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\begin{document}

\title{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
\author{A. Badri \and J.-C. Charr \and R. Couturier \and A. Giersch}
\maketitle

\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.

\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.

\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.

\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}

\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.
\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}
\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}
  \textit{Program Time} = \max_{i=1,2,\dots,N} T_i
\end{equation}
where $T_i$ is the execution time of process $i$.

\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}
  P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
\end{equation}
The static power $P_{static}$ captures the leakage power consumption as well as
the power consumption of peripheral devices like the I/O subsystem.
\begin{equation}
  \label{eq:ps}
  P_{static}  = V \cdot N \cdot K_{design} \cdot I_{leak}
\end{equation}
where V is the supply voltage, N is the number of transistors, $K_{design}$ is a
design dependent parameter and $I_{leak}$ is a technology-dependent
parameter. Energy consumed by an individual processor $E_{ind}$ is the summation
of the dynamic and the static power multiply by the execution time for example
see~\cite{36,15} .
\begin{equation}
  \label{eq:eind}
  E_{ind} = ( P_{dyn} + P_{static} ) \cdot T
\end{equation}
The dynamic voltage and frequency scaling (DVFS) is a process that allowed in
modern processors to reduce the dynamic power by scaling down the voltage and
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}}
\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 = P_{dyn} \cdot S_1^{-2} \cdot
    \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
    P_{static} \cdot T_1 \cdot S_1 \cdot N
 \hfill
\end{equation}
Where \emph N is the number of parallel nodes, $T_1 $ is the time of the slower
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,\dots,F} S_i
\end{equation}
\begin{equation}
  \label{eq:si}
  S_i = S \cdot \frac{T_1}{T_i}
      = \frac{F_{max}}{F_{new}} \cdot \frac{T_1}{T_i}
\end{equation}
Where $F$ is the number of available frequencies. In this paper we depend on
Rauber's energy model EQ~(\ref{eq:energy}) for two reasons : 1-this model used
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} \cdot \frac{P_{dyn}}{P_{static}} \cdot
    \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
\end{equation}

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

\section{Performance Prediction Verification}

In this section we evaluate the precision of our performance prediction methods
on the NAS benchmark. We use the EQ~(\ref{eq:tnew}) that predicts the execution
time for any scale value. The NAS programs run the class B for comparing the
real execution time with the predicted execution time. Each program runs offline
with all available scaling factors on 8 or 9 nodes to produce real execution
time values. These scaling factors are computed by dividing the maximum
frequency by the new one see EQ~(\ref{eq:s}). In all tests, we use the simulator
Simgrid/SMPI v3.10 to run the NAS programs.
\begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
  \centering
  \includegraphics[scale=0.60]{cg_per.eps}
  \includegraphics[scale=0.60]{mg_pre.eps}
  \includegraphics[scale=0.60]{bt_pre.eps}
  \includegraphics[scale=0.60]{lu_pre.eps}
  \caption{Fitting Predicted to Real Execution Time}
  \label{fig:pred}
\end{figure}
%see Figure~\ref{fig:pred}
In our cluster there are 18 available frequency states for each processor from
2.5 GHz to 800 MHz, there is 100 MHz difference between two successive
frequencies. For more details on the characteristics of the platform refer to
table~(\ref{table:platform}). This lead to 18 run states for each program. We
use seven MPI programs of the NAS parallel benchmarks : CG, MG, EP, FT, BT, LU
and SP. The average normalized errors between the predicted execution time and
the real time (Simgrid time) for all programs is between 0.0032 to 0.0133. AS an
example, we are present the execution times of the NAS benchmarks as in the
figure~(\ref{fig:pred}).

\section{Performance to Energy Competition}
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} = \frac{E_{Reduced}}{E_{Orginal}}
          = \frac{ P_{dyn} \cdot S_i^{-2} \cdot
               \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
               P_{static} \cdot T_1 \cdot S_i \cdot N  }{
              P_{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
              P_{static} \cdot T_1\, \cdot N }
\end{equation}
By the same way we can normalize the performance as follows :
\begin{equation}
  \label{eq:pnorm}
  P_{Norm} = \frac{T_{New}}{T_{Old}}
          = \frac{T_{\textit{Max Comp Old}} \cdot S +
              T_{\textit{Max Comm Old}}}{T_{Old}}
\end{equation}
The second problem is the optimization operation for both energy and performance
is not in the same direction. In other words, the normalized energy and the
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}
  P^{-1}_{Norm} = \frac{T_{Old}}{T_{New}}
               = \frac{T_{Old}}{T_{\textit{Max Comp Old}} \cdot S +
                 T_{\textit{Max Comm Old}}}
\end{equation}
\begin{figure}
  \centering
  \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}
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}
  \textit{MaxDist} = \max (\overbrace{P^{-1}_{Norm}}^{\text{Maximize}} -
                           \overbrace{E_{Norm}}^{\text{Minimize}} )
\end{equation}
Then we can select the optimal scaling factor that satisfy the
EQ~(\ref{eq:max}).  Our objective function can works with any energy model or
static power values stored in a data file. Moreover, this function works in
optimal way when the energy function has a convex form with frequency scaling
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.

\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.
\begin{algorithm}[t]
  \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\par
               $P_{NormInv}=T_{old}/T_{new}$ as in EQ~(\ref{eq:pnorm_en}).
      \If{  $(P_{NormInv}-E_{Norm} > Dist$) }
        \State $S_{optimal} = S$
        \State $Dist = P_{NormInv} - E_{Norm}$
      \EndIf
    \EndFor
    \State  $ Return \; \;  (S_{optimal})$
  \end{algorithmic}
\end{algorithm}
The proposed EPSA algorithm works online during the execution time of the MPI
program. It selects the optimal scaling factor by gathering some information
from the program after one iteration. This algorithm has small execution time
(between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ for 32 nodes). The data
required by this algorithm is the computation time and the communication time
for each task from the first iteration only. When these times are measured, the
MPI program calls the EPSA algorithm to choose the new frequency using the
optimal scaling factor. Then the program set the new frequency to the
system. The algorithm is called just one time during the execution of the
program. The following example shows where and when the EPSA algorithm is called
in the MPI program :
\begin{minipage}{\textwidth}
\begin{lstlisting}[frame=tb]
FOR J:=1 to Some_iterations Do
   -Computations Section.
   -Communications Section.
   IF (J==1) THEN
     -Gather all times of computation and communication
      from each node.
     -Call EPSA with these times.
     -Calculate the new frequency from optimal scale.
     -Set the new frequency to the system.
   ENDIF
ENDFOR
\end{lstlisting}
\end{minipage}
After obtaining the optimal scale factor from the EPSA algorithm. The program
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} \cdot T_i}{S_{optimal} \cdot T_{max}}
\end{equation}
According to this equation all the nodes may have the same frequency value if
they have balanced workloads. Otherwise, they take different frequencies when
have imbalanced workloads. Then EQ~(\ref{eq:fi}) works in adaptive way to change
the freguency according to the nodes workloads.

\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]
  \caption{Platform File Parameters}
  % title of Table
  \centering
  \begin{tabular}{ | l | l | l |l | l |l |l |  p{2cm} |}
    \hline
    Max & Min & Backbone & Backbone&Link &Link& Sharing  \\
    Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline
    2.5 &800 & 2.25 GBps &5E-7 s & 1 GBps & 5E-5 s&Full  \\
    GHz& MHz&  & & &  &Duplex  \\\hline
  \end{tabular}
  \label{table:platform}
\end{table}
Depending on the EQ~(\ref{eq:energy}), we measure the energy consumption for all
the NAS MPI programs while assuming the power dynamic is equal to 20W and the
power static is equal to 4W for all experiments. We run the proposed ESPA
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]
  \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}
\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} |}
    \hline
    Program & Optimal & Energy  & Performance&Energy-Perf.\\
    Name & Scaling Factor& Saving \%&Degradation \% &Distance  \\ \hline
    CG & 1.56 &39.23 & 14.88 & 24.35\\ \hline
    MG & 1.47 &34.97&21.7& 13.27   \\ \hline
    EP & 1.04 &22.14&20.73 &1.41\\ \hline
    LU & 1.388 &35.83&22.49 &13.34\\ \hline
    BT & 1.315 &29.6&21.28 &8.32\\ \hline
    SP & 1.388 &33.48 &21.36&12.12\\ \hline
    FT & 1.47 &34.72 &19&15.72\\ \hline
  \end{tabular}
  \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
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).

\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.
\begin{table}[ht]
  \caption{Comparing Results for  The NAS Class A}
  % title of Table
  \centering
  \begin{tabular}{ | l | l | l |l | l |l|  }
    \hline
    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
    name &name&value& Saving \%&Degradation \% &Distance
    \\ \hline
    % \rowcolor[gray]{0.85}
    EPSA&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
    $Rauber_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.5\\ \hline
    $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline

    EPSA&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&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]
  \caption{Comparing Results for The NAS Class B}
  % title of Table
  \centering
  \begin{tabular}{ | l | l | l |l | l |l|  }
    \hline
    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
    name &name&value& Saving \%&Degradation \% &Distance
    \\ \hline
    % \rowcolor[gray]{0.85}
    EPSA&CG & 1.66 &39.23&16.63&22.6   \\ \hline
    $Rauber_{E-P}$&CG &2.15 &45.34&27.6&17.74\\ \hline
    $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline

    EPSA&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
    $Rauber_{E}$&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}

\begin{table}[ht]
  \caption{Comparing Results for The NAS Class C}
  % title of Table
  \centering
  \begin{tabular}{ | l | l | l |l | l |l|  }
    \hline
    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
    name &name&value& Saving \%&Degradation \% &Distance
    \\ \hline
    % \rowcolor[gray]{0.85}
    EPSA&CG & 1.56 &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
    $Rauber_{E}$&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}
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}

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

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