X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/c9875e5d70672da61c5cdc75fac90dad56cb1b04..93680d5e578f3b8fb3410499de0446561b2893a3:/Heter_paper.tex?ds=sidebyside

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 2fc8549..38481af 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -8,7 +8,7 @@
 \usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
-
+\usepackage{multirow}
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
@@ -53,7 +53,7 @@
 \newcommand{\Told}{\Xsub{T}{Old}} 
 \begin{document} 
 
-\title{Energy Consumption Reduction in a Heterogeneous Architecture Using DVFS}
+\title{Energy Consumption Reduction for Message Passing Iterative  Applications in Heterogeneous Architecture Using DVFS}
  
 \author{% 
   \IEEEauthorblockN{%
@@ -76,141 +76,123 @@
 \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 try to give 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 is 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. Finally, the algorithm is compared to an existing method and the comparison results show that it outperforms the latter.
+
 \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, 
-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.  
+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 L-CSC from the GSI Helmholtz Center which  
+became the top of the Green500 list in November 2014 \cite{Green500_List}. 
+This heterogeneous platform executes more than 5  GFLOPS per watt while consumed 57.15 kilowatts.
+
+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  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. In \cite{Our_first_paper},  a frequency selecting algorithm 
+was proposed to reduce the energy consumption of message passing iterative applications running over homogeneous platforms. The  results of the experiments showed significant energy consumption reductions. In this paper,  a new frequency selecting algorithm  adapted for heterogeneous platform  is presented. It selects the vector of frequencies, for a heterogeneous platform running a message passing iterative application,  that simultaneously tries to give the maximum energy reduction and minimum performance degradation ratio. 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 shows the results of running three 
+different power scenarios and comparing them. Moreover, it also shows the comparison results
+between the proposed method and an existing method.
+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 developed online methods that compute the new frequency while executing the application, such as ~\cite{Hao_Learning.based.DVFS,Dhiman_Online.Learning.Power.Management}. 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, ... 
+
+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: 
+\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 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 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 minimizes the value of $energy*delay^2$ (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 severs  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 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
@@ -227,7 +209,7 @@ task which have the highest computation time and no slack time.
   
  \begin{figure}[t]
   \centering
-    \includegraphics[scale=0.6]{fig/commtasks}
+   \includegraphics[scale=0.5]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
@@ -239,7 +221,7 @@ and consequently its computing power, the execution time of a program running
 over that scaled down processor might increase, especially if the program is 
 compute bound.  The frequency reduction process can be  expressed by the scaling 
 factor S which is the ratio between  the maximum and the new frequency of a CPU 
-as in EQ (\ref{eq:s}).
+as in (\ref{eq:s}).
 \begin{equation}
   \label{eq:s}
  S = \frac{F_\textit{max}}{F_\textit{new}}
@@ -253,7 +235,7 @@ as in EQ (\ref{eq:s}).
  communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}. 
  The communication time for a task is the summation of  periods of 
  time that begin with an MPI call for sending or receiving   a message 
- till the message is synchronously sent or received.
+ until the message is synchronously sent or received.
 
 Since in a heterogeneous platform, each node has different characteristics,
 especially different frequency gears, when applying DVFS operations on these
@@ -264,24 +246,29 @@ applications running over a heterogeneous platform, for different vectors of
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
-vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
+vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
  \textit  T_\textit{new} = 
- \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) +  MinTcm
+ \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) +  MinTcm 
+\end{equation}
+Where:\\
+\begin{equation}
+\label{eq:perf}
+ MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
 \end{equation}
 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 
 slowest node, it means  only the  communication time without any slack time. 
-Therefore, we can consider the execution time of the iterative application is 
-equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied 
+Therefore, the execution time of the iterative application is 
+equal to the execution time of one iteration as in (\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}. 
-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.
 
@@ -294,7 +281,7 @@ two power metrics: the static and the dynamic power.  While the first one is
 consumed as long as the computing unit is turned on, the latter is only consumed during
 computation times.  The dynamic power $Pd$ is related to the switching
 activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
-operational frequency $F$, as shown in EQ(\ref{eq:pd}).
+operational frequency $F$, as shown in (\ref{eq:pd}).
 \begin{equation}
   \label{eq:pd}
   Pd = \alpha \cdot C_L \cdot V^2 \cdot F
@@ -313,23 +300,23 @@ to execute a given program can be computed as:
    E_\textit{ind} =  Pd \cdot Tcp + Ps \cdot T
 \end{equation}
 where $T$ is the execution time of the program, $Tcp$ is the computation
-time and $Tcp \leq T$.  $Tcp$ may be equal to $T$ if there is no
+time and $Tcp \le T$.  $Tcp$ may be equal to $T$ if there is no
 communication and no slack time.
 
 The main objective of DVFS operation is to reduce the overall energy consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  
 The operational frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot F$ with some
-constant $\beta$.  This equation is used to study the change of the dynamic
+constant $\beta$.~This equation is used to study the change of the dynamic
 voltage with respect to various frequency values in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction
 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}).
+ratio between the maximum and the new frequency as in (\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 (\ref{eq:s}) can be calculated as follows:
 \begin{equation}
   \label{eq:fnew}
    F_\textit{new} = S^{-1} \cdot F_\textit{max}
 \end{equation}
-Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following 
+Replacing $F_{new}$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following 
 equation for dynamic power consumption:
 \begin{multline}
   \label{eq:pdnew}
@@ -339,7 +326,7 @@ equation for dynamic power consumption:
 where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the 
 new frequency and the maximum frequency respectively.
 
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 
+According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 
 reducing the frequency by a factor of $S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is proportional 
 to the frequency of a CPU, the computation time is increased proportionally to $S$.  
 The new dynamic energy is the  dynamic power multiplied by the new time of computation 
@@ -350,10 +337,10 @@ 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}, 
-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 
+According to the execution time model in (\ref{eq:perf}), the execution time of the program 
 is the summation of the computation and the communication times. The computation time is linearly related  
 to the frequency scaling factor, while this scaling factor does not affect the communication time. 
 The static energy of a processor after scaling its frequency is computed as follows: 
@@ -370,7 +357,7 @@ in order to decrease the overall energy consumption of the application and reduc
 The communication time of a processor $i$ is noted as $Tcm_{i}$ and could contain slack times 
 if it is communicating with slower 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 EQ(\ref{eq:Edyn}), the static energy is 
+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 of each processor multiplied by its static power. 
 The overall energy consumption of a message passing  distributed application executed over a 
 heterogeneous platform during one iteration is the summation of all dynamic and static energies 
@@ -385,8 +372,8 @@ 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
-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 (\ref{eq:energy}) 
 multiplied by the number of iterations of that application.
 
 
@@ -415,10 +402,10 @@ 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}.  
-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:
+frequency for all nodes) as follows:
 \begin{multline}
   \label{eq:pnorm}
   P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
@@ -427,7 +414,7 @@ frequency for all nodes,) as follows:
 \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}
@@ -436,21 +423,20 @@ while scaling down the frequency and the consumed energy with maximum frequency
  \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
  \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
 \end{multline} 
-Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
+Where $E_\textit{Reduced}$ and $E_\textit{Original}$ are computed using (\ref{eq:energy}) and
+  $T_{New}$ and $T_{Old}$ are computed as in (\ref{eq:pnorm}).
 
- While the main 
+While the main 
 goal is to optimize the energy and execution time at the same time, the normalized 
 energy and execution time curves are not in the same direction. According 
-to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the vector  of frequency
+to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the vector  of frequency
 scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
 time simultaneously.  But the main objective is to produce maximum energy
 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}}\\
@@ -462,19 +448,20 @@ normalized execution time which gives the normalized performance equation, as fo
 \begin{figure}
   \centering
   \subfloat[Homogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
-  \qquad%
+    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+  
+  
   \subfloat[Heterogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
+    \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}
 
-Then, we can model our objective function 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
+Then, the objective function can be modeled   as finding 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 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}
@@ -483,23 +470,24 @@ function has the following form:
       (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
        \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 
+where $N$ is the number of nodes and $F$ is the  number of available frequencies for each node. 
+Then, the optimal set of scaling factors that satisfies (\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.  
 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 EQ(\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs 
+function (\ref{eq:max}). The program apply 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.
@@ -508,7 +496,7 @@ 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. 
-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 
@@ -519,7 +507,7 @@ computation time of the node $i$ as follows:
   \label{eq:Scp}
  Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
 \end{equation}
-Using the initial  frequency scaling factors computed in EQ(\ref{eq:Scp}), the algorithm computes 
+Using the initial  frequency scaling factors computed in (\ref{eq:Scp}), the algorithm computes 
 the initial frequencies for all nodes as a ratio between the maximum frequency of node $i$  
 and the computation scaling factor $Scp_i$ as follows:
 \begin{equation}
@@ -538,11 +526,11 @@ scaling factors starts the search method from these initial frequencies and take
 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 EQ(\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of  
+according to (\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 EQ(\ref{eq:max}).
+function (\ref{eq:max}).
 
 The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an 
 application running on a homogeneous platform and a heterogeneous platform respectively while increasing the 
@@ -586,8 +574,8 @@ which results in bigger energy savings.
     \EndIf 
     \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
     \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
-    \State $Dist \gets 0$
     \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
+    \State $Dist \gets 0 $
     \While {(all nodes not reach their  minimum  frequency)}
         \If{(not the last freq. \textbf{and} not the slowest node)}
         \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
@@ -605,7 +593,7 @@ which results in bigger energy savings.
     \EndWhile
     \State  Return $Sopt_1,Sopt_2,\dots,Sopt_N$
   \end{algorithmic}
-  \caption{Heterogeneous scaling algorithm}
+  \caption{frequency scaling factors selection algorithm}
   \label{HSA}
 \end{algorithm}
 
@@ -618,7 +606,7 @@ which results in bigger energy savings.
       \If {$(k=1)$}
         \State Gather all times of computation and\newline\hspace*{3em}%
                communication from each node.
-        \State Call algorithm from Figure~\ref{HSA} with these times.
+        \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.
@@ -629,16 +617,40 @@ which results in bigger energy savings.
   \label{dvfs}
 \end{algorithm}
 
+\subsection{The evaluation of the proposed algorithm}
+\label{sec.verif.algo}
+The precision of the proposed algorithm mainly depends on the execution time prediction model defined in 
+(\ref{eq:perf}) and the energy model computed by (\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 works 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 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  \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
-available frequencies and the computational power, see table (\ref{table:platform}). The characteristics 
+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 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, 
 for example if  a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors 
@@ -660,16 +672,16 @@ Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwi
                   &           & GHz      & GHz          &GHz             &              &       \\
     \hline
     1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
-                  &           &          &              &                &              &  \\
+         
     \hline
     2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
-                  &           &          &              &                &              &  \\
+                  
     \hline
     3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
-                  &           &          &              &                &              &  \\
+                  
     \hline
     4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
-                  &           &          &              &                &              &  \\
+                  
     \hline
   \end{tabular}
   \label{table:platform}
@@ -698,7 +710,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &  64.64        & 3560.39        &34.16        &6.72               &27.44       \\
@@ -725,7 +737,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &36.11    	   &3263.49	        &31.25	      &7.12	               &24.13     \\
@@ -752,7 +764,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &31.74	       &4373.90	        &26.29	      &9.57	               &16.72          \\
@@ -779,7 +791,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &32.35	       &6704.21	        &16.15	      &5.30	               &10.85           \\
@@ -806,7 +818,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &46.65	       &17521.83	    &8.13	      &1.68	               &6.45           \\
@@ -834,7 +846,7 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance     \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
     \hline
     CG         &56.92	       &41163.36        &4.00	      &1.10	               &2.90          \\
@@ -855,13 +867,12 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \label{table:res_128n}
 \end{table}
 The overall energy consumption was computed for each instance according to the energy 
-consumption  model EQ(\ref{eq:energy}), with and without applying the algorithm. The 
+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}, \ref{table:res_8n}, \ref{table:res_16n}, 
+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 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 reduce significantly the energy 
 consumption (up to 35\%) and tries to limit the performance degradation. They also show that 
@@ -887,17 +898,17 @@ compared to the communication times.
 \begin{figure}
   \centering
   \subfloat[Energy saving]{%
-    \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%
-  \quad%
+    \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
+  
   \subfloat[Performance degradation ]{%
-    \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}
+    \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}
   \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
 \end{figure}
 
 Plots (\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 are decreased linearly  when the the 
+the energy saving percentages of the benchmarks MG, LU, BT and FT are decreased linearly  when the  
 number of nodes is increased. While for the  EP and SP benchmarks, the energy saving percentage is not 
 affected by the increase of the number of computing nodes, because in these benchmarks there are little or 
 no communications. Finally, the energy saving of the GC benchmark  is significantly decreased when the number 
@@ -910,7 +921,7 @@ down the frequencies of some nodes have less effect on the performance.
 
 
 \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  
@@ -922,9 +933,9 @@ are the following:
 \item 90\% dynamic power  and 10\% static power
 \end{itemize}
 
-The NAS parallel benchmarks were executed again over processors that follow the 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 \ref{table:res_s2}). These tables show that the energy saving percentage of the 70\%-30\% 
+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 \ref{table:res_s2}. These tables show that the energy saving percentage of the 70\%-30\% 
 scenario is less for all benchmarks compared to the energy saving of the 90\%-10\% scenario. Indeed, in the latter 
 more dynamic power is consumed when nodes are running on their maximum frequencies, thus, scaling down the frequency 
 of the nodes results in higher energy savings than in the 70\%-30\% scenario. On the other hand,  the performance 
@@ -932,12 +943,12 @@ degradation percentage is less in the 70\%-30\% scenario  compared to the 90\%-1
 higher static power percentage in the first scenario which makes it more relevant in the overall consumed energy. 
 Indeed, the static energy is related to the execution time and if the performance is  degraded the total consumed 
 static energy is directly increased. Therefore, the proposed algorithm do not scales 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 .
+nodes  in order to limit the increase of the execution time and thus limiting the effect of the consumed static energy.
 
 The two 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 because at maximum frequency the dynamic  energy is the the most relevant 
+when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the  most relevant 
 in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, 
 the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in 
 the overall consumed energy and lowering the frequency do not returns big energy savings.
@@ -955,7 +966,7 @@ results in less energy saving but less performance degradation.
   \centering
   \begin{tabular}{|*{6}{l|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
+    Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
     \hline
     CG         &4144.21          &22.42        &7.72                &14.70         \\
@@ -984,7 +995,7 @@ results in less energy saving but less performance degradation.
   \centering
   \begin{tabular}{|*{6}{l|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
+    Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
     \hline
     CG         &2812.38	         &36.36	       &6.80                &29.56         \\
@@ -1008,46 +1019,83 @@ results in less energy saving but less performance degradation.
 
 \begin{figure}
   \centering
-  \subfloat[Comparison the average of the results on 8 nodes]{%
-    \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
-  \quad%
+  \subfloat[Comparison  of the results on 8 nodes]{%
+    \includegraphics[width=.30\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
+
   \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
-    \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
+    \includegraphics[width=.34\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
 \end{figure}  
 
 
 
-\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}).
+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
 
-\section{Conclusion}
-\label{sec.concl}
+In this section, the scaling  factors selection algorithm
+is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}. 
+They developed a green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Enegry*Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
+ 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 consumptions 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 gives 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 term of energy and performance tradeoff, 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. 
+
+
+\begin{table}[h]
+ \caption{Comparing the proposed algorithm}
+ \centering
+\begin{tabular}{|l|l|l|l|l|l|l|l|}
+\hline
+\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Program \\ name\end{tabular}}} & \multicolumn{2}{l|}{Energy saving \%} & \multicolumn{2}{l|}{Perf.  degradation \%} & \multicolumn{2}{l|}{Distance} \\ \cline{3-8} 
+\multicolumn{2}{|l|}{}                                                                         & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
+\multicolumn{2}{|l|}{CG}                                                                       & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
+\multicolumn{2}{|l|}{MG}                                                                       & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
+\multicolumn{2}{|l|}{LU}                                                                       & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
+\multicolumn{2}{|l|}{EP}                                                                       & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
+\multicolumn{2}{|l|}{BT}                                                                       & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
+\multicolumn{2}{|l|}{SP}                                                                       & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
+\multicolumn{2}{|l|}{FT}                                                                       & 27.03           & 31.02            & 2.75           & 2.54              & 24.28        & 28.48           \\ \hline
+
+\end{tabular}
+\label{table:compare_EDP}
+\end{table}
+
+
+
+\begin{figure}[t]
+  \centering
+   \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
+  \caption{Tradeoff comparison for NAS benchmarks class C}
+  \label{fig:compare_EDP}
+\end{figure}
+
+
+\section{Conclusion}
+\label{sec.concl} 
+In this paper, a new online frequency selecting algorithm has 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. Finally, the algorithm was compared to Spiliopoulos et al. algorithm and the results showed that it 
+ outperforms their algorithm in term of energy-time tradeoff.
+
+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}
 
+This work has been partially supported by the Labex
+ACTION project (contract âANR-11-LABX-01-01â). As a PhD student,
+Mr. Ahmed Fanfakh, would like to thank the University of
+Babylon (Iraq) for supporting his work.
+
 
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