X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/42a388f1b7b2acb29caec469d31d0d533e41a673..8c643c738580db77edb260d118e0a2dc4af78ab2:/Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 8aeea6b..ca66e17 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{%
@@ -63,8 +63,7 @@
     Arnaud Giersch
   } 
   \IEEEauthorblockA{%
-    FEMTO-ST Institute\\
-    University of Franche-ComtÃ©\\
+    FEMTO-ST Institute, University of Franche-Comte\\
     IUT de Belfort-MontbÃ©liard,
     19 avenue du MarÃ©chal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % RaphaÃ«l
@@ -76,141 +75,155 @@
 \maketitle
 
 \begin{abstract}
-  
+Computing platforms  are consuming  more and more  energy due to  the increasing
+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 which tries  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   show  that  it  reduces  the  energy
+consumption by up to 35\% while  limiting the performance degradation as much as
+possible.   Finally,  the algorithm  is  compared  to  an existing  method,  the
+comparison results showing 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 
-\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.  
+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 platforms 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  million 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, the price of the energy consumed by
+the Tianhe-2  platform is approximately more  than \$10 million  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 consuming
+57.15 kilowatts.
+
+Besides platform  improvements, there are many software  and hardware techniques
+to lower  the energy consumption of  these platforms, such  as scheduling, DVFS,
+...   DVFS is  a  widely used  process to  reduce  the energy  consumption of  a
+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 use
+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 show  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 offer 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 ends with a summary and some future works.
 
 \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 used in modern processors to scale down both the voltage and
+the  frequency  of the  CPU  while  computing, in  order  to  reduce the  energy
+consumption of  the processor. DVFS is also  allowed in  GPUs  to achieve the
+same goal. Reducing the frequency of  a processor lowers its number of FLOPS and
+might  degrade the  performance of  the application  running on  that processor,
+especially if it is compute bound. Therefore selecting the appropriate frequency
+for a processor to satisfy some objectives while taking into account all the
+constraints,  is  not a  trivial  operation.   Many  researchers used  different
+strategies to  tackle this problem. Some  of them developed  online methods that
+compute   the  new   frequency  while   executing  the   application,   such  as
+~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}. Others
+used  offline methods  that might  need to  run the  application and  profile it
+before       selecting       the        new       frequency,       such       as
+~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The
+methods could  be heuristics, exact or  brute force methods  that satisfy varied
+objectives such as  energy reduction or performance. They  also could be adapted
+to  the  execution's  environment  and  the  type of  the  application  such  as
+sequential, parallel  or distributed architecture,  homogeneous or heterogeneous
+platform, synchronous or asynchronous application, ...
+
+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 computing intensive parallel tasks are executed on the  GPUs and the rest are executed 
+on the CPUs.  Luley et al.
 ~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed  a heterogeneous 
-cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal 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 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 servers  while respecting given time constraints. This approach 
+had considerable overhead.
+In contrast to the above described papers, this paper presents the following contributions :
 \begin{enumerate}
-\item 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 take into account  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 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
@@ -223,11 +236,11 @@ heterogeneous computation power of the computing nodes, slack times might occur
 when fast nodes have to  wait, during synchronous communications, for  the slower 
 nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
 Therefore,  the overall execution time  of the program is the execution time of the slowest
-task which have the highest computation time and no slack time.
+task which has the highest computation time and no slack time.
   
  \begin{figure}[t]
   \centering
-    \includegraphics[scale=0.6]{fig/commtasks}
+   \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
@@ -239,7 +252,7 @@ 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}}
@@ -253,9 +266,9 @@ 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,
+Since in a heterogeneous platform each node has different characteristics,
 especially different frequency gears, when applying DVFS operations on these
 nodes, they may get different scaling factors represented by a scaling vector:
 $(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
@@ -264,26 +277,31 @@ 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 $TcpOld_i$ 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 
-by the number of iterations of that application.
-
-This prediction model is developed from our 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 
-energy consumption and performance of iterative methods, which is presented in the 
-following sections.
+Where:\\
+\begin{equation}
+\label{eq:perf2}
+ 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 is taken into
+account.  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  the model to predict the execution time
+of     message    passing     distributed    applications     for    homogeneous
+architectures~\cite{Our_first_paper}.   The execution  time prediction  model is
+used in  the method  to optimize both the energy consumption and the performance of
+iterative methods, which is presented in the following sections.
 
 
 \subsection{Energy model for heterogeneous platform}
@@ -294,42 +312,42 @@ 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
 \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}
 \end{equation}
 where V is the supply voltage, $N_{trans}$ is the number of transistors,
 $K_{design}$ is a design dependent parameter and $I_{leak}$ is a
-technology-dependent parameter.  The energy consumed by an individual processor
+technology dependent parameter.  The energy consumed by an individual processor
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}
    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 +357,7 @@ 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 
@@ -350,11 +368,11 @@ 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 
-is the summation of the computation and the communication times. The computation time is linearly related  
+According to the execution time model in (\ref{eq:perf}), the execution time of the program 
+is the sum 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: 
 \begin{equation}
@@ -362,19 +380,22 @@ The static energy of a processor after scaling its frequency is computed as foll
  E_\textit{s} = Ps \cdot (Tcp \cdot S  + Tcm)
 \end{equation}
 
-In the considered heterogeneous platform, each processor $i$ might have different dynamic and 
-static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed 
-message passing iterative application is load balanced, the computation time of each CPU $i$ 
-noted $Tcp_{i}$ might be different and different frequency  scaling factors might be computed 
-in order to decrease the overall energy consumption of the application and reduce the slack times. 
-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 
-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 
-for each  processor.  It is computed as follows:
+In  the  considered  heterogeneous  platform,  each  processor  $i$  might  have
+different   dynamic  and  static   powers,  noted   as  $Pd_{i}$   and  $Ps_{i}$
+respectively.  Therefore,  even if  the  distributed  message passing  iterative
+application  is  load balanced,  the  computation time  of  each  CPU $i$  noted
+$Tcp_{i}$ might  be different and  different frequency scaling factors  might be
+computed in order to decrease  the overall energy consumption of the application
+and reduce slack  times.  The communication time of a processor  $i$ is noted as
+$Tcm_{i}$  and could  contain slack  times when  communicating  with slower
+nodes,  see figure(\ref{fig:heter}).  Therefore,  all nodes  do  not have  equal
+communication  times. While  the dynamic  energy  is computed  according to  the
+frequency  scaling   factor  and   the  dynamic  power   of  each  node   as  in
+(\ref{eq:Edyn}), the static energy is computed  as the sum of the execution time
+of  one iteration multiplied  by the static  power of  each processor.   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 for each processor.  It is computed as follows:
 \begin{multline}
   \label{eq:energy}
  E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot  Tcp_i)} + {} \\
@@ -385,40 +406,45 @@ 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.
 
 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 
-Using the lowest frequency for each processor does not necessarily gives the most energy 
-efficient execution of an application. Indeed, even though the dynamic power is reduced 
-while scaling down the frequency of a processor, its computation power is proportionally 
-decreased and thus the execution time might be drastically increased during which dynamic 
-and static powers are being consumed. Therefore,  it might cancel any gains achieved by 
-scaling down the frequency of all nodes to the minimum  and the overall energy consumption 
-of the application might not be the optimal one. It is not trivial to select the appropriate 
-frequency scaling factor for each processor while considering the characteristics of each processor 
-(computation power, range of frequencies, dynamic and static powers) and the task executed 
-(computation/communication ratio) in order to reduce the overall energy consumption and not 
-significantly increase the execution time. In our previous work~\cite{Our_first_paper}, we  proposed a method 
-that selects the optimal frequency scaling factor for a homogeneous cluster executing a message 
-passing iterative synchronous application while giving the best trade-off  between the energy 
-consumption and the performance for such applications. In this work we are interested in 
-heterogeneous clusters as described above. Due to the heterogeneity of the processors, not 
-one but a  vector of scaling factors should be selected and it must  give the best trade-off 
-between energy consumption and performance. 
-
-The relation between the energy consumption and the execution time for an application is 
-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 
-scaling down the frequencies of some processors) and the initial one (with maximum 
-frequency for all nodes,) as follows:
+Using the lowest frequency for each processor does not necessarily give the most
+energy efficient  execution of an  application. Indeed, even though  the dynamic
+power  is  reduced  while  scaling  down  the  frequency  of  a  processor,  its
+computation power  is proportionally decreased. Hence, the  execution time might
+be drastically  increased and  during that time,  dynamic and static  powers are
+being consumed.  Therefore,  it might cancel any gains  achieved by scaling down
+the frequency of all nodes to  the minimum and the overall energy consumption of
+the application might not  be the optimal one.  It is not  trivial to select the
+appropriate frequency  scaling factor for  each processor while  considering the
+characteristics  of each  processor  (computation power,  range of  frequencies,
+dynamic  and static  powers)  and the  task executed  (computation/communication
+ratio). The  aim being  to reduce  the overall energy  consumption and  to avoid
+increasing    significantly    the    execution    time.   In    our    previous
+work~\cite{Our_first_paper},  we  proposed a  method  that  selects the  optimal
+frequency scaling factor  for a homogeneous cluster executing  a message passing
+iterative synchronous  application while giving  the best trade-off  between the
+energy consumption and  the performance for such applications.   In this work we
+are  interested  in heterogeneous  clusters  as  described  above.  Due  to  the
+heterogeneity of the processors, a vector of scaling factors should
+be selected and  it must give the best trade-off  between energy consumption and
+performance.
+
+The  relation between  the  energy consumption  and  the execution  time for  an
+application  is complex  and nonlinear,  Thus, unlike  the relation  between the
+execution time and  the scaling factor, the relation between  the energy and the
+frequency   scaling    factors   is   nonlinear,   for    more   details   refer
+to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.   Moreover,  these relations
+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}
   \label{eq:pnorm}
   P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
@@ -427,7 +453,7 @@ 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 +462,20 @@ 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 following 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 +487,20 @@ 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 in order to find the maximum distance
+between the energy curve (\ref{eq:enorm}) and the  performance
+curve (\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
 represents the minimum energy consumption with minimum execution time (maximum 
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then 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 +509,24 @@ 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 
-(static and dynamic powers). However, the most energy reduction gain can be achieved when 
+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 important 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 applies DVFS operations to change the frequencies of the CPUs 
 according to the computed scaling factors.  This algorithm is called just once during the execution 
 of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called 
 in the iterative MPI program.
@@ -508,7 +535,7 @@ 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 
@@ -519,41 +546,48 @@ 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}
   \label{eq:Fint}
  F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
 \end{equation}
-If the computed initial frequency for a node is not available in the gears of that node, the computed 
-initial frequency is replaced by the nearest available frequency. In  figure (\ref{fig:st_freq}), 
-the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster 
-nodes are scaled down according to the computed initial frequency scaling factors. The resulting new 
-frequencies are colored in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered 
-as a higher bound for the search space of the optimal vector of frequencies because selecting frequency 
-scaling factors higher than the higher bound will not improve the performance of the application and 
-it will increase its overall energy consumption. Therefore the algorithm that selects the frequency 
-scaling factors starts the search method from these initial frequencies and takes a downward search direction 
-toward lower frequencies. The algorithm iterates on all left frequencies, from the higher bound until all 
-nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select 
-the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node 
-according to EQ(\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}).
-
-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 
-scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor 
-should be started from the maximum frequency because the performance and the consumed energy is decreased since  
-the beginning of the plot. On the other hand, in  the heterogeneous platform the performance is  maintained at 
-the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes 
-have computing powers lower than the slowest node. In other words, until they reach the higher bound. It can 
-also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger 
-the maximum distance between the energy curve and the performance curve is while varying the scaling factors 
-which results in bigger energy savings. 
+If the computed  initial frequency for a  node is not available in  the gears of
+that  node,  it  is replaced  by  the  nearest  available frequency.  In  figure
+(\ref{fig:st_freq}), the nodes are sorted by their computing power in ascending
+order and the  frequencies of the faster nodes are scaled  down according to the
+computed initial  frequency scaling factors.  The resulting new  frequencies are
+colored in  blue in figure (\ref{fig:st_freq}).  This set of  frequencies can be
+considered  as a higher  bound for  the search  space of  the optimal  vector of
+frequencies because  selecting frequency scaling factors higher  than the higher
+bound will not  improve the performance of the application  and it will increase
+its  overall  energy  consumption.  Therefore  the algorithm  that  selects  the
+frequency  scaling   factors  starts  the  search  method   from  these  initial
+frequencies and takes a downward  search direction toward lower frequencies. The
+algorithm  iterates on all  left frequencies,  from the  higher bound  until all
+nodes  reach  their  minimum   frequencies,  to  compute  their  overall  energy
+consumption and  performance, and select  the optimal frequency  scaling factors
+vector. At each iteration the algorithm determines the slowest node according to
+the equation (\ref{eq:perf}) and keeps  its frequency unchanged, while it lowers
+the  frequency  of  all  other  nodes  by one  gear.   The  new  overall  energy
+consumption  and  execution time  are  computed  according  to the  new  scaling
+factors.  The optimal set of frequency scaling factors is the set that gives the
+highest distance according to the objective function (\ref{eq:max}).
+
+Figures~\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 scaling factors. It can
+be noticed  that in a  homogeneous platform the  search for the  optimal scaling
+factor should start  from the maximum frequency because  the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand,
+in the heterogeneous platform the  performance is maintained at the beginning of
+the plot  even if the  frequencies of the  faster nodes decrease  until the
+computing power of scaled down  nodes are lower than the slowest  node. In other
+words, until they reach the higher bound. It can also be noticed that the higher
+the difference between the faster nodes  and the slower nodes is, the bigger the
+maximum distance  between the  energy curve and  the performance curve  is while
+ the scaling factors are varying which results in bigger energy savings.
 \begin{figure}[t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}
@@ -586,8 +620,8 @@ 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 +639,7 @@ 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 +652,7 @@ 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.
@@ -629,24 +663,56 @@ 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 to  the
+execution time  and the dynamic energy  is related to the  computation time. So,
+all the works presented  in this paper are 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 it does not test all the possible solutions (vectors of scaling factors) 
+in the search space. To prove its efficiency, it was compared on small instances to a brute force search algorithm 
+that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with 
+different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical 
+and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time: 
+for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in 
+table~\ref{table:platform}, it takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes 
+to compute the best scaling factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number 
+of iterations and $N$ is the number of computing nodes. The algorithm needs  from 12 to 20 iterations to select the best 
+vector of frequency scaling factors that gives the results of the next sections.
+
 \section{Experimental results}
 \label{sec.expe}
-To evaluate the efficiency and the overall energy consumption reduction of algorithm~(\ref{HSA}), 
-it was applied to the NAS parallel benchmarks NPB v3.3  \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 
-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 
-of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were 
-chosen proportionally to its computing power (FLOPS).  In the initial heterogeneous platform,  while computing 
-with highest frequency, each node  consumed power proportional to its computing power which 80\% of it was 
-dynamic power and the rest was 20\% for the static power, the same assumption  was made in \cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. 
-Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
+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 of  CPUs do not specify the  dynamic and the static  power of their
+CPUs, for  each type of  node they were  chosen proportionally to  its computing
+power  (FLOPS).  In  the initial  heterogeneous platform,  while  computing with
+highest frequency,  each node  consumed an amount  of power proportional  to its
+computing  power  (which  corresponds to  80\%  of  its  dynamic power  and  the
+remaining  20\%  to  the  static   power),  the  same  assumption  was  made  in
+\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.    Finally,  These
+nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
 
 
 \begin{table}[htb]
@@ -660,16 +726,16 @@ 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}
@@ -683,12 +749,14 @@ Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwi
 \label{sec.res}
 
 
-The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) 
-and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in 
-this paper, only the results of the biggest class, C, are presented while being run on different number 
-of nodes, ranging  from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the 
-benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes. 
-The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
+MG, FT, BT, LU and SP) and  the benchmarks were executed with the three classes:
+A, B and C. However, due to the lack of space in this paper, only the results of
+the  biggest class,  C, are  presented while  being run  on different  number of
+nodes,  ranging from 4  to 128  or 144  nodes depending  on the  benchmark being
+executed. Indeed, the benchmarks CG, MG, LU, EP and FT had to be executed on $1,
+2, 4, 8, 16, 32, 64, 128$ nodes.   The other benchmarks such as BT and SP had to
+be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
 
  
  
@@ -698,7 +766,7 @@ 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 +793,7 @@ 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 +820,7 @@ 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 +847,7 @@ 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 +874,7 @@ 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 +902,7 @@ 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          \\
@@ -854,108 +922,129 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \end{tabular}
   \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 
-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}, 
-\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 
-the  energy saving percentage is decreased  when the number of the computing nodes is increased. 
-This reduction is due to the increase of the communication times compared to the execution times 
-when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C, 
-are executed on different number of nodes, so the computation required for each iteration is divided 
-by the number of computing nodes.   On the other hand, more communications are required when increasing 
-the number of nodes so the static energy is increased linearly according to the communication time and 
-the dynamic power is less relevant in the overall energy consumption. Therefore, reducing the frequency 
-with algorithm~(\ref{HSA}) have less effect in reducing the overall energy savings. It can also be 
-noticed that for the benchmarks EP and SP that contain little or no communications,  the energy savings 
-are not significantly affected with the high number of nodes. No experiments were conducted using bigger 
-classes such as D, because they require a lot of memory(more than 64GB) when being executed by the simulator 
-on one machine. The maximum distance between the normalized energy curve and the normalized performance 
-for each instance is also shown in the result tables. It is decreased in the same way as the energy 
-saving percentage. The tables also show that the performance degradation percentage is not significantly 
-increased when the number of computing nodes is increased because the computation times are small when 
-compared to the communication times.  
+The overall energy  consumption was computed for each  instance according to the
+energy  consumption  model  (\ref{eq:energy}),  with and  without  applying  the
+algorithm. The execution time was also measured for all these experiments. Then,
+the energy saving and performance degradation percentages were computed for each
+instance.    The   results   are   presented  in   Tables   (\ref{table:res_4n},
+\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
+significantly reduces the energy consumption (up to 35\%) and tries to limit the
+performance  degradation.  They  also  show that  the  energy saving  percentage
+decreases when the  number of computing nodes increases.   This reduction is due
+to the increase of the communication  times compared to the execution times when
+the benchmarks are run over a high number of nodes.  Indeed, the benchmarks with
+the  same  class,  C,  are  executed  on different  numbers  of  nodes,  so  the
+computation required  for each iteration is  divided by the  number of computing
+nodes.  On the other hand,  more communications are required when increasing the
+number  of  nodes so  the  static energy  increases  linearly  according to  the
+communication time and the dynamic power  is less relevant in the overall energy
+consumption.   Therefore, reducing the  frequency with  algorithm~(\ref{HSA}) is
+less effective  in reducing the overall  energy savings. It can  also be noticed
+that for the benchmarks EP and  SP that contain little or no communications, the
+energy savings are  not significantly affected by the high  number of nodes.  No
+experiments were conducted  using bigger classes than D,  because they require a
+lot  of memory (more  than 64GB)  when being  executed by  the simulator  on one
+machine.   The maximum  distance between  the  normalized energy  curve and  the
+normalized performance for each instance is  also shown in the result tables. It
+decrease in the same way as  the energy saving percentage.  The tables also show
+that the performance degradation  percentage is not significantly increased when
+the number  of computing  nodes is increased  because the computation  times are
+small when 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}
+  \caption{The energy and performance for all NAS benchmarks running with a different 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 
-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 
-of nodes is increased because  this benchmark has more communications than the others. The second plot 
-shows that the performance degradation percentages of most of the benchmarks are decreased when they 
-run on a big number of nodes because they spend more time communicating than computing, thus, scaling 
-down the frequencies of some nodes have less effect on the performance. 
+Figures  \ref{fig:energy} and  \ref{fig:per_deg} present  the energy  saving and
+performance  degradation respectively for  all the  benchmarks according  to the
+number of used nodes. As shown  in the first plot, the energy saving percentages
+of the benchmarks MG,  LU, BT and FT decrease linearly when  the number of nodes
+increase. 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  significantly  decrease  when the  number  of nodes  increase
+because this benchmark has more  communications than the others. The second plot
+shows that  the performance  degradation percentages of  most of  the benchmarks
+decrease when  they run on a  big number of  nodes because they spend  more time
+communicating than computing,  thus, scaling down the frequencies  of some nodes
+has less effect on the performance.
 
 
 
 
 \subsection{The results for different power consumption scenarios}
-
-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  
-algorithm adapts itself according to the static and dynamic power values.  The two new power scenarios 
-are the following: 
+\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 of 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
+algorithm adapts itself  according to the static and  dynamic power values.  The
+two new power scenarios are the following:
 
 \begin{itemize}
-\item 70\% dynamic power  and 30\% static power
-\item 90\% dynamic power  and 10\% static power
+\item 70\% of dynamic power  and 30\% of static power
+\item 90\% of dynamic power  and 10\% of 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\% 
-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 
-degradation percentage is less in the 70\%-30\% scenario  compared to the 90\%-10\%  scenario. This is due to the 
-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 .
-
-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 
-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.
-Moreover, the average of the performance degradation is decreased when using a higher ratio for static power 
-(e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed algorithm optimizes the energy consumption 
-when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in 
-more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens 
-when using a higher ratio for  static  power, the algorithm proportionally  selects  smaller scaling values which 
-results in less energy saving but less performance degradation. 
+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  smaller 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 degradation  percentage is smaller
+in the 70\%-30\% scenario compared to the 90\%-10\% scenario. This is due to the
+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 amount of consumed
+static  energy directly  increas.  Therefore,  the proposed  algorithm  does not
+really significantly  scale down much the  frequencies of the nodes  in order to
+limit the  increase of the  execution time and  thus limiting the effect  of the
+consumed static energy.
+
+Both   new  power   scenarios   are  compared   to   the  old   one  in   figure
+(\ref{fig:sen_comp}). It  shows the average of the  performance degradation, the
+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  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 decreases  when the 70\%-30\%
+scenario is  used because  the dynamic  energy is less  relevant in  the overall
+consumed energy and  lowering the frequency does not  return big energy savings.
+Moreover, the average  of the performance degradation is  decreased when using a
+higher  ratio   for  static  power  (e.g.   70\%-30\%   scenario  and  80\%-20\%
+scenario). Since  the proposed algorithm  optimizes the energy  consumption when
+using a  higher ratio for dynamic  power the algorithm  selects bigger frequency
+scaling  factors that result  in more  energy saving  but less  performance, for
+example see  Figure (\ref{fig:scales_comp}). The  opposite happens when  using a
+higher  ratio for  static power,  the algorithm  proportionally  selects smaller
+scaling  values which result  in less  energy saving  but also  less performance
+degradation.
 
 
  \begin{table}[htb]
-  \caption{The results of 70\%-30\% powers scenario}
+  \caption{The results of the 70\%-30\% power scenario}
   % title of Table
   \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         \\
@@ -979,12 +1068,12 @@ results in less energy saving but less performance degradation.
 
 
 \begin{table}[htb]
-  \caption{The results of 90\%-10\% powers scenario}
+  \caption{The results of the 90\%-10\% power scenario}
   % title of Table
   \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 +1097,104 @@ 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  between the results on 8 nodes]{%
+    \includegraphics[width=.33\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=.33\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}
+In this section, the scaling  factors selection algorithm, called MaxDist,
+is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP. 
+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 provides better energy savings than Spiliopoulos et al. algorithm, 
+on average it results in 29.76\% energy saving while their algorithm returns just 25.75\%. The average of performance degradation percentage is approximately the same for both algorithms, about 4\%. 
+
+
+For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
+terms of  energy and  performance 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}
+
+
 
-\section{Conclusion}
-\label{sec.concl}
 
 
+\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  platforms. 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 terms 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 platforms 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  other tasks  to finish  their  works.  The
+development of such a 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. 
+
 
 % trigger a \newpage just before the given reference
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@@ -1058,7 +1205,7 @@ vector of frequency scaling factors that gives the results of the section (\ref{
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-
+   
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@@ -1067,6 +1214,6 @@ vector of frequency scaling factors that gives the results of the section (\ref{
 %%% End:
 
 % LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords:  CMOS EQ EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex
+% LocalWords:  CMOS EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex
 % LocalWords:  de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
 % LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
