\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
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}).
\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.
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}).
-
-
-
-
-
-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.
+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}
+
+
\begin{algorithm}
\begin{algorithmic}[1]
% \footnotesize
\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{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 & & & & & \\
\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.
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.
\subsection{The results for different power consumption 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}
-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 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.
\section{Conclusion}
\label{sec.concl}
