Modified a paragraph

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 7bd1de4e79cfeaec3964a2c2c2fdd3d813c0d1b0..7416888376c895b4944ab97e4008033a1320ae1d 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -141,7 +141,7 @@ vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
  \textit  T_\textit{new} = 
 \begin{equation}
   \label{eq:perf}
  \textit  T_\textit{new} = 

- \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  MinTcm
+ \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) +  MinTcm

 \end{equation}
 where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $MinTcm$ is the communication time of the slowest processor from the first iteration.  The model computes the maximum computation time 
  with scaling factor from each node  added to the communication time of the slowest node, it means  only the
 \end{equation}
 where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $MinTcm$ is the communication time of the slowest processor from the first iteration.  The model computes the maximum computation time 
  with scaling factor from each node  added to the communication time of the slowest node, it means  only the


@@ -262,7 +262,7 @@ In the same way, we normalize the energy by computing the ratio between the cons
   E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
   {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
  \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
   E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
   {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
  \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +

- \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
+ \sum_{i=1}^{N} {(Ps_i@+eYd162 \cdot T_{Old})}}

 \end{multline} 
 Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
 
 \end{multline} 
 Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
 


@@ -323,8 +323,8 @@ However, the most energy reduction gain can be achieved when the energy curve ha
 \section{The scaling factors selection algorithm for heterogeneous platforms }
 \label{sec.optim}
 
 \section{The scaling factors selection algorithm for heterogeneous platforms }
 \label{sec.optim}
 

-In this section we  propose algorithm~\ref{HSA}) which selects the frequency scaling factors vector that gives the best trade-off between minimizing the energy consumption  and maximizing the performance of a message passing synchronous iterative application executed on a heterogeneous platform.  
-IT works online during the execution time of the iterative message passing program.  It uses information gathered during the first iteration such as the computation time and the communication time in one iteration for each node. The algorithm is executed  after the first iteration and returns a vector of optimal frequency scaling factors   that satisfies the objective function EQ(\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs according to the computed scaling factors.  This algorithm is called just once during the execution of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called in the iterative MPI program.
+In this section we  propose algorithm~(\ref{HSA}) which selects the frequency scaling factors vector that gives the best trade-off between minimizing the energy consumption  and maximizing the performance of a message passing synchronous iterative application executed on a heterogeneous platform.  
+It works online during the execution time of the iterative message passing program.  It uses information gathered during the first iteration such as the computation time and the communication time in one iteration for each node. The algorithm is executed  after the first iteration and returns a vector of optimal frequency scaling factors   that satisfies the objective function EQ(\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs according to the computed scaling factors.  This algorithm is called just once during the execution of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called in the iterative MPI program.

 
 
 The nodes in a heterogeneous platform have different computing powers, thus while executing message passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). These periods are called idle or slack times.
 
 
 The nodes in a heterogeneous platform have different computing powers, thus while executing message passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). These periods are called idle or slack times.


@@ -340,25 +340,10 @@ 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 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}).
+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}).

 
 

-  
-
-
-
-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. 
-
-
-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.
+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}


@@ -368,6 +353,8 @@ frequencies until all node reach their minimum frequencies.
 
 
 
 
 
 

+
+

 \begin{algorithm}
   \begin{algorithmic}[1]
     % \footnotesize
 \begin{algorithm}
   \begin{algorithmic}[1]
     % \footnotesize


@@ -443,6 +430,16 @@ available frequencies and the computational power, see table
 
 
 \textbf{modify the characteristics table by replacing the similar column with the computing power of the different types of nodes in flops}
 
 
 \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
 \begin{table}[htb]
   \caption{Heterogeneous nodes characteristics}
   % title of Table


@@ -450,7 +447,7 @@ available frequencies and the computational power, see table
   \begin{tabular}{|*{7}{l|}}
     \hline
     Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\
   \begin{tabular}{|*{7}{l|}}
     \hline
     Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\

-    type     & to          & Freq. GHz  & Freq. GHz    & Freq GHz       & power        & power \\
+    type     & to          & Freq. GHz  & Freq. GHz    & Freq. GHz       & power        & power \\

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


@@ -475,8 +472,9 @@ available frequencies and the computational power, see table
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 

-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 and FT should be executed on $2^1, 2^2, 2^4 or 2^8$ nodes. The other benchmarks such as BT and SP should be executed on $2^1, 2^2, 2^4 or 2^9$ nodes.
+\textbf{there must be an error in the number of nodes }

  
  
 \begin{table}[htb]
  
  
 \begin{table}[htb]


@@ -641,8 +639,10 @@ 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.  
 
 
 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.  
 
 


@@ -658,12 +658,8 @@ The maximum distance between the normalized energy curve and the normalized perf
   \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
 \end{figure}
 
   \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
 \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
+ 
+   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
 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


@@ -673,20 +669,22 @@ the same. In BT and SP benchmarks, the average of the  energy saving is not decr
 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
 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

-benchmarks such as the computation to communication ratio that has.
+benchmark such as the computations to communications ratio that has.

 
 

+\textbf{All the previous paragraph should be deleted, we need to talk about it}

 \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{should explain the tables more}
+
+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}


@@ -754,15 +752,17 @@ running the NAS benchmarks class C on 8 or 9 nodes are place in the tables
     \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
     \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}

-\end{figure}
+\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 thid model, we are compare 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. 
+\textbf{should put the paragraph about the overhead here}

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