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
-Figure~(\ref{fig:r2}). While the main goal is to optimize the energy and
-performance in the same time. According to the equations~(\ref{eq:enorm})
-and~(\ref{eq:pnorm}), the scaling factor $S$ 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 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 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 follows:
+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
+equations~(\ref{eq:enorm}) and~(\ref{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
+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
+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
+follows:
\begin{equation}
\label{eq:pnorm_en}
P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
\subfloat[Converted relation.]{%
\includegraphics[width=.5\linewidth]{fig/file}\label{fig:r1}}
\caption{The energy and performance relation}
+ \label{fig:rel}
\end{figure}
Then, we can model 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
represents the minimum energy consumption with minimum execution time (better
-performance) at the same time, see Figure~(\ref{fig:r1}). Then our objective
-function has the following form:
+performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}. Then
+our objective function has the following form:
\begin{equation}
\label{eq:max}
\textit{Max Dist} = \max_{j=1,2,\dots,F}
and communication times from the program after one iteration. Then the program
changes the new frequencies of the CPUs according to the computed scaling
factors. In our experiments over a homogeneous cluster described in
-section~\ref{sec.expe}, this algorithm has a small execution time. It takes
+Section~\ref{sec.expe}, this algorithm has a small execution time. It takes
\np[$\mu$s]{1.52} on average for 4 nodes and \np[$\mu$s]{6.65} on average 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 algorithm
-is called just once during the execution of the program. The DVFS
-algorithm~(Fig.~\ref{dvfs}) shows where and when the algorithm is called in the
-MPI program.
+is called just once during the execution of the program. The DVFS algorithm on
+Figure~\ref{dvfs} shows where and when the algorithm is called in the MPI
+program.
%\begin{table}[htb]
% \caption{Platform file parameters}
% % title of Table
%see Figure~\ref{fig:pred}
In our cluster there are 18 available frequency states for each processor. This
leads 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. Figure~(\ref{fig:pred})
+parallel benchmarks: CG, MG, EP, FT, BT, LU and SP. Figure~\ref{fig:pred}
presents plots of the real execution times and the simulated ones. The maximum
normalized error between these two execution times varies between \np{0.0073} to
\np{0.031} dependent on the executed benchmark. The smallest prediction error
(EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and C).
For each instance the benchmarks were executed on a number of processors
proportional to the size of the class. Each class represents the problem size
-ascending from 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
+ascending from 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 that the dynamic power with
-the highest frequency is equal to \np[W]{20} and the power static is equal to
-\np[W]{4} for all experiments. These power values were also used by Rauber and
-Rünger in~\cite{3}. The results showed that the algorithm selected different
-scaling factors for each program depending on the communication features of the
-program as in the plots~(\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 communication features for each
-benchmark. When there are little or no communications, the inverted
-performance 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 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 performance
-degradation percentage at the same time from all available scaling factors.
+consumption for all the NAS MPI programs while assuming that the dynamic power
+with the highest frequency is equal to \np[W]{20} and the power static is equal
+to \np[W]{4} for all experiments. These power values were also used by Rauber
+and Rünger in~\cite{3}. The results showed that the algorithm selected
+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
+communication features for each benchmark. When there are little or no
+communications, the inverted performance 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
+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
+performance degradation percentage at the same time from all available scaling
+factors.
\begin{figure*}[t]
\centering
\includegraphics[width=.33\linewidth]{fig/ep}\hfill%
\label{fig:nas}
\end{figure*}
-As shown in Table~(\ref{table:compareC}), when the optimal scaling
-factor has a big value we can gain more energy savings as in CG and
-FT benchmarks. The opposite happens when the optimal scaling factor has a
-small value as in BT and EP benchmarks. Our algorithm selects a big scaling
-factor value when the communication and the other slacks times are big and smaller
-ones in opposite cases. In EP there are no communication inside the iterations.
-This leads our algorithm to select smaller scaling factor values (inducing smaller energy
+As shown in Table~\ref{table:compareC}, when the optimal scaling factor has a
+big value we can gain more energy savings as in CG and FT benchmarks. The
+opposite happens when the optimal scaling factor has a small value as in BT and
+EP benchmarks. Our algorithm selects a big scaling factor value when the
+communication and the other slacks times are big and smaller ones in opposite
+cases. In EP there are no communication inside the iterations. This leads our
+algorithm to select smaller scaling factor values (inducing smaller energy
savings).
\subsection{Results comparison}
(when the scale $S=1$) to keep the performance from degradation as mush as
possible. We refer to this scenario as $R_{E-P}$. While we refer to our
algorithm as EPSA (Energy to Performance Scaling Algorithm). The comparison is
-made in Table ~\ref{table:compareC}. This table shows the results of our method and
+made in Table~\ref{table:compareC}. This table shows the results of our method and
Rauber and Rünger scenarios for all the NAS benchmarks programs for class C.
\begin{table}
