Merge ssh://info.iut-bm.univ-fcomte.fr/mpi-energy

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 00d85a2d1fde3e5502f5711de6949f73be900727..47be6a49b02f4eeed48cff7698dee082d7db966a 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -16,7 +16,6 @@
 \usepackage{xspace}
 \usepackage[textsize=footnotesize]{todonotes}
 \newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
 \usepackage{xspace}
 \usepackage[textsize=footnotesize]{todonotes}
 \newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}

-\newcommand{\JC}[2][inline]{\todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}

 
 \begin{document}
 
 
 \begin{document}
 


@@ -32,16 +31,14 @@
   \IEEEauthorblockA{%
     FEMTO-ST Institute\\
     University of Franche-Comté\\
   \IEEEauthorblockA{%
     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\\
-    Email: \{jean-claude.charr, ahmed.fanfakh\_badri\_muslim, raphael.couturier, arnaud.giersch\}@univ-fcomte.fr
+    IUT de Belfort-Montb\'{e}liard, Rue Engel Gros, BP 27, 90016 Belfort, France\\
+   Fax  : (+33)~3~84~58~77~32\\
+   Email: \{jean-claude.charr, raphael.couturier, ahmed.fanfakh\_badri\_muslim, arnaud.giersch\}@univ-fcomte.fr

    }
   }
 
 \maketitle
 
    }
   }
 
 \maketitle
 

-\AG{Is the fax number correct? Shall we add a telephone number?}
-\JC{Use Capital letters for only the first letter in the title of a section, table, figure, ...} 

 \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
 \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


@@ -50,7 +47,7 @@ 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
 frequency is selected. The performance degradation ratio can even be higher than
 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
 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 tradeoff
+the saved energy ratio. Therefore, the chosen scaling factor must give the best possible tradeoff

 between energy reduction and performance. 
 
 In this paper we present an algorithm
 between energy reduction and performance. 
 
 In this paper we present an algorithm


@@ -104,19 +101,17 @@ benchmarks (NPB v3.3) developed by NASA~\cite{44}. Our experiments are executed
 using the simulator SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183}
 over an homogeneous distributed memory architecture. Furthermore, we compare the
 proposed algorithm with Rauber and Rünger methods~\cite{3}.
 using the simulator SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183}
 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 tradeoff.
+The comparison's results show that our algorithm gives better energy-time tradeoff.

 
 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 tradeoffs
 
 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 tradeoffs

-objective function. Section~\ref{sec.optim} demonstrates the proposed
-energy-performance algorithm. Section~\ref{sec.expe} verifies the performance prediction
+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}.
 model and presents the results of the proposed algorithm.  Also, It shows the comparison results. Finally,
 we conclude in Section~\ref{sec.concl}.

-\section{Related Works}
+\section{Related works}

 \label{sec.relwork}
 
 \AG{Consider introducing the models sec.~\ref{sec.exe} maybe before related works}
 \label{sec.relwork}
 
 \AG{Consider introducing the models sec.~\ref{sec.exe} maybe before related works}


@@ -152,9 +147,9 @@ 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 et al. 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 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. 

 To maintain the performance of the parallel program , they
 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 pf the processors. Although this model was built for parallel architectures, it can be adapted  to distributed architectures by taking into account the communications. 
+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 :
 \begin{enumerate}
 \item Based on Rauber's analytical model to predict the energy consumption and the execution time of the application with different frequency gears. 
 The primary contribution of this paper is presenting a new online scaling factor selection method which has the following characteristics :
 \begin{enumerate}
 \item Based on Rauber's analytical model to predict the energy consumption and the execution time of the application with different frequency gears. 


@@ -167,10 +162,10 @@ The primary contribution of this paper is presenting a new online scaling factor
 \end{enumerate}
 
 
 \end{enumerate}
 
 

-\section{Execution and Energy of Parallel Tasks on Homogeneous Platform} 
+\section{Execution and energy of parallel tasks on homogeneous platform} 

 \label{sec.exe}
 \label{sec.exe}

-\JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'', can be deleted if we need space, we can just say we are interested in this paper in homogeneous clusters}
-\subsection{Parallel Tasks Execution on Homogeneous Platform}
+%\AG{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'', can be deleted if we need space, we can just say we are interested in this paper in homogeneous clusters}
+\subsection{Parallel tasks execution on homogeneous platform}

 A homogeneous cluster consists of identical nodes in terms of hardware and software. 
 Each node has its own memory and at least one processor which can
 be a multi-core. The nodes are connected via a high bandwidth network. Tasks
 A homogeneous cluster consists of identical nodes in terms of hardware and software. 
 Each node has its own memory and at least one processor which can
 be a multi-core. The nodes are connected via a high bandwidth network. Tasks


@@ -179,9 +174,9 @@ we consider execution of the synchronous tasks on distributed homogeneous
 platform. These tasks can exchange the data via synchronous message passing.
 \begin{figure*}[t]
   \centering
 platform. These tasks can exchange the data via synchronous message passing.
 \begin{figure*}[t]
   \centering

-  \subfloat[Sync. Imbalanced Communications]{\includegraphics[scale=0.67]{commtasks}\label{fig:h1}}
-  \subfloat[Sync. Imbalanced Computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
-  \caption{Parallel Tasks on Homogeneous Platform}
+  \subfloat[Sync. imbalanced communications]{\includegraphics[scale=0.67]{commtasks}\label{fig:h1}}
+  \subfloat[Sync. imbalanced computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
+  \caption{Parallel tasks on homogeneous platform}

   \label{fig:homo}
 \end{figure*}
 Therefore, the execution time of a task consists of the computation time and the
   \label{fig:homo}
 \end{figure*}
 Therefore, the execution time of a task consists of the computation time and the


@@ -193,19 +188,19 @@ with different number of nodes. Another source of idle times is the imbalanced c
 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 

-of the program is the execution time of the slowest task as in equation \ref{eq:T1}.
+of the program is the execution time of the slowest task as in EQ~(\ref{eq:T1}).

 \begin{equation}
   \label{eq:T1}
   \textit{Program Time} = \max_{i=1,2,\dots,N} T_i
 \end{equation}
 where $T_i$ is the execution time of task $i$ and all the tasks are executed concurrently on different processors.
 
 \begin{equation}
   \label{eq:T1}
   \textit{Program Time} = \max_{i=1,2,\dots,N} T_i
 \end{equation}
 where $T_i$ is the execution time of task $i$ and all the tasks are executed concurrently on different processors.
 

-\subsection{Energy Model for Homogeneous Platform}
+\subsection{Energy model for homogeneous platform}

 
 Many researchers~\cite{9,3,15,26} divide the power consumed by a processor to two power metrics: the
  static and the dynamic power. While the first one is consumed as long as the computing unit is on, the latter is only consumed during computation times. The dynamic power
 $P_{dyn}$  is related to the switching activity $\alpha$, load capacitance $C_L$,
 
 Many researchers~\cite{9,3,15,26} divide the power consumed by a processor to two power metrics: the
  static and the dynamic power. While the first one is consumed as long as the computing unit is on, the latter is only consumed during computation times. The dynamic power
 $P_{dyn}$  is related to the switching activity $\alpha$, load capacitance $C_L$,

-the supply voltage $V$ and operational frequency $f$, as shown in equation ~\ref{eq:pd}.
+the supply voltage $V$ and operational frequency $f$, as shown in EQ~(\ref{eq:pd}).

 \begin{equation}
   \label{eq:pd}
   P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
 \begin{equation}
   \label{eq:pd}
   P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f


@@ -243,7 +238,6 @@ lower a core's frequency.  This factor reduces
 quadratically the dynamic power which may cause degradation in performance and thus, the increase of the static energy because the execution time is increased~\cite{36}. If the tasks are sorted according to their execution times before scaling in a descending order,   the total energy consumption model for a parallel
 homogeneous platform, as  presented by Rauber et al.~\cite{3}, can be written as a function of   the scaling factor \emph S,   as in EQ~(\ref{eq:energy}).
 
 quadratically the dynamic power which may cause degradation in performance and thus, the increase of the static energy because the execution time is increased~\cite{36}. If the tasks are sorted according to their execution times before scaling in a descending order,   the total energy consumption model for a parallel
 homogeneous platform, as  presented by Rauber et al.~\cite{3}, can be written as a function of   the scaling factor \emph S,   as in EQ~(\ref{eq:energy}).
 

-\JC{Are you sure of the following equation}

 \begin{equation}
   \label{eq:energy}
   E = P_\textit{dyn} \cdot S_1^{-2} \cdot
 \begin{equation}
   \label{eq:energy}
   E = P_\textit{dyn} \cdot S_1^{-2} \cdot


@@ -253,29 +247,26 @@ homogeneous platform, as  presented by Rauber et al.~\cite{3}, can be written as
 \end{equation}
 where \emph N is the number of parallel nodes,  $T_i  \  and  \  S_i \  for \  i=1,...,N$ are the execution times and scaling factors of the sorted tasks. Therefore,  $T1$ is the time of the slowest task,  and $S_1$ its scaling factor which should be the highest because they are proportional to
 the time values $T_i$.  The scaling factors are computed as in EQ~(\ref{eq:si}).
 \end{equation}
 where \emph N is the number of parallel nodes,  $T_i  \  and  \  S_i \  for \  i=1,...,N$ are the execution times and scaling factors of the sorted tasks. Therefore,  $T1$ is the time of the slowest task,  and $S_1$ its scaling factor which should be the highest because they are proportional to
 the time values $T_i$.  The scaling factors are computed as in EQ~(\ref{eq:si}).

-\JC{This equation does not make sense either, what's S? there is no F}

 \begin{equation}
   \label{eq:si}
   S_i = S \cdot \frac{T_1}{T_i}
       = \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}
 \end{equation}
 \begin{equation}
   \label{eq:si}
   S_i = S \cdot \frac{T_1}{T_i}
       = \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}
 \end{equation}

-\JC{The Rauber model was used for a parallel machine not a homogeneous platform}
-where $F$ is the number of available frequencies. In this paper we depend on
+In this paper we depend on

 Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this
 Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this

-model is used for homogeneous platform that we work on in this paper, and (2) we
+model is used for any number of concurrent tasks, and (2) we

 compare our algorithm with Rauber and Rünger scaling factor selection method which is based on
 EQ~(\ref{eq:energy}). The optimal scaling factor is computed by minimizing the derivation for this equation which produces EQ~(\ref{eq:sopt}).
 compare our algorithm with Rauber and Rünger scaling factor selection method which is based on
 EQ~(\ref{eq:energy}). The optimal scaling factor is computed by minimizing the derivation for this equation which produces EQ~(\ref{eq:sopt}).

-\JC{what's the small n in the equation}

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

-  S_\textit{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot
+  S_\textit{opt} = \sqrt[3]{\frac{2}{N} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot

     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 
     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 

-\JC{The following 2 sections can be merged easily}
+\AG{The following 2 sections can be merged easily}

 
 

-\section{Performance Evaluation of MPI Programs}
+\section{Performance evaluation of MPI programs}

 \label{sec.mpip}
 
 The performance (execution time) of  parallel synchronous MPI applications depend on
 \label{sec.mpip}
 
 The performance (execution time) of  parallel synchronous MPI applications depend on


@@ -296,8 +287,7 @@ the new scaling factor as in EQ~(\ref{eq:tnew}).
 \end{equation}
  In this paper, this prediction method is used to select the best scaling factor for each processor as presented in the next section.
 
 \end{equation}
  In this paper, this prediction method is used to select the best scaling factor for each processor as presented in the next section.
 

-
-\section{Performance to Energy Competition}
+\section{Performance to energy competition}

 \label{sec.compet}
 
 This section demonstrates our approach for choosing the optimal scaling
 \label{sec.compet}
 
 This section demonstrates our approach for choosing the optimal scaling


@@ -312,9 +302,9 @@ without scaled frequency:
 \begin{multline}
   \label{eq:enorm}
   E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
 \begin{multline}
   \label{eq:enorm}
   E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\

-        {} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot
+        {} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot

                \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
                \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +

-               P_\textit{static} \cdot T_1 \cdot S_i \cdot N  }{
+               P_\textit{static} \cdot T_1 \cdot S_1 \cdot N  }{

               P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
               P_\textit{static} \cdot T_1 \cdot N }
 \end{multline}
               P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
               P_\textit{static} \cdot T_1 \cdot N }
 \end{multline}


@@ -349,13 +339,13 @@ performance as follows:
 \end{equation}
 \begin{figure*}
   \centering
 \end{equation}
 \begin{figure*}
   \centering

-  \subfloat[Converted Relation.]{%
+  \subfloat[Converted relation.]{%

     \includegraphics[width=.4\textwidth]{file.eps}\label{fig:r1}}%
   \qquad%
     \includegraphics[width=.4\textwidth]{file.eps}\label{fig:r1}}%
   \qquad%

-  \subfloat[Real Relation.]{%
+  \subfloat[Real relation.]{%

     \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}}
   \label{fig:rel}
     \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}}
   \label{fig:rel}

-  \caption{The Energy and Performance Relation}
+  \caption{The energy and performance relation}

 \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


@@ -365,16 +355,16 @@ at the same time, see figure~(\ref{fig:r1}). Then our objective function has the
 following form:
 \begin{equation}
   \label{eq:max}
 following form:
 \begin{equation}
   \label{eq:max}

-  \textit{MaxDist} = \max (\overbrace{P^{-1}_\textit{Norm}}^{\text{Maximize}} -
-                           \overbrace{E_\textit{Norm}}^{\text{Minimize}} )
+ 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}
 \end{equation}

-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 satisfy

 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
 factors as shown in~\cite{15,3,19}. 
 
 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
 factors as shown in~\cite{15,3,19}. 
 

-\section{Optimal Scaling Factor for Performance and Energy}
+\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.
 \begin{algorithm}[tp]
 \label{sec.optim}
  Algorithm~\ref{EPSA} compute the optimal scaling factor according to the objective function described above.
 \begin{algorithm}[tp]


@@ -386,22 +376,23 @@ factors as shown in~\cite{15,3,19}.
     \State Set $P_{states}$ to the number of available frequencies.
     \State Set the variable $F_{new}$ to max. frequency,  $F_{new} = F_{max} $
     \State Set the variable $F_{diff}$ to the difference between two successive frequencies.
     \State Set $P_{states}$ to the number of available frequencies.
     \State Set the variable $F_{new}$ to max. frequency,  $F_{new} = F_{max} $
     \State Set the variable $F_{diff}$ to the difference between two successive frequencies.

-    \For {$i=1$   to   $P_{states} $}
+    \For {$j:=1$   to   $P_{states} $}

       \State - $F_{new}=F_{new} - F_{diff} $
       \State - $S = \frac{F_\textit{max}}{F_\textit{new}}$
       \State - $F_{new}=F_{new} - F_{diff} $
       \State - $S = \frac{F_\textit{max}}{F_\textit{new}}$

-      \State - $S_i = S \cdot \frac{T_1}{T_i}= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i} \  for \  i=1,...,N$
-      \State - $E_\textit{Norm} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot
+      \State - $S_i = S \cdot \frac{T_1}{T_i}= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i} \                   
+       for \  i=1,...,N$
+      \State - $E_\textit{Norm} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot

                \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
                \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +

-               P_\textit{static} \cdot T_1 \cdot S_i \cdot N  }{
+               P_\textit{static} \cdot T_1 \cdot S_1 \cdot N  }{

               P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
               P_\textit{static} \cdot T_1 \cdot N }$
       \State -  $P_{NormInv}=T_{old}/T_{new}$ 
       \If{  $(P_{NormInv}-E_{Norm} > Dist$) }
               P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
               P_\textit{static} \cdot T_1 \cdot N }$
       \State -  $P_{NormInv}=T_{old}/T_{new}$ 
       \If{  $(P_{NormInv}-E_{Norm} > Dist$) }

-        \State $S_{optimal} = S$
+        \State $S_{opt} = S$

         \State $Dist = P_{NormInv} - E_{Norm}$
       \EndIf
     \EndFor
         \State $Dist = P_{NormInv} - E_{Norm}$
       \EndIf
     \EndFor

-    \State  Return $S_{optimal}$
+    \State  Return $S_{opt}$

   \end{algorithmic}
 \end{algorithm}
 
   \end{algorithmic}
 \end{algorithm}
 


@@ -418,13 +409,13 @@ in the MPI program.
   \caption{DVFS}
   \label{dvfs}
   \begin{algorithmic}[1]
   \caption{DVFS}
   \label{dvfs}
   \begin{algorithmic}[1]

- \For {$J:=1$ to $Some-Iterations \; $}
+ \For {$k:=1$ to $Some-Iterations \; $}

   \State -Computations Section.
    \State -Communications Section.
   \State -Computations Section.
    \State -Communications Section.

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

-     \State -Compute the new frequency from the returned optimal scaling factor.
+     \State -Compute the new frequency from the \par\hspace{13 pt} returned optimal scaling factor.

      \State -Set the new frequency to the CPU.
    \EndIf
 \EndFor
      \State -Set the new frequency to the CPU.
    \EndIf
 \EndFor


@@ -442,10 +433,10 @@ According to this equation all the nodes may have the same frequency value if
 they have balanced workloads, otherwise, they take different frequencies when
 having imbalanced workloads. Thus, EQ~(\ref{eq:fi}) adapts the frequency of the CPU to the nodes' workloads to maintain performance.
 
 they have balanced workloads, otherwise, they take different frequencies when
 having imbalanced workloads. Thus, EQ~(\ref{eq:fi}) adapts the frequency of the CPU to the nodes' workloads to maintain performance.
 

-\section{Experimental Results}
+\section{Experimental results}

 \label{sec.expe}
 Our experiments are executed on the simulator SimGrid/SMPI
 \label{sec.expe}
 Our experiments are executed on the simulator SimGrid/SMPI

-v3.10. We configure the simulator to use a a homogeneous cluster with one core per
+v3.10. We configure the simulator to use a homogeneous cluster with one core per

 node. The
 detailed characteristics of our platform file are shown in the
 table~(\ref{table:platform}).
 node. The
 detailed characteristics of our platform file are shown in the
 table~(\ref{table:platform}).


@@ -454,7 +445,7 @@ 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]
 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}
+  \caption{Platform file parameters}

   % title of Table
   \centering
   \begin{tabular}{|*{7}{l|}}
   % title of Table
   \centering
   \begin{tabular}{|*{7}{l|}}


@@ -466,7 +457,7 @@ The backbone of the cluster simulates a high performance switch.
   \end{tabular}
   \label{table:platform}
 \end{table}
   \end{tabular}
   \label{table:platform}
 \end{table}

-\subsection{Performance Prediction Verification}
+\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
 real execution time with the predicted execution time. Each program runs offline
 
 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
 real execution time with the predicted execution time. Each program runs offline


@@ -475,10 +466,10 @@ time values. These scaling factors are computed by dividing the maximum
 frequency by the new one see EQ~(\ref{eq:s}). 
 \begin{figure*}[t]
   \centering
 frequency by the new one see EQ~(\ref{eq:s}). 
 \begin{figure*}[t]
   \centering

-  \includegraphics[width=.4\textwidth]{cg_per.eps}\qquad%
-  \includegraphics[width=.4\textwidth]{mg_pre.eps}
-  \includegraphics[width=.4\textwidth]{bt_pre.eps}\qquad%
-  \includegraphics[width=.4\textwidth]{lu_pre.eps}
+  \includegraphics[width=.328\textwidth]{cg_per.eps}\hfill%
+  \includegraphics[width=.328\textwidth]{mg_pre.eps}\hfill%
+ % \includegraphics[width=.4\textwidth]{bt_pre.eps}\qquad%
+  \includegraphics[width=.328\textwidth]{lu_pre.eps}\hfill%

   \caption{Comparing predicted to real execution time}
   \label{fig:pred}
 \end{figure*}
   \caption{Comparing predicted to real execution time}
   \label{fig:pred}
 \end{figure*}


@@ -486,10 +477,8 @@ frequency by the new one see EQ~(\ref{eq:s}).
 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
  NAS parallel benchmarks: CG, MG, EP, FT, BT, LU
 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
  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 average normalized errors between the predicted execution time and
-the real time (SimGrid time) for all programs is between 0.0032 to 0.0133. 
-\JC{why compute the average error not the max}
-\subsection{The  experimental results}
+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 }

 The proposed algorithm was applied to seven MPI programs of the NAS
 benchmarks (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and
 C). For each instance the benchmarks were executed on a number of processors
 The proposed algorithm was applied to seven MPI programs of the NAS
 benchmarks (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and
 C). For each instance the benchmarks were executed on a number of processors


@@ -518,17 +507,17 @@ energy saving percent and the minimum performance degradation percent at the
 same time from all available scaling factors.
 \begin{figure*}[t]
   \centering
 same time from all available scaling factors.
 \begin{figure*}[t]
   \centering

-  \includegraphics[width=.33\textwidth]{ep.eps}\hfill%
-  \includegraphics[width=.33\textwidth]{cg.eps}\hfill%
-  \includegraphics[width=.33\textwidth]{sp.eps}
-  \includegraphics[width=.33\textwidth]{lu.eps}\hfill%
-  \includegraphics[width=.33\textwidth]{bt.eps}\hfill%
-  \includegraphics[width=.33\textwidth]{ft.eps}
-  \caption{Optimal scaling factors for The parallel NAS benchmarks}
+  \includegraphics[width=.328\textwidth]{ep.eps}\hfill%
+  \includegraphics[width=.328\textwidth]{cg.eps}\hfill%
+  \includegraphics[width=.328\textwidth]{sp.eps}
+  \includegraphics[width=.328\textwidth]{lu.eps}\hfill%
+  \includegraphics[width=.328\textwidth]{bt.eps}\hfill%
+  \includegraphics[width=.328\textwidth]{ft.eps}
+  \caption{Optimal scaling factors for the predicted energy and performance of NAS benchmarks}

   \label{fig:nas}
 \end{figure*}
 \begin{table}[htb]
   \label{fig:nas}
 \end{figure*}
 \begin{table}[htb]

-  \caption{The Scaling Factors Results}
+  \caption{The scaling factors results}

   % title of Table
   \centering
   \begin{tabular}{|l|*{4}{r|}}
   % title of Table
   \centering
   \begin{tabular}{|l|*{4}{r|}}


@@ -552,9 +541,9 @@ FT. The opposite happens when the optimal scaling factor is small value as
 example BT and EP. Our algorithm selects big scaling factor value when the
 communication and the other slacks times are big and smaller ones in opposite
 cases. In EP there are no communications inside the iterations. This make our
 example BT and EP. Our algorithm selects big scaling factor value when the
 communication and the other slacks times are big and smaller ones in opposite
 cases. In EP there are no communications inside the iterations. This make our

-EPSA to selects smaller scaling factor values (inducing smaller energy savings).
+algorithm to selects smaller scaling factor values (inducing smaller energy savings).

 
 

-\subsection{Comparing Results}
+\subsection{Results comparison}

 
 In this section, we compare our scaling factor selection method with Rauber and Rünger
 methods~\cite{3}. They had two scenarios, the first is to reduce energy to the
 
 In this section, we compare our scaling factor selection method with Rauber and Rünger
 methods~\cite{3}. They had two scenarios, the first is to reduce energy to the


@@ -562,12 +551,11 @@ 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 $R_{E-P}$. The comparison is made in tables~(\ref{table:compare
-  Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
+scenario as $R_{E-P}$. While we refer to our algorithm as EPSA. The comparison is made in tables~(\ref{table:compareA},\ref{table:compareB},\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.
 \begin{table}[p]
 tables show the results of our method and Rauber and Rünger scenarios for all the
 NAS benchmarks programs for classes A,B and C.
 \begin{table}[p]

-  \caption{Comparing Results for  The NAS Class A}
+  \caption{Comparing results for  the NAS class A}

   % title of Table
   \centering
   \begin{tabular}{|l|l|*{4}{r|}}
   % title of Table
   \centering
   \begin{tabular}{|l|l|*{4}{r|}}


@@ -608,7 +596,7 @@ NAS benchmarks programs for classes A,B and C.
   % is used to refer this table in the text
 \end{table}
 \begin{table}[p]
   % is used to refer this table in the text
 \end{table}
 \begin{table}[p]

-  \caption{Comparing Results for The NAS Class B}
+  \caption{Comparing results for the NAS class B}

   % title of Table
   \centering
   \begin{tabular}{|l|l|*{4}{r|}}
   % title of Table
   \centering
   \begin{tabular}{|l|l|*{4}{r|}}


@@ -650,7 +638,7 @@ NAS benchmarks programs for classes A,B and C.
 \end{table}
 
 \begin{table}[p]
 \end{table}
 
 \begin{table}[p]

-  \caption{Comparing Results for The NAS Class C}
+  \caption{Comparing results for the NAS class C}

   % title of Table
   \centering
   \begin{tabular}{|l|l|*{4}{r|}}
   % title of Table
   \centering
   \begin{tabular}{|l|l|*{4}{r|}}


@@ -699,17 +687,17 @@ while keeping the performance degradation as low as possible. Our algorithm alwa
 gives the highest positive energy to performance tradeoffs while Rauber and Rünger method
 ($R_{E-P}$) gives in some time negative tradeoffs such as in BT and
 EP.
 gives the highest positive energy to performance tradeoffs while Rauber and Rünger method
 ($R_{E-P}$) gives in some time negative tradeoffs such as in BT and
 EP.

-\begin{figure}[t]
+\begin{figure*}[t]

   \centering
   \centering

-  \includegraphics[width=.33\textwidth]{compare_class_A.pdf}
-  \includegraphics[width=.33\textwidth]{compare_class_B.pdf}
-  \includegraphics[width=.33\textwidth]{compare_class_c.pdf}
-  \caption{Comparing our method to Rauber and Rünger Methods}
+  \includegraphics[width=.328\textwidth]{compare_class_A.pdf}
+  \includegraphics[width=.328\textwidth]{compare_class_B.pdf}
+  \includegraphics[width=.328\textwidth]{compare_class_c.pdf}
+  \caption{Comparing our method to Rauber and Rünger methods}

   \label{fig:compare}
   \label{fig:compare}

-\end{figure}
+\end{figure*}

 \section{Conclusion}
 \label{sec.concl}
 \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 tradeoff 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's methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber's methods in terms of energy-performance ratio. 
+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 tradeoff 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. 

 
 In the near future, we would like to adapt this scaling factor selection method to heterogeneous platforms where each node has different characteristics. In particular, each CPU has different available frequencies, energy consumption and performance. It would be also interesting to develop a new energy model for asynchronous parallel iterative methods where the number of iterations is not known in advance and depends on the global convergence of the iterative system.
 
 
 In the near future, we would like to adapt this scaling factor selection method to heterogeneous platforms where each node has different characteristics. In particular, each CPU has different available frequencies, energy consumption and performance. It would be also interesting to develop a new energy model for asynchronous parallel iterative methods where the number of iterations is not known in advance and depends on the global convergence of the iterative system.
 


@@ -718,7 +706,6 @@ In the near future, we would like to adapt this scaling factor selection method
 As a PhD student, M. Ahmed Fanfakh, would like to thank the University of
 Babylon (Iraq) for supporting his work.
 
 As a PhD student, M. Ahmed Fanfakh, would like to thank the University of
 Babylon (Iraq) for supporting his work.
 

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