Merge branch 'master' of ssh://info.iut-bm.univ-fcomte.fr/mpi-energy

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 61d95c6232a8e08cf3b0fff75438afee2ffdb09b..e60d17b12394fd04f692ff516231af37f8680aa3 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -36,21 +36,20 @@
     FEMTO-ST Institute\\
     University of Franche-Comté\\
     IUT de Belfort-Montbéliard, 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
     FEMTO-ST Institute\\
     University of Franche-Comté\\
     IUT de Belfort-Montbéliard, 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\

-    Fax  : +33~3~84~58~77~32\\
+   % Fax  : +33~3~84~58~77~32\\

     Email: \email{{jean-claude.charr,raphael.couturier,ahmed.fanfakh_badri_muslim,arnaud.giersch}@univ-fcomte.fr}
    }
   }
 
 \maketitle
 
     Email: \email{{jean-claude.charr,raphael.couturier,ahmed.fanfakh_badri_muslim,arnaud.giersch}@univ-fcomte.fr}
    }
   }
 
 \maketitle
 

-\AG{Is the fax number correct? Shall we add a telephone number?}

 \begin{abstract}
   Dynamic Voltage Frequency Scaling (DVFS) can be applied to modern CPUs. 
 This technique is usually used to reduce the energy consumed by a CPU while
 computing . Indeed, power consumption by a processor at a given instant is
 exponentially related to its frequency. Thus, decreasing the frequency reduces
 the power consumed by the CPU. However, it can also significantly affect the
 \begin{abstract}
   Dynamic Voltage Frequency Scaling (DVFS) can be applied to modern CPUs. 
 This technique is usually used to reduce the energy consumed by a CPU while
 computing . Indeed, power consumption by a processor at a given instant is
 exponentially related to its frequency. Thus, decreasing the frequency reduces
 the power consumed by the CPU. However, it can also significantly affect the

-performance of the executed program if it is compute bound and a low CPU
+performance of the executed program if it is compute bound and if a low CPU

 frequency is selected. The performance degradation ratio can even be higher than
 the saved energy ratio. Therefore, the chosen scaling factor must give the best possible trade-off
 between energy reduction and performance. 
 frequency is selected. The performance degradation ratio can even be higher than
 the saved energy ratio. Therefore, the chosen scaling factor must give the best possible trade-off
 between energy reduction and performance. 


@@ -108,14 +107,14 @@ over an homogeneous distributed memory architecture. Furthermore, we compare the
 proposed algorithm with Rauber and Rünger methods~\cite{3}.
 The comparison's results show that our algorithm gives better energy-time trade-off.
 
 proposed algorithm with Rauber and Rünger methods~\cite{3}.
 The comparison's results show that our algorithm gives better energy-time trade-off.
 

-This paper is organized as follows: Section~\ref{sec.relwork} presents related works
-from other authors.  Section~\ref{sec.exe} shows the execution of parallel
-tasks and sources of idle times.  It resumes the energy
-model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance
-of MPI program.  Section~\ref{sec.compet} presents the energy-performance trade-offs
-objective function. Section~\ref{sec.optim} demonstrates the proposed energy-performance algorithm. Section~\ref{sec.expe} verifies the performance prediction
-model and presents the results of the proposed algorithm.  Also, It shows the comparison results. Finally,
-we conclude in Section~\ref{sec.concl}.
+This paper is organized as follows: Section~\ref{sec.relwork} presents some related works
+from other authors.  Section~\ref{sec.exe} explains the execution of parallel
+tasks and the sources of slack times.  It also presents an energy
+model for homogeneous platforms. Section~\ref{sec.mpip} describes how the performance
+of MPI programs can be predicted .  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 energy-performance algorithm. Section~\ref{sec.expe} verifies the accuracy of the performance prediction
+model and presents the results of the proposed algorithm.  It also shows the comparison results between our method and other existing methods. Finally,
+we conclude in Section~\ref{sec.concl} with a summary and some future works.

 \section{Related works}
 \label{sec.relwork}
 
 \section{Related works}
 \label{sec.relwork}
 


@@ -152,16 +151,16 @@ The main drawback for these methods is that they all require executing a part or
 The online scaling factor selection methods are executed during the runtime of the program. They are usually integrated into iterative programs where the same block of instructions is executed many times.  During the first few iterations, many informations are measured such as the execution time, the energy consumed using a multimeter, the slack times, ... Then a  method will exploit these measurements to compute the scaling factor values for each processor. This operation, measurements and computing new scaling factor, can be repeated as much as needed if the iterations are not regular. Kimura, Peraza, Yu-Liang et al.  ~\cite{11,2,31} used learning methods to select the appropriate scaling factor values  to eliminate the slack times during runtime. However, as seen in ~\cite{39,19}, machine learning methods can take a lot of time to converge when the number of available gears is big. To reduce the impact of slack times,  in~\cite{1}, Lim et al. developed an algorithm that detects the
 communication sections and changes the frequency during these sections
 only. This approach might change the frequency of each processor many times per iteration if an iteration
 The online scaling factor selection methods are executed during the runtime of the program. They are usually integrated into iterative programs where the same block of instructions is executed many times.  During the first few iterations, many informations are measured such as the execution time, the energy consumed using a multimeter, the slack times, ... Then a  method will exploit these measurements to compute the scaling factor values for each processor. This operation, measurements and computing new scaling factor, can be repeated as much as needed if the iterations are not regular. Kimura, Peraza, Yu-Liang et al.  ~\cite{11,2,31} used learning methods to select the appropriate scaling factor values  to eliminate the slack times during runtime. However, as seen in ~\cite{39,19}, machine learning methods can take a lot of time to converge when the number of available gears is big. To reduce the impact of slack times,  in~\cite{1}, Lim et al. developed an algorithm that detects the
 communication sections and changes the frequency during these sections
 only. This approach might change the frequency of each processor many times per iteration if an iteration

-contains more than one communication section. In ~\cite{3}, Rauber and Rünger used an analytical model that after measuring the energy consumed and the execution time with the highest frequency gear, it can predict the energy consumed and the execution time for every frequency gear .   These predictions may be used to choose the optimal gear for each processor executing the parallel program to reduce energy consumption. 
+contains more than one communication section. In ~\cite{3}, Rauber and Rünger used an analytical model that can predict the consumed energy  and the execution time for every frequency gear after measuring the consumed energy  and the execution time with the highest frequency gear.   These predictions may be used to choose the optimal gear for each processor executing the parallel program to reduce energy consumption. 

 To maintain the performance of the parallel program , they
 set the  processor with the biggest load to the highest gear and then compute the scaling  factor values for the rest of the processors. Although this model was built for parallel architectures, it can be adapted  to distributed architectures by taking into account the communications. 
 To maintain the performance of the parallel program , they
 set the  processor with the biggest load to the highest gear and then compute the scaling  factor values for the rest of the processors. Although this model was built for parallel architectures, it can be adapted  to distributed architectures by taking into account the communications. 

-The primary contribution of this paper is presenting a new online scaling factor selection method which has the following characteristics :
+The primary contribution of our paper is presenting a new online scaling factor selection method which has the following characteristics :

 \begin{enumerate}
 \begin{enumerate}

-\item Based on Rauber and Rünger analytical model to predict the energy consumption and the execution time of the application with different frequency gears. 
-\item Selects the frequency scaling factor for simultaneously optimizing energy reduction and maintaining performance.
-\item Well adapted to distributed architectures because it takes into account the communication time.
-\item Well adapted to distributed applications with imbalanced tasks.
-\item Has very small footprint when compared to other
+\item It is based on Rauber and Rünger analytical model to predict the energy consumption and the execution time of the application with different frequency gears. 
+\item It selects the frequency scaling factor for simultaneously optimizing energy reduction and maintaining performance.
+\item It is well adapted to distributed architectures because it takes into account the communication time.
+\item It is well adapted to distributed applications with imbalanced tasks.
+\item it has very small footprint when compared to other

   methods (e.g.,~\cite{19}) and does not require profiling or training as
   in~\cite{38,34}.
 \end{enumerate}
   methods (e.g.,~\cite{19}) and does not require profiling or training as
   in~\cite{38,34}.
 \end{enumerate}


@@ -189,7 +188,7 @@ communication time. Moreover, the synchronous communications between tasks can
 lead to slack times while tasks wait at the synchronization barrier for other tasks to
 finish their tasks (see figure~(\ref{fig:h1})). The imbalanced communications 
 happen when nodes have to send/receive different amount of data or they communicate
 lead to slack times while tasks wait at the synchronization barrier for other tasks to
 finish their tasks (see figure~(\ref{fig:h1})). The imbalanced communications 
 happen when nodes have to send/receive different amount of data or they communicate

-with different number of nodes. Another source of idle times is the imbalanced computations.  
+with different number of nodes. Another source of slack times is the imbalanced computations.  

 This happens when processing different amounts of data on each processor  (see figure~(\ref{fig:h2})).  
 In this case the fastest tasks have to wait at the synchronization barrier for the 
 slowest ones to begin the next task. In both cases the overall execution time 
 This happens when processing different amounts of data on each processor  (see figure~(\ref{fig:h2})).  
 In this case the fastest tasks have to wait at the synchronization barrier for the 
 slowest ones to begin the next task. In both cases the overall execution time 


@@ -217,12 +216,13 @@ The static power $P_{static}$ captures the leakage power as follows:
 \end{equation}
 where V is the supply voltage, $N_{trans}$ is the number of transistors, $K_{design}$ is a
 design dependent parameter and $I_{leak}$ is a technology-dependent
 \end{equation}
 where V is the supply voltage, $N_{trans}$ is the number of transistors, $K_{design}$ is a
 design dependent parameter and $I_{leak}$ is a technology-dependent

-parameter. Energy consumed by an individual processor $E_{ind}$ is the summation
-of the dynamic and the static powers multiplied by the execution time~\cite{36,15}.
+parameter. The energy consumed by an individual processor to execute a given program can be computed as:

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

-   E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T
+   E_\textit{ind} =  P_\textit{dyn} \cdot T_{Comp} + P_\textit{static} \cdot T

 \end{equation}
 \end{equation}

+where $T$ is the execution time of the program, $T_{Comp}$ is the computation time and $T_{Comp} \le T$. $T_{Comp}$ may be equal to $T$ if there is no communications, no slack times and no synchronizations. 
+

 DVFS is a process that is allowed in
 modern processors to reduce the dynamic power by scaling down the voltage and
 frequency. Its main objective is to reduce the overall energy
 DVFS is a process that is allowed in
 modern processors to reduce the dynamic power by scaling down the voltage and
 frequency. Its main objective is to reduce the overall energy


@@ -339,7 +339,8 @@ performance as follows:
 \begin{equation}
   \label{eq:pnorm_en}
   P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
 \begin{equation}
   \label{eq:pnorm_en}
   P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}

-               = \frac{ T_\textit{Old}}{T_\textit{Max Comp Old} \cdot S +
+               = \frac{T_\textit{Max Comp Old} +
+                 T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} \cdot S +

                  T_\textit{Max Comm Old}}
 \end{equation}
 \begin{figure*}
                  T_\textit{Max Comm Old}}
 \end{equation}
 \begin{figure*}


@@ -354,7 +355,7 @@ performance as follows:
 \end{figure*}
 Then, we can modelize our objective function as finding the maximum distance
 between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance
 \end{figure*}
 Then, we can modelize our objective function as finding the maximum distance
 between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance

-curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent
+curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represents

 the minimum energy consumption with minimum execution time (better performance)
 at the same time, see figure~(\ref{fig:r1}). Then our objective function has the
 following form:
 the minimum energy consumption with minimum execution time (better performance)
 at the same time, see figure~(\ref{fig:r1}). Then our objective function has the
 following form:


@@ -363,7 +364,7 @@ following form:
  Max Dist = \max_{j=1,2,\dots,F} (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
                         \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
 \end{equation}
  Max Dist = \max_{j=1,2,\dots,F} (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
                         \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
 \end{equation}

-where F is the number of available frequencies. Then we can select the optimal scaling factor that satisfy
+where F is the number of available frequencies. Then we can select the optimal scaling factor that satisfies

 EQ~(\ref{eq:max}).  Our objective function can work with any energy model or
 static power values stored in a data file. Moreover, this function works in
 optimal way when the energy curve has a convex form over the available frequency scaling
 EQ~(\ref{eq:max}).  Our objective function can work with any energy model or
 static power values stored in a data file. Moreover, this function works in
 optimal way when the energy curve has a convex form over the available frequency scaling


@@ -371,7 +372,7 @@ factors as shown in~\cite{15,3,19}.
 
 \section{Optimal scaling factor for performance and energy}
 \label{sec.optim}
 
 \section{Optimal scaling factor for performance and energy}
 \label{sec.optim}

- Algorithm~\ref{EPSA} compute the optimal scaling factor according to the objective function described above.
+ Algorithm~\ref{EPSA} computes the optimal scaling factor according to the objective function described above.

 \begin{algorithm}[tp]
   \caption{Scaling factor selection algorithm}
   \label{EPSA}
 \begin{algorithm}[tp]
   \caption{Scaling factor selection algorithm}
   \label{EPSA}


@@ -403,11 +404,24 @@ factors as shown in~\cite{15,3,19}.
 
 The proposed algorithm works online during the execution time of the MPI
 program. It selects the optimal scaling factor after gathering the computation and communication times 
 
 The proposed algorithm works online during the execution time of the MPI
 program. It selects the optimal scaling factor after gathering the computation and communication times 

-from the program after one iteration. Then the program changes the new frequencies of the CPUs according to the computed scaling factors. This algorithm has small execution time
-(between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ for 32 nodes). The algorithm complexity is O(F$\cdot$N), 
+from the program after one iteration. Then the program changes the new frequencies of the CPUs according to the computed scaling factors. This algorithm has a small execution time: for a homogeneous cluster composed of nodes having the characteristics presented in table~\ref{table:platform}, it takes  0.00152 $ms$ on average for 4 nodes and 0.00665 $ms$ on average for 32 nodes. The algorithm complexity is O(F$\cdot$N), 

 where F is the number of available frequencies and N is the number of computing nodes.  The algorithm is called just 
 once during the execution of the program. The DVFS algorithm~(\ref{dvfs}) shows where and when the algorithm is called
 in the MPI program.
 where F is the number of available frequencies and N is the number of computing nodes.  The algorithm is called just 
 once during the execution of the program. The DVFS algorithm~(\ref{dvfs}) shows where and when the algorithm is called
 in the MPI program.

+\begin{table}[htb]
+  \caption{Platform file parameters}
+  % title of Table
+  \centering
+  \begin{tabular}{|*{7}{l|}}
+    \hline
+    Max & Min & Backbone & Backbone&Link &Link& Sharing  \\
+    Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline
+    \np{2.5} & \np{800} & \np[GBps]{2.25} &\np[$\mu$s]{0.5}& \np[GBps]{1} & \np[$\mu$s]{50} &Full  \\
+    GHz& MHz&  & & &  &Duplex  \\\hline
+  \end{tabular}
+  \label{table:platform}
+\end{table}
+

 %\begin{minipage}{\textwidth}
 
 \begin{algorithm}[tp]
 %\begin{minipage}{\textwidth}
 
 \begin{algorithm}[tp]


@@ -415,8 +429,8 @@ in the MPI program.
   \label{dvfs}
   \begin{algorithmic}[1]
  \For {$k:=1$ to $Some-Iterations \; $}
   \label{dvfs}
   \begin{algorithmic}[1]
  \For {$k:=1$ to $Some-Iterations \; $}

-  \State -Computations Section.
-   \State -Communications Section.
+  \State -Computations section.
+   \State -Communications section.

    \If {$(k=1)$} 
      \State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
      \State -Call algorithm~\ref{EPSA} with these times.
    \If {$(k=1)$} 
      \State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
      \State -Call algorithm~\ref{EPSA} with these times.


@@ -449,19 +463,7 @@ Each node in the cluster has 18 frequency values
 from 2.5 GHz to 800 MHz with 100 MHz difference between each two successive
 frequencies. The simulated network link is 1 GB Ethernet (TCP/IP). 
 The backbone of the cluster simulates a high performance switch.
 from 2.5 GHz to 800 MHz with 100 MHz difference between each two successive
 frequencies. The simulated network link is 1 GB Ethernet (TCP/IP). 
 The backbone of the cluster simulates a high performance switch.

-\begin{table}[htb]
-  \caption{Platform file parameters}
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Max & Min & Backbone & Backbone&Link &Link& Sharing  \\
-    Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline
-    \np{2.5} & \np{800} & \np[GBps]{2.25} &\np[$\mu$s]{0.5}& \np[GBps]{1} & \np[$\mu$s]{50} &Full  \\
-    GHz& MHz&  & & &  &Duplex  \\\hline
-  \end{tabular}
-  \label{table:platform}
-\end{table}
+

 \subsection{Performance prediction verification}
 
 In this section we evaluate the precision of our performance prediction method based on EQ~(\ref{eq:tnew}) by applying it the NAS benchmarks. The NAS programs are executed with the class B option for comparing the
 \subsection{Performance prediction verification}
 
 In this section we evaluate the precision of our performance prediction method based on EQ~(\ref{eq:tnew}) by applying it the NAS benchmarks. The NAS programs are executed with the class B option for comparing the


@@ -480,7 +482,7 @@ frequency by the new one see EQ~(\ref{eq:s}).
 \end{figure*}
 %see Figure~\ref{fig:pred}
 In our cluster there are 18 available frequency states for each processor. 
 \end{figure*}
 %see Figure~\ref{fig:pred}
 In our cluster there are 18 available frequency states for each processor. 

-This lead to 18 run states for each program. We use seven MPI programs of the
+This leads to 18 run states for each program. We use seven MPI programs of the

  NAS parallel benchmarks: CG, MG, EP, FT, BT, LU
 and SP. Figure~(\ref{fig:pred}) presents plots of the real execution times and the simulated ones. The maximum normalized error between the predicted execution time and the real time (SimGrid time) for all programs is between 0.0073 to 0.031. The  better case is for CG and the worse case is for LU. 
 \subsection{The experimental results for the scaling algorithm }
  NAS parallel benchmarks: CG, MG, EP, FT, BT, LU
 and SP. Figure~(\ref{fig:pred}) presents plots of the real execution times and the simulated ones. The maximum normalized error between the predicted execution time and the real time (SimGrid time) for all programs is between 0.0073 to 0.031. The  better case is for CG and the worse case is for LU. 
 \subsection{The experimental results for the scaling algorithm }


@@ -556,11 +558,11 @@ optimal level without considering the performance as in EQ~(\ref{eq:sopt}). We
 refer to this scenario as $R_{E}$. The second scenario is similar to the first
 except setting the slower task to the maximum frequency (when the scale $S=1$)
 to keep the performance from degradation as mush as possible. We refer to this
 refer to this scenario as $R_{E}$. The second scenario is similar to the first
 except setting the slower task to the maximum frequency (when the scale $S=1$)
 to keep the performance from degradation as mush as possible. We refer to this

-scenario as $R_{E-P}$. While we refer to our algorithm as EPSA. The comparison
+scenario as $R_{E-P}$. While we refer to our algorithm as EPSA (Energy to Performance Scaling Algorithm). The comparison

 is made in tables \ref{table:compareA}, \ref{table:compareB},
 and~\ref{table:compareC}. These
 tables show the results of our method and Rauber and Rünger scenarios for all the
 is made in tables \ref{table:compareA}, \ref{table:compareB},
 and~\ref{table:compareC}. These
 tables show the results of our method and Rauber and Rünger scenarios for all the

-NAS benchmarks programs for classes A,B and C.
+NAS benchmarks programs for classes A, B and C.

 \begin{table}[p]
   \caption{Comparing results for  the NAS class A}
   % title of Table
 \begin{table}[p]
   \caption{Comparing results for  the NAS class A}
   % title of Table


@@ -685,23 +687,19 @@ NAS benchmarks programs for classes A,B and C.
 \label{table:compareC}
 % is used to refer this table in the text
 \end{table}
 \label{table:compareC}
 % is used to refer this table in the text
 \end{table}

-As shown in tables~\ref{table:compareA},~\ref{table:compareB} and~\ref{table:compareC}, the ($R_{E-P}$) method outperforms the ($R_{E}$) method in terms of performance and energy reduction. The ($R_{E-P}$) method also gives better energy savings than our method. However, although our scaling factor is not optimal for energy reduction, the results in these tables prove that our algorithm returns the best scaling factor that satisfy our objective method : the largest distance between energy reduction and performance degradation. 
-
-Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving
-percent and the performance degradation percent.  
-Negative values mean that one of the two objectives (energy or performance) have been degraded more than the other. The positive trade-offs with the highest values lead to maximum energy savings
+As shown in tables~\ref{table:compareA},~\ref{table:compareB} and~\ref{table:compareC}, the ($R_{E-P}$) method outperforms the ($R_{E}$) method in terms of performance and energy reduction. The ($R_{E-P}$) method also gives better energy savings than our method. However, although our scaling factor is not optimal for energy reduction, the results in these tables prove that our algorithm returns the best scaling factor that satisfy our objective method : the largest distance between energy reduction and performance degradation. Negative values in the energy-performance column mean that one of the two objectives (energy or performance) have been degraded more than the other. The positive trade-offs with the highest values lead to maximum energy savings

 while keeping the performance degradation as low as possible. Our algorithm always
 gives the highest positive energy to performance trade-offs while Rauber and Rünger method
 ($R_{E-P}$) gives in some time negative trade-offs such as in BT and
 EP.
 while keeping the performance degradation as low as possible. Our algorithm always
 gives the highest positive energy to performance trade-offs while Rauber and Rünger method
 ($R_{E-P}$) gives in some time negative trade-offs such as in BT and
 EP.

-\begin{figure*}[t]
-  \centering
-  \includegraphics[width=.328\textwidth]{fig/compare_class_A}
-  \includegraphics[width=.328\textwidth]{fig/compare_class_B}
-  \includegraphics[width=.328\textwidth]{fig/compare_class_C}
-  \caption{Comparing our method to Rauber and Rünger methods}
-  \label{fig:compare}
-\end{figure*}
+%\begin{figure*}[t]
+%  \centering
+%  \includegraphics[width=.328\textwidth]{fig/compare_class_A}
+%  \includegraphics[width=.328\textwidth]{fig/compare_class_B}
+%  \includegraphics[width=.328\textwidth]{fig/compare_class_C}
+%  \caption{Comparing our method to Rauber and Rünger methods}
+%  \label{fig:compare}
+%\end{figure*}

 \section{Conclusion}
 \label{sec.concl}
 In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed application running on an homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy consumption and the performance of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best trade-off between energy reduction and performance which is the maximum distance between the energy and the inverted performance curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber and Rünger methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber and Rünger methods in terms of energy-performance ratio. 
 \section{Conclusion}
 \label{sec.concl}
 In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed application running on an homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy consumption and the performance of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best trade-off between energy reduction and performance which is the maximum distance between the energy and the inverted performance curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber and Rünger methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber and Rünger methods in terms of energy-performance ratio.