conclusion corrected

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 8aeea6b8dd30ec61d3af183d47cd723e6879b524..9f2d19222b51e19bccde6f6ad1e9b3631ba782b5 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -76,141 +76,127 @@
 \maketitle
 
 \begin{abstract}
 \maketitle
 
 \begin{abstract}

-  
+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}
 \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. 
 \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}. 
 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
 
 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.
 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}
 
 \section{Related works}
 \label{sec.relwork}

-Energy reduction process for a high performance clusters recently performed using 
-dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique enabled 
-in a 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 a heterogeneous cluster to 
-simultaneously optimize both the energy and the execution time  without adding a big 
-overhead. This work is developed from our previous work of a 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 a GPUs and the others executed 
-on a 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 
 ~\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 a CPU or a 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 this 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 of 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}
 \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}
 
       
 \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
 
 \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
 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


@@ -273,15 +259,36 @@ vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
 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 
 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.method}
+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. 
 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.
 
 equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied 
 by the number of iterations of that application.
 

-This prediction model is developed from our model for predicting the execution time of 
+This prediction model is developed from the model for predicting the execution time of 

 message passing distributed applications for homogeneous architectures~\cite{Our_first_paper}. 
 message passing distributed applications for homogeneous architectures~\cite{Our_first_paper}. 

-The execution time prediction model is used in our method for optimizing both 
+The execution time prediction model is used in the method for optimizing both 

 energy consumption and performance of iterative methods, which is presented in the 
 following sections.
 
 energy consumption and performance of iterative methods, which is presented in the 
 following sections.
 


@@ -299,7 +306,7 @@ operational frequency $F$, as shown in EQ(\ref{eq:pd}).
   \label{eq:pd}
   Pd = \alpha \cdot C_L \cdot V^2 \cdot F
 \end{equation}
   \label{eq:pd}
   Pd = \alpha \cdot C_L \cdot V^2 \cdot F
 \end{equation}

-The static power $P_{s}$ captures the leakage power as follows:
+The static power $Ps$ captures the leakage power as follows:

 \begin{equation}
   \label{eq:ps}
    Ps  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
 \begin{equation}
   \label{eq:ps}
    Ps  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}


@@ -323,8 +330,8 @@ voltage with respect to various frequency values in~\cite{Rauber_Analytical.Mode
 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
 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}
 \begin{equation}
   \label{eq:fnew}
    F_\textit{new} = S^{-1} \cdot F_\textit{max}


@@ -350,7 +357,7 @@ and is given by the following equation:
 \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}, 
 \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 
 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 


@@ -385,8 +392,8 @@ for each  processor.  It is computed as follows:
 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
 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.
 
 
 multiplied by the number of iterations of that application.
 
 


@@ -415,8 +422,8 @@ The relation between the energy consumption and the execution time for an applic
 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}.  
 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}
 scaling down the frequencies of some processors) and the initial one (with maximum 
 frequency for all nodes,) as follows:
 \begin{multline}


@@ -427,7 +434,7 @@ frequency for all nodes,) as follows:
 \end{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}
 while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
 \begin{multline}
   \label{eq:enorm}


@@ -448,9 +455,9 @@ reduction with minimum execution time reduction.
 
  
   
 
  
   

-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}}\\
 \begin{multline}
   \label{eq:pnorm_inv}
   P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\


@@ -470,11 +477,11 @@ normalized execution time which gives the normalized performance equation, as fo
   \caption{The energy and performance relation}
 \end{figure}
 
   \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 
 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 

-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then our objective
+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}
 function has the following form:
 \begin{equation}
   \label{eq:max}


@@ -484,15 +491,16 @@ function has the following form:
        \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. 
        \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}
 
 (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.  
 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.  


@@ -508,7 +516,7 @@ The nodes in a heterogeneous platform have different computing powers, thus whil
 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. 
 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. 

-Our algorithm takes into account this problem and tries to reduce these slack times when selecting the 
+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 
 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 


@@ -629,15 +637,39 @@ which results in bigger energy savings.
   \label{dvfs}
 \end{algorithm}
 
   \label{dvfs}
 \end{algorithm}
 

+\subsection{The verifications of the proposed algorithm}
+\label{sec.verif.algo}
+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}), 
 \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, 
 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, 


@@ -910,7 +942,7 @@ down the frequencies of some nodes have less effect on the performance.
 
 
 \subsection{The results for different power consumption scenarios}
 
 
 \subsection{The results for different power consumption scenarios}

-
+\label{sec.compare}

 The results of the previous section were obtained while using processors that consume during computation 
 an overall power which is 80\% composed of  dynamic power and 20\% of static power. In this section, 
 these ratios are changed and two new power scenarios are considered in order to evaluate how the proposed  
 The results of the previous section were obtained while using processors that consume during computation 
 an overall power which is 80\% composed of  dynamic power and 20\% of static power. In this section, 
 these ratios are changed and two new power scenarios are considered in order to evaluate how the proposed  


@@ -1019,36 +1051,24 @@ results in less energy saving but less performance degradation.
 
 
 
 
 
 

-\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 section (\ref{sec.res}).

 
 \section{Conclusion}
 
 \section{Conclusion}

-\label{sec.concl}
+\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 that gives the maximum distance (optimal tradeoff) between the predicted energy and 
+the predicted performance curves for a heterogeneous platform. This algorithm uses a new energy model for measuring  
+and predicting the energy of distributed iterative applications running over heterogeneous 
+platform. To evaluate the proposed method, it  was  applied on the NAS parallel benchmarks and executed over a heterogeneous platform simulated by  Simgrid. The results of the experiments showed that the algorithm reduces up to 35\% the energy consumption of a message passing iterative method while limiting the degradation of the performance. The algorithm also  selects different scaling factors   according to the percentage of the computing and communication times, and according to the values of  the static and  dynamic powers of the CPUs. 

 
 

+In the near future, this method will be applied to real heterogeneous platforms to evaluate its performance in a real study case. It would also be interesting to evaluate its scalability over large scale heterogeneous platform and measure the energy consumption reduction it can produce. Afterward, We would like  to develop a similar method that is adapted to asynchronous  iterative applications 
+where each task does not wait for others tasks to finish there works. The development of such method might require a new 
+energy model because the number of iterations is not 
+known in advance and depends on the global convergence of the iterative system.

 
 \section*{Acknowledgment}
 
 
 
 \section*{Acknowledgment}
 
 

+

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