\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
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.
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 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
+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 there computations to
-communications ratio is not affected by the increase of the number of local communications.
+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
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 which is more powerful.
-Therefore, the energy saving of EP benchmarks are bigger in the two site scenario due
+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
+
+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\%.
+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
+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
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 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}
+\textcolor{red}{The transition between the execution times to the performance degradation is not clear}
+
+
+The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to
+the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its
+performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
+The rest of the benchmarks showed different performance degradation percentages, which decrease
+when the communication times increase and vice versa.
+Figure \ref{fig:dist} presents the distance percentage between the energy saving and the performance degradation for each benchmark over both scenarios. The tradeoff distance percentage can be
+computed as in equation \ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance
+tradeoff, on average it is equal to 26\%. The one site scenario using both 16 and 32 nodes had better energy and performance
+tradeoff comparing to the two sites scenario because the former has high speed local communications
+which increase the computations to communications ratio and the latter uses long distance communications which decrease this ratio.
-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:
+ 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}
+
+
+\subsection{The experimental results of multi-cores clusters}
\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
+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
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.
+the two sites multi-cores scenario than those over the two sites one core scenario. In the two sites multi-cores scenario, There are three types of communications :
+\begin{itemize}
+\item between cores on the same node via shared memory
+\item between cores from distinct nodes but belonging to the same cluster or site via local network
+\item between cores from distinct sites via long distance network
+\end{itemize}
+The latency of the communications increases from shared memory to LAN to WAN.
+Therefore, using multi-cores communicating via shared memory
+has reduced the communication times, and thus the overall execution time is also decreased.
-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
+
+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!}
+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 when compared to the ratios of the other scenarios.
+More energy reduction was achieved when this ratio is increased because the proposed scaling algorithm selects smaller frequencies that decrease the dynamic power consumption.
+
+
+ \textcolor{red}{ The next sentence is completely false! It is impossible to have these results! 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
+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.
+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!}
+
+The energy saving percentages of all NAS benchmarks running over these four scenarios are presented in the figure \ref{fig:eng-s-mc}. It shows that the energy saving percentages over the two sites multi-cores scenario
+and over the two sites one core scenario are on average equal to 22\% and 18\%
+respectively. The energy saving percentages are higher in the former scenario because its computations to communications ratio is higher than the ratio of the latter scenario 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}.
+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.
-It indicates that the performance
-degradation percentages for the NAS benchmarks are higher over the two sites
+The performance degradation percentages of the NAS benchmarks are presented in
+figure \ref{fig:per-d-mc}. It shows 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.
+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 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 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 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.
-
+site one core scenario, their performance degradation percentages are equal on average
+to 10\% and are higher than those executed over the one site multi-cores scenario,
+which on average is equal to 7\%.
+\textcolor{red}{the next sentence is completely false!
+The higher performance degradation percentages over the first scenario is due to the use of multi-cores which
+decreases the computations to communications ratio. Therefore, selecting small
+frequencies by the scaling algorithm do not increase the execution time significantly. }
+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 verify which scenario is the best in terms of energy reduction and performance. The figure shows that using muti-cores in both of the one site and two sites scenarios gives bigger tradeoff distance percentages, on overage equal to 17.6\% and 15.3\% respectively, than using one core per node in both of one site and two sites scenarios, on average equal to 14.7\% and 13.3\% respectively.
\begin{table}[]
\centering
\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
+\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
-frequencies selecting algorithm when the static power consumption is changed.
-
+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 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:
+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}
-
-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.
+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
\begin{figure}
\centering
\includegraphics[scale=0.47]{fig/three_scenarios.pdf}
- \caption{Comparing the selected frequencies of MG benchmarks for three static power scenarios}
+ \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 showed the 10\% of static power scenario
+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
-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.
+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, because
+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 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
+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 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.
+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}
\label{sec.concl}
