\end{multline}
Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
- The normalized energy and execution time curves are not in the same direction. While the main
-goal is to optimize the energy and execution time at the same time. According
+ 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}
