\maketitle
\begin{abstract}
-Green computing emphasizes the importance of energy conservation, minimizing the negative impact
-on the environment while achieving high performance and minimizing operating costs. So, energy reduction
-process in a high performance clusters it can be archived using dynamic voltage and frequency
-scaling (DVFS) technique, through reducing the frequency of a CPU. Using DVFS to lower levels
-result in a high increase in performance degradation ratio. Therefore selecting the best frequencies
-must give the best possible tradeoff between the energy and the performance of parallel program.
-
-In this paper we present a new online heterogeneous scaling algorithm that selects the best vector
-of frequency scaling factors. These factors give the best tradeoff between the energy saving and the
-performance degradation. The algorithm has small overhead and works without training and profiling.
-We developed a new energy model for distributed iterative application running on heterogeneous cluster.
-The proposed algorithm experimented on Simgrid simulator that applying the NAS parallel benchmarks.
-It reduces the energy consumption up to 35\% while limits the performance degradation as much as possible.
+Computing platforms are consuming more and more energy due to the increase of the number of nodes composing them.
+To minimize the operating costs of these platforms many techniques have been used. Dynamic voltage and frequency
+scaling (DVFS) is one of them, it reduces the frequency of a CPU to lower its energy consumption. However,
+lowering the frequency of a CPU might increase the execution time of an application running on that processor.
+Therefore, the frequency that gives the best tradeoff between the energy consumption and the performance of an
+application must be selected.
+
+In this paper, a new online frequencies selecting algorithm for heterogeneous platforms is presented.
+It selects the frequency that gives the best tradeoff between energy saving and performance degradation,
+for each node computing the message passing iterative application. The algorithm has a small overhead and
+works without training or profiling. It uses a new energy model for message passing iterative applications
+running on a heterogeneous platform. The proposed algorithm was evaluated on the Simgrid simulator while
+running the NAS parallel benchmarks. The experiments demonstrated that it reduces the energy consumption
+up to 35\% while limiting the performance degradation as much as possible.
\end{abstract}
\section{Introduction}
\label{sec.intro}
-Modern processors continue increasing in performance,
-the CPUs constructors are competing to achieve maximum number
-of floating point operations per second (FLOPS).
-Thus, the energy consumption and the heat dissipation are increased
-drastically according to this increase. Because the number of FLOPS
-is more related to the power consumption of a CPU
-~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}.
-As an example of the most power hungry cluster, Tianhe-2 became in
-the top of the Top500 list in June 2014 \cite{TOP500_Supercomputers_Sites}.
-It has more than 3 millions of cores and consumed more than 17.8 megawatts.
-Moreover, according to the U.S. annual energy outlook 2014
+The need for more computing power is continually increasing. To partially satisfy this need, most supercomputers
+constructors just put more computing nodes in their platform. The resulting platform might achieve higher floating
+point operations per second (FLOPS), but the energy consumption and the heat dissipation are also increased.
+As an example, the chinese supercomputer Tianhe-2 had the highest FLOPS in November 2014 according to the Top500
+list \cite{TOP500_Supercomputers_Sites}. However, it was also the most power hungry platform with its over 3 millions
+cores consuming around 17.8 megawatts. Moreover, according to the U.S. annual energy outlook 2014
\cite{U.S_Annual.Energy.Outlook.2014}, the price of energy for 1 megawatt-hour
was approximately equal to \$70.
-Therefore, we can consider the price of the energy consumption for the
-Tianhe-2 platform is approximately more than \$10 millions for
-one year. For this reason, the heterogeneous clusters must be offer more
-energy efficiency due to the increase in the energy cost and the environment
-influences. Therefore, a green computing clusters with maximum number of
-FLOPS per watt are required nowadays. For example, the GSIC center of Tokyo,
+Therefore, the price of the energy consumed by the
+Tianhe-2 platform is approximately more than \$10 millions each year.
+The computing platforms must be more energy efficient and offer the highest number of FLOPS per watt possible,
+such as the TSUBAME-KFC at the GSIC center of Tokyo which
became the top of the Green500 list in June 2014 \cite{Green500_List}.
-This heterogeneous platform has more than four thousand of MFLOPS per watt. Dynamic
-voltage and frequency scaling (DVFS) is a process used widely to reduce the energy
-consumption of the processor. In heterogeneous clusters enabled DVFS, many researchers
-used DVFS in a different ways. DVFS can be minimized the energy consumption
-but it leads to a disadvantage due to the increase in performance degradation.
-Therefore, researchers used different optimization strategies to overcame
-this problem. The best tradeoff relation between the energy reduction and
-performance degradation ratio is became a key challenges in a heterogeneous
-platforms. In this paper we are propose a heterogeneous scaling algorithm
-that selects the optimal vector of the frequency scaling factors for distributed
-iterative application, producing maximum energy reduction against minimum
-performance degradation ratio simultaneously. The algorithm has very small
-overhead, works online and not needs for any training or profiling.
+This heterogeneous platform executes more than four GFLOPS per watt.
+
+Besides hardware improvements, there are many software 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 processor by lowering
+its frequency \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also the reduces the number of FLOPS
+executed by the processor which might increase the execution time of the application running over that processor.
+Therefore, researchers used different optimization strategies to select the frequency that gives the best tradeoff
+between the energy reduction and
+performance degradation ratio. \textbf{In our previous paper \cite{Our_first_paper}, a frequency selecting algorithm
+was proposed for distributed iterative application running over homogeneous platform. While in this paper the algorithm is significantly adapted to run over a heterogeneous platform. This platform is a collection of heterogeneous computing nodes interconnected via a high speed homogeneous network.}
+
+The proposed frequency selecting algorithm selects the vector of frequencies for a heterogeneous platform that runs a message passing iterative application, that gives the maximum energy reduction and minimum
+performance degradation ratio simultaneously. The algorithm has a very small
+overhead, works online and does not need any training or profiling.
This paper is organized as follows: Section~\ref{sec.relwork} presents some
related works from other authors. Section~\ref{sec.exe} describes how the
-execution time of MPI programs can be predicted. It also presents an energy
-model for heterogeneous platforms. Section~\ref{sec.compet} presents
+execution time of message passing programs can be predicted. It also presents an energy
+model that predicts the energy consumption of an application running over a heterogeneous platform. Section~\ref{sec.compet} presents
the energy-performance objective function that maximizes the reduction of energy
consumption while minimizing the degradation of the program's performance.
-Section~\ref{sec.optim} details the proposed heterogeneous scaling algorithm.
-Section~\ref{sec.expe} presents the results of running the NAS benchmarks on
-the proposed heterogeneous platform. It also shows the comparison of three
-different power scenarios and it verifies the precision of the proposed algorithm.
-Finally, we conclude in Section~\ref{sec.concl} with a summary and some future works.
+Section~\ref{sec.optim} details the proposed frequency selecting algorithm then the precision of the proposed algorithm is verified.
+Section~\ref{sec.expe} presents the results of applying the algorithm on the NAS parallel benchmarks and executing them
+on a heterogeneous platform. It also shows the results of running three
+different power scenarios and comparing them.
+Finally, in Section~\ref{sec.concl} the paper is ended with a summary and some future works.
\section{Related works}
\label{sec.relwork}
-Energy reduction process for high performance clusters recently performed using
-dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique enabled
-in modern processors to scaled down both of the voltage and the frequency of
-the CPU while it is in the computing mode to reduce the energy consumption. DVFS is
-also allowed in the graphical processors GPUs, to achieved the same goal. Applying
-DVFS has a dramatical side effect if it is applied to minimum levels to gain more
-energy reduction, producing a high percentage of performance degradations for the
-parallel applications. Many researchers used different strategies to solve this
-nonlinear problem for example in
-~\cite{Hao_Learning.based.DVFS,Dhiman_Online.Learning.Power.Management}, their methods
-add big overheads to the algorithm to select the suitable frequency.
-In this paper we present a method
-to find the optimal set of frequency scaling factors for heterogeneous cluster to
-simultaneously optimize both the energy and the execution time without adding big
-overhead. This work is developed from our previous work of homogeneous cluster~\cite{Our_first_paper}.
-Therefore we are interested to present some works that concerned the heterogeneous clusters
-enabled DVFS. In general, the heterogeneous cluster works fall into two categorizes:
-GPUs-CPUs heterogeneous clusters and CPUs-CPUs heterogeneous clusters. In GPUs-CPUs
-heterogeneous clusters some parallel tasks executed on GPUs and the others executed
-on CPUs. As an example of this works, Luley et al.
+DVFS is a technique enabled
+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 the 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 and while taking into account all the constraints, is not a trivial operation. Many researchers used different strategies to tackle this problem. Some of them used online methods that compute the new frequency while executing the application \textbf{add a reference for an online method here}. Others used offline methods that might need to run the application and profile it before selecting the new frequency \textbf{add a reference for an offline method}. 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, ...
+
+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 it can be classified into two types of heterogeneous platforms:
+\begin{itemize}
+
+\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
+\item the platform is only composed of heterogeneous CPUs.
+
+\end{itemize}
+
+For the first type of platform, the compute 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 is to determined the
-energy efficiency as a function of performance per watt, the best tradeoff is done when the
-performance per watt function is maximized. In the work of Kia Ma et al.
-~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, they developed a scheduling
-algorithm to distributed different workloads proportional to the computing power of the node
-to be executed on CPU or GPU, emphasize all tasks must be finished in the same time.
-Recently, Rong et al.~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, Their study explain that
-a heterogeneous clusters enabled DVFS using GPUs and CPUs gave better energy and performance
-efficiency than other clusters composed of only CPUs.
-The CPUs-CPUs heterogeneous clusters consist of number of computing nodes all of the type CPU.
-Our work in this paper can be classified to this type of the clusters.
-As an example of these works see Naveen et al.~\cite{Naveen_Power.Efficient.Resource.Scaling} work,
-They developed a policy to dynamically assigned the frequency to a heterogeneous cluster.
-The goal is to minimizing a fixed metric of $energy*delay^2$. Where our proposed method is automatically
-optimized the relation between the energy and the delay of the iterative applications.
-Other works such as Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling},
-their algorithm divided the executed tasks into two types: the critical and
-non critical tasks. The algorithm scaled down the frequency of the non critical tasks
-as function to the amount of the slack and communication times that
-have with maximum of performance degradation percentage less than 10\%. In our method there is no
-fixed bounds for performance degradation percentage and the bound is dynamically computed
-according to the energy and the performance tradeoff relation of the executed application.
-There are some approaches used a heterogeneous cluster composed from two different types
-of Intel and AMD processors such as~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}
-and \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, they predicated both the energy
-and the performance for each frequency gear, then the algorithm selected the best gear that gave
-the best tradeoff. In contrast our algorithm works over a heterogeneous platform composed of
-four different types of processors. Others approaches such as
-\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks},
-they are selected the best frequencies for a specified heterogeneous clusters offline using some
-heuristic methods. While our proposed algorithm works online during the execution time of
-iterative application. Greedy dynamic approach used by Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements},
-minimized the power consumption of a heterogeneous severs with time/space complexity, this approach
-had considerable overhead. In our proposed scaling algorithm has very small overhead and
-it is works without any previous analysis for the application time complexity. The primary
-contributions of our paper are :
+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 minimize the value of $energy*delay^2$ by dynamically assigning new frequencies to the CPUs of the heterogeneous cluster. \textbf{should define the delay} Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} propose
+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}
+and \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, a heterogeneous cluster composed of two types
+of Intel and AMD processors. The consumed energy
+and the performance for each frequency gear were predicted, then the algorithm selected the best gear that gave
+the best tradeoff. \textbf{what energy model they used? what method they used? }
+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 approach to
+minimize the power consumption of heterogeneous severs with time/space complexity \textbf{what does it mean}. This approach
+had considerable overhead.
+In contrast to the above described papers, this paper presents the following contributions :
\begin{enumerate}
-\item It is presents a new online heterogeneous scaling algorithm which has very small
- overhead and not need for any training and profiling.
-\item It is develops a new energy model for iterative distributed applications running over
- a heterogeneous clusters, taking into account the communication and slack times.
-\item The proposed scaling algorithm predicts both the energy and the execution time
- of the iterative application.
-\item It demonstrates a new optimization function which maximize the performance and
- minimize the energy consumption simultaneously.
+\item two new energy and performance models for message passing iterative synchronous applications running over
+ a heterogeneous platform. Both models takes into account the communication and slack times. The models can predict the required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous platforms. The algorithm has a very small
+ overhead and does not need for any training or profiling. It uses a new optimization function which simultaneously maximizes the performance and minimizes the energy consumption of a message passing iterative synchronous application .
+
\end{enumerate}
\section{The performance and energy consumption measurements on heterogeneous architecture}
\label{sec.exe}
-% \JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'',
-% can be deleted if we need space, we can just say we are interested in this
-% paper in homogeneous clusters}
+
\subsection{The execution time of message passing distributed
iterative applications on a heterogeneous platform}
In this paper, we are interested in reducing the energy consumption of message
passing distributed iterative synchronous applications running over
-heterogeneous platforms. We define a heterogeneous platform as a collection of
+heterogeneous platforms. A heterogeneous platform is defined as a collection of
heterogeneous computing nodes interconnected via a high speed homogeneous
network. Therefore, each node has different characteristics such as computing
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
where $TcpOld_i$ is the computation time of processor $i$ during the first
iteration and $MinTcm$ is the communication time of the slowest processor from
the first iteration. The model computes the maximum computation time
-with scaling factor from each node added to the communication time of the
+with scaling factor from each node added to the communication time of the \subsection{The verifications of the proposed method}
+\label{sec.verif}
+The precision of the proposed algorithm mainly depends on the execution time prediction model defined in
+EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}).
+The energy model is also significantly dependent on the execution time model because the static energy is
+linearly related the execution time and the dynamic energy is related to the computation time. So, all of
+the work presented in this paper is based on the execution time model. To verify this model, the predicted
+execution time was compared to the real execution time over Simgrid for all the NAS parallel benchmarks
+running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is 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 and do not test all the possible solutions (vectors of scaling factors)
+in the search space and 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 sections (\ref{sec.res}) and (\ref{sec.compare}).
slowest node, it means only the communication time without any slack time.
-Therefore, we can consider the execution time of the iterative application is
+Therefore, the execution time of the iterative application is
equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied
by the number of iterations of that application.
process of the frequency can be expressed by the scaling factor $S$ which is the
ratio between the maximum and the new frequency as in EQ(\ref{eq:s}).
The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. we can calculate the new frequency
-$F_{new}$ from EQ(\ref{eq:s}) as follow:
+system's kernel to lower a core's frequency. The new frequency
+$F_{new}$ from EQ(\ref{eq:s}) can be calculated as follows:
\begin{equation}
\label{eq:fnew}
F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
The static power is related to the power leakage of the CPU and is consumed during computation
and even when idle. As in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
-we assume that the static power of a processor is constant
+ the static power of a processor is considered as constant
during idle and computation periods, and for all its available frequencies.
The static energy is the static power multiplied by the execution time of the program.
According to the execution time model in EQ(\ref{eq:perf}), the execution time of the program
Reducing the frequencies of the processors according to the vector of
scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
application and thus, increase the static energy because the execution time is
-increased~\cite{Kim_Leakage.Current.Moore.Law}. We can measure the overall energy consumption for the iterative
-application by measuring the energy consumption for one iteration as in EQ(\ref{eq:energy})
+increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption for the iterative
+application can be measured by measuring the energy consumption for one iteration as in EQ(\ref{eq:energy})
multiplied by the number of iterations of that application.
complex and nonlinear, Thus, unlike the relation between the execution time
and the scaling factor, the relation of the energy with the frequency scaling
factors is nonlinear, for more details refer to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.
-Moreover, they are not measured using the same metric. To solve this problem, we normalize the
-execution time by computing the ratio between the new execution time (after
+Moreover, they are not measured using the same metric. To solve this problem, the
+execution time is normalized by computing the ratio between the new execution time (after
scaling down the frequencies of some processors) and the initial one (with maximum
frequency for all nodes,) as follows:
\begin{multline}
\end{multline}
-In the same way, we normalize the energy by computing the ratio between the consumed energy
+In the same way, the energy is normalized by computing the ratio between the consumed energy
while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
\begin{multline}
\label{eq:enorm}
-Our solution for this problem is to make the optimization process for energy and
-execution time follow the same direction. Therefore, we inverse the equation of the
-normalized execution time which gives the normalized performance equation, as follows:
+This problem can be solved by making the optimization process for energy and
+execution time follow the same direction. Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
\caption{The energy and performance relation}
\end{figure}
-Then, we can model our objective function as finding the maximum distance
+Then, the objective function can be modeled as finding the maximum distance
between the energy curve EQ~(\ref{eq:enorm}) and the performance
curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors. This
represents the minimum energy consumption with minimum execution time (maximum
\overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
\end{equation}
where $N$ is the number of nodes and $F$ is the number of available frequencies for each nodes.
-Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).
-Our objective function can work with any energy model or any power values for each node
+Then, the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}) can be selected.
+The objective function can work with any energy model or any power values for each node
(static and dynamic powers). However, the most energy reduction gain can be achieved when
the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
\section{The scaling factors selection algorithm for heterogeneous platforms }
\label{sec.optim}
-In this section we propose algorithm~(\ref{HSA}) which selects the frequency scaling factors
+\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.
\label{dvfs}
\end{algorithm}
+\subsection{The verifications of the proposed algorithm}
+\label{sec.verif}
+The precision of the proposed algorithm mainly depends on the execution time prediction model defined in
+EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}).
+The energy model is also significantly dependent on the execution time model because the static energy is
+linearly related the execution time and the dynamic energy is related to the computation time. So, all of
+the work presented in this paper is based on the execution time model. To verify this model, the predicted
+execution time was compared to the real execution time over SimGrid/SMPI simulator, v3.10~\cite{casanova+giersch+legrand+al.2014.versatile},
+for all the NAS parallel benchmarks NPB v3.3
+\cite{NAS.Parallel.Benchmarks}, running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is 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 and do not test all the possible solutions (vectors of scaling factors)
+in the search space and 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 \cite{NAS.Parallel.Benchmarks}. The experiments were executed
-on the simulator SimGrid/SMPI v3.10~\cite{casanova+giersch+legrand+al.2014.versatile} 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
+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 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,
-\subsection{The verifications of the proposed method}
-\label{sec.verif}
-The precision of the proposed algorithm mainly depends on the execution time prediction model defined in
-EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}).
-The energy model is also significantly dependent on the execution time model because the static energy is
-linearly related the execution time and the dynamic energy is related to the computation time. So, all of
-the work presented in this paper is based on the execution time model. To verify this model, the predicted
-execution time was compared to the real execution time over Simgrid for all the NAS parallel benchmarks
-running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is 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 and do not test all the possible solutions (vectors of scaling factors)
-in the search space and 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 sections (\ref{sec.res}) and (\ref{sec.compare}) .
\section{Conclusion}
-\label{sec.concl}
-In this paper, we have presented a new online heterogeneous scaling algorithm
-that selects the best possible vector of frequency scaling factors. This vector
-gives the maximum distance (optimal tradeoff) between the normalized energy and
-the performance curves. In addition, we developed a new energy model for measuring
+\label{sec.concl}
+In this paper, a new online frequency selecting algorithm have been presented. It selects the best possible vector of frequency scaling factors for a heterogeneous platform.
+This vector gives the maximum distance (optimal tradeoff) between the predicted energy and
+the predicted performance curves. In addition, we developed a new energy model for measuring
and predicting the energy of distributed iterative applications running over heterogeneous
cluster. The proposed method evaluated on Simgrid/SMPI simulator to built a heterogeneous
platform to executes NAS parallel benchmarks. The results of the experiments showed the ability of
