hline
-    4       & core-i7       & 3.4         & 1.6          & 0.133         & 35~w         &7~w    \\
-            & 2600s         &             &              &               &              &  \\
-    \hline
-  \end{tabular}
-  \label{table:platform}
-\end{table}
+where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured peak power values, 
+$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
+Therefore, the dynamic power of one core is computed as the difference between the maximum 
+measured value in peak powers vector and the minimum measured value in the idle powers vector.
 
- 
-%\subsection{Performance prediction verification}
+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  np[\%]{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}).
 
-\subsection{The experimental results of the scaling algorithm}
-\label{sec.res}
+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, 
+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}.
 
-The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.3 
-\cite{44},  which were run with three classes (A, B and C).
-In this experiments we are interested to run the class C, the biggest class compared to A and B, on different number of 
-nodes, from 4 to 128 or 144 nodes according to the type of the iterative MPI program.  Depending on the proposed energy consumption model EQ(\ref{eq:energy}),
- we are measure the energy consumption for all NAS MPI programs. The dynamic and static power values are used under the same assumption used by \cite{45,3}, we are used a percentage of 80\% for dynamic power and 20\% for static of the total power consumption of a CPU. The heterogeneous nodes in table (\ref{table:platform}) have different  simulated computing power (FLOPS), ranked from the node of type 1 with smaller computing power (FLOPS) to the highest computing power (FLOPS) for node of type 4. Therefore, the power values are used proportionally increased from nodes of type 1 to nodes of type 4 that with highest computing power. Then, we are used an assumption that the power consumption is increased linearly when the computing power (FLOPS) of the processor is increased, see \cite{48}.   
- 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 4 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &  64.64        & 3560.39        &34.16        &6.72               &27.44       \\
-    \hline 
-    MG         & 18.89         & 1074.87	    &35.37	      &4.34	              &31.03       \\
-   \hline
-    EP         &79.73	       &5521.04	        &26.83   	  &3.04               &23.79      \\
-   \hline
-    LU         &308.65	       &21126.00	   &34.00	      &6.16	              &27.84      \\
-    \hline
-    BT         &360.12         &21505.55	   &35.36         &8.49               &26.87     \\
-   \hline
-    SP         &234.24	       &13572.16	   &35.22         &5.70	              &29.52    \\
-   \hline
-    FT         &81.58          &4151.48        &35.58         &0.99        	      &34.59    \\
-\hline 
-  \end{tabular}
-  \label{table:res_4n}
-\end{table}
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 8 and 9 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &36.11    	   &3263.49	        &31.25	      &7.12	               &24.13     \\
-    \hline 
-    MG         &8.99 	       &953.39	        &33.78	      &6.41	               &27.37     \\
-   \hline
-    EP         &40.39	       &5652.81	        &27.04	      &0.49	               &26.55     \\
-   \hline
-    LU         &218.79  	   &36149.77	    &28.23        &0.01   	           &28.22      \\
-    \hline
-    BT         &166.89 	       &23207.42	    &32.32	      &7.89	               &24.43      \\
-   \hline
-    SP         &104.73	       &18414.62	    &24.73	      &2.78	               &21.95      \\
-   \hline
-    FT         &51.10	       &4913.26	        &31.02	      &2.54	               &28.48      \\
-\hline 
-  \end{tabular}
-  \label{table:res_8n}
-\end{table}
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 16 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &31.74	       &4373.90	        &26.29	      &9.57	               &16.72          \\
-    \hline 
-    MG         &5.71	       &1076.19         &32.49	      &6.05	               &26.44         \\
-   \hline
-    EP         &20.11	       &5638.49	        &26.85	      &0.56	               &26.29         \\
-   \hline
-    LU         &144.13	       &42529.06	    &28.80	      &6.56	               &22.24         \\
-    \hline
-    BT         &97.29	       &22813.86	    &34.95   	  &5.80	               &29.15         \\
-   \hline
-    SP         &66.49	       &20821.67 	    &22.49	      &3.82	               &18.67         \\
-   \hline
-    FT     	   &37.01          &5505.60	        &31.59	      &6.48	               &25.11         \\
-\hline 
-  \end{tabular}
-  \label{table:res_16n}
-\end{table}
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 32 and 36 nodes }
-  % title of Table
+\begin{figure}[!t]
   \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &32.35	       &6704.21	        &16.15	      &5.30	               &10.85           \\
-    \hline 
-    MG         &4.30	       &1355.58	        &28.93	      &8.85	               &20.08          \\
-   \hline
-    EP         &9.96           &5519.68	        &26.98	      &0.02	               &26.96          \\
-   \hline
-    LU         &99.93	       &67463.43	    &23.60	      &2.45	               &21.15          \\
-    \hline
-    BT         &48.61	       &23796.97	    &34.62	      &5.83	               &28.79          \\
-   \hline
-    SP         &46.01	       &27007.43	    &22.72	      &3.45	               &19.27           \\
-   \hline
-    FT     	   &28.06     	   &7142.69	        &23.09	      &2.90	               &20.19           \\
-\hline 
-  \end{tabular}
-  \label{table:res_32n}
-\end{table}
+  \includegraphics[scale=1]{fig/grid5000}
+  \caption{The selected two sites of grid'5000}
+  \label{fig:grid5000}
+\end{figure}
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 64 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &46.65	       &17521.83	    &8.13	      &1.68	               &6.45           \\
-    \hline 
-    MG         &3.27	       &1534.70	        &29.27	      &14.35	           &14.92          \\
-   \hline
-    EP         &5.05           &5471.1084	    &27.12	      &3.11    	           &24.01         \\
-   \hline
-    LU         &73.92	       &101339.16	    &21.96	      &3.67	               &18.29         \\
-    \hline
-    BT         &39.99 	       &27166.71	    &32.02	      &12.28	           &19.74         \\
-   \hline
-    SP         &52.00	       &49099.28	    &24.84	      &0.03	               &24.81         \\
-   \hline
-    FT         &25.97	       &10416.82        &20.15	      &4.87	               &15.28         \\
-\hline 
-  \end{tabular}
-  \label{table:res_64n}
-\end{table}
+
+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{table}[htb]
-  \caption{Running NAS benchmarks on 128 and 144 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
-    \hline
-    CG         &56.92	       &41163.36        &4.00	      &1.10	               &2.90          \\
-    \hline 
-    MG         &3.55           &2843.33         &18.77	      &10.38	           &8.39          \\
-   \hline
-    EP         &2.67           &5669.66	        &27.09	      &0.03	               &27.06         \\
-   \hline
-    LU         &51.23	       &144471.90   	&16.67	      &2.36	               &14.31         \\
-    \hline
-    BT         &37.96          &44243.82	    &23.18	      &1.28	               &21.90         \\
-   \hline
-    SP         &64.53	       &115409.71 	    &26.72	      &0.05	               &26.67         \\
-   \hline
-    FT         &25.51	       &18808.72	    &12.85	      &2.84	               &10.01         \\
-\hline 
-  \end{tabular}
-  \label{table:res_128n}
-\end{table}
 
-The results of applying the proposed scaling algorithm to the NAS benchmarks is demonstrated in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}). These tables are show the experimental results for running the NAS benchmarks on different number of nodes. In general the energy saving percent is decreased  when the number of the computing nodes is increased. Also the distance is decreased by the same direction of the energy saving. This because when we are run the iterative MPI programs on a big number of nodes the communications times is increased, so the static energy is increased linearly to these times. The tables also show that the performance degradation percent still approximately the same ratio or decreased in a very small present when the number of computing nodes is increased. This is gives a good prove that the proposed algorithm keeping the performance degradation as mush as possible is the same. 
 
-\begin{figure}
+\begin{figure}[!t]
   \centering
-  \subfloat[CG, MG, LU and FT benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
-  \quad%
-  \subfloat[BT and SP benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
-  \label{fig:avg}
-  \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
+  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
+  \caption{The power consumption by one core from Taurus cluster}
+  \label{fig:power_cons}
 \end{figure}
 
-In the NAS benchmarks there are some programs executed on different number of
-nodes. The benchmarks CG, MG, LU and FT executed on 2 to a power of (1, 2, 4, 8,
-\dots{}) of nodes. The other benchmarks such as BT and SP executed on 2 to a
-power of (1, 2, 4, 9, \dots{}) of nodes. We are take the average of energy
-saving, performance degradation and distances for all results of NAS
-benchmarks. The average of values of these three objectives are plotted to the number of
-nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}).  In CG, MG, LU, and
-FT benchmarks the average of energy saving is decreased when the number of nodes
-is increased because the communication times is increased as mentioned
-before. Thus, the average of distances (our objective function) is decreased
-linearly with energy saving while keeping the average of performance degradation approximately is 
-the same. In BT and SP benchmarks, the average of the  energy saving is not decreased
-significantly compare to other benchmarks when the number of nodes is
-increased. Nevertheless, the average of performance degradation approximately
-still the same ratio. This difference is depends on the characteristics of the
-benchmarks such as the computation to communication ratio that has.
 
-\subsection{The results for different power consumption scenarios}
 
-The results of the previous section are obtained using a percentage of 80\% for
-dynamic power and 20\% for static power of the total power consumption of a CPU. In this
-section we are change these ratio by using two others power scenarios. Because is
-interested to measure the ability of the proposed algorithm when these power ratios are changed. 
-In fact, we are used two different scenarios for dynamic and static power ratios in addition to the previous
-scenario in section (\ref{sec.res}). Therefore, we have three different
-scenarios for three different dynamic and static power ratios refer to these as: 
-70\%-20\%, 80\%-20\% and 90\%-10\% scenario respectively. The results of these scenarios
-running the NAS benchmarks class C on 8 or 9 nodes are place in the tables
-(\ref{table:res_s1} and \ref{table:res_s2}).
-
- \begin{table}[htb]
-  \caption{The results of 70\%-30\% powers scenario}
+  
+\begin{table}[!t]
+  \caption{CPUs characteristics of the selected clusters}
   % title of Table
   \centering
-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{7}{c|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
+    Cluster     & CPU         & Max   & Min   & Diff. & no. of cores    & dynamic power   \\
+    Name        & model       & Freq. & Freq. & Freq. & per CPU         & of one core     \\
+                &             & GHz   & GHz   & GHz   &                 &           \\
     \hline
-    CG         &4144.21          &22.42        &7.72                &14.70         \\
-    \hline 
-    MG         &1133.23          &24.50        &5.34                &19.16          \\
-   \hline
-    EP         &6170.30	        &16.19	       &0.02	            &16.17          \\
-   \hline
-    LU         &39477.28        &20.43	       &0.07	            &20.36          \\
+    Taurus      & Intel       & 2.3  & 1.2  & 0.1     & 6               & \np[W]{35} \\
+                & Xeon        &       &       &       &                 &            \\
+                & E5-2630     &       &       &       &                 &            \\         
     \hline
-    BT         &26169.55	    &25.34	       &6.62	            &18.71          \\
-   \hline
-    SP         &19620.09	    &19.32	       &3.66	            &15.66          \\
-   \hline
-    FT         &6094.07	        &23.17	       &0.36	            &22.81          \\
-\hline 
-  \end{tabular}
-  \label{table:res_s1}
-\end{table}
-
-
-
-\begin{table}[htb]
-  \caption{The results of 90\%-10\% powers scenario}
-  % title of Table
-  \centering
-  \begin{tabular}{|*{6}{l|}}
+    Graphene    & Intel       & 2.53  & 1.2   & 0.133 & 4               & \np[W]{23} \\
+                & Xeon        &       &       &       &                 &            \\
+                & X3440       &       &       &       &                 &            \\    
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
+    Griffon     & Intel       & 2.5   & 2     & 0.5   & 4               & \np[W]{46} \\
+                & Xeon        &       &       &       &                 &            \\
+                & L5420       &       &       &       &                 &            \\  
     \hline
-    CG         &2812.38	         &36.36	       &6.80                &29.56         \\
-    \hline 
-    MG         &825.427	         &38.35	       &6.41	            &31.94         \\
-   \hline
-    EP         &5281.62	         &35.02	       &2.68	            &32.34         \\
-   \hline
-    LU         &31611.28	     &39.15        &3.51	            &35.64        \\
+    Graphite    & Intel       & 2     & 1.2   & 0.1   & 8               & \np[W]{35} \\
+                & Xeon        &       &       &       &                 &            \\
+                & E5-2650     &       &       &       &                 &            \\  
     \hline
-    BT         &21296.46	     &36.70	       &6.60	            &30.10       \\
-   \hline
-    SP         &15183.42	     &35.19	       &11.76	            &23.43        \\
-   \hline
-    FT         &3856.54	         &40.80	       &5.67	            &35.13        \\
-\hline 
   \end{tabular}
-  \label{table:res_s2}
-\end{table}
+  \label{table:grid5000}
+\end{table} 
 
 
-\begin{figure}
-  \centering
-  \subfloat[Comparison the average of the results on 8 nodes]{%
-    \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
-  \quad%
-  \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
-    \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
-  \label{fig:comp}
-  \caption{The comparison of the three power scenarios}
-\end{figure}
 
-To compare the results of these three powers scenarios, we are take the average of the performance degradation, the energy saving and the distances for all NAS benchmarks running on 8 or 9 nodes of class C, as in figure (\ref{fig:sen_comp}). Thus, according to the average of these results, the energy saving ratio is increased when using a higher percentage for dynamic power (e.g. 90\%-10\% scenario), due to increase in dynamic energy. While the average of energy saving is decreased in 70\%-30\% scenario. Because the static energy consumption is increase. Moreover, the average of distances is more related to energy saving changes. The average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). The raison behind these relations, that the proposed algorithm optimize both energy consumption and performance in the same time. Therefore, when using a higher ratio for dynamic power the algorithm selecting bigger frequency scaling factors values, more energy saving versus more performance degradation, for example see the figure (\ref{fig:scales_comp}). The inverse happen when using a higher ratio for  static  power, the algorithm proportionally  selects a smaller scaling values, less energy saving versus less performance degradation. This is because the  
-algorithm is optimizes the  static energy consumption that is always related to the execution time. 
 
-\subsection{The verifications of the proposed method}
-\label{sec.verif}
-The precision of the proposed algorithm mainly depends on the execution time prediction model EQ(\ref{eq:perf}) and the energy model EQ(\ref{eq:energy}). The energy model is significantly depends on the execution time model, that the static energy is related linearly. So, our work is depends mainly on execution time model. To verifying thid model, we are compare the predicted execution time with the real execution time (Simgrid time) values that gathered  offline from the NAS benchmarks class B executed on 8 or 9 nodes. The execution time model can predicts the real execution time by maximum normalized error equal to 0.03 for all the NAS benchmarks. The second verification that we are made is for the proposed scaling algorithm to prove its ability to selects the best vector of the frequency scaling factors. Therefore, we are expand the algorithm to test at each iteration the frequency scaling factor of the slowest node with the all available scaling factors of the other nodes, all possible solutions. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the expanded algorithms and the proposed algorithm are identical. While the proposed algorithm is runs  by 10 times faster on average compare to the expanded algorithm.
+\subsection{The experimental results of the scaling algorithm}
+\label{sec.res}
+
+\subsection{The experimental results of multi-cores clusters}
+\label{sec.res}
+
+\subsection{The results for different power consumption scenarios}
+\label{sec.compare}
+
+
+
+
+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
+
+
 
 \section{Conclusion}
 \label{sec.concl}
 
 
+
 \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  MÃ©socentre de  calcul de  Franche-ComtÃ©. As  a  PhD student,
+Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
+supporting his work.
 
 % trigger a \newpage just before the given reference
 % number - used to balance the columns on the last page
@@ -792,6 +721,6 @@ The precision of the proposed algorithm mainly depends on the execution time pre
 %%% End:
 
 % LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords:  CMOS EQ EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex
-% LocalWords:  de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
-% LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
+% LocalWords:  CMOS EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex GPU
+% LocalWords:  de badri muslim MPI SimGrid GFlops Xeon EP BT GPUs CPUs AMD
+%  LocalWords:  Spiliopoulos scalability
