X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/88ca69b0273314e1fb1b061bd1bf75accff40205..d509f4d19eb0a9dfb0cb8fcc367e63e0e172fd42:/Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index b7e61b8..af3d660 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -5,9 +5,9 @@
 \usepackage[english]{babel}
 \usepackage{algpseudocode}
 \usepackage{graphicx}
+\usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
-
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
@@ -23,48 +23,53 @@
 \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}
+\newcommand{\Told}{\Xsub{T}{Old}} 
 
-\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
+\begin{document} 
 
-\author{%
+\title{Energy Consumption Reduction for Message Passing Iterative  Applications in Heterogeneous Architecture Using DVFS}
+ 
+\author{% 
   \IEEEauthorblockN{%
     Jean-Claude Charr,
     RaphaÃ«l Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch
-  }
+  } 
   \IEEEauthorblockA{%
-    FEMTO-ST Institute\\
-    University of Franche-ComtÃ©\\
+    FEMTO-ST Institute, University of Franche-Comte\\
     IUT de Belfort-MontbÃ©liard,
     19 avenue du MarÃ©chal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % RaphaÃ«l
@@ -76,60 +81,215 @@
 \maketitle
 
 \begin{abstract}
-  
+Computing platforms  are consuming  more and more  energy due to  the increasing
+number  of nodes  composing  them.  To  minimize  the operating  costs of  these
+platforms many techniques have been  used. Dynamic voltage and frequency scaling
+(DVFS) is  one of them. It  reduces the frequency of  a CPU to  lower its energy
+consumption.  However,  lowering the  frequency  of  a  CPU might  increase  the
+execution  time of  an application  running on  that processor.   Therefore, the
+frequency that  gives the best 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 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 35\% while  limiting the performance degradation as much as
+possible.   Finally,  the algorithm  is  compared  to  an existing  method,  the
+comparison results showing that it outperforms the latter.
+
 \end{abstract}
 
 \section{Introduction}
 \label{sec.intro}
-
+The  need for  more  computing  power is  continually  increasing. To  partially
+satisfy  this need,  most supercomputers  constructors just  put  more computing
+nodes in their  platform. The resulting platforms might  achieve higher floating
+point operations  per second  (FLOPS), but the  energy consumption and  the heat
+dissipation  are  also increased.   As  an  example,  the Chinese  supercomputer
+Tianhe-2 had  the highest FLOPS  in November 2014  according to the  Top500 list
+\cite{TOP500_Supercomputers_Sites}.  However, it was  also the most power hungry
+platform  with  its  over  3  million cores  consuming  around  17.8  megawatts.
+Moreover,    according   to    the    U.S.    annual    energy   outlook    2014
+\cite{U.S_Annual.Energy.Outlook.2014}, the  price of energy  for 1 megawatt-hour
+was approximately equal to \$70.  Therefore, the price of the energy consumed by
+the Tianhe-2  platform is approximately more  than \$10 million  each year.  The
+computing platforms must  be more energy efficient and  offer the highest number
+of FLOPS  per watt  possible, such as  the L-CSC  from the GSI  Helmholtz Center
+which became the top of the Green500 list in November 2014 \cite{Green500_List}.
+This heterogeneous platform executes more than 5 GFLOPS per watt while consuming
+57.15 kilowatts.
+
+Besides platform  improvements, there are many software  and hardware techniques
+to lower  the energy consumption of  these platforms, such  as scheduling, DVFS,
+\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 might increase the execution
+time of the application running over that processor.  Therefore, researchers use
+different optimization  strategies to select  the frequency that gives  the best
+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 might
+degrade the performance of the application running on that processor, especially
+if it is compute bound. Therefore selecting the appropriate frequency for a
+processor to satisfy some objectives while taking into account all the
+constraints, is not a trivial operation.  Many researchers used different
+strategies to tackle this problem. Some of them developed online methods that
+compute the new frequency while executing the application, such
+as~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}.
+Others used offline methods that might need to run the application and profile
+it before selecting the new frequency, such
+as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
+The methods could be heuristics, exact or brute force methods that satisfy
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+sequential, parallel or distributed architecture, homogeneous or heterogeneous
+platform, synchronous or asynchronous application, \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
+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 execution time of message passing distributed iterative applications on a heterogeneous platform}
+
+\subsection{The execution time of message passing distributed 
+                iterative applications on a heterogeneous platform}
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over
-heterogeneous platforms. We define a heterogeneous platform as a collection of
+heterogeneous platforms. A heterogeneous platform is defined as a collection of
 heterogeneous computing nodes interconnected via a high speed homogeneous
 network. Therefore, each node has different characteristics such as computing
 power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
 have the same network bandwidth and latency.
 
-
-\begin{figure}[t]
+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 might occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
+overall execution time of the program is the execution time of the slowest task
+which has the highest computation time and no slack time.
+  
+ \begin{figure}[!t]
   \centering
-    \includegraphics[scale=0.6]{fig/commtasks}
+   \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 
- 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 might occur when fast nodes have to
- wait, during synchronous communications, for  the slower nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
- Therefore,  the overall execution time  of the program is the execution time of the slowest
- task which have the highest computation time and no slack time.
- 
-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 might increase, especially if the program is compute bound.  The frequency reduction process can be  expressed by the scaling factor S which is the ratio between  the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
+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 might increase, especially if the program is 
+compute bound.  The frequency reduction process can be  expressed by the scaling 
+factor S which is the ratio between  the maximum and the new frequency of a CPU 
+as in (\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 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{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.
-
-Since in a heterogeneous platform, each node has different characteristics,
+ 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
@@ -138,220 +298,321 @@ applications running over a heterogeneous platform, for different vectors of
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
-vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
+vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
- \textit  T_\textit{new} = 
- \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  TcmOld_{j}
+  \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) +  \MinTcm
 \end{equation}
-where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $TcmOld_j$ is the communication time of the slowest processor $j$.
-  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. 
-
-This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. The execution time prediction model is used in our method for optimizing both energy consumption and performance of iterative methods, which is presented in the following sections.
+Where:
+\begin{equation}
+\label{eq:perf2}
+ \MinTcm = \min_{i=1,2,\dots,N} (\Tcm[i])
+\end{equation}
+where  $\TcpOld[i]$ is  the computation  time of  processor $i$  during  the first
+iteration and $\MinTcm$  is the communication time of  the slowest processor from
+the  first iteration.   The model  computes  the maximum  computation time  with
+scaling factor  from each node  added to the  communication time of  the slowest
+node. It means only the communication  time without any slack time is taken into
+account.  Therefore, the execution time of the iterative application is equal to
+the  execution time of  one iteration  as in  (\ref{eq:perf}) multiplied  by the
+number of iterations of that application.
+
+This prediction model is developed from  the model to predict the execution time
+of     message    passing     distributed    applications     for    homogeneous
+architectures~\cite{Our_first_paper}.   The execution  time prediction  model is
+used in  the method  to optimize both the energy consumption and the performance of
+iterative methods, which is presented in the following sections.
 
 
 \subsection{Energy model for heterogeneous platform}
-
-Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
+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 $P_{d}$ 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}).
+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
+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{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
+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 EQ~(\ref{eq:s}).
+ratio between the maximum and the new frequency as in (\ref{eq:s}).
 The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. we can calculate the new frequency 
-$F_{new}$ from EQ(\ref{eq:s}) as follow:
+system's kernel to lower a core's frequency. The new frequency 
+$\Fnew$ from (\ref{eq:s}) can be calculated as follows:
 \begin{equation}
   \label{eq:fnew}
-   F_\textit{new} = S^{-1} \cdot F_\textit{max}
+   \Fnew = S^{-1} \cdot \Fmax
 \end{equation}
-Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives 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{dNew} = \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_{dOld} \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}
-where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the new frequency and the maximum frequency respectively.
-
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 
-reducing the frequency by a factor of $S$~\cite{3}. 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:
+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{dNew} = P_{dOld} \cdot S^{-3} \cdot (T_{cp} \cdot S)= S^{-2}\cdot P_{dOld} \cdot  T_{cp} 
+   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp 
 \end{equation}
-The static power is related to the power leakage of the CPU and is consumed during computation and even when idle. As in~\cite{3,46}, we assume that the static power of a processor is constant during idle and computation periods, and for all its available frequencies. 
-The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}), 
-the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related  
-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: 
+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 (T_{cp} \cdot S  + T_{cm})
+  \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
 \end{equation}
 
-In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $P_{di}$ and $P_{si}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $T_{cpi}$ might be different and different frequency  scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $T_{cmi}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the  frequency scaling factor and the dynamic power of each node as in EQ(\ref{eq:Edyn}),  the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing  distributed application  executed over a heterogeneous platform is the summation of all dynamic and static energies for each  processor.  It is computed as follows:
+In  the  considered  heterogeneous  platform,  each  processor  $i$  might  have
+different   dynamic  and  static   powers,  noted   as  $\Pd[i]$   and  $\Ps[i]$
+respectively.  Therefore,  even if  the  distributed  message passing  iterative
+application  is  load balanced,  the  computation time  of  each  CPU $i$  noted
+$\Tcp[i]$ might  be different and  different frequency scaling factors  might be
+computed in order to decrease  the overall energy consumption of the application
+and reduce slack  times.  The communication time of a processor  $i$ is noted as
+$\Tcm[i]$  and could  contain slack  times when  communicating  with slower
+nodes,  see Figure~\ref{fig:heter}.  Therefore,  all nodes  do  not have  equal
+communication  times. While  the dynamic  energy  is computed  according to  the
+frequency  scaling   factor  and   the  dynamic  power   of  each  node   as  in
+(\ref{eq:Edyn}), the static energy is computed  as the sum of the execution time
+of  one iteration multiplied  by the static  power of  each processor.   The overall
+energy consumption of a message  passing distributed application executed over a
+heterogeneous platform during one iteration  is the summation of all dynamic and
+static energies for each processor.  It is computed as follows:
 \begin{multline}
   \label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot P_{di} \cdot  T_{cpi})} + {} \\
- \sum_{i=1}^{N} (P_{si} \cdot (\max_{i=1,2,\dots,N} (T_{cpi} \cdot S_{i}) +
- \min_{i=1,2,\dots,N} {T_{cmi}))}
+ 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 the frequencies of the processors according to the vector of
+
+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{36}.
+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,
-$Sopt_1,Sopt_2,\dots,Sopt_N$, 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})}}
+  \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 $T_{New}$ 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
+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 are not in the same direction. 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.  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:
+reduction with minimum execution time reduction.  
+  
+This problem can be solved by making the optimization process for energy and 
+execution time following the same direction.  Therefore, the equation of the 
+normalized execution time is inverted which gives the normalized performance equation, as follows:
 \begin{multline}
   \label{eq:pnorm_inv}
-  P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
-          = \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}\label{fig:r1}}%
-  \qquad%
+    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+  
+  
   \subfloat[Heterogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
+    \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}
 
-Then, we can model our objective function as finding the maximum distance
-between the energy curve EQ~(\ref{eq:enorm}) and the  performance
-curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
-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:
+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}} )
+  \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 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}.
+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 heterogeneous scaling algorithm }
+\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 trade-off (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]
+\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.
+
+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 colored in blue in Figure~\ref{fig:st_freq}.  This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting frequency scaling factors higher
+than the higher bound will not improve the performance of the application and it
+will increase its overall energy consumption.  Therefore the algorithm that
+selects the frequency scaling factors starts the search method from these
+initial frequencies and takes a downward search direction toward lower
+frequencies. The algorithm iterates on all left frequencies, from the higher
+bound until all nodes reach their minimum frequencies, to compute their overall
+energy consumption and performance, and select the optimal frequency scaling
+factors vector. At each iteration the algorithm determines the slowest node
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+while it lowers the frequency of all other nodes by one gear.  The new overall
+energy consumption and execution time are computed according to the new scaling
+factors.  The optimal set of frequency scaling factors is the set that gives the
+highest distance according to the objective function (\ref{eq:max}).
+
+Figures~\ref{fig:r1} and \ref{fig:r2}  illustrate the normalized performance and
+consumed  energy for  an application  running on  a homogeneous  platform  and a
+heterogeneous platform respectively while increasing the scaling factors. It can
+be noticed  that in a  homogeneous platform the  search for the  optimal scaling
+factor should start  from the maximum frequency because  the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand,
+in the heterogeneous platform the  performance is maintained at the beginning of
+the plot  even if the  frequencies of the  faster nodes decrease  until the
+computing power of scaled down  nodes are lower than the slowest  node. In other
+words, until they reach the higher bound. It can also be noticed that the higher
+the difference between the faster nodes  and the slower nodes is, the bigger the
+maximum distance  between the  energy curve and  the performance curve  is while
+ the scaling factors are varying which results in bigger energy savings.
+\begin{figure}[!t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}
   \caption{Selecting the initial frequencies}
@@ -359,80 +620,55 @@ frequencies until the last available ones.
 \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 difference 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 $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
     \EndIf 
-    \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
-    \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
-    \State $Dist \gets 0$
-    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
+    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
+    % \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 $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
-        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,\dots,N.$
+        \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,\dots,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 trade-off.  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}}
@@ -441,7 +677,7 @@ and when the algorithm is 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.
@@ -450,49 +686,86 @@ and when the algorithm is called in the MPI program.
   \end{algorithmic}
   \caption{DVFS algorithm}
   \label{dvfs}
-\end{figure}
+\end{algorithm}
+
+\subsection{The evaluation of the proposed algorithm}
+\label{sec.verif.algo}
+The precision  of the  proposed algorithm mainly  depends on the  execution time
+prediction model  defined in  (\ref{eq:perf}) and the  energy model  computed by
+(\ref{eq:energy}).   The energy  model is  also significantly  dependent  on the
+execution  time model  because  the static  energy  is linearly  related to  the
+execution time  and the dynamic energy  is related to the  computation time. So,
+all the works presented  in this paper are based on the  execution time model. To
+verify  this  model, the  predicted  execution time  was  compared  to the  real
+execution          time           over          SimGrid/SMPI          simulator,
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile},   for   all   the   NAS
+parallel benchmarks NPB v3.3  \cite{NAS.Parallel.Benchmarks}, running class B on
+8 or  9 nodes. The comparison showed  that the proposed execution  time model is
+very precise, the maximum  normalized difference between the predicted execution
+time and the real execution time is equal to 0.03 for all the NAS benchmarks.
+
+Since the proposed algorithm is not an exact method it does not test all the
+possible solutions (vectors of scaling factors) in the search space. To prove
+its efficiency, it was compared on small instances to a brute force search
+algorithm that tests all the possible solutions. The brute force algorithm was
+applied to different NAS benchmarks classes with different number of nodes. The
+solutions returned by the brute force algorithm and the proposed algorithm were
+identical and the proposed algorithm was on average 10 times faster than the
+brute force algorithm. It has a small execution time: for a heterogeneous
+cluster composed of four different types of nodes having the characteristics
+presented in Table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4
+nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling
+factors vector.  The algorithm complexity is $O(F\cdot N)$, where $F$
+is the number of iterations and $N$ is the number of computing nodes. The
+algorithm needs from 12 to 20 iterations to select the best vector of frequency
+scaling factors that gives the results of the next sections.
 
 \section{Experimental results}
 \label{sec.expe}
-
-The experiments of this work are executed on the simulator SimGrid/SMPI
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. 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]
+To  evaluate the  efficiency and  the  overall energy  consumption reduction  of
+Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The
+experiments were executed on the  simulator SimGrid/SMPI which offers easy tools
+to create a heterogeneous platform and run message passing applications over it.
+The heterogeneous  platform that was used  in the experiments, had  one core per
+node because just one process was executed per node.  The heterogeneous platform
+was  composed  of  four  types  of  nodes. Each  type  of  nodes  had  different
+characteristics  such as  the maximum  CPU  frequency, the  number of  available
+frequencies  and the  computational power,  see Table~\ref{table:platform}. The
+characteristics  of  these  different  types  of nodes  are  inspired  from  the
+specifications of real  Intel processors.  The heterogeneous platform  had up to
+144 nodes and had nodes from the four types in equal proportions, for example if
+a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the
+constructors of  CPUs do not specify the  dynamic and the static  power of their
+CPUs, for  each type of  node they were  chosen proportionally to  its computing
+power  (FLOPS).  In  the initial  heterogeneous platform,  while  computing with
+highest frequency,  each node  consumed an amount  of power proportional  to its
+computing  power  (which  corresponds to  80\%  of  its  dynamic power  and  the
+remaining  20\%  to  the  static   power),  the  same  assumption  was  made  in
+\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.    Finally,  These
+nodes were connected via an Ethernet network with 1 Gbit/s bandwidth.
+
+
+\begin{table}[!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}
@@ -505,19 +778,26 @@ connected via an Ethernet network with 1 Gbit/s 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}.   
+
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
+MG, FT, BT, LU and SP) and  the benchmarks were executed with the three classes:
+A, B and C. However, due to the lack of space in this paper, only the results of
+the  biggest class,  C, are  presented while  being run  on different  number of
+nodes,  ranging from 4  to 128  or 144  nodes depending  on the  benchmark being
+executed. Indeed, the benchmarks CG, MG, LU, EP and FT had to be executed on $1,
+2, 4, 8, 16, 32, 64, 128$ nodes.   The other benchmarks such as BT and SP had to
+be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
+
+ 
  
-\begin{table}[htb]
+\begin{table}[!t]
   \caption{Running NAS benchmarks on 4 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
+    \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       \\
@@ -536,15 +816,17 @@ nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Dep
 \hline 
   \end{tabular}
   \label{table:res_4n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+\medskip
+% \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      \\
+     \hspace{-2.2084pt}%
+    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
     CG         &36.11    	   &3263.49	        &31.25	      &7.12	               &24.13     \\
@@ -563,15 +845,17 @@ nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Dep
 \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      \\
+    \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          \\
@@ -590,15 +874,17 @@ nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Dep
 \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      \\
+    \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           \\
@@ -617,15 +903,17 @@ nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Dep
 \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      \\
+    \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           \\
@@ -644,16 +932,17 @@ nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Dep
 \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     \\
+    \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          \\
@@ -673,59 +962,129 @@ nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Dep
   \end{tabular}
   \label{table:res_128n}
 \end{table}
+The overall energy  consumption was computed for each  instance according to the
+energy  consumption  model  (\ref{eq:energy}),  with and  without  applying  the
+algorithm. The execution time was also measured for all these experiments. Then,
+the energy saving and performance degradation percentages were computed for each
+instance.    The   results   are   presented  in   Tables~\ref{table:res_4n},
+\ref{table:res_8n},           \ref{table:res_16n},          \ref{table:res_32n},
+\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
+values  from many experiments  for energy  savings and  performance degradation.
+The tables show the experimental results for running the NAS parallel benchmarks
+on  different  number  of  nodes.   The  experiments  show  that  the  algorithm
+significantly reduces the energy consumption (up to 35\%) and tries to limit the
+performance  degradation.  They  also  show that  the  energy saving  percentage
+decreases when the  number of computing nodes increases.   This reduction is due
+to the increase of the communication  times compared to the execution times when
+the benchmarks are run over a high number of nodes.  Indeed, the benchmarks with
+the  same  class,  C,  are  executed  on different  numbers  of  nodes,  so  the
+computation required  for each iteration is  divided by the  number of computing
+nodes.  On the other hand,  more communications are required when increasing the
+number  of  nodes so  the  static energy  increases  linearly  according to  the
+communication time and the dynamic power  is less relevant in the overall energy
+consumption.   Therefore, reducing the  frequency with  Algorithm~\ref{HSA} is
+less effective  in reducing the overall  energy savings. It can  also be noticed
+that for the benchmarks EP and  SP that contain little or no communications, the
+energy savings are  not significantly affected by the high  number of nodes.  No
+experiments were conducted  using bigger classes than D,  because they require a
+lot  of memory (more  than 64GB)  when being  executed by  the simulator  on one
+machine.   The maximum  distance between  the  normalized energy  curve and  the
+normalized performance for each instance is  also shown in the result tables. It
+decrease in the same way as  the energy saving percentage.  The tables also show
+that the performance degradation  percentage is not significantly increased when
+the number  of computing  nodes is increased  because the computation  times are
+small when compared to the communication times.
 
-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[CG, MG, LU and FT Benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
-  \quad%
-  \subfloat[BT and SP benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_neq}\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,
-\dots{}) of nodes. The other benchmarks such as BT and SP executed on 2 to a
-power of (1, 2, 4, 9, \dots{}) 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
-behavior 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}
+Figures~\ref{fig:energy} and  \ref{fig:per_deg} present  the energy  saving and
+performance  degradation respectively for  all the  benchmarks according  to the
+number of used nodes. As shown  in the first plot, the energy saving percentages
+of the benchmarks MG,  LU, BT and FT decrease linearly when  the number of nodes
+increase. While  for the EP and  SP benchmarks, the energy  saving percentage is
+not affected by the increase of  the number of computing nodes, because in these
+benchmarks there are little or  no communications. Finally, the energy saving of
+the  GC benchmark  significantly  decrease  when the  number  of nodes  increase
+because this benchmark has more  communications than the others. The second plot
+shows that  the performance  degradation percentages of  most of  the benchmarks
+decrease when  they run on a  big number of  nodes because they spend  more time
+communicating than computing,  thus, scaling down the frequencies  of some nodes
+has less effect on the performance.
+
+
+
+
+\subsection{The results for different power consumption scenarios}
+\label{sec.compare}
+The results  of the previous section  were obtained while  using processors that
+consume during  computation an overall power  which is 80\%  composed of dynamic
+power and of 20\% of static power. In this section, these ratios are changed and
+two new  power scenarios are  considered in order  to evaluate how  the proposed
+algorithm adapts itself  according to the static and  dynamic power values.  The
+two new power scenarios are the following:
+
+\begin{itemize}
+\item 70\% of dynamic power  and 30\% of static power
+\item 90\% of dynamic power  and 10\% of static power
+\end{itemize}
+
+The NAS parallel benchmarks were  executed again over processors that follow the
+new power scenarios.   The class C of each  benchmark was run over 8  or 9 nodes
+and   the    results   are   presented   in    Tables~\ref{table:res_s1}   and
+\ref{table:res_s2}. These tables  show that the energy saving  percentage of the
+70\%-30\% scenario is  smaller for all benchmarks compared  to the energy saving
+of the 90\%-10\% scenario. Indeed, in  the latter more dynamic power is consumed
+when  nodes are running  on their  maximum frequencies,  thus, scaling  down the
+frequency of  the nodes results in  higher energy savings than  in the 70\%-30\%
+scenario. On the  other hand, the performance degradation  percentage is smaller
+in the 70\%-30\% scenario compared to the 90\%-10\% scenario. This is due to the
+higher  static  power percentage  in  the first  scenario  which  makes it  more
+relevant in the  overall consumed energy.  Indeed, the  static energy is related
+to the execution time and if  the performance is degraded the amount of consumed
+static  energy directly  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 90\%-10\% scenario
+because at  maximum frequency  the dynamic  energy is the  most relevant  in the
+overall consumed  energy and can  be reduced by  lowering the frequency  of some
+processors. On  the other hand, the  energy saving decreases  when the 70\%-30\%
+scenario is  used because  the dynamic  energy is less  relevant in  the overall
+consumed energy and  lowering the frequency does not  return big energy savings.
+Moreover, the average  of the performance degradation is  decreased when using a
+higher  ratio   for  static  power  (e.g.   70\%-30\%   scenario  and  80\%-20\%
+scenario). Since  the proposed algorithm  optimizes the energy  consumption when
+using a  higher ratio for dynamic  power the algorithm  selects bigger frequency
+scaling  factors that result  in more  energy saving  but less  performance, for
+example see  Figure~\ref{fig:scales_comp}. The  opposite happens when  using a
+higher  ratio for  static power,  the algorithm  proportionally  selects smaller
+scaling  values which result  in less  energy saving  but also  less performance
+degradation.
+
+
+ \begin{table}[!t]
+  \caption{The results of the 70\%-30\% power scenario}
   % title of Table
   \centering
-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
+    Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
     \hline
     CG         &4144.21          &22.42        &7.72                &14.70         \\
@@ -748,13 +1107,13 @@ running NAS benchmarks class C on 8 or 9 nodes are place in the tables
 
 
 
-\begin{table}[htb]
-  \caption{The results of 90\%-10\% powers scenario}
+\begin{table}[!t]
+  \caption{The results of the 90\%-10\% power scenario}
   % title of Table
   \centering
-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}
     \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
+    Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
     \hline
     CG         &2812.38	         &36.36	       &6.80                &29.56         \\
@@ -775,33 +1134,113 @@ running NAS benchmarks class C on 8 or 9 nodes are place in the tables
   \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            & 0.0            & 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}\label{fig:sen_comp}}%
-  \quad%
-  \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
-    \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
+  \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}  
+
+\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}
 
-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. 
 
-\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 29.76\% energy saving
+while their algorithm returns just 25.75\%. The average of performance
+degradation percentage is approximately the same for both algorithms, about 4\%.
+
+
+For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
+terms of  energy and  performance 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}
 
+\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 35\% the energy consumption of a message
+passing iterative method while limiting  the degradation of the performance. The
+algorithm also selects different scaling  factors according to the percentage of
+the computing and communication times, and according to the values of the static
+and  dynamic  powers  of the  CPUs.   Finally,  the  algorithm was  compared  to
+Spiliopoulos et al.  algorithm and  the results showed that it outperforms their
+algorithm in terms of energy-time 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 âANR-11-LABX-01-01â). As a PhD student, 
+Mr. Ahmed Fanfakh, would like to thank the University of
+Babylon (Iraq) for supporting his work. 
+
 
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@@ -812,7 +1251,7 @@ the real execution time by maximum normalized error 0.03 of all NAS benchmarks.
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-
+   
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@@ -821,6 +1260,6 @@ the real execution time by maximum normalized error 0.03 of all NAS benchmarks.
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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:  de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
-% LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
+% 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
