corrections

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 024cc2920e68006095b0f7dcfa672736effc5ff2..76c2bc43047f62e57a5a7e3c0b97a2ed01454a89 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -151,7 +151,6 @@ This prediction model is based on our model for predicting the execution time of
 
 
 \subsection{Energy model for heterogeneous platform}
 
 
 \subsection{Energy model for heterogeneous platform}

-

 Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
 two power metrics: the static and the dynamic power.  While the first one is
 consumed as long as the computing unit is turned on, the latter is only consumed during
 Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
 two power metrics: the static and the dynamic power.  While the first one is
 consumed as long as the computing unit is turned on, the latter is only consumed during


@@ -340,7 +339,10 @@ maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as fol
  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.
  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}[t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}


@@ -426,37 +428,27 @@ 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.
 
 
 (\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.
 
 

-\textbf{modify the characteristics table by replacing the similar column with the computing power of the different types of nodes in flops}
-
-
- The proposed scaling 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 on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144
-nodes to compute the best scaling factors vector.  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  from 12 to 20 iterations to select the best vector of frequency scaling factors that gives the results of the next section.
-

 \begin{table}[htb]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
 \begin{table}[htb]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline

-    Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\
-    type     & to          & Freq. GHz  & Freq. GHz    & Freq. GHz       & power        & power \\
+    Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
+    type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
+                  &           & GHz      & GHz          &GHz             &              &       \\

     \hline
     \hline

-    1       & core-i3       & 2.5         & 1.2          & 0.1           & 20~w         &4~w    \\
-            &  2100T        &             &              &               &              &  \\
+    1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
+                  &           &          &              &                &              &  \\

     \hline
     \hline

-    2       & Xeon          & 2.66        & 1.6          & 0.133         & 25~w         &5~w    \\
-            & 7542          &             &              &               &              &  \\
+    2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
+                  &           &          &              &                &              &  \\

     \hline
     \hline

-    3       & core-i5       & 2.9         & 1.2          & 0.1           & 30~w         &6~w    \\
-            & 3470s         &             &              &               &              &  \\
+    3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
+                  &           &          &              &                &              &  \\

     \hline
     \hline

-    4       & core-i7       & 3.4         & 1.6          & 0.133         & 35~w         &7~w    \\
-            & 2600s         &             &              &               &              &  \\
+    4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
+                  &           &          &              &                &              &  \\

     \hline
   \end{tabular}
   \label{table:platform}
     \hline
   \end{tabular}
   \label{table:platform}


@@ -469,8 +461,8 @@ needs  from 12 to 20 iterations to select the best vector of frequency scaling f
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 

-<<<<<<< HEAD
-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. 
+
+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. Indeed, the benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes. The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.

 
  
  
 
  
  


@@ -636,52 +628,43 @@ The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
   \end{tabular}
   \label{table:res_128n}
 \end{table}
   \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 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}). All these results are the average values from many experiments for  energy savings and performance degradation.
+
+The 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.  
 
 
  
 \begin{figure}
   \centering
 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.  
 
 
  
 \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%
   \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}
   \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}
 
 \end{figure}
 

-In the NAS benchmarks there are some programs executed on different number of
-nodes. The benchmarks CG, MG, LU and FT executed on 2 to a power of (1, 2, 4, 8,
-\dots{}) of nodes. The other benchmarks such as BT and SP executed on 2 to a
-power of (1, 2, 4, 9, \dots{}) of nodes. We are take the average of energy
-saving, performance degradation and distances for all results of NAS
-benchmarks. 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{ 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}
 
 
 \subsection{The results for different power consumption scenarios}
 

-The results of the previous section are obtained using a percentage of 80\% for
-dynamic power and 20\% for static power of the total power consumption of a CPU. In this
-section we are change these ratio by using two others power scenarios. Because is
-interested to measure the ability of the proposed algorithm when these power ratios are changed. 
-In fact, we are used two different scenarios for dynamic and static power ratios in addition to the previous
-scenario in section (\ref{sec.res}). Therefore, we have three different
-scenarios for three different dynamic and static power ratios refer to these as: 
-70\%-20\%, 80\%-20\% and 90\%-10\% scenario respectively. The results of these scenarios
-running the NAS benchmarks class C on 8 or 9 nodes are place in the tables
-(\ref{table:res_s1} and \ref{table:res_s2}).
+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
+section, these ratios are changed and two new power scenarios are considered in order to evaluate how the proposed  algorithm adapts itself according to the static and dynamic power values.  The two new power scenarios are the following: 
+\begin{itemize}
+\item 70\% dynamic power  and 30\% static power
+\item 90\% dynamic power  and 10\% static power
+\end{itemize}
+The NAS parallel benchmarks were executed again over processors that follow the the new power scenarios. The class C of each benchmark was run over 8 or 9 nodes and the results are presented in  tables (\ref{table:res_s1} and \ref{table:res_s2}). \textbf{These tables show that the energy saving percentage of the 70\%-30\% scenario is less for all benchmarks compared to the energy saving of the 90\%-10\% scenario, because this scenario uses higher percentage of dynamic dynamic power that is quadratically related to scaling factors. While the performance degradation percentage is less in 70\%-30\% scenario  compared to 90\%-10\%  scenario, because the first scenario used higher percentage for static power consumption that is linearly related to scaling factors and thus the execution time. }
+
+The two new power scenarios are compared to the old one  in figure (\ref{fig:sen_comp}). It shows the average of the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. The comparison shows that  the energy saving ratio is proportional to the dynamic power ratio: it is increased when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the the most relevant in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in the overall consumed energy and lowering the frequency do not returns big energy savings.
+Moreover, the average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed algorithm optimizes the energy consumption when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens when using a higher ratio for  static  power, the algorithm proportionally  selects  smaller scaling values which results in less energy saving but less performance degradation. 
+

 
  \begin{table}[htb]
   \caption{The results of 70\%-30\% powers scenario}
 
  \begin{table}[htb]
   \caption{The results of 70\%-30\% powers scenario}


@@ -745,18 +728,23 @@ running the NAS benchmarks class C on 8 or 9 nodes are place in the tables
   \subfloat[Comparison the average of the results on 8 nodes]{%
     \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
   \quad%
   \subfloat[Comparison the average of the results on 8 nodes]{%
     \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
   \quad%

-  \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
+  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%

     \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
 \end{figure}  
 
     \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
 \end{figure}  
 

-To compare the results of these three powers scenarios, we are take the average of the performance degradation, the energy saving and the distances for all NAS benchmarks running on 8 or 9 nodes of class C, as in figure (\ref{fig:sen_comp}). Thus, according to the average of these results, the energy saving ratio is increased when using a higher percentage for dynamic power (e.g. 90\%-10\% scenario), due to increase in dynamic energy. While the average of energy saving is decreased in 70\%-30\% scenario. Because the static energy consumption is increase. Moreover, the average of distances is more related to energy saving changes. The average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). The raison behind these relations, that the proposed algorithm optimize both energy consumption and performance in the same time. Therefore, when using a higher ratio for dynamic power the algorithm selecting bigger frequency scaling factors values, more energy saving versus more performance degradation, for example see the figure (\ref{fig:scales_comp}). The inverse happen when using a higher ratio for  static  power, the algorithm proportionally  selects a smaller scaling values, less energy saving versus less performance degradation. This is because the  
-algorithm is optimizes the  static energy consumption that is always related to the execution time. 
+

 
 \subsection{The verifications of the proposed method}
 \label{sec.verif}
 
 \subsection{The verifications of the proposed method}
 \label{sec.verif}

-The precision of the proposed algorithm mainly depends on the execution time prediction model EQ(\ref{eq:perf}) and the energy model EQ(\ref{eq:energy}). The energy model is significantly depends on the execution time model, that the static energy is related linearly. So, our work is depends mainly on execution time model. To verifying this model, we are compared the predicted execution time with the real execution time (Simgrid time) values that gathered  offline from the NAS benchmarks class B executed on 8 or 9 nodes. The execution time model can predicts the real execution time by maximum normalized error equal to 0.03 for all the NAS benchmarks. The second verification that we are made is for the proposed scaling algorithm to prove its ability to selects the best vector of the frequency scaling factors. Therefore, we are expand the algorithm to test at each iteration the frequency scaling factor of the slowest node with the all available scaling factors of the other nodes, all possible solutions. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the expanded algorithms and the proposed algorithm are identical. While the proposed algorithm is runs  by 10 times faster on average compare to the expanded algorithm.
+The precision of the proposed algorithm mainly depends on the execution time prediction model defined in EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}). 
+The energy model is also significantly dependent  on the execution time model because the static energy is linearly related the execution time and the dynamic energy is related to the computation time. So, all of the work presented in this paper is based on the execution time model. To verify this model, the predicted execution time was compared to  the real execution time over Simgrid for all  the NAS parallel benchmarks running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise, the maximum normalized difference between  the predicted execution time  and the real execution time is equal to 0.03 for all the NAS benchmarks.
+
+Since  the proposed algorithm is not an exact method and do not test all the possible solutions (vectors of scaling factors) in the search space and to prove its efficiency, it was compared on small instances to a brute force search algorithm that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical and the proposed algorithm was on average 10 times faster than the brute force algorithm. It 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 on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling factors vector.  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  from 12 to 20 iterations to select the best vector of frequency scaling factors that gives the results of the section (\ref{sec.res}).

 
 \section{Conclusion}
 \label{sec.concl}
 
 \section{Conclusion}
 \label{sec.concl}