\newcommand{\Pdyn}{\Xsub{P}{dyn}}
\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
\newcommand{\Pnorm}{\Xsub{P}{Norm}}
+\newcommand{\Tnorm}{\Xsub{T}{Norm}}
\newcommand{\Pstates}{\Xsub{P}{states}}
\newcommand{\Pstatic}{\Xsub{P}{static}}
\newcommand{\Sopt}{\Xsub{S}{opt}}
predict both energy consumption and execution time over all available scaling
factors. The prediction achieved depends on some computing time information,
gathered at the beginning of the runtime. We apply this algorithm to the NAS parallel benchmarks (NPB v3.3)~\cite{44}. Our experiments are executed using the simulator
-SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over an homogeneous
+SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over a homogeneous
distributed memory architecture. Furthermore, we compare the proposed algorithm
with Rauber and Rünger methods~\cite{3}. The comparison's results show that our
algorithm gives better energy-time trade-off.
% paper in homogeneous clusters}
-\section{Energy model for an homogeneous platform}
+\section{Energy model for a homogeneous platform}
\label{sec.exe}
Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
two power metrics: the static and the dynamic power. While the first one is
\section{Performance evaluation of MPI programs}
\label{sec.mpip}
-The performance (execution time) of parallel synchronous MPI applications depends
+The execution time of parallel synchronous MPI applications depends
on the time of the slowest task. If there is no
communication and the application is not data bounded, the execution time of a
parallel program is linearly proportional to the operational frequency and any
This section presents our approach for choosing the optimal scaling factor.
This factor gives maximum energy reduction while taking into account the execution
-times for both computation and communication. The relation between the performance
-and the energy is nonlinear and complex. Thus, unlike the relation between the performance and the scaling factor, the relation of energy with the scaling factor is nonlinear, for more details refer to~\cite{17}. Moreover, they are not measured using the same metric. To
+times for both computation and communication. The relation between the execution time
+and the energy is nonlinear and complex. Thus, unlike the relation between the execution time and the scaling factor, the relation of energy with the scaling factor is nonlinear, for more details refer to~\cite{17}. Moreover, they are not measured using the same metric. To
solve this problem, we normalize the energy by calculating the ratio between
the consumed energy with scaled frequency and the consumed energy without scaled
frequency:
\Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
\Pstatic \cdot T_1 \cdot N }
\end{multline}
-In the same way we can normalize the performance as follows:
+In the same way we can normalize the time as follows:
\begin{equation}
\label{eq:pnorm}
- \Pnorm = \frac{\Tnew}{\Told}
+ \Tnorm = \frac{\Tnew}{\Told}
= \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{
\TmaxCompOld + \TmaxCommOld}
\end{equation}
The second problem is that 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 at the same direction see
+execution time is not in the same direction. In other words, the normalized energy
+and the execution time curves are not at the same direction see
Figure~\ref{fig:rel}\subref{fig:r2}. While the main goal is to optimize the
-energy and performance in the same time. According to the
+energy and execution time in the same time. According to the
equations~\eqref{eq:enorm} and~\eqref{eq:pnorm}, the scaling factor $S$ reduce
-both the energy and the performance simultaneously. But the main objective is
-to produce maximum energy reduction with minimum performance reduction. Many
+both the energy and the execution time simultaneously. But the main objective is
+to produce maximum energy reduction with minimum execution time reduction. Many
researchers used different strategies to solve this nonlinear problem for
example see~\cite{19,42}, their methods add big overheads to the algorithm to
select the suitable frequency. In this paper we present a method to find the
-optimal scaling factor $S$ to optimize both energy and performance
+optimal scaling factor $S$ to optimize both energy and execution time
simultaneously without adding a big overhead. Our solution for this problem is
-to make the optimization process for energy and performance follow the same
-direction. Therefore, we inverse the equation of the normalized performance as
+to make the optimization process for energy and execution time follow the same
+direction. Therefore, we inverse the equation of the normalized execution time as
follows:
\begin{equation}
\label{eq:pnorm_en}
- \Pnorm^{-1} = \frac{ \Told}{ \Tnew}
+ \Pnorm = \frac{ \Told}{ \Tnew}
= \frac{\TmaxCompOld +
\TmaxCommOld}{\TmaxCompOld \cdot S +
\TmaxCommOld}
\label{fig:rel}
\end{figure}
Then, we can model our objective function as finding the maximum distance
-between the energy curve EQ~\eqref{eq:enorm} and the inverse of performance
+between the energy curve EQ~\eqref{eq:enorm} and the inverse of execution time (performance)
curve EQ~\eqref{eq:pnorm_en} over all available scaling factors. This
represents the minimum energy consumption with minimum execution time (better
performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}. Then
\begin{equation}
\label{eq:max}
\MaxDist = \max_{j=1,2,\dots,F}
- (\overbrace{\Pnorm^{-1}(S_j)}^{\text{Maximize}} -
+ (\overbrace{\Pnorm(S_j)}^{\text{Maximize}} -
\overbrace{\Enorm(S_j)}^{\text{Minimize}} )
\end{equation}
where $F$ is the number of available frequencies. Then we can select the optimal
\Pdyn \cdot
\left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
\Pstatic \cdot T_1 \cdot N }$
- \State $\PnormInv \gets \Told / \Tnew$
- \If{$(\PnormInv - \Enorm > \Dist)$}
+ \State $\Pnorm \gets \Told / \Tnew$
+ \If{$(\Pnorm - \Enorm > \Dist)$}
\State $\Sopt \gets S$
- \State $\Dist \gets \PnormInv - \Enorm$
+ \State $\Dist \gets \Pnorm - \Enorm$
\EndIf
\EndFor
\State Return $\Sopt$
from \np[GHz]{2.5} to \np[MHz]{800} with \np[MHz]{100} difference between each
two successive frequencies. The nodes are connected via an ethernet network with 1Gbit/s bandwidth.
-\subsection{Performance prediction verification}
+\subsection{Execution time prediction verification}
-In this section we evaluate the precision of our performance prediction method
+In this section we evaluate the precision of our execution time prediction method
based on EQ~\eqref{eq:tnew} by applying it to the NAS benchmarks. The NAS programs
are executed with the class B option to compare the real execution time with
the predicted execution time. Each program runs offline with all available
different scaling factors for each program depending on the communication
features of the program as in the plots from Figure~\ref{fig:nas}. These plots
illustrate that there are different distances between the normalized energy and
-the normalized inverted performance curves, because there are different
+the normalized inverted execution time curves, because there are different
communication features for each benchmark. When there are little or no
-communications, the inverted performance curve is very close to the energy
+communications, the inverted execution time curve is very close to the energy
curve. Then the distance between the two curves is very small. This leads to
small energy savings. The opposite happens when there are a lot of
communication, the distance between the two curves is big. This leads to more
energy savings (e.g. CG and FT), see Table~\ref{table:compareC}. All discovered
-frequency scaling factors optimize both the energy and the performance
+frequency scaling factors optimize both the energy and the execution time
simultaneously for all NAS benchmarks. In Table~\ref{table:compareC}, we record
all optimal scaling factors results for each benchmark running class C. These
scaling factors give the maximum energy saving percentage and the minimum
In this section, we compare our scaling factor selection method with Rauber and
Rünger methods~\cite{3}. They had two scenarios, the first is to reduce energy
-to the optimal level without considering the performance as in
+to the optimal level without considering the execution time as in
EQ~\eqref{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
In this paper, we have presented a new online scaling factor selection method
that optimizes simultaneously the energy and performance of a distributed
-application running on an homogeneous cluster. It uses the computation and
+application running on a homogeneous cluster. It uses the computation and
communication times measured at the first iteration to predict energy
-consumption and the performance of the parallel application at every available
+consumption and the execution time of the parallel application at every available
frequency. Then, it selects the scaling factor that gives the best trade-off
between energy reduction and performance which is the maximum distance between
-the energy and the inverted performance curves. To evaluate this method, we
+the energy and the inverted eexecution time curves. To evaluate this method, we
have applied it to the NAS benchmarks and it was compared to Rauber and Rünger
methods while being executed on the simulator SimGrid. The results showed that
our method, outperforms Rauber and Rünger's methods in terms of energy-performance
