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}
-
+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.
+Therefore, there is a small distance between the energy and
+the performance curves in a homogeneous cluster compare to heterogeneous one, for example see the figure(\ref{fig:r1}) and figure(\ref{fig:r2}) . Then the
+algorithm starts to search for the optimal vector of the frequency scaling factors from the selected initial
+frequencies until all node reach their minimum frequencies.
+\begin{figure}[t]
+ \centering
+ \includegraphics[scale=0.5]{fig/start_freq}
+ \caption{Selecting the initial frequencies}
+ \label{fig:st_freq}
+\end{figure}
\section{Experimental results}
\label{sec.expe}
+To evaluate the efficiency and the overall energy consumption reduction of algorithm~\ref{HSA}), it was applied to the NAS parallel benchmarks NPB v3.3
+\cite{44}. The experiments were executed on the simulator SimGrid/SMPI
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile} which offers easy tools to create a heterogeneous platform and run message passing applications over it. The heterogeneous platform that was used in the experiments, had one core per node because just one process was executed per node. The heterogeneous platform was composed of four types of nodes. Each type of nodes had different characteristics such as the maximum CPU frequency, the number of
+available frequencies and the computational power, see table
+(\ref{table:platform}). The characteristics of these different types of nodes are inspired from the specifications of real Intel processors. The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, for example if a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were chosen proportionally to its computing power (FLOPS). In the initial heterogeneous platform, while computing with highest frequency, each node consumed power proportional to its computing power which 80\% of it was dynamic power and the rest was 20\% was static power, the same assumption was made in \cite{45,3}. Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
-The experiments of this work are executed on the simulator SimGrid/SMPI
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We are configure the
-simulator to use a heterogeneous cluster with one core per node. The proposed
-heterogeneous cluster has four different types of nodes. Each node in the cluster
-has different characteristics such as the maximum frequency speed, the number of
-available frequencies and dynamic and static powers values, see table
-(\ref{table:platform}). These different types of processing nodes are simulate some
-real Intel processors. The maximum number of nodes that supported by the cluster
-is 144 nodes according to characteristics of some MPI programs of the NAS
-benchmarks that used. We are use the same number from each type of nodes when we
-run the iterative MPI programs, for example if we are execute the program on 8 node, there
-are 2 nodes from each type participating in the computation. The dynamic and
-static power values is different from one type to other. Each node has a dynamic
-and static power values proportional to their computing power (FLOPS), for more
-details see the Intel data sheets in \cite{47}. Each node has a percentage of
-80\% for dynamic power and 20\% for static power of the total power
-consumption of a CPU, the same assumption is made in \cite{45,3}. These nodes are
-connected via an ethernet network with 1 Gbit/s bandwidth.
+
+\textbf{modify the characteristics table by replacing the similar column with the computing power of the different types of nodes in flops}
\begin{table}[htb]
\caption{Heterogeneous nodes characteristics}
% title of Table
\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
-The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.3
-\cite{44}, which were run with three classes (A, B and C).
-In this experiments we are interested to run the class C, the biggest class compared to A and B, on different number of
-nodes, from 4 to 128 or 144 nodes according to the type of the iterative MPI program. Depending on the proposed energy consumption model EQ(\ref{eq:energy}),
- we are measure the energy consumption for all NAS MPI programs. The dynamic and static power values are used under the same assumption used by \cite{45,3}, we are used a percentage of 80\% for dynamic power and 20\% for static of the total power consumption of a CPU. The heterogeneous nodes in table (\ref{table:platform}) have different simulated computing power (FLOPS), ranked from the node of type 1 with smaller computing power (FLOPS) to the highest computing power (FLOPS) for node of type 4. Therefore, the power values are used proportionally increased from nodes of type 1 to nodes of type 4 that with highest computing power. Then, we are used an assumption that the power consumption is increased linearly when the computing power (FLOPS) of the processor is increased, see \cite{48}.
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in this paper, only the results of the biggest class, C, are presented while being run on different number of nodes, ranging from 4 to 128 or 144 nodes depending on the benchmark being executed.
+
+
\begin{table}[htb]
\caption{Running NAS benchmarks on 4 nodes }
\end{tabular}
\label{table:res_128n}
\end{table}
+The overall energy consumption was computed for each instance according to the energy consumption model EQ(\ref{eq:energy}), with and without applying the algorithm. The execution time was also measured for all these experiments. Then, the energy saving and performance degradation percentages were computed for each instance. The results are presented in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}).
+These tables show the experimental results for running the NAS parallel benchmarks on different number of nodes. The experiments show that the algorithm reduce significantly the energy consumption (up to 35\%) and tries to limit the performance degradation. They also show that the energy saving percentage is decreased when the number of the computing nodes is increased. This reduction is due to the increase of the communication times compared to the execution times when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C, are executed on different number of nodes, so the computation required for each iteration is divided by the number of computing nodes. On the other hand, more communications are required when increasing the number of nodes so the static energy is increased linearly according to the communication time and the dynamic power is less relevant in the overall energy consumption. Therefore, reducing the frequency with algorithm~\ref{HSA}) have less effect in reducing the overall energy savings. It can also be noticed that for the benchmarks EP and SP that contain little or no communications, the energy savings are not significantly affected with the high number of nodes. No experiments were conducted using bigger classes such as D, because they require a lot of memory(more than 64GB) when being executed by the simulator on one machine.
+The maximum distance between the normalized energy curve and the normalized performance for each instance is also shown in the result tables. It is decreased in the same way as the energy saving percentage. The tables also show that the performance degradation percentage is not significantly increased when the number of computing nodes is increased because the computation times are small when compared to the communication times.
-The results of applying the proposed scaling algorithm to the NAS benchmarks is demonstrated in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}). These tables are show the experimental results for running the NAS benchmarks on different number of nodes. In general the energy saving percent is decreased when the number of the computing nodes is increased. Also the distance is decreased by the same direction of the energy saving. This because when we are run the iterative MPI programs on a big number of nodes the communications times is increased, so the static energy is increased linearly to these times. The tables also show that the performance degradation percent still approximately the same ratio or decreased in a very small present when the number of computing nodes is increased. This is gives a good prove that the proposed algorithm keeping the performance degradation as mush as possible is the same.
+
\begin{figure}
\centering
\subfloat[CG, MG, LU and FT benchmarks]{%
