F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
\end{equation}
If the computed initial frequency for a node is not available in the gears of that node, the computed initial frequency is replaced by the nearest available frequency.
-In figure (\ref{fig:st_freq}), the nodes are sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are coloured in blue in figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal vector of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the frequency selecting factors algorithm starts its search method from these initial frequencies and takes a downward search direction. If the algorithm starts to search from the first frequencies of all nodes, regardless the higher bound frequencies, at each step the predicted performance and energy are degreased together, then the best distance be unreachable. This case is similar to homogeneous scaling algorithm when all nodes in the cluster has the same computing power, therefore there is a smaller distance between the performance and the energy curves, while in a heterogeneous cluster the distance is bigger and the energy saving against smaller execution time is higher, as an example see figure~(\ref{fig:r1} and \ref{fig:r2}). The algorithm iterates on all left frequencies, from the higher bound until all nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node according to EQ(\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of all other nodes by one gear. The new overall energy consumption and execution time are computed according to the new scaling factors. The optimal set of frequency scaling factors is the set that gives the highest distance according to the objective function EQ(\ref{eq:max}).
+In figure (\ref{fig:st_freq}), the nodes are sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are colored in blue in figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal vector of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the algorithm that selects the frequency scaling factors starts the search method from these initial frequencies and takes a downward search direction toward lower frequencies. The algorithm iterates on all left frequencies, from the higher bound until all nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node according to EQ(\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of all other nodes by one gear.
+The new overall energy consumption and execution time are computed according to the new scaling factors. The optimal set of frequency scaling factors is the set that gives the highest distance according to the objective function EQ(\ref{eq:max}).
+
+The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an application running on a homogeneous platform and a heterogeneous platform respectively while increasing the scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor should be started from the maximum frequency because the performance and the consumed energy is decreased since the beginning of the plot. On the other hand, in the heterogeneous platform the performance is maintained at the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes have computing powers lower than the slowest node. In other words, until they reach the higher bound. It can also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger the maximum distance between the energy curve and the performance curve is while varying the scaling factors which results in bigger energy savings.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{fig/start_freq}
\begin{figure}
\centering
- \subfloat[CG, MG, LU and FT benchmarks]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
+ \subfloat[Energy saving]{%
+ \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%
\quad%
- \subfloat[BT and SP benchmarks]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
+ \subfloat[Performance degradation ]{%
+ \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}
\label{fig:avg}
- \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
+ \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
\end{figure}
- The average of values of these three objectives are plotted to the number of
-nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}). In CG, MG, LU, and
-FT benchmarks the average of energy saving is decreased when the number of nodes
-is increased because the communication times is increased as mentioned
-before. Thus, the average of distances (our objective function) is decreased
-linearly with energy saving while keeping the average of performance degradation approximately is
-the same. In BT and SP benchmarks, the average of the energy saving is not decreased
-significantly compare to other benchmarks when the number of nodes is
-increased. Nevertheless, the average of performance degradation approximately
-still the same ratio. This difference is depends on the characteristics of the
-benchmark such as the computations to communications ratio that has.
-
-\textbf{All the previous paragraph should be deleted, we need to talk about it}
+ \textbf{ The energy saving and performance degradation of all benchmarks are plotted to the number of
+nodes as in plots (\ref{fig:energy} and \ref{fig:per_deg}). A shown in the plots, the energy saving percentage of the benchmarks MG, LU, BT and FT is decreased linearly when the the number of nodes increased. While in EP benchmarks the energy saving percentage is approximately the same percentage when the number of computing nodes is increased, because in this benchmarks there is no communications. In the SP benchmarks the energy saving percentage is decreased when it run on a small number of nodes, while this percentage is increased when it runs on a big number of nodes. The energy saving of the GC benchmarks is significantly decreased when the number of nodes is increased, because this benchmarks has more communications compared to other benchmarks. The performance degradation percentage of the benchmarks CG, EP, LU and BT is decreased when they run on a big number of nodes. While in MG benchmarks has a higher percentage of performance degradation when it runs on a big number of nodes. The inverse happen in SP benchmarks has smaller performance degradation percentage when it runs on a big number of nodes.}
+
+
\subsection{The results for different power consumption scenarios}
The results of the previous section were obtained while using processors that consume during computation an overall power which is 80\% composed of dynamic power and 20\% of static power. In this
