objective function changes

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
+++ b/paper.tex
@@ -1,5 +1,4 @@
-\documentclass[12pt]{article}
-%\documentclass[12pt,twocolumn]{article}
+\documentclass[conference]{IEEEtran}

 
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}


@@ -9,35 +8,66 @@
 \usepackage{subfig}
 \usepackage{listings}
 \usepackage{colortbl}
 \usepackage{subfig}
 \usepackage{listings}
 \usepackage{colortbl}

-% \usepackage{sectsty}
-% \usepackage{titlesec}
-% \usepackage{secdot}
-%\usepackage[font={footnotesize,bt}]{caption}
-%\usepackage[font=scriptsize,labelfont=bf]{caption}
-\usepackage{lmodern}
+\usepackage{amsmath}

 
 

-\usepackage{todonotes}
-\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\small\textbf{AG:} #2}}
+\usepackage[autolanguage,np]{numprint}
+\renewcommand*\npunitcommand[1]{\text{#1}}
+
+\usepackage{xspace}
+\usepackage[textsize=footnotesize]{todonotes}
+\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}

 
 \begin{document}
 
 \title{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
 
 \begin{document}
 
 \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}
+
+\author{%
+  \IEEEauthorblockN{%
+    Jean-Claude Charr,
+    Raphaël Couturier,
+    Ahmed Fanfakh and
+    Arnaud Giersch
+  }
+  \IEEEauthorblockA{%
+    FEMTO-ST Institute\\
+    University of Franche-Comté
+  }
+}
+

 \maketitle
 
 \maketitle
 

+\AG{``Optimal'' is a bit pretentious in the title.\\
+  Complete affiliation, add an email address, etc.}
+
+\begin{abstract}
+  The important technique for energy reduction of parallel systems is CPU
+  frequency scaling. This operation is used by many researchers to reduce energy
+  consumption in many ways. Frequency scaling operation also has a big impact on
+  the performances. In some cases, the performance degradation ratio is bigger
+  than energy saving ratio when the frequency is scaled to low level. Therefore,
+  the trade offs between the energy and performance becomes more important topic
+  when using this technique. In this paper we developed an algorithm that select
+  the frequency scaling factor for both energy and performance simultaneously.
+  This algorithm takes into account the communication times when selecting the
+  frequency scaling factor. It works online without training or profiling to
+  have a very small overhead.  The algorithm has better energy-performance trade
+  offs compared to other methods.
+\end{abstract}
+

 \section{Introduction}
 \section{Introduction}

+\label{sec.intro}

 
 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
 constructors have been regularly increasing the number of computing cores in
 
 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
 constructors have been regularly increasing the number of computing cores in

-supercomputers (for example in November 2013, according to the top 500
+supercomputers (for example in November 2013, according to the TOP500

 list~\cite{43}, the Tianhe-2 was the fastest supercomputer. It has more than 3
 millions of cores and delivers more than 33 Tflop/s while consuming 17808
 kW). This large increase in number of computing cores has led to large energy
 consumption by these architectures. Moreover, the price of energy is expected to
 continue its ascent according to the demand. For all these reasons energy
 reduction became an important topic in the high performance computing field. To
 list~\cite{43}, the Tianhe-2 was the fastest supercomputer. It has more than 3
 millions of cores and delivers more than 33 Tflop/s while consuming 17808
 kW). This large increase in number of computing cores has led to large energy
 consumption by these architectures. Moreover, the price of energy is expected to
 continue its ascent according to the demand. For all these reasons energy
 reduction became an important topic in the high performance computing field. To

-tackle this problem, many researchers used DVFS (Dynamic Voltage Frequency
+tackle this problem, many researchers used DVFS (Dynamic Voltage and Frequency

 Scaling) operations which reduce dynamically the frequency and voltage of cores
 and thus their energy consumption. However, this operation also degrades the
 performance of computation. Therefore researchers try to reduce the frequency to
 Scaling) operations which reduce dynamically the frequency and voltage of cores
 and thus their energy consumption. However, this operation also degrades the
 performance of computation. Therefore researchers try to reduce the frequency to


@@ -52,317 +82,335 @@ of the performance. It is important to notice that lowering the frequency to
 minimum value does not always give the most efficient execution due to energy
 leakage. The best scaling factor might be chosen during execution (online) or
 during a pre-execution phase.  In this paper we emphasize to develop an
 minimum value does not always give the most efficient execution due to energy
 leakage. The best scaling factor might be chosen during execution (online) or
 during a pre-execution phase.  In this paper we emphasize to develop an

-algorithm that selects the optimal frequency scaling factor that takes into
-consideration simultaneously the energy consumption and the performance. The
+algorithm that selects a frequency scaling factor that simultaneously takes into
+consideration  the energy consumption and the performance. The

 main objective of HPC systems is to run the application with less execution
 main objective of HPC systems is to run the application with less execution

-time. Therefore, our algorithm selects the optimal scaling factor online with
+time. Therefore, our algorithm selects the scaling factor online with

 very small footprint. The proposed algorithm takes into account the
 very small footprint. The proposed algorithm takes into account the

-communication times of the MPI programs to choose the scaling factor. This
+communication times of the MPI program to choose the scaling factor. This

 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
 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

-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.
-
+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}.
+The comparison's results show that our
+algorithm gives better energy-time trade off.
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents the works
+from other authors.  Section~\ref{sec.exe} shows the execution of parallel
+tasks and sources of idle times. Also, it resumes the energy
+model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance
+of MPI program.  Section~\ref{sec.compet} presents the energy-performance trade offs
+objective function. Section~\ref{sec.optim} demonstrates the proposed
+energy-performance algorithm. Section~\ref{sec.expe} verifies the performance prediction
+model and presents the results of the proposed algorithm.  Also, It shows the comparison results. Finally,
+we conclude in Section~\ref{sec.concl}.

 \section{Related Works}
 \section{Related Works}

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

 
 \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
 
 \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

-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
-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.
+during the compilation phases as for example in Azevedo et al.~\cite{40}. They
+use dynamic voltage scaling (DVS) algorithm to choose the DVS 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 to save energy with
+time limits. Another approach gathers and stores the runtime information for
+each DVFS state, then selects the suitable DVFS offline to optimize energy-time
+trade offs. As an example, Rountree et al.~\cite{8} use liner programming
+algorithm, while in~\cite{38,34}, Cochran et al. use multi logistic regression
+algorithm for the same goal. The offline study that shows the DVFS impact on the
+communication time of the MPI program is~\cite{17}, where Freeh et al. show that
+these times do not change when the frequency is scaled down.

 
 \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
-predicting approaches for the energy-time trade offs developed by
-~\cite{11,2,31}. These works select the best DVFS setting depending on the slack
-times. These times happen when the processors have to wait for data from other
-processors to compute their task. For example, during the synchronous
-communication time that take place in the MPI programs, the processors are
-idle. The optimal DVFS can be selected using the learning methods. Therefore, in
-~\cite{39,19} used machine learning to converge to the suitable DVFS
-configuration. Their learning algorithms have big time to converge when the
-number of available frequencies is high. Also, the communication time of the MPI
-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
-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
-for maintaining performance.  In this paper we compare our algorithm with
-Rauber's model~\cite{3}, because his model can be used for any number of
-concurrent tasks for homogeneous platform and this is the same direction of this
-paper.  However, the primary contributions of this paper are:
+The objective of the online DVFS orientations is to dynamically compute and set
+the frequency of the CPU for saving energy during the runtime of the
+programs. Estimating and predicting approaches for the energy-time trade offs
+are developed by Kimura, Peraza, Yu-Liang et al.  ~\cite{11,2,31}. These works
+select the best DVFS setting depending on the slack times. These times happen
+when the processors have to wait for data from other processors to compute their
+task. For example, during the synchronous communications that take place in MPI
+programs, some processors are idle. The optimal DVFS can be selected using
+learning methods. Therefore, in Dhiman, Hao Shen et al.  ~\cite{39,19} used
+machine learning to converge to the suitable DVFS configuration. Their learning
+algorithms take big time to converge when the number of available frequencies is
+high. Also, the communication sections of the MPI program can be used to save
+energy. 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 of analytical modeling
+can also be used for choosing the optimal frequency as in Rauber and
+Rünger~\cite{3}. They developed an analytical mathematical model to determine
+the optimal frequency scaling factor for any number of concurrent tasks. They
+set the slowest task to maximum frequency for maintaining performance.  In this
+paper we compare our algorithm with Rauber and Rünger model~\cite{3}, because
+their model can be used for any number of concurrent tasks for homogeneous
+platforms. The primary contributions of this paper are:

 \begin{enumerate}
 \begin{enumerate}

-\item Selecting the optimal frequency scaling factor for energy and performance
-  simultaneously. While taking into account the communication time.
-\item Adapting our scale factor to taking into account the imbalanced tasks.
+\item Selecting the frequency scaling factor for simultaneously optimizing energy and performance,
+   while taking into account the communication time.
+\item Adapting our scaling factor to take into account the imbalanced tasks.

 \item The execution time of our algorithm is very small when compared to other
   methods (e.g.,~\cite{19}).
 \item The proposed algorithm works online without profiling or training as
   in~\cite{38,34}.
 \end{enumerate}
 \item The execution time of our algorithm is very small when compared to other
   methods (e.g.,~\cite{19}).
 \item The proposed algorithm works online without profiling or training as
   in~\cite{38,34}.
 \end{enumerate}

+\section{Execution and Energy of Parallel Tasks on Homogeneous Platform} 
+\label{sec.exe}

 
 

-\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
+\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
 executed on this model can be either synchronous or asynchronous. In this paper
 we consider execution of the synchronous tasks on distributed homogeneous
 be a multi-core. The nodes are connected via a high bandwidth network. Tasks
 executed on this model can be either synchronous or asynchronous. In this paper
 we consider execution of the synchronous tasks on distributed homogeneous

-platform. These tasks can exchange the data via synchronous memory passing.
-\begin{figure}[h]
+platform. These tasks can exchange the data via synchronous message passing.
+\begin{figure*}[t]

   \centering
   \centering

-  \subfloat[Synch. Imbalanced Communications]{\includegraphics[scale=0.67]{synch_tasks}\label{fig:h1}}
-  \subfloat[Synch. Imbalanced Computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
+  \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}
   \caption{Parallel Tasks on Homogeneous Platform}
   \label{fig:homo}

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

 Therefore, the execution time of a task consists of the computation time and the
 communication time. Moreover, the synchronous communications between tasks can
 Therefore, the execution time of a task consists of the computation time and the
 communication time. Moreover, the synchronous communications between tasks can

-lead to idle time while tasks wait at the synchronous point for others tasks to
-finish their communications see figure~(\ref{fig:h1}).  Another source for idle
-times is the imbalanced computations. This happen when processing different
-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 :
-\begin{equation}  \label{eq:T1}
-  Program Time=MAX_{i=1,2,..,N} (T_i) \hfill
+lead to idle time while tasks wait at the synchronization barrier for other tasks to
+finish their communications (see figure~(\ref{fig:h1})). The imbalanced communications 
+happen when nodes have to send/receive different amount of data or each node is communicates 
+with different number of nodes. Another source for idle times is the imbalanced computations.  
+This happens when processing different amounts of data on each processor  (see figure~(\ref{fig:h2})).  
+In this case the fastest tasks have to wait at the synchronization barrier for the 
+slowest tasks to finish their job. In both cases the overall execution time 
+of the program is the execution time of the slowest task as: 
+\begin{equation}
+  \label{eq:T1}
+  \textit{Program Time} = \max_{i=1,2,\dots,N} T_i

 \end{equation}
 \end{equation}

-where $T_i$ is the execution time of process $i$.
+where $T_i$ is the execution time of task $i$.

 
 

-\section{Energy Model for Homogeneous Platform}
+\subsection{Energy Model for Homogeneous Platform}

 
 

-The energy consumption by the processor consists of two powers metric: the
+The energy consumption by the processor consists of two power metrics: the

 dynamic and the static power. This general power formulation is used by many
 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~\cite{9,3,15,26}. The dynamic power of the CMOS processors

 $P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
 $P_{dyn}$ 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
+the supply voltage $V$ and operational frequency $f$ respectively as follow:
+\begin{equation}
+  \label{eq:pd}
+  P_\textit{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_\textit{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
 \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
+of the dynamic and the static power multiplied by the execution time~\cite{36,15}.
+\begin{equation}
+  \label{eq:eind}
+   E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T

 \end{equation}
 \end{equation}

-The dynamic voltage and frequency scaling (DVFS) is a process that allowed in
+The dynamic voltage and frequency scaling (DVFS) is a process that is allowed in

 modern processors to reduce the dynamic power by scaling down the voltage and
 frequency. Its main objective is to reduce the overall energy
 consumption~\cite{37}. The operational frequency \emph f depends linearly on the
 modern processors to reduce the dynamic power by scaling down the voltage and
 frequency. Its main objective is to reduce the overall energy
 consumption~\cite{37}. The operational frequency \emph f depends linearly on the

-supply voltage $V$, i.e., $V = \beta . f$ with some constant $\beta$. This
+supply voltage $V$, i.e., $V = \beta \cdot f$ with some constant $\beta$. This

 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
 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
+expressed by the scaling factor \emph S. This scaling factor is the ratio between the

 maximum and the new frequency as in EQ~(\ref{eq:s}).
 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_\textit{max}}{F_\textit{new}}

 \end{equation}
 \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
-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
-static energy linearly because the execution time is increased~\cite{36}.  The
-energy model, depending on the frequency scaling factor, of homogeneous platform
-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}).
-
-\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
+The value of the scale $S$ is greater than 1 when changing the frequency to any
+new frequency value~(\emph {P-state}) in governor, the CPU governor is an
+interface driver supplied by the operating system kernel (e.g. Linux) to
+lowering core's frequency.  The scaling factor is equal to 1 when the new frequency is 
+set to the maximum frequency.  The energy consumption model for parallel
+homogeneous platform depends on the scaling factor \emph S. This factor reduces
+quadratically the dynamic power.  Also, this factor increases the static energy
+linearly because the execution time is increased~\cite{36}.  The energy model
+depending on the frequency scaling factor for homogeneous platform for any
+number of concurrent tasks was developed by Rauber and Rünger~\cite{3}. This
+model considers the two power metrics for measuring the energy of the parallel
+tasks as in EQ~(\ref{eq:energy}):
+\begin{equation}
+  \label{eq:energy}
+  E = P_\textit{dyn} \cdot S_1^{-2} \cdot
+    \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
+    P_\textit{static} \cdot T_1 \cdot S_1 \cdot N

  \hfill
 \end{equation}
  \hfill
 \end{equation}

-Where \emph N is the number of parallel nodes, $T_1 $ is the time of the slower
+where \emph N is the number of parallel nodes, $T_1 $ is the time of the slowest

 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}).
 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}).

-\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,N} 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_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}

 \end{equation}
 \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
-for homogeneous platform that we work on in this paper. 2-we are compare our
-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}).
-\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
+where $N$ is the number of nodes. In this paper we depend on
+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
+compare our algorithm with Rauber and Rünger scaling model.  Rauber and Rünger
+scaling factor that reduce energy consumption derived from the
+EQ~(\ref{eq:energy}). They take 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_\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}
 \end{equation}

-%[\Big 3]

 
 \section{Performance Evaluation of MPI Programs}
 
 \section{Performance Evaluation of MPI Programs}

+\label{sec.mpip}

 
 

-The performance (execution time) of the parallel MPI applications are depends on
+The performance (execution time) of  parallel MPI applications depend on

 the time of the slowest task as in figure~(\ref{fig:homo}). Normally the
 execution time of the parallel programs are proportional to the operational
 the time of the slowest task as in figure~(\ref{fig:homo}). Normally the
 execution time of the parallel programs are proportional to the operational

-frequency. Therefore, any DVFS operation for the energy reduction increase the
+frequency. Therefore, any DVFS operation for the energy reduction increases the

 execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) 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
+energy is 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
 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
-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
+to EQ~(\ref{eq:s}), the value of the scale $S$ has inverse relation with
+new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decreasing the
+frequency value, the execution time increases. Then the new frequency value has
+inverse relation with time ($F_{new} \propto \frac{1}{T}$). This leads to the

 frequency scaling factor $S$ proportional linearly with execution time ($S
 \propto T$). Large scale MPI applications such as NAS benchmarks have
 considerable amount of communications embedded in these programs. During the
 frequency scaling factor $S$ proportional linearly with execution time ($S
 \propto T$). Large scale MPI applications such as NAS benchmarks have
 considerable amount of communications embedded in these programs. During the

-communication process the processor remain idle until the communication has
+communication process the processors remain idle until the communication has

 finished. For that reason any change in the frequency has no impact on the time
 of communication but it has obvious impact on the time of
 finished. For that reason any change in the frequency has no impact on the time
 of communication but it has obvious impact on the time of

-computation~\cite{17}. We are made many tests on real cluster to prove that the
-frequency scaling factor \emph S has a linear relation with computation time
-only also see~\cite{41}. To predict the execution time of MPI program, firstly
-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}).
-\begin{equation}  \label{eq:tnew}
-  \displaystyle T_{new}= T_{Max \:Comp \:Old} \; . \:S \;+ \;T_{Max\: Comm\: Old}
-  \hfill
+computation~\cite{17}. To predict the execution time of MPI program, the communication time and 
+the computation time for the slower task must be first precisely specified. Secondly, 
+these times are used to predict the execution time for any MPI program as a function of 
+the new scaling factor as in EQ~(\ref{eq:tnew}).
+\begin{equation}
+  \label{eq:tnew}
+ \textit  T_\textit{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
-method as we will show in the coming sections. In the next section we make an
-investigation study for the EQ~(\ref{eq:tnew}).
+value. Depending on this prediction we can produce our energy-performance scaling
+method as we will show in the coming sections. In section~\ref{sec.expe} we make an
+investigation study for EQ~(\ref{eq:tnew}).

 
 

-\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
-time for any scale value. The NAS programs run the class B for comparing the
-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
-Simgrid/SMPI v3.10 to run the NAS programs.
-\begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
-  \centering
-  \includegraphics[scale=0.60]{cg_per.eps}
-  \includegraphics[scale=0.60]{mg_pre.eps}
-  \includegraphics[scale=0.60]{bt_pre.eps}
-  \includegraphics[scale=0.60]{lu_pre.eps}
-  \caption{Fitting Predicted to Real Execution Time}
-  \label{fig:pred}
-\end{figure}
-%see Figure~\ref{fig:pred}
-In our cluster there are 18 available frequency states for each processor from
-2.5 GHz to 800 MHz, there is 100 MHz difference between two successive
-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
-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}).

 
 \section{Performance to Energy Competition}
 
 \section{Performance to Energy Competition}

+\label{sec.compet}
+

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

 between the energy and the performance are nonlinear and complex, because the
 relation of the energy with scaling factor is nonlinear and with the performance
 it is linear see~\cite{17}. The relation between the energy and the performance
 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
 between the energy and the performance are nonlinear and complex, because the
 relation of the energy with scaling factor is nonlinear and with the performance
 it is linear see~\cite{17}. The relation between the energy and the performance
 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 :
-\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   }
-\end{equation}
-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}}    \;\;
+without scaled frequency:
+\begin{multline}
+  \label{eq:enorm}
+  E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
+        {} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
+               \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+               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}
+By the same way we can normalize the performance as follows:
+\begin{equation}
+  \label{eq:pnorm}
+  P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}
+          = \frac{T_\textit{Max Comp Old} \cdot S +
+           T_\textit{Max Comm Old}}{ T_\textit{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
 performance curves are not in the same direction see figure~(\ref{fig:r2}).
 While the main goal is to optimize the energy and performance in the same
 \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
 performance curves are not in the same direction see figure~(\ref{fig:r2}).
 While the main goal is to optimize the energy and performance in the same

-time. According to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}) the
+time. According to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the

 scaling factor \emph S reduce both the energy and the performance
 simultaneously. But the main objective is to produce maximum energy reduction
 with minimum performance reduction. Many researchers used different strategies
 to solve this nonlinear problem for example see~\cite{19,42}, their methods add
 big overhead to the algorithm for selecting the suitable frequency. In this
 scaling factor \emph S reduce both the energy and the performance
 simultaneously. But the main objective is to produce maximum energy reduction
 with minimum performance reduction. Many researchers used different strategies
 to solve this nonlinear problem for example see~\cite{19,42}, their methods add
 big overhead to the algorithm for selecting the suitable frequency. In this

-paper we are present a method to find the optimal scaling factor \emph S for
+paper we present a method to find the optimal scaling factor \emph S for

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

 \end{equation}
 \end{equation}

-\begin{figure}
+\begin{figure*}

   \centering
   \centering

-  \subfloat[Converted Relation.]{\includegraphics[scale=0.70]{file.eps}\label{fig:r1}}
-  \subfloat[Real Relation.]{\includegraphics[scale=0.70]{file3.eps}\label{fig:r2}}
+  \subfloat[Converted Relation.]{%
+    \includegraphics[width=.4\textwidth]{file.eps}\label{fig:r1}}%
+  \qquad%
+  \subfloat[Real Relation.]{%
+    \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}}

   \label{fig:rel}
   \label{fig:rel}

-  \caption{The Energy and Performance Relation}
-\end{figure}
+  \caption{The Relation of Energy and 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
 Then, we can modelize our objective function as finding the maximum distance
 between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance

-curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent
-the minimum energy consumption with minimum execution time (better performance)
+curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors $S_j$. This represent
+the minimum energy consumption with minimum execution time (better performwhere F is the number of available frequenciesance)

 in the same time, see figure~(\ref{fig:r1}). Then our objective function has the
 following form:
 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}
+  S_\textit{optimal} = \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 the
+where F is the number of available frequencies. 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
 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
-objective of this paper and we choose Rauber's model as an example with two
+factor as shown in~\cite{15,3,19}. Energy measurement model is not the
+objective of this paper and we choose Rauber and Rünger model as an example with two

 reasons that mentioned before.
 
 \section{Optimal Scaling Factor for Performance and Energy}
 reasons that mentioned before.
 
 \section{Optimal Scaling Factor for Performance and Energy}

+\label{sec.optim}

 
 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.
 
 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.

-\begin{algorithm}[t]
+\begin{algorithm}[tp]

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


@@ -371,92 +419,132 @@ scaling factor for both energy and performance at the same time.
     \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 scale value between each two 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 scale value between each two frequencies.

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

       \State - Calculate the new frequency as $F_{new}=F_{new} - F_{diff} $
       \State - Calculate the scale factor $S$ as in EQ~(\ref{eq:s}).
       \State - Calculate the new frequency as $F_{new}=F_{new} - F_{diff} $
       \State - Calculate the scale factor $S$ as in EQ~(\ref{eq:s}).

-      \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 all available scales $S_i$  depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}).
+      \State - Select the maximum scale factor $S_1$ from the set\par\hspace{1 pt} of scales $S_i$.
+      \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ 
+               \par\hspace{1 pt}  as in EQ~(\ref{eq:enorm}).
+      \State - Calculate the normalize inverse of performance\par\hspace{1 pt}
+               $P_{NormInv}=T_{old}/T_{new}$ 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}
 The proposed EPSA algorithm works online during the execution time of the MPI
   \end{algorithmic}
 \end{algorithm}
 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}
-\begin{lstlisting}[frame=tb]
-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}
-\end{minipage}
-After obtaining the optimal scale factor from the EPSA algorithm. The program
+program. It selects the optimal scaling factor by gathering the computation and communication times 
+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 algorithm complexity is O(F$\cdot$N), 
+where F is the number of available frequencies and N is the number of computing 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 changes the new frequency of the system. The algorithm is called just 
+one time during the execution of the program. The DVFS algorithm~(\ref{dvfs}) 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}[tp]
+  \caption{DVFS}
+  \label{dvfs}
+  \begin{algorithmic}[1]
+ \For {$K:=1$ to $Some-Iterations \; $}
+  \State -Computations Section.
+   \State -Communications Section.
+   \If {$(K=1)$} 
+     \State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
+     \State -Call EPSA with these times.
+     \State -Calculate the new frequency from optimal scale.
+     \State -Change the new frequency of the system.
+   \EndIf
+\EndFor
+\end{algorithmic}
+\end{algorithm}
+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
 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
+value $T_i$. By substitution of  EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we
+can calculate the new frequency $F_i$ as follows:
+\begin{equation}
+  \label{eq:fi}
+  F_i = \frac{F_\textit{max} \cdot T_i}{S_\textit{optimal} \cdot T_\textit{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.
+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
-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
+\label{sec.expe}
+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.
-\begin{table}[ht]
-  \caption{Platform File Parameters}
+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. Each core simulates the real Intel core i5-3210M processor. 
+This processor has frequencies from 2.5 GHz to 1.2 GHz with 100 MHz difference between each two successive 
+frequencies. We increased this range to verify the EPSA algorithm takes small execution 
+time while it has a big number of available 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{SimGrid Platform File Parameters}

   % title of Table
   \centering
   % title of Table
   \centering

-  \begin{tabular}{ | l | l | l |l | l |l |l |  p{2cm} |}
+  \begin{tabular}{|*{7}{l|}}

     \hline
     Max & Min & Backbone & Backbone&Link &Link& Sharing  \\
     Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline
     \hline
     Max & Min & Backbone & Backbone&Link &Link& Sharing  \\
     Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline

-    2.5 &800 & 2.25 GBps &5E-7 s & 1 GBps & 5E-5 s&Full  \\
+    \np{2.5} & \np{800} & \np[GBps]{2.25} &\np[$\mu$s]{0.5}& \np[GBps]{1} & \np[$\mu$s]{50} &Full  \\

     GHz& MHz&  & & &  &Duplex  \\\hline
   \end{tabular}
   \label{table:platform}
 \end{table}
     GHz& MHz&  & & &  &Duplex  \\\hline
   \end{tabular}
   \label{table:platform}
 \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
+\subsection{Performance Prediction Verification}
+
+In this section we evaluate the precision of our performance prediction methods
+on the NAS benchmarks. We use EQ~(\ref{eq:tnew}) that predicts the execution
+time for any scale value. The NAS programs run the class B for comparing the
+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}). 
+\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}
+  \caption{Fitting Predicted to Real Execution Time}
+  \label{fig:pred}
+\end{figure*}
+%see Figure~\ref{fig:pred}
+In our cluster there are 18 available frequency states for each processor. 
+This lead to 18 run states for each program. We use seven MPI programs of the
+ 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
+example, we present the execution times of the NAS benchmarks as in the
+figure~(\ref{fig:pred}).
+
+\subsection{The EPSA Results}
+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
+respectively. 
+Depending on EQ~(\ref{eq:energy}), we measure the energy consumption for all
+the NAS MPI programs while assuming the power dynamic is equal to \np[W]{20} and
+the power static is equal to \np[W]{4} for all experiments. These power values 
+used by Rauber and Rünger~\cite{3}.  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


@@ -464,46 +552,45 @@ there are different distances between the normalized energy and the normalized
 inversed performance curves, because there are different communication features
 for each MPI program.  When there are little or not communications, the inversed
 performance curve is very close to the energy curve. Then the distance between
 inversed performance curves, because there are different communication features
 for each MPI program.  When there are little or not communications, the inversed
 performance curve is very close to the energy curve. Then the distance between

-the two curves is very small. This lead to small energy savings. The opposite
+the two curves is very small. This leads to small energy savings. The opposite

 happens when there are a lot of communication, the distance between the two
 happens when there are a lot of communication, the distance between the two

-curves is big.  This lead to more energy savings (e.g. CG and FT), see
+curves is big.  This leads to more energy savings (e.g. CG and FT), see

 table~(\ref{table:factors results}). All discovered frequency scaling factors
 optimize both the energy and the performance simultaneously for all the NAS
 programs. In table~(\ref{table:factors results}), we record all optimal scaling
 factors results for each program on class C. These factors give the maximum
 energy saving percent and the minimum performance degradation percent in the
 same time over all available scales.
 table~(\ref{table:factors results}). All discovered frequency scaling factors
 optimize both the energy and the performance simultaneously for all the NAS
 programs. In table~(\ref{table:factors results}), we record all optimal scaling
 factors results for each program on class C. These factors give the maximum
 energy saving percent and the minimum performance degradation percent in the
 same time over all available scales.

-\begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
+\begin{figure*}[t]

   \centering
   \centering

-  \includegraphics[scale=0.47]{ep.eps}
-  \includegraphics[scale=0.47]{cg.eps}
-  \includegraphics[scale=0.47]{sp.eps}
-  \includegraphics[scale=0.47]{lu.eps}
-  \includegraphics[scale=0.47]{bt.eps}
-  \includegraphics[scale=0.47]{ft.eps}
-  \caption{Optimal scaling factors for The NAS MPI Programs}
+  \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{The Discovered  scaling factors for NAS MPI Programs}

   \label{fig:nas}
   \label{fig:nas}

-\end{figure}
-\begin{table}[width=\textwidth,height=\textheight,keepaspectratio]
-  \caption{Optimal Scaling Factors Results}
+\end{figure*}
+\begin{table}[htb]
+  \caption{The EPSA Scaling Factors Results}

   % title of Table
   \centering
   % title of Table
   \centering

-  \begin{tabular}{ | l | l | l |l | l | p{2cm} |}
+  \begin{tabular}{|l|*{4}{r|}}

     \hline
     Program & Optimal & Energy  & Performance&Energy-Perf.\\
     Name & Scaling Factor& Saving \%&Degradation \% &Distance  \\ \hline
     \hline
     Program & Optimal & Energy  & Performance&Energy-Perf.\\
     Name & Scaling Factor& Saving \%&Degradation \% &Distance  \\ \hline

-    CG & 1.56 &39.23 & 14.88 & 24.35\\ \hline
-    MG & 1.47 &34.97&21.7& 13.27   \\ \hline
+    CG & 1.56 &39.23&14.88 &24.35\\ \hline
+    MG & 1.47 &34.97&21.70 &13.27 \\ \hline

     EP & 1.04 &22.14&20.73 &1.41\\ \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
-  \end{tabular}
+    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}
   \label{table:factors results}
   % is used to refer this table in the text
 \end{table}

-

 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
 FT. The opposite happens when the optimal scaling factor is small value as
 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
 FT. The opposite happens when the optimal scaling factor is small value as


@@ -512,148 +599,148 @@ 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).
 

-\section{Comparing Results}
+\subsection{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
-level without considering the performance as in EQ~(\ref{eq:sopt}). We refer to
-this scenario as $Rauber_{E}$. The second scenario is similar to the first
+In this section, we compare our EPSA algorithm results with Rauber and Rünger
+methods~\cite{3}. They had two scenarios, the first is to reduce energy to
+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
 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 $Rauber_{E-P}$. The comparison is made in tables~(\ref{table:compare
+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
   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.
-\begin{table}[ht]
+tables show the results of our EPSA 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}
   % title of Table
   \centering
   \caption{Comparing Results for  The NAS Class A}
   % title of Table
   \centering

-  \begin{tabular}{ | l | l | l |l | l |l|  }
+  \begin{tabular}{|l|l|*{4}{r|}}

     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\
     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\

-    name &name&value& Saving \%&Degradation \% &Distance
+    Name &Name&Value& Saving \%&Degradation \% &Distance

     \\ \hline
     % \rowcolor[gray]{0.85}
     \\ \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
-    $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
+    $EPSA$&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
+    $R_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline
+    $R_{E}$&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
+    $EPSA$&MG & 1.47 &27.66&16.82&10.84\\ \hline
+    $R_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
+    $R_{E}$&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
-    $Rauber_{E}$&EP&2.05&42.09&57.59&-15.5\\ \hline
+    $EPSA$&EP &1.19 &25.32&20.79&4.53\\ \hline
+    $R_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
+    $R_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline

 
 

-    EPSA&LU&1.56& 39.55 &19.38& 20.17\\ \hline
-    $Rauber_{E-P}$&LU&2.14&45.62&27&18.62 \\ \hline
-    $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
+    $EPSA$&LU&1.56& 39.55 &19.38& 20.17\\ \hline
+    $R_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline
+    $R_{E}$&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
+    $R_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline
+    $R_{E}$&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
-    $Rauber_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
+    $EPSA$&SP&1.38& 33.51&15.65&17.86 \\ \hline
+    $R_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline
+    $R_{E}$&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
+    $R_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
+    $R_{E}$&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{table}
   \end{tabular}
   \label{table:compare Class A}
   % is used to refer this table in the text
 \end{table}

-\begin{table}[ht]
+\begin{table}[p]

   \caption{Comparing Results for The NAS Class B}
   % title of Table
   \centering
   \caption{Comparing Results for The NAS Class B}
   % title of Table
   \centering

-  \begin{tabular}{ | l | l | l |l | l |l|  }
+  \begin{tabular}{|l|l|*{4}{r|}}

     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\
     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\

-    name &name&value& Saving \%&Degradation \% &Distance
+    Name &Name&Value& Saving \%&Degradation \% &Distance

     \\ \hline
     % \rowcolor[gray]{0.85}
     \\ \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
-    $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
+    $EPSA$&CG & 1.66 &39.23&16.63&22.60   \\ \hline
+    $R_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline
+    $R_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline

 
 

-    EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline
-    $Rauber_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
-    $Rauber_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
+    $EPSA$ &MG & 1.47 &34.98&18.35&16.63\\ \hline
+    $R_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
+    $R_{E}$&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
+    $EPSA$&EP &1.08 &20.29&17.15&3.14 \\ \hline
+    $R_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline
+    $R_{E}$&EP&2.00&39.73&59.94&-20.21\\ \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
+    $EPSA$&LU&1.47&38.57&21.34&17.23 \\ \hline
+    $R_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline
+    $R_{E}$&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
+    $R_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline
+    $R_{E}$&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
-    $Rauber_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
+    $EPSA$&SP&1.38&33.44&19.24&14.20 \\ \hline
+    $R_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline
+    $R_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline

 
 

-    EPSA&FT&1.388&34.4&14.57&19.83 \\ \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
+    $EPSA$&FT&1.38&34.40&14.57&19.83 \\ \hline
+    $R_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
+    $R_{E}$&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{table}
 
   \end{tabular}
   \label{table:compare Class B}
   % is used to refer this table in the text
 \end{table}
 

-\begin{table}[ht]
+\begin{table}[p]

   \caption{Comparing Results for The NAS Class C}
   % title of Table
   \centering
   \caption{Comparing Results for The NAS Class C}
   % title of Table
   \centering

-  \begin{tabular}{ | l | l | l |l | l |l|  }
+  \begin{tabular}{|l|l|*{4}{r|}}

     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\
     \hline
     Method&Program&Factor& Energy& Performance &Energy-Perf.\\

-    name &name&value& Saving \%&Degradation \% &Distance
+    Name &Name&Value& Saving \%&Degradation \% &Distance

     \\ \hline
     % \rowcolor[gray]{0.85}
     \\ \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
+    $EPSA$&CG & 1.56 &39.23&14.88&24.35  \\ \hline
+    $R_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline
+    $R_{E}$&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
-    $Rauber_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
+    $EPSA$&MG & 1.47 &34.97&21.69&13.27\\ \hline
+    $R_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline
+    $R_{E}$&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
+    $EPSA$&EP &1.04 &22.14&20.73&1.41 \\ \hline
+    $R_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline
+    $R_{E}$&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
+    $R_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline
+    $R_{E}$&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
+    $R_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline
+    $R_{E}$&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
+    $R_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline
+    $R_{E}$&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
+    $R_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
+    $R_{E}$&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}
 As shown in these tables our scaling factor is not optimal for energy saving
   \end{tabular}
 \label{table:compare Class C}
 % is used to refer this table in the text
 \end{table}
 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
+such as Rauber and Rünger 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
 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
+compared to Rauber and Rünger energy-performance method ($R_{E-P}$). Also, in
+($R_{E-P}$) 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
 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


@@ -662,25 +749,40 @@ 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
 same resultant of our objective function EQ~(\ref{eq:max}). Our algorithm always
 Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving
 percent and the performance degradation percent. Therefore, this means it is the
 same resultant of our objective function EQ~(\ref{eq:max}). Our algorithm always

-gives positive energy to performance trade offs while Rauber's method
-($Rauber_{E-P}$) gives in some time negative trade offs such as in BT and
+gives positive energy to performance trade offs while Rauber and Rünger method
+($R_{E-P}$) gives in some time negative trade offs such as in BT and

 EP. The positive trade offs with highest values lead to maximum energy savings
 concatenating with less performance degradation and this the objective of this
 paper. While the negative trade offs refers to improving energy saving (or may
 be the performance) while degrading the performance (or may be the energy) more
 than the first.
 EP. The positive trade offs with highest values lead to maximum energy savings
 concatenating with less performance degradation and this the objective of this
 paper. While the negative trade offs refers to improving energy saving (or may
 be the performance) while degrading the performance (or may be the energy) more
 than the first.

-\begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
+\begin{figure}[t]

   \centering
   \centering

-  \includegraphics[scale=0.60]{compare_class_A.pdf}
-  \includegraphics[scale=0.60]{compare_class_B.pdf}
-  \includegraphics[scale=0.60]{compare_class_c.pdf}
-  % use scale 35 for all to be in the same line
-  \caption{Comparing Our EPSA with Rauber's Methods}
+  \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 EPSA with Rauber and Rünger Methods}

   \label{fig:compare}
 \end{figure}
   \label{fig:compare}
 \end{figure}

+\section{Conclusion}
+\label{sec.concl}
+In this paper we developed the simultaneous energy-performance algorithm. It works based on the trade off relation between the energy and performance. The results showed that when the scaling factor is big value refer to more energy saving. Also, when the scaling factor is smaller value, then it has bigger impact on performance than energy. The algorithm optimizes the energy saving and performance in the same time to have positive trade off. The optimal trade off represents the maximum distance between the energy and the inversed performance curves. Also, the results explained when setting the slowest task to maximum frequency usually not have a big improvement on performance. In future, we will apply the EPSA algorithm on heterogeneous platform.
+
+\section*{Acknowledgment}
+Computations have been performed on the supercomputer facilities of the
+Mésocentre de calcul de Franche-Comté.
+As a PhD student, M. Ahmed Fanfakh, would like to thank the University of
+Babylon (Iraq) for supporting his work.

 
 

-\bibliographystyle{plain}
-\bibliography{my_reference}
+
+% trigger a \newpage just before the given reference
+% number - used to balance the columns on the last page
+% adjust value as needed - may need to be readjusted if
+% the document is modified later
+%\IEEEtriggeratref{15}
+
+\bibliographystyle{IEEEtran}
+\bibliography{IEEEabrv,my_reference}

 \end{document}
 
 %%% Local Variables:
 \end{document}
 
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@@ -689,3 +791,6 @@ than the first.
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 %%% End:
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:

+
+%  LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
+%  LocalWords:  CMOS EQ EPSA Franche Comté Tflop Rünger