\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
-\newcommand{\Pmax}[1][]{\Xsub{P}{max}_{#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,
\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}
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 $Pmax[jx]$. 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}).
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} (Pmax[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 $j$,
Therefore, the dynamic power of one core is computed as the difference between the maximum
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}
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 one site with
-16 or 32 nodes is on average equal to 3\% or 10\% respectively.
+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.
- \textcolor{red}{please correct the following paragraph because I do not understand it at all! Stop using we, this because, effected, while, ...}
-
-
+
+ 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).
- 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 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 benchmark gives bigger performance degradation percentage, because there is no
+communications and no slack times in this benchmark which their performance controlled by
+the computing powers of the nodes. \textcolor{red}{les deux phrases précédentes n'ont pas de sens}
+
+
+Figure \ref{fig:dist} presents the distance between the energy consumption reduction and the performance degradation for all benchmarks over both scenarios. This distance can be computed as in the tradeoff function \ref{eq:max}. The one site scenario with 16 and 32 nodes had the best tradeoff distance 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 gives the best energy and performance tradeoff which is on average equal to 26\%. \textcolor{red}{distance is a percentage}
+
+ Therefore, the tradeoff distance is linearly related to the energy saving
+percentage. 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.
+
+\textcolor{red}{compare the two scenarios}
\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 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 represented
+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 of NAS benchmarks over the one site multicores scenario
-is higher than the execution time of those running over one site multicores scenario.
-This because 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. While, 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 because using multicores decrease the communications,
-while the cores shared same nodes' link but the communications between the cores
-are 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 gives smaller execution times
-comparing to other scenarios. This because 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 communication
-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, because the dynamic power consumption are 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, this because the the computation
-times in the two sites multicores scenario is higher than the computation times
-of the two sites one core scenario, 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\%. This
-because in the 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, on average is equal to 7\%, gives more performance degradation percentage
-than two sites one core scenario, which on average is equal to 4\%.
-This because when using the two sites multicores scenario increased
-the computations to communications ratio, which may be increased the effect
-on 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
+
+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 site multi-cores scenario than those 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. \textcolor{red}{this is not true}
+
+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.
+The computations to communications ratios of the NAS benchmarks are higher over the one site one core scenario than the other scenarios \textcolor{red}{ then the new scaled frequencies are decreased the dynamic energy
+consumption which is decreased exponentially
+with the new frequency scaling factors. I do not understand this sentence}
+\textcolor{red}{It is useless to use multi-cores then!}
+
+
+ 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.
+
+The energy saving percentages of all NAS benchmarks running over these four scenarios are presented in figure \ref{fig:eng-s-mc}. The figure
+shows that the energy saving percentages are higher over the two sites multi-cores scenario
+than over the two sites one core scenario, because the computation
+times are higher in the first scenario than in the latter, thus, more dynamic energy can be saved by applying the frequency scaling algorithm. \textcolor{red}{why the computation times are higher!}
+
+
+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.
+
+
+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 sit one core,
-which on average is equal to 7\%. This because in one site
+is equal to 10\%, are higher than those executed over one site one core,
+which on average is equal to 7\%. \textcolor{red}{You are comparing the one
+site one core scenario to itself! Please rewrite all the following paragraphs because they are full of mistakes! Look how I modified the previous parts, discover your mistakes and stop making the same mistakes.}
+
+So, in one site
multicores scenario the computations to communications ratio is decreased
-as mentioned before, thus selecting new frequencies are less effect
-on the overall execution time. The tradeoff distances of all NAS
+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\%. This because the one site multicores scenario
+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 is because using multicores are increased the computations to communications
-ratio in two sites scenario and thus the energy saving increased over the performance degradation, whereas decreased this ratio
-in one site scenario causing the performance degradation decreased over the energy saving.
+
+Finally, using multi-cores 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.
\label{fig:dist-mc}
\end{figure}
-\subsection{The results for different power consumption scenarios}
-\label{sec.compare}
+\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}
+
+\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 scaling algorithm }
+\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}
