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+\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\small\textbf{AG:} #2}}
+
\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
+\AG{``Optimal'' is a bit pretentious in the title}
+
+\begin{abstract}
+ \AG{FIXME}
+\end{abstract}
+
\section{Introduction}
The need for computing power is still increasing and it is not expected to slow
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.
+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. The comparison's results show that our
+algorithm gives better energy-time trade off.
\section{Related Works}
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
+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 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},
+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
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 $i$.
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
+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
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]
\section{Performance Evaluation of MPI Programs}
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
+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
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
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
+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}).
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.
+SimGrid/SMPI v3.10 to run the NAS programs.
+\AG{Fig.~\ref{fig:pred} is hard to read when printed in black and white,
+ especially the ``Normalize Real Perf.'' curve.}
\begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
\centering
\includegraphics[scale=0.60]{cg_per.eps}
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
+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}).
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_{Original}}
+ = \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}
+\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 :
-\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
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
+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.
\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})$
+ \State Return $S_{optimal}$
\end{algorithmic}
\end{algorithm}
The proposed EPSA algorithm works online during the execution time of the MPI
program. The following example shows where and when the EPSA algorithm is called
in the MPI program :
\begin{minipage}{\textwidth}
+\AG{Use the same format as for Algorithm~\ref{EPSA}}
\begin{lstlisting}[frame=tb]
FOR J:=1 to Some_iterations Do
-Computations Section.
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.
+the frequency 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
+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
+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.
+detailed characteristics of our platform file are shown in the
+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.
\begin{table}[ht]
\caption{Platform File Parameters}
% title of Table
\centering
+ \AG{Use e.g. $5\times 10^{-7}$ instead of 5E-7}
\begin{tabular}{ | l | l | l |l | l |l |l | p{2cm} |}
\hline
Max & Min & Backbone & Backbone&Link &Link& Sharing \\
\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
+power static is equal to 4W for all experiments. 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
\caption{Optimal 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} |}
\hline
Program & Optimal & Energy & Performance&Energy-Perf.\\
\label{fig:compare}
\end{figure}
+\AG{\texttt{bibtex} gives many errors, please correct them}
\bibliographystyle{plain}
\bibliography{my_reference}
\end{document}
%%% fill-column: 80
%%% ispell-local-dictionary: "american"
%%% End:
+
+% LocalWords: Badri Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
+% LocalWords: CMOS EQ EPSA
