From: Arnaud Giersch Date: Mon, 15 Dec 2014 12:34:38 +0000 (+0100) Subject: Misc corrections. X-Git-Tag: pdsec15_submission~27 X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/commitdiff_plain/8635fccbf588d4a9dd25d07624be34eae4ab66ad?ds=inline Misc corrections. * remove double spaces * use \dots{} instead of ... * tidy \ref to figures, tables, etc. * fix overfull hboxes in tables * reformat some parts of source code (use git diff --word-diff to see changed words) --- diff --git a/Heter_paper.tex b/Heter_paper.tex index dc87dc8..014ad60 100644 --- a/Heter_paper.tex +++ b/Heter_paper.tex @@ -118,7 +118,7 @@ This heterogeneous platform executes more than 5 GFLOPS per watt while consuming Besides platform improvements, there are many software and hardware techniques to lower the energy consumption of these platforms, such as scheduling, DVFS, -... DVFS is a widely used process to reduce the energy consumption of a +\dots{} DVFS is a widely used process to reduce the energy consumption of a processor by lowering its frequency \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces the number of FLOPS executed by the processor which might increase the execution @@ -151,24 +151,24 @@ Finally, in Section~\ref{sec.concl} the paper ends with a summary and some futur \section{Related works} \label{sec.relwork} DVFS is a technique used in modern processors to scale down both the voltage and -the frequency of the CPU while computing, in order to reduce the energy -consumption of the processor. DVFS is also allowed in GPUs to achieve the -same goal. Reducing the frequency of a processor lowers its number of FLOPS and -might degrade the performance of the application running on that processor, -especially if it is compute bound. Therefore selecting the appropriate frequency -for a processor to satisfy some objectives while taking into account all the -constraints, is not a trivial operation. Many researchers used different -strategies to tackle this problem. Some of them developed online methods that -compute the new frequency while executing the application, such as -~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}. Others -used offline methods that might need to run the application and profile it -before selecting the new frequency, such as -~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The -methods could be heuristics, exact or brute force methods that satisfy varied -objectives such as energy reduction or performance. They also could be adapted -to the execution's environment and the type of the application such as -sequential, parallel or distributed architecture, homogeneous or heterogeneous -platform, synchronous or asynchronous application, ... +the frequency of the CPU while computing, in order to reduce the energy +consumption of the processor. DVFS is also allowed in GPUs to achieve the same +goal. Reducing the frequency of a processor lowers its number of FLOPS and might +degrade the performance of the application running on that processor, especially +if it is compute bound. Therefore selecting the appropriate frequency for a +processor to satisfy some objectives while taking into account all the +constraints, is not a trivial operation. Many researchers used different +strategies to tackle this problem. Some of them developed online methods that +compute the new frequency while executing the application, such +as~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}. +Others used offline methods that might need to run the application and profile +it before selecting the new frequency, such +as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. +The methods could be heuristics, exact or brute force methods that satisfy +varied objectives such as energy reduction or performance. They also could be +adapted to the execution's environment and the type of the application such as +sequential, parallel or distributed architecture, homogeneous or heterogeneous +platform, synchronous or asynchronous application, \dots{} In this paper, we are interested in reducing energy for message passing iterative synchronous applications running over heterogeneous platforms. Some works have already been done for such platforms and they can be classified into two types of heterogeneous platforms: @@ -179,30 +179,44 @@ Some works have already been done for such platforms and they can be classified \end{itemize} -For the first type of platform, the computing intensive parallel tasks are executed on the GPUs and the rest are executed -on the CPUs. Luley et al. -~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a heterogeneous -cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal was to maximize the -energy efficiency of the platform during computation by maximizing the number of FLOPS per watt generated. -In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et al. developed a scheduling -algorithm that distributes workloads proportional to the computing power of the nodes which could be a GPU or a CPU. All the tasks must be completed at the same time. -In~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, Rong et al. showed that -a heterogeneous (GPUs and CPUs) cluster that enables DVFS gave better energy and performance -efficiency than other clusters only composed of CPUs. +For the first type of platform, the computing intensive parallel tasks are +executed on the GPUs and the rest are executed on the CPUs. Luley et +al.~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a +heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main +goal was to maximize the energy efficiency of the platform during computation by +maximizing the number of FLOPS per watt generated. +In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et +al. developed a scheduling algorithm that distributes workloads proportional to +the computing power of the nodes which could be a GPU or a CPU. All the tasks +must be completed at the same time. In~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, +Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables +DVFS gave better energy and performance efficiency than other clusters only +composed of CPUs. -The work presented in this paper concerns the second type of platform, with heterogeneous CPUs. -Many methods were conceived to reduce the energy consumption of this type of platform. Naveen et al.~\cite{Naveen_Power.Efficient.Resource.Scaling} -developed a method that minimizes the value of $energy\cdot delay^2$ (the delay is the sum of slack times that happen during synchronous communications) by dynamically assigning new frequencies to the CPUs of the heterogeneous cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed -an algorithm that divides the executed tasks into two types: the critical and -non critical tasks. The algorithm scales down the frequency of non critical tasks proportionally to their slack and communication times while limiting the performance degradation percentage to less than 10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed - a heterogeneous cluster composed of two types -of Intel and AMD processors. They use a gradient method to predict the impact of DVFS operations on performance. -In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, - the best frequencies for a specified heterogeneous cluster are selected offline using some -heuristic. Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic programming approach to -minimize the power consumption of heterogeneous servers while respecting given time constraints. This approach -had considerable overhead. -In contrast to the above described papers, this paper presents the following contributions : +The work presented in this paper concerns the second type of platform, with +heterogeneous CPUs. Many methods were conceived to reduce the energy +consumption of this type of platform. Naveen et +al.~\cite{Naveen_Power.Efficient.Resource.Scaling} developed a method that +minimizes the value of $energy\cdot delay^2$ (the delay is the sum of slack +times that happen during synchronous communications) by dynamically assigning +new frequencies to the CPUs of the heterogeneous cluster. Lizhe et +al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed an algorithm +that divides the executed tasks into two types: the critical and non critical +tasks. The algorithm scales down the frequency of non critical tasks +proportionally to their slack and communication times while limiting the +performance degradation percentage to less than +10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a +heterogeneous cluster composed of two types of Intel and AMD processors. They +use a gradient method to predict the impact of DVFS operations on performance. +In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and +\cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best +frequencies for a specified heterogeneous cluster are selected offline using +some heuristic. Chen et +al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic +programming approach to minimize the power consumption of heterogeneous servers +while respecting given time constraints. This approach had considerable +overhead. In contrast to the above described papers, this paper presents the +following contributions : \begin{enumerate} \item two new energy and performance models for message passing iterative synchronous applications running over a heterogeneous platform. Both models take into account communication and slack times. The models can predict the required energy and the execution time of the application. @@ -228,14 +242,14 @@ network. Therefore, each node has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all have the same network bandwidth and latency. -The overall execution time of a distributed iterative synchronous application -over a heterogeneous platform consists of the sum of the computation time and -the communication time for every iteration on a node. However, due to the -heterogeneous computation power of the computing nodes, slack times might occur -when fast nodes have to wait, during synchronous communications, for the slower -nodes to finish their computations (see Figure~(\ref{fig:heter})). -Therefore, the overall execution time of the program is the execution time of the slowest -task which has the highest computation time and no slack time. +The overall execution time of a distributed iterative synchronous application +over a heterogeneous platform consists of the sum of the computation time and +the communication time for every iteration on a node. However, due to the +heterogeneous computation power of the computing nodes, slack times might occur +when fast nodes have to wait, during synchronous communications, for the slower +nodes to finish their computations (see Figure~\ref{fig:heter}). Therefore, the +overall execution time of the program is the execution time of the slowest task +which has the highest computation time and no slack time. \begin{figure}[!t] \centering @@ -387,7 +401,7 @@ $Tcp_{i}$ might be different and different frequency scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce slack times. The communication time of a processor $i$ is noted as $Tcm_{i}$ and could contain slack times when communicating with slower -nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal +nodes, see Figure~\ref{fig:heter}. Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the frequency scaling factor and the dynamic power of each node as in (\ref{eq:Edyn}), the static energy is computed as the sum of the execution time @@ -495,12 +509,12 @@ normalized execution time is inverted which gives the normalized performance equ \caption{The energy and performance relation} \end{figure} -Then, the objective function can be modeled in order to find the maximum distance -between the energy curve (\ref{eq:enorm}) and the performance -curve (\ref{eq:pnorm_inv}) over all available sets of scaling factors. This -represents the minimum energy consumption with minimum execution time (maximum -performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then the objective -function has the following form: +Then, the objective function can be modeled in order to find the maximum +distance between the energy curve (\ref{eq:enorm}) and the performance curve +(\ref{eq:pnorm_inv}) over all available sets of scaling factors. This +represents the minimum energy consumption with minimum execution time (maximum +performance) at the same time, see Figure~\ref{fig:r1} or +Figure~\ref{fig:r2}. Then the objective function has the following form: \begin{equation} \label{eq:max} Max Dist = @@ -518,29 +532,33 @@ the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynam \label{sec.optim} \subsection{The algorithm details} -In this section, algorithm \ref{HSA} is presented. It selects the frequency scaling factors -vector that gives the best trade-off between minimizing the energy consumption and maximizing -the performance of a message passing synchronous iterative application executed on a heterogeneous -platform. It works online during the execution time of the iterative message passing program. -It uses information gathered during the first iteration such as the computation time and the -communication time in one iteration for each node. The algorithm is executed after the first -iteration and returns a vector of optimal frequency scaling factors that satisfies the objective -function (\ref{eq:max}). The program applies DVFS operations to change the frequencies of the CPUs -according to the computed scaling factors. This algorithm is called just once during the execution -of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called -in the iterative MPI program. - -The nodes in a heterogeneous platform have different computing powers, thus while executing message -passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their -computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). -These periods are called idle or slack times. -The algorithm takes into account this problem and tries to reduce these slack times when selecting the -frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase -the execution times of fast nodes and minimize the differences between the computation times of -fast and slow nodes. The value of the initial frequency scaling factor for each node is inversely -proportional to its computation time that was gathered from the first iteration. These initial frequency -scaling factors are computed as a ratio between the computation time of the slowest node and the -computation time of the node $i$ as follows: +In this section, Algorithm~\ref{HSA} is presented. It selects the frequency +scaling factors vector that gives the best trade-off between minimizing the +energy consumption and maximizing the performance of a message passing +synchronous iterative application executed on a heterogeneous platform. It works +online during the execution time of the iterative message passing program. It +uses information gathered during the first iteration such as the computation +time and the communication time in one iteration for each node. The algorithm is +executed after the first iteration and returns a vector of optimal frequency +scaling factors that satisfies the objective function (\ref{eq:max}). The +program applies DVFS operations to change the frequencies of the CPUs according +to the computed scaling factors. This algorithm is called just once during the +execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed +scaling algorithm is called in the iterative MPI program. + +The nodes in a heterogeneous platform have different computing powers, thus +while executing message passing iterative synchronous applications, fast nodes +have to wait for the slower ones to finish their computations before being able +to synchronously communicate with them as in Figure~\ref{fig:heter}. These +periods are called idle or slack times. The algorithm takes into account this +problem and tries to reduce these slack times when selecting the frequency +scaling factors vector. At first, it selects initial frequency scaling factors +that increase the execution times of fast nodes and minimize the differences +between the computation times of fast and slow nodes. The value of the initial +frequency scaling factor for each node is inversely proportional to its +computation time that was gathered from the first iteration. These initial +frequency scaling factors are computed as a ratio between the computation time +of the slowest node and the computation time of the node $i$ as follows: \begin{equation} \label{eq:Scp} Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} @@ -552,25 +570,25 @@ and the computation scaling factor $Scp_i$ as follows: \label{eq:Fint} F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N} \end{equation} -If the computed initial frequency for a node is not available in the gears of -that node, it is replaced by the nearest available frequency. In figure -(\ref{fig:st_freq}), the nodes are sorted by their computing power in ascending -order and the frequencies of the faster nodes are scaled down according to the -computed initial frequency scaling factors. The resulting new frequencies are -colored in blue in figure (\ref{fig:st_freq}). This set of frequencies can be -considered as a higher bound for the search space of the optimal vector of -frequencies because selecting frequency scaling factors higher than the higher -bound will not improve the performance of the application and it will increase -its overall energy consumption. Therefore the algorithm that selects the -frequency scaling factors starts the search method from these initial -frequencies and takes a downward search direction toward lower frequencies. The -algorithm iterates on all left frequencies, from the higher bound until all -nodes reach their minimum frequencies, to compute their overall energy -consumption and performance, and select the optimal frequency scaling factors -vector. At each iteration the algorithm determines the slowest node according to -the equation (\ref{eq:perf}) and keeps its frequency unchanged, while it lowers -the frequency of all other nodes by one gear. The new overall energy -consumption and execution time are computed according to the new scaling +If the computed initial frequency for a node is not available in the gears of +that node, it is replaced by the nearest available frequency. In +Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in +ascending order and the frequencies of the faster nodes are scaled down +according to the computed initial frequency scaling factors. The resulting new +frequencies are colored in blue in Figure~\ref{fig:st_freq}. This set of +frequencies can be considered as a higher bound for the search space of the +optimal vector of frequencies because selecting frequency scaling factors higher +than the higher bound will not improve the performance of the application and it +will increase its overall energy consumption. Therefore the algorithm that +selects the frequency scaling factors starts the search method from these +initial frequencies and takes a downward search direction toward lower +frequencies. The algorithm iterates on all left frequencies, from the higher +bound until all nodes reach their minimum frequencies, to compute their overall +energy consumption and performance, and select the optimal frequency scaling +factors vector. At each iteration the algorithm determines the slowest node +according to the equation (\ref{eq:perf}) and keeps its frequency unchanged, +while it lowers the frequency of all other nodes by one gear. The new overall +energy consumption and execution time are computed according to the new scaling factors. The optimal set of frequency scaling factors is the set that gives the highest distance according to the objective function (\ref{eq:max}). @@ -651,7 +669,7 @@ maximum distance between the energy curve and the performance curve is while \If {$(k=1)$} \State Gather all times of computation and\newline\hspace*{3em}% communication from each node. - \State Call algorithm \ref{HSA}. + \State Call Algorithm \ref{HSA}. \State Compute the new frequencies from the\newline\hspace*{3em}% returned optimal scaling factors. \State Set the new frequencies to nodes. @@ -678,28 +696,33 @@ parallel benchmarks NPB v3.3 \cite{NAS.Parallel.Benchmarks}, running class B on very precise, the maximum normalized difference between the predicted execution time and the real execution time is equal to 0.03 for all the NAS benchmarks. -Since the proposed algorithm is not an exact method it does not test all the possible solutions (vectors of scaling factors) -in the search space. To prove its efficiency, it was compared on small instances to a brute force search algorithm -that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with -different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical -and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time: -for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in -table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4 nodes and \np[ms]{0.15} on average for 144 nodes -to compute the best scaling factors vector. The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number -of iterations and $N$ is the number of computing nodes. The algorithm needs from 12 to 20 iterations to select the best -vector of frequency scaling factors that gives the results of the next sections. +Since the proposed algorithm is not an exact method it does not test all the +possible solutions (vectors of scaling factors) in the search space. To prove +its efficiency, it was compared on small instances to a brute force search +algorithm that tests all the possible solutions. The brute force algorithm was +applied to different NAS benchmarks classes with different number of nodes. The +solutions returned by the brute force algorithm and the proposed algorithm were +identical and the proposed algorithm was on average 10 times faster than the +brute force algorithm. It has a small execution time: for a heterogeneous +cluster composed of four different types of nodes having the characteristics +presented in Table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4 +nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling +factors vector. The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ +is the number of iterations and $N$ is the number of computing nodes. The +algorithm needs from 12 to 20 iterations to select the best vector of frequency +scaling factors that gives the results of the next sections. \section{Experimental results} \label{sec.expe} To evaluate the efficiency and the overall energy consumption reduction of -algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The +Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The experiments were executed on the simulator SimGrid/SMPI which offers easy tools to create a heterogeneous platform and run message passing applications over it. The heterogeneous platform that was used in the experiments, had one core per node because just one process was executed per node. The heterogeneous platform was composed of four types of nodes. Each type of nodes had different characteristics such as the maximum CPU frequency, the number of available -frequencies and the computational power, see Table \ref{table:platform}. The +frequencies and the computational power, see Table~\ref{table:platform}. The characteristics of these different types of nodes are inspired from the specifications of real Intel processors. The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, for example if @@ -765,6 +788,7 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes. \centering \begin{tabular}{|*{7}{r|}} \hline + \hspace{-2.2084pt}% Program & Execution & Energy & Energy & Performance & Distance \\ name & time/s & consumption/J & saving\% & degradation\% & \\ \hline @@ -793,6 +817,7 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes. \centering \begin{tabular}{|*{7}{r|}} \hline + \hspace{-2.2084pt}% Program & Execution & Energy & Energy & Performance & Distance \\ name & time/s & consumption/J & saving\% & degradation\% & \\ \hline @@ -821,6 +846,7 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes. \centering \begin{tabular}{|*{7}{r|}} \hline + \hspace{-2.2084pt}% Program & Execution & Energy & Energy & Performance & Distance \\ name & time/s & consumption/J & saving\% & degradation\% & \\ \hline @@ -849,6 +875,7 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes. \centering \begin{tabular}{|*{7}{r|}} \hline + \hspace{-2.2084pt}% Program & Execution & Energy & Energy & Performance & Distance \\ name & time/s & consumption/J & saving\% & degradation\% & \\ \hline @@ -877,6 +904,7 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes. \centering \begin{tabular}{|*{7}{r|}} \hline + \hspace{-2.2084pt}% Program & Execution & Energy & Energy & Performance & Distance \\ name & time/s & consumption/J & saving\% & degradation\% & \\ \hline @@ -905,6 +933,7 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes. \centering \begin{tabular}{|*{7}{r|}} \hline + \hspace{-2.2084pt}% Program & Execution & Energy & Energy & Performance & Distance \\ name & time/s & consumption/J & saving\% & degradation\% & \\ \hline @@ -929,9 +958,9 @@ The overall energy consumption was computed for each instance according to the energy consumption model (\ref{eq:energy}), with and without applying the algorithm. The execution time was also measured for all these experiments. Then, the energy saving and performance degradation percentages were computed for each -instance. The results are presented in Tables (\ref{table:res_4n}, +instance. The results are presented in Tables~\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n}, -\ref{table:res_64n} and \ref{table:res_128n}). All these results are the average +\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average values from many experiments for energy savings and performance degradation. The tables show the experimental results for running the NAS parallel benchmarks on different number of nodes. The experiments show that the algorithm @@ -945,7 +974,7 @@ computation required for each iteration is divided by the number of computing nodes. On the other hand, more communications are required when increasing the number of nodes so the static energy increases linearly according to the communication time and the dynamic power is less relevant in the overall energy -consumption. Therefore, reducing the frequency with algorithm~(\ref{HSA}) is +consumption. Therefore, reducing the frequency with Algorithm~\ref{HSA} is less effective in reducing the overall energy savings. It can also be noticed that for the benchmarks EP and SP that contain little or no communications, the energy savings are not significantly affected by the high number of nodes. No @@ -971,7 +1000,7 @@ small when compared to the communication times. \caption{The energy and performance for all NAS benchmarks running with a different number of nodes} \end{figure} -Figures \ref{fig:energy} and \ref{fig:per_deg} present the energy saving and +Figures~\ref{fig:energy} and \ref{fig:per_deg} present the energy saving and performance degradation respectively for all the benchmarks according to the number of used nodes. As shown in the first plot, the energy saving percentages of the benchmarks MG, LU, BT and FT decrease linearly when the number of nodes @@ -1004,7 +1033,7 @@ two new power scenarios are the following: The NAS parallel benchmarks were executed again over processors that follow the new power scenarios. The class C of each benchmark was run over 8 or 9 nodes -and the results are presented in Tables \ref{table:res_s1} and +and the results are presented in Tables~\ref{table:res_s1} and \ref{table:res_s2}. These tables show that the energy saving percentage of the 70\%-30\% scenario is smaller for all benchmarks compared to the energy saving of the 90\%-10\% scenario. Indeed, in the latter more dynamic power is consumed @@ -1020,8 +1049,8 @@ really significantly scale down much the frequencies of the nodes in order to limit the increase of the execution time and thus limiting the effect of the consumed static energy. -Both new power scenarios are compared to the old one in figure -(\ref{fig:sen_comp}). It shows the average of the performance degradation, the +Both new power scenarios are compared to the old one in +Figure~\ref{fig:sen_comp}. It shows the average of the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. The comparison shows that the energy saving ratio is proportional to the dynamic power ratio: it is increased when applying the 90\%-10\% scenario @@ -1035,7 +1064,7 @@ higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed algorithm optimizes the energy consumption when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in more energy saving but less performance, for -example see Figure (\ref{fig:scales_comp}). The opposite happens when using a +example see Figure~\ref{fig:scales_comp}. The opposite happens when using a higher ratio for static power, the algorithm proportionally selects smaller scaling values which result in less energy saving but also less performance degradation. @@ -1120,12 +1149,18 @@ They developed a green governor that regularly applies an online frequency selec To fairly compare both algorithms, the same energy and execution time models, equations (\ref{eq:energy}) and (\ref{eq:fnew}), were used for both algorithms to predict the energy consumption and the execution times. Also Spiliopoulos et al. algorithm was adapted to start the search from the initial frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm is an exhaustive search algorithm that minimizes the EDP and has the initial frequencies values as an upper bound. -Both algorithms were applied to the parallel NAS benchmarks to compare their efficiency. Table \ref{table:compare_EDP} presents the results of comparing the execution times and the energy consumption for both versions of the NAS benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous nodes. The results show that our algorithm provides better energy savings than Spiliopoulos et al. algorithm, -on average it results in 29.76\% energy saving while their algorithm returns just 25.75\%. The average of performance degradation percentage is approximately the same for both algorithms, about 4\%. +Both algorithms were applied to the parallel NAS benchmarks to compare their +efficiency. Table~\ref{table:compare_EDP} presents the results of comparing the +execution times and the energy consumption for both versions of the NAS +benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous +nodes. The results show that our algorithm provides better energy savings than +Spiliopoulos et al. algorithm, on average it results in 29.76\% energy saving +while their algorithm returns just 25.75\%. The average of performance +degradation percentage is approximately the same for both algorithms, about 4\%. For all benchmarks, our algorithm outperforms Spiliopoulos et al. algorithm in -terms of energy and performance trade-off, see figure (\ref{fig:compare_EDP}), +terms of energy and performance trade-off, see Figure~\ref{fig:compare_EDP}, because it maximizes the distance between the energy saving and the performance degradation values while giving the same weight for both metrics.