-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 set 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. 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}).
-
-
-
-
-
-This algorithm has a small
-execution time: for a heterogeneous cluster composed of four different types of
-nodes having the characteristics presented in table~(\ref{table:platform}), it
-takes \np[ms]{0.04} on average for 4 nodes and \np[ms]{0.15} on average for 144
-nodes. The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the
-number of iterations and $N$ is the number of computing nodes. The algorithm
-needs on average from 12 to 20 iterations to selects the best vector of frequency scaling factors that give the results of the next section. \textbf{put the lst paragraph in experiments}
-
+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}).
+\begin{figure}[t]
+ \centering
+ \includegraphics[scale=0.5]{fig/start_freq}
+ \caption{Selecting the initial frequencies}
+ \label{fig:st_freq}
+\end{figure}
