\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
-\newcommand{\Ppeak}[1][]{\Xsub{P}{peak}_{#1}}
+\newcommand{\Pmax}[1][]{\Xsub{P}{max}_{\fxheight{#1}}}
\newcommand{\Pidle}[1][]{\Xsub{P}{idle}_{\fxheight{#1}}}
\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
\newcommand{\Tnew}{\Xsub{T}{New}}
of iterative methods, which is presented in the following sections.
-\subsection{Energy model for heterogeneous platform}
+\subsection{Energy model for heterogeneous grid platform}
Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
\end{equation}
Where $\Ereduced$ is computed using (\ref{eq:energy}) and $\Eoriginal$ is
-computed as in ().
+computed as in (\ref{eq:eorginal}).
+
-\textcolor{red}{A reference is missing}
\begin{equation}
\label{eq:eorginal}
\Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot \Tcp[ij]) +
\label{dvfs}
\end{algorithm}
-\subsection{The algorithm details}
-\textcolor{red}{Delete the subsection if there's only one.}
In this section, the scaling factors selection algorithm for grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency
scaling factors that gives the best trade-off between minimizing the
\section{Experimental results}
\label{sec.expe}
-While in~\cite{pdsec2015} the energy model and the scaling factors selection algorithm were applied to a heterogeneous cluster and evaluated over the SimGrid simulator~\cite{SimGrid.org},
+While in~\cite{pdsec2015} the energy model and the scaling factors selection algorithm were applied to a heterogeneous cluster and evaluated over the SimGrid simulator~\cite{SimGrid},
in this paper real experiments were conducted over the grid'5000 platform.
\subsection{Grid'5000 architature and power consumption}
which is the French National Telecommunication Network for Technology.
Each site of the grid is composed of few heterogeneous
computing clusters and each cluster contains many homogeneous nodes. In total,
- grid'5000 has about one thousand heterogeneous nodes and eight thousand cores. In each site,
+grid'5000 has about one thousand heterogeneous nodes and eight thousand cores. In each site,
the clusters and their nodes are connected via high speed local area networks.
Two types of local networks are used, Ethernet or Infiniband networks which have different characteristics in terms of bandwidth and latency.
Since grid'5000 is dedicated for testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture
the power consumption for each node in those sites. The measured power is the overall consumed power by by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ... For more details refer to
\cite{Energy_measurement}. To just measure the CPU power of one core in a node $j$,
- firstly, the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $P\max[jx]$. The difference between the two measured power consumption represents the
+ firstly, the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumption represents the
dynamic power consumption of that core with the maximum frequency, see figure(\ref{fig:power_cons}).
-\textcolor{red}{why maximum and minimum, change peak in the equation and the figure}
The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
\begin{equation}
\label{eq:pdyn}
- \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (P\max[jx]) - \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
+ \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx]) - \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
\end{equation}
where $\Pd[j]$ is the dynamic power consumption for one core of node $j$,
-$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured peak power values,
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured maximum power values,
$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured idle power values.
Therefore, the dynamic power of one core is computed as the difference between the maximum
-measured value in peak powers vector and the minimum measured value in the idle powers vector.
+measured value in maximum powers vector and the minimum measured value in the idle powers vector.
-On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that the static power represents a ratio of the dynamic power, the value of the static power is assumed as np[\%]{20} of dynamic power consumption of the core.
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that the static power represents a ratio of the dynamic power, the value of the static power is assumed as 20\% of dynamic power consumption of the core.
In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
-In this section, the results of the the application of the scaling factors selection algorithm \ref{HSA}
+In this section, the results of the application of the scaling factors selection algorithm \ref{HSA}
to the NAS parallel benchmarks are presented.
As mentioned previously, the experiments
-were conducted over two sites of grid'5000, Lyon and Nancy sites.
+were conducted over two sites of grid'5000, Lyon and Nancy sites.
Two scenarios were considered while selecting the clusters from these two sites :
\begin{itemize}
\item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected
-are connected via a long distance network.
-\item In the second scenario nodes from three clusters that are
-located in one site, Nancy site.
+ via a long distance network.
+\item In the second scenario nodes from three clusters that are located in one site, Nancy site.
\end{itemize}
The main reason
behind using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the
-scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio
+scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio
is very low due to the higher communication times which reduces the effect of DVFS operations.
The NAS parallel benchmarks are executed over
\centering
\begin{tabular}{|*{4}{c|}}
\hline
-\multirow{2}{*}{Scenario name} & \multicolumn{2}{c|} {The participating clusters} \\ \cline{2-4}
+\multirow{2}{*}{Scenario name} & \multicolumn{3}{c|} {The participating clusters} \\ \cline{2-4}
& Cluster & Site & No. of nodes \\
\hline
\multirow{3}{*}{Two sites / 16 nodes} & Taurus & Lyon & 5 \\ \cline{2-4}
\label{fig:time_sen}
\end{figure}
-The NAS parallel benchmarks are executed over these two platform
+The NAS parallel benchmarks are executed over these two platforms
with different number of nodes, as in Table \ref{tab:sc}.
The overall energy consumption of all the benchmarks solving the class D instance and
using the proposed frequency selection algorithm is measured
For the majority of the benchmarks, the energy consumed while executing the NAS benchmarks over one site scenario
for 16 and 32 nodes is lower than the energy consumed while using two sites.
-The long distance communications between the two distributed sites increase the idle time which leads to more static energy consumption.
- The execution times of these benchmarks
+The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption.
+The execution times of these benchmarks
over one site with 16 and 32 nodes are also lower when compared to those of the two sites
scenario.
-
-
However, the execution times and the energy consumptions of EP and MG benchmarks, which have no or small communications, are not significantly affected
in both scenarios. Even when the number of nodes is doubled. On the other hand, the communications of the rest of the benchmarks increases when using long distance communications between two sites or increasing the number of computing nodes.
energy consumption, equation (\ref{eq:energy}), and the original energy consumption,
equation (\ref{eq:eorginal}), for all benchmarks as in figure \ref{fig:eng_s}.
This figure shows that the energy saving percentages of one site scenario for
-16 and 32 nodes are bigger than those of the two sites scenario. This is because
-the computations to communications ratio in one site scenario is higher
-than the ratio of the two sites scenarios, due to the increase in the communication
-times. Moreover, the frequencies selecting algorithm selects smaller frequencies, bigger
-scaling factors, when the computations times are higher than communication times,
-producing smaller energy consumption, because the dynamic energy consumption
-is decreased linearly with computation times that decreased exponentially with
-scaling factors. On the other side, the increase in the number of computing nodes can be
-increase the communication times and thus producing less energy saving depending on the
-benchmarks being executed. The benchmarks CG, MG, BT and FT show more
-energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While
-the benchmarks LU and SP showed the inverse, because there computations to
-communications ratio is not effected to the increase in local site communications.
-While all benchmarks are effected by the long distance communications in the two sites
-scenarios, except EP benchmarks. In EP benchmark there is no communications
-in their iterations, then it is independent from the effect of local and long
-distance communications. Therefore, the energy saving percentage of this benchmarks is
-depend on differences between the computing powers of the computing nodes, for example
+16 and 32 nodes are bigger than those of the two sites scenario which is due
+to the higher computations to communications ratio in the first scenario
+than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which
+results in a lower energy consumption. Indeed, the dynamic consumed power
+is exponentially related to the CPU's frequency value. On the other side, the increase in the number of computing nodes can
+increase the communication times and thus produces less energy saving depending on the
+benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more
+energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While, LU and SP consume more energy with 16 nodes than 32 in one site because their computations to communications ratio is not affected by the increase of the number of local communications.
+
+
+The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites
+scenario, except for the EP benchmark which has no communications. Therefore, the energy saving percentage of this benchmark is
+dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example
in the one site scenario, the graphite cluster is selected but in the two sits scenario
-this cluster is replaced with Taurus cluster that be more powerful in computing power.
-Therefore, the energy saving of EP benchmarks are bigger in the two site scenario due
-to increase in the differences between the computing powers of the nodes. This means, the higher
-differences between the nodes' computing powers make the proposed frequencies selecting
-algorithm to selects smaller frequencies in the nodes of the higher computing power,
-producing less energy consumption and thus more energy saving.
-The best energy saving percentage was for one site scenario with 16 nodes, on average it
-saves the energy consumption up to 30\%.
-
-Figure \ref{fig:per_d}, presents the performance degradation percentages for all benchmarks.
-It shows that the performance degradation percentages of the one site scenario with
-32 nodes, on average equal to 10\%, is higher than the performance degradation of one 16 nodes,
-which on average equal to 3\%. This because selecting smaller frequencies in the one site scenarios,
-when the computations grater than the communications , increase the number of the critical nodes
-when the number of nodes increased. The inverse happens in the tow sites scenario,
-this due to the lower computations to communications ratio that decreased with highest
-communications. Therefore, the number of the critical nodes are decreased. The average performance
-degradation for the two sites scenario with 16 nodes is equal to 8\% and for 32 nodes is equal to 4\%.
-The EP benchmarks is gives the bigger performance degradation ratio, because there is no
-communications and no slack times in this benchmarks that is always their performance effected
-by selecting big or small frequencies.
-The tradeoff between these scenarios can be computed as in the trade-off function \ref{eq:max}.
-Figure \ref{fig:dist}, presents the tradeoff distance for all benchmarks over all
-platform scenarios. The one site scenario with 16 and 32 nodes had the best tradeoff distance
-compared to the two sites scenarios, because the increase in the communications as mentioned before.
-The one site scenario with 16 nodes is the best scenario in term of energy and performance tradeoff,
-which on average is up 26\%. Then, the tradeoff distance is related linearly to the energy saving
-percentage. Finally, the best energy and performance tradeoff depends on the increase in all of:
-1) the computations to communications ratio, 2) the differences in computing powers
-between the computing nodes and 3) the differences in static and the dynamic powers of the nodes.
+this cluster is replaced with Taurus cluster which is more powerful.
+Therefore, the energy saving of EP benchmarks are bigger in the two sites scenario due
+to the higher maximum difference between the computing powers of the nodes.
+
+In fact, high differences between the nodes' computing powers make the proposed frequencies selecting
+algorithm select smaller frequencies for the powerful nodes which
+produces less energy consumption and thus more energy saving.
+The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.
+
+
+Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
+The performance degradation percentage for the benchmarks running on two sites with
+16 or 32 nodes is on average equal to 8\% or 4\% respectively.
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller computations to communications ratio. On the other hand, the performance degradation percentage for the benchmarks running on one site with
+16 or 32 nodes is on average equal to 3\% or 10\% respectively. In opposition to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing
+nodes when the communications occur in high speed network does not decrease the computations to
+communication ratio.
+
+
+ Figure \ref{fig:time_sen} presents the execution times for all the benchmarks over the two scenarios. For most of the benchmarks running over the one site scenario, their execution times are approximately divided by two when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).
+
+
+\textcolor{blue}{
+The performance degradation percentage of EP benchmark is the higher when it is compared with
+the other benchmarks. There are no communication and slack times in this benchmark and its
+performance degradation percentage depends on the frequency value selected in the computing node.
+The rest of the benchmarks showed different performance degradation percentages, which are decreased
+when the communication times are increased and vice versa.}
+
+\textcolor{blue}{Figure \ref{fig:dist} presents the tradeoff distance percentage between the energy saving and the performance degradation for all benchmarks over both scenarios. The tradeoff distance percentage can be
+computed as in the tradeoff function \ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance
+tradeoff, on average is equal to 26\%. As a result, one site scenario using both 16 and 32 nodes had better energy and performance
+tradeoff comparing to the two sites scenario. This because the former used high speed local communications
+which increased the computations to communications ratio and the latter used long distance communications which decreased this ratio. } \textcolor{red}{The last paragraph has compared the two scenarios}
+
+
+ Finally, the best energy and performance tradeoff depends on all of the following:
+1) the computations to communications ratio when there are communications and slack times, 2) the heterogeneity of the computing powers of the nodes and 3) the heterogeneity of the consumed static and dynamic powers of the nodes.
+
+
+
\subsection{The experimental results of multi-cores clusters}
-\label{sec.res}
+\label{sec.res-mc}
+The clusters of grid'5000 have different number of cores embedded in their nodes
+as shown in Table \ref{table:grid5000}. The cores of each node can exchange
+data via the shared memory \cite{rauber_book}. In
+this section, the proposed scaling algorithm is evaluated over the grid'5000 grid while using multi-core nodes
+selected according to the two platform scenarios described in the section \ref{sec.res}.
+The two platform scenarios, the two sites and one site scenarios, use 32
+cores from multi-cores nodes instead of 32 distinct nodes. For example if
+the participating number of cores from a certain cluster is equal to 12,
+in the multi-core scenario the selected nodes is equal to 3 nodes while using
+4 cores from each node. The platforms with one
+core per node and multi-cores nodes are shown in Table \ref{table:sen-mc}.
+The energy consumptions and execution times of running the NAS parallel
+benchmarks, class D, over these four different scenarios are presented
+in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+
+The execution times for most of the NAS benchmarks are higher over the one site multi-cores per node scenario
+ than the execution time of those running over one site single core per node scenario. Indeed,
+ the communication times are higher in the one site multi-cores scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. On the other hand, the execution times for most of the NAS benchmarks are lower over
+the two sites multi-cores scenario than those over the two sites one core scenario.
+
+\textcolor{blue}{Furthermore, in two sites multi-cores per node scenario part of the communications happened via shared memory
+and the rest via long distance network. According to the high latency in the long distance network, the
+communication times are smaller compared to the communication times of the shared memory.
+Therefore, using the shared memory communications mixed with the long distance communications
+has decreased the communication times, and thus the overall execution time is decreased.}
+
+The experiments showed that for most of the NAS benchmarks and between the four scenarios,
+the one site one core scenario gives the best execution times because the communication times are the lowest.
+Indeed, in this scenario each core has a dedicated network link and all the communications are local.
+Moreover, the energy consumptions of the NAS benchmarks are lower over the
+one site one core scenario than over the one site multi-cores scenario because
+the first scenario had less execution time than the latter which results in less static energy being consumed.
+
+\textcolor{blue}{
+Therefore, the computations to communications ratios of the NAS benchmarks are higher over
+the one site one core scenario compared to the other scenarios.
+More energy reduction has achieved when this ratio increased, because the proposed scaling algorithm selecting smaller frequencies that decreased the dynamic power consumption. Whereas, the energy consumption in the two sites multi-cores scenario is higher than the energy consumption
+of the two sites one core scenario. Actually, using multi-cores in this scenario decreased the communication times that decreased the static energy consumption.}
+
+
+These experiments also showed that the energy
+consumption and the execution times of the EP and MG benchmarks do not change significantly over these four
+scenarios because there are no or small communications,
+which could increase or decrease the static power consumptions. Contrary to EP and MG, the energy consumptions
+and the execution times of the rest of the benchmarks vary according to the communication times that are different from one scenario to the other.
+
+\textcolor{blue}{
+The energy saving percentages of all NAS benchmarks running over these four scenarios are presented in the figure \ref{fig:eng-s-mc}. This figure
+shows that the energy saving percentages are higher over the two sites multi-cores scenario
+than over the two sites one core scenario, on average they are equal to 22\% and 18\%
+respectively. This is according to the increase or decrease in the computations to communications ratio as mentioned previously.}
+
+
+In contrast, in the one site one
+core and one site multi-cores scenarios the energy saving percentages
+are approximately equivalent, on average they are up to 25\%. In both scenarios there
+are a small difference in the computations to communications ratios, which leads
+the proposed scaling algorithm to select similar frequencies for both scenarios.
+
+The performance degradation percentages of the NAS benchmarks are presented in
+figure \ref{fig:per-d-mc}.
+
+It indicates that the performance degradation percentages for the NAS benchmarks are higher over the two sites
+multi-cores scenario than over the two sites one core scenario, equal on average to 7\% and 4\% respectively.
+Moreover, using the two sites multi-cores scenario increased
+the computations to communications ratio, which may increase
+the overall execution time when the proposed scaling algorithm is applied and the frequencies scaled down.
+
+\textcolor{blue}{
+When the benchmarks are executed over the one
+site one core scenario their performance degradation percentages, on average
+is equal to 10\%, are higher than those executed over one site multi-cores,
+which on average is equal to 7\%. This because using multi-cores in one site scenario
+decreased the computations to communications ratio. Therefore, selecting small
+frequencies by the scaling algorithm do not increase the execution time significantly.}
+
+\textcolor{blue}{
+The tradeoff distance percentages of the NAS
+benchmarks over all scenarios are presented in the figure \ref{fig:dist-mc}.
+These tradeoff distance percentages are used to verified which scenario is the best in term of the energy and performance ratio. The figure indicates that using muti-cores in both of the one site and two sites scenarios gives bigger tradeoff distance percentages, on overage they are equal to 17.6\% and 15.3\% respectively. On the contrary, using one core per node in both of one site and two sites scenarios gives lower tradeoff distance percentages, on average they are equal to 14.7\% and 13.3\% respectively. }
+
+\begin{table}[]
+\centering
+\caption{The multicores scenarios}
-\subsection{The results for different power consumption scenarios}
-\label{sec.compare}
+\begin{tabular}{|*{4}{c|}}
+\hline
+Scenario name & Cluster name & \begin{tabular}[c]{@{}c@{}}No. of nodes\\ in each cluster\end{tabular} &
+ \begin{tabular}[c]{@{}c@{}}No. of cores\\ for each node\end{tabular} \\ \hline
+\multirow{3}{*}{Two sites/ one core} & Taurus & 10 & 1 \\ \cline{2-4}
+ & Graphene & 10 & 1 \\ \cline{2-4}
+ & Griffon & 12 & 1 \\ \hline
+\multirow{3}{*}{Two sites/ multicores} & Taurus & 3 & 3 or 4 \\ \cline{2-4}
+ & Graphene & 3 & 3 or 4 \\ \cline{2-4}
+ & Griffon & 3 & 4 \\ \hline
+\multirow{3}{*}{One site/ one core} & Graphite & 4 & 1 \\ \cline{2-4}
+ & Graphene & 12 & 1 \\ \cline{2-4}
+ & Griffon & 12 & 1 \\ \hline
+\multirow{3}{*}{One site/ multicores} & Graphite & 3 & 3 or 4 \\ \cline{2-4}
+ & Graphene & 3 & 3 or 4 \\ \cline{2-4}
+ & Griffon & 3 & 4 \\ \hline
+\end{tabular}
+\label{table:sen-mc}
+\end{table}
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/eng_con.eps}
+ \caption{Comparing the energy consumptions of running NAS benchmarks over one core and multicores scenarios }
+ \label{fig:eng-cons-mc}
+\end{figure}
+ \begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/time.eps}
+ \caption{Comparing the execution times of running NAS benchmarks over one core and multicores scenarios }
+ \label{fig:time-mc}
+\end{figure}
-\subsection{The comparison of the proposed scaling algorithm }
-\label{sec.compare_EDP}
+ \begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/eng_s_mc.eps}
+ \caption{The energy saving of running NAS benchmarks over one core and multicores scenarios }
+ \label{fig:eng-s-mc}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/per_d_mc.eps}
+ \caption{The performance degradation of running NAS benchmarks over one core and multicores scenarios }
+ \label{fig:per-d-mc}
+\end{figure}
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/dist_mc.eps}
+ \caption{The tradeoff distance of running NAS benchmarks over one core and multicores scenarios }
+ \label{fig:dist-mc}
+\end{figure}
+
+\subsection{The results of using different static power consumption scenarios}
+\label{sec.pow_sen}
+\textcolor{blue}{
+The static power consumption for one core is the leakage power
+consumption when it is idle. The measured static power of the node,
+as in section \ref{sec.grid5000}, had a collection of power values such as
+all cores static powers and the power consumptions of the other devices. Furthermore, the static power for one core is hard to measured precisely. On the other hand, the core has consumed the static power during
+the communication and computation times. However, the static power consumption becomes more important when the execution time is
+increased using DVFS. Therefore, the objective of this section is to verify the ability of the proposed
+scaling algorithm to select the best frequencies when the static power consumption is changing.
+All the results obtained in the previous sections depend on the measured dynamic power
+consumptions as in table \ref{table:grid5000}. Moreover, the static power consumption for one core is represented by 20\% of the measured dynamic power consumption.
+This assumption is extended in this section to taking into account other ratios for the static power consumption.
+In addition to the previous ratio of the static power consumption, two other static power ratios are used, which are 10\% and 30\% of the measured dynamic power of the core.
+As a result, all of these static power scenarios is denoted as follow:
+\begin{itemize}
+\item 10\% of static power scenario
+\item 20\% of static power scenario
+\item 30\% of static power scenario
+\end{itemize}
+The NAS parallel benchmarks, class D, are executed over Nancy site.
+The number of computing nodes used is 16 nodes distributed between three cluster, which are Graphite, Graphene and Griffon. The NAS benchmarks rerun
+with these two new static power scenarios over one site scenario
+using one core per node. }
+
+ \begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/eng_pow.eps}
+ \caption{The energy saving percentages for NAS benchmarks of the three power scenario}
+ \label{fig:eng-pow}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/per_pow.eps}
+ \caption{The performance degradation percentages for NAS benchmarks of the three power scenario}
+ \label{fig:per-pow}
+\end{figure}
+
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/dist_pow.eps}
+ \caption{The tradeoff distance for NAS benchmarks of the three power scenario}
+ \label{fig:dist-pow}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.47]{fig/three_scenarios.pdf}
+ \caption{Comparing the selected frequency scaling factors of MG benchmark for three static power scenarios}
+ \label{fig:fre-pow}
+\end{figure}
+
+\textcolor{blue}{
+The energy saving percentages of NAS benchmarks with these three static power scenarios are presented
+in figure \ref{fig:eng_sen}. This figure shows that 10\% of static power scenario
+gives the biggest energy saving percentage comparing to 20\% and 30\% static power
+scenarios. The smaller ratio of the static power consumption makes the proposed
+scaling algorithm to select smaller frequencies, bigger scaling factors.
+These smaller frequencies has reduced the dynamic energy consumption and thus the
+overall energy consumption is decreased.
+The energy saving percentages of 30\% static power scenario is the smallest between the other scenarios, because of the scaling algorithm selects bigger frequencies, smaller scaling factors, that increased the energy consumption. For example, figure \ref{fig:fre-pow}, illustrates that the proposed scaling algorithm is proportionally selected the best frequency scaling factors according to the static power consumption ratio being used.
+Furthermore, the proposed scaling algorithm tries to limit selecting smaller frequencies, which increased the execution time. Hence, the increase in the execution time is relatively increased the static energy consumption.
+The performance degradation percentages are presented in the figure \ref{fig:per-pow},
+the 30\% of static power scenario had less performance degradation percentage. This because
+bigger frequencies was selected due to the big ratio in the static power consumption.
+The inverse happens in the 20\% and 30\% scenarios, the scaling algorithm is selecting
+smaller frequencies, bigger scaling factors, according to the ratio of the static power.
+The tradeoff distance percentage for the NAS benchmarks with these three static power scenarios
+are presented in the figure \ref{fig:dist}. It shows that the tradeoff
+distance percentage is the best when the 10\% of static power scenario is used, and this percentage
+is decreased for the other two scenarios propositionally to their static power ratios.
+In EP benchmark, the results of energy saving, performance degradation and tradeoff
+distance are showed small differences when the these static power scenarios were used.
+The absent of the communications in this benchmark made the proposed scaling algorithm to select equivalent frequencies even if the static power values are different. While, the
+inverse has been shown for the rest of the benchmarks, which have different communication times
+that increased the static energy consumption proportionally. Therefore, the scaling algorithm relatively selects
+different frequencies for each benchmark when these static power scenarios are used. }
+
+
+\subsection{The comparison of the proposed frequencies selecting algorithm }
+\label{sec.compare_EDP}
+\textcolor{blue}{
+The tradeoff between the energy consumption and the performance of the parallel
+applications had significant importance in the domain of the research.
+Many researchers, \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs},
+have optimized the tradeoff between the energy and the performance using the well known energy and delay product, $EDP=energy \times delay$.
+This model is also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS},
+the objective is to select the frequencies that minimized EDP product for the multi-cores
+architecture when DVFS is used. Moreover, their algorithm is applied online, which synchronously optimized the energy consumption
+and the execution time. Both energy consumption and execution time of a processor are predicted by the their algorithm.
+In this section the proposed frequencies selection algorithm, called Maxdist is compared with Spiliopoulos et al. algorithm, called EDP.
+To make both of the algorithms follow the same direction and fairly comparing them, the same energy model, equation \ref{eq:energy} and
+the execution time model, equation \ref{eq:perf}, are used in the prediction process to select the best vector of the frequencies.
+In contrast, the proposed algorithm starts the search space from the lower bound computed as in equation the \ref{eq:Fint}. Also, the algorithm
+stops the search process when it is reached to the lower bound as mentioned before. In the same way, the EDP algorithm is developed to start from the
+same upper bound used in Maxdist algorithm, and it stops the search process when a minimum available frequencies is reached.
+Finally, the resulting EDP algorithm is an exhaustive search algorithm that test all possible frequencies, starting from the initial frequencies,
+and selecting those minimized the EDP product.
+Both algorithms were applied to NAS benchmarks, class D, over 16 nodes selected from grid'5000 clusters.
+The participating computing nodes are distributed between two sites and one site to have two different scenarios that used in the section \ref{sec.res}.
+The experimental results: the energy saving, performance degradation and tradeoff distance percentages are
+presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/edp_eng}
+ \caption{Comparing of the energy saving for the proposed method with EDP method}
+ \label{fig:edp-eng}
+\end{figure}
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/edp_per}
+ \caption{Comparing of the performance degradation for the proposed method with EDP method}
+ \label{fig:edp-perf}
+\end{figure}
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{fig/edp_dist}
+ \caption{Comparing of the tradeoff distance for the proposed method with EDP method}
+ \label{fig:edp-dist}
+\end{figure}
+As shown form these figures, the proposed frequencies selection algorithm, Maxdist, outperform the EDP algorithm in term of energy and performance for all of the benchmarks executed over the two scenarios.
+Generally, the proposed algorithm gives better results for all benchmarks because it is
+optimized the distance between the energy saving and the performance degradation in the same time.
+Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
+Whereas, the EDP algorithm gives some times negative tradeoff values for some benchmarks in the two sites scenarios.
+These negative tradeoff values mean that the performance degradation percentage is higher than energy saving percentage.
+The higher positive value of the tradeoff distance percentage mean that the energy saving percentage is much higher than the performance degradation percentage.
+The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and
+$O(N \cdot M \cdot F^2)$ respectively. Where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the
+maximum number of available frequencies. The proposed algorithm, Maxdist, has selected the best frequencies in a small execution time,
+on average is equal to 0.01 $ms$, when it is executed over 32 nodes distributed between Nancy and Lyon sites.
+While the EDP algorithm was slower than Maxdist algorithm by ten times over the same number of nodes and same distribution, its execution time on average
+is equal to 0.1 $ms$.
+}
\section{Conclusion}
