corrections

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 7262696d4780b7ab299f91366cbe3ebc7ccd3e26..df0bb8fd4f408d28085056d2e0901053a20fd14d 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -9,6 +9,7 @@
 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{url}
 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{url}

+

 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage[autolanguage,np]{numprint}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage[autolanguage,np]{numprint}


@@ -53,8 +54,6 @@
 \newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
 \newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
 \newcommand{\Tcp}[1][]{\Xsub{T}{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}}
 \newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Told}{\Xsub{T}{Old}}


@@ -85,18 +84,168 @@
 \maketitle
 
 \begin{abstract}
 \maketitle
 
 \begin{abstract}

-  
+  Computing platforms are 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 reduces the frequency of a CPU to lower its
+  energy consumption.  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 heterogeneous
+  platforms (heterogeneous CPUs) is presented.  It selects the frequencies and tries to give the best
+  trade-off between energy saving and performance degradation, for each node
+  computing the message passing iterative application. The algorithm has a small
+  overhead and works without training or profiling. It uses a new energy model
+  for message passing iterative applications running on a heterogeneous
+  platform. The proposed algorithm is evaluated on the SimGrid simulator while
+  running the NAS parallel benchmarks.  The experiments show that it reduces the
+  energy consumption by up to \np[\%]{34} while limiting the performance
+  degradation as much as possible.  Finally, the algorithm is compared to an
+  existing method, the comparison results show that it outperforms the
+  latter, on average it saves \np[\%]{4} more energy while keeping the same performance. 

 
 \end{abstract}
 
 \section{Introduction}
 \label{sec.intro}
 
 
 \end{abstract}
 
 \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 November 2014 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 2014
+\cite{U.S_Annual.Energy.Outlook.2014}, 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 L-CSC from the GSI Helmholtz Center
+which became the top of the Green500 list in November 2014 \cite{Green500_List}.
+This heterogeneous platform executes more than 5 GFLOPS per watt while consuming
+57.15 kilowatts.
+
+Besides platform improvements, there are many software and hardware techniques
+to lower the energy consumption of these platforms, such as scheduling, DVFS,
+\dots{} DVFS is a widely used process to reduce the energy consumption of a
+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}, a frequency selecting algorithm was proposed to reduce
+the energy consumption of message passing iterative applications running over
+homogeneous platforms.  The results of the experiments show significant energy
+consumption reductions. In this paper, a new frequency selecting algorithm
+adapted for heterogeneous platform is presented. It selects the vector of
+frequencies, for a heterogeneous platform running a message passing iterative
+application, that simultaneously tries to offer the maximum energy reduction and
+minimum performance degradation ratio. The algorithm has a very small overhead,
+works online and does not need any training or profiling.
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents some
+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 heterogeneous 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 frequency selecting algorithm then
+the precision of the proposed algorithm is verified.  Section~\ref{sec.expe}
+presents the results of applying the algorithm on the NAS parallel benchmarks
+and executing them on a heterogeneous platform. It shows the results of running
+three different power scenarios and comparing them. Moreover, it also 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, \dots{}
+
+In this paper, we are interested in reducing energy for message passing
+iterative synchronous applications running over heterogeneous 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}
+
+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 workloads 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.  In contrast to the above described papers, this paper presents the
+following contributions :
+\begin{enumerate}
+\item two new energy and performance models for message passing iterative
+  synchronous applications running over a heterogeneous platform. Both models
+  take into account communication and slack times. The models can predict the
+  required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous
+  platforms. The algorithm has a very small overhead and does not need any
+  training or profiling. It uses a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing iterative synchronous application.
+
+\end{enumerate}

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


@@ -106,9 +255,11 @@
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications 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 platforms. A heterogeneous platform is defined 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.

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


@@ -117,13 +268,13 @@ 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 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
 the communication time for every iteration on a node. However, due to the

-heterogeneous computation power of the computing clusters, slack times may occur
+heterogeneous computation power of the computing nodes, 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
 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 
+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
 which has the highest computation time and no slack time.
 
 Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in


@@ -149,39 +300,40 @@ 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.
 
 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
+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

 be able to predict the execution time of message passing synchronous iterative
 be able to predict the execution time of message passing synchronous iterative

-applications running over a heterogeneous grid, for different vectors of
+applications running over a heterogeneous platform, for different vectors of

 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
 vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
 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}).
 \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 = \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) +  \MinTcm

 \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 $i$ 
-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
+Where:
+\begin{equation}
+  \label{eq:perf2}
+  \MinTcm = \min_{i=1,2,\dots,N} (\Tcm[i])
+\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 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
 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
+of message passing distributed applications for homogeneous
+architectures~\cite{Our_first_paper}.  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.
 
 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{Malkowski_energy.efficient.high.performance.computing,
 \subsection{Energy model for heterogeneous platform}
 
 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,


@@ -264,32 +416,31 @@ processor after scaling its frequency is computed as follows:
   \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
 \end{equation}
 
   \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
 \end{equation}
 

-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
+In the considered heterogeneous platform, each processor $i$ may 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
+$\Tcp[i]$ may be different and 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
-$\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
+and reduce slack times.  The communication time of a processor $i$ is noted as
+$\Tcm[i]$ 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
 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:
+heterogeneous platform during one iteration is the summation of all dynamic and
+static energies for each processor.  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} {(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
 \end{multline}
 
 Reducing the frequencies of the processors according to the vector of scaling

-factors $(S_{11}, S_{12},\dots, S_{NM})$ may degrade the performance of the application
+factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the application

 and thus, increase the static energy because the execution time is
 increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
 for the iterative application can be measured by measuring the energy
 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


@@ -312,11 +463,11 @@ 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
 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
+work~\cite{Our_first_paper}, 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
 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
+are interested in heterogeneous clusters 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.
 
 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.
 


@@ -329,58 +480,49 @@ 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}
+\begin{multline}

   \label{eq:pnorm}
   \label{eq:pnorm}

-  \Pnorm = \frac{\Tnew}{\Told}                 
-\end{equation}
-
+  \Pnorm = \frac{\Tnew}{\Told}\\
+       {} = \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}

 
 

-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
 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}
+maximum frequency for all nodes:
+\begin{multline}

   \label{eq:enorm}
   \label{eq:enorm}

-  \Enorm = \frac{\Ereduced}{\Eoriginal} 
-\end{equation}
-
-Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
-computed as in ().
-
-\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}
+  \Enorm = \frac{\Ereduced}{\Eoriginal} \\
+  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}
+\end{multline}
+Where $\Ereduced$ and $\Eoriginal$ are computed using (\ref{eq:energy}) and
+$\Tnew$ and $\Told$ are computed as in (\ref{eq:pnorm}).

 
 While the main goal is to optimize the energy and execution time at the same
 
 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
+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
 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
+execution time follow the same evolution according to the vector of scaling factors. Therefore, the equation of the

 normalized execution time is inverted which gives the normalized performance
 equation, as follows:
 normalized execution time is inverted which gives the normalized performance
 equation, as follows:

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

   \label{eq:pnorm_inv}
   \label{eq:pnorm_inv}

-  \Pnorm = \frac{\Told}{\Tnew}          
-\end{equation}
+  \Pnorm = \frac{\Told}{\Tnew}\\
+          = \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}

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

-  \subfloat[Homogeneous cluster]{%
+  \subfloat[Homogeneous platform]{%

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

-  \subfloat[Heterogeneous grid]{%
+  \subfloat[Heterogeneous platform]{%

     \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}
   \caption{The energy and performance relation}
     \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}
   \caption{The energy and performance relation}


@@ -395,19 +537,19 @@ Figure~\ref{fig:r2}. Then the objective function has the following form:
 \begin{equation}
   \label{eq:max}
   \MaxDist =
 \begin{equation}
   \label{eq:max}
   \MaxDist =

-\mathop{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
-      (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
-       \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
+  \mathop{\max_{i=1,\dots F}}_{j=1,\dots,N}
+      (\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_{ij})}^{\text{Minimize}} )

 \end{equation}
 \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 
-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 
+where $N$ is the number of nodes 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}.
 
 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 }
+\section{The scaling factors selection algorithm for heterogeneous platforms }

 \label{sec.optim}
 
 \begin{algorithm}
 \label{sec.optim}
 
 \begin{algorithm}


@@ -415,44 +557,44 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     % \footnotesize
     \Require ~
     \begin{description}
     % \footnotesize
     \Require ~
     \begin{description}

-    \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.
+    \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 differences between two successive frequencies for all nodes.

     \end{description}
     \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 $\Sopt[1],\Sopt[2] \dots, \Sopt[N]$ is a vector of optimal scaling factors

 
 

-    \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.
+    \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.

     \If{(not the first frequency)}
     \If{(not the first frequency)}

-          \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$

     \EndIf
     \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 $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
+    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}$
+    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \Told)}$
+    \State $\Sopt[i] \gets 1,~i=1,\dots,N. $

     \State $\Dist \gets 0 $
     \State $\Dist \gets 0 $

-    \While {(all nodes have not reached their  minimum   \newline\hspace*{2.5em} frequency \textbf{or}  $\Pnorm - \Enorm < 0 $)}
+    \While {(all nodes not reach their  minimum  frequency)}

         \If{(not the last freq. \textbf{and} not the slowest node)}
         \If{(not the last freq. \textbf{and} not the slowest node)}

-        \State $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
-        \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
+        \State $F_i \gets F_i - \Fdiff[i],~i=1,\dots,N.$
+        \State $S_i \gets \frac{\Fmax[i]}{F_i},~i=1,\dots,N.$

         \EndIf
         \EndIf

-       \State $\Tnew \gets $ computed as  in equations (\ref{eq:perf}). 
-       \State $\Ereduced \gets $ computed as  in equations (\ref{eq:energy}). 
+       \State $\Tnew \gets \max_{i=1,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm $
+%       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + \sum_{i=1}^{N} {(\Ps[i] \cdot \rlap{\Tnew)}} $
+       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \rlap{\Tnew)}} $

        \State $\Pnorm \gets \frac{\Told}{\Tnew}$
        \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
        \State $\Pnorm \gets \frac{\Told}{\Tnew}$
        \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}

-        \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
+        \State $\Sopt[i] \gets S_{i},~i=1,\dots,N. $

         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile
         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile

-    \State  Return $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$
+    \State  Return $\Sopt[1],\Sopt[2],\dots,\Sopt[N]$

   \end{algorithmic}
   \end{algorithmic}

-  \caption{Scaling factors selection algorithm}
+  \caption{frequency scaling factors selection algorithm}

   \label{HSA}
 \end{algorithm}
 
   \label{HSA}
 \end{algorithm}
 


@@ -478,12 +620,10 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 
 \subsection{The algorithm details}
 
 
 \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
+In this section, Algorithm~\ref{HSA} is presented. It 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
 energy consumption and maximizing the performance of a message passing

-synchronous iterative application executed on a  grid. It works
+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
 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


@@ -496,18 +636,18 @@ scaling algorithm is called in the iterative MPI program.
 
 \begin{figure}[!t]
   \centering
 
 \begin{figure}[!t]
   \centering

-  \includegraphics[scale=0.45]{fig/init_freq}
+  \includegraphics[scale=0.5]{fig/start_freq}

   \caption{Selecting the initial frequencies}
   \label{fig:st_freq}
 \end{figure}
 
   \caption{Selecting the initial frequencies}
   \label{fig:st_freq}
 \end{figure}
 

-Nodes from distinct clusters in a grid have different computing powers, thus
+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.  The algorithm takes into account this
 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
+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
 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


@@ -516,7 +656,7 @@ 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[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}

 \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


@@ -524,28 +664,24 @@ maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ 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_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}

 \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

-Figure~\ref{fig:st_freq}, the nodes are sorted by their computing powers in
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power 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
 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
+optimal vector of frequencies because selecting 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 algorithm that
 selects the frequency scaling factors starts the search method from these
 initial frequencies and takes a downward search direction toward lower
 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
+frequencies. The algorithm iterates on all remaining 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 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
 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


@@ -553,154 +689,541 @@ 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
 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
+consumed energy for an application running on a homogeneous platform and a
+heterogeneous platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous platform 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
 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
+the heterogeneous 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
 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. 
+maximum distance between the energy curve and the performance curve is while the
+scaling factors are varying which results in bigger energy savings. 
+Finally, in a homogeneous platform the energy consumption is increased when the scaling factor is very high. 
+Indeed, the dynamic energy  saved  by reducing the frequency of the processor is compensated by the significant increase of the execution time and thus the increased of the static energy. On the other hand, in a heterogeneous platform this is not the case.
+
+\subsection{The evaluation of the proposed algorithm}
+\label{sec.verif.algo}
+
+The precision of the proposed algorithm mainly depends on the execution time
+prediction model defined in (\ref{eq:perf}) and the energy model computed by
+(\ref{eq:energy}).  The energy model is also significantly dependent on the
+execution time model because the static energy is linearly related to the
+execution time and the dynamic energy is related to the computation time. So,
+all the works presented in this paper are based on the execution time model. To
+verify this model, the predicted execution time was compared to the real
+execution time over SimGrid/SMPI simulator,
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}, for all the NAS
+parallel benchmarks NPB v3.3 \cite{NAS.Parallel.Benchmarks}, running class B on
+8 or 9 nodes. The comparison showed that the proposed execution time model is
+very precise, the maximum normalized difference between the predicted execution
+time and the real execution time is equal to 0.03 for all the NAS benchmarks.
+
+Since the proposed algorithm is not an exact method, it does not test all the
+possible solutions (vectors of scaling factors) in the search space. To prove
+its efficiency, it was compared on small instances to a brute force search
+algorithm that tests all the possible solutions. The brute force algorithm was
+applied to different NAS benchmarks classes with different number of nodes. The
+solutions returned by the brute force algorithm and the proposed algorithm were
+identical and the proposed algorithm was on average 10 times faster than the
+brute force algorithm. It 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 on average \np[ms]{0.04} for 4
+nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling
+factors vector.  The algorithm complexity is $O(F\cdot N)$, where $F$ is the
+maximum number of available frequencies, and $N$ is the number of computing
+nodes. The algorithm needs from 12 to 20 iterations to select the best vector of
+frequency scaling factors that gives the results of the next sections.

 
 

+\begin{table}[!t]
+  \caption{Heterogeneous nodes characteristics}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    Node   & Simulated & Max   & Min   & Diff. & Dynamic    & Static    \\
+    type   & GFLOPS    & Freq. & Freq. & Freq. & power      & power     \\
+           &           & GHz   & GHz   & GHz   &            &           \\
+    \hline
+    1      & 40        & 2.50  & 1.20  & 0.100 & \np[W]{20} & \np[W]{4} \\
+    \hline
+    2      & 50        & 2.66  & 1.60  & 0.133 & \np[W]{25} & \np[W]{5} \\
+    \hline
+    3      & 60        & 2.90  & 1.20  & 0.100 & \np[W]{30} & \np[W]{6} \\
+    \hline
+    4      & 70        & 3.40  & 1.60  & 0.133 & \np[W]{35} & \np[W]{7} \\
+    \hline
+  \end{tabular}
+  \label{table:platform}
+\end{table}

 
 \section{Experimental results}
 \label{sec.expe}
 
 \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.  
+To evaluate the efficiency and the overall energy consumption reduction of
+Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3 which 
+is composed of synchronous message passing applications. The 
+experiments were executed on the simulator SimGrid/SMPI which offers easy tools
+to create a heterogeneous platform and run message passing applications over it.
+The heterogeneous platform that was used in the experiments, had one core per
+node because just one process was executed per node.  The heterogeneous platform
+was composed of four types of nodes. Each type of nodes had different
+characteristics such as the maximum CPU frequency, the number of available
+frequencies and the computational power, see Table~\ref{table:platform}. The
+characteristics of these different types of nodes are inspired from the
+specifications of real Intel processors.  The heterogeneous platform had up to
+144 nodes and had nodes from the four types in equal proportions, for example if
+a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the
+constructors of CPUs do not specify the dynamic and the static power of their
+CPUs, for each type of node they were chosen proportionally to its computing
+power (FLOPS).  In the initial heterogeneous platform, while computing with
+highest frequency, each node consumed an amount of power proportional to its
+computing power (which corresponds to \np[\%]{80} of its dynamic power and the
+remaining \np[\%]{20} to the static power), the same assumption was made in
+\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.  Finally, These
+nodes were connected via an Ethernet network with \np[Gbit/s]{1} bandwidth.

 
 

-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. 

 
 

+\subsection{The experimental results of the scaling algorithm}
+\label{sec.res}

 
 

+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
+MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes:
+A, B and C. However, due to the lack of space in this paper, only the results of
+the biggest class, C, are presented while being run on different number of
+nodes, ranging from 8 to 128 or 144 nodes depending on the benchmark being
+executed. Indeed, the benchmarks CG, MG, LU, EP and FT had to be executed on 1,
+2, 4, 8, 16, 32, 64, or 128 nodes.  The other benchmarks such as BT and SP had
+to be executed on 1, 4, 9, 16, 36, 64, or 144 nodes.

 
 

-we are interested  to run NAS parallel v3.3 \cite{NAS.Parallel.Benchmarks} over grid'5000.
-We are used seven benchmarks, CG, MG, EP, LU, BT, SP and FT. These benchmarks used seven different types of classes.
-These classes are S, W, A, B, C, D, E, where S represents the smaller problem size that used by benchmark and
-E is represents the biggest class. In this work, the class D is used for all benchmarks in all the experiments that will 
-be showed in the coming sections.
-Moreover, the NAS parallel benchmarks have different computations and communications ratios, then it is interested 
-to study their energy consumption and their performance on real testbed such as grid'5000.
-In this work, the NAS benchmarks are executed over two sites, Lyon and Nancy sites, of grid'5000.
-These two sites had seven different types of computing clusters as in figure (\ref{fig:grid5000}).
+\begin{table}[!t]
+  
+% \end{table}

 
 

-\begin{figure}[!t]
+  
+% \begin{table}[!t]
+  \caption{Running NAS benchmarks on 8 and 9 nodes }
+  % title of Table

   \centering
   \centering

-  \includegraphics[scale=1]{fig/grid5000}
-  \caption{The selected two sites of grid'5000}
-  \label{fig:grid5000}
-\end{figure}
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & 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}

 
 

-Four clusters from the two sites are selected in the experiments, one cluster from 
-Lyon site, Taurus cluster, and three clusters from Nancy site where are Graphene, 
-Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while their nodes are
-different from the nodes of other clusters in many aspects such as: computing power, power consumption, available 
-frequencies ranges and the network features, the bandwidth and the latency. The Table \ref{table:grid5000} shows 
-the details characteristics of these four clusters.
+  \medskip
+% \begin{table}[!t]
+  \caption{Running NAS benchmarks on 16 nodes }
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & 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}[!t]
-  \caption{CPUs characteristics of the selected clusters}
+  \medskip
+% \begin{table}[!t]
+  \caption{Running NAS benchmarks on 32 and 36 nodes }

   % title of Table
   \centering
   % title of Table
   \centering

-  \begin{tabular}{|*{7}{c|}}
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & 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
     \hline

-    Cluster     & CPU         & Max   & Min   & Diff. & no. of cores    & dynamic power   \\
-    Name        & model       & Freq. & Freq. & Freq. & per CPU         & of one core     \\
-                &             & GHz   & GHz   & GHz   &                 &           \\
+    MG      &  4.30     &  1355.58      & 28.93    & 8.85          & 20.08    \\

     \hline
     \hline

-    Taurus      & Intel       & 2.3  & 1.2  & 0.1     & 6               & \np[W]{35} \\
-                & Xeon        &       &       &       &                 &            \\
-                & E5-2630     &       &       &       &                 &            \\         
+    EP      &  9.96     &  5519.68      & 26.98    & 0.02          & 26.96    \\

     \hline
     \hline

-    Graphene    & Intel       & 2.53  & 1.2   & 0.133 & 4               & \np[W]{23} \\
-                & Xeon        &       &       &       &                 &            \\
-                & X3440       &       &       &       &                 &            \\    
+    LU      & 99.93     & 67463.43      & 23.60    & 2.45          & 21.15    \\

     \hline
     \hline

-    Griffon     & Intel       & 2.5   & 2     & 0.5   & 4               & \np[W]{46} \\
-                & Xeon        &       &       &       &                 &            \\
-                & L5420       &       &       &       &                 &            \\  
+    BT      & 48.61     & 23796.97      & 34.62    & 5.83          & 28.79    \\

     \hline
     \hline

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

-  \label{table:grid5000}
-\end{table} 
-
-The grid'5000 testbed provided some monitoring and measurements features to captured 
-the power consumption values for each node in any cluster of Lyon and Nancy sites. 
-The power consumed for each node from the selected four clusters is measured. 
-While the power consumed by any computing node is a collection of the powers  consumed by 
-hard drive, main-board, memory and node's computing cores, for more detail refer to
-\cite{Energy_measurement}. Therefore, the dynamic power consumed
-by one core is not allowed to measured alone. To overcome this problem, firstly, 
-we measured the power consumed by one node when there is no computation, when 
-the CPU is in the idle state. The second step, we run EP benchmark, there is no communications 
-in this benchmarks, over one core with maximum frequency of the desired node and 
-capturing the power consumed by a node, this representing the peak power of the node with one core.  
-The difference between the peak power  and the idle power representing the 
-dynamic power consumption of that core with maximum frequency, for example see figure(\ref{fig:power_cons}). 
-The $\Ppeak[jx]$ is the peak power value in time $x$ with maximum frequency for one core of node $j$, 
-and $\Pidle[jy]$ is the idle power value in time $y$ for the one  core of the node $j$ . 
-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} (\Ppeak[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
-\end{equation}
+  \label{table:res_32n}
+% \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.
-We are computed the dynamic powers, using the equation (\ref{eq:pdyn}), for all nodes in the 
-selected clusters, which is recorded in table  \ref{table:grid5000}.
-On the other side, the static power consumption by one core is embedded with whole idle power consumption of the node.
-Indeed, the static power is represents as ratio from  dynamic power. So, we supposed  
-the static power consumption represented  as \np[\%]{20} of dynamic power consumption of the core, 
-the same assumption was made in \cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy}.
-\begin{figure}[!t]
+  \medskip
+% \begin{table}[!t]
+  \caption{Running NAS benchmarks on 64 nodes }
+  % title of Table

   \centering
   \centering

-  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
-  \caption{The power consumption by one core from Taurus cluster}
-  \label{fig:power_cons}
-\end{figure}
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & 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     &   5471.11     & 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}
+
+  \medskip
+% \begin{table}[!t]
+  \caption{Running NAS benchmarks on 128 and 144 nodes }
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    \hspace{-2.2084pt}%
+    Program & 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}

 
 

+\begin{figure}[!t]
+  \centering
+  \subfloat[Energy saving (\%)]{%
+    \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%

 
 

-\subsection{The experimental results of the scaling algorithm}
-\label{sec.res}
+  \subfloat[Performance degradation (\%)]{%
+    \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
+  \label{fig:avg}
+  \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}
+\end{figure}

 
 

-\subsection{The experimental results of multi-cores clusters}
-\label{sec.res}
+The overall energy consumption was computed for each instance according to the
+energy consumption model (\ref{eq:energy}), with and without applying the
+algorithm. The execution time was also measured for all these experiments. Then,
+the energy saving and performance degradation percentages were computed for each
+instance.  The results are presented in Tables 
+\ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n},
+\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
+values from many experiments for energy savings and performance degradation.
+The tables show the experimental results for running the NAS parallel benchmarks
+on different numbers of nodes.  The experiments show that the algorithm
+significantly reduces the energy consumption (up to \np[\%]{34}) and tries to
+limit the performance degradation.  They also show that the energy saving
+percentage decreases when the number of computing nodes increases.  This
+reduction is due to the increase of the communication times compared to the
+execution times when the benchmarks are run over a higher number of nodes.
+Indeed, the benchmarks with the same class, C, are executed on different numbers
+of nodes, so the computation required for each iteration is divided by the
+number of computing nodes.  On the other hand, more communications are required
+when increasing the number of nodes so the static energy increases linearly
+according to the communication time and the dynamic power is less relevant in
+the overall energy consumption.  Therefore, reducing the frequency with
+Algorithm~\ref{HSA} is less effective in reducing the overall energy savings. It
+can also be noticed that for the benchmarks EP and SP that contain little or no
+communications, the energy savings are not significantly affected by the high
+number of nodes.  No experiments were conducted using bigger classes than D,
+because they require a lot of memory (more than \np[GB]{64}) when being executed
+by the simulator on one machine.  The maximum distance between the normalized
+energy curve and the normalized performance for each instance is also shown in
+the result tables. It decrease in the same way as the energy saving percentage.
+The tables also show that the performance degradation percentage is not
+significantly increased when the number of computing nodes is increased because
+the computation times are small when compared to the communication times.
+
+Figures~\ref{fig:energy} and \ref{fig:per_deg} present the energy saving and
+performance degradation respectively for all the benchmarks according to the
+number of used nodes. As shown in the first plot, the energy saving percentages
+of the benchmarks MG, LU, BT and FT decrease linearly when the number of nodes
+increase. While for the EP and SP benchmarks, the energy saving percentage is
+not affected by the increase of the number of computing nodes, because in these
+benchmarks there are little or no communications. Finally, the energy saving of
+the CG benchmark significantly decreases when the number of nodes increase
+because this benchmark has more communications than the others. The second plot
+shows that the performance degradation percentages of most of the benchmarks
+decrease when they run on a big number of nodes because they spend more time
+communicating than computing, thus, scaling down the frequencies of some nodes
+has less effect on the performance.

 
 \subsection{The results for different power consumption scenarios}
 \label{sec.compare}
 
 
 \subsection{The results for different power consumption scenarios}
 \label{sec.compare}
 

+The results of the previous section were obtained while using processors that
+consume during computation an overall power which is \np[\%]{80} composed of
+dynamic power and of \np[\%]{20} of static power. In this section, these ratios
+are changed and two new power scenarios are considered in order to evaluate how
+the proposed algorithm adapts itself according to the static and dynamic power
+values.  The two new power scenarios are the following:
+
+\begin{itemize}
+\item \np[\%]{70} of dynamic power and \np[\%]{30} of static power
+\item \np[\%]{90} of dynamic power and \np[\%]{10} of static power
+\end{itemize}
+
+The NAS parallel benchmarks were executed again over processors that follow the
+new power scenarios.  The class C of each benchmark was run over 8 or 9 nodes
+and the results are presented in Tables~\ref{table:res_s1} and
+\ref{table:res_s2}. These tables show that the energy saving percentage of the
+\np[\%]{70}-\np[\%]{30} scenario is smaller for all benchmarks compared to the
+energy saving of the \np[\%]{90}-\np[\%]{10} scenario.  Indeed, in the latter
+more dynamic power is consumed when nodes are running on their maximum
+frequencies, thus, scaling down the frequency of the nodes results in higher
+energy savings than in the \np[\%]{70}-\np[\%]{30} scenario. On the other hand,
+the performance degradation percentage is smaller in the \np[\%]{70}-\np[\%]{30}
+scenario compared to the \np[\%]{90}-\np[\%]{10} scenario. This is due to the
+higher static power percentage in the first scenario which makes it more
+relevant in the overall consumed energy.  Indeed, the static energy is related
+to the execution time and if the performance is degraded the amount of consumed
+static energy directly increases.  Therefore, the proposed algorithm does not
+really significantly scale down much the frequencies of the nodes in order to
+limit the increase of the execution time and thus limiting the effect of the
+consumed static energy.
+
+Both new power scenarios are compared to the old one in
+Figure~\ref{fig:sen_comp}. It shows the average of the performance degradation,
+the energy saving and the distances for all NAS benchmarks of class C running on
+8 or 9 nodes.  The comparison shows that the energy saving ratio is proportional
+to the dynamic power ratio: it is increased when applying the
+\np[\%]{90}-\np[\%]{10} scenario because at maximum frequency the dynamic energy
+is the most relevant in the overall consumed energy and can be reduced by
+lowering the frequency of some processors. On the other hand, the energy saving
+decreases when the \np[\%]{70}-\np[\%]{30} scenario is used because the dynamic
+energy is less relevant in the overall consumed energy and lowering the
+frequency does not return big energy savings.  Moreover, the average of the
+performance degradation is decreased when using a higher ratio for static power
+(e.g.  \np[\%]{70}-\np[\%]{30} scenario and \np[\%]{80}-\np[\%]{20}
+scenario). Since the proposed algorithm optimizes the energy consumption when
+using a higher ratio for dynamic power the algorithm selects bigger frequency
+scaling factors that result in more energy saving but less performance, for
+example see Figure~\ref{fig:scales_comp}. The opposite happens when using a
+higher ratio for static power, the algorithm proportionally selects smaller
+scaling values which result in less energy saving but also less performance
+degradation.
+
+\begin{table}[!t]
+  \caption{The results of the \np[\%]{70}-\np[\%]{30} power scenario}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{6}{r|}}
+    \hline
+    Program & Energy        & Energy   & Performance   & Distance \\
+    name    & consumption/J & saving\% & degradation\% &          \\
+    \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    \\
+    \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}[!t]
+  \caption{The results of the \np[\%]{90}-\np[\%]{10} power scenario}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{6}{r|}}
+    \hline
+    Program & Energy          & Energy      & Performance   & Distance \\
+    name    & consumption/J   & saving\%    & degradation\% &          \\
+    \hline
+    CG      &  2812.38        & 36.36       &  6.80         & 29.56    \\
+    \hline
+    MG      &   825.43        & 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    \\
+    \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}
+
+\begin{table}[!t]
+  \caption{Comparing the proposed algorithm}
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    Program & \multicolumn{2}{c|}{Energy saving \%}
+            & \multicolumn{2}{c|}{Perf.  degradation \%}
+            & \multicolumn{2}{c|}{Distance} \\
+    \cline{2-7}
+    name    & EDP   & MaxDist & EDP  & MaxDist & EDP   & MaxDist \\
+    \hline
+    CG      & 27.58 & 31.25   & 5.82 & 7.12    & 21.76 & 24.13   \\
+    \hline
+    MG      & 29.49 & 33.78   & 3.74 & 6.41    & 25.75 & 27.37   \\
+    \hline
+    LU      & 19.55 & 28.33   & 0.00 & 0.01    & 19.55 & 28.22   \\
+    \hline
+    EP      & 28.40 & 27.04   & 4.29 & 0.49    & 24.11 & 26.55   \\
+    \hline
+    BT      & 27.68 & 32.32   & 6.45 & 7.87    & 21.23 & 24.43   \\
+    \hline
+    SP      & 20.52 & 24.73   & 5.21 & 2.78    & 15.31 & 21.95   \\
+    \hline
+    FT      & 27.03 & 31.02   & 2.75 & 2.54    & 24.28 & 28.48   \\
+    \hline
+  \end{tabular}
+  \label{table:compare_EDP}
+\end{table}
+
+\begin{figure}[!t]
+  \centering
+  \subfloat[Comparison  between the results on 8 nodes]{%
+    \includegraphics[width=.33\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%

 
 

+  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
+    \includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
+  \label{fig:comp}
+  \caption{The comparison of the three power scenarios}
+\end{figure}
+
+\begin{figure}[!t]
+  \centering
+   \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
+  \caption{Trade-off comparison for NAS benchmarks class C}
+  \label{fig:compare_EDP}
+\end{figure}

 
 
 \subsection{The comparison of the proposed scaling algorithm }
 \label{sec.compare_EDP}
 
 
 
 \subsection{The comparison of the proposed scaling algorithm }
 \label{sec.compare_EDP}
 

-
+In this section, the scaling factors selection algorithm, called MaxDist, is
+compared to Spiliopoulos et al. algorithm
+\cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP.  They developed a
+green governor that regularly applies an online frequency selecting algorithm to
+reduce the energy consumed by a multicore architecture without degrading much
+its performance. The algorithm selects the frequencies that minimize the energy
+and delay products, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$ using
+the predicted overall energy consumption and execution time delay for each
+frequency.  To fairly compare both algorithms, the same energy and execution
+time models, equations (\ref{eq:energy}) and (\ref{eq:fnew}), were used for both
+algorithms to predict the energy consumption and the execution times. Also
+Spiliopoulos et al. algorithm was adapted to start the search from the initial
+frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm
+is an exhaustive search algorithm that minimizes the EDP and has the initial
+frequencies values as an upper bound.
+
+Both algorithms were applied to the parallel NAS benchmarks to compare their
+efficiency. Table~\ref{table:compare_EDP} presents the results of comparing the
+execution times and the energy consumption for both versions of the NAS
+benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
+nodes. The results show that our algorithm provides better energy savings than
+Spiliopoulos et al. algorithm, on average it results in \np[\%]{29.76} energy
+saving while their algorithm returns just \np[\%]{25.75}. The average of
+performance degradation percentage is approximately the same for both
+algorithms, about \np[\%]{4}.
+
+For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
+terms of  energy and  performance trade-off, see  Figure~\ref{fig:compare_EDP},
+because it maximizes the distance  between the energy saving and the performance
+degradation values while giving the same weight for both metrics.

 
 \section{Conclusion}
 \label{sec.concl}
 
 
 \section{Conclusion}
 \label{sec.concl}
 

-
+In this paper, a new online frequency selecting algorithm has been presented. It
+selects the best possible vector of frequency scaling factors that gives the
+maximum distance (optimal trade-off) between the predicted energy and the
+predicted performance curves for a heterogeneous platform. This algorithm uses a
+new energy model for measuring and predicting the energy of distributed
+iterative applications running over heterogeneous platforms. To evaluate the
+proposed method, it was applied on the NAS parallel benchmarks and executed over
+a heterogeneous platform simulated by SimGrid. The results of the experiments
+showed that the algorithm reduces up to \np[\%]{34} the energy consumption of a
+message passing iterative method while limiting the degradation of the
+performance. The algorithm also selects different scaling factors according to
+the percentage of the computing and communication times, and according to the
+values of the static and dynamic powers of the CPUs.  Finally, the algorithm was
+compared to Spiliopoulos et al.  algorithm and the results showed that it
+outperforms their algorithm in terms of energy-time trade-off.
+
+In the near future, this method  will be applied to real heterogeneous platforms
+to evaluate its  performance in a real study case. It  would also be interesting
+to evaluate its scalability over large scale heterogeneous platforms and measure
+the energy  consumption reduction it can  produce.  Afterward, we  would like to
+develop a similar method that  is adapted to asynchronous iterative applications
+where  each task  does not  wait for  other tasks  to finish  their  works.  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.

 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}