X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/2c04063b3d9996634b93b35dc89827eb103370d7..HEAD:/Heter_paper.tex?ds=inline

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 3b2b238..df0bb8f 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -5,10 +5,11 @@
 \usepackage[english]{babel}
 \usepackage{algpseudocode}
 \usepackage{graphicx}
+\usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
-
 \usepackage{url}
+
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage[autolanguage,np]{numprint}
@@ -23,37 +24,45 @@
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 
-\newcommand{\Xsub}[2]{\ensuremath{#1_\textit{#2}}}
+\newcommand{\Xsub}[2]{{\ensuremath{#1_\mathit{#2}}}}
+
+%% used to put some subscripts lower, and make them more legible
+\newcommand{\fxheight}[1]{\ifx#1\relax\relax\else\rule{0pt}{1.52ex}#1\fi}
 
-\newcommand{\Dist}{\textit{Dist}}
+\newcommand{\CL}{\Xsub{C}{L}}
+\newcommand{\Dist}{\mathit{Dist}}
+\newcommand{\EdNew}{\Xsub{E}{dNew}}
 \newcommand{\Eind}{\Xsub{E}{ind}}
 \newcommand{\Enorm}{\Xsub{E}{Norm}}
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
-\newcommand{\Fdiff}{\Xsub{F}{diff}}
-\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Es}{\Xsub{E}{S}}
+\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
+\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}
 \newcommand{\Fnew}{\Xsub{F}{new}}
 \newcommand{\Ileak}{\Xsub{I}{leak}}
 \newcommand{\Kdesign}{\Xsub{K}{design}}
-\newcommand{\MaxDist}{\textit{Max Dist}}
+\newcommand{\MaxDist}{\mathit{Max}\Dist}
+\newcommand{\MinTcm}{\mathit{Min}\Tcm}
 \newcommand{\Ntrans}{\Xsub{N}{trans}}
-\newcommand{\Pdyn}{\Xsub{P}{dyn}}
-\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
+\newcommand{\PdNew}{\Xsub{P}{dNew}}
+\newcommand{\PdOld}{\Xsub{P}{dOld}}
 \newcommand{\Pnorm}{\Xsub{P}{Norm}}
-\newcommand{\Tnorm}{\Xsub{T}{Norm}}
-\newcommand{\Pstates}{\Xsub{P}{states}}
-\newcommand{\Pstatic}{\Xsub{P}{static}}
-\newcommand{\Sopt}{\Xsub{S}{opt}}
-\newcommand{\Tcomp}{\Xsub{T}{comp}}
-\newcommand{\TmaxCommOld}{\Xsub{T}{Max Comm Old}}
-\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
-\newcommand{\Tmax}{\Xsub{T}{max}}
+\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
+\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
+\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
+\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
+\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
+\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Told}{\Xsub{T}{Old}}
 
 \begin{document}
 
-\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
+\title{Energy Consumption Reduction with DVFS for \\
+  Message Passing Iterative Applications on \\
+  Heterogeneous Architectures}
 
 \author{%
   \IEEEauthorblockN{%
@@ -63,8 +72,7 @@
     Arnaud Giersch
   }
   \IEEEauthorblockA{%
-    FEMTO-ST Institute\\
-    University of Franche-ComtÃ©\\
+    FEMTO-ST Institute, University of Franche-ComtÃ©\\
     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
@@ -76,314 +84,521 @@
 \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 may increase
+  the execution time of an application running on that processor.  Therefore,
+  the frequency that gives the best trade-off between the energy consumption and
+  the performance of an application must be selected.
+
+  In this paper, a new online frequency selecting algorithm for heterogeneous
+  platforms (heterogeneous CPUs) is presented.  It selects the frequencies and tries to give the best
+  trade-off 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 \np[\%]{34} while limiting the performance
+  degradation as much as possible.  Finally, the algorithm is compared to an
+  existing method, the comparison results show that it outperforms the
+  latter, on average it saves \np[\%]{4} more energy while keeping the same performance. 
+
 \end{abstract}
 
 \section{Introduction}
 \label{sec.intro}
 
+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 may 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,
+\dots{} DVFS is a widely used process to reduce the energy consumption of a
+processor by lowering its frequency
+\cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
+the number of FLOPS executed by the processor which may 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
+trade-off 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 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 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}
 
-
-
-
+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 may
+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 may need to run the application and profile
+it before selecting the new frequency, such
+as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
+The methods could be heuristics, exact or brute force methods that satisfy
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+sequential, parallel or distributed architecture, homogeneous or heterogeneous
+platform, synchronous or asynchronous application, \dots{}
+
+In this paper, we are interested in reducing energy for message passing
+iterative synchronous applications running over heterogeneous platforms.  Some
+works have already been done for such platforms and they can be classified into
+two types of heterogeneous platforms:
+\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 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 $\mathit{energy}\times \mathit{delay}^2$ (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 \np[\%]{10}.
+In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
+heterogeneous cluster composed of two types of Intel and AMD processors. They
+use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
+\cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
+frequencies for a specified heterogeneous cluster are selected offline using
+some heuristic.  Chen et
+al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
+programming approach to minimize the power consumption of heterogeneous servers
+while respecting given time constraints.  This approach had considerable
+overhead.  In contrast to the above described papers, this paper presents the
+following contributions :
+\begin{enumerate}
+\item two new energy and performance models for message passing iterative
+  synchronous applications running over a heterogeneous platform. Both models
+  take into account communication and slack times. The models can predict the
+  required energy and the execution time of the application.
+
+\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 performance of parallel tasks on heterogeneous cluster}
+\subsection{The execution time of message passing distributed iterative
+  applications on a heterogeneous platform}
 
-The  heterogeneous cluster is a collection of non identical computing nodes. Each node in 
-a cluster is connected via a high speed network. The communication capabilities between nodes  
-are  identical or different. In this work we are interested in identical communications. While each 
-node has different processing capabilities such as CPU speeds and memory. Tasks executed
-on this model can be either synchronous or asynchronous. In this paper we are consider execution of 
-the synchronous tasks on distributed heterogeneous platform. These tasks can exchange
-the data via synchronous message passing. 
+In this paper, we are interested in reducing the energy consumption of message
+passing distributed iterative synchronous applications running over
+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
+have the same network bandwidth and latency.
 
-\begin{figure}[t]
+\begin{figure}[!t]
   \centering
-    \includegraphics[scale=0.6]{fig/commtasks}
-  \caption{Parallel tasks on heterogeneous platform}
+  \includegraphics[scale=0.6]{fig/commtasks}
+  \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 
- Therefore, the  execution time of a task consists of the computation time and
- the communication time. Due to  heterogeneous computations can lead to slack times while the tasks 
- wait at the synchronization barrier for other tasks to finish  their jobs (see Figure~(\ref{fig:heter})). 
- In this case the fastest tasks  have to wait at the synchronization barrier for the slowest ones  to begin
- the next task. Therefore,  the overall execution time  of the program is the execution time of the slowest
- task as in  EQ~(\ref{eq:T1}).
-\begin{equation}
-  \label{eq:T1}
-  \textit{Program Time} = \max_{i=1,2,\dots,N} T_i
-\end{equation}
- where $T_i$ is the execution time of the task $i$ and all the tasks are executed concurrently on different processors. DVFS is a process that is allowed in modern processors to reduce the dynamic
-power by scaling down the voltage and frequency. Then any DVFS operation used to reduce energy of the processor has direct affect on the execution time of the MPI program. 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}).
+The overall execution time of a distributed iterative synchronous application
+over a heterogeneous platform consists of the sum of the computation time and
+the communication time for every iteration on a node. However, due to the
+heterogeneous computation power of the computing nodes, slack times may occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
+overall execution time of the program is the execution time of the slowest task
+which has the highest computation time and no slack time.
+
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
+modern processors, that reduces the energy consumption of a CPU by scaling
+down its voltage and frequency.  Since DVFS lowers the frequency of a CPU
+and consequently its computing power, the execution time of a program running
+over that scaled down processor may 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 (\ref{eq:s}).
 \begin{equation}
   \label{eq:s}
- S = \frac{F_\textit{max}}{F_\textit{new}}
+  S = \frac{\Fmax}{\Fnew}
 \end{equation}
- The execution time of a parallel program is linearly proportional to the frequency scaling factor $S$. 
- However, in most  MPI applications the processes exchange data. During these  communications the
- processors involved remain idle until the  communications are finished. For that reason, any change in
- the frequency has no impact on the time of communication~\cite{17}. 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.
- Each node has different DVFS features such as frequency values and the number of available frequencies 
- (Pstates) for each node. By contrast there are different frequency scaling factors for each node $S_1, S_2,..., S_N$. To be able to predict the execution time of MPI program, the  communication time and the computation time for the slowest
- task must be measured before scaling. These times are used to  predict the execution time for any MPI program running on heterogeneous cluster as a function
- of the new scaling factors as in EQ (\ref{eq:perf}). The model is computes the maximum production of  computation time 
- with scaling factor from each node  added to the minimum communication time of the slowest node, it means  only the
- communication time without slack times, because in MPI the slack times is measured with communication times. 
-\begin{multline}
+The execution time of a compute bound sequential program is linearly
+proportional to the frequency scaling factor $S$.  On the other hand, message
+passing distributed applications consist of two parts: computation and
+communication.  The execution time of the computation part is linearly
+proportional to the frequency scaling factor $S$ but the communication time is
+not affected by the scaling factor because the processors involved remain idle
+during the 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 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
+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
+be able to predict the execution time of message passing synchronous iterative
+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 (\ref{eq:perf}).
+\begin{equation}
   \label{eq:perf}
- \textit  T_\textit{new} = \\ 
- {} \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  
- \min_{i=1,2,\dots,N} Tcm Old_{i}
-\end{multline}
-This prediction modal is developed from our model for predicting the execution time of parallel task on homogeneous architecture~\cite{45}. The execution time predicting model is useful to used in our method for optimizing both energy and performance of iterative methods as in the coming sections.
-
+  \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) +  \MinTcm
+\end{equation}
+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}
 
-Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
-two power metrics: the static and the dynamic power.  While the first one is
-consumed as long as the computing unit is on, the latter is only consumed during
-computation times.  The dynamic power $P_{d}$ 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}).
+Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
+  Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
+  Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
+a processor into 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 $\CL$, the supply voltage $V$
+and operational frequency $F$, as shown in (\ref{eq:pd}).
 \begin{equation}
   \label{eq:pd}
-  P_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F
+  \Pd = \alpha \cdot \CL \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}
-   P_\textit{s}  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
+   \Ps  = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak
 \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
+where V is the supply voltage, $\Ntrans$ is the number of transistors,
+$\Kdesign$ is a design dependent parameter and $\Ileak$ is a
+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} =  P_\textit{d} \cdot T_{cp} + P_\textit{s} \cdot T
+  \Eind =  \Pd \cdot \Tcp + \Ps \cdot T
 \end{equation}
-where $T$ is the execution time of the program, $T_{cp}$ is the computation
-time and $T_{cp} \leq T$.  $T_{cp}$ may be equal to $T$ if there is no
-communication, no slack time and no synchronization.
-
-The main objective of DVFS operation is to
-reduce the overall energy consumption~\cite{37}.  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
-voltage with respect to various frequency values in~\cite{3}.  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}).
-The value of the scaling factor $S$ is greater than 1 when changing the
-frequency of the CPU to any new frequency value~(\emph{P-state}) in the
-governor.  The CPU governor is an interface driver 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:
+where $T$ is the execution time of the program, $\Tcp$ is the computation
+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 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 (\ref{eq:s}).  The CPU governors are
+power schemes supplied by the operating system's kernel to lower a core's
+frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
+follows:
 \begin{equation}
   \label{eq:fnew}
-   F_\textit{new} = S^{-1} .  F_\textit{max}
+   \Fnew = S^{-1} \cdot \Fmax
 \end{equation}
-By substituting this equation in EQ(\ref{eq:pd}) results the following equation for dynamic 
-power consumption:
+Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
+equation for dynamic power consumption:
 \begin{multline}
   \label{eq:pdnew}
-   {P}_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
-   = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{d} \cdot S^{-3}
+   \PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 \\
+   {} = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}
 \end{multline}
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduce by a factor of $S^{-3}$ when 
-reducing the frequency by a factor of $S$~\cite{3}. The dynamic energy is the energy consumed by a CPU when its in the computation mode. 
-So, the dynamic energy is the  dynamic power multiply by the time of computations. While the time of computation is decreased by a factor of $S$. Therefore the 
-the dynamic energy  is decreased by a factor of $S^{-2}$ as follow:
+where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
+new frequency and the maximum frequency respectively.
+
+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 and is given by the following equation:
 \begin{equation}
   \label{eq:Edyn}
-   E_\textit{d} = P_{d} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{d} \cdot  Tcp 
+   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp
 \end{equation}
-The static power is related to leakage power consumption, its mean the CPU continue consumes energy 
-whereas in idle state. Therefore, we are make an assumption that the static power is constant as in~\cite{3,46}.
-The static energy is the static power multiply by the execution time of the program. Moreover, the CPU consumes static
-energy in all times of the program such as computation, communication and slacks times. 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 related  
-to frequency scaling factor linearly, while this scaling factor not affecting on the time of communication~\cite{17}, then  the static energy 
-of individual processor is as follow: 
-
+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},
+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 (\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}
   \label{eq:Estatic}
- E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)
+  \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
 \end{equation}
-In heterogeneous architecture there is a  number of different processors $P_1, P_2,...,P_N$, where $N$ is the number of nodes . Moreover, each processor perhaps has different frequency  scaling factor, so there are a set of frequency scaling factors for such platform  $S_1,S_2,...,S_N$. According  to these  different 
-scaling factors producing different computation time, $Tcp_1,Tcp_2,...,Tcp_N$, because these times linearly related to these scaling factors. In MPI program the communication times is measured with slacks times. The slack times also has linear relation with the scaling factors. So, there are different mesured communication times $Tcm_1,Tcm_2,...,Tcm_N$ even if its identical communications, e.g. see figure(\ref{fig:heter}). The energy modal of an  heterogeneous architecture represents the summation of all dynamic and static energies from each  processors, each processor has its dynamic and static powers, for example, in the hole architecture their are: $Pd_1,Pd_2,...,Pd_N$ and $Ps_1,Ps_2,...,Ps_N$.  The dynamic energy is computes as in EQ(\ref{eq:Edyn}) with regarding to the frequency scaling factor and the dynamic power of each node. While the static energy is computes using EQ(\ref{eq:perf}) multiplied  by the static power of each processor. So, the energy modal of an heterogeneous platform has the following form:
+
+In the considered heterogeneous platform, each processor $i$ may 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
+$\Tcp[i]$ may be different and different frequency scaling factors may 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)} +\\ 
- {}\sum_{i=1}^{N} {(Ps_i \cdot (\max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +} 
- {}\min_{i=1,2,\dots,N} {Tcm_{i}))} 
- \end{multline}
- 
-These set of frequency scaling factors $S_i$ reduce quadratically the dynamic power which may cause degradation in performance and thus, the
-increase of the static energy because the execution time is increased~\cite{36}.
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + {} \\
+ \sum_{i=1}^{N} (\Ps[i] \cdot (\max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +
+  {\MinTcm))}
+\end{multline}
+
+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}. 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}
-Applying DVFS to lower level  not surly reducing the energy consumption to minimum level. Also, a big scaling for the frequency produces high performance degradation percent. Moreover, by considering the  drastically  increase in execution time of parallel program, the static energy is related to this time 
-and it also increased by the same ratio. Thus, the opportunity for gaining more energy reduction is restricted. For that choosing frequency scaling factors is very important process to taking into account both energy and performance. In our previous work~\cite{45}, we are proposed a method that selects the optimal frequency scaling factor for an homogeneous cluster, depending on the trade-off relation between the energy and performance. In this work we have an  heterogeneous cluster, at each node there is different scaling factors, so our goal is to selects the optimal set of frequency scaling factors, $Sopt_1,Sopt_2,...,Sopt_N$, that gives the best trade-off  between energy consumption and performance. The relation between the energy and the execution time is complex and nonlinear, Thus, unlike the relation between the performance and  the scaling factor,  the relation of the energy with the frequency scaling factors is nonlinear, for more details refer to~\cite{17}.  Moreover, they are not measured using the same metric.  To solve this problem, we normalize the execution time by calculating the ratio between the new execution time (the scaled execution time) and the old one as follow:
+
+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}}\\
-          = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}} 
-           {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
+  \Pnorm = \frac{\Tnew}{\Told}\\
+       {} = \frac{ \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +\MinTcm}
+           {\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
 \end{multline}
 
-
-By the same way, we are normalize the energy by calculating the ratio between the consumed energy with scaled frequency and the consumed energy without scaled frequency:
+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}
-  E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
-   = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
- \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 $T_{Old}$ is computed as in EQ(\ref{eq:pnorm}). The second problem 
-is that the optimization operation for both energy and performance is not in the same direction.  
-In other words, the normalized energy and the normalized execution time curves are not at the same direction.  
-While the main goal is to optimize the energy and execution time in the same time.  According to the 
-equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the set of frequency scaling factors $S_1,S_2,...,S_N$ reduce both the energy and the
-execution time simultaneously.  But the main objective is to produce maximum energy
-reduction with minimum execution time reduction.  Many researchers used different
-strategies to solve this nonlinear problem for example see~\cite{19,42}, their
-methods add big overheads to the algorithm to select the suitable frequency.
-In this paper we are present a method to find the optimal set of frequency scaling factors to optimize both energy and execution time simultaneously
-without adding a big overhead.  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, the normalized performance,  as follows:
-
+  \Enorm = \frac{\Ereduced}{\Eoriginal} \\
+  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}
+\end{multline}
+Where $\Ereduced$ and $\Eoriginal$ are computed using (\ref{eq:energy}) and
+$\Tnew$ and $\Told$ are computed as in (\ref{eq:pnorm}).
+
+While the main goal is to optimize the energy and execution time at the same
+time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. According 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.
+
+This problem can be solved by making the optimization process for energy and
+execution time follow the same evolution according to the vector of scaling factors. 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}}\\
-          = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
-            { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}} 
+  \Pnorm = \frac{\Told}{\Tnew}\\
+          = \frac{\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
+            { \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm}
 \end{multline}
 
-
-\begin{figure}
+\begin{figure}[!t]
   \centering
   \subfloat[Homogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/homo.eps}\label{fig:r1}}%
-  \qquad%
+    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+
   \subfloat[Heterogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/heter.eps}\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
-represents the minimum energy consumption with minimum execution time (better
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}) .  Then our objective
-function has the following form:
-\begin{multline}
+Then, the objective function can be modeled in order to find the maximum
+distance between the energy curve (\ref{eq:enorm}) and the performance curve
+(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
+represents the minimum energy consumption with minimum execution time (maximum
+performance) at the same time, see Figure~\ref{fig:r1} or
+Figure~\ref{fig:r2}. Then the objective function has the following form:
+\begin{equation}
   \label{eq:max}
-  Max Dist = 
-  \max_{i=1,\dots F, j=1,\dots,N}
-      (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
-       \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
-\end{multline}
-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 energy values stored in a data file.
-Moreover, this function works in optimal way when the energy curve has a convex
-form over the available frequency scaling factors as shown in~\cite{15,3,19}.
-
-\section{The heterogeneous scaling algorithm }
+  \MaxDist =
+  \mathop{\max_{i=1,\dots F}}_{j=1,\dots,N}
+      (\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_{ij})}^{\text{Minimize}} )
+\end{equation}
+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 proposed an heterogeneous scaling algorithm, (figure~\ref{HSA}), that selects the optimal set of scaling factors from each node. 
-The algorithm is numerates the suitable range of available scaling factors for each node in the heterogeneous cluster, returns a set of optimal frequency scaling factors for each node. Using heterogeneous cluster is produces different workloads for each node. Therefore, the fastest nodes waiting at the barrier for the slowest nodes to finish there work as in figure (\ref{fig:heter}). Our algorithm takes into account these imbalanced workloads when is starts to search for selecting the best scaling factors. So, the algorithm is selecting the initial frequencies values for each node proportional to the times of computations that gathered from the first iteration. As an example in figure (\ref{fig:st_freq}), the algorithm don't test the first frequencies of the fastest nodes until it converge their frequencies to the frequency of the slowest node. If the algorithm is starts test changing the frequency of the slowest nodes from beginning, we are loosing performance and then not selecting the best tradeoff (the distance). This case will be similar to the homogeneous cluster when all nodes scales their frequencies together from the  beginning. In this case there is a small distance between energy and performance curves, for example see the figure(\ref{fig:r1}).  Then the algorithm searching for optimal frequency  scaling factor from the selected frequencies until the last available ones. 
-\begin{figure}[t]
-  \centering
-    \includegraphics[scale=0.5]{fig/start_freq.pdf}
-  \caption{Selecting the initial frequencies}
-  \label{fig:st_freq}
-\end{figure}
-
 
-To compute the initial frequencies, the algorithm firstly needs to compute the computation scaling factors $Scp_i$ for each node. Each one of these factors represent a ratio between the computation time of the slowest node and the computation time of the node $i$ as follow:
-\begin{equation}
-  \label{eq:Scp}
- Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
-\end{equation}
-Depending on the initial computation scaling factors EQ(\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 follow:
-\begin{equation}
-  \label{eq:Fint}
- F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
-\end{equation}
-\begin{figure}[tp]
+\begin{algorithm}
   \begin{algorithmic}[1]
     % \footnotesize
     \Require ~
     \begin{description}
-    \item[$Tcp_i$] array of all computation times for all nodes during one iteration and with highest frequency.
-    \item[$Tcm_i$] array of all communication times for all nodes during one iteration and with highest frequency.
-    \item[$Fmax_i$] array of the maximum frequencies for all nodes.
-    \item[$Pd_i$] array of the dynamic powers for all nodes.
-    \item[$Ps_i$] array of the static powers for all nodes.
-    \item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
+    \item[{$\Tcp[i]$}] array of all computation times for all nodes during one iteration and with highest frequency.
+    \item[{$\Tcm[i]$}] array of all communication times for all nodes during one iteration and with highest frequency.
+    \item[{$\Fmax[i]$}] array of the maximum frequencies for all nodes.
+    \item[{$\Pd[i]$}] array of the dynamic powers for all nodes.
+    \item[{$\Ps[i]$}] array of the static powers for all nodes.
+    \item[{$\Fdiff[i]$}] array of the differences between two successive frequencies for all nodes.
     \end{description}
-    \Ensure $Sopt_1, \dots ,Sopt_N$ is a set of optimal scaling factors 
+    \Ensure $\Sopt[1],\Sopt[2] \dots, \Sopt[N]$ is a vector of optimal scaling factors
 
-    \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
-    \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
+    \State $\Scp[i] \gets \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]} $
+    \State $F_{i} \gets  \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\cdots,N}$
     \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
     \If{(not the first frequency)}
-          \State $F_i \gets F_i+Fdiff_i,~i=1,...,N.$
-    \EndIf 
-    \State $T_\textit{Old} \gets max_{~i=1,...,N } (Tcp_i+Tcm_i)$
-    \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,...,N. $
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
+    \EndIf
+    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
+    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}$
+    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \Told)}$
+    \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 $F_i \gets F_i - Fdiff_i,~i=1,...,N.$ 
-        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,...,N.$ 
+        \State $F_i \gets F_i - \Fdiff[i],~i=1,\dots,N.$
+        \State $S_i \gets \frac{\Fmax[i]}{F_i},~i=1,\dots,N.$
         \EndIf
-       \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + min_\textit{~i=1,\dots,N}(Tcm_i) $
-       \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} + $  \hspace*{43 mm} 
-               $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
-       \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
-       \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
+       \State $\Tnew \gets \max_{i=1,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm $
+%       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + \sum_{i=1}^{N} {(\Ps[i] \cdot \rlap{\Tnew)}} $
+       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \rlap{\Tnew)}} $
+       \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+       \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
-        \State $Sopt_{i} \gets S_{i},~i=1,...,N. $
+        \State $\Sopt[i] \gets S_{i},~i=1,\dots,N. $
         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile
-    \State  Return $Sopt_1,Sopt_2, \dots ,Sopt_N$
+    \State  Return $\Sopt[1],\Sopt[2],\dots,\Sopt[N]$
   \end{algorithmic}
-  \caption{Heterogeneous scaling algorithm}
+  \caption{frequency scaling factors selection algorithm}
   \label{HSA}
-\end{figure}
-When the initial frequencies are computed the algorithm numerates all available scaling factors starting from these frequencies  until all nodes reach their 
-minimum frequencies. At each iteration the algorithm remains the frequency of the slowest node without change and scaling the frequency of the other nodes. This is gives better performance and energy tradeoff. 
-The proposed algorithm works online during the execution time of the MPI
-program.  Its returns a set of optimal frequency scaling factors $Sopt_i$ depending on the objective function EQ(\ref{eq:max}). The program changes the new frequencies of the CPUs according to the computed scaling factors.  This algorithm has a small execution time: 
-for an heterogeneous  cluster composed of four different types of nodes having the characteristics presented in
-table~(\ref{table:platform}), it takes \np[ms]{0.04} on average for 4 nodes and
-\np[ms]{0.1} on average for 128 nodes.  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 on average from 12 to 20 iterations for all the NAS benchmark on class C to selects the best set of frequency scaling factors. Its called just once during the execution of the program.  The DVFS figure~(\ref{dvfs}) shows where and when the algorithm is
-called in the MPI program.
-\begin{figure}[tp]
+\end{algorithm}
+
+\begin{algorithm}
   \begin{algorithmic}[1]
     % \footnotesize
     \For {$k=1$ to \textit{some iterations}}
@@ -392,7 +607,7 @@ called in the MPI program.
       \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.
@@ -401,316 +616,622 @@ called in the MPI program.
   \end{algorithmic}
   \caption{DVFS algorithm}
   \label{dvfs}
+\end{algorithm}
+
+\subsection{The algorithm details}
+
+In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
+scaling factors vector that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+synchronous iterative application executed on a heterogeneous platform. It works
+online during the execution time of the iterative message passing program.  It
+uses information gathered during the first iteration such as the computation
+time and the communication time in one iteration for each node. The algorithm is
+executed after the first iteration and returns a vector of optimal frequency
+scaling factors that satisfies the objective function (\ref{eq:max}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors.  This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
+scaling algorithm is called in the iterative MPI program.
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.5]{fig/start_freq}
+  \caption{Selecting the initial frequencies}
+  \label{fig:st_freq}
 \end{figure}
 
-\section{Experimental results}
-\label{sec.expe}
-The experiments  of this work are executed on the simulator Simgrid/SMPI v3.10. We configure the simulator to use a heterogeneous cluster 
-with one core per node. The proposed heterogeneous cluster has four different types of nodes. Each node in cluster has different characteristics 
-such as the maximum frequency speed, the number of available frequencies and dynamic and static powers values, see table (\ref{table:platform}). These different types of processing nodes simulate some real Intel processors. The maximum number of nodes that supported by the cluster is 144 nodes according to  characteristics of some MPI programs of the NAS benchmarks that used. We are use the same number from each type of nodes when running the MPI programs, for example if we execute the program on 8 node, there are 2 nodes from each type participating in the computing. The dynamic and static power values is different from one type to other. Each node has a dynamic and static power values proportional to their performance/GFlops, for more details see the Intel data sheets in \cite{47}.  Each node has a percentage of  80\% for dynamic power and 20\% for static power from the hole power consumption, the same assumption is made in \cite{45,3}. These nodes are connected via an ethernet network with 1 Gbit/s bandwidth.
-\begin{table}[htb]
+The nodes in a heterogeneous platform have different computing powers, thus
+while executing message passing iterative synchronous applications, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
+periods are called idle or slack times.  The algorithm takes into account this
+problem and tries to reduce these slack times when selecting the frequency
+scaling factors vector. At first, it selects initial frequency scaling factors
+that increase the execution times of fast nodes and minimize the differences
+between the computation times of fast and slow nodes. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
+\begin{equation}
+  \label{eq:Scp}
+  \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}
+\end{equation}
+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,\dots,N}
+\end{equation}
+If the computed initial frequency for a node is not available in the gears of
+that node, it is replaced by the nearest available frequency.  In
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors.  The resulting new
+frequencies are highlighted 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 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 remaining 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. 
+Finally, in a homogeneous platform the energy consumption is increased when the scaling factor is very high. 
+Indeed, the dynamic energy  saved  by reducing the frequency of the processor is compensated by the significant increase of the execution time and thus the increased of the static energy. On the other hand, in a heterogeneous platform this is not the case.
+
+\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 $O(F\cdot N)$, where $F$ is the
+maximum number of available frequencies, 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.
+
+\begin{table}[!t]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
-    Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\
-    type     & to          & Freq. GHz  & Freq. GHz    & Freq GHz       & power        & power \\
+    Node   & Simulated & Max   & Min   & Diff. & Dynamic    & Static    \\
+    type   & GFLOPS    & Freq. & Freq. & Freq. & power      & power     \\
+           &           & GHz   & GHz   & GHz   &            &           \\
     \hline
-    1       & core-i3       & 2.5         & 1.2          & 0.1           & 20~w         &4~w    \\
-            &  2100T        &             &              &               &              &  \\
+    1      & 40        & 2.50  & 1.20  & 0.100 & \np[W]{20} & \np[W]{4} \\
     \hline
-    2       & Xeon          & 2.66        & 1.6          & 0.133         & 25~w         &5~w    \\
-            & 7542          &             &              &               &              &  \\
+    2      & 50        & 2.66  & 1.60  & 0.133 & \np[W]{25} & \np[W]{5} \\
     \hline
-    3       & core-i5       & 2.9         & 1.2          & 0.1           & 30~w         &6~w    \\
-            & 3470s         &             &              &               &              &  \\
+    3      & 60        & 2.90  & 1.20  & 0.100 & \np[W]{30} & \np[W]{6} \\
     \hline
-    4       & core-i7       & 3.4         & 1.6          & 0.133         & 35~w         &7~w    \\
-            & 2600s         &             &              &               &              &  \\
+    4      & 70        & 3.40  & 1.60  & 0.133 & \np[W]{35} & \np[W]{7} \\
     \hline
   \end{tabular}
   \label{table:platform}
 \end{table}
 
- 
-%\subsection{Performance prediction verification}
+\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 which 
+is composed of synchronous message passing applications. 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 \np[\%]{80} of its dynamic power and the
+remaining \np[\%]{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 \np[Gbit/s]{1} bandwidth.
 
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 
-The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.
-\cite{44},  which were run with three classes (A, B and C).
-In this experiments we are focus on running of the class C, the biggest class compared to A and B, on different number of 
-nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Depending on the proposed energy consumption model EQ(\ref{eq:energy}),
- we are measure the energy consumption for all NAS MPI programs. The dynamic and static power values used under the same assumption used by \cite{45,3}. We used a percentage of 80\% for dynamic power from all power consumption of the CPU and 20\% for static power. The heterogeneous nodes in table (\ref{table:platform}) have different  simulated performance, ranked from the node of type 1 with smaller performance/GFlops to the highest performance/GFlops for node 4. Therefore, the power values used proportionally increased from node 1 to node 4 that with highest performance. Then we used an assumption that the power consumption increases linearly with the performance/GFlops of the processor see \cite{48}.   
- 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 4 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 8 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, or 128 nodes.  The other benchmarks such as BT and SP had
+to be executed on 1, 4, 9, 16, 36, 64, or 144 nodes.
+
+\begin{table}[!t]
+  
+% \end{table}
+
+  
+% \begin{table}[!t]
+  \caption{Running NAS benchmarks on 8 and 9 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    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       \\
-    \hline 
-    MG         & 18.89         & 1074.87	    &35.37	      &4.34	              &31.03       \\
-   \hline
-    EP         &79.73	       &5521.04	        &26.83   	  &3.04               &23.79      \\
-   \hline
-    LU         &308.65	       &21126.00	   &34.00	      &6.16	              &27.84      \\
+    CG      &  36.11    &  3263.49      & 31.25    & 7.12          & 24.13    \\
     \hline
-    BT         &360.12         &21505.55	   &35.36         &8.49               &26.87     \\
-   \hline
-    SP         &234.24	       &13572.16	   &35.22         &5.70	              &29.52    \\
-   \hline
-    FT         &81.58          &4151.48        &35.58         &0.99        	      &34.59    \\
-\hline 
-  \end{tabular}
-  \label{table:res_4n}
-\end{table}
-
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 8 and 9 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
+    MG      &   8.99    &   953.39      & 33.78    & 6.41          & 27.37    \\
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    EP      &  40.39    &  5652.81      & 27.04    & 0.49          & 26.55    \\
     \hline
-    CG         &36.11    	   &3263.49	        &31.25	      &7.12	               &24.13     \\
-    \hline 
-    MG         &8.99 	       &953.39	        &33.78	      &6.41	               &27.37     \\
-   \hline
-    EP         &40.39	       &5652.81	        &27.04	      &0.49	               &26.55     \\
-   \hline
-    LU         &218.79  	   &36149.77	    &28.23        &0.01   	           &28.22      \\
+    LU      & 218.79    & 36149.77      & 28.23    & 0.01          & 28.22    \\
+    \hline
+    BT      & 166.89    & 23207.42      & 32.32    & 7.89          & 24.43    \\
+    \hline
+    SP      & 104.73    & 18414.62      & 24.73    & 2.78          & 21.95    \\
+    \hline
+    FT      &  51.10    &  4913.26      & 31.02    & 2.54          & 28.48    \\
     \hline
-    BT         &166.89 	       &23207.42	    &32.32	      &7.89	               &24.43      \\
-   \hline
-    SP         &104.73	       &18414.62	    &24.73	      &2.78	               &21.95      \\
-   \hline
-    FT         &51.10	       &4913.26	        &31.02	      &2.54	               &28.48      \\
-\hline 
   \end{tabular}
   \label{table:res_8n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+  \medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 16 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    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          \\
-    \hline 
-    MG         &5.71	       &1076.19         &32.49	      &6.05	               &26.44         \\
-   \hline
-    EP         &20.11	       &5638.49	        &26.85	      &0.56	               &26.29         \\
-   \hline
-    LU         &144.13	       &42529.06	    &28.80	      &6.56	               &22.24         \\
+    CG      &  31.74    &  4373.90      & 26.29    & 9.57          & 16.72    \\
+    \hline
+    MG      &   5.71    &  1076.19      & 32.49    & 6.05          & 26.44    \\
+    \hline
+    EP      &  20.11    &  5638.49      & 26.85    & 0.56          & 26.29    \\
+    \hline
+    LU      & 144.13    & 42529.06      & 28.80    & 6.56          & 22.24    \\
+    \hline
+    BT      &  97.29    & 22813.86      & 34.95    & 5.80          & 29.15    \\
+    \hline
+    SP      &  66.49    & 20821.67      & 22.49    & 3.82          & 18.67    \\
+    \hline
+    FT      &  37.01    &  5505.60      & 31.59    & 6.48          & 25.11    \\
     \hline
-    BT         &97.29	       &22813.86	    &34.95   	  &5.80	               &29.15         \\
-   \hline
-    SP         &66.49	       &20821.67 	    &22.49	      &3.82	               &18.67         \\
-   \hline
-    FT     	   &37.01          &5505.60	        &31.59	      &6.48	               &25.11         \\
-\hline 
   \end{tabular}
   \label{table:res_16n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+  \medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 32 and 36 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    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           \\
-    \hline 
-    MG         &4.30	       &1355.58	        &28.93	      &8.85	               &20.08          \\
-   \hline
-    EP         &9.96           &5519.68	        &26.98	      &0.02	               &26.96          \\
-   \hline
-    LU         &99.93	       &67463.43	    &23.60	      &2.45	               &21.15          \\
+    CG      & 32.35     &  6704.21      & 16.15    & 5.30          & 10.85    \\
+    \hline
+    MG      &  4.30     &  1355.58      & 28.93    & 8.85          & 20.08    \\
+    \hline
+    EP      &  9.96     &  5519.68      & 26.98    & 0.02          & 26.96    \\
+    \hline
+    LU      & 99.93     & 67463.43      & 23.60    & 2.45          & 21.15    \\
+    \hline
+    BT      & 48.61     & 23796.97      & 34.62    & 5.83          & 28.79    \\
+    \hline
+    SP      & 46.01     & 27007.43      & 22.72    & 3.45          & 19.27    \\
+    \hline
+    FT      & 28.06     &  7142.69      & 23.09    & 2.90          & 20.19    \\
     \hline
-    BT         &48.61	       &23796.97	    &34.62	      &5.83	               &28.79          \\
-   \hline
-    SP         &46.01	       &27007.43	    &22.72	      &3.45	               &19.27           \\
-   \hline
-    FT     	   &28.06     	   &7142.69	        &23.09	      &2.90	               &20.19           \\
-\hline 
   \end{tabular}
   \label{table:res_32n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+  \medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 64 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    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           \\
-    \hline 
-    MG         &3.27	       &1534.70	        &29.27	      &14.35	           &14.92          \\
-   \hline
-    EP         &5.05           &5471.1084	    &27.12	      &3.11    	           &24.01         \\
-   \hline
-    LU         &73.92	       &101339.16	    &21.96	      &3.67	               &18.29         \\
+    CG      & 46.65     &  17521.83     &  8.13    &  1.68         &  6.45    \\
+    \hline
+    MG      &  3.27     &   1534.70     & 29.27    & 14.35         & 14.92    \\
+    \hline
+    EP      &  5.05     &   5471.11     & 27.12    &  3.11         & 24.01    \\
+    \hline
+    LU      & 73.92     & 101339.16     & 21.96    &  3.67         & 18.29    \\
+    \hline
+    BT      & 39.99     &  27166.71     & 32.02    & 12.28         & 19.74    \\
+    \hline
+    SP      & 52.00     &  49099.28     & 24.84    &  0.03         & 24.81    \\
+    \hline
+    FT      & 25.97     &  10416.82     & 20.15    &  4.87         & 15.28    \\
     \hline
-    BT         &39.99 	       &27166.71	    &32.02	      &12.28	           &19.74         \\
-   \hline
-    SP         &52.00	       &49099.28	    &24.84	      &0.03	               &24.81         \\
-   \hline
-    FT         &25.97	       &10416.82        &20.15	      &4.87	               &15.28         \\
-\hline 
   \end{tabular}
   \label{table:res_64n}
-\end{table}
-
+% \end{table}
 
-\begin{table}[htb]
+  \medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 128 and 144 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
+    \hspace{-2.2084pt}%
+    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          \\
-    \hline 
-    MG         &3.55           &2843.33         &18.77	      &10.38	           &8.39          \\
-   \hline
-    EP         &2.67           &5669.66	        &27.09	      &0.03	               &27.06         \\
-   \hline
-    LU         &51.23	       &144471.90   	&16.67	      &2.36	               &14.31         \\
+    CG      & 56.92     &  41163.36     &  4.00    &  1.10         &  2.90    \\
+    \hline
+    MG      &  3.55     &   2843.33     & 18.77    & 10.38         &  8.39    \\
+    \hline
+    EP      &  2.67     &   5669.66     & 27.09    &  0.03         & 27.06    \\
+    \hline
+    LU      & 51.23     & 144471.90     & 16.67    &  2.36         & 14.31    \\
+    \hline
+    BT      & 37.96     &  44243.82     & 23.18    &  1.28         & 21.90    \\
+    \hline
+    SP      & 64.53     & 115409.71     & 26.72    &  0.05         & 26.67    \\
+    \hline
+    FT      & 25.51     &  18808.72     & 12.85    &  2.84         & 10.01    \\
     \hline
-    BT         &37.96          &44243.82	    &23.18	      &1.28	               &21.90         \\
-   \hline
-    SP         &64.53	       &115409.71 	    &26.72	      &0.05	               &26.67         \\
-   \hline
-    FT         &25.51	       &18808.72	    &12.85	      &2.84	               &10.01         \\
-\hline 
   \end{tabular}
   \label{table:res_128n}
 \end{table}
 
-The results of applying the proposed scaling algorithm to NAS benchmarks is demonstrated 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}). These tables show the results for running the NAS benchmarks on different number of nodes. In general the energy saving percent is decreased  when the number of the computing nodes is increased. While the distance is decreased by the same direction. This because when we are run the MPI programs on a big number of nodes the communications is biggest than the computations, so the static energy is increased linearly to these times. The tables also show that performance degradation percent still approximately the same or decreased a little when the number of computing nodes is increased. This  gives good a prove that the proposed algorithm keeping as mush as possible the performance degradation. 
-
-\begin{figure}
+\begin{figure}[!t]
   \centering
-  \subfloat[Balanced nodes type scenario]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_eq.eps}\label{fig:avg_eq}}%
-  \quad%
-  \subfloat[Imbalanced nodes type scenario]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_neq.eps}\label{fig:avg_neq}}
+  \subfloat[Energy saving (\%)]{%
+    \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
+
+  \subfloat[Performance degradation (\%)]{%
+    \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}
-  \caption{The average of 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}
 
-In the NAS benchmarks there are some programs executed on different number of nodes. The benchmarks CG, MG, LU and FT executed on 2 to a power of (1, 2, 4, 8, ...) of nodes. The other benchmarks such as BT and SP executed on 2 to a power of (1, 2, 4, 9, ...) of nodes. We are take the average of energy saving, performance degradation and distances for all results of NAS benchmarks. The average of these three objectives are plotted to the number of nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}).  In CG, MG, LU, and FT benchmarks the average of energy saving is decreased when the number of nodes is increased due to the increasing in the communication times as mentioned before. Thus, the average of distances (our objective function) is decreased linearly with energy saving while keeping the average of performance degradation the same. In BT and SP benchmarks, the average of energy saving is not decreased significantly compare to other benchmarks when the number of nodes is increased. Nevertheless, the average of performance  degradation approximately still the same ratio. This difference is depends on the characteristics of the benchmarks such as the computation to communication ratio that has. 
-
-\subsection{The results for different powers scenarios}
-The results of the previous section are obtained using a percentage of 80\% for dynamic power and 20\% for static power of total power consumption. In this section we are change these ratio by using two others scenarios. Because is  interested to measure the ability  of the proposed algorithm to changes it   behaviour when these power ratios are changed. In fact, we are use two different scenarios for dynamic and static power ratios in addition to the previous scenario in section (\ref{sec.res}). Therefore, we have three different scenarios for three different dynamic and static power ratios refer to as: 70\%-20\%, 80\%-20\% and 90\%-10\% scenario. The results of these scenarios running NAS benchmarks class C on 8 or 9 nodes are place in the tables (\ref{table:res_s1} and \ref{table:res_s2}).
-
- \begin{table}[htb]
-  \caption{The results of 70\%-30\% powers scenario}
+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_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 numbers of nodes.  The experiments show that the algorithm
+significantly reduces the energy consumption (up to \np[\%]{34}) 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 higher 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 \np[GB]{64}) 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.
+
+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 CG benchmark significantly decreases 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}
+\label{sec.compare}
+
+The results of the previous section were obtained while using processors that
+consume during computation an overall power which is \np[\%]{80} composed of
+dynamic power and of \np[\%]{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 \np[\%]{70} of dynamic power and \np[\%]{30} of static power
+\item \np[\%]{90} of dynamic power and \np[\%]{10} of static power
+\end{itemize}
+
+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
+\np[\%]{70}-\np[\%]{30} scenario is smaller for all benchmarks compared to the
+energy saving of the \np[\%]{90}-\np[\%]{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 \np[\%]{70}-\np[\%]{30} scenario. On the other hand,
+the performance degradation percentage is smaller in the \np[\%]{70}-\np[\%]{30}
+scenario compared to the \np[\%]{90}-\np[\%]{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 increases.  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
+\np[\%]{90}-\np[\%]{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 \np[\%]{70}-\np[\%]{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.  \np[\%]{70}-\np[\%]{30} scenario and \np[\%]{80}-\np[\%]{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}[!t]
+  \caption{The results of the \np[\%]{70}-\np[\%]{30} power scenario}
   % title of Table
   \centering
-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}
+    \hline
+    Program & Energy        & Energy   & Performance   & Distance \\
+    name    & consumption/J & saving\% & degradation\% &          \\
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
+    CG      &  4144.21      & 22.42    & 7.72          & 14.70    \\
     \hline
-    CG         &4144.21          &22.42        &7.72                &14.70         \\
-    \hline 
-    MG         &1133.23          &24.50        &5.34                &19.16          \\
+    MG      &  1133.23      & 24.50    & 5.34          & 19.16    \\
    \hline
-    EP         &6170.30	        &16.19	       &0.02	            &16.17          \\
+    EP      &  6170.30      & 16.19    & 0.02          & 16.17    \\
    \hline
-    LU         &39477.28        &20.43	       &0.07	            &20.36          \\
+    LU      & 39477.28      & 20.43    & 0.07          & 20.36    \\
     \hline
-    BT         &26169.55	    &25.34	       &6.62	            &18.71          \\
+    BT      & 26169.55      & 25.34    & 6.62          & 18.71    \\
    \hline
-    SP         &19620.09	    &19.32	       &3.66	            &15.66          \\
+    SP      & 19620.09      & 19.32    & 3.66          & 15.66    \\
    \hline
-    FT         &6094.07	        &23.17	       &0.36	            &22.81          \\
-\hline 
+    FT      &  6094.07      & 23.17    & 0.36          & 22.81    \\
+    \hline
   \end{tabular}
   \label{table:res_s1}
 \end{table}
 
-
-
-\begin{table}[htb]
-  \caption{The results of 90\%-10\% powers scenario}
+\begin{table}[!t]
+  \caption{The results of the \np[\%]{90}-\np[\%]{10} power scenario}
   % title of Table
   \centering
-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
+    Program & Energy          & Energy      & Performance   & Distance \\
+    name    & consumption/J   & saving\%    & degradation\% &          \\
     \hline
-    CG         &2812.38	         &36.36	       &6.80                &29.56         \\
-    \hline 
-    MG         &825.427	         &38.35	       &6.41	            &31.94         \\
-   \hline
-    EP         &5281.62	         &35.02	       &2.68	            &32.34         \\
-   \hline
-    LU         &31611.28	     &39.15        &3.51	            &35.64        \\
+    CG      &  2812.38        & 36.36       &  6.80         & 29.56    \\
+    \hline
+    MG      &   825.43        & 38.35       &  6.41         & 31.94    \\
+    \hline
+    EP      &  5281.62        & 35.02       &  2.68         & 32.34    \\
+    \hline
+    LU      & 31611.28        & 39.15       &  3.51         & 35.64    \\
+    \hline
+    BT      & 21296.46        & 36.70       &  6.60         & 30.10    \\
+    \hline
+    SP      & 15183.42        & 35.19       & 11.76         & 23.43    \\
+    \hline
+    FT      &  3856.54        & 40.80       &  5.67         & 35.13    \\
     \hline
-    BT         &21296.46	     &36.70	       &6.60	            &30.10       \\
-   \hline
-    SP         &15183.42	     &35.19	       &11.76	            &23.43        \\
-   \hline
-    FT         &3856.54	         &40.80	       &5.67	            &35.13        \\
-\hline 
   \end{tabular}
   \label{table:res_s2}
 \end{table}
 
+\begin{table}[!t]
+  \caption{Comparing the proposed algorithm}
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    Program & \multicolumn{2}{c|}{Energy saving \%}
+            & \multicolumn{2}{c|}{Perf.  degradation \%}
+            & \multicolumn{2}{c|}{Distance} \\
+    \cline{2-7}
+    name    & EDP   & MaxDist & EDP  & MaxDist & EDP   & MaxDist \\
+    \hline
+    CG      & 27.58 & 31.25   & 5.82 & 7.12    & 21.76 & 24.13   \\
+    \hline
+    MG      & 29.49 & 33.78   & 3.74 & 6.41    & 25.75 & 27.37   \\
+    \hline
+    LU      & 19.55 & 28.33   & 0.00 & 0.01    & 19.55 & 28.22   \\
+    \hline
+    EP      & 28.40 & 27.04   & 4.29 & 0.49    & 24.11 & 26.55   \\
+    \hline
+    BT      & 27.68 & 32.32   & 6.45 & 7.87    & 21.23 & 24.43   \\
+    \hline
+    SP      & 20.52 & 24.73   & 5.21 & 2.78    & 15.31 & 21.95   \\
+    \hline
+    FT      & 27.03 & 31.02   & 2.75 & 2.54    & 24.28 & 28.48   \\
+    \hline
+  \end{tabular}
+  \label{table:compare_EDP}
+\end{table}
 
-\begin{figure}
+\begin{figure}[!t]
   \centering
-  \subfloat[Comparison the average of the results on 8 nodes]{%
-    \includegraphics[width=.22\textwidth]{fig/sen_comp.pdf}\label{fig:sen_comp}}%
-  \quad%
-  \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
-    \includegraphics[width=.24\textwidth]{fig/three_scenarios.pdf}\label{fig:scales_comp}}
-  \label{fig:avg}
+  \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=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
+  \label{fig:comp}
   \caption{The comparison of the three power scenarios}
 \end{figure}
 
-To compare the results of these three powers scenarios, we are take the average of the performance degradation, the energy saving and the distances for all NAS benchmarks running on 8 or 9 nodes of class C, as in figure (\ref{fig:sen_comp}). Thus, according to the average of the results, the energy saving ratio is increased when using the a higher percentage for dynamic power (e.g. 90\%-10\% scenario). While the average of energy saving is decreased in 70\%-30\% scenario. 
-Because the static energy consumption is increase. Moreover, the average of distances is more related to energy saving changes. 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). The raison behind these relations, that the proposed algorithm optimize both energy consumption and performance in the same time. Therefore, when using a higher ratio for dynamic power the algorithm selecting bigger frequency scaling factors values, more energy saving versus more performance degradation, for example see the figure (\ref{fig:scales_comp}). The inverse happen when using a higher ratio for  static  power, the algorithm proportionally  selects a smaller scaling values, less energy saving versus less performance degradation. This is because the  
-algorithm also keeps as much as possible the static energy consumption that is always related to execution time. 
+\begin{figure}[!t]
+  \centering
+   \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
+  \caption{Trade-off comparison for NAS benchmarks class C}
+  \label{fig:compare_EDP}
+\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 and the energy model. The energy model is significantly depends on the execution time model EQ(\ref{eq:perf}), that the energy static related linearly. So, we are compare the predicted execution time with the real execution time (Simgrid time)  values that gathered  offline for the NAS benchmarks class B executed on 8 or 9 nodes. The execution time model can predicts 
-the real execution time by maximum normalized error 0.03 of all NAS benchmarks. The second verification that we are make is for the scaling algorithm to prove its ability to selects the best set of frequency scaling factors. Therefore, we are expand the algorithm to test at each iteration the frequency scaling factor of the slowest node with the all scaling factors available of the other nodes. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the two algorithms is identical. While the proposed algorithm is runs faster, 10 times faster on average than the expanded algorithm.
+\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, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$ 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 consumption for both versions of the NAS
+benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
+nodes. The results show that our algorithm provides better energy savings than
+Spiliopoulos et al. algorithm, on average it results in \np[\%]{29.76} energy
+saving while their algorithm returns just \np[\%]{25.75}. The average of
+performance degradation percentage is approximately the same for both
+algorithms, about \np[\%]{4}.
+
+For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
+terms of  energy and  performance trade-off, see  Figure~\ref{fig:compare_EDP},
+because it maximizes the distance  between the energy saving and the performance
+degradation values while giving the same weight for both metrics.
 
 \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 trade-off) 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 \np[\%]{34} 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 trade-off.
+
+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
+``ANR-11-LABX-01-01'').  Computations  have been performed  on the supercomputer
+facilities  of the  MÃ©socentre de  calcul de  Franche-ComtÃ©. As  a  PhD student,
+Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
+supporting his work.
 
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-%  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:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
+% LocalWords:  CMOS EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex GPU
+% LocalWords:  de badri muslim MPI SimGrid GFlops Xeon EP BT GPUs CPUs AMD
+%  LocalWords:  Spiliopoulos scalability
