\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
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}
(\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
- 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
- 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
- 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
- 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
- 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}
\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. 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 }
+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.
+
\begin{table}[htb]
\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 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.
\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}
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}
+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}
