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 \begin{document}
 \begin{center}
 
 \begin{document}
 \begin{center}

-\Large
-\title*\textbf{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
-  \end{center}
-\parskip 0pt 
+  \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}
 \linespread{1.18}
 \normalsize
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 \renewcommand*{\@seccntformat}[1]{\csname the#1\endcsname\hspace{0.01cm}}
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 \sectionfont{\large}

+

 \section{.~Introduction }
 \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 constructors have been regularly increasing the number of computing cores in supercomputers (for example in November 2013, according to the top 500 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 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 minimum when processors are idle (waiting for data from other processors or communicating with other processors). Moreover, depending on their objectives they use heuristics to find the best scaling factor during the computation. If they aim for performance they choose the best scaling factor that reduces the consumed energy while affecting as little as possible the performance. On the other hand, if they aim for energy reduction, the chosen scaling factor must produce the most energy efficient execution without considering the degradation 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 algorithm that selects the optimal frequency scaling factor that takes into consideration simultaneously the energy consumption and the performance. The main objective of HPC systems is to run the application with less execution time. Therefore, our algorithm selects the optimal scaling factor online with very small footprint. The proposed algorithm takes into account the communication times of the MPI programs 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 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.  
+
+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
+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
+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
+minimum when processors are idle (waiting for data from other processors or
+communicating with other processors). Moreover, depending on their objectives
+they use heuristics to find the best scaling factor during the computation. If
+they aim for performance they choose the best scaling factor that reduces the
+consumed energy while affecting as little as possible the performance. On the
+other hand, if they aim for energy reduction, the chosen scaling factor must
+produce the most energy efficient execution without considering the degradation
+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
+algorithm that selects the optimal frequency scaling factor that takes into
+consideration simultaneously the energy consumption and the performance. The
+main objective of HPC systems is to run the application with less execution
+time. Therefore, our algorithm selects the optimal scaling factor online with
+very small footprint. The proposed algorithm takes into account the
+communication times of the MPI programs 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
+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}
 \sectionfont{\large}

+

 \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. 
- \sectionfont{\large} 
+
+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 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. 
-\sectionfont{\large} 
-\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:
-\\1-Selecting the optimal frequency scaling factor for energy and performance 
-    simultaneously. While taking into account the communication time.
-\\2-Adapting our scale factor to taking into account the imbalanced tasks.
-\\3-The execution time of our algorithm is very small when compared  to other methods (e.g.,~\cite{19}).
-\\4-The proposed algorithm works online without profiling or training as in~\cite{38,34}.
+
+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.

 \sectionfont{\large}
 \sectionfont{\large}

+
+\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:
+\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 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}
+\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 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.
+
+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
+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]
 \begin{figure}[h]

-\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}}
-    \caption{Parallel Tasks on Homogeneous Platform}
-     \label{fig:homo}
+  \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}}
+  \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 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 
+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

 \end{equation}
 where $T_i$ is the execution time of process $i$.
 \sectionfont{\large}
 \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 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 :
-\begin{equation}  \label{eq:pd} 
-\displaystyle  P_{dyn} = \alpha . C_L . V^2 . f
+
+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
+$P_{dyn}$ 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

 \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}
+  \displaystyle  P_{static}  = V . N . K_{design} . I_{leak}

 \end{equation}
 \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
+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

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

 \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 \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

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

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

 \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\: .\:(\frac{T_1}{T_i})=\: (\frac{F_{max}}{F_{new}}).(\frac{T_1}{T_i}) \hfill
+\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

 \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  
-\end{equation} 

 %[\Big 3]
 \sectionfont{\large}
 %[\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 execution time of the parallel programs are proportional to the operational 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 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 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 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 
- \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}).
+
+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
+execution time of the parallel programs are proportional to the operational
+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
+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
+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
+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
+\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}).

 \sectionfont{\large}
 \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 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.  
+
+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
 \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}
+  \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}
   \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}).
+%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}).

 \sectionfont{\large}
 \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 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} 
+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
+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 :
 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}}    \;\;
- \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 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 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}}
-\end{equation} 
+\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}}    \;\;
+\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
+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
+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}}
+\end{equation}

 \begin{figure}
 \begin{figure}

-\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}}  
-   \label{fig:rel}
-    \caption{The Energy and Performance Relation}
+  \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}}
+  \label{fig:rel}
+  \caption{The Energy and Performance Relation}

 \end{figure}
 \end{figure}

-Then, we can modelize our objective function as finding the maximum distance between the energy curve  EQ~(\ref{eq:enorm}) and the inverse of performance curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent 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} \;) 
-\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 objective of this paper and we choose Rauber's model as an example with two reasons that mentioned before. 
+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)
+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} \;)
+\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
+objective of this paper and we choose Rauber's model as an example with two
+reasons that mentioned before.

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

-\caption{EPSA}
-\label{EPSA}
-\begin{algorithmic}[1]
-\State  Initialize the variable $Dist=0$ 
-\State Set dynamic and static power values.
-\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} $} 
-\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 $Dist = P_{NormInv} - E_{Norm}$
-    \EndIf 
-\EndFor 
-\State  $ Return \; \;  (S_{optimal})$
-\end{algorithmic}
+  \caption{EPSA}
+  \label{EPSA}
+  \begin{algorithmic}[1]
+    \State  Initialize the variable $Dist=0$
+    \State Set dynamic and static power values.
+    \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} $}
+      \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 $Dist = P_{NormInv} - E_{Norm}$
+      \EndIf
+    \EndFor
+    \State  $ Return \; \;  (S_{optimal})$
+  \end{algorithmic}

 \end{algorithm}
 \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
+\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
 \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. 
+     -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}
      -Call EPSA with these times.
      -Calculate the new frequency from optimal scale.
      -Set the new frequency to the system.
    ENDIF
 ENDFOR
 \end{lstlisting}

-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 
+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

 \end{equation}
 \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.
+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}
 \sectionfont{\large}

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

-\caption{Platform File Parameters}
-% title of Table
-\centering
-    \begin{tabular}{ | l | l | l |l | l |l |l |  p{2cm} |}
+  \caption{Platform File Parameters}
+  % title of Table
+  \centering
+  \begin{tabular}{ | l | l | l |l | l |l |l |  p{2cm} |}

     \hline
     \hline

-    Max & Min & Backbone & Backbone&Link &Link& Sharing  \\ 
+    Max & Min & Backbone & Backbone&Link &Link& Sharing  \\

     Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline
     Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline

-    2.5 &800 & 2.25 GBps &5E-7 s & 1 GBps & 5E-5 s&Full  \\ 
+    2.5 &800 & 2.25 GBps &5E-7 s & 1 GBps & 5E-5 s&Full  \\

     GHz& MHz&  & & &  &Duplex  \\\hline
     GHz& MHz&  & & &  &Duplex  \\\hline

-    \end{tabular}
-\label{table:platform}
+  \end{tabular}
+  \label{table:platform}

 \end{table}
 \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 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 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 the two curves is very small. This lead to small energy savings. The opposite 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 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.
+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
+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
+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
+the two curves is very small. This lead to small energy savings. The opposite
+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
+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}[width=\textwidth,height=\textheight,keepaspectratio]

-\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}
-     \label{fig:nas}    
+  \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}
+  \label{fig:nas}

 \end{figure}
 \linespread{1.1}
 \begin{table}[width=\textwidth,height=\textheight,keepaspectratio]
 \end{figure}
 \linespread{1.1}
 \begin{table}[width=\textwidth,height=\textheight,keepaspectratio]

-\caption{Optimal Scaling Factors Results}
-% title of Table
-\centering
-    \begin{tabular}{ | l | l | l |l | l | p{2cm} |}
+  \caption{Optimal Scaling Factors Results}
+  % title of Table
+  \centering
+  \begin{tabular}{ | l | l | l |l | l | p{2cm} |}

     \hline
     \hline

-    Program & Optimal & Energy  & Performance&Energy-Perf.\\ 
+    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
     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


@@ -246,159 +511,200 @@ When there are little or not communications, the inversed performance curve is v
     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
     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}
-\label{table:factors results}
-% is used to refer this table in the text
+  \end{tabular}
+  \label{table:factors results}
+  % is used to refer this table in the text

 \end{table}
 \linespread{1.2}
 
 \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 FT. The opposite happens when the optimal scaling factor is small value as example BT and EP. Our algorithm selects big scaling factor value when the communication and the other slacks times are big and smaller ones in opposite cases. In EP there are no communications inside the iterations. This make our EPSA to selects smaller scaling factor values (inducing smaller energy savings).   
-  
+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
+example BT and EP. Our algorithm selects big scaling factor value when the
+communication and the other slacks times are big and smaller ones in opposite
+cases. In EP there are no communications inside the iterations. This make our
+EPSA to selects smaller scaling factor values (inducing smaller energy savings).
+

 % \clearpage
 \sectionfont{\large}
 % \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 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 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 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.
+
+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
+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
+  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]
 %\linespread{1}
 \begin{table}[ht]

-\caption{Comparing Results for  The NAS Class A}
-% title of Table
-\centering
-    \begin{tabular}{ | l | l | l |l | l |l|  }
+  \caption{Comparing Results for  The NAS Class A}
+  % title of Table
+  \centering
+  \begin{tabular}{ | l | l | l |l | l |l|  }

     \hline
     \hline

-    Method&Program&Factor& Energy& Performance &Energy-Perf.\\ 
-    name &name&value& Saving \%&Degradation \% &Distance   
+    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
+    name &name&value& Saving \%&Degradation \% &Distance

     \\ \hline
     \\ \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  
+    % \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
     $Rauber_{E}$&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 
-     
-     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&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&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&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
     $Rauber_{E-P}$&LU&2.14&45.62&27&18.62 \\ \hline
     $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline

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

 \end{table}
 \begin{table}[ht]
 \end{table}
 \begin{table}[ht]

-\caption{Comparing Results for The NAS Class B}
-% title of Table
-\centering
-    \begin{tabular}{ | l | l | l |l | l |l|  }
+  \caption{Comparing Results for The NAS Class B}
+  % title of Table
+  \centering
+  \begin{tabular}{ | l | l | l |l | l |l|  }

     \hline
     \hline

-    Method&Program&Factor& Energy& Performance &Energy-Perf.\\ 
-    name &name&value& Saving \%&Degradation \% &Distance   
+    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
+    name &name&value& Saving \%&Degradation \% &Distance

     \\ \hline
     \\ \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  
+    % \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
     $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline

-   
-     EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline
-     $Rauber_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \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
-     
-     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&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&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&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&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&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&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&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&FT&1.388&34.4&14.57&19.83 \\ \hline
     $Rauber_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \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
-    \end{tabular}
-\label{table:compare Class B}
-% is used to refer this table in the text
+    $Rauber_{E}$&FT&2.13&43.04&37.9&5.14\\ \hline
+  \end{tabular}
+  \label{table:compare Class B}
+  % is used to refer this table in the text

 \end{table}
 \end{table}

-     
- \begin{table}[ht]
-\caption{Comparing Results for The NAS Class C}
-% title of Table
-\centering
-    \begin{tabular}{ | l | l | l |l | l |l|  }
+
+\begin{table}[ht]
+  \caption{Comparing Results for The NAS Class C}
+  % title of Table
+  \centering
+  \begin{tabular}{ | l | l | l |l | l |l|  }

     \hline
     \hline

-    Method&Program&Factor& Energy& Performance &Energy-Perf.\\ 
-    name &name&value& Saving \%&Degradation \% &Distance   
+    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
+    name &name&value& Saving \%&Degradation \% &Distance

     \\ \hline
     \\ \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&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&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&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&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&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
-     
+    % \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&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&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&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&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&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&FT&1.47&34.72&19&15.72 \\ \hline
     $Rauber_{E-P}$&FT&2.04&39.4&37.1&2.3\\ \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
-    \end{tabular}
+    $Rauber_{E}$&FT&2.04&39.35&37.7&1.65\\ \hline
+  \end{tabular}

 \label{table:compare Class C}
 % is used to refer this table in the text
 \end{table}
 %\linespread{1.2}
 \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. 
- 
-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 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]
-\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}
-   \label{fig:compare}
-    
+\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.
+
+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
+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]
+  \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}
+  \label{fig:compare}

 \end{figure}
 \end{figure}

- \clearpage
+
+\clearpage

 \bibliographystyle{plain}
 \bibliography{my_reference}
 \end{document}
 \bibliographystyle{plain}
 \bibliography{my_reference}
 \end{document}

+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% fill-column: 80
+%%% ispell-local-dictionary: "american"
+%%% End: