correcting the last sections

[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 79c35112930a50ac8bd1db4a8fec160f012c0603..1774eba61645015f7900a0f4a4ea93ed408824c7 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -9,6 +9,7 @@
 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{url}
 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{url}

+\usepackage{multirow}

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


@@ -53,7 +54,7 @@
 \newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
 \newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
 \newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
 \newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
 \newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
 \newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}

-\newcommand{\Ppeak}[1][]{\Xsub{P}{peak}_{#1}}
+\newcommand{\Pmax}[1][]{\Xsub{P}{max}_{\fxheight{#1}}}

 \newcommand{\Pidle}[1][]{\Xsub{P}{idle}_{\fxheight{#1}}}
 \newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Pidle}[1][]{\Xsub{P}{idle}_{\fxheight{#1}}}
 \newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}


@@ -106,8 +107,9 @@
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over

-heterogeneous grid platforms. A heterogeneous grid platform is defined as a collection of
-heterogeneous computing clusters interconnected via a long distance network (the internet network). Each computing cluster in the grid composed from homogeneous nodes, where are connected together via high speed homogeneous network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
+heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
+heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
+and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.

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


@@ -164,7 +166,7 @@ vector of scaling factors can be predicted using (\ref{eq:perf}).
   +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
 \end{equation}
 
   +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
 \end{equation}
 

-where $N$ is the number of the clusters in the grid, $M$ is the number of the nodes in
+where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in

 each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
 first  iteration. The model computes the maximum computation time with scaling factor 
 each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
 and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
 first  iteration. The model computes the maximum computation time with scaling factor 


@@ -180,7 +182,8 @@ of message passing distributed applications for homogeneous and heterogeneous cl
 used in the method to optimize both the energy consumption and the performance
 of iterative methods, which is presented in the following sections.
 
 used in the method to optimize both the energy consumption and the performance
 of iterative methods, which is presented in the following sections.
 

-\subsection{Energy model for heterogeneous platform}
+
+\subsection{Energy model for heterogeneous grid platform}

 
 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
   Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
 
 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
   Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,


@@ -347,7 +350,8 @@ maximum frequency for all  nodes:
 \end{equation}
 
 Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
 \end{equation}
 
 Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 

-computed as in ().
+computed as in (\ref{eq:eorginal}).
+

 
 \begin{equation}
   \label{eq:eorginal}
 
 \begin{equation}
   \label{eq:eorginal}


@@ -404,7 +408,7 @@ values for each node (static and dynamic powers). However, the most important
 energy reduction gain can be achieved when the energy curve has a convex form as shown 
 in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
 
 energy reduction gain can be achieved when the energy curve has a convex form as shown 
 in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
 

-\section{The scaling factors selection algorithm for heterogeneous grid platforms }
+\section{The scaling factors selection algorithm for  grids }

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


@@ -473,12 +477,12 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
   \label{dvfs}
 \end{algorithm}
 
   \label{dvfs}
 \end{algorithm}
 

-\subsection{The algorithm details}

 
 

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

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

 online during the execution time of the iterative message passing program.  It
 uses information gathered during the first iteration such as the computation
 time and the communication time in one iteration for each node. The algorithm is
 online during the execution time of the iterative message passing program.  It
 uses information gathered during the first iteration such as the computation
 time and the communication time in one iteration for each node. The algorithm is


@@ -496,13 +500,13 @@ scaling algorithm is called in the iterative MPI program.
   \label{fig:st_freq}
 \end{figure}
 
   \label{fig:st_freq}
 \end{figure}
 

-The nodes in a heterogeneous grid have different computing powers, thus
+Nodes from distinct clusters in a grid have different computing powers, thus

 while executing message passing iterative synchronous applications, fast nodes
 have to wait for the slower ones to finish their computations before being able
 to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
 periods are called idle or slack times.  The algorithm takes into account this
 while executing message passing iterative synchronous applications, fast nodes
 have to wait for the slower ones to finish their computations before being able
 to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
 periods are called idle or slack times.  The algorithm takes into account this

-problem and tries to reduce these slack times when selecting the frequency
-scaling factors vector. At first, it selects initial frequency scaling factors
+problem and tries to reduce these slack times when selecting the vector of the frequency
+scaling factors. At first, it selects initial frequency scaling factors

 that increase the execution times of fast nodes and minimize the differences
 between the computation times of fast and slow nodes. The value of the initial
 frequency scaling factor for each node is inversely proportional to its
 that increase the execution times of fast nodes and minimize the differences
 between the computation times of fast and slow nodes. The value of the initial
 frequency scaling factor for each node is inversely proportional to its


@@ -523,25 +527,24 @@ follows:
 \end{equation}
 If the computed initial frequency for a node is not available in the gears of
 that node, it is replaced by the nearest available frequency.  In
 \end{equation}
 If the computed initial frequency for a node is not available in the gears of
 that node, it is replaced by the nearest available frequency.  In

-Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing powers in

 ascending order and the frequencies of the faster nodes are scaled down
 according to the computed initial frequency scaling factors.  The resulting new
 frequencies are highlighted in Figure~\ref{fig:st_freq}.  This set of
 frequencies can be considered as a higher bound for the search space of the
 ascending order and the frequencies of the faster nodes are scaled down
 according to the computed initial frequency scaling factors.  The resulting new
 frequencies are highlighted in Figure~\ref{fig:st_freq}.  This set of
 frequencies can be considered as a higher bound for the search space of the

-optimal vector of frequencies because selecting scaling factors higher
+optimal vector of frequencies because selecting higher frequencies

 than the higher bound will not improve the performance of the application and it
 will increase its overall energy consumption.  Therefore the algorithm that
 selects the frequency scaling factors starts the search method from these
 initial frequencies and takes a downward search direction toward lower
 than the higher bound will not improve the performance of the application and it
 will increase its overall energy consumption.  Therefore the algorithm that
 selects the frequency scaling factors starts the search method from these
 initial frequencies and takes a downward search direction toward lower

-frequencies or reaching to the lower bound. The lower bound is used to  stop 
-the algorithm search process when the new computed distance between the energy and performance is less than zero.
-The new negative distance is  mean that the performance degradation ratio is higher than energy saving ratio.
-Therefore, the algorithm must stop the iterations before reaching to the end of the search space, the minimum frequencies, 
-because the all the coming new distances are negative values.
-The algorithm iterates on all remaining frequencies, from the higher
-bound until all nodes reach their minimum frequencies or to the lower bound, to compute their overall
-energy consumption and performance, and select the optimal frequency scaling
-factors vector. At each iteration the algorithm determines the slowest node
+frequencies until reaching the nodes' minimum frequencies or lower bounds. A node's frequency is considered its lower bound if the computed distance between the energy and performance at this frequency is less than zero.
+A negative distance means that the performance degradation ratio is higher than the energy saving ratio.
+In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances. 
+
+Therefore, the algorithm iterates on all remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
+energy consumption and performance and selects the optimal vector of the frequency scaling
+factors. At each iteration the algorithm determines the slowest node

 according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
 while it lowers the frequency of all other nodes by one gear.  The new overall
 energy consumption and execution time are computed according to the new scaling
 according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
 while it lowers the frequency of all other nodes by one gear.  The new overall
 energy consumption and execution time are computed according to the new scaling


@@ -549,48 +552,66 @@ factors.  The optimal set of frequency scaling factors is the set that gives the
 highest distance according to the objective function (\ref{eq:max}).
 
 Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
 highest distance according to the objective function (\ref{eq:max}).
 
 Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and

-consumed energy for an application running on a homogeneous platform and a
-heterogeneous grid platform respectively while increasing the scaling factors. It can
-be noticed that in a homogeneous platform the search for the optimal scaling
+consumed energy for an application running on a homogeneous cluster and a
+ grid platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous cluster the search for the optimal scaling

 factor should start from the maximum frequency because the performance and the
 consumed energy decrease from the beginning of the plot. On the other hand, in
 factor should start from the maximum frequency because the performance and the
 consumed energy decrease from the beginning of the plot. On the other hand, in

-the heterogeneous grid platform the performance is maintained at the beginning of the
+the  grid platform the performance is maintained at the beginning of the

 plot even if the frequencies of the faster nodes decrease until the computing
 power of scaled down nodes are lower than the slowest node. In other words,
 until they reach the higher bound. It can also be noticed that the higher the
 difference between the faster nodes and the slower nodes is, the bigger the
 plot even if the frequencies of the faster nodes decrease until the computing
 power of scaled down nodes are lower than the slowest node. In other words,
 until they reach the higher bound. It can also be noticed that the higher the
 difference between the faster nodes and the slower nodes is, the bigger the

-maximum distance between the energy curve and the performance curve is while the
-scaling factors are varying which results in bigger energy savings. 
+maximum distance between the energy curve and the performance curve is, which results in bigger energy savings. 

 
 
 \section{Experimental results}
 \label{sec.expe}
 
 
 \section{Experimental results}
 \label{sec.expe}

+While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid}, 
+in this paper real experiments were conducted over the grid'5000 platform. 

 
 \subsection{Grid'5000 architature and power consumption}
 \label{sec.grid5000}
 
 \subsection{Grid'5000 architature and power consumption}
 \label{sec.grid5000}

-The grid'5000 is a large-scale testbed found in France \cite{grid5000}.
-The grid infrastructure consist of ten sites distributed over all France 
-metropolitan regions. Each site in the grid'5000 composed from number of heterogeneous 
-computing clusters, while each cluster includes a collection of homogeneous nodes.
-In general, the grid'5000 had one thousand of heterogeneous nodes and eight thousand of cores. 
-All the sites are connected together via special long distance network called RENATER,
-which is the French National Telecommunication Network for Technology. Whereas inside each site 
-the clusters and their nodes are connected throw high speed local area networks. 
-There are different types of local networks used such as Ethernet and Infiniband netwoks,  
-which allowed different gigabits bandwidth and latencies.  On the other hand, the nodes inside each cluster 
-are homogeneous, while they are different from the nodes of the other clusters. Therefore, there are
-a wide diversity of processors in grid'5000, that mainly had different processors families 
-such as  Intel Xeon and AMD Opteron families. 
-
-In this paper we are interested  to run NAS parallel v3.3 \cite{NAS.Parallel.Benchmarks} over grid'5000.
-We are used seven benchmarks, CG, MG, EP, LU, BT, SP and FT. These benchmarks used seven different types of classes.
-These classes are S, W, A, B, C, D, E, where S represents the smaller problem size that used by benchmark and
-E is represents the biggest class. In this work, the class D is used for all benchmarks in all the experiments that will 
-be showed in the coming sections.
-Moreover, the NAS parallel benchmarks have different computations and communications ratios, then it is interested 
-to study their energy consumption and their performance on real testbed such as grid'5000.
-In this work, the NAS benchmarks are executed over two sites, Lyon and Nancy sites, of grid'5000.
-These two sites had seven different types of computing clusters as in figure (\ref{fig:grid5000}).
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed over all metropolitan France and Luxembourg. All the sites are connected together via        a special long distance network called RENATER,
+which is the French National Telecommunication Network for Technology.
+Each site of the grid is composed of few heterogeneous 
+computing clusters and each cluster contains many homogeneous nodes. In total,
+grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
+the clusters and their nodes are connected via  high speed local area networks. 
+Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
+
+Since grid'5000 is dedicated for testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture 
+the power consumption  for each node in those sites. The measured power is the overall consumed power by by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ...  For more details refer to
+\cite{Energy_measurement}. To just measure the CPU power of one core in a node $j$, 
+ firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumption represents the 
+dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 
+
+
+The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
+\begin{equation}
+  \label{eq:pdyn}
+    \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
+\end{equation}
+
+where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured maximum power values, 
+$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
+Therefore, the dynamic power of one core is computed as the difference between the maximum 
+measured value in maximum powers vector and the minimum measured value in the idle powers vector.
+
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.
+
+In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
+
+Four clusters from the two sites were selected in the experiments: one cluster from 
+Lyon's site, Taurus cluster, and three clusters from Nancy's site, Graphene, 
+Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
+frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 
+the details characteristics of these four clusters. Moreover, the dynamic powers were computed  using the equation (\ref{eq:pdyn}) for all the nodes in the 
+selected clusters and are presented in table  \ref{table:grid5000}.
+
+
+

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


@@ -599,12 +620,22 @@ These two sites had seven different types of computing clusters as in figure (\r
   \label{fig:grid5000}
 \end{figure}
 
   \label{fig:grid5000}
 \end{figure}
 

-Four clusters from the two sites are selected in the experiments, one cluster from 
-Lyon site, Taurus cluster, and three clusters from Nancy site where are Graphene, 
-Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while their nodes are
-different from the nodes of other clusters in many aspects such as: computing power, power consumption, available 
-frequencies ranges and the network features, the bandwidth and the latency. The Table \ref{table:grid5000} shows 
-the details characteristics of these four clusters.
+
+The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 
+
+
+
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
+  \caption{The power consumption by one core from Taurus cluster}
+  \label{fig:power_cons}
+\end{figure}
+
+

 
   
 \begin{table}[!t]
 
   
 \begin{table}[!t]


@@ -637,61 +668,461 @@ the details characteristics of these four clusters.
   \label{table:grid5000}
 \end{table} 
 
   \label{table:grid5000}
 \end{table} 
 

-The grid'5000 testbed provided some monitoring and measurements features to captured 
-the power consumption values for each node in any cluster of Lyon and Nancy sites. 
-The power consumed for each node from the selected four clusters is measured. 
-While the power consumed by any computing node is a collection of the powers  consumed by 
-hard drive, main-board, memory and node's computing cores, for more detail refer to
-\cite{Energy_measurement}. Therefore, the dynamic power consumed
-by one core is not allowed to measured alone. To overcome this problem, firstly, 
-we measured the power consumed by one node when there is no computation, when 
-the CPU is in the idle state. The second step, we run EP benchmark, there is no communications 
-in this benchmarks, over one core with maximum frequency of the desired node and 
-capturing the power consumed by a node, this representing the peak power of the node with one core.  
-The difference between the peak power  and the idle power representing the 
-dynamic power consumption of that core with maximum frequency, for example see figure(\ref{fig:power_cons}). 
-The $\Ppeak[jx]$ is the peak power value in time $x$ with maximum frequency for one core of node $j$, 
-and $\Pidle[jy]$ is the idle power value in time $y$ for the one  core of the node $j$ . 
-The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
-\begin{equation}
-  \label{eq:pdyn}
-    \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Ppeak[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
-\end{equation}

 
 

-where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
-$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured peak power values, 
-$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
-Therefore, the dynamic power of one core is computed as the difference between the maximum 
-measured value in peak powers vector and the minimum measured value in the idle powers vector.
-We are computed the dynamic powers, using the equation (\ref{eq:pdyn}), for all nodes in the 
-selected clusters, which is recorded in table  \ref{table:grid5000}.
-On the other side, the static power consumption by one core is embedded with whole idle power consumption of the node.
-Indeed, the static power is represents as ratio from  dynamic power. So, we supposed  
-the static power consumption represented  as \np[\%]{20} of dynamic power consumption of the core, 
-the same assumption was made in \cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy}.
-\begin{figure}[!t]
+
+\subsection{The experimental results of the scaling algorithm}
+\label{sec.res}
+In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 
+to the NAS parallel benchmarks are presented. 
+
+As mentioned previously, the experiments 
+were conducted over two sites of grid'5000,  Lyon and Nancy sites. 
+Two scenarios were considered while selecting the clusters from these two sites :
+\begin{itemize}
+\item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected 
+ via a long distance network.
+\item In the second scenario nodes from three clusters that are located in one site, Nancy site.  
+\end{itemize}
+
+The main reason 
+behind using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
+scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 
+is very low due to the higher communication times which reduces the effect of DVFS operations.
+
+The NAS parallel benchmarks are executed over 
+16 and 32 nodes for each scenario. The number of participating computing nodes form each cluster 
+are different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
+Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 
+
+\begin{table}[h]
+
+\caption{The different clusters scenarios}
+\centering
+\begin{tabular}{|*{4}{c|}}
+\hline
+\multirow{2}{*}{Scenario name}        & \multicolumn{3}{c|} {The participating clusters} \\ \cline{2-4} 
+                                      & Cluster & Site           & No. of  nodes     \\ 
+\hline
+\multirow{3}{*}{Two sites / 16 nodes} & Taurus & Lyon                & 5                      \\ \cline{2-4} 
+                                      & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
+                                      & Griffon       & Nancy        & 6                      \\ 
+\hline
+\multirow{3}{*}{Tow sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
+                                      & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
+                                      & Griffon     &Nancy           & 12                     \\ 
+\hline
+\multirow{3}{*}{One site / 16 nodes}  & Graphite    & Nancy            & 4                      \\ \cline{2-4} 
+                                      & Graphene     & Nancy           & 6                      \\ \cline{2-4} 
+                                      & Griffon         & Nancy        & 6                      \\ 
+\hline
+\multirow{3}{*}{One site / 32 nodes}  & Graphite   & Nancy             & 4                      \\ \cline{2-4} 
+                                      & Graphene      & Nancy          & 12                     \\ \cline{2-4} 
+                                      & Griffon          & Nancy       & 12                       \\ 
+\hline
+\end{tabular}
+ \label{tab:sc}
+\end{table}
+
+\begin{figure}

   \centering
   \centering

-  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
-  \caption{The power consumption by one core from Taurus cluster}
-  \label{fig:power_cons}
+  \includegraphics[scale=0.5]{fig/eng_con_scenarios.eps}
+  \caption{The energy consumptions of NAS benchmarks over different scenarios }
+  \label{fig:eng_sen}

 \end{figure}
 
 
 \end{figure}
 
 

-\subsection{The experimental results of the scaling algorithm}
-\label{sec.res}
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/time_scenarios.eps}
+  \caption{The execution times of NAS benchmarks over different scenarios }
+  \label{fig:time_sen}
+\end{figure}
+
+The NAS parallel benchmarks are executed over these two platforms
+ with different number of nodes, as in Table \ref{tab:sc}. 
+The overall energy consumption of all the benchmarks solving the class D instance and
+using the proposed frequency selection algorithm is measured 
+using the equation of the reduced energy consumption, equation 
+(\ref{eq:energy}). This model uses the measured dynamic and static 
+power values  showed in Table \ref{table:grid5000}. The execution
+time is measured for all the benchmarks over these different scenarios.  
+
+The energy consumptions  and the execution times for all the benchmarks are 
+presented in the plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 
+
+For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
+for  16 and 32 nodes is lower than the energy consumed while using two sites. 
+The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption. 
+The execution times of these benchmarks 
+over one site with 16 and 32 nodes are also lower when  compared to those of the  two sites 
+scenario.
+
+However, the  execution times and the energy consumptions of EP and MG benchmarks, which have no or small communications, are not significantly affected 
+ in both scenarios. Even when the number of nodes is doubled. On the other hand, the communications of the rest of the benchmarks increases when using long distance communications between two sites or increasing the number of computing nodes.
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/eng_s.eps}
+  \caption{The energy saving of NAS benchmarks over different scenarios }
+  \label{fig:eng_s}
+\end{figure}
+
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/per_d.eps}
+  \caption{The performance degradation of NAS benchmarks over different scenarios }
+  \label{fig:per_d}
+\end{figure}
+
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/dist.eps}
+  \caption{The tradeoff distance of NAS benchmarks over different scenarios }
+  \label{fig:dist}
+\end{figure}
+
+The energy saving percentage is computed as the ratio between the reduced 
+energy consumption, equation (\ref{eq:energy}), and the original energy consumption,
+equation (\ref{eq:eorginal}), for all benchmarks as in figure \ref{fig:eng_s}. 
+This figure shows that the energy saving percentages of one site scenario for
+16 and 32 nodes are bigger than those of the two sites scenario which is due
+to the higher  computations to communications ratio in the first scenario   
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which 
+results in  a lower energy consumption. Indeed, the dynamic  consumed power
+is exponentially related to the CPU's frequency value. On the other side, the increase in the number of computing nodes can 
+increase the communication times and thus produces less energy saving depending on the 
+benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more 
+energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While, LU and SP consume more energy with 16 nodes than 32 in one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 
+
+
+The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 
+scenario, except for the   EP benchmark which has  no communications. Therefore, the energy saving percentage of this benchmark is 
+dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 
+in the one site scenario, the graphite cluster is selected but in the two sits scenario 
+this cluster is replaced with Taurus cluster which is more powerful. 
+Therefore, the energy saving of EP benchmarks are bigger in the two sites scenario due 
+to the higher maximum difference between the computing powers of the nodes. 
+
+In fact, high differences between the nodes' computing powers make the proposed frequencies selecting  
+algorithm  select smaller frequencies for the powerful nodes which 
+produces less energy consumption and thus more energy saving.
+The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
+
+
+Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
+The performance degradation percentage for the benchmarks running on two sites  with
+16 or 32  nodes is on average equal to 8\% or 4\% respectively. 
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with
+16 or 32  nodes is on average equal to 3\% or 10\% respectively. In opposition to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
+nodes when the communications occur in high speed network does not decrease the computations to 
+communication ratio. 
+
+
+ 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{blue}{
+The performance degradation percentage of EP benchmark is the higher when it is compared with 
+the other benchmarks. There are no communication and slack times in this benchmark and its 
+performance degradation percentage depends on the frequency value selected in the computing node.
+The rest of the benchmarks showed different performance degradation percentages, which are decreased
+when the communication times are increased and vice versa.}
+
+\textcolor{blue}{Figure \ref{fig:dist} presents the tradeoff distance percentage between the energy saving  and the performance degradation for all benchmarks  over both  scenarios. The tradeoff distance percentage can be 
+computed as in the tradeoff function \ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
+tradeoff, on average is equal to  26\%. As a result, one site scenario using both 16 and 32 nodes had better energy and performance 
+tradeoff comparing to the two sites scenario. This because the former used high speed local communications 
+which increased the computations to communications ratio  and the latter used long distance communications which decreased this ratio. }   \textcolor{red}{The last paragraph has compared the two scenarios}
+
+
+ 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}
 
 \subsection{The experimental results of multi-cores clusters}

-\label{sec.res}
+\label{sec.res-mc}
+The  clusters of grid'5000 have different number of cores embedded in their nodes
+as shown in Table \ref{table:grid5000}. The cores of each node can exchange 
+data via the shared memory \cite{rauber_book}. In 
+this section, the proposed scaling algorithm is evaluated over the grid'5000 grid while using multi-core nodes 
+selected according to the two  platform scenarios described in the section \ref{sec.res}.
+The two platform scenarios, the two sites and one site scenarios, use  32 
+cores from multi-cores nodes instead of 32 distinct nodes. For example if 
+the participating number of cores from a certain cluster is equal to 12, 
+in the multi-core scenario the selected nodes is equal to 3 nodes while using 
+4 cores from each node. The platforms with one  
+core per node and  multi-cores nodes are  shown in Table \ref{table:sen-mc}. 
+The energy consumptions and execution times of running the NAS parallel 
+benchmarks, class D, over these four different scenarios are presented 
+in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+
+The execution times for most of  the NAS  benchmarks are higher over the one site multi-cores per node scenario 
+ than the execution time of those running over one site single core per node  scenario. Indeed,  
+   the communication times  are higher in the one site multi-cores scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. 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. 
+
+\textcolor{blue}{Furthermore,  in two sites multi-cores per node scenario part of the communications happened via shared memory
+and the rest via long distance network. According to the high latency in the long distance network, the
+communication times are smaller compared to the  communication times of the shared memory. 
+Therefore, using the shared memory communications mixed with the long distance communications 
+has decreased the communication times, and thus the overall execution time is 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.
+
+\textcolor{blue}{ 
+Therefore, the computations to communications ratios of the NAS benchmarks are higher over 
+the one site one core scenario  compared to the other scenarios. 
+More energy reduction has achieved when this ratio increased, because the proposed scaling algorithm selecting smaller frequencies that decreased the dynamic power consumption.  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.
+
+\textcolor{blue}{
+The energy saving percentages of all NAS benchmarks running over these four scenarios are presented in the figure \ref{fig:eng-s-mc}. This figure 
+shows that  the energy saving percentages are higher  over the two sites multi-cores scenario 
+than over the two sites one core scenario, on average they are equal to 22\% and 18\%
+respectively. This is according to the increase or decrease in the computations to communications  ratio  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 indicates 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.  
+
+\textcolor{blue}{
+When the benchmarks are executed  over the one 
+site one core scenario their performance degradation percentages, on average
+is equal to 10\%, are higher than those executed over one site multi-cores, 
+which on average is equal to 7\%. This because using  multi-cores in one site scenario
+decreased the computations to communications ratio. Therefore, selecting small 
+frequencies by the scaling algorithm do not increase the execution time significantly.}
+
+\textcolor{blue}{
+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 verified which scenario is the best in term of the energy and performance ratio. The figure indicates that using muti-cores in both of the one site and two sites scenarios gives bigger  tradeoff distance percentages, on overage they are equal to 17.6\% and 15.3\% respectively. On the contrary, using one core per node in both of one site and two sites scenarios gives lower  tradeoff distance percentages, on average they are 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}

 
 

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

 
 

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

 
 

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

 
 

+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/dist_pow.eps}
+  \caption{The tradeoff distance for NAS benchmarks of the three power scenario}
+  \label{fig:dist-pow}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.47]{fig/three_scenarios.pdf}
+  \caption{Comparing the selected frequency scaling factors of MG benchmark for three static power scenarios}
+  \label{fig:fre-pow}
+\end{figure}
+
+\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. }
+
+ 
+\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. 
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/edp_eng}
+  \caption{Comparing of the energy saving for the proposed method with EDP method}
+  \label{fig:edp-eng}
+\end{figure}
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/edp_per}
+  \caption{Comparing of the performance degradation for the proposed method with EDP method}
+  \label{fig:edp-perf}
+\end{figure}
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{fig/edp_dist}
+  \caption{Comparing of the tradeoff distance for the proposed method with EDP method}
+  \label{fig:edp-dist}
+\end{figure}
+As shown 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}
 
 
 \section{Conclusion}