adding fig

[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 a7aa66d049ffbc06bbbf724587b4f503edcde042..d157b79df7cde05aa2f49f94dc9d82ff634e7ce7 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -85,21 +85,184 @@
 
 \maketitle
 
 
 \maketitle
 

+

 \begin{abstract}
 \begin{abstract}

-  

 
 

+  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} only for the same instance. 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}
 
 \end{abstract}
 

+

 \section{Introduction}
 \label{sec.intro}
 \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{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 +475,15 @@ 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
+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
 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.
 
 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 +644,6 @@ 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 +776,6 @@ 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 +783,6 @@ 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. 


@@ -672,22 +834,21 @@ The benchmarks have seven different classes, S, W, A, B, C, D and E, that repres
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}

-In this section, the results of the the application of the scaling factors selection algorithm \ref{HSA} 
+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 
 to the NAS parallel benchmarks are presented. 
 
 As mentioned previously, the experiments 

-were conducted over two sites of grid'5000,   Lyon and Nancy sites. 
+were conducted over two sites of grid'5000,  Lyon and Nancy sites. 

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

-are connected via a long distance network.
-\item In the second scenario nodes from three clusters that are 
-located in one site, Nancy site.  
+ via a long distance network.
+\item In the second scenario nodes from three clusters 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 
 \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 
+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 
 is very low due to the higher communication times which reduces the effect of DVFS operations.
 
 The NAS parallel benchmarks are executed over 


@@ -754,10 +915,11 @@ 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. 
 
 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 
+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 
 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.


@@ -791,13 +953,12 @@ equation (\ref{eq:eorginal}), for all benchmarks as in figure \ref{fig:eng_s}.
 This figure shows that the energy saving percentages of one site scenario for
 16 and 32 nodes are bigger than those of the two sites scenario which is due
 to the higher  computations to communications ratio in the first scenario   
 This figure shows that the energy saving percentages of one site scenario for
 16 and 32 nodes are bigger than those of the two sites scenario which is due
 to the higher  computations to communications ratio in the first scenario   

-than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are higher than the communication times which 
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the 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 
 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 there computations to 
-communications ratio is not affected by the increase of the number of local communications. 
+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 
 
 
 The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 


@@ -805,137 +966,119 @@ scenario, except for the   EP benchmark which has  no communications. Therefore,
 dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 
 in the one site scenario, the graphite cluster is selected but in the two sits scenario 
 this cluster is replaced with Taurus cluster which is more powerful. 
 dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 
 in the one site scenario, the graphite cluster is selected but in the two sits scenario 
 this cluster is replaced with Taurus cluster which is more powerful. 

-Therefore, the energy saving of EP benchmarks are bigger in the two site scenario due 
+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. 
 to the higher maximum difference between the computing powers of the nodes. 

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

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

-The best energy saving percentage was obtained in the one site scenario with 16 nodes, The energy consumption was on average reduced up to 30\%.
+The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.

 
 
 
 

-Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks.
-The performance degradation percentage  for the benchmarks running  on two sites  with
+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. 
 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.

 
 

- \textcolor{red}{ 
-The proposed scaling algorithm selecting smaller frequencies in two sites scenario,
-due to decreasing in the computations to communications ratio when the number of nodes is increased and
-leads to less performance degradation percentage.  
-In contrast, the performance degradation percentage  for the benchmarks running  on one site  with
-16 or 32  nodes is on average equal to 3\% or 10\% respectively. 
-The inverse is happens in this scenario when the number of computing nodes is increased 
-the performance degradation percentage is decreased. So, using double number of computing 
-nodes when the communications occur in high speed network not decreased the computations to 
-communication ratio. Moreover, as shown in the figure \ref{fig:time_sen}, the execution time of one site scenario with 32 nodes 
-are less by approximately double, linear speed-up, for most of the benchmarks comparing to the one site with 16 nodes scenario. 
-This leads to increased the number of the critical nodes which any one of them may increased the overall the execution time of the benchmarks.
-The EP benchmarks is gives the bigger performance degradation ratio, because there is no 
-communications and no slack times in this benchmarks which their performance controlled by 
-the computing powers of the nodes.
-The tradeoff between these scenarios can be computed as in the tradeoff function \ref{eq:max}.
-Figure \ref{fig:dist}, presents the tradeoff distance for all benchmarks  over all 
-platform scenarios.  The one site scenario with 16 and 32 nodes had the best tradeoff distance 
-compared to the two sites scenarios, due to the increase or decreased in the communications as mentioned before.
-The one site scenario with 16 nodes is the best scenario in term of energy and performance tradeoff,
-which on average is up 26\%.  Therefore, the tradeoff distance is related linearly to the  energy saving  
-percentage. Finally, the best energy and performance tradeoff depends on the all of the following:
-1) the computations to communications ratio when there is a communications and slack times, 2) the differences in computing powers 
-between the computing nodes and 3) the differences in static and the dynamic powers of the nodes.}
-
-
-
-\subsection{The experimental results of multicores clusters}
+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-mc}
 \label{sec.res-mc}

-The grid'5000 clusters have different number of cores embedded in their nodes
-as in the Table \ref{table:grid5000}. Moreover, the cores of each node are 
-connected via shared memory model, the data transfer between cores' local 
-memories achieved via the global memory \cite{rauber_book}. Therefore, in 
-this section the proposed scaling algorithm is implemented over the  grid'5000 
-clusters which are included multicores in the selected nodes as same as the 
-two previous platform scenarios that mentioned in the section \ref{sec.res}.
-The two platform scenarios, the two sites and one site scenarios,  with 32 
-nodes are reconfigured to used multicores for each node. For example if 
-the participating number of nodes from a certain cluster is equal to 12 nodes, 
-in the multicores scenario the selected nodes is equal to 3 nodes with using 
-4 cores for each of them to produced 12 cores. These scenarios with one  
-core and  multicores are  demonstrated in Table \ref{table:sen-mc}. 
+The  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 
 The energy consumptions and execution times of running the NAS parallel 

-benchmarks, class D, over these four different scenarios are represented 
+benchmarks, class D, over these four different scenarios are presented 

 in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
 in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.

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


@@ -997,36 +1140,15 @@ Scenario name                          & Cluster name & \begin{tabular}[c]{@{}c@
   \label{fig:dist-mc}
 \end{figure}
 
   \label{fig:dist-mc}
 \end{figure}
 

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

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

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

 
 

-These three scenarios represented the ratio of the static power consumption that can be computed from 
-the dynamic power consumption of the core. The NAS benchmarks of class D are executed over 16 nodes
-in the Nancy site using three clusters: Graphite, Graphene and Griffon. As same as used before, the one site 16 nodes
-platform scenario explained in the last experiments, as in table \ref{tab:sc}, is uses to run 
-the NAS benchmarks with these static power scenarios. 
+In section \ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 
+
+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 and 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 sites, Graphite, Graphene and Griffon, where used in this experiment.  

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


@@ -1053,122 +1175,120 @@ the NAS benchmarks with these static power scenarios.
 \begin{figure}
   \centering
   \includegraphics[scale=0.47]{fig/three_scenarios.pdf}
 \begin{figure}
   \centering
   \includegraphics[scale=0.47]{fig/three_scenarios.pdf}

-  \caption{Comparing the selected frequencies of MG benchmarks for three static power scenarios}
+  \caption{Comparing the selected frequency scaling factors of MG benchmark for three static power scenarios}

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

-The energy saving percentages of NAS benchmarks with these three static power scenarios are presented 
-in figure \ref{fig:eng_sen}. This figure showed the 10\% of static power scenario 
-gives the biggest energy saving percentage comparing to 20\% and 30\% static power 
-scenario. When using smaller ratio of static power consumption, the proposed 
-frequencies selecting algorithm selects smaller frequencies, bigger scaling factors, 
-because the static energy consumption not increased significantly the overall energy 
-consumption. Therefore, more energy reduction can be achieved  when the frequencies are scaled down.
-For example figure \ref{fig:fre-pow}, illustrated that the proposed algorithm  
-proportionally scaled down the new computed frequencies with the overall predicted energy 
-consumption. The results of 30\% static power scenario gives the smallest energy saving percentages 
-because the new selected frequencies produced smaller ratio in the reduced energy consumption. 
-Furthermore, The proposed algorithm tries to limit selecting smaller frequencies that increased 
-the static energy consumption if the static power consumption is increased.
+
+The 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 percentage in comparison to the 20\% and 30\% static power 
+scenarios. The small value of 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.
+
+\textcolor{blue}{ 

 The performance degradation percentages are presented in the figure \ref{fig:per-pow},
 The performance degradation percentages are presented in the figure \ref{fig:per-pow},

-the 30\% of static power scenario had less performance degradation percentage, because
-bigger frequencies was selected due to the big ratio in the static power consumption.
-The inverse was happens in the 20\% and 30\% scenario, the algorithm was selected 
-biggest frequencies, smaller scaling factors, according to this increased in the static power ratios.
-The tradoff distance for the NAS benchmarks with these three static powers scenarios 
-are presented in the figure \ref{fig:dist}. The results showed that the tradeoff
-distance is the best when the  10\% of static power scenario is used, and this percentage 
-is decreased for the other two scenarios propositionally to their static power ratios.
-In EP benchmarks, the results of energy saving, performance degradation and tradeoff 
-distance are showed small differences when the these static power scenarios were used, 
-because this benchmark not has communications. The proposed algorithm is selected 
-same frequencies in this benchmark when all these static power scenarios are used. 
-The small differences in the results are due to the static power is consumed during the computation 
-times side by side to the dynamic power consumption, knowing that the dynamic power consumption 
-representing the highest ratio in the total power consumption of the core, then any change in 
-the static power during these times have less affect on the overall energy consumption. While the 
-inverse was happens for the rest of the benchmarks which have the communications 
-that increased the static energy consumption linearly to the mount of communications 
-in these benchmarks.
+the 30\% of static power scenario had less performance degradation percentage. This  because
+bigger frequencies are selected for the CPUs by the scaling algorithm. While, 
+the inverse happens in the 20\% and 30\% scenarios, because the scaling algorithm selects  bigger 
+frequencies. 
+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 because of  different frequencies have being selected by the scaling algorithm.
+In EP benchmark, the results of energy saving, performance degradation and tradeoff 
+distance are showed small differences when the these static power scenarios are used.
+In this benchmark there are no communications which leads  the proposed scaling algorithm to select similar 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.
+This makes the scaling algorithm proportionally selects big or small frequencies for each benchmark,
+because the communication times  proportionally increase or decrease the static energy consumption. }

 
  
 
  

-

 \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 
 The tradeoff between the energy consumption and the performance of the parallel 

-application had significant importance in the domain of the research. 
+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},
 Many researchers, \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs},

-are optimized the tradeoff between the energy and performance using the energy and delay product, $EDP=energy \times delay$. 
-This model is used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS},
-the  objective is to selects the suitable frequencies that minimized EDP product for the multicores 
-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 frequency selection algorithm, called Maxdist is compared with Spiliopoulos et al. algorithm, called EDP.
+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
 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  reaching to the lower bound as mentioned before. While, the EDP algorithm is developed to start from the 
-same upper bound until it reach to the minimum available frequencies. Finally, resulting the algorithm is an exhaustive search algorithm that
-test all possible frequencies, starting from the initial frequencies, and selecting those minimized the EDP products.
-
-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 to had two different scenarios. 
-These scenarios are two sites and one site scenarios that explained previously. 
-The experimental results of the energy saving, performance degradation and tradeoff distance are 
+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. 
 presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively. 

-
-In one site scenario the proposed frequencies selection algorithm outperform the EDP algorithm 
-in term of energy and performance for all of the benchmarks. While, the compassion results from the two sites scenario 
-showed that the proposed algorithm outperform EDP algorithm for all benchmarks except MG benchmark.
-In case of MG benchmark the are small communications and bigger frequencies selected in EDP algorithm 
-decreased the performance degradation more than the frequencies selected by Maxdist algorithm. 
-While the energy saving percentage are higher for Maxdist algorithm.
-
-Generally, the proposed algorithm gives better results for all benchmarks because it
-optimized the distance between the energy saving and the performance degradation. 
-Whereas, in EDP algorithm gives negative tradeoff for some benchmarks in the two sites scenarios.
-These negative tradeoffs mean the performance degradation percentage is higher than energy saving percentage.
-The higher positive value for tradeoff distance is mean the best energy and performance tradeoff is achieved synchronously, when
-the energy saving percentage is much higher than the performance degradation percentage 
-The time complexity of the proposed algorithm is $O(N \cdot M \cdot F)$, where $N$ is the number of the clusters,
-$M$ is the number of nodes and $F$ is the maximum number of available frequencies. The algorithm is selected 
-the best frequencies in small execution time, on average is equal to  0.01 $ms$ when it works over 32 nodes.
-While the EDP algorithm was slower than Maxdist algorithm by ten times, where their execution time  on average 
-takes 0.1 $ms$  to selects the suitable frequencies over 32 nodes. 
-The time complexity of this algorithm is  $O(N^2 \cdot M^2 \cdot F)$.
-
-
-  
- 
-
-
-

 \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_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_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}
 \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 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. 
+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.
+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 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 $ms$. 
+}
+

 
 

-  

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