X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/7851a05d8c696e72f32cc2b36170ff31a213bfc1..7f9ca96425114b39b06dc8c85ff608e87facf26f:/mpi-energy2-extension/Heter_paper.tex?ds=inline

diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index f5917fa..f00d065 100644
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -54,7 +54,7 @@
 \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{\Pmax}[1][]{\Xsub{P}{max}_{\fxheight{#1}}}
 \newcommand{\Pidle}[1][]{\Xsub{P}{idle}_{\fxheight{#1}}}
 \newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
@@ -85,21 +85,183 @@
 
 \maketitle
 
-\begin{abstract}
-  
 
+\begin{abstract}
+\textcolor{blue}{
+  In recent years, green computing topic  has being became an important topic in 
+  the domain of the research. The increase in computing power of the computing 
+  platforms is increased the energy consumption and the carbon dioxide emissions.
+  Many techniques have being used to minimize the cost of the energy consumption 
+  and reduce environmental pollution. Dynamic voltage and frequency scaling (DVFS) 
+  is one of these techniques. It used to reduce the power consumption of the CPU 
+  while computing by lowering its frequency. Moreover, 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
+  grid (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
+  grid. The proposed algorithm is evaluated on real testbed, grid'5000 platform, while
+  running the NAS parallel benchmarks.  The experiments show that it reduces the
+  energy consumption on average up to \np[\%]{30} while declines the performance
+  on average by \np[\%]{3}. Finally, the algorithm is 
+  compared to an existing method, the comparison results show that it outperforms the
+  latter in term of energy and performance trade-off.}
 \end{abstract}
 
+
 \section{Introduction}
 \label{sec.intro}
-
-
+\textcolor{blue}{
+The need for more computing power is continually increasing. To partially
+satisfy this need, most supercomputers constructors just put more computing
+nodes in their platform. The resulting platforms may achieve higher floating
+point operations per second (FLOPS), but the energy consumption and the heat
+dissipation are also increased.  As an example, the Chinese supercomputer
+Tianhe-2 had the highest FLOPS in June 2015  according to the Top500 list
+\cite{TOP500_Supercomputers_Sites}.  However, it was also the most power hungry
+platform with its over 3 million cores consuming around 17.8 megawatts.
+Moreover, according to the U.S.  annual energy outlook 2015
+\cite{U.S_Annual.Energy.Outlook.2015}, the price of energy for 1 megawatt-hour
+was approximately equal to \$70.  Therefore, the price of the energy consumed by
+the Tianhe-2 platform is approximately more than \$10 million each year.  The
+computing platforms must be more energy efficient and offer the highest number
+of FLOPS per watt possible, such as the Shoubu-ExaScaler from RIKEN
+which became the top of the Green500 list in June 2015 \cite{Green500_List}.
+This heterogeneous platform executes more than 7 GFLOPS per watt while consuming
+50.32 kilowatts.
+}
+
+\textcolor{blue}{
+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} and \cite{pdsec2015} , a frequencies selecting algorithm was proposed to reduce
+the energy consumption of message passing iterative applications running over
+homogeneous  and heterogeneous clusters respectively.  
+The results of the experiments show significant energy
+consumption reductions. All the experimental results were conducted over 
+Simgrid simulator \cite{SimGrid}, which offers easy tools to create a homogeneous and heterogeneous platforms. In this paper, a new frequencies selecting algorithm
+adapted for heterogeneous grid platform is presented and executed over real testbed, 
+the grid'5000 platform \cite{grid5000}. It selects the vector of
+frequencies, for a heterogeneous grid 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.}
+
+\textcolor{blue}{
+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 grid. Section~\ref{sec.compet} presents the
+energy-performance objective function that maximizes the reduction of energy
+consumption while minimizing the degradation of the program's performance.
+Section~\ref{sec.optim} details the proposed frequencies selecting algorithm.
+Section~\ref{sec.expe} presents the results of applying the algorithm on the 
+NAS parallel benchmarks and executing them on a grid'5000 testbed. 
+It shows the results of running different scenarios using multi-cores and one core per node 
+and comparing them. It also shows the results of running
+three different power scenarios and comparing them. Moreover, it shows the
+comparison results between the proposed method and an existing method.  Finally,
+in Section~\ref{sec.concl} the paper ends with a summary and some future works.}
 
 \section{Related works}
 \label{sec.relwork}
 
-
-\section{The performance and energy consumption measurements on heterogeneous architecture}
+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 grid platforms.  Some
+works have already been done for such platforms and they can be classified into
+two types of heterogeneous platforms:
+\begin{itemize}
+\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
+\item the platform is only composed of heterogeneous CPUs.
+\end{itemize}
+
+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 grid platform. Both models
+  take into account communication and slack times. The models can predict the
+  required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous grid
+  platforms. The algorithm has a very small overhead and does not need any
+  training or profiling. It uses a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing iterative synchronous application.
+
+\end{enumerate}
+
+
+
+\section{The performance and energy consumption measurements on heterogeneous grid architecture}
 \label{sec.exe}
 
 \subsection{The execution time of message passing distributed iterative
@@ -183,7 +345,7 @@ 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}
+\subsection{Energy model for heterogeneous grid platform}
 
 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
   Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
@@ -312,12 +474,15 @@ 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
+increasing significantly the execution time.
+\textcolor{blue}{  In our previous
+works~\cite{Our_first_paper} and \cite{pdsec2015}, we proposed a methods that select the optimal
+frequency scaling factors for a homogeneous and a heterogeneous clusters respectively. 
+Both of the two methods 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
+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.
 
@@ -350,9 +515,9 @@ maximum frequency for all  nodes:
 \end{equation}
 
 Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
-computed as in ().
+computed as in (\ref{eq:eorginal}).
+
 
-\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])  + 
@@ -477,9 +642,6 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
   \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
@@ -569,7 +731,7 @@ maximum distance between the energy curve and the performance curve is, which re
 
 \section{Experimental results}
 \label{sec.expe}
-While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid.org}, 
+While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid}, 
 in this paper real experiments were conducted over the grid'5000 platform. 
 
 \subsection{Grid'5000 architature and power consumption}
@@ -578,31 +740,30 @@ Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites di
 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,
+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 
+ firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumption represents the 
 dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 
 
-\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])
+    \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
 \end{equation}
 
 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, 
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured maximum 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.
+measured value in maximum powers vector and the minimum measured value in the idle powers vector.
 
-On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  np[\%]{20} of dynamic power consumption of the core.
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.
 
 In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
 
@@ -614,8 +775,6 @@ the details characteristics of these four clusters. Moreover, the dynamic powers
 selected clusters and are presented in table  \ref{table:grid5000}.
 
 
-
-
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=1]{fig/grid5000}
@@ -623,7 +782,6 @@ selected clusters and are presented in table  \ref{table:grid5000}.
   \label{fig:grid5000}
 \end{figure}
 
-
 The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
 The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
 The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 
@@ -675,52 +833,52 @@ The benchmarks have seven different classes, S, W, A, B, C, D and E, that repres
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
-In this section, the scaling factor selection algorithm \ref{HSA}, is applied 
-to NAS parallel benchmarks. Seven benchmarks, CG, MG, EP, LU, BT, SP and FT, of the class D
-are executed over grid'5000 computing clusters. As mentioned previously, the experiments 
-of this paper obtained from a collection of many clusters distributed in two sites, Lyon and Nancy sites, 
-of grid'5000. Four different clusters are selected from these two sites to generate two 
-different scenarios. Each of these two scenarios used three clusters. The first scenario,
-is composed from three clusters that located in two sites, Lyon and Nancy sites. One of these three
-clusters is from Lyon site, Taurus cluster and the other two clusters are form Nancy site, 
-Graphene and Griffon clusters. The second scenario, is composed from three clusters that are 
-located in one site, Nancy site. These cluster are Graphite, Graphene and griffon. The main reason 
-behind using these two scenarios is because the first one is executing the NAS parllel benchmarks over 
-two sites that are connected via long distance network, then the computations to communications ratio 
-is very low due to the increase in communication times, while in the second scenario, all of the three clusters are 
-located in one site and they are connected via high speed local area networks, where the computations 
-to communications ratio is higher. Therefore, it is very interested to know the performance behaviour 
-and the energy consumption of NAS parallel benchmarks using the proposed method, when they run 
-over these two different platform scenarios. Moreover, The NAS parallel benchmarks are executed over 
+In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 
+to the NAS parallel benchmarks are presented. 
+
+As mentioned previously, the experiments 
+were conducted over two sites of grid'5000,  Lyon and Nancy sites. 
+Two scenarios were considered while selecting the clusters from these two sites :
+\begin{itemize}
+\item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected 
+ via a long distance network.
+\item In the second scenario nodes from three clusters that are located in one site, Nancy site.  
+\end{itemize}
+
+The main reason 
+behind using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
+scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 
+is very low due to the higher communication times which reduces the effect of DVFS operations.
+
+The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes form each cluster 
-are different, this  depends on the available number of nodes in each cluster. 
-Table \ref{tab:sc} shows the details of these two scenarios and the number of nodes 
-used from each cluster.
+are different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
+Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
 
 \begin{table}[h]
 
 \caption{The different clusters scenarios}
 \centering
-\begin{tabular}{|*{3}{c|}}
+\begin{tabular}{|*{4}{c|}}
 \hline
-\multirow{2}{*}{Scenario name}        & \multicolumn{2}{c|} {The participating clusters} \\ \cline{2-3} 
-                                      & Cluster name           & No. of  nodes of each cluster     \\ 
+\multirow{2}{*}{Scenario name}        & \multicolumn{3}{c|} {The participating clusters} \\ \cline{2-4} 
+                                      & Cluster & Site           & No. of  nodes     \\ 
 \hline
-\multirow{3}{*}{Two sites / 16 nodes} & Taurus                 & 5                      \\ \cline{2-3} 
-                                      & Graphene               & 5                      \\ \cline{2-3} 
-                                      & Griffon                & 6                      \\ 
+\multirow{3}{*}{Two sites / 16 nodes} & Taurus & Lyon                & 5                      \\ \cline{2-4} 
+                                      & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
+                                      & Griffon       & Nancy        & 6                      \\ 
 \hline
-\multirow{3}{*}{Tow sites / 32 nodes} & Taurus                 & 10                     \\ \cline{2-3} 
-                                      & Graphene               & 10                     \\ \cline{2-3} 
-                                      & Griffon                & 12                     \\ 
+\multirow{3}{*}{Tow sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
+                                      & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
+                                      & Griffon     &Nancy           & 12                     \\ 
 \hline
-\multirow{3}{*}{One site / 16 nodes}  & Graphite               & 4                      \\ \cline{2-3} 
-                                      & Graphene               & 6                      \\ \cline{2-3} 
-                                      & Griffon                & 6                      \\ 
+\multirow{3}{*}{One site / 16 nodes}  & Graphite    & Nancy            & 4                      \\ \cline{2-4} 
+                                      & Graphene     & Nancy           & 6                      \\ \cline{2-4} 
+                                      & Griffon         & Nancy        & 6                      \\ 
 \hline
-\multirow{3}{*}{One site / 32 nodes}  & Graphite               & 4                      \\ \cline{2-3} 
-                                      & Graphene               & 12                     \\ \cline{2-3} 
-                                      & Griffon                & 12                       \\ 
+\multirow{3}{*}{One site / 32 nodes}  & Graphite   & Nancy             & 4                      \\ \cline{2-4} 
+                                      & Graphene      & Nancy          & 12                     \\ \cline{2-4} 
+                                      & Griffon          & Nancy       & 12                       \\ 
 \hline
 \end{tabular}
  \label{tab:sc}
@@ -742,31 +900,28 @@ used from each cluster.
   \label{fig:time_sen}
 \end{figure}
 
-The NAS parallel benchmarks are executed over these two platform
-scenarios with different number of nodes, as in Table \ref{tab:sc}. 
-The overall energy consumption of all benchmark, class D, with 
-applying the proposed frequency selection algorithm is measured 
+The NAS parallel benchmarks are executed over these two platforms
+ with different number of nodes, as in Table \ref{tab:sc}. 
+The overall energy consumption of all the benchmarks solving the class D instance and
+using the proposed frequency selection algorithm is measured 
 using the equation of the reduced energy consumption, equation 
 (\ref{eq:energy}). This model uses the measured dynamic and static 
-power values that showed in Table \ref{table:grid5000}. The execution
-time is measured for all benchmarks over these different scenarios.  
-The energy consumptions  and the execution times for all benchmarks are 
-demonstrated in the plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 
-In general, the energy consumptions of NAS benchmarks over one site scenario 
-for  16 and 32 nodes are less than those executed over the two sites 
-scenarios. This because in the two sites scenario the communication times 
-are higher, due to long distance communications between the two distributed sites. 
-This leading to more static energy consumption which is linearly related to the 
-increased in the communication time. The execution times of these benchmarks 
-over one sites for 16 and 32 nodes are less  comparing to the two sites 
-scenario according to the increase in communications times.
-
-The EP and MG benchmarks, where there are no or small communications, showed 
-that their execution times and the energy consumptions are not effected 
-significantly in both scenarios and when the number of nodes is increase, 
-while the other benchmarks showed the inverse, because they have more communications 
-that proportionally increase the communication times if there are slow 
-communications or using more number of nodes or both of them.
+power values  showed in Table \ref{table:grid5000}. The execution
+time is measured for all the benchmarks over these different scenarios.  
+
+The energy consumptions  and the execution times for all the benchmarks are 
+presented in the plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 
+
+For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
+for  16 and 32 nodes is lower than the energy consumed while using two sites. 
+The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption. 
+
+The execution times of these benchmarks 
+over one site with 16 and 32 nodes are also lower when  compared to those of the  two sites 
+scenario. Moreover, most of the benchmarks running over the one site scenario their execution times  are approximately divided by two  when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).  
+
+However, the  execution times and the energy consumptions of EP and MG benchmarks, which have no or small communications, are not significantly affected 
+ in both scenarios. Even when the number of nodes is doubled. On the other hand, the communications of the rest of the benchmarks increases when using long distance communications between two sites or increasing the number of computing nodes.
 
 \begin{figure}
   \centering
@@ -795,73 +950,337 @@ The energy saving percentage is computed as the ratio between the reduced
 energy consumption, equation (\ref{eq:energy}), and the original energy consumption,
 equation (\ref{eq:eorginal}), for all benchmarks as in figure \ref{fig:eng_s}. 
 This figure shows that the energy saving percentages of one site scenario for
-16 and 32 nodes are bigger than those of the two sites scenario. This is because
-the computations to communications ratio in  one site scenario is higher 
-than the ratio of the two sites scenarios, due to the increase in the communication
-times. Moreover, the frequencies selecting algorithm selects smaller frequencies, bigger
-scaling factors, when the computations times are higher than communication times, 
-producing  smaller energy consumption, because the dynamic energy consumption 
-is decreased linearly with computation times that decreased exponentially with 
-scaling factors. On the other side, the increase in the number of computing nodes can be 
-increase the communication times and thus producing less energy saving depending on the 
-benchmarks being executed. The benchmarks CG, MG, BT and FT show more 
-energy saving percentage in one site scenario when executed  over 16 nodes comparing to 32 nodes. While 
-the benchmarks  LU and SP showed the inverse, because there computations to 
-communications ratio is not effected to the increase in local site communications. 
-While all benchmarks are effected by the long distance communications in the two sites 
-scenarios, except  EP benchmarks. In EP benchmark  there is no communications 
-in their iterations, then it is independent from the effect of local and long 
-distance communications. Therefore, the energy saving percentage of this benchmarks is 
-depend on differences between the computing powers of the computing nodes, for example 
+16 and 32 nodes are bigger than those of the two sites scenario which is due
+to the higher  computations to communications ratio in the first scenario   
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which 
+results in  a lower energy consumption. Indeed, the dynamic  consumed power
+is exponentially related to the CPU's frequency value. On the other side, the increase in the number of computing nodes can 
+increase the communication times and thus produces less energy saving depending on the 
+benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more 
+energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While, LU and SP consume more energy with 16 nodes than 32 in one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 
+
+
+The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 
+scenario, except for the   EP benchmark which has  no communications. Therefore, the energy saving percentage of this benchmark is 
+dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 
 in the one site scenario, the graphite cluster is selected but in the two sits scenario 
-this cluster is replaced with Taurus cluster that be more powerful in computing power. 
-Therefore, the energy saving of EP benchmarks are bigger in the two site scenario due 
-to increase in the differences between the computing powers of the nodes. This means, the higher 
-differences between the nodes' computing powers make the proposed frequencies selecting  
-algorithm  to selects smaller frequencies in the nodes of the higher computing power, 
-producing less energy consumption and thus more energy saving.
-The best energy saving percentage was for one site scenario with 16 nodes, on average it
-saves the energy consumption up to 30\%. 
-
-Figure \ref{fig:per_d}, presents the performance degradation percentages for all benchmarks.
-It shows that the performance degradation percentages  of the one site scenario with
-32 nodes, on average equal to 10\%, is higher than the performance degradation of one 16 nodes, 
-which on average equal to 3\%. This because selecting smaller frequencies in the one site scenarios,
-when the computations grater than the communications , increase the number of the critical nodes 
-when the number of nodes increased. The inverse happens in the tow sites scenario,
-this due to the lower computations to communications ratio that decreased with highest 
-communications. Therefore, the number of the critical nodes are decreased. The average performance 
-degradation for the two sites scenario with 16 nodes is equal to 8\% and for 32 nodes is equal to 4\%.
-The EP benchmarks is gives the bigger performance degradation ratio, because there is no 
-communications and no slack times in this benchmarks that is always their performance effected 
-by selecting big or small frequencies.
-The tradeoff between these scenarios can be computed as in the trade-off function \ref{eq:max}.
-Figure \ref{fig:dist}, presents the tradeoff distance for all benchmarks  over all 
-platform scenarios.  The one site scenario with 16 and 32 nodes had the best tradeoff distance 
-compared to the two sites scenarios, because the increase in the communications as mentioned before.
-The one site scenario with 16 nodes is the best scenario in term of energy and performance tradeoff,
-which on average is up 26\%.  Then, the tradeoff distance is related linearly to the  energy saving  
-percentage. Finally, the best energy and performance tradeoff depends on the increase in all of:
-1) the computations to communications ratio, 2) the differences in computing powers 
-between the computing nodes and 3) the differences in static and the dynamic powers of the nodes.
+this cluster is replaced with Taurus cluster which is more powerful. 
+Therefore, the energy saving of EP benchmarks are bigger in the two sites scenario due 
+to the higher maximum difference between the computing powers of the nodes. 
+
+In fact, high differences between the nodes' computing powers make the proposed frequencies selecting  
+algorithm  select smaller frequencies for the powerful nodes which 
+produces less energy consumption and thus more energy saving.
+The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
+
+
+Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
+The performance degradation percentage for the benchmarks running on two sites  with
+16 or 32  nodes is on average equal to 8\% or 4\% respectively. 
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with
+16 or 32  nodes is on average equal to 3\% or 10\% respectively. In opposition to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
+nodes when the communications occur in high speed network does not decrease the computations to 
+communication ratio. 
+
+The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to 
+the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its 
+performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
+The rest of the benchmarks showed different performance degradation percentages, which decrease
+when the communication times increase and vice versa.
+
+Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The tradeoff distance percentage can be 
+computed as in equation \ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
+tradeoff, on average it is equal to  26\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
+tradeoff comparing to the two sites scenario  because the former has high speed local communications 
+which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 
+
+
+ Finally, the best energy and performance tradeoff depends on all of the following:
+1) the computations to communications ratio when there are  communications and slack times, 2) the heterogeneity of the computing powers of the nodes and 3) the heterogeneity of the consumed  static and dynamic powers of the nodes.
+
+
+
 
 \subsection{The experimental results of multi-cores clusters}
-\label{sec.res}
+\label{sec.res-mc}
+The  clusters of grid'5000 have different number of cores embedded in their nodes
+as shown in Table \ref{table:grid5000}. The cores of each node can exchange 
+data via the shared memory \cite{rauber_book}. In 
+this section, the proposed scaling algorithm is evaluated over the grid'5000 grid while using multi-core nodes 
+selected according to the two  platform scenarios described in the section \ref{sec.res}.
+The two platform scenarios, the two sites and one site scenarios, use  32 
+cores from multi-cores nodes instead of 32 distinct nodes. For example if 
+the participating number of cores from a certain cluster is equal to 12, 
+in the multi-core scenario the selected nodes is equal to 3 nodes while using 
+4 cores from each node. The platforms with one  
+core per node and  multi-cores nodes are  shown in Table \ref{table:sen-mc}. 
+The energy consumptions and execution times of running the NAS parallel 
+benchmarks, class D, over these four different scenarios are presented 
+in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+
+The execution times for most of  the NAS  benchmarks are higher over the one site multi-cores per node scenario 
+ than the execution time of those running over one site single core per node  scenario. Indeed,  
+   the communication times  are higher in the one site multi-cores scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. 
+   
+ \textcolor{blue}{On the other hand,  the execution times for most of the NAS benchmarks  are lower over 
+the two sites  multi-cores scenario than those over the two sites one core scenario.   ???????
+}
+
+The experiments showed that for most of the NAS benchmarks and between the four scenarios,  
+the one site one core scenario gives the best execution times because the communication times are the lowest. 
+Indeed, in this scenario each core has a dedicated network link and all the communications are local.  
+Moreover, the energy consumptions of the NAS benchmarks are lower over the 
+one site one core scenario  than over the one site multi-cores scenario because 
+the first scenario had less execution time than the latter which results in less static energy being consumed.
+
+The computations to communications ratios of the NAS benchmarks are higher over 
+the one site one core scenario  when compared to the ratios of the other scenarios. 
+More energy reduction was achieved when this ratio is increased because the proposed scaling algorithm selects smaller frequencies that decrease the dynamic power consumption. 
+
+  \textcolor{blue}{ Whereas, the energy consumption in the two sites one core scenario is higher than the energy consumption of the two sites multi-core scenario. This is according to the increase in the execution time of the two sites one core scenario. }
+
+
+These experiments also showed that the energy 
+consumption and the execution times of the EP and MG benchmarks do not change significantly over these four 
+scenarios  because there are no or small communications,  
+which could increase or decrease the static power consumptions. Contrary to EP and MG, the  energy consumptions 
+and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
+
+
+The energy saving percentages of all NAS benchmarks running over these four scenarios are presented in the figure \ref{fig:eng-s-mc}. It shows that  the energy saving percentages   over the two sites multi-cores scenario 
+and over the two sites one core scenario are on average  equal to 22\% and 18\%
+respectively. The energy saving percentages   are higher in the former scenario because  its computations to communications  ratio is higher than the ratio of the latter scenario  as mentioned previously.
+
+In contrast, in the one site one 
+core and one site multi-cores scenarios the energy saving percentages 
+are approximately equivalent, on average they are up to 25\%. In both scenarios there 
+are a small difference  in the computations to communications ratios, which leads 
+the proposed scaling algorithm to select similar frequencies for both scenarios.  
+
+The performance degradation percentages of the NAS benchmarks are presented in
+figure \ref{fig:per-d-mc}. It shows that the performance degradation percentages for the NAS benchmarks are higher over the two sites 
+multi-cores scenario than over the  two sites  one core scenario, equal on average to 7\% and 4\% respectively. 
+Moreover, using the two sites multi-cores scenario increased 
+the computations to communications ratio, which may increase 
+the overall execution time  when the proposed scaling algorithm is applied and the frequencies scaled down.  
+
+
+When the benchmarks are executed  over the one 
+site one core scenario, their performance degradation percentages are equal  on average
+to 10\% and are higher than those executed over the one site multi-cores scenario, 
+which on average is equal to 7\%. 
+
+\textcolor{blue}{
+The performance degradation percentages over one site multi-cores is lower because  the computations to communications ratio is decreased. Therefore, selecting bigger 
+frequencies by the scaling algorithm are proportional to this ratio, and thus the execution time do not increase significantly.}
+
+
+The tradeoff distance percentages of the NAS 
+benchmarks over all scenarios are presented in the figure \ref{fig:dist-mc}.
+These  tradeoff distance percentages are used to verify which scenario is the best in terms of energy reduction and performance. The figure shows that using muti-cores in both of the one site and two sites scenarios gives bigger  tradeoff distance percentages, on overage equal to 17.6\% and 15.3\% respectively, than using one core per node in both of one site and two sites scenarios,  on average  equal to 14.7\% and 13.3\% respectively. 
+
+\begin{table}[]
+\centering
+\caption{The multicores scenarios}
+
+\begin{tabular}{|*{4}{c|}}
+\hline
+Scenario name                          & Cluster name & \begin{tabular}[c]{@{}c@{}}No. of  nodes\\ in each cluster\end{tabular} & 
+                                       \begin{tabular}[c]{@{}c@{}}No. of  cores\\ for each node\end{tabular}  \\ \hline
+\multirow{3}{*}{Two sites/ one core}   & Taurus       & 10              & 1                   \\ \cline{2-4}
+                                       & Graphene     & 10              & 1                   \\ \cline{2-4}
+                                       & Griffon      & 12              & 1                   \\ \hline
+\multirow{3}{*}{Two sites/ multicores} & Taurus       & 3               & 3 or 4              \\ \cline{2-4}
+                                       & Graphene     & 3               & 3 or 4              \\  \cline{2-4}
+                                       & Griffon      & 3               & 4                   \\ \hline
+\multirow{3}{*}{One site/ one core}    & Graphite     & 4               & 1                   \\  \cline{2-4}
+                                       & Graphene     & 12              & 1                   \\  \cline{2-4}
+                                       & Griffon      & 12              & 1                   \\ \hline
+\multirow{3}{*}{One site/ multicores}  & Graphite     & 3               & 3 or 4              \\  \cline{2-4}
+                                       & Graphene     & 3               & 3 or 4              \\  \cline{2-4}
+                                       & Griffon      & 3               & 4                   \\ \hline
+\end{tabular}
+\label{table:sen-mc}
+\end{table}
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/eng_con.eps}
+  \caption{Comparing the  energy consumptions of running NAS benchmarks over one core and multicores scenarios }
+  \label{fig:eng-cons-mc}
+\end{figure}
+
+
+  \begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/time.eps}
+  \caption{Comparing the  execution times of running NAS benchmarks over one core and multicores scenarios }
+  \label{fig:time-mc}
+\end{figure}
+
+ \begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/eng_s_mc.eps}
+  \caption{The energy saving of running NAS benchmarks over one core and multicores scenarios }
+  \label{fig:eng-s-mc}
+\end{figure}
 
-\subsection{The results for different power consumption scenarios}
-\label{sec.compare}
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/per_d_mc.eps}
+  \caption{The performance degradation of running NAS benchmarks over one core and multicores scenarios }
+  \label{fig:per-d-mc}
+\end{figure}
 
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/dist_mc.eps}
+  \caption{The tradeoff distance of running NAS benchmarks over one core and multicores scenarios }
+  \label{fig:dist-mc}
+\end{figure}
+
+\subsection{Experiments with different static and dynamic powers consumption scenarios}
+\label{sec.pow_sen}
+
+In section \ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 
 
+The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
+In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.
+The experiments have been executed with these two new static power scenarios  over the one site one core per node scenario.
+In these experiments, the class D of the NAS parallel benchmarks are executed over Nancy's site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, where used in this experiment. 
+
+ \begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/eng_pow.eps}
+  \caption{The energy saving percentages for NAS benchmarks of the three power scenario}
+  \label{fig:eng-pow}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/per_pow.eps}
+  \caption{The performance degradation percentages for NAS benchmarks of the three power scenario}
+  \label{fig:per-pow}
+\end{figure}
+
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/dist_pow.eps}
+  \caption{The tradeoff distance for NAS benchmarks of the three power scenario}
+  \label{fig:dist-pow}
+\end{figure}
 
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.47]{fig/three_scenarios.pdf}
+  \caption{Comparing the selected frequency scaling factors of MG benchmark for three static power scenarios}
+  \label{fig:fre-pow}
+\end{figure}
 
-\subsection{The comparison of the proposed scaling algorithm }
+The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 
+in figure \ref{fig:eng_sen}. This figure shows that the  10\% of static power scenario 
+gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
+scenarios. The small value of the static power consumption makes the proposed 
+scaling algorithm  select smaller frequencies for the CPUs. 
+These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives 		less overall energy consumption. 
+The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors   according to the static power consumption ratio being used.
+
+The performance degradation percentages are presented in the figure \ref{fig:per-pow}.
+The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
+had  selected big frequencies for the CPUs. While, 
+the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The tradeoff distance percentage for the NAS benchmarks with these three static power scenarios 
+are presented in the figure \ref{fig:dist}. 
+It shows that the best  tradeoff
+distance percentage is obtained with  the  10\% static power scenario  and this percentage 
+is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.
+
+In the EP benchmark, the energy saving, performance degradation and tradeoff 
+distance percentages for the these static power scenarios are not significantly different because there is no communication in this benchmark. Therefore, the static power is only consumed during computation and   the proposed scaling algorithm selects similar frequencies for the three scenarios.  On the other hand,  for the rest of the benchmarks,  the scaling algorithm  selects  the values of the frequencies according to the communication times of each benchmark because the static energy consumption increases  proportionally to the  communication times.
+
+
+ 
+\subsection{The comparison of the proposed frequencies selecting algorithm }
 \label{sec.compare_EDP}
 
+Finding the frequencies that gives the best tradeoff between the energy consumption and the performance for a parallel 
+application is not a trivial task.  Many algorithms have been proposed to tackle this problem.  
+In this section, the proposed frequencies selecting algorithm is compared to well known  energy and delay product method, $EDP=energy \times delay$, that have been used by many researchers  \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs}. 
+This method  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-cores 
+architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
+
+To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  equation \ref{eq:energy} and
+execution time model, equation \ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
+Moreover, both algorithms start the search space from the upper bound computed as in equation   \ref{eq:Fint}.
+Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
+and selects the vector of frequencies that minimize the EDP product.
+
+Both algorithms were applied to the class D of the NAS benchmarks over 16 nodes.
+The participating computing nodes are distributed  according to the two scenarios described in  section \ref{sec.res}. 
+The experimental results, the energy saving, performance degradation and tradeoff distance percentages, are 
+presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
+
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/edp_eng}
+  \caption{Comparing of the energy saving for the proposed method with EDP method}
+  \label{fig:edp-eng}
+\end{figure}
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/edp_per}
+  \caption{Comparing of the performance degradation for the proposed method with EDP method}
+  \label{fig:edp-perf}
+\end{figure}
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/edp_dist}
+  \caption{Comparing of the tradeoff distance for the proposed method with EDP method}
+  \label{fig:edp-dist}
+\end{figure}
+
+
+
+As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
+The proposed algorithm gives better results than EDP  because it 
+maximizes the energy saving and the performance at the same time. 
+Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
+Whereas, the EDP algorithm gives sometimes negative tradeoff values for some benchmarks in the two sites scenarios.
+These negative tradeoff values mean that the performance degradation percentage is higher than energy saving percentage.
+The high positive values of the tradeoff distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
+The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and 
+$O(N \cdot M \cdot F^2)$ respectively, where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the 
+maximum number of available frequencies. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average  $0.01 ms$  to compute the best frequencies while EDP is on average ten times slower over the same architecture.  
+
 
 
 \section{Conclusion}
 \label{sec.concl}
-
+\textcolor{blue}{
+This paper has been  presented a new online frequencies selection algorithm.
+It works based on objective function that maximized the tradeoff distance 
+between the predicted energy consumption and the predicted execution time of the distributed 
+iterative applications running over heterogeneous grid. The algorithm selects the best vector of the 
+frequencies which maximized the objective function has been used. A new energy model 
+used by the proposed algorithm for measuring and predicting the energy consumption 
+of the distributed iterative message passing application running over grid architecture.
+To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
+NAS parallel benchmarks  class D instance  and executed over grid'5000 testbed platform. 
+The experimental results showed that the algorithm saves the energy consumptions on average 
+for all NAS benchmarks up to 30\%  while gives only 3\% percentage on average for the performance 
+degradation for the same instance. The algorithm also selecting different frequencies according to the 
+computations and communication times ratio, and according to the values of the static and measured dynamic power of the CPUs. The computations to communications ratio was varied between different scenarios have been used, concerning to the distribution of the computing nodes between different clusters' sites and using one core or multi-cores per node.
+Finally, the proposed algorithm was compared to other algorithm which it
+used the will known energy and delay product as an objective function. The comparison results showed 
+that the proposed algorithm outperform the other one in term of energy-time tradeoff.
+In the near future, 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}
