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\author{%
\IEEEauthorblockN{%
- Ahmed Badri,
Jean-Claude Charr,
- Raphaël Couturier and
+ Raphaël Couturier,
+ Ahmed Fanfakh and
Arnaud Giersch
}
\IEEEauthorblockA{%
Complete affiliation, add an email address, etc.}
\begin{abstract}
- \AG{complete the abstract\dots}
+ The important technique for energy reduction of parallel systems is CPU
+ frequency scaling. This operation is used by many researchers to reduce energy
+ consumption in many ways. Frequency scaling operation also has a big impact on
+ the performances. In some cases, the performance degradation ratio is bigger
+ than energy saving ratio when the frequency is scaled to low level. Therefore,
+ the trade offs between the energy and performance becomes more important topic
+ when using this technique. In this paper we developed an algorithm that select
+ the frequency scaling factor for both energy and performance simultaneously.
+ This algorithm takes into account the communication times when selecting the
+ frequency scaling factor. It works online without training or profiling to
+ have a very small overhead. The algorithm has better energy-performance trade
+ offs compared to other methods.
\end{abstract}
\section{Introduction}
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
minimum value does not always give the most efficient execution due to energy
leakage. The best scaling factor might be chosen during execution (online) or
during a pre-execution phase. In this paper we emphasize to develop an
-algorithm that selects the optimal frequency scaling factor that takes into
-consideration simultaneously the energy consumption and the performance. The
+algorithm that selects a frequency scaling factor that simultaneously takes into
+consideration the energy consumption and the performance. The
main objective of HPC systems is to run the application with less execution
-time. Therefore, our algorithm selects the optimal scaling factor online with
+time. Therefore, our algorithm selects the scaling factor online with
very small footprint. The proposed algorithm takes into account the
-communication times of the MPI programs to choose the scaling factor. This
+communication times of the MPI program to choose the scaling factor. This
algorithm has ability to predict both energy consumption and execution time over
all available scaling factors. The prediction achieved depends on some
computing time information, gathered at the beginning of the runtime. We apply
benchmarks (NPB v3.3) developed by NASA~\cite{44}. Our experiments are executed
using the simulator SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183}
over an homogeneous distributed memory architecture. Furthermore, we compare the
-proposed algorithm with Rauber's methods.
-\AG{Add citation for Rauber's methods. Moreover, Rauber was not alone to to this work (use ``Rauber et al.'', or ``Rauber and Gudula'', or \dots)}
+proposed algorithm with Rauber and Rünger methods~\cite{3}.
The comparison's results show that our
algorithm gives better energy-time trade off.
-%
-\AG{Correctly reword the following}%
-In Section~\ref{sec.relwork} we present works from other
-authors. Then, in Sections~\ref{sec.ptasks} and~\ref{sec.energy}, we
-introduce our model. [\dots] Finally, we conclude in
-Section~\ref{sec.concl}.
+This paper is organized as follows: Section~\ref{sec.relwork} presents the works
+from other authors. Section~\ref{sec.exe} shows the execution of parallel
+tasks and sources of idle times. Also, it resumes the energy
+model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance
+of MPI program. Section~\ref{sec.compet} presents the energy-performance trade offs
+objective function. Section~\ref{sec.optim} demonstrates the proposed
+energy-performance algorithm. Section~\ref{sec.expe} verifies the performance prediction
+model and presents the results of the proposed algorithm. Also, It shows the comparison results. Finally,
+we conclude in Section~\ref{sec.concl}.
\section{Related Works}
\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.
\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
-\AG{what is an ``intra-task algorithm''?}
-to choose the DVFS setting when there are dependency points
-between tasks. While in~\cite{29}, Xie et al. used breadth-first search
-algorithm to do that. Their goal is saving energy with time limits. Another
-approaches gathers and stores the runtime information for each DVFS state, then
-used their methods offline to select the suitable DVFS that optimize energy-time
-trade offs. As an example~\cite{8}, Rountree et al. used liner programming
-algorithm, while in~\cite{38,34}, Cochran et al. used multi logistic regression
-algorithm for the same goal. The offline study that shown the DVFS impact on the
-communication time of the MPI program is~\cite{17}, Freeh et al. show that these
-times not changed when the frequency is scaled down.
+during the compilation phases as for example in Azevedo et al.~\cite{40}. They
+use dynamic voltage scaling (DVS) algorithm to choose the DVS setting when there
+are dependency points between tasks. While in~\cite{29}, Xie et al. used
+breadth-first search algorithm to do that. Their goal is to save energy with
+time limits. Another approach gathers and stores the runtime information for
+each DVFS state, then selects the suitable DVFS offline to optimize energy-time
+trade offs. As an example, Rountree et al.~\cite{8} use liner programming
+algorithm, while in~\cite{38,34}, Cochran et al. use multi logistic regression
+algorithm for the same goal. The offline study that shows the DVFS impact on the
+communication time of the MPI program is~\cite{17}, where Freeh et al. show that
+these times do not change when the frequency is scaled down.
\subsection{The online DVFS orientations}
-The objective of these works is to dynamically compute and set the frequency of
-the CPU during the runtime of the program for saving energy. Estimating and
-predicting approaches for the energy-time trade offs developed by
-~\cite{11,2,31}. These works select the best DVFS setting depending on the slack
-times. These times happen when the processors have to wait for data from other
-processors to compute their task. For example, during the synchronous
-communication time that take place in the MPI programs, the processors are
-idle. The optimal DVFS can be selected using the learning methods. Therefore, in
-~\cite{39,19} used machine learning to converge to the suitable DVFS
-configuration. Their learning algorithms have big time to converge when the
-number of available frequencies is high. Also, the communication time of the MPI
-program used online for saving energy as in~\cite{1}, Lim et al. developed an
-algorithm that detects the communication sections and changes the frequency
-during these sections only. This approach changes the frequency many times
-because an iteration may contain more than one communication section. The domain
-of analytical modeling used for choosing the optimal frequency as in~\cite{3},
-Rauber et al. developed an analytical mathematical model for determining the
-optimal frequency scaling factor for any number of concurrent tasks, without
-considering communication times. They set the slowest task to maximum frequency
-for maintaining performance. In this paper we compare our algorithm with
-Rauber's model~\cite{3}, because his model can be used for any number of
-concurrent tasks for homogeneous platform and this is the same direction of this
-paper. However, the primary contributions of this paper are:
+The objective of the online DVFS orientations is to dynamically compute and set
+the frequency of the CPU for saving energy during the runtime of the
+programs. Estimating and predicting approaches for the energy-time trade offs
+are developed by Kimura, Peraza, Yu-Liang et al. ~\cite{11,2,31}. These works
+select the best DVFS setting depending on the slack times. These times happen
+when the processors have to wait for data from other processors to compute their
+task. For example, during the synchronous communications that take place in MPI
+programs, some processors are idle. The optimal DVFS can be selected using
+learning methods. Therefore, in Dhiman, Hao Shen et al. ~\cite{39,19} used
+machine learning to converge to the suitable DVFS configuration. Their learning
+algorithms take big time to converge when the number of available frequencies is
+high. Also, the communication sections of the MPI program can be used to save
+energy. In~\cite{1}, Lim et al. developed an algorithm that detects the
+communication sections and changes the frequency during these sections
+only. This approach changes the frequency many times because an iteration may
+contain more than one communication section. The domain of analytical modeling
+can also be used for choosing the optimal frequency as in Rauber and
+Rünger~\cite{3}. They developed an analytical mathematical model to determine
+the optimal frequency scaling factor for any number of concurrent tasks. They
+set the slowest task to maximum frequency for maintaining performance. In this
+paper we compare our algorithm with Rauber and Rünger model~\cite{3}, because
+their model can be used for any number of concurrent tasks for homogeneous
+platforms. The primary contributions of this paper are:
\begin{enumerate}
-\item Selecting the optimal frequency scaling factor for energy and performance
- simultaneously. While taking into account the communication time.
-\item Adapting our scale factor to taking into account the imbalanced tasks.
+\item Selecting the frequency scaling factor for simultaneously optimizing energy and performance,
+ while taking into account the communication time.
+\item Adapting our scaling factor to take into account the imbalanced tasks.
\item The execution time of our algorithm is very small when compared to other
methods (e.g.,~\cite{19}).
\item The proposed algorithm works online without profiling or training as
in~\cite{38,34}.
\end{enumerate}
+\section{Execution and Energy of Parallel Tasks on Homogeneous Platform}
+\label{sec.exe}
-\section{Parallel Tasks Execution on Homogeneous Platform}
-\label{sec.ptasks}
-
-A homogeneous cluster consists of identical nodes in terms of the hardware and
-the software. Each node has its own memory and at least one processor which can
+\subsection{Parallel Tasks Execution on Homogeneous Platform}
+A homogeneous cluster consists of identical nodes in terms of hardware and software.
+Each node has its own memory and at least one processor which can
be a multi-core. The nodes are connected via a high bandwidth network. Tasks
executed on this model can be either synchronous or asynchronous. In this paper
we consider execution of the synchronous tasks on distributed homogeneous
-platform. These tasks can exchange the data via synchronous memory passing.
+platform. These tasks can exchange the data via synchronous message passing.
\begin{figure*}[t]
\centering
- \subfloat[Sync. Imbalanced Communications]{\includegraphics[scale=0.67]{synch_tasks}\label{fig:h1}}
+ \subfloat[Sync. Imbalanced Communications]{\includegraphics[scale=0.67]{commtasks}\label{fig:h1}}
\subfloat[Sync. Imbalanced Computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
\caption{Parallel Tasks on Homogeneous Platform}
\label{fig:homo}
\end{figure*}
-\AG{On fig.~\ref{fig:h1}, how can there be a synchronization point without communications just before ?\\
-Use ``Sync.'' to abbreviate ``Synchronization''}
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 :
+lead to idle time while tasks wait at the synchronization barrier for other tasks to
+finish their communications (see figure~(\ref{fig:h1})). The imbalanced communications
+happen when nodes have to send/receive different amount of data or each node is communicates
+with different number of nodes. Another source for idle times is the imbalanced computations.
+This happens when processing different amounts of data on each processor (see figure~(\ref{fig:h2})).
+In this case the fastest tasks have to wait at the synchronization barrier for the
+slowest tasks to finish their job. In both cases the overall execution time
+of the program is the execution time of the slowest task as:
\begin{equation}
\label{eq:T1}
\textit{Program Time} = \max_{i=1,2,\dots,N} T_i
\end{equation}
-where $T_i$ is the execution time of process $i$.
+where $T_i$ is the execution time of task $i$.
-\section{Energy Model for Homogeneous Platform}
-\label{sec.energy}
+\subsection{Energy Model for Homogeneous Platform}
-The energy consumption by the processor consists of two powers metric: the
+The energy consumption by the processor consists of two power metrics: the
dynamic and the static power. This general power formulation is used by many
-researchers see~\cite{9,3,15,26}. The dynamic power of the CMOS processors
+researchers~\cite{9,3,15,26}. The dynamic power of the CMOS processors
$P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
-the supply voltage $V$ and operational frequency $f$ respectively as follow :
+the supply voltage $V$ and operational frequency $f$ respectively as follow:
\begin{equation}
\label{eq:pd}
- \textit P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
+ P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
\end{equation}
The static power $P_{static}$ captures the leakage power consumption as well as
the power consumption of peripheral devices like the I/O subsystem.
\begin{equation}
\label{eq:ps}
- \textit P_{static} = V \cdot N \cdot K_{design} \cdot I_{leak}
+ P_\textit{static} = V \cdot N \cdot K_{design} \cdot I_{leak}
\end{equation}
where V is the supply voltage, N is the number of transistors, $K_{design}$ is a
design dependent parameter and $I_{leak}$ is a technology-dependent
parameter. Energy consumed by an individual processor $E_{ind}$ is the summation
-of the dynamic and the static power multiply 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}
- \textit E_{ind} = ( P_{dyn} + P_{static} ) \cdot T
+ E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T
\end{equation}
-The dynamic voltage and frequency scaling (DVFS) is a process that allowed in
+The dynamic voltage and frequency scaling (DVFS) is a process that is allowed in
modern processors to reduce the dynamic power by scaling down the voltage and
frequency. Its main objective is to reduce the overall energy
consumption~\cite{37}. The operational frequency \emph f depends linearly on the
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}
- S = \frac{F_{max}}{F_{new}}
+ S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
-The value of the scale $S$ is greater than 1 when changing the frequency to
-any new frequency value (\emph {P-state}) in governor.
-\AG{Explain what's a 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}).
-
+The value of the scale $S$ is greater than 1 when changing the frequency to any
+new frequency value~(\emph {P-state}) in governor, the CPU governor is an
+interface driver supplied by the operating system kernel (e.g. Linux) to
+lowering core's frequency. The scaling factor is equal to 1 when the new frequency is
+set to the maximum frequency. The energy consumption model for parallel
+homogeneous platform depends on the scaling factor \emph S. This factor reduces
+quadratically the dynamic power. Also, this factor increases the static energy
+linearly because the execution time is increased~\cite{36}. The energy model
+depending on the frequency scaling factor for homogeneous platform for any
+number of concurrent tasks was developed by Rauber and Rünger~\cite{3}. This
+model considers the two power metrics for measuring the energy of the parallel
+tasks as in EQ~(\ref{eq:energy}):
\begin{equation}
\label{eq:energy}
- E = P_{dyn} \cdot S_1^{-2} \cdot
+ E = P_\textit{dyn} \cdot S_1^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
- P_{static} \cdot T_1 \cdot S_1 \cdot N
+ P_\textit{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
+where \emph N is the number of parallel nodes, $T_1 $ is the time of the slowest
task, $T_i$ is the time of the task $i$ and $S_1$ is the maximum scaling factor
for the slower task. The scaling factor $S_1$, as in EQ~(\ref{eq:s1}), selects
from the set of scales values $S_i$. Each of these scales are proportional to
the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}).
\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_{max}}{F_{new}} \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
-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}).
+where $N$ is the number of nodes. In this paper we depend on
+Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this
+model is used for homogeneous platform that we work on in this paper, and (2) we
+compare our algorithm with Rauber and Rünger scaling model. Rauber and Rünger
+scaling factor that reduce energy consumption derived from the
+EQ~(\ref{eq:energy}). They take the derivation for this equation (to be
+minimized) and set it to zero to produce the scaling factor as in
+EQ~(\ref{eq:sopt}).
\begin{equation}
\label{eq:sopt}
- \textit S_{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_{dyn}}{P_{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}
-The performance (execution time) of the parallel MPI applications are depends on
+The performance (execution time) of parallel MPI applications depend on
the time of the slowest task as in figure~(\ref{fig:homo}). Normally the
execution time of the parallel programs are proportional to the operational
-frequency. Therefore, any DVFS operation for the energy reduction increase the
+frequency. Therefore, any DVFS operation for the energy reduction increases the
execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) the
-energy affected by the scaling factor $S$. This factor also has a great impact
+energy is affected by the scaling factor $S$. This factor also has a great impact
on the performance. When scaling down the frequency to the new value according
-to EQ~(\ref{eq:s}) lead to the value of the scale $S$ has inverse relation with
-new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decrease the
-frequency value, the execution time increase. Then the new frequency value has
-inverse relation with time ($F_{new} \propto \frac{1}{T}$). This lead to the
+to EQ~(\ref{eq:s}), the value of the scale $S$ has inverse relation with
+new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decreasing the
+frequency value, the execution time increases. Then the new frequency value has
+inverse relation with time ($F_{new} \propto \frac{1}{T}$). This leads to the
frequency scaling factor $S$ proportional linearly with execution time ($S
\propto T$). Large scale MPI applications such as NAS benchmarks have
considerable amount of communications embedded in these programs. During the
-communication process the processor remain idle until the communication has
+communication process the processors remain idle until the communication has
finished. For that reason any change in the frequency has no impact on the time
of communication but it has obvious impact on the time of
-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}).
+computation~\cite{17}. To predict the execution time of MPI program, the communication time and
+the computation time for the slower task must be first precisely specified. Secondly,
+these times are used to predict the execution time for any MPI program as a function of
+the new scaling factor as in EQ~(\ref{eq:tnew}).
\begin{equation}
\label{eq:tnew}
- \textit T_{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 an
-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}
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 :
+without scaled frequency:
\begin{multline}
\label{eq:enorm}
- \textit E_{Norm} = \frac{\textit E_{Reduced}}{\textit E_{Original}} \\
- {} = \frac{ P_{dyn} \cdot S_i^{-2} \cdot
+ E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
+ {} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_{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 }
+ 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}
-\AG{Use \texttt{\textbackslash{}text\{xxx\}} or
- \texttt{\textbackslash{}textit\{xxx\}} for all subscripted words in equations
- (e.g. \mbox{\texttt{E\_\{\textbackslash{}text\{Norm\}\}}}).
-
- Don't hesitate to define new commands :
- \mbox{\texttt{\textbackslash{}newcommand\{\textbackslash{}ENorm\}\{E\_\{\textbackslash{}text\{Norm\}\}\}}}
-}
-By the same way we can normalize the performance as follows :
+By the same way we can normalize the performance as follows:
\begin{equation}
\label{eq:pnorm}
-\textit P_{Norm} = \frac{\textit T_{New}}{\textit T_{Old}}
- = \frac{T_{\textit{Max Comp Old}} \cdot S +
- T_{\textit{Max Comm Old}}}{\textit T_{Old}}
+ P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}
+ = \frac{T_\textit{Max Comp Old} \cdot S +
+ T_\textit{Max Comm Old}}{ T_\textit{Old}}
\end{equation}
The second problem is the optimization operation for both energy and performance
is not in the same direction. In other words, the normalized energy and the
performance curves are not in the same direction see figure~(\ref{fig:r2}).
While the main goal is to optimize the energy and performance in the same
-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
-performance as follows :
+performance as follows:
\begin{equation}
\label{eq:pnorm_en}
-\textit P^{-1}_{Norm} = \frac{\textit T_{Old}}{\textit T_{New}}
- = \frac{\textit T_{Old}}{T_{\textit{Max Comp Old}} \cdot S +
- T_{\textit{Max Comm Old}}}
+ P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
+ = \frac{ T_\textit{Old}}{T_\textit{Max Comp Old} \cdot S +
+ T_\textit{Max Comm Old}}
\end{equation}
\begin{figure*}
\centering
\subfloat[Converted Relation.]{%
- \includegraphics[width=.33\textwidth]{file.eps}\label{fig:r1}}%
+ \includegraphics[width=.4\textwidth]{file.eps}\label{fig:r1}}%
\qquad%
\subfloat[Real Relation.]{%
- \includegraphics[width=.33\textwidth]{file3.eps}\label{fig:r2}}
+ \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{\textit P^{-1}_{Norm}}^{\text{Maximize}} -
- \overbrace{\textit E_{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
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
+objective of this paper and we choose Rauber and Rünger model as an example with two
reasons that mentioned before.
\section{Optimal Scaling Factor for Performance and Energy}
\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 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
+ \State - Calculate all available scales $S_i$ depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}).
+ \State - Select the maximum scale factor $S_1$ from the set\par\hspace{1 pt} of scales $S_i$.
+ \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$
+ \par\hspace{1 pt} as in EQ~(\ref{eq:enorm}).
+ \State - Calculate the normalize inverse of performance\par\hspace{1 pt}
$P_{NormInv}=T_{old}/T_{new}$ as in EQ~(\ref{eq:pnorm_en}).
\If{ $(P_{NormInv}-E_{Norm} > Dist$) }
\State $S_{optimal} = S$
\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}}
+%\AG{Use the same format as for Algorithm~\ref{$EPSA$}}
\begin{algorithm}[tp]
\caption{DVFS}
\label{dvfs}
- \begin{algorithmic}
- \For {$J:=1$ to $Some-Iterations \; $}
+ \begin{algorithmic}[1]
+ \For {$K:=1$ to $Some-Iterations \; $}
\State -Computations Section.
\State -Communications Section.
- \If {$(J==1)$}
- \State -Gather all times of computation and communication from\par each node.
+ \If {$(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
-can calculate the new frequency $F_i$ as follows :
+value $T_i$. By substitution of EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we
+can calculate the new frequency $F_i$ as follows:
\begin{equation}
\label{eq:fi}
- F_i = \frac{F_{max} \cdot T_i}{S_{optimal} \cdot T_{max}}
+ F_i = \frac{F_\textit{max} \cdot T_i}{S_\textit{optimal} \cdot T_\textit{max}}
\end{equation}
According to this equation all the nodes may have the same frequency value if
they have balanced workloads. Otherwise, they take different frequencies when
\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}{ | l | l | l |l | l |l |l | p{2cm} |}
+ \begin{tabular}{|*{7}{l|}}
\hline
Max & Min & Backbone & Backbone&Link &Link& Sharing \\
Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy \\ \hline
- 2.5 &800 & 2.25 GBps &$5\times 10^{-7} s$& 1 GBps & $5\times 10^{-5} s$ &Full \\
+ \np{2.5} & \np{800} & \np[GBps]{2.25} &\np[$\mu$s]{0.5}& \np[GBps]{1} & \np[$\mu$s]{50} &Full \\
GHz& MHz& & & & &Duplex \\\hline
\end{tabular}
\label{table:platform}
\end{table}
-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
+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 lead to more energy savings (e.g. CG and FT), see
+curves is big. This leads to more energy savings (e.g. CG and FT), see
table~(\ref{table:factors results}). All discovered frequency scaling factors
optimize both the energy and the performance simultaneously for all the NAS
programs. In table~(\ref{table:factors results}), we record all optimal scaling
\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]
- \caption{Optimal Scaling Factors Results}
+ \caption{The EPSA Scaling Factors Results}
% title of Table
\centering
- \AG{Use the same number of decimals for all numbers in a column,
- and vertically align the numbers along the decimal points.
- The same for all the following tables.}
- \begin{tabular}{ | l | l | l |l | l | p{2cm} |}
+ \begin{tabular}{|l|*{4}{r|}}
\hline
Program & Optimal & Energy & Performance&Energy-Perf.\\
Name & Scaling Factor& Saving \%&Degradation \% &Distance \\ \hline
\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's
-methods~\cite{3}. He had two scenarios, the first is to reduce energy to optimal
-level without considering the performance as in EQ~(\ref{eq:sopt}). We refer to
-this scenario as $Rauber_{E}$. The second scenario is similar to the first
+In this section, we compare our EPSA algorithm results with Rauber and Rünger
+methods~\cite{3}. They had two scenarios, the first is to reduce energy to
+optimal level without considering the performance as in EQ~(\ref{eq:sopt}). We
+refer to this scenario as $R_{E}$. The second scenario is similar to the first
except setting the slower task to the maximum frequency (when the scale $S=1$)
to keep the performance from degradation as mush as possible. We refer to this
-scenario as $Rauber_{E-P}$. The comparison is made in tables~(\ref{table:compare
+scenario as $R_{E-P}$. The comparison is made in tables~(\ref{table:compare
Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
-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*}[p]
+tables show the results of our EPSA and Rauber and Rünger scenarios for all the
+NAS benchmarks programs for classes A,B and C.
+\begin{table}[p]
\caption{Comparing Results for The NAS Class A}
% title of Table
\centering
- \begin{tabular}{ | l | l | l |l | l | l| }
+ \begin{tabular}{|l|l|*{4}{r|}}
\hline
Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- name &name&value& Saving \%&Degradation \% &Distance
+ Name &Name&Value& Saving \%&Degradation \% &Distance
\\ \hline
% \rowcolor[gray]{0.85}
- EPSA&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
- $Rauber_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline
- $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
+ $EPSA$&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
+ $R_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline
+ $R_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
- EPSA&MG & 1.47 &27.66&16.82&10.84\\ \hline
- $Rauber_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
- $Rauber_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline
+ $EPSA$&MG & 1.47 &27.66&16.82&10.84\\ \hline
+ $R_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
+ $R_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline
- EPSA&EP &1.19 &25.32&20.79&4.53\\ \hline
- $Rauber_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
- $Rauber_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline
+ $EPSA$&EP &1.19 &25.32&20.79&4.53\\ \hline
+ $R_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
+ $R_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline
- EPSA&LU&1.56& 39.55 &19.38& 20.17\\ \hline
- $Rauber_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline
- $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
+ $EPSA$&LU&1.56& 39.55 &19.38& 20.17\\ \hline
+ $R_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline
+ $R_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
- EPSA&BT&1.31& 29.60&20.53&9.07 \\ \hline
- $Rauber_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline
- $Rauber_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline
+ $EPSA$&BT&1.31& 29.60&20.53&9.07 \\ \hline
+ $R_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline
+ $R_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline
- EPSA&SP&1.38& 33.51&15.65&17.86 \\ \hline
- $Rauber_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline
- $Rauber_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
+ $EPSA$&SP&1.38& 33.51&15.65&17.86 \\ \hline
+ $R_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline
+ $R_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
- EPSA&FT&1.25&25.00&10.80&14.20 \\ \hline
- $Rauber_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
- $Rauber_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline
+ $EPSA$&FT&1.25&25.00&10.80&14.20 \\ \hline
+ $R_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
+ $R_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline
\end{tabular}
\label{table:compare Class A}
% is used to refer this table in the text
-\end{table*}
-\begin{table*}[p]
+\end{table}
+\begin{table}[p]
\caption{Comparing Results for The NAS Class B}
% title of Table
\centering
- \begin{tabular}{ | l | l | l |l | l |l| }
+ \begin{tabular}{|l|l|*{4}{r|}}
\hline
Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- name &name&value& Saving \%&Degradation \% &Distance
+ Name &Name&Value& Saving \%&Degradation \% &Distance
\\ \hline
% \rowcolor[gray]{0.85}
- EPSA&CG & 1.66 &39.23&16.63&22.60 \\ \hline
- $Rauber_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline
- $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
+ $EPSA$&CG & 1.66 &39.23&16.63&22.60 \\ \hline
+ $R_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline
+ $R_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
- EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline
- $Rauber_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
- $Rauber_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
+ $EPSA$ &MG & 1.47 &34.98&18.35&16.63\\ \hline
+ $R_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
+ $R_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
- EPSA&EP &1.08 &20.29&17.15&3.14 \\ \hline
- $Rauber_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline
- $Rauber_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline
+ $EPSA$&EP &1.08 &20.29&17.15&3.14 \\ \hline
+ $R_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline
+ $R_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline
- EPSA&LU&1.47&38.57&21.34&17.23 \\ \hline
- $Rauber_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline
- $Rauber_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline
+ $EPSA$&LU&1.47&38.57&21.34&17.23 \\ \hline
+ $R_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline
+ $R_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline
- EPSA&BT&1.31& 29.59&20.88&8.71\\ \hline
- $Rauber_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline
- $Rauber_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline
+ $EPSA$&BT&1.31& 29.59&20.88&8.71\\ \hline
+ $R_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline
+ $R_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline
- EPSA&SP&1.38&33.44&19.24&14.20 \\ \hline
- $Rauber_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline
- $Rauber_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
+ $EPSA$&SP&1.38&33.44&19.24&14.20 \\ \hline
+ $R_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline
+ $R_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
- EPSA&FT&1.38&34.40&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&37.90&5.14\\ \hline
+ $EPSA$&FT&1.38&34.40&14.57&19.83 \\ \hline
+ $R_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
+ $R_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline
\end{tabular}
\label{table:compare Class B}
% is used to refer this table in the text
-\end{table*}
+\end{table}
-\begin{table*}[p]
+\begin{table}[p]
\caption{Comparing Results for The NAS Class C}
% title of Table
\centering
- \begin{tabular}{ | l | l | l |l | l |l| }
+ \begin{tabular}{|l|l|*{4}{r|}}
\hline
Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- name &name&value& Saving \%&Degradation \% &Distance
+ 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&26.70&18.66\\ \hline
+ $EPSA$&CG & 1.56 &39.23&14.88&24.35 \\ \hline
+ $R_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline
+ $R_{E}$&CG &2.15 &45.36&26.70&18.66\\ \hline
- EPSA&MG & 1.47 &34.97&21.69&13.27\\ \hline
- $Rauber_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline
- $Rauber_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
+ $EPSA$&MG & 1.47 &34.97&21.69&13.27\\ \hline
+ $R_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline
+ $R_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
- EPSA&EP &1.04 &22.14&20.73&1.41 \\ \hline
- $Rauber_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline
- $Rauber_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline
+ $EPSA$&EP &1.04 &22.14&20.73&1.41 \\ \hline
+ $R_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline
+ $R_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline
- EPSA&LU&1.38&35.83&22.49&13.34 \\ \hline
- $Rauber_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline
- $Rauber_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline
+ $EPSA$&LU&1.38&35.83&22.49&13.34 \\ \hline
+ $R_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline
+ $R_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline
- EPSA&BT&1.31& 29.60&21.28&8.32\\ \hline
- $Rauber_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline
- $Rauber_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline
+ $EPSA$&BT&1.31& 29.60&21.28&8.32\\ \hline
+ $R_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline
+ $R_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline
- EPSA&SP&1.38&33.48&21.35&12.12\\ \hline
- $Rauber_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline
- $Rauber_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline
+ $EPSA$&SP&1.38&33.48&21.35&12.12\\ \hline
+ $R_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline
+ $R_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline
- EPSA&FT&1.47&34.72&19.00&15.72 \\ \hline
- $Rauber_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
- $Rauber_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline
+ $EPSA$&FT&1.47&34.72&19.00&15.72 \\ \hline
+ $R_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
+ $R_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline
\end{tabular}
\label{table:compare Class C}
% is used to refer this table in the text
-\end{table*}
+\end{table}
As shown in these tables our scaling factor is not optimal for energy saving
-such as Rauber's scaling factor EQ~(\ref{eq:sopt}), but it is optimal for both
+such as Rauber and Rünger scaling factor EQ~(\ref{eq:sopt}), but it is optimal for both
the energy and the performance simultaneously. Our EPSA optimal scaling factors
has better simultaneous optimization for both the energy and the performance
-compared to Rauber's energy-performance method ($Rauber_{E-P}$). Also, in
-($Rauber_{E-P}$) method when setting the frequency to maximum value for the
+compared to Rauber and Rünger energy-performance method ($R_{E-P}$). Also, in
+($R_{E-P}$) method when setting the frequency to maximum value for the
slower task lead to a small improvement of the performance. Also the results
show that this method keep or improve energy saving. Because of the energy
consumption decrease when the execution time decreased while the frequency value
Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving
percent and the performance degradation percent. Therefore, this means it is the
same resultant of our objective function EQ~(\ref{eq:max}). Our algorithm always
-gives positive energy to performance trade offs while Rauber's method
-($Rauber_{E-P}$) gives in some time negative trade offs such as in BT and
+gives positive energy to performance trade offs while Rauber and Rünger method
+($R_{E-P}$) gives in some time negative trade offs such as in BT and
EP. The positive trade offs with highest values lead to maximum energy savings
concatenating with less performance degradation and this the objective of this
paper. While the negative trade offs refers to improving energy saving (or may
\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's Methods}
+ \caption {Comparing Our EPSA with Rauber and Rünger Methods}
\label{fig:compare}
\end{figure}
-
\section{Conclusion}
\label{sec.concl}
-
-\AG{the conclusion needs to be written\dots{} one day}
+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 work.
+
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-% LocalWords: Badri Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords: CMOS EQ EPSA Franche Comté Tflop
+% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
+% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger
