Figure captions modified

[mpi-energy2.git] / mpi-energy2-extension / Heter_paper.tex

diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index bc489f480ed288ab79eb5bcd867d43b1fdd424b0..4d36639818c236a4ba107d1f0630ece062c57e5d 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -85,21 +85,182 @@
 
 \maketitle
 
 
 \maketitle
 

+

 \begin{abstract}
 \begin{abstract}

-  

 
 

+  In recent years, green computing topic  has  become an important topic 
+  in the supercomputing research domain. However, the 
+  computing platforms are still  consuming more and
+more energy due to the increasing number of nodes composing
+them. To minimize the operating costs of these platforms many
+techniques have been used. Dynamic voltage and frequency
+scaling (DVFS) is one of them. It can be used to reduce the power consumption of the CPU 
+  while computing, by lowering its frequency. However, lowering the frequency of 
+  a CPU may increase the execution time of an application running on that 
+  processor. Therefore, the frequency that gives the best trade-off between 
+  the energy consumption and the performance of an application must be selected. 
+  In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.  
+  It selects the frequencies and tries to give the best
+  trade-off between energy saving and performance degradation, for each node
+  computing the message passing 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  grid. 
+  The proposed algorithm is evaluated on a real grid , the grid'5000 platform, while
+  running the NAS parallel benchmarks.  The experiments show that it reduces the
+  energy consumption on average by \np[\%]{30} while  the performance  is only degraded
+  on average by \np[\%]{3}. Finally, the algorithm is 
+  compared to an existing method. The comparison results show that it outperforms the
+  latter in terms of energy consumption reduction and performance.

 \end{abstract}
 
 \end{abstract}
 

+

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

-
-
+\textcolor{red}{did you verify that these informations are still accurate before changing the years to 2015?}
+The need for more computing power is continually increasing. To partially
+satisfy this need, most supercomputers constructors just put more computing
+nodes in their platform. The resulting platforms may achieve higher floating
+point operations per second (FLOPS), but the energy consumption and the heat
+dissipation are also increased.  As an example, the Chinese supercomputer
+Tianhe-2 had the highest FLOPS in June 2015  according to the Top500 list
+\cite{TOP500_Supercomputers_Sites}.  However, it was also the most power hungry
+platform with its over 3 million cores consuming around 17.8 megawatts.
+Moreover, according to the U.S.  annual energy outlook 2015
+\cite{U.S_Annual.Energy.Outlook.2015}, the price of energy for 1 megawatt-hour
+was approximately equal to \$70.  Therefore, the price of the energy consumed by
+the Tianhe-2 platform is approximately more than \$10 million each year.  The
+computing platforms must be more energy efficient and offer the highest number
+of FLOPS per watt possible, such as the Shoubu-ExaScaler from RIKEN
+which became the top of the Green500 list in June 2015 \cite{Green500_List}.
+This heterogeneous platform executes more than 7 GFLOPS per watt while consuming
+50.32 kilowatts.
+
+Besides platform improvements, there are many software and hardware techniques
+to lower the energy consumption of these platforms, such as 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 showed significant energy
+consumption reductions. All the experimental results were conducted over 
+Simgrid simulator \cite{SimGrid}, which offers easy tools to create a homogeneous and heterogeneous platforms and run message passing parallel applications over them. In this paper, a new frequencies selecting algorithm,
+adapted to  grid platforms composed of heterogeneous clusters, is presented. It is applied to the NAS parallel benchmarks and evaluated over a real testbed, 
+the grid'5000 platform \cite{grid5000}. It selects  for a grid platform running a message passing iterative
+application the vector of
+frequencies  that simultaneously tries to offer the maximum energy reduction and
+minimum performance degradation ratios. The algorithm has a very small overhead,
+works online and does not need any training or profiling.
+
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents some
+related works from other authors.  Section~\ref{sec.exe} describes how the
+execution time of message passing programs can be predicted.  It also presents
+an energy model that predicts the energy consumption of an application running
+over a grid platform. Section~\ref{sec.compet} presents the
+energy-performance objective function that maximizes the reduction of energy
+consumption while minimizing the degradation of the program's performance.
+Section~\ref{sec.optim} details the proposed frequencies selecting algorithm.
+Section~\ref{sec.expe} presents the results of applying the algorithm on the 
+NAS parallel benchmarks and executing them on the grid'5000 testbed. 
+%It shows the results of running different scenarios using multi-cores and one core per node and comparing them. 
+It also evaluates the algorithm over three different power scenarios. Moreover, it shows the
+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 grid platforms.  Some
+works have already been done for such platforms and they can be classified into
+two types of heterogeneous platforms:
+\begin{itemize}
+\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
+\item the platform is only composed of heterogeneous CPUs.
+\end{itemize}

 
 

-\section{The performance and energy consumption measurements on heterogeneous architecture}
+For the first type of platform, the computing intensive parallel tasks are
+executed on the GPUs and the rest are executed on the CPUs.  Luley et
+al.~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a
+heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
+goal was to maximize the energy efficiency of the platform during computation by
+maximizing the number of FLOPS per watt generated.
+In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
+al. developed a scheduling algorithm that distributes 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
 \label{sec.exe}
 
 \subsection{The execution time of message passing distributed iterative


@@ -312,12 +473,15 @@ 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
 characteristics of each processor (computation power, range of frequencies,
 dynamic and static powers) and the task executed (computation/communication
 ratio). The aim being to reduce the overall energy consumption and to avoid

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

 heterogeneity of the processors, a vector of scaling factors should be selected
 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.
 


@@ -478,7 +642,6 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 \end{algorithm}
 
 
 \end{algorithm}
 
 

-

 In this section, the scaling factors selection algorithm for  grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency
 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing
 In this section, the scaling factors selection algorithm for  grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency
 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing


@@ -611,8 +774,6 @@ the details characteristics of these four clusters. Moreover, the dynamic powers
 selected clusters and are presented in table  \ref{table:grid5000}.
 
 
 selected clusters and are presented in table  \ref{table:grid5000}.
 
 

-
-

 \begin{figure}[!t]
   \centering
   \includegraphics[scale=1]{fig/grid5000}
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=1]{fig/grid5000}


@@ -620,7 +781,6 @@ selected clusters and are presented in table  \ref{table:grid5000}.
   \label{fig:grid5000}
 \end{figure}
 
   \label{fig:grid5000}
 \end{figure}
 

-

 The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
 The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
 The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 
 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. 


@@ -631,7 +791,7 @@ The benchmarks have seven different classes, S, W, A, B, C, D and E, that repres
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/power_consumption.pdf}
 \begin{figure}[!t]
   \centering
   \includegraphics[scale=0.6]{fig/power_consumption.pdf}

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

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


@@ -726,7 +886,7 @@ Table \ref{tab:sc} shows the number of nodes used from each cluster for each sce
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/eng_con_scenarios.eps}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/eng_con_scenarios.eps}

-  \caption{The energy consumptions of NAS benchmarks over different scenarios }
+  \caption{The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios }

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


@@ -735,7 +895,7 @@ Table \ref{tab:sc} shows the number of nodes used from each cluster for each sce
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/time_scenarios.eps}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/time_scenarios.eps}

-  \caption{The execution times of NAS benchmarks over different scenarios }
+  \caption{The execution times of the NAS benchmarks over different scenarios }

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


@@ -754,9 +914,10 @@ 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. 
 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 
 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.
+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.
 
 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.


@@ -764,7 +925,7 @@ However, the  execution times and the energy consumptions of EP and MG benchmark
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/eng_s.eps}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/eng_s.eps}

-  \caption{The energy saving of NAS benchmarks over different scenarios }
+  \caption{The energy reduction while executing the NAS benchmarks over different scenarios }

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


@@ -772,7 +933,7 @@ However, the  execution times and the energy consumptions of EP and MG benchmark
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/per_d.eps}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/per_d.eps}

-  \caption{The performance degradation of NAS benchmarks over different scenarios }
+  \caption{The performance degradation of the NAS benchmarks over different scenarios }

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


@@ -780,7 +941,7 @@ However, the  execution times and the energy consumptions of EP and MG benchmark
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/dist.eps}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/dist.eps}

-  \caption{The tradeoff distance of NAS benchmarks over different scenarios }
+  \caption{The tradeoff distance between the energy reduction and the performance of the NAS benchmarks over different scenarios }

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


@@ -820,11 +981,6 @@ For this scenario, the proposed scaling algorithm selects smaller frequencies fo
 nodes when the communications occur in high speed network does not decrease the computations to 
 communication ratio. 
 
 nodes when the communications occur in high speed network does not decrease the computations to 
 communication ratio. 
 

-
- Figure \ref{fig:time_sen} presents the execution times for all the benchmarks over the two scenarios. For 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).  
-\textcolor{red}{The transition between the execution times to the performance degradation is not clear}
-
-

 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 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.


@@ -844,192 +1000,161 @@ which increase the computations to communications ratio  and the latter uses lon
 
 
 
 
 
 

-\subsection{The experimental results of multi-cores clusters}
-\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. 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. In the two sites multi-cores scenario, There are three types of communications :
-\begin{itemize}
-\item between cores on the same node via shared memory 
-\item between cores from distinct nodes but belonging to the same cluster or site via local network
-\item between cores from distinct sites via long distance network
-\end{itemize}
-The latency of the communications increases from shared memory to LAN to WAN.
-Therefore, using multi-cores communicating via shared memory
-has reduced the communication times, and thus the overall execution time is also decreased.
-
-
-
-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{red}{ The next sentence is completely false! It is impossible to have these results!  Whereas, the energy consumption in the two sites multi-cores scenario is higher than the energy consumption 
-of the two sites one core scenario. 
-Actually, using multi-cores in this scenario decreased the communication times that decreased the static energy consumption.}
-
-
-These experiments also showed that the energy 
-consumption and the execution times of the EP and MG benchmarks do not change significantly over these 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{red}{the next sentence is completely false! 
-The higher performance degradation percentages over the first scenario is due to the use of multi-cores which 
-decreases the computations to communications ratio. Therefore, selecting small 
-frequencies by the scaling algorithm do not increase the execution time 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}
-
-\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}
+%\subsection{The experimental results of multi-cores clusters}
+%\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}. 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 and. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck. 
+%   
+%
+%The experiments showed that for most of the NAS benchmarks 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 memory bus 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}
+%
+%\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}

 
 

-\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}
+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. 

 
 

-\subsection{The results of using different static power consumption scenarios}
-\label{sec.pow_sen}
-\textcolor{blue}{
-The static power consumption for one core is the leakage power
-consumption when it is idle. The measured static  power of the node, 
-as in section \ref{sec.grid5000}, had a collection of power values such as 
-all cores static powers and the power consumptions of the other devices. Furthermore, the static power for one core is hard to measured precisely. On the other hand,  the core has consumed the static power during 
-the communication and computation times.  However, the static power consumption becomes more important  when the execution time is
-increased using DVFS. Therefore, the objective of this section is to verify the ability of the proposed 
-scaling algorithm to select the best frequencies when the static power consumption is changing. 
-All the results obtained in the previous sections depend on the measured dynamic power 
-consumptions as in table \ref{table:grid5000}. Moreover, the static power consumption for one core is represented by 20\% of the measured dynamic power consumption. 
-This assumption is extended in this section to taking into account other ratios for the static power consumption.
-In addition to the previous ratio of the static power consumption, two other static power ratios are used, which are 10\% and 30\% of the measured dynamic power of the core.
-As a result, all of these static power scenarios is denoted as follow: 
-\begin{itemize}
-\item 10\% of static power scenario 
-\item 20\% of static power scenario 
-\item 30\% of static power scenario 
-\end{itemize}
-The NAS parallel benchmarks, class D, are executed over Nancy site. 
-The number of computing nodes used is 16 nodes distributed between three cluster, which are Graphite, Graphene and Griffon.  The NAS benchmarks rerun
-with these two new static power scenarios over one site scenario
-using one core per node. }
+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}
 
  \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/eng_pow.eps}

-  \caption{The energy saving percentages for NAS benchmarks of the three power scenario}
+  \caption{The energy saving percentages for the nodes executing the NAS benchmarks over the three power scenarios}

   \label{fig:eng-pow}
 \end{figure}
 
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/per_pow.eps}
   \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}
+  \caption{The performance degradation percentages for the NAS benchmarks over the three power scenarios}

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


@@ -1037,111 +1162,126 @@ using one core per node. }
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/dist_pow.eps}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/dist_pow.eps}

-  \caption{The tradeoff distance for NAS benchmarks of the three power scenario}
+  \caption{The tradeoff distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios}

   \label{fig:dist-pow}
 \end{figure}
 
 \begin{figure}
   \centering
   \includegraphics[scale=0.47]{fig/three_scenarios.pdf}
   \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}
+  \caption{Comparing the selected frequency scaling factors for the MG benchmark over the three static power scenarios}

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

-\textcolor{blue}{
-The energy saving percentages of NAS benchmarks with these three static power scenarios are presented 
-in figure \ref{fig:eng_sen}. This figure shows that  10\% of static power scenario 
-gives the biggest energy saving percentage comparing to 20\% and 30\% static power 
-scenarios. The smaller ratio of the static power consumption makes the proposed 
-scaling algorithm to select smaller frequencies, bigger scaling factors. 
-These smaller frequencies has reduced the dynamic energy consumption and thus the 
-overall energy consumption is decreased. 
-The energy saving percentages of 30\% static power scenario is the smallest between the other scenarios, because of the scaling algorithm selects bigger frequencies, smaller scaling factors, that increased the energy consumption. For example, figure \ref{fig:fre-pow}, illustrates that the proposed scaling algorithm is proportionally selected the best frequency scaling factors according to the static power consumption ratio being used.
-Furthermore, the proposed scaling algorithm tries to limit selecting smaller frequencies, which increased the execution time. Hence, the increase in the execution time is relatively increased the static energy consumption.
-The performance degradation percentages are presented in the figure \ref{fig:per-pow},
-the 30\% of static power scenario had less performance degradation percentage. This  because
-bigger frequencies was selected due to the big ratio in the static power consumption.
-The inverse happens in the 20\% and 30\% scenarios, the scaling algorithm is selecting
-smaller frequencies, bigger scaling factors, according to the ratio of the static power.
-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 tradeoff
-distance percentage is the best when the  10\% of static power scenario is used, and this percentage 
-is decreased for the other two scenarios propositionally to their static power ratios.
-In EP benchmark, the results of energy saving, performance degradation and tradeoff 
-distance are showed small differences when the these static power scenarios were used.
-The absent of the communications in this benchmark made the proposed scaling algorithm to select equivalent frequencies even if the static power values are different. While, the 
-inverse has been shown  for the rest of the benchmarks, which have  different communication times
-that increased the static energy consumption proportionally. Therefore, the scaling algorithm relatively selects 
-different frequencies for each benchmark when these static power scenarios are used. }
+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}
 
  
 \subsection{The comparison of the proposed frequencies selecting algorithm }
 \label{sec.compare_EDP}

-\textcolor{blue}{
-The tradeoff between the energy consumption and the performance of the parallel 
-applications had significant importance in the domain of the research. 
-Many researchers, \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs},
-have optimized the tradeoff between the energy and the performance using the well known  energy and delay product, $EDP=energy \times delay$. 
-This model is also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS},
-the  objective is to select the frequencies that minimized EDP product for the multi-cores 
-architecture when DVFS is used. Moreover, their algorithm is applied online, which synchronously optimized the energy consumption 
-and the execution time. Both energy consumption and execution time of a processor are predicted by the their algorithm.
-In this section the proposed frequencies selection algorithm, called Maxdist is compared with Spiliopoulos et al. algorithm, called EDP.
-To make both of the algorithms follow the same direction and  fairly  comparing them, the same energy model,  equation \ref{eq:energy} and
-the execution time model, equation \ref{eq:perf}, are used in the prediction process to select the best vector of the frequencies. 
-In contrast, the proposed algorithm starts the search space from the lower bound computed as in equation the  \ref{eq:Fint}. Also, the algorithm
-stops  the search process when it is reached to the lower bound as mentioned before. In the same way, the EDP algorithm is developed to start from the 
-same upper bound used in Maxdist algorithm, and it stops the search process when  a minimum available frequencies is reached. 
-Finally, the resulting EDP algorithm is an exhaustive search algorithm that test all possible frequencies, starting from the initial frequencies, 
-and selecting those minimized the EDP product.
-Both algorithms were applied to NAS benchmarks, class D, over 16 nodes selected from grid'5000 clusters.
-The participating computing nodes are distributed between two sites and one site to have two different scenarios that used in the 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. 
+
+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 a method that uses the well known  energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers  \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs}. 
+This objective function  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-cores 
+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}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/edp_eng}

-  \caption{Comparing of the energy saving for the proposed method with EDP method}
+  \caption{The energy reduction induced by  the Maxdist method and the EDP method}

   \label{fig:edp-eng}
 \end{figure}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/edp_per}
   \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}
+  \caption{The performance degradation induced by  the Maxdist method and the EDP method}

   \label{fig:edp-perf}
 \end{figure}
 \begin{figure}
   \centering
   \includegraphics[scale=0.5]{fig/edp_dist}
   \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}
+  \caption{The tradeoff distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method}

   \label{fig:edp-dist}
 \end{figure}
   \label{fig:edp-dist}
 \end{figure}

-As shown form these figures, the proposed frequencies selection algorithm, Maxdist, outperform the EDP algorithm in term of energy and performance for all of the benchmarks executed over the two scenarios. 
-Generally, the proposed algorithm gives better results for all benchmarks because it is
-optimized the distance between the energy saving and the performance degradation in the same time. 
+
+
+
+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.
 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.

-Whereas, the EDP algorithm gives some times negative tradeoff values for some benchmarks in the two sites scenarios.
+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.
 These negative tradeoff values mean that the performance degradation percentage is higher than energy saving percentage.

-The higher positive value of the tradeoff distance percentage mean that the  energy saving percentage is much higher than the performance degradation 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 
 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. The proposed algorithm, Maxdist, has selected the best frequencies in a small execution time, 
-on average is equal to  0.01 $ms$, when it is executed over 32 nodes distributed between Nancy and Lyon sites.
-While the EDP algorithm was slower than Maxdist algorithm by ten times over the same number of nodes and same distribution, its execution time on average 
-is equal to 0.1 $ms$. 
-}
+$O(N \cdot M \cdot F^2)$ 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}
 
 
 \section{Conclusion}
 \label{sec.concl}

+This paper has presented a new online frequencies selection algorithm.
+ The algorithm selects the best vector of 
+frequencies that maximizes  the tradeoff distance 
+between the predicted energy consumption and the predicted execution time of the distributed 
+iterative applications running over a heterogeneous grid. A new energy model 
+is used by the proposed algorithm to predict the energy consumption 
+of the distributed iterative message passing application running over a grid architecture.
+To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
+ NAS parallel benchmarks   and the  class D instance was executed over the  grid'5000 testbed platform. 
+ The experimental results showed that the algorithm reduces  on average 30\% of the energy consumption
+for all the NAS benchmarks   while  only degrading by 3\% on average  the performance. 
+The Maxdist algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or in the values of the consumed static power. The algorithm selects different vector of frequencies according to the 
+computations and communication times ratios, and  the values of the static and measured dynamic powers of the CPUs. 
+Finally, the proposed algorithm was compared to another method that uses
+the well known energy and delay product as an objective function. The comparison results showed 
+that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better tradeoff  between energy consumption reduction and performance. 
+
+In the near future, we would like to develop a similar method that is adapted to
+asynchronous iterative applications where iterations are not synchronized and communications are overlapped with computations. 
+ The development of
+such a method might require a new energy model because the
+number of iterations is not known in advance and depends on
+the global convergence of the iterative system.

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

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

 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.
 
 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.