change tables values to be all of them in the same long

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index e6d674576c2498e3775bf478dd47334162588acf..c38368cd7b263869d14db8f8717eb45cd114a144 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -1,33 +1,37 @@
 \documentclass[12pt]{article}
 %\documentclass[12pt,twocolumn]{article}
 \documentclass[12pt]{article}
 %\documentclass[12pt,twocolumn]{article}

-\DeclareMathSizes{40}{4000}{200}{2000}
+

 \usepackage[T1]{fontenc}
 \usepackage[T1]{fontenc}

+\usepackage[utf8]{inputenc}

 \usepackage[english]{babel}
 \usepackage{algorithm,algorithmicx,algpseudocode}
 \usepackage{graphicx,graphics}
 \usepackage{subfig}
 \usepackage{listings}
 \usepackage{colortbl}
 \usepackage[english]{babel}
 \usepackage{algorithm,algorithmicx,algpseudocode}
 \usepackage{graphicx,graphics}
 \usepackage{subfig}
 \usepackage{listings}
 \usepackage{colortbl}

-\usepackage{sectsty}
-\usepackage{titlesec}
-\usepackage{secdot}
+\usepackage{amsmath}
+% \usepackage{sectsty}
+% \usepackage{titlesec}
+% \usepackage{secdot}

 %\usepackage[font={footnotesize,bt}]{caption}
 %\usepackage[font=scriptsize,labelfont=bf]{caption}
 %\usepackage[font={footnotesize,bt}]{caption}
 %\usepackage[font=scriptsize,labelfont=bf]{caption}

+\usepackage{lmodern}
+\usepackage{todonotes}
+\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\small\textbf{AG:} #2}}

 
 \begin{document}
 
 \begin{document}

-\begin{center}
-  \Large
-  \title*\textbf{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
-\end{center}
-\parskip 0pt
-\linespread{1.18}
-\normalsize
-\makeatletter
-\renewcommand*{\@seccntformat}[1]{\csname the#1\endcsname\hspace{0.01cm}}
-\makeatother
-\sectionfont{\large}
-
-\section{.~Introduction }
+
+\title{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
+\author{A. Badri \and J.-C. Charr \and R. Couturier \and A. Giersch}
+\maketitle
+
+\AG{``Optimal'' is a bit pretentious in the title}
+
+\begin{abstract}
+  \AG{FIXME}
+\end{abstract}
+
+\section{Introduction}

 
 The need for computing power is still increasing and it is not expected to slow
 down in the coming years. To satisfy this demand, researchers and supercomputers
 
 The need for computing power is still increasing and it is not expected to slow
 down in the coming years. To satisfy this demand, researchers and supercomputers


@@ -64,20 +68,18 @@ algorithm has ability to predict both energy consumption and execution time over
 all available scaling factors.  The prediction achieved depends on some
 computing time information, gathered at the beginning of the runtime.  We apply
 this algorithm to seven MPI benchmarks. These MPI programs are the NAS parallel
 all available scaling factors.  The prediction achieved depends on some
 computing time information, gathered at the beginning of the runtime.  We apply
 this algorithm to seven MPI benchmarks. These MPI programs are the NAS parallel

-penchmarks (NPB v3.3) developed by NASA~\cite{44}. Our experiments are executed
-using the simulator Simgrid/SMPI v3.10~\cite{45} over an homogeneous distributed
-memory architecture. Furthermore, we compare the proposed algorithm with
-Rauber's methods. The comparison's results show that our algorithm gives better
-energy-time trade off.
-\sectionfont{\large}
+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's methods. The comparison's results show that our
+algorithm gives better energy-time trade off.

 
 

-\section{.~Related Works }
+\section{Related Works}

 
 In the this section some heuristics, to compute the scaling factor, are
 presented and classified in two parts : offline and online methods.
 
 In the this section some heuristics, to compute the scaling factor, are
 presented and classified in two parts : offline and online methods.

- \sectionfont{\large}

 
 

-\subsection{~The offline DVFS orientations}
+\subsection{The offline DVFS orientations}

 
 The DVFS offline methods are static and are not executed during the runtime of
 the program. Some approaches used heuristics to select the best DVFS state
 
 The DVFS offline methods are static and are not executed during the runtime of
 the program. Some approaches used heuristics to select the best DVFS state


@@ -85,16 +87,15 @@ during the compilation phases as an example in Azevedo et al.~\cite{40}. He used
 intra-task algorithm to choose the DVFS setting when there are dependency points
 between tasks. While in~\cite{29}, Xie et al. used breadth-first search
 algorithm to do that. Their goal is saving energy with time limits. Another
 intra-task algorithm to choose the DVFS setting when there are dependency points
 between tasks. While in~\cite{29}, Xie et al. used breadth-first search
 algorithm to do that. Their goal is saving energy with time limits. Another

-approaches gathers and stores the runtime information for each DVFS state , then
+approaches gathers and stores the runtime information for each DVFS state, then

 used their methods offline to select the suitable DVFS that optimize energy-time
 trade offs. As an example~\cite{8}, Rountree et al. used liner programming
 algorithm, while in~\cite{38,34}, Cochran et al. used multi logistic regression
 algorithm for the same goal. The offline study that shown the DVFS impact on the
 communication time of the MPI program is~\cite{17}, Freeh et al. show that these
 times not changed when the frequency is scaled down.
 used their methods offline to select the suitable DVFS that optimize energy-time
 trade offs. As an example~\cite{8}, Rountree et al. used liner programming
 algorithm, while in~\cite{38,34}, Cochran et al. used multi logistic regression
 algorithm for the same goal. The offline study that shown the DVFS impact on the
 communication time of the MPI program is~\cite{17}, Freeh et al. show that these
 times not changed when the frequency is scaled down.

-\sectionfont{\large}

 
 

-\subsection{~The online DVFS orientations}
+\subsection{The online DVFS orientations}

 
 The objective of these works is to dynamically compute and set the frequency of
 the CPU during the runtime of the program for saving energy. Estimating and
 
 The objective of these works is to dynamically compute and set the frequency of
 the CPU during the runtime of the program for saving energy. Estimating and


@@ -111,7 +112,7 @@ program used online for saving energy as in~\cite{1}, Lim et al. developed an
 algorithm that detects the communication sections and changes the frequency
 during these sections only. This approach changes the frequency many times
 because an iteration may contain more than one communication section. The domain
 algorithm that detects the communication sections and changes the frequency
 during these sections only. This approach changes the frequency many times
 because an iteration may contain more than one communication section. The domain

-of analytical modeling used for choosing the optimal frequency as in ~\cite{3},
+of analytical modeling used for choosing the optimal frequency as in~\cite{3},

 Rauber et al. developed an analytical mathematical model for determining the
 optimal frequency scaling factor for any number of concurrent tasks, without
 considering communication times. They set the slowest task to maximum frequency
 Rauber et al. developed an analytical mathematical model for determining the
 optimal frequency scaling factor for any number of concurrent tasks, without
 considering communication times. They set the slowest task to maximum frequency


@@ -128,9 +129,8 @@ paper.  However, the primary contributions of this paper are:
 \item The proposed algorithm works online without profiling or training as
   in~\cite{38,34}.
 \end{enumerate}
 \item The proposed algorithm works online without profiling or training as
   in~\cite{38,34}.
 \end{enumerate}

-\sectionfont{\large}

 
 

-\section{.~Parallel Tasks Execution on Homogeneous Platform}
+\section{Parallel Tasks Execution on Homogeneous Platform}

 
 A homogeneous cluster consists of identical nodes in terms of the hardware and
 the software. Each node has its own memory and at least one processor which can
 
 A homogeneous cluster consists of identical nodes in terms of the hardware and
 the software. Each node has its own memory and at least one processor which can


@@ -154,34 +154,37 @@ amounts of data on each processor as an example see figure~(\ref{fig:h2}). In
 this case the fastest tasks have to wait at the synchronous barrier for the
 slowest tasks to finish their job. In both two cases the overall execution time
 of the program is the execution time of the slowest task as :
 this case the fastest tasks have to wait at the synchronous barrier for the
 slowest tasks to finish their job. In both two cases the overall execution time
 of the program is the execution time of the slowest task as :

-\begin{equation}  \label{eq:T1}
-  Program Time=MAX_{i=1,2,..,N} (T_i) \hfill
+\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 process $i$.
 \end{equation}
 where $T_i$ is the execution time of process $i$.

-\sectionfont{\large}

 
 

-\section{.~Energy Model for Homogeneous Platform}
+\section{Energy Model for Homogeneous Platform}

 
 The energy consumption by the processor consists of two powers metric: the
 dynamic and the static power. This general power formulation is used by many
 
 The energy consumption by the processor consists of two powers metric: the
 dynamic and the static power. This general power formulation is used by many

-researchers see ~\cite{9,3,15,26}. The dynamic power of the CMOS processors
+researchers see~\cite{9,3,15,26}. The dynamic power of the CMOS processors

 $P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
 the supply voltage $V$ and operational frequency $f$ respectively as follow :
 $P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
 the supply voltage $V$ and operational frequency $f$ respectively as follow :

-\begin{equation}  \label{eq:pd}
-  \displaystyle  P_{dyn} = \alpha . C_L . V^2 . f
+\begin{equation}
+  \label{eq:pd}
+  P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f

 \end{equation}
 The static power $P_{static}$ captures the leakage power consumption as well as
 the power consumption of peripheral devices like the I/O subsystem.
 \end{equation}
 The static power $P_{static}$ captures the leakage power consumption as well as
 the power consumption of peripheral devices like the I/O subsystem.

-\begin{equation}  \label{eq:ps}
-  \displaystyle  P_{static}  = V . N . K_{design} . I_{leak}
+\begin{equation}
+  \label{eq:ps}
+  P_{static}  = V \cdot N \cdot K_{design} \cdot I_{leak}

 \end{equation}
 where V is the supply voltage, N 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 power multiply by the execution time for example
 see~\cite{36,15} .
 \end{equation}
 where V is the supply voltage, N 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 power multiply by the execution time for example
 see~\cite{36,15} .

-\begin{equation}  \label{eq:eind}
-  \displaystyle  E_{ind} = (P_{dyn} + P_{static} ) . T
+\begin{equation}
+  \label{eq:eind}
+  E_{ind} = ( P_{dyn} + P_{static} ) \cdot T

 \end{equation}
 The dynamic voltage and frequency scaling (DVFS) is a process that allowed in
 modern processors to reduce the dynamic power by scaling down the voltage and
 \end{equation}
 The dynamic voltage and frequency scaling (DVFS) is a process that allowed in
 modern processors to reduce the dynamic power by scaling down the voltage and


@@ -192,11 +195,12 @@ equation is used to study the change of the dynamic voltage with respect to
 various frequency values in~\cite{3}. The reduction process of the frequency are
 expressed by scaling factor \emph S. The scale \emph S is the ratio between the
 maximum and the new frequency as in EQ~(\ref{eq:s}).
 various frequency values in~\cite{3}. The reduction process of the frequency are
 expressed by scaling factor \emph S. The scale \emph S is the ratio between the
 maximum and the new frequency as in EQ~(\ref{eq:s}).

-\begin{equation}  \label{eq:s}
-  S=\:\frac{F_{max}}{F_{new}} \hfill  \newline
+\begin{equation}
+  \label{eq:s}
+  S = \frac{F_{max}}{F_{new}}

 \end{equation}
 The value of the scale \emph S is grater than 1 when changing the frequency to
 \end{equation}
 The value of the scale \emph S is grater than 1 when changing the frequency to

-any new frequency value(\emph {P-state}) in governor.~It is equal to 1 when the
+any new frequency value(\emph {P-state}) in governor. It is equal to 1 when the

 frequency are set to the maximum frequency.  The energy consumption model for
 parallel homogeneous platform is depending on the scaling factor \emph S. This
 factor reduces quadratically the dynamic power.  Also, this factor increases the
 frequency are set to the maximum frequency.  The energy consumption model for
 parallel homogeneous platform is depending on the scaling factor \emph S. This
 factor reduces quadratically the dynamic power.  Also, this factor increases the


@@ -206,8 +210,11 @@ for any number of concurrent tasks develops by Rauber~\cite{3}. This model
 consider the two powers metric for measuring the energy of the parallel tasks as
 in EQ~(\ref{eq:energy}).
 
 consider the two powers metric for measuring the energy of the parallel tasks as
 in EQ~(\ref{eq:energy}).
 

-\begin{equation}  \label{eq:energy}
-  E= \displaystyle \;P_{dyn}\,.\,S_1^{-2}\;.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_1\;\,.\,N
+\begin{equation}
+  \label{eq:energy}
+  E = P_{dyn} \cdot S_1^{-2} \cdot
+    \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
+    P_{static} \cdot T_1 \cdot S_1 \cdot N

  \hfill
 \end{equation}
 Where \emph N is the number of parallel nodes, $T_1 $ is the time of the slower
  \hfill
 \end{equation}
 Where \emph N is the number of parallel nodes, $T_1 $ is the time of the slower


@@ -215,11 +222,14 @@ task, $T_i$ is the time of the task $i$ and $S_1$ is the maximum scaling factor
 for the slower task. The scaling factor $S_1$, as in EQ~(\ref{eq:s1}), selects
 from the set of scales values $S_i$. Each of these scales are proportional to
 the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}).
 for the slower task. The scaling factor $S_1$, as in EQ~(\ref{eq:s1}), selects
 from the set of scales values $S_i$. Each of these scales are proportional to
 the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}).

-\begin{equation}  \label{eq:s1}
-  S_1=MAX_{i=1,2,..,F} (S_i) \hfill
+\begin{equation}
+  \label{eq:s1}
+  S_1 = \max_{i=1,2,\dots,F} S_i

 \end{equation}
 \end{equation}

-\begin{equation}  \label{eq:si}
-  S_i=\:S\: .\:(\frac{T_1}{T_i})=\: (\frac{F_{max}}{F_{new}}).(\frac{T_1}{T_i}) \hfill
+\begin{equation}
+  \label{eq:si}
+  S_i = S \cdot \frac{T_1}{T_i}
+      = \frac{F_{max}}{F_{new}} \cdot \frac{T_1}{T_i}

 \end{equation}
 Where $F$ is the number of available frequencies. In this paper we depend on
 Rauber's energy model EQ~(\ref{eq:energy}) for two reasons : 1-this model used
 \end{equation}
 Where $F$ is the number of available frequencies. In this paper we depend on
 Rauber's energy model EQ~(\ref{eq:energy}) for two reasons : 1-this model used


@@ -228,13 +238,13 @@ algorithm with Rauber's scaling model.  Rauber's optimal scaling factor for
 optimal energy reduction derived from the EQ~(\ref{eq:energy}). He takes the
 derivation for this equation (to be minimized) and set it to zero to produce the
 scaling factor as in EQ~(\ref{eq:sopt}).
 optimal energy reduction derived from the EQ~(\ref{eq:energy}). He takes the
 derivation for this equation (to be minimized) and set it to zero to produce the
 scaling factor as in EQ~(\ref{eq:sopt}).

-\begin{equation}  \label{eq:sopt}
-  S_{opt}= {\sqrt [3~]{\frac{2}{n} \frac{P_{dyn}}{P_{static}}  \Big(1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^3}\Big) }} \hfill
+\begin{equation}
+  \label{eq:sopt}
+  S_{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_{dyn}}{P_{static}} \cdot
+    \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }

 \end{equation}
 \end{equation}

-%[\Big 3]
-\sectionfont{\large}

 
 

-\section{.~Performance Evaluation of MPI Programs}
+\section{Performance Evaluation of MPI Programs}

 
 The performance (execution time) of the parallel MPI applications are depends on
 the time of the slowest task as in figure~(\ref{fig:homo}). Normally the
 
 The performance (execution time) of the parallel MPI applications are depends on
 the time of the slowest task as in figure~(\ref{fig:homo}). Normally the


@@ -243,7 +253,7 @@ frequency. Therefore, any DVFS operation for the energy reduction increase the
 execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) the
 energy affected by the scaling factor $S$. This factor also has a great impact
 on the performance. When scaling down the frequency to the new value according
 execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) the
 energy affected by the scaling factor $S$. This factor also has a great impact
 on the performance. When scaling down the frequency to the new value according

-to EQ(~\ref{eq:s}) lead to the value of the scale $S$ has inverse relation with
+to EQ~(\ref{eq:s}) lead to the value of the scale $S$ has inverse relation with

 new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decrease the
 frequency value, the execution time increase. Then the new frequency value has
 inverse relation with time ($F_{new} \propto \frac{1}{T}$). This lead to the
 new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decrease the
 frequency value, the execution time increase. Then the new frequency value has
 inverse relation with time ($F_{new} \propto \frac{1}{T}$). This lead to the


@@ -260,21 +270,20 @@ must be precisely specifying communication time and the computation time for the
 slower task. Secondly, we use these times for predicting the execution time for
 any MPI program as a function of the new scaling factor as in the
 EQ~(\ref{eq:tnew}).
 slower task. Secondly, we use these times for predicting the execution time for
 any MPI program as a function of the new scaling factor as in the
 EQ~(\ref{eq:tnew}).

-\begin{equation}  \label{eq:tnew}
-  \displaystyle T_{new}= T_{Max \:Comp \:Old} \; . \:S \;+ \;T_{Max\: Comm\: Old}
-  \hfill
+\begin{equation}
+  \label{eq:tnew}
+  T_{new} = T_{\textit{Max Comp Old}} \cdot S + T_{\textit{Max Comm Old}}

 \end{equation}
 The above equation shows that the scaling factor \emph S has linear relation
 with the computation time without affecting the communication time. The
 communication time consists of the beginning times which an MPI calls for
 sending or receiving till the message is synchronously sent or received. In this
 paper we predict the execution time of the program for any new scaling factor
 \end{equation}
 The above equation shows that the scaling factor \emph S has linear relation
 with the computation time without affecting the communication time. The
 communication time consists of the beginning times which an MPI calls for
 sending or receiving till the message is synchronously sent or received. In this
 paper we predict the execution time of the program for any new scaling factor

-value. Depending on this prediction we can produce our energy-performace scaling
+value. Depending on this prediction we can produce our energy-performance scaling

 method as we will show in the coming sections. In the next section we make an
 investigation study for the EQ~(\ref{eq:tnew}).
 method as we will show in the coming sections. In the next section we make an
 investigation study for the EQ~(\ref{eq:tnew}).

-\sectionfont{\large}

 
 

-\section{.~Performance Prediction Verification }
+\section{Performance Prediction Verification}

 
 In this section we evaluate the precision of our performance prediction methods
 on the NAS benchmark. We use the EQ~(\ref{eq:tnew}) that predicts the execution
 
 In this section we evaluate the precision of our performance prediction methods
 on the NAS benchmark. We use the EQ~(\ref{eq:tnew}) that predicts the execution


@@ -283,7 +292,9 @@ real execution time with the predicted execution time. Each program runs offline
 with all available scaling factors on 8 or 9 nodes to produce real execution
 time values. These scaling factors are computed by dividing the maximum
 frequency by the new one see EQ~(\ref{eq:s}). In all tests, we use the simulator
 with all available scaling factors on 8 or 9 nodes to produce real execution
 time values. These scaling factors are computed by dividing the maximum
 frequency by the new one see EQ~(\ref{eq:s}). In all tests, we use the simulator

-Simgrid/SMPI v3.10 to run the NAS programs.
+SimGrid/SMPI v3.10 to run the NAS programs.
+\AG{Fig.~\ref{fig:pred} is hard to read when printed in black and white,
+  especially the ``Normalize Real Perf.'' curve.}

 \begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
   \centering
   \includegraphics[scale=0.60]{cg_per.eps}
 \begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
   \centering
   \includegraphics[scale=0.60]{cg_per.eps}


@@ -300,12 +311,11 @@ frequencies. For more details on the characteristics of the platform refer to
 table~(\ref{table:platform}). 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. The average normalized errors between the predicted execution time and
 table~(\ref{table:platform}). 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. 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. AS an
+the real time (SimGrid time) for all programs is between 0.0032 to 0.0133. AS an

 example, we are present the execution times of the NAS benchmarks as in the
 figure~(\ref{fig:pred}).
 example, we are present the execution times of the NAS benchmarks as in the
 figure~(\ref{fig:pred}).

-\sectionfont{\large}

 
 

-\section{.~Performance to Energy Competition}
+\section{Performance to Energy Competition}

 This section demonstrates our approach for choosing the optimal scaling
 factor. This factor gives maximum energy reduction taking into account the
 execution time for both computation and communication times . The relation
 This section demonstrates our approach for choosing the optimal scaling
 factor. This factor gives maximum energy reduction taking into account the
 execution time for both computation and communication times . The relation


@@ -316,12 +326,28 @@ is not straightforward. Moreover, they are not measured using the same metric.
 For solving this problem, we normalize the energy by calculating the ratio
 between the consumed energy with scaled frequency and the consumed energy
 without scaled frequency :
 For solving this problem, we normalize the energy by calculating the ratio
 between the consumed energy with scaled frequency and the consumed energy
 without scaled frequency :

-\begin{equation}  \label{eq:enorm}
-  E_{Norm}=\displaystyle\frac{E_{Reduced}}{E_{Orginal}}= \frac{\displaystyle \;P_{dyn}\,.\,S_i^{-2}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_i\;\,.\,N  }{\displaystyle \;P_{dyn}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,\,.\,N   }
+\begin{equation}
+  \label{eq:enorm}
+  E_\textit{Norm} = \frac{E_{Reduced}}{E_{Original}}
+          = \frac{ P_{dyn} \cdot S_i^{-2} \cdot
+               \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+               P_{static} \cdot T_1 \cdot S_i \cdot N  }{
+              P_{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+              P_{static} \cdot T_1 \cdot N }

 \end{equation}
 \end{equation}

+\AG{Use \texttt{\textbackslash{}text\{xxx\}} or
+  \texttt{\textbackslash{}textit\{xxx\}} for all subscripted words in equations
+  (e.g. \mbox{\texttt{E\_\{\textbackslash{}text\{Norm\}\}}}).
+
+  Don't hesitate to define new commands :
+  \mbox{\texttt{\textbackslash{}newcommand\{\textbackslash{}ENorm\}\{E\_\{\textbackslash{}text\{Norm\}\}\}}}
+}

 By the same way we can normalize the performance as follows :
 By the same way we can normalize the performance as follows :

-\begin{equation}  \label{eq:pnorm}
-  P_{Norm}=\displaystyle \frac{T_{New}}{T_{Old}}=\frac{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}}{T_{Old}}    \;\;
+\begin{equation}
+  \label{eq:pnorm}
+  P_{Norm} = \frac{T_{New}}{T_{Old}}
+          = \frac{T_{\textit{Max Comp Old}} \cdot S +
+              T_{\textit{Max Comm Old}}}{T_{Old}}

 \end{equation}
 The second problem is the optimization operation for both energy and performance
 is not in the same direction. In other words, the normalized energy and the
 \end{equation}
 The second problem is the optimization operation for both energy and performance
 is not in the same direction. In other words, the normalized energy and the


@@ -338,8 +364,11 @@ optimize both energy and performance simultaneously without adding big
 overheads.  Our solution for this problem is to make the optimization process
 have the same direction. Therefore, we inverse the equation of normalize
 performance as follows :
 overheads.  Our solution for this problem is to make the optimization process
 have the same direction. Therefore, we inverse the equation of normalize
 performance as follows :

-\begin{equation}  \label{eq:pnorm_en}
-  \displaystyle P^{-1}_{Norm}= \frac{T_{Old}}{T_{New}}=\frac{T_{Old}}{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}}
+\begin{equation}
+  \label{eq:pnorm_en}
+  P^{-1}_{Norm} = \frac{T_{Old}}{T_{New}}
+               = \frac{T_{Old}}{T_{\textit{Max Comp Old}} \cdot S +
+                 T_{\textit{Max Comm Old}}}

 \end{equation}
 \begin{figure}
   \centering
 \end{equation}
 \begin{figure}
   \centering


@@ -354,27 +383,27 @@ curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent
 the minimum energy consumption with minimum execution time (better performance)
 in 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)
 in the same time, see figure~(\ref{fig:r1}). Then our objective function has the
 following form:

-\begin{equation}  \label{eq:max}
-  \displaystyle MaxDist = Max \;(\;\overbrace{P^{-1}_{Norm}}^{Maximize}\; -\; \overbrace{E_{Norm}}^{Minimize} \;)
+\begin{equation}
+  \label{eq:max}
+  \textit{MaxDist} = \max (\overbrace{P^{-1}_{Norm}}^{\text{Maximize}} -
+                           \overbrace{E_{Norm}}^{\text{Minimize}} )

 \end{equation}
 Then we can select the optimal scaling factor that satisfy the
 EQ~(\ref{eq:max}).  Our objective function can works with any energy model or
 static power values stored in a data file. Moreover, this function works in
 optimal way when the energy function has a convex form with frequency scaling
 \end{equation}
 Then we can select the optimal scaling factor that satisfy the
 EQ~(\ref{eq:max}).  Our objective function can works with any energy model or
 static power values stored in a data file. Moreover, this function works in
 optimal way when the energy function has a convex form with frequency scaling

-factor as shown in ~\cite{15,3,19}. Energy measurement model is not the
+factor as shown in~\cite{15,3,19}. Energy measurement model is not the

 objective of this paper and we choose Rauber's model as an example with two
 reasons that mentioned before.
 objective of this paper and we choose Rauber's model as an example with two
 reasons that mentioned before.

-\sectionfont{\large}

 
 

-\section{.~Optimal Scaling Factor for Performance and Energy }
+\section{Optimal Scaling Factor for Performance and Energy}

 
 In the previous section we described the objective function that satisfy our
 goal in discovering optimal scaling factor for both performance and energy at
 the same time. Therefore, we develop an energy to performance scaling algorithm
 (EPSA). This algorithm is simple and has a direct way to calculate the optimal
 
 In the previous section we described the objective function that satisfy our
 goal in discovering optimal scaling factor for both performance and energy at
 the same time. Therefore, we develop an energy to performance scaling algorithm
 (EPSA). This algorithm is simple and has a direct way to calculate the optimal

-scaling factor for both energy and performance at the same time.  \clearpage
-\linespread{1}
-\begin{algorithm}[width=\textwidth,height=\textheight,keepaspectratio]
+scaling factor for both energy and performance at the same time.
+\begin{algorithm}[t]

   \caption{EPSA}
   \label{EPSA}
   \begin{algorithmic}[1]
   \caption{EPSA}
   \label{EPSA}
   \begin{algorithmic}[1]


@@ -389,73 +418,81 @@ scaling factor for both energy and performance at the same time.  \clearpage
       \State - Calculate all available scales $S_i$  depend on $S$ as in EQ~(\ref{eq:si}).
       \State - Select the maximum scale factor $S_1$ from the set of scales $S_i$.
       \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ as in EQ~(\ref{eq:enorm}).
       \State - Calculate all available scales $S_i$  depend on $S$ as in EQ~(\ref{eq:si}).
       \State - Select the maximum scale factor $S_1$ from the set of scales $S_i$.
       \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ as in EQ~(\ref{eq:enorm}).

-      \State - Calculate the normalize inverse of performance $P_{NormInv}=T_{old}/T_{new}$
-
-      as in EQ~(\ref{eq:pnorm_en}).
-      \If{  $(P_{NormInv}-E_{Norm}$ $>$ $Dist$) }
-        \State $S_{optimal}=S$
+      \State - Calculate the normalize inverse of performance\par
+               $P_{NormInv}=T_{old}/T_{new}$ as in EQ~(\ref{eq:pnorm_en}).
+      \If{  $(P_{NormInv}-E_{Norm} > Dist$) }
+        \State $S_{optimal} = S$

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

-    \State  $ Return \; \;  (S_{optimal})$
+    \State  Return $S_{optimal}$

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

-\linespread{1.2} The proposed EPSA algorithm works online during the execution
-time of the MPI program. It selects the optimal scaling factor by gathering some
-information from the program after one iteration. This algorithm has small
-execution time (between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ for 32
-nodes). The data required by this algorithm is the computation time and the
-communication time for each task from the first iteration only. When these times
-are measured, the MPI program calls the EPSA algorithm to choose the new
-frequency using the optimal scaling factor. Then the program set the new
-frequency to the system. The algorithm is called just one time during the
-execution of the program. The following example shows where and when the EPSA
-algorithm is called in the MPI program : \clearpage
-\begin{lstlisting}
-FOR J:=1 to Some_iterations Do
-   -Computations Section.
-   -Communications Section.
-   IF (J==1) THEN
-     -Gather all times of computation and communication
-      from each node.
-     -Call EPSA with these times.
-     -Calculate the new frequency from optimal scale.
-     -Set the new frequency to the system.
-   ENDIF
-ENDFOR
-\end{lstlisting}
+The proposed EPSA algorithm works online during the execution time of the MPI
+program. It selects the optimal scaling factor by gathering some information
+from the program after one iteration. This algorithm has small execution time
+(between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ for 32 nodes). The data
+required by this algorithm is the computation time and the communication time
+for each task from the first iteration only. When these times are measured, the
+MPI program calls the EPSA algorithm to choose the new frequency using the
+optimal scaling factor. Then the program set the new frequency to the
+system. The algorithm is called just one time during the execution of the
+program. The following example shows where and when the EPSA algorithm is called
+in the MPI program :
+%\begin{minipage}{\textwidth}
+%\AG{Use the same format as for Algorithm~\ref{EPSA}}
+
+\begin{algorithm}[d]
+  \caption{DVFS}
+  \label{dvfs}
+  \begin{algorithmic}
+ \For {$J:=1$ to $Some_iterations Do$}
+  \State -Computations Section.
+   \State -Communications Section.
+   \If {$(J==1)$} 
+     \State -Gather all times of computation and\par
+        \State      communication from each node.
+     \State -Call EPSA with these times.
+     \State -Calculate the new frequency from optimal scale.
+     \State -Set the new frequency to the system.
+   \EndIf
+\EndFor
+\end{algorithmic}
+\end{algorithm}
+\clearpage

 After obtaining the optimal scale factor from the EPSA algorithm. The program
 calculates the new frequency $F_i$ for each task proportionally to its time
 value $T_i$. By substitution of the EQ~(\ref{eq:s}) in the EQ~(\ref{eq:si}), we
 can calculate the new frequency $F_i$ as follows :
 After obtaining the optimal scale factor from the EPSA algorithm. The program
 calculates the new frequency $F_i$ for each task proportionally to its time
 value $T_i$. By substitution of the EQ~(\ref{eq:s}) in the EQ~(\ref{eq:si}), we
 can calculate the new frequency $F_i$ as follows :

-\begin{equation}  \label{eq:fi}
-  F_i=\frac{F_{max} \; . \;T_i}{S_{optimal} \; . \;T_{max}} \hfill
+\begin{equation}
+  \label{eq:fi}
+  F_i = \frac{F_{max} \cdot T_i}{S_{optimal} \cdot T_{max}}

 \end{equation}
 According to this equation all the nodes may have the same frequency value if
 they have balanced workloads. Otherwise, they take different frequencies when
 have imbalanced workloads. Then EQ~(\ref{eq:fi}) works in adaptive way to change
 \end{equation}
 According to this equation all the nodes may have the same frequency value if
 they have balanced workloads. Otherwise, they take different frequencies when
 have imbalanced workloads. Then EQ~(\ref{eq:fi}) works in adaptive way to change

-the freguency according to the nodes workloads.
-\sectionfont{\large}
+the frequency according to the nodes workloads.

 
 

-\section{.~Experimental Results}
+\section{Experimental Results}

 
 

-The proposed ESPA algorithm was applied to seven MPI programs of the NAS
-benchmarks (EP ,CG , MG ,FT , BT, LU and SP). We work on three classes (A, B and
+The proposed EPSA algorithm was applied to seven MPI programs of the NAS
+benchmarks (EP, CG, MG, FT, BT, LU and SP). We work on three classes (A, B and

 C) for each program. Each program runs on specific number of processors
 proportional to the size of the class.  Each class represents the problem size
 ascending from the class A to C. Additionally, depending on some speed up points
 for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes
 C) for each program. Each program runs on specific number of processors
 proportional to the size of the class.  Each class represents the problem size
 ascending from the class A to C. Additionally, depending on some speed up points
 for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes

-respectively. Our experiments are executed on the simulator Simgrid/SMPI
+respectively. Our experiments are executed on the simulator SimGrid/SMPI

 v3.10. We design a platform file that simulates a cluster with one core per
 node. This cluster is a homogeneous architecture with distributed memory. The
 v3.10. We design a platform file that simulates a cluster with one core per
 node. This cluster is a homogeneous architecture with distributed memory. The

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

 \begin{table}[ht]
   \caption{Platform File Parameters}
   % title of Table
   \centering
 \begin{table}[ht]
   \caption{Platform File Parameters}
   % title of Table
   \centering

+  \AG{Use e.g. $5\times 10^{-7}$ instead of 5E-7}

   \begin{tabular}{ | l | l | l |l | l |l |l |  p{2cm} |}
     \hline
     Max & Min & Backbone & Backbone&Link &Link& Sharing  \\
   \begin{tabular}{ | l | l | l |l | l |l |l |  p{2cm} |}
     \hline
     Max & Min & Backbone & Backbone&Link &Link& Sharing  \\


@@ -467,7 +504,7 @@ successive frequencies.
 \end{table}
 Depending on the EQ~(\ref{eq:energy}), we measure the energy consumption for all
 the NAS MPI programs while assuming the power dynamic is equal to 20W and the
 \end{table}
 Depending on the EQ~(\ref{eq:energy}), we measure the energy consumption for all
 the NAS MPI programs while assuming the power dynamic is equal to 20W and the

-power static is equal to 4W for all experiments. We run the proposed ESPA
+power static is equal to 4W for all experiments. We run the proposed EPSA

 algorithm for all these programs. The results showed that the algorithm selected
 different scaling factors for each program depending on the communication
 features of the program as in the figure~(\ref{fig:nas}). This figure shows that
 algorithm for all these programs. The results showed that the algorithm selected
 different scaling factors for each program depending on the communication
 features of the program as in the figure~(\ref{fig:nas}). This figure shows that


@@ -495,11 +532,13 @@ same time over all available scales.
   \caption{Optimal scaling factors for The NAS MPI Programs}
   \label{fig:nas}
 \end{figure}
   \caption{Optimal scaling factors for The NAS MPI Programs}
   \label{fig:nas}
 \end{figure}

-\linespread{1.1}

 \begin{table}[width=\textwidth,height=\textheight,keepaspectratio]
   \caption{Optimal Scaling Factors Results}
   % title of Table
   \centering
 \begin{table}[width=\textwidth,height=\textheight,keepaspectratio]
   \caption{Optimal Scaling Factors Results}
   % title of Table
   \centering

+  \AG{Use the same number of decimals for all numbers in a column,
+    and vertically align the numbers along the decimal points.
+    The same for all the following tables.}

   \begin{tabular}{ | l | l | l |l | l | p{2cm} |}
     \hline
     Program & Optimal & Energy  & Performance&Energy-Perf.\\
   \begin{tabular}{ | l | l | l |l | l | p{2cm} |}
     \hline
     Program & Optimal & Energy  & Performance&Energy-Perf.\\


@@ -507,15 +546,14 @@ same time over all available scales.
     CG & 1.56 &39.23 & 14.88 & 24.35\\ \hline
     MG & 1.47 &34.97&21.7& 13.27   \\ \hline
     EP & 1.04 &22.14&20.73 &1.41\\ \hline
     CG & 1.56 &39.23 & 14.88 & 24.35\\ \hline
     MG & 1.47 &34.97&21.7& 13.27   \\ \hline
     EP & 1.04 &22.14&20.73 &1.41\\ \hline

-    LU & 1.388 &35.83&22.49 &13.34\\ \hline
-    BT & 1.315 &29.6&21.28 &8.32\\ \hline
-    SP & 1.388 &33.48 &21.36&12.12\\ \hline
-    FT & 1.47 &34.72 &19&15.72\\ \hline
+    LU & 1.38 &35.83&22.49 &13.34\\ \hline
+    BT & 1.31 &29.60&21.28 &8.32\\ \hline
+    SP & 1.38 &33.48 &21.36&12.12\\ \hline
+    FT & 1.47 &34.72 &19.00&15.72\\ \hline

   \end{tabular}
   \label{table:factors results}
   % is used to refer this table in the text
 \end{table}
   \end{tabular}
   \label{table:factors results}
   % is used to refer this table in the text
 \end{table}

-\linespread{1.2}

 
 As shown in the table~(\ref{table:factors results}), when the optimal scaling
 factor has big value we can gain more energy savings for example as in CG and
 
 As shown in the table~(\ref{table:factors results}), when the optimal scaling
 factor has big value we can gain more energy savings for example as in CG and


@@ -525,10 +563,7 @@ 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).
 
 cases. In EP there are no communications inside the iterations. This make our
 EPSA to selects smaller scaling factor values (inducing smaller energy savings).
 

-% \clearpage
-\sectionfont{\large}
-
-\section{.~Comparing Results}
+\section{Comparing Results}

 
 In this section, we compare our EPSA algorithm results with Rauber's
 methods~\cite{3}. He had two scenarios, the first is to reduce energy to optimal
 
 In this section, we compare our EPSA algorithm results with Rauber's
 methods~\cite{3}. He had two scenarios, the first is to reduce energy to optimal


@@ -540,44 +575,43 @@ scenario as $Rauber_{E-P}$. The comparison is made in tables~(\ref{table:compare
   Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
 tables show the results of our EPSA and Rauber's two scenarios for all the NAS
 benchmarks programs for classes A,B and C.
   Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
 tables show the results of our EPSA and Rauber's two scenarios for all the NAS
 benchmarks programs for classes A,B and C.

-%\linespread{1}

 \begin{table}[ht]
   \caption{Comparing Results for  The NAS Class A}
   % title of Table
   \centering
 \begin{table}[ht]
   \caption{Comparing Results for  The NAS Class A}
   % title of Table
   \centering

-  \begin{tabular}{ | l | l | l |l | l |l|  }
+  \begin{tabular}{ | l | l | l |l | l | l|  }

     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\
     name &name&value& Saving \%&Degradation \% &Distance
     \\ \hline
     % \rowcolor[gray]{0.85}
     EPSA&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\
     name &name&value& Saving \%&Degradation \% &Distance
     \\ \hline
     % \rowcolor[gray]{0.85}
     EPSA&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline

-    $Rauber_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.5\\ \hline
+    $Rauber_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline

     $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
 
     EPSA&MG & 1.47 &27.66&16.82&10.84\\ \hline
     $Rauber_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
     $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
 
     EPSA&MG & 1.47 &27.66&16.82&10.84\\ \hline
     $Rauber_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline

-    $Rauber_{E}$&MG &2.14&34.48&33.65&0.8 \\ \hline
+    $Rauber_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline

 
     EPSA&EP &1.19 &25.32&20.79&4.53\\ \hline
     $Rauber_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
 
     EPSA&EP &1.19 &25.32&20.79&4.53\\ \hline
     $Rauber_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline

-    $Rauber_{E}$&EP&2.05&42.09&57.59&-15.5\\ \hline
+    $Rauber_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline

 
     EPSA&LU&1.56& 39.55 &19.38& 20.17\\ \hline
 
     EPSA&LU&1.56& 39.55 &19.38& 20.17\\ \hline

-    $Rauber_{E-P}$&LU&2.14&45.62&27&18.62 \\ \hline
+    $Rauber_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline

     $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
 
     $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
 

-    EPSA&BT&1.315& 29.6&20.53&9.07 \\ \hline
-    $Rauber_{E-P}$&BT&2.1&45.53&49.63&-4.1\\ \hline
-    $Rauber_{E}$&BT&2.1&43.93&52.86&-8.93\\ \hline
+    EPSA&BT&1.31& 29.60&20.53&9.07 \\ \hline
+    $Rauber_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline
+    $Rauber_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline

 
 

-    EPSA&SP&1.388& 33.51&15.65&17.86 \\ \hline
-    $Rauber_{E-P}$&SP&2.11&45.62&42.52&3.1\\ \hline
+    EPSA&SP&1.38& 33.51&15.65&17.86 \\ \hline
+    $Rauber_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline

     $Rauber_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
 
     $Rauber_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
 

-    EPSA&FT&1.25& 25&10.8&14.2 \\ \hline
-    $Rauber_{E-P}$&FT&2.1&39.29&34.3&4.99 \\ \hline
-    $Rauber_{E}$&FT&2.1&37.56&38.21&-0.65\\ \hline
+    EPSA&FT&1.25&25.00&10.80&14.20 \\ \hline
+    $Rauber_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
+    $Rauber_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline

   \end{tabular}
   \label{table:compare Class A}
   % is used to refer this table in the text
   \end{tabular}
   \label{table:compare Class A}
   % is used to refer this table in the text


@@ -592,8 +626,8 @@ benchmarks programs for classes A,B and C.
     name &name&value& Saving \%&Degradation \% &Distance
     \\ \hline
     % \rowcolor[gray]{0.85}
     name &name&value& Saving \%&Degradation \% &Distance
     \\ \hline
     % \rowcolor[gray]{0.85}

-    EPSA&CG & 1.66 &39.23&16.63&22.6   \\ \hline
-    $Rauber_{E-P}$&CG &2.15 &45.34&27.6&17.74\\ \hline
+    EPSA&CG & 1.66 &39.23&16.63&22.60   \\ \hline
+    $Rauber_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline

     $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
 
     EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline
     $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
 
     EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline


@@ -601,24 +635,24 @@ benchmarks programs for classes A,B and C.
     $Rauber_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
 
     EPSA&EP &1.08 &20.29&17.15&3.14 \\ \hline
     $Rauber_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
 
     EPSA&EP &1.08 &20.29&17.15&3.14 \\ \hline

-    $Rauber_{E-P}$&EP&2&42.38&56.88&-14.5\\ \hline
-    $Rauber_{E}$&EP&2&39.73&59.94&-20.21\\ \hline
+    $Rauber_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline
+    $Rauber_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline

 
     EPSA&LU&1.47&38.57&21.34&17.23 \\ \hline
 
     EPSA&LU&1.47&38.57&21.34&17.23 \\ \hline

-    $Rauber_{E-P}$&LU&2.1&43.62&36.51&7.11 \\ \hline
-    $Rauber_{E}$&LU&2.1&43.61&38.54&5.07 \\ \hline
+    $Rauber_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline
+    $Rauber_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline

 
 

-    EPSA&BT&1.315& 29.59&20.88&8.71\\ \hline
-    $Rauber_{E-P}$&BT&2.1&44.53&53.05&-8.52\\ \hline
-    $Rauber_{E}$&BT&2.1&42.93&52.806&-9.876\\ \hline
+    EPSA&BT&1.31& 29.59&20.88&8.71\\ \hline
+    $Rauber_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline
+    $Rauber_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline

 
 

-    EPSA&SP&1.388&33.44&19.24&14.2 \\ \hline
-    $Rauber_{E-P}$&SP&2.15&45.69&43.2&2.49\\ \hline
+    EPSA&SP&1.38&33.44&19.24&14.20 \\ \hline
+    $Rauber_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline

     $Rauber_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
 
     $Rauber_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
 

-    EPSA&FT&1.388&34.4&14.57&19.83 \\ \hline
+    EPSA&FT&1.38&34.40&14.57&19.83 \\ \hline

     $Rauber_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
     $Rauber_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline

-    $Rauber_{E}$&FT&2.13&43.04&37.9&5.14\\ \hline
+    $Rauber_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline

   \end{tabular}
   \label{table:compare Class B}
   % is used to refer this table in the text
   \end{tabular}
   \label{table:compare Class B}
   % is used to refer this table in the text


@@ -636,46 +670,45 @@ benchmarks programs for classes A,B and C.
     % \rowcolor[gray]{0.85}
     EPSA&CG & 1.56 &39.23&14.88&24.35  \\ \hline
     $Rauber_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline
     % \rowcolor[gray]{0.85}
     EPSA&CG & 1.56 &39.23&14.88&24.35  \\ \hline
     $Rauber_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline

-    $Rauber_{E}$&CG &2.15 &45.36&26.7&18.66\\ \hline
+    $Rauber_{E}$&CG &2.15 &45.36&26.70&18.66\\ \hline

 
 

-    EPSA&MG & 1.47 &34.97&21.697&13.273\\ \hline
-    $Rauber_{E-P}$&MG &2.15&43.65&40.45&3.2 \\ \hline
+    EPSA&MG & 1.47 &34.97&21.69&13.27\\ \hline
+    $Rauber_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline

     $Rauber_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
 
     EPSA&EP &1.04 &22.14&20.73&1.41 \\ \hline
     $Rauber_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
 
     EPSA&EP &1.04 &22.14&20.73&1.41 \\ \hline

-    $Rauber_{E-P}$&EP&1.92&39.4&56.33&-16.93\\ \hline
-    $Rauber_{E}$&EP&1.92&38.1&56.35&-18.25\\ \hline
+    $Rauber_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline
+    $Rauber_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline

 
 

-    EPSA&LU&1.388&35.83&22.49&13.34 \\ \hline
-    $Rauber_{E-P}$&LU&2.15&44.97&41&3.97 \\ \hline
-    $Rauber_{E}$&LU&2.15&44.97&41.8&3.17 \\ \hline
+    EPSA&LU&1.38&35.83&22.49&13.34 \\ \hline
+    $Rauber_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline
+    $Rauber_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline

 
 

-    EPSA&BT&1.315& 29.6&21.28&8.32\\ \hline
-    $Rauber_{E-P}$&BT&2.13&45.6&49.84&-4.24\\ \hline
-    $Rauber_{E}$&BT&2.13&44.9&55.16&-10.26\\ \hline
+    EPSA&BT&1.31& 29.60&21.28&8.32\\ \hline
+    $Rauber_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline
+    $Rauber_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline

 
 

-    EPSA&SP&1.388&33.48&21.35&12.12\\ \hline
-    $Rauber_{E-P}$&SP&2.1&45.69&43.6&2.09\\ \hline
-    $Rauber_{E}$&SP&2.1&45.75&44.1&1.65\\ \hline
+    EPSA&SP&1.38&33.48&21.35&12.12\\ \hline
+    $Rauber_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline
+    $Rauber_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline

 
 

-    EPSA&FT&1.47&34.72&19&15.72 \\ \hline
-    $Rauber_{E-P}$&FT&2.04&39.4&37.1&2.3\\ \hline
-    $Rauber_{E}$&FT&2.04&39.35&37.7&1.65\\ \hline
+    EPSA&FT&1.47&34.72&19.00&15.72 \\ \hline
+    $Rauber_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
+    $Rauber_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline

   \end{tabular}
 \label{table:compare Class C}
 % is used to refer this table in the text
 \end{table}
   \end{tabular}
 \label{table:compare Class C}
 % is used to refer this table in the text
 \end{table}

-%\linespread{1.2}
-\clearpage As shown in these tables our scaling factor is not optimal for energy
-saving such as Rauber's scaling factor EQ~(\ref{eq:sopt}), but it is optimal for
-both the energy and the performance simultaneously. Our EPSA optimal scaling
-factors has better simultaneous optimization for both the energy and the
-performance compared to Rauber's energy-performance method
-($Rauber_{E-P}$). Also, in ($Rauber_{E-P}$) method when setting the frequency to
-maximum value for the slower task lead to a small improvement of the
-performance. Also the results show that this method keep or improve energy
-saving. Because of the energy consumption decrease when the execution time
-decreased while the frequency value increased.
+As shown in these tables our scaling factor is not optimal for energy saving
+such as Rauber's scaling factor EQ~(\ref{eq:sopt}), but it is optimal for both
+the energy and the performance simultaneously. Our EPSA optimal scaling factors
+has better simultaneous optimization for both the energy and the performance
+compared to Rauber's energy-performance method ($Rauber_{E-P}$). Also, in
+($Rauber_{E-P}$) method when setting the frequency to maximum value for the
+slower task lead to a small improvement of the performance. Also the results
+show that this method keep or improve energy saving. Because of the energy
+consumption decrease when the execution time decreased while the frequency value
+increased.

 
 Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving
 percent and the performance degradation percent. Therefore, this means it is the
 
 Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving
 percent and the performance degradation percent. Therefore, this means it is the


@@ -697,6 +730,7 @@ than the first.
   \label{fig:compare}
 \end{figure}
 
   \label{fig:compare}
 \end{figure}
 

+\AG{\texttt{bibtex} gives many errors, please correct them}

 \clearpage
 \bibliographystyle{plain}
 \bibliography{my_reference}
 \clearpage
 \bibliographystyle{plain}
 \bibliography{my_reference}


@@ -706,5 +740,8 @@ than the first.
 %%% mode: latex
 %%% TeX-master: t
 %%% fill-column: 80
 %%% mode: latex
 %%% TeX-master: t
 %%% fill-column: 80

-%%% ispell-local-dictionary: "american"
+%%%ispell-local-dictionary: "american"

 %%% End:
 %%% End:

+
+%  LocalWords:  Badri Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
+%  LocalWords:  CMOS EQ EPSA