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\begin{document}
-\begin{center}
- \Large
- \title*\textbf{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
-\end{center}
-\parskip 0pt
-\linespread{1.18}
-\normalsize
-\makeatletter
-\renewcommand*{\@seccntformat}[1]{\csname the#1\endcsname\hspace{0.01cm}}
-\makeatother
-\sectionfont{\large}
-
-\section{.~Introduction }
+
+\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
memory architecture. Furthermore, we compare the proposed algorithm with
Rauber's methods. The comparison's results show that our algorithm gives better
energy-time trade off.
-\sectionfont{\large}
-\section{.~Related Works }
+\section{Related Works}
In the this section some heuristics, to compute the scaling factor, are
presented and classified in two parts : offline and online methods.
- \sectionfont{\large}
-\subsection{~The offline DVFS orientations}
+\subsection{The offline DVFS orientations}
The DVFS offline methods are static and are not executed during the runtime of
the program. Some approaches used heuristics to select the best DVFS state
algorithm for the same goal. The offline study that shown the DVFS impact on the
communication time of the MPI program is~\cite{17}, Freeh et al. show that these
times not changed when the frequency is scaled down.
-\sectionfont{\large}
-\subsection{~The online DVFS orientations}
+\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
\item The proposed algorithm works online without profiling or training as
in~\cite{38,34}.
\end{enumerate}
-\sectionfont{\large}
-\section{.~Parallel Tasks Execution on Homogeneous Platform}
+\section{Parallel Tasks Execution on Homogeneous Platform}
A homogeneous cluster consists of identical nodes in terms of the hardware and
the software. Each node has its own memory and at least one processor which can
this case the fastest tasks have to wait at the synchronous barrier for the
slowest tasks to finish their job. In both two cases the overall execution time
of the program is the execution time of the slowest task as :
-\begin{equation} \label{eq:T1}
- Program Time=MAX_{i=1,2,..,N} (T_i) \hfill
+\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 $i$.
-\sectionfont{\large}
-\section{.~Energy Model for Homogeneous Platform}
+\section{Energy Model for Homogeneous Platform}
The energy consumption by the processor consists of two powers metric: the
dynamic and the static power. This general power formulation is used by many
researchers see ~\cite{9,3,15,26}. The dynamic power of the CMOS processors
$P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
the supply voltage $V$ and operational frequency $f$ respectively as follow :
-\begin{equation} \label{eq:pd}
- \displaystyle P_{dyn} = \alpha . C_L . V^2 . f
+\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}
- \displaystyle P_{static} = V . N . K_{design} . I_{leak}
+\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}
- \displaystyle E_{ind} = (P_{dyn} + P_{static} ) . T
+\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
various frequency values in~\cite{3}. The reduction process of the frequency are
expressed by scaling factor \emph S. The scale \emph S is the ratio between the
maximum and the new frequency as in EQ~(\ref{eq:s}).
-\begin{equation} \label{eq:s}
- S=\:\frac{F_{max}}{F_{new}} \hfill \newline
+\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
+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
consider the two powers metric for measuring the energy of the parallel tasks as
in EQ~(\ref{eq:energy}).
-\begin{equation} \label{eq:energy}
- E= \displaystyle \;P_{dyn}\,.\,S_1^{-2}\;.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_1\;\,.\,N
+\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
for the slower task. The scaling factor $S_1$, as in EQ~(\ref{eq:s1}), selects
from the set of scales values $S_i$. Each of these scales are proportional to
the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}).
-\begin{equation} \label{eq:s1}
- S_1=MAX_{i=1,2,..,F} (S_i) \hfill
+\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\: .\:(\frac{T_1}{T_i})=\: (\frac{F_{max}}{F_{new}}).(\frac{T_1}{T_i}) \hfill
+\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
optimal energy reduction derived from the EQ~(\ref{eq:energy}). He takes the
derivation for this equation (to be minimized) and set it to zero to produce the
scaling factor as in EQ~(\ref{eq:sopt}).
-\begin{equation} \label{eq:sopt}
- S_{opt}= {\sqrt [3~]{\frac{2}{n} \frac{P_{dyn}}{P_{static}} \Big(1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^3}\Big) }} \hfill
+\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}
-%[\Big 3]
-\sectionfont{\large}
-\section{.~Performance Evaluation of MPI Programs}
+\section{Performance Evaluation of MPI Programs}
The performance (execution time) of the parallel MPI applications are depends on
the time of the slowest task as in figure~(\ref{fig:homo}). Normally the
slower task. Secondly, we use these times for predicting the execution time for
any MPI program as a function of the new scaling factor as in the
EQ~(\ref{eq:tnew}).
-\begin{equation} \label{eq:tnew}
- \displaystyle T_{new}= T_{Max \:Comp \:Old} \; . \:S \;+ \;T_{Max\: Comm\: Old}
- \hfill
+\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
value. Depending on this prediction we can produce our energy-performace scaling
method as we will show in the coming sections. In the next section we make an
investigation study for the EQ~(\ref{eq:tnew}).
-\sectionfont{\large}
-\section{.~Performance Prediction Verification }
+\section{Performance Prediction Verification}
In this section we evaluate the precision of our performance prediction methods
on the NAS benchmark. We use the EQ~(\ref{eq:tnew}) that predicts the execution
the real time (Simgrid time) for all programs is between 0.0032 to 0.0133. AS an
example, we are present the execution times of the NAS benchmarks as in the
figure~(\ref{fig:pred}).
-\sectionfont{\large}
-\section{.~Performance to Energy Competition}
+\section{Performance to Energy Competition}
This section demonstrates our approach for choosing the optimal scaling
factor. This factor gives maximum energy reduction taking into account the
execution time for both computation and communication times . The relation
For solving this problem, we normalize the energy by calculating the ratio
between the consumed energy with scaled frequency and the consumed energy
without scaled frequency :
-\begin{equation} \label{eq:enorm}
- E_{Norm}=\displaystyle\frac{E_{Reduced}}{E_{Orginal}}= \frac{\displaystyle \;P_{dyn}\,.\,S_i^{-2}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_i\;\,.\,N }{\displaystyle \;P_{dyn}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,\,.\,N }
+\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}=\displaystyle \frac{T_{New}}{T_{Old}}=\frac{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}}{T_{Old}} \;\;
+\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
overheads. Our solution for this problem is to make the optimization process
have the same direction. Therefore, we inverse the equation of normalize
performance as follows :
-\begin{equation} \label{eq:pnorm_en}
- \displaystyle P^{-1}_{Norm}= \frac{T_{Old}}{T_{New}}=\frac{T_{Old}}{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}}
+\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
the minimum energy consumption with minimum execution time (better performance)
in the same time, see figure~(\ref{fig:r1}). Then our objective function has the
following form:
-\begin{equation} \label{eq:max}
- \displaystyle MaxDist = Max \;(\;\overbrace{P^{-1}_{Norm}}^{Maximize}\; -\; \overbrace{E_{Norm}}^{Minimize} \;)
+\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
factor as shown in ~\cite{15,3,19}. Energy measurement model is not the
objective of this paper and we choose Rauber's model as an example with two
reasons that mentioned before.
-\sectionfont{\large}
-\section{.~Optimal Scaling Factor for Performance and Energy }
+\section{Optimal Scaling Factor for Performance and Energy}
In the previous section we described the objective function that satisfy our
goal in discovering optimal scaling factor for both performance and energy at
the same time. Therefore, we develop an energy to performance scaling algorithm
(EPSA). This algorithm is simple and has a direct way to calculate the optimal
-scaling factor for both energy and performance at the same time. \clearpage
-\linespread{1}
-\begin{algorithm}[width=\textwidth,height=\textheight,keepaspectratio]
+scaling factor for both energy and performance at the same time.
+\begin{algorithm}[t]
\caption{EPSA}
\label{EPSA}
\begin{algorithmic}[1]
\State - Calculate all available scales $S_i$ depend on $S$ as in EQ~(\ref{eq:si}).
\State - Select the maximum scale factor $S_1$ from the set of scales $S_i$.
\State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ as in EQ~(\ref{eq:enorm}).
- \State - Calculate the normalize inverse of performance $P_{NormInv}=T_{old}/T_{new}$
-
- as in EQ~(\ref{eq:pnorm_en}).
- \If{ $(P_{NormInv}-E_{Norm}$ $>$ $Dist$) }
- \State $S_{optimal}=S$
+ \State - 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}
-\linespread{1.2} The proposed EPSA algorithm works online during the execution
-time of the MPI program. It selects the optimal scaling factor by gathering some
-information from the program after one iteration. This algorithm has small
-execution time (between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ for 32
-nodes). The data required by this algorithm is the computation time and the
-communication time for each task from the first iteration only. When these times
-are measured, the MPI program calls the EPSA algorithm to choose the new
-frequency using the optimal scaling factor. Then the program set the new
-frequency to the system. The algorithm is called just one time during the
-execution of the program. The following example shows where and when the EPSA
-algorithm is called in the MPI program : \clearpage
-\begin{lstlisting}
+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.
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} \; . \;T_i}{S_{optimal} \; . \;T_{max}} \hfill
+\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.
-\sectionfont{\large}
-\section{.~Experimental Results}
+\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
\caption{Optimal scaling factors for The NAS MPI Programs}
\label{fig:nas}
\end{figure}
-\linespread{1.1}
\begin{table}[width=\textwidth,height=\textheight,keepaspectratio]
\caption{Optimal Scaling Factors Results}
% title of Table
\label{table:factors results}
% is used to refer this table in the text
\end{table}
-\linespread{1.2}
As shown in the table~(\ref{table:factors results}), when the optimal scaling
factor has big value we can gain more energy savings for example as in CG and
cases. In EP there are no communications inside the iterations. This make our
EPSA to selects smaller scaling factor values (inducing smaller energy savings).
-% \clearpage
-\sectionfont{\large}
-
-\section{.~Comparing Results}
+\section{Comparing Results}
In this section, we compare our EPSA algorithm results with Rauber's
methods~\cite{3}. He had two scenarios, the first is to reduce energy to optimal
Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
tables show the results of our EPSA and Rauber's two scenarios for all the NAS
benchmarks programs for classes A,B and C.
-%\linespread{1}
\begin{table}[ht]
\caption{Comparing Results for The NAS Class A}
% title of Table
\label{table:compare Class C}
% is used to refer this table in the text
\end{table}
-%\linespread{1.2}
-\clearpage As shown in these tables our scaling factor is not optimal for energy
-saving such as Rauber's scaling factor EQ~(\ref{eq:sopt}), but it is optimal for
-both the energy and the performance simultaneously. Our EPSA optimal scaling
-factors has better simultaneous optimization for both the energy and the
-performance compared to Rauber's energy-performance method
-($Rauber_{E-P}$). Also, in ($Rauber_{E-P}$) method when setting the frequency to
-maximum value for the slower task lead to a small improvement of the
-performance. Also the results show that this method keep or improve energy
-saving. Because of the energy consumption decrease when the execution time
-decreased while the frequency value increased.
+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
\label{fig:compare}
\end{figure}
-\clearpage
\bibliographystyle{plain}
\bibliography{my_reference}
\end{document}
