\section{Optimization of both energy consumption and performance}
\label{sec.compet}
-Applying DVFS to lower level not surly reducing the energy consumption to
-minimum level. Also, a big scaling for the frequency produces high performance
-degradation percent. Moreover, by considering the drastically increase in
-execution time of parallel program, the static energy is related to this time
-and it also increased by the same ratio. Thus, the opportunity for gaining more
-energy reduction is restricted. For that choosing frequency scaling factors is
-very important process to taking into account both energy and performance. In
-our previous work~\cite{45}, we are proposed a method that selects the optimal
-frequency scaling factor for an homogeneous cluster, depending on the trade-off
-relation between the energy and performance. In this work we have an
-heterogeneous cluster, at each node there is different scaling factors, so our
-goal is to selects the optimal set of frequency scaling factors,
-$Sopt_1,Sopt_2,\dots,Sopt_N$, that gives the best trade-off between energy
-consumption and performance. The relation between the energy and the execution
-time is complex and nonlinear, Thus, unlike the relation between the performance
+Using the lowest frequency for each processor does not necessarily gives the most energy efficient execution of an application. Indeed, even though the dynamic power is reduced while scaling down the frequency of a processor, its computation power is proportionally decreased and thus the execution time might be drastically increased during which dynamic and static powers are being consumed. Therefore, it might cancel any gains achieved by scaling down the frequency of all nodes to the minimum and the overall energy consumption of the application might not be the optimal one. It is not trivial to select the appropriate frequency scaling factor for each processor while considering the characteristics of each processor (computation power, range of frequencies, dynamic and static powers) and the task executed (computation/communication ratio) in order to reduce the overall energy consumption and not significantly increase the execution time. In
+our previous work~\cite{45}, we proposed a method that selects the optimal
+frequency scaling factor for a homogeneous cluster executing a message passing iterative synchronous application while giving the best trade-off
+ between the energy consumption and the performance for such applications. In this work we are interested in
+heterogeneous clusters as described above. Due to the heterogeneity of the processors, not one but a set of scaling factors should be selected and it must give the best trade-off between energy
+consumption and performance.
+
+The relation between the energy consumption and the execution
+time for an application is complex and nonlinear, Thus, unlike the relation between the performance
and the scaling factor, the relation of the energy with the frequency scaling
factors 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
-execution time by calculating the ratio between the new execution time (the
-scaled execution time) and the old one as follow:
+execution time by computing the ratio between the new execution time (after scaling down the frequencies of some processors) and the initial one (with maximum frequency for all nodes,) as follows:
\begin{multline}
\label{eq:pnorm}
P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
\end{multline}
-By the same way, we are normalize the energy by calculating the ratio between the consumed energy with scaled frequency and the consumed energy without scaled frequency:
+In the same way, we normalize the energy by computing the ratio between the consumed energy while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
\begin{multline}
\label{eq:enorm}
E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
\sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot Tcp_i)} +
\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
\end{multline}
-Where $T_{New}$ and $T_{Old}$ is computed as in EQ(\ref{eq:pnorm}). 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
-normalized execution time curves are not at the same direction. While the main
-goal is to optimize the energy and execution time in the same time. According
+Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
+
+ While the main
+goal is to optimize the energy and execution time at the same time, the normalized energy and execution time curves are not in the same direction. According
to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the set of frequency
scaling factors $S_1,S_2,\dots,S_N$ reduce 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
+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 are present a method to find the optimal
-set of frequency scaling factors to optimize both energy and execution time
-simultaneously without adding a big overhead. Our solution for this problem is
+in~\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
+set of frequency scaling factors to simultaneously optimize both energy and execution time
+ without adding a big overhead. \textbf{put the last two phrases in the related work section}
+
+
+ Our solution for this problem is
to make the optimization process for energy and execution time follow the same
-direction. Therefore, we inverse the equation of the normalized execution time,
-the normalized performance, as follows:
+direction. Therefore, we inverse the equation of the normalized execution time which gives
+the normalized performance equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
\end{equation}
where $N$ is the number of nodes and $F$ is the number of available frequencies for each nodes.
Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}). Our objective function can
-work with any energy model or energy values stored in a data file.
-Moreover, this function works in optimal way when the energy curve has a convex
-form over the available frequency scaling factors as shown in~\cite{15,3,19}.
+work with any energy model or any power values for each node (static and dynamic powers).
+However, the most energy reduction gain can be achieved the energy curve has a convex form as shown in~\cite{15,3,19}.
\section{The heterogeneous scaling algorithm }
\label{sec.optim}
\begin{figure}
\centering
- \subfloat[Balanced nodes type scenario]{%
+ \subfloat[CG, MG, LU and FT Benchmarks]{%
\includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
\quad%
- \subfloat[Imbalanced nodes type scenario]{%
+ \subfloat[BT and SP benchmarks]{%
\includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
\label{fig:avg}
\caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
