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
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
-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
-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}.
-
\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.
\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
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 $i$.
-\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
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
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}
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
+lowering core's frequency. The scaling factor is equal to 1 when the new frequency is
+set to the maximum frequency. The energy consumption model for parallel
homogeneous platform depends on the scaling factor \emph S. This factor reduces
quadratically the dynamic power. Also, this factor increases the static energy
linearly because the execution time is increased~\cite{36}. The energy model
number of concurrent tasks was developed by Rauber and Rünger~\cite{3}. This
model considers the two power metrics for measuring the energy of the parallel
tasks as in EQ~(\ref{eq:energy}):
-
\begin{equation}
\label{eq:energy}
E = P_\textit{dyn} \cdot S_1^{-2} \cdot
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}
-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
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 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}
- \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
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}
\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) +
- 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}
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
-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
\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
-curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent
-the minimum energy consumption with minimum execution time (better performance)
+curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors $S_j$. This represent
+the minimum energy consumption with minimum execution time (better performwhere F is the number of available frequenciesance)
in the same time, see figure~(\ref{fig:r1}). Then our objective function has the
following form:
\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}
-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
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}
\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}).
\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
+program. It selects the optimal scaling factor by gathering the computation and communication times
+from the program after one iteration.
+ This algorithm has small execution time
+(between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ for 32 nodes). The algorithm complexity is O(F$\cdot$N),
+where F is the number of available frequencies and N is the number of computing nodes. The data required
+by this algorithm is the computation time and the communication time for each task from the first iteration only.
+When these times are measured, the MPI program calls the EPSA algorithm to choose the new frequency using the
+optimal scaling factor. Then the program changes the new frequency of the system. The algorithm is called just
+one time during the execution of the program. The DVFS algorithm~(\ref{dvfs}) shows where and when the EPSA algorithm is called
in the MPI program.
%\begin{minipage}{\textwidth}
%\AG{Use the same format as for Algorithm~\ref{$EPSA$}}
\caption{DVFS}
\label{dvfs}
\begin{algorithmic}[1]
- \For {$J:=1$ to $Some-Iterations \; $}
+ \For {$K:=1$ to $Some-Iterations \; $}
\State -Computations Section.
\State -Communications Section.
- \If {$(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 -Set the new frequency to the system.
+ \State -Change the new frequency of the system.
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
-
-After obtaining the optimal scale factor from the EPSA algorithm. The program
+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
+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}
\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
-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
-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]
- \caption{Platform File Parameters}
+ \caption{SimGrid Platform File Parameters}
% title of Table
\centering
\begin{tabular}{|*{7}{l|}}
\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
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
\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{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
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
-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
-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
% 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
\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}
+ \caption {Comparing Our EPSA with Rauber and Rünger Methods}
\label{fig:compare}
\end{figure}
\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}
-
-\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
-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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