\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}).
\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
The energy consumptions and the execution times for all the benchmarks are
presented in the plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively.
-In general, the energy consumed while executing the NAS benchmarks over one site scenario
-for 16 and 32 nodes is lower than the energy consumed while executing over the two sites.
-The long distance communications between the two distributed sites increases the idle time which leads to more static energy consumption.
+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
over one site with 16 and 32 nodes are also lower when compared to those of the two sites
scenario.
-
-
-The EP and MG benchmarks, where there are no or small communications, showed
-that their execution times and the energy consumptions are not effected
-significantly in both scenarios and when the number of nodes is increase,
-while the other benchmarks showed the inverse, because they have more communications
-that proportionally increase the communication times if there are slow
-communications or using more number of nodes or both of them.
+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.
\begin{figure}
\centering
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 higher 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 there 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.
+this cluster is replaced with Taurus cluster which is more powerful.
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
+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 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\%.
+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.
+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.
+
+ \textcolor{red}{
+The proposed scaling algorithm selecting smaller frequencies in two sites scenario,
+due to decreasing in the computations to communications ratio when the number of nodes is increased and
+leads to less performance degradation percentage.
+In contrast, 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.
+The inverse is happens in this scenario when the number of computing nodes is increased
+the performance degradation percentage is decreased. So, using double number of computing
+nodes when the communications occur in high speed network not decreased the computations to
+communication ratio. Moreover, as shown in the figure \ref{fig:time_sen}, the execution time of one site scenario with 32 nodes
+are less by approximately double, linear speed-up, for most of the benchmarks comparing to the one site with 16 nodes scenario.
+This leads to increased the number of the critical nodes which any one of them may increased the overall the execution time of the benchmarks.
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}.
+communications and no slack times in this benchmarks which their performance controlled by
+the computing powers of the nodes.
+The tradeoff between these scenarios can be computed as in the tradeoff 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.
+compared to the two sites scenarios, due to the increase or decreased 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.
+which on average is up 26\%. Therefore, the tradeoff distance is related linearly to the energy saving
+percentage. Finally, the best energy and performance tradeoff depends on the all of the following:
+1) the computations to communications ratio when there is a communications and slack times, 2) the differences in computing powers
+between the computing nodes and 3) the differences in static and the dynamic powers of the nodes.}
+
+
+
+\subsection{The experimental results of multicores clusters}
+\label{sec.res-mc}
+The grid'5000 clusters have different number of cores embedded in their nodes
+as in the Table \ref{table:grid5000}. Moreover, the cores of each node are
+connected via shared memory model, the data transfer between cores' local
+memories achieved via the global memory \cite{rauber_book}. Therefore, in
+this section the proposed scaling algorithm is implemented over the grid'5000
+clusters which are included multicores in the selected nodes as same as the
+two previous platform scenarios that mentioned in the section \ref{sec.res}.
+The two platform scenarios, the two sites and one site scenarios, with 32
+nodes are reconfigured to used multicores for each node. For example if
+the participating number of nodes from a certain cluster is equal to 12 nodes,
+in the multicores scenario the selected nodes is equal to 3 nodes with using
+4 cores for each of them to produced 12 cores. These scenarios with one
+core and multicores are demonstrated 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 represented
+in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.
+The execution times of NAS benchmarks over the one site multicores scenario
+is higher than the execution time of those running over one site multicores scenario.
+The reason in the one site multicores scenario the communication is increased significantly,
+and all node's cores share the same node network link which increased
+the communication times. Whereas, the execution times of the NAS benchmarks over
+the two site multicores scenario is less than those executed over the two
+sites one core scenario. This goes back when using multicores is decreasing the communications.
+As explained previously, the cores shared same nodes' linkbut the communications between the cores
+are still less than the communication times between the nodes over the long distance
+networks, and thus the over all execution time decreased. Generally, executing
+the NAS benchmarks over the one site one core scenario gives smaller execution times
+comparing to other scenarios. This due to each node in this scenario has it's
+dedicated network link that used independently by one core, while in the other
+scenarios the communication times are higher when using long distance communications
+link or using the shared link communications between cores of each node.
+On the other hand, the energy consumptions of the NAS benchmarks over the
+one site one cores is less than the one site multicores scenario because
+this scenario had less execution time as mentioned before. Also, in the
+one site one core scenario the computations to communications ratio is
+higher, then the new scaled frequencies are decreased the dynamic energy
+consumption which is decreased exponentially
+with the new frequency scaling factors. These experiments also showed, the energy
+consumption and the execution times of EP and MG benchmarks over these four
+scenarios are not change a lot, because there are no or small communications
+which are increase or decrease the static power consumptions.
+The other benchmarks were showed that their energy consumptions and execution times
+are changed according to the decreasing or increasing in the communication
+times that are different from scenario to other or due to the amount of
+communications in each of them.
+
+The energy saving percentages of all NAS benchmarks, as in figure
+\ref{fig:eng-s-mc}, running over these four scenarios are presented. The figure
+showed the energy saving percentages of NAS benchmarks over two sites multicores scenario is higher
+than two sites once core scenario, because the computation
+times in this scenario is higher than the other one, then the more reduction in the
+dynamic energy can be obtained as mentioned previously. In contrast, in the one site one
+core and one site multicores scenarios the energy saving percentages
+are approximately equivalent, on average they are up to 25\%. In these both scenarios there are a small difference in the
+computations to communications ratio, leading the proposed scaling algorithm
+to selects the frequencies proportionally to these ratios and keeping
+as much as possible the energy saving percentages the same. The
+performance degradation percentages of NAS benchmarks are presented in
+figure \ref{fig:per-d-mc}. This figure indicates that performance
+degradation percentages of running NAS benchmarks over two sites
+multocores scenario, on average is equal to 7\%, gives more performance degradation percentage
+than two sites one core scenario, which on average is equal to 4\%.
+Moreover, using the two sites multicores scenario increased
+the computations to communications ratio, which may be increased
+the overall execution time when the proposed scaling algorithm is applied and scaling down the frequencies.
+The inverse was happened when the benchmarks are executed over one
+site one core scenario their performance degradation percentages, on average
+is equal to 10\%, are higher than those executed over one sit one core,
+which on average is equal to 7\%. So, in one site
+multicores scenario the computations to communications ratio is decreased
+as mentioned before, thus selecting new frequencies are not increased
+the overall execution time. The tradeoff distances of all NAS
+benchmarks over all scenarios are presented in the figure \ref{fig:dist-mc}.
+These tradeoff distances are used to verified which scenario is the best in term of
+energy and performance ratio. The one sites multicores scenario is the best scenario in term of
+energy and performance tradeoff, on average is equal to 17.6\%, when comparing to the one site one core
+scenario, one average is equal to 15.3\%. The one site multicores scenario
+has the same energy saving percentages of the one site one core scenario but
+with less performance degradation. The two sites multicores scenario is gives better
+energy and performance tradeoff, one average is equal to 14.7\%, than the two sites
+one core, on average is equal to 13.3\%.
+Finally, using multicore in both scenarios increased the energy and performance tradeoff
+distance. This generally due to using multicores was increased the computations to communications
+ratio in two sites scenario and thus the energy saving percentage increased over the performance degradation percentage, whereas this ratio was decreased
+in one site scenario causing the performance degradation percentage decreased over the energy saving percentage.
+
+
+
+
+
+\begin{table}[]
+\centering
+\caption{The multicores scenarios}
-\subsection{The experimental results of multi-cores clusters}
-\label{sec.res}
+\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}
-\subsection{The results for different power consumption scenarios}
-\label{sec.compare}
+\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}
+ \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}
+The static power consumption for one core of the computing node is the leakage power
+consumption when this core is in the idle state. The node's idle state power value that measured
+as in section \ref{sec.grid5000} had many power consumptions embedded such as
+all cores static powers in addition to the power consumption of the other devices. So, the static power for one core
+can't measured precisely. On the other hand, while the static power consumption of
+one core representing the core's power when there is no any computation, thus
+the majority of ratio of the total power consumption is depends on the dynamic power consumption.
+Despite that, the static power consumption is becomes more important when the execution time
+increased using DVFS. Therefore, the objective of this section is to verify the ability of the proposed
+frequencies selecting algorithm when the static power consumption is changed.
+
+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 is assumed for
+one core represents 20\% of the measured dynamic power of that core.
+This assumption is extended in this section to taking into account others ratios for the static power consumption.
+In addition to the previous ratio of the static power consumption, two other scenarios are used which
+all of these scenarios can be 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}
+
+These three scenarios represented the ratio of the static power consumption that can be computed from
+the dynamic power consumption of the core. The NAS benchmarks of class D are executed over 16 nodes
+in the Nancy site using three clusters: Graphite, Graphene and Griffon. As same as used before, the one site 16 nodes
+platform scenario explained in the last experiments, as in table \ref{tab:sc}, is uses to run
+the NAS benchmarks with these static power scenarios.
+
+ \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}
-\subsection{The comparison of the proposed scaling algorithm }
+\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 frequencies of MG benchmarks for three static power scenarios}
+ \label{fig:fre-pow}
+\end{figure}
+
+The energy saving percentages of NAS benchmarks with these three static power scenarios are presented
+in figure \ref{fig:eng_sen}. This figure showed the 10\% of static power scenario
+gives the biggest energy saving percentage comparing to 20\% and 30\% static power
+scenario. When using smaller ratio of static power consumption, the proposed
+frequencies selecting algorithm selects smaller frequencies, bigger scaling factors,
+because the static energy consumption not increased significantly the overall energy
+consumption. Therefore, more energy reduction can be achieved when the frequencies are scaled down.
+For example figure \ref{fig:fre-pow}, illustrated that the proposed algorithm
+proportionally scaled down the new computed frequencies with the overall predicted energy
+consumption. The results of 30\% static power scenario gives the smallest energy saving percentages
+because the new selected frequencies produced smaller ratio in the reduced energy consumption.
+Furthermore, The proposed algorithm tries to limit selecting smaller frequencies that increased
+the static energy consumption if the static power consumption is increased.
+The performance degradation percentages are presented in the figure \ref{fig:per-pow},
+the 30\% of static power scenario had less performance degradation percentage, because
+bigger frequencies was selected due to the big ratio in the static power consumption.
+The inverse was happens in the 20\% and 30\% scenario, the algorithm was selected
+biggest frequencies, smaller scaling factors, according to this increased in the static power ratios.
+The tradoff distance for the NAS benchmarks with these three static powers scenarios
+are presented in the figure \ref{fig:dist}. The results showed that the tradeoff
+distance 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 benchmarks, the results of energy saving, performance degradation and tradeoff
+distance are showed small differences when the these static power scenarios were used,
+because this benchmark not has communications. The proposed algorithm is selected
+same frequencies in this benchmark when all these static power scenarios are used.
+The small differences in the results are due to the static power is consumed during the computation
+times side by side to the dynamic power consumption, knowing that the dynamic power consumption
+representing the highest ratio in the total power consumption of the core, then any change in
+the static power during these times have less affect on the overall energy consumption. While the
+inverse was happens for the rest of the benchmarks which have the communications
+that increased the static energy consumption linearly to the mount of communications
+in these benchmarks.
+
+
+
+\subsection{The comparison of the proposed frequencies selecting algorithm }
\label{sec.compare_EDP}
+The tradeoff between the energy consumption and the performance of the parallel
+application had significant importance in the domain of the research.
+Many researchers, \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs},
+are optimized the tradeoff between the energy and performance using the energy and delay product, $EDP=energy \times delay$.
+This model is used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS},
+the objective is to selects the suitable frequencies that minimized EDP product for the multicores
+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 frequency 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 reaching to the lower bound as mentioned before. While, the EDP algorithm is developed to start from the
+same upper bound until it reach to the minimum available frequencies. Finally, resulting the algorithm is an exhaustive search algorithm that
+test all possible frequencies, starting from the initial frequencies, and selecting those minimized the EDP products.
+
+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 to had two different scenarios.
+These scenarios are two sites and one site scenarios that explained previously.
+The experimental results of the energy saving, performance degradation and tradeoff distance are
+presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
+
+In one site scenario the proposed frequencies selection algorithm outperform the EDP algorithm
+in term of energy and performance for all of the benchmarks. While, the compassion results from the two sites scenario
+showed that the proposed algorithm outperform EDP algorithm for all benchmarks except MG benchmark.
+In case of MG benchmark the are small communications and bigger frequencies selected in EDP algorithm
+decreased the performance degradation more than the frequencies selected by Maxdist algorithm.
+While the energy saving percentage are higher for Maxdist algorithm.
+
+Generally, the proposed algorithm gives better results for all benchmarks because it
+optimized the distance between the energy saving and the performance degradation.
+Whereas, in EDP algorithm gives negative tradeoff for some benchmarks in the two sites scenarios.
+These negative tradeoffs mean the performance degradation percentage is higher than energy saving percentage.
+The higher positive value for tradeoff distance is mean the best energy and performance tradeoff is achieved synchronously, when
+the energy saving percentage is much higher than the performance degradation percentage
+The time complexity of the proposed algorithm is $O(N \cdot M \cdot F)$, where $N$ is the number of the clusters,
+$M$ is the number of nodes and $F$ is the maximum number of available frequencies. The algorithm is selected
+the best frequencies in small execution time, on average is equal to 0.01 $ms$ when it works over 32 nodes.
+While the EDP algorithm was slower than Maxdist algorithm by ten times, where their execution time on average
+takes 0.1 $ms$ to selects the suitable frequencies over 32 nodes.
+The time complexity of this algorithm is $O(N^2 \cdot M^2 \cdot F)$.
+
+
+
+
+
+\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}
+
+
\section{Conclusion}
\label{sec.concl}
