corrections

[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 d4eef92f4baafb3429f2c81ec64a8734c90ec60e..0973f102920fffb8b120050e6579f17d31a5ffc5 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -107,9 +107,9 @@
 
 
 
 
 
 

-\title{Optimizing Energy Consumption with DVFS for Message \\
-         Passing Applications \textcolor{blue}{with iterations} on \\
-                    Grid Architectures} 
+\title{Optimizing the Energy Consumption \\ 
+of Message Passing Applications with Iterations \\ 
+Executed over Grids} 

   
 
 
   
 
 


@@ -143,10 +143,10 @@ scaling (DVFS) is one of them. It can be used to reduce the power consumption of
   In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.  
   It selects the frequencies and tries to give the best
   trade-off between energy saving and performance degradation, for each node
   In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.  
   It selects the frequencies and tries to give the best
   trade-off between energy saving and performance degradation, for each node

-  computing the message passing  application \textcolor{blue}{with iterations}. 
+  computing the message passing  application with iterations. 

   The algorithm has a small
   overhead and works without training or profiling. It uses a new energy model
   The algorithm has a small
   overhead and works without training or profiling. It uses a new energy model

-  for message passing  applications \textcolor{blue}{with iterations} running on a  grid. 
+  for message passing  applications with iterations running on a  grid. 

   The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
   running the NAS parallel benchmarks.  The experiments on 16 nodes, distributed on three clusters, show that it reduces  on average the
   energy consumption  by \np[\%]{30} while  the performance  is on average only degraded
   The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
   running the NAS parallel benchmarks.  The experiments on 16 nodes, distributed on three clusters, show that it reduces  on average the
   energy consumption  by \np[\%]{30} while  the performance  is on average only degraded


@@ -192,7 +192,7 @@ This heterogeneous platform executes more than 7 GFlops per watt while consuming
 50.32 kilowatts.
 
 Besides platform improvements, there are many software and hardware techniques
 50.32 kilowatts.
 
 Besides platform improvements, there are many software and hardware techniques

-to lower the energy consumption of these platforms, such as DVFS, scheduling \textcolor{blue}{and other techniques}.
+to lower the energy consumption of these platforms, such as DVFS, scheduling and other techniques.

  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
  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


@@ -202,7 +202,7 @@ 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 frequency selecting algorithm
 was proposed to reduce the energy consumption of message passing 
 trade-off between the energy reduction and performance degradation ratio. In
 \cite{Our_first_paper} and \cite{pdsec2015}, a frequency selecting algorithm
 was proposed to reduce the energy consumption of message passing 

-applications \textcolor{blue}{with iterations} running over homogeneous and heterogeneous clusters respectively.
+applications with iterations running over homogeneous and heterogeneous clusters respectively.

 The results of the experiments showed significant energy consumption
 reductions. All the experimental results were conducted over the SimGrid
 simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous  platforms, and to simulate the execution of message passing parallel
 The results of the experiments showed significant energy consumption
 reductions. All the experimental results were conducted over the SimGrid
 simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous  platforms, and to simulate the execution of message passing parallel


@@ -212,7 +212,7 @@ In this paper, a new frequency selecting algorithm, adapted to grid platforms
 composed of heterogeneous clusters, is presented. It is applied to the NAS
 parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
 \cite{grid5000}. It selects for a grid platform running a message passing
 composed of heterogeneous clusters, is presented. It is applied to the NAS
 parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
 \cite{grid5000}. It selects for a grid platform running a message passing

- application \textcolor{blue}{with iterations} the vector of frequencies that simultaneously tries to
+ application with iterations the vector of frequencies that simultaneously tries to

 offer the maximum energy reduction and minimum performance degradation
 ratios. The algorithm has a very small overhead, works online and does not need
 any training or profiling.
 offer the maximum energy reduction and minimum performance degradation
 ratios. The algorithm has a very small overhead, works online and does not need
 any training or profiling.


@@ -255,8 +255,8 @@ 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.
 
 sequential, parallel or distributed architecture, homogeneous or heterogeneous
 platform, synchronous or asynchronous application.
 

-In this paper, we are interested in reducing energy for message passing
- synchronous applications \textcolor{blue}{with iterations} running over heterogeneous grid platforms.  Some
+In this paper, we are interested in reducing the energy consumption of message passing
+ synchronous applications with iterations 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}
 works have already been done for such platforms and they can be classified into
 two types of heterogeneous platforms:
 \begin{itemize}


@@ -304,7 +304,7 @@ 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 
 following contributions :
 \begin{enumerate}
 \item two new energy and performance models for message passing 

-  synchronous applications \textcolor{blue}{with iterations} running over a heterogeneous grid platform. Both models
+  synchronous applications with iterations 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.
 
   take into account communication and slack times. The models can predict the
   required energy and the execution time of the application.
 


@@ -312,7 +312,7 @@ following contributions :
   platforms. The algorithm has a very small overhead and does not need any
   training nor profiling. It uses a new optimization function which
   simultaneously maximizes the performance and minimizes the energy consumption
   platforms. The algorithm has a very small overhead and does not need any
   training nor profiling. It uses a new optimization function which
   simultaneously maximizes the performance and minimizes the energy consumption

-  of a message passing  synchronous application \textcolor{blue}{with iterations}.
+  of a message passing  synchronous application with iterations.

 
 \end{enumerate}
 
 
 \end{enumerate}
 


@@ -322,25 +322,25 @@ following contributions :
 \label{sec.exe}
 
 \subsection{The execution time of message passing distributed 
 \label{sec.exe}
 
 \subsection{The execution time of message passing distributed 

-  applications \textcolor{blue}{with iterations} on a heterogeneous platform}
+  applications with iterations on a heterogeneous platform}

 
 In this paper, we are interested in reducing the energy consumption of message
 
 In this paper, we are interested in reducing the energy consumption of message

-passing distributed  synchronous applications \textcolor{blue}{with iterations} running over
+passing distributed  synchronous applications with iterations running over

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

-The overall execution time of a distributed  synchronous application  \textcolor{blue}{with iterations} 
+The overall execution time of a distributed  synchronous application  with iterations running 

 over a heterogeneous grid consists of the sum of the computation time and 
 the communication time for every iteration on a node. 
 over a heterogeneous grid consists of the sum of the computation time and 
 the communication time for every iteration on a node. 

-\textcolor{blue}{However, nodes from distinct clusters in a grid have different computing powers, thus
-while executing message passing \textcolor{blue}{with iterations} synchronous applications, fast nodes
+However, nodes from distinct clusters in a grid have different computing powers, thus
+while the application, 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
 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. } 
+periods are called idle or slack times. 

 Therefore, the
 overall execution time of the program is the execution time of the slowest task 
 Therefore, the
 overall execution time of the program is the execution time of the slowest task 

-which has the highest computation time and no slack time. \textcolor{blue}{For example, in Figure \ref{fig:heter}  the task 1 is the slower task which has no slack time (not waits for the other nodes) and it is only has the communication times.}
+which has the highest computation time and almost no slack time. For example, in Figure \ref{fig:heter}, task 1 is the slower task and it does not have to wait for the other nodes to communicate with them because they all finish their computations before it.  

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


@@ -361,7 +361,8 @@ as in (\ref{eq:s}).
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}

-\textcolor{blue}{Where $\Fmax$ is the maximum frequency before applying DVFS and $\Fnew$  is the new frequency after applying DVFS.}
+where $\Fmax$ is the maximum frequency before applying any DVFS and $\Fnew$  is the new frequency after applying DVFS.
+

 The execution time of a compute bound sequential program is linearly
 proportional to the frequency scaling factor $S$.  On the other hand, message
 passing distributed applications consist of two parts: computation and
 The execution time of a compute bound sequential program is linearly
 proportional to the frequency scaling factor $S$.  On the other hand, message
 passing distributed applications consist of two parts: computation and


@@ -378,7 +379,7 @@ especially different frequency gears, when applying DVFS operations on the nodes
 of these clusters, they may get different scaling factors represented by a scaling vector:
 $(S_{11}, S_{12},\dots, S_{NM_i})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
 be able to predict the execution time of message passing synchronous 
 of these clusters, they may get different scaling factors represented by a scaling vector:
 $(S_{11}, S_{12},\dots, S_{NM_i})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
 be able to predict the execution time of message passing synchronous 

-applications \textcolor{blue}{with iterations} running over a heterogeneous grid, for different vectors of
+applications with iterations running over a heterogeneous grid, for different vectors of

 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any


@@ -394,10 +395,10 @@ where $N$ is the number of  clusters in the grid, $M_i$ is the number of  nodes
  cluster $i$, $\TcpOld[ij]$ 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 execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
  cluster $i$, $\TcpOld[ij]$ 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 execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors

-and \textcolor{blue}{the  communication time of the slower node without slack time during one iteration.
-The slower node $h$ is the node that gives maximum execution time in all clusters befor scaling its frequency.}
+and the  communication time of the slowest node without slack time during one iteration.
+ The slowest node $h$ is the node which takes the  maximum execution time to execute an iteration  before scaling down its  frequency.

 It means that only the communication time without any slack time is taken into account.
 It means that only the communication time without any slack time is taken into account.

-Therefore, the execution time of the  application \textcolor{blue}{with iterations} is equal to
+Therefore, the execution time of the  application is equal to

 the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the
 number of iterations of that application.
 
 the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the
 number of iterations of that application.
 


@@ -546,8 +547,8 @@ frequency scaling factors for a homogeneous and a heterogeneous cluster respecti
 Both methods selects the frequencies that gives the best trade-off between 
 energy consumption reduction and performance for  message passing
  synchronous applications \textcolor{blue}{with iterations}.   In this work we
 Both methods selects the frequencies that gives the best trade-off between 
 energy consumption reduction and performance for  message passing
  synchronous applications \textcolor{blue}{with iterations}.   In this work we

-are interested in grids that are composed of heterogeneous clusters, \textcolor{blue}{where} the nodes 
-have different characteristics such  as  dynamic power, static power, computation power, 
+are interested in grids that are composed of heterogeneous clusters. The nodes from distinct clusters may have 
+ different characteristics such  as  dynamic power, static power, computation power, 

 frequencies range, network latency and bandwidth. 
 Due to the heterogeneity of the processors, a vector of scaling factors should be selected
 and it must give the best trade-off between energy consumption and performance.
 frequencies range, network latency and bandwidth. 
 Due to the heterogeneity of the processors, a vector of scaling factors should be selected
 and it must give the best trade-off between energy consumption and performance.


@@ -750,7 +751,7 @@ of the slowest node and the computation time of the node $i$ as follows:
 \end{equation}
 Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
 algorithm computes the initial frequencies for all nodes as a ratio between the
 \end{equation}
 Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
 algorithm computes the initial frequencies for all nodes as a ratio between the

-maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+maximum frequency of node and its computed scaling factor as

 follows:
 \begin{equation}
   \label{eq:Fint}
 follows:
 \begin{equation}
   \label{eq:Fint}


@@ -800,7 +801,7 @@ in this paper real experiments were conducted over the Grid'5000 platform.
 
 \subsection{Grid'5000 architecture and power consumption}
 \label{sec.grid5000}
 
 \subsection{Grid'5000 architecture and power consumption}
 \label{sec.grid5000}

-Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed all over  metropolitan France and Luxembourg. All the sites are connected together via       a special long distance network called RENATER,
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed all over  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 a few heterogeneous 
 computing clusters and each cluster contains many homogeneous nodes. In total,
 which is the French National Telecommunication Network for Technology.
 Each site of the grid is composed of a few heterogeneous 
 computing clusters and each cluster contains many homogeneous nodes. In total,


@@ -809,7 +810,7 @@ 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 to  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 
 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 to  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 all the components of a node at a given instant, such as CPU, hard drive, main-board and memory. For more details refer to
+the power consumption  for each node in those sites. The measured power is the overall consumed power  by all the components of a node at a given instant. For more details refer to

 \cite{Energy_measurement}. In order to correctly 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 consumptions represents the 
 dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 
 \cite{Energy_measurement}. In order to correctly 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 consumptions represents the 
 dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 


@@ -858,7 +859,7 @@ The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT.
 the same block of operations several times, starting from the initial solution until reaching
 the acceptable approximation of the exact solution.}
  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 same block of operations several times, starting from the initial solution until reaching
 the acceptable approximation of the exact solution.}
  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,  class D was used for all benchmarks in all the experiments presented in the next sections. 
+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 the next sections, the  class D was used for all the benchmarks in all the experiments. 

 
 
   
 
 
   


@@ -911,7 +912,7 @@ Two scenarios were considered while selecting the clusters from these two sites
 The main reason 
 for 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 
 The main reason 
 for 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 reduce the effect of DVFS operations.
+is very low due to the higher communication times which reduces the effect of the DVFS operations.

 
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
 
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 


@@ -931,7 +932,7 @@ Table~\ref{tab:sc} shows the number of nodes used from each cluster for each sce
                                       & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                       & Griffon       & Nancy        & 6                      \\ 
 \hline
                                       & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                       & Griffon       & Nancy        & 6                      \\ 
 \hline

-\multirow{3}{*}{Tow sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
+\multirow{3}{*}{Two sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 

                                       & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                       & Griffon     &Nancy           & 12                     \\ 
 \hline
                                       & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                       & Griffon     &Nancy           & 12                     \\ 
 \hline


@@ -997,7 +998,7 @@ results in  a lower energy consumption. Indeed, the dynamic  consumed power
 is exponentially related to the CPU's frequency value. On the other hand, 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 benchmarks CG, MG, BT and FT show more 
 is exponentially related to the CPU's frequency value. On the other hand, 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 benchmarks CG, MG, BT and FT show more 

-energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes.  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. 
+energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes.  LU and SP consume more energy with 16 nodes than 32 nodes on one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 

 
 \begin{figure*}[!h]
   \centering
 
 \begin{figure*}[!h]
   \centering


@@ -1031,7 +1032,7 @@ The best energy saving percentage was obtained in the one site scenario with 16
 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 8.3\% or 4.7\% respectively. 
 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 8.3\% or 4.7\% 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
+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 high 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 3.2\% and 10.6\% respectively. In contrary 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. 
 16 or 32  nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary 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. 


@@ -1229,8 +1230,8 @@ The experimental results, the energy saving, performance degradation and trade-o
 presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
 
 As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
 presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
 
 As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 

-The proposed algorithm gives better results than the EDP method because it 
-maximizes the energy saving and the performance at the same time. 
+The proposed algorithm gives better results than the EDP method because the former selects the set of frequencies that  
+gives the best tradeoff between energy saving and performance. 

 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
 These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
 These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.