some changes

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 8039f59d1c098b1df5081f3ee0ec978ce698df2d..781b3e2168cf63dc183b26cf950666b6f8c22c5d 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -19,7 +19,7 @@
 
 \begin{document}
 
 
 \begin{document}
 

-\title{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
+\title{The Simultaneous Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}

 
 \author{%
   \IEEEauthorblockN{%
 
 \author{%
   \IEEEauthorblockN{%


@@ -36,8 +36,7 @@
 
 \maketitle
 
 
 \maketitle
 

-\AG{``Optimal'' is a bit pretentious in the title.\\
-  Complete affiliation, add an email address, etc.}
+\AG{Complete affiliation, add an email address, etc.}

 
 \begin{abstract}
   The important technique for energy reduction of parallel systems is CPU
 
 \begin{abstract}
   The important technique for energy reduction of parallel systems is CPU


@@ -67,7 +66,7 @@ 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
 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


@@ -100,21 +99,18 @@ 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
 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.ptasks} shows the execution of parallel
-tasks and sources of idle times. Section~\ref{sec.energy} resumes the energy
+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
 model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance

-of MPI program.  Section~\ref{sec.verif} verifies the performance prediction
-model. Section~\ref{sec.compet} presents the energy-performance trade offs
+of MPI program.  Section~\ref{sec.compet} presents the energy-performance trade offs

 objective function. Section~\ref{sec.optim} demonstrates the proposed
 objective function. Section~\ref{sec.optim} demonstrates the proposed

-energy-performance algorithm. Section~\ref{sec.expe} presents the results of our
-experiments.  Section~\ref{sec.compare} shows the comparison results. Finally,
+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}.
 we conclude in Section~\ref{sec.concl}.

-

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

-\AG{Consider introducing the models (sec.~\ref{sec.ptasks},
-  maybe~\ref{sec.energy}) before related works}
+\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.


@@ -169,10 +165,10 @@ platforms. The primary contributions of this paper are:
 \item The proposed algorithm works online without profiling or training as
   in~\cite{38,34}.
 \end{enumerate}
 \item The proposed algorithm works online without profiling or training as
   in~\cite{38,34}.
 \end{enumerate}

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

 
 

-\section{Parallel Tasks Execution on Homogeneous Platform}
-\label{sec.ptasks}
-
+\subsection{Parallel Tasks Execution on Homogeneous Platform}

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


@@ -189,19 +185,20 @@ platform. These tasks can exchange the data via synchronous message passing.
 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 synchronization barrier for other tasks to
 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 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 happen 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:
+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}
 where $T_i$ is the execution time of task $i$.
 
 \begin{equation}
   \label{eq:T1}
   \textit{Program Time} = \max_{i=1,2,\dots,N} T_i
 \end{equation}
 where $T_i$ is the execution time of task $i$.
 

-\section{Energy Model for Homogeneous Platform}
-\label{sec.energy}
+\subsection{Energy Model for Homogeneous Platform}

 
 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
 
 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


@@ -221,8 +218,7 @@ the power consumption of peripheral devices like the I/O subsystem.
 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
 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 multiplied by the execution time for example
-see~\cite{36,15}.
+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
 \begin{equation}
   \label{eq:eind}
    E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T


@@ -234,7 +230,7 @@ consumption~\cite{37}. The operational frequency \emph f depends linearly on the
 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
 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

-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}).
 \begin{equation}
   \label{eq:s}
 maximum and the new frequency as in EQ~(\ref{eq:s}).
 \begin{equation}
   \label{eq:s}


@@ -243,16 +239,14 @@ maximum and the new frequency as in EQ~(\ref{eq:s}).
 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
 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 frequency
-set is 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
+lowering core's frequency.  The scaling factor is equal to 1 when the new frequency is 
+set to the maximum frequency.   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
 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}):
-
+tasks as in EQ~(\ref{eq:energy}). This factor reduces
+quadratically the dynamic power.  Also, it increases the static energy
+linearly because the execution time is increased~\cite{36}.

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


@@ -267,14 +261,14 @@ 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}
 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,\dots,F} S_i
+  S_1 = \max_{i=1,2,\dots,N} S_i

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

-where $F$ is the number of available frequencies. In this paper we depend on
+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
 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


@@ -284,14 +278,14 @@ minimized) and set it to zero to produce the scaling factor as in
 EQ~(\ref{eq:sopt}).
 \begin{equation}
   \label{eq:sopt}
 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
+  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}
 
 \section{Performance Evaluation of MPI Programs}
 \label{sec.mpip}
 
     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 
 \section{Performance Evaluation of MPI Programs}
 \label{sec.mpip}
 

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


@@ -308,15 +302,13 @@ considerable amount of communications embedded in these programs. During the
 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
 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

-computation~\cite{17}. We have made many tests on a real cluster to prove that the
-frequency scaling factor \emph S has a linear relation with computation time
-only. To predict the execution time of MPI program, the communication time and 
+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 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 the EQ~(\ref{eq:tnew}).
+the new scaling factor as in EQ~(\ref{eq:tnew}).

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

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


@@ -324,39 +316,10 @@ 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-performance scaling
 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-performance scaling

-method as we will show in the coming sections. In the next section we make to finishan
-investigation study for the EQ~(\ref{eq:tnew}).
+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}
-\label{sec.verif}

 
 

-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*}[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 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}
 \label{sec.compet}
 
 \section{Performance to Energy Competition}
 \label{sec.compet}


@@ -374,9 +337,9 @@ without scaled frequency:
 \begin{multline}
   \label{eq:enorm}
   E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
 \begin{multline}
   \label{eq:enorm}
   E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\

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

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

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

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


@@ -391,13 +354,13 @@ 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
 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


@@ -416,20 +379,20 @@ performance as follows:
   \subfloat[Real Relation.]{%
     \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}}
   \label{fig:rel}
   \subfloat[Real Relation.]{%
     \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}}
   \label{fig:rel}

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

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

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


@@ -443,7 +406,7 @@ reasons that mentioned before.
 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
 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
+(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}[tp]
   \caption{EPSA}
 scaling factor for both energy and performance at the same time.
 \begin{algorithm}[tp]
   \caption{EPSA}


@@ -454,7 +417,7 @@ 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 all available scales $S_i$  depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}).
       \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\par\hspace{1 pt} in EQ~(\ref{eq:si}).


@@ -472,16 +435,12 @@ scaling factor for both energy and performance at the same time.
   \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 DVFS algorithm~(\ref{dvfs}) shows where and when the EPSA algorithm is called
-in the MPI program.
+program. 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. It selects the optimal scaling factor by gathering the computation and communication times 
+from the program after one iteration. 
+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{minipage}{\textwidth}
 %\AG{Use the same format as for Algorithm~\ref{$EPSA$}}
 


@@ -489,22 +448,21 @@ in the MPI program.
   \caption{DVFS}
   \label{dvfs}
   \begin{algorithmic}[1]
   \caption{DVFS}
   \label{dvfs}
   \begin{algorithmic}[1]

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

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

-   \If {$(J==1)$} 
+   \If {$(K=1)$} 

      \State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
      \State -Call EPSA with these times.
      \State -Calculate the new frequency from optimal scale.
      \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 -Set the new frequency to the system.
+     \State -Change the new frequency of the system.

    \EndIf
 \EndFor
 \end{algorithmic}
 \end{algorithm}
    \EndIf
 \EndFor
 \end{algorithmic}
 \end{algorithm}

-
-After obtaining the optimal scale factor from the EPSA algorithm. The program
+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
+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}
 can calculate the new frequency $F_i$ as follows:
 \begin{equation}
   \label{eq:fi}


@@ -517,22 +475,20 @@ the frequency according to the nodes workloads.
 
 \section{Experimental Results}
 \label{sec.expe}
 
 \section{Experimental Results}
 \label{sec.expe}

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

-table~(\ref{table:platform}). Each node in the cluster has 18 frequency values
+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
 from 2.5 GHz to 800 MHz with 100 MHz difference between each two successive

-frequencies.
+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]
 \begin{table}[htb]

-  \caption{Platform File Parameters}
+  \caption{SimGrid Platform File Parameters}

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


@@ -544,9 +500,45 @@ frequencies.
   \end{tabular}
   \label{table:platform}
 \end{table}
   \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 EPSA
+\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


@@ -554,9 +546,9 @@ 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
-happens when there are a lot of communication, theto finish distance between the two
-curves is big.  This lead to more energy savings (e.g. CG and FT), see
+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
+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
 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


@@ -571,7 +563,7 @@ same time over all available scales.
   \includegraphics[width=.33\textwidth]{lu.eps}\hfill%
   \includegraphics[width=.33\textwidth]{bt.eps}\hfill%
   \includegraphics[width=.33\textwidth]{ft.eps}
   \includegraphics[width=.33\textwidth]{lu.eps}\hfill%
   \includegraphics[width=.33\textwidth]{bt.eps}\hfill%
   \includegraphics[width=.33\textwidth]{ft.eps}

-  \caption{Optimal scaling factors for The NAS MPI Programs}
+  \caption{The Discovered  scaling factors for NAS MPI Programs}

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


@@ -593,7 +585,6 @@ same time over all available scales.
   \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


@@ -602,19 +593,18 @@ 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}
-\label{sec.compare}
+\subsection{Comparing Results}

 
 In this section, we compare our EPSA algorithm results with Rauber and Rünger
 
 In this section, we compare our EPSA algorithm results with Rauber and Rünger

-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 $R_{E}$. The second scenario is similar to the first
+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
 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
 except setting the slower task to the maximum frequency (when the scale $S=1$)
 to keep the performance from degradation as mush as possible. We refer to this
 scenario as $R_{E-P}$. The comparison is made in tables~(\ref{table:compare
   Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These

-tables show the results of our EPSA and Rauber and Rünger scenarios for all the NAS
-benchmarks programs for classes A,B and C.
+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
 \begin{table}[p]
   \caption{Comparing Results for  The NAS Class A}
   % title of Table


@@ -740,8 +730,8 @@ benchmarks programs for classes A,B and 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
 % 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
-the energy and the performance simultaneously. Our $EPSA$ optimal scaling factors
+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
 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
 has better simultaneous optimization for both the energy and the performance
 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


@@ -749,7 +739,14 @@ 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.
 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.

-
+\begin{figure}[t]
+  \centering
+  \includegraphics[width=.33\textwidth]{compare_class_A.pdf}
+  \includegraphics[width=.33\textwidth]{compare_class_B.pdf}
+  \includegraphics[width=.33\textwidth]{compare_class_c.pdf}
+  \caption {Comparing Our EPSA with Rauber and Rünger Methods}
+  \label{fig:compare}
+\end{figure}

 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


@@ -760,27 +757,17 @@ 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.
 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}[t]
-  \centering
-  \includegraphics[width=.33\textwidth]{compare_class_A.pdf}
-  \includegraphics[width=.33\textwidth]{compare_class_B.pdf}
-  \includegraphics[width=.33\textwidth]{compare_class_c.pdf}
-  \caption{Comparing Our EPSA with Rauber and Rünger Methods}
-  \label{fig:compare}
-\end{figure}
+

 \section{Conclusion}
 \label{sec.concl}
 \section{Conclusion}
 \label{sec.concl}

-In this paper we develop the simultaneous energy-performance algorithm. It is works based on the trade off relation between the energy and performance. The results showed that when the scaling factor is big value leads to more energy saving. Also, it show that when the the scaling factor is small value leads to the fact that the scaling factor has bigger impact on performance than energy. Then the algorithm optimize the energy saving and performance in the same time to have positive trade off. The optimal trade off refer to 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 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}
 
 \section*{Acknowledgment}

-
-\AG{Right?}

 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
 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 scholarship program that allows him to work on
-this paper.
-\AG{What about simply: ``[...] for supporting his work.''}
+Babylon (Iraq) for supporting his work.
+

 
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