Correct definition for F in algorithmic complexity.

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index a31c0fbec3daac05e494d832b2bd23499cbca224..a9a1e6da497ddbd13099d0c00c389ec2b6a52213 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -25,6 +25,9 @@
 
 \newcommand{\Xsub}[2]{{\ensuremath{#1_\mathit{#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{\CL}{\Xsub{C}{L}}
 \newcommand{\Dist}{\mathit{Dist}}
 \newcommand{\EdNew}{\Xsub{E}{dNew}}
 \newcommand{\CL}{\Xsub{C}{L}}
 \newcommand{\Dist}{\mathit{Dist}}
 \newcommand{\EdNew}{\Xsub{E}{dNew}}


@@ -33,40 +36,42 @@
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
 \newcommand{\Es}{\Xsub{E}{S}}
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
 \newcommand{\Es}{\Xsub{E}{S}}

-\newcommand{\Fdiff}{\Xsub{F}{diff}}
-\newcommand{\Fmax}{\Xsub{F}{max}}
+\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}{\mathit{Max}\Dist}
 \newcommand{\MinTcm}{\mathit{Min}\Tcm}
 \newcommand{\Ntrans}{\Xsub{N}{trans}}
 \newcommand{\Fnew}{\Xsub{F}{new}}
 \newcommand{\Ileak}{\Xsub{I}{leak}}
 \newcommand{\Kdesign}{\Xsub{K}{design}}
 \newcommand{\MaxDist}{\mathit{Max}\Dist}
 \newcommand{\MinTcm}{\mathit{Min}\Tcm}
 \newcommand{\Ntrans}{\Xsub{N}{trans}}

-\newcommand{\Pd}{\Xsub{P}{d}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}

 \newcommand{\PdNew}{\Xsub{P}{dNew}}
 \newcommand{\PdOld}{\Xsub{P}{dOld}}
 \newcommand{\Pnorm}{\Xsub{P}{Norm}}
 \newcommand{\PdNew}{\Xsub{P}{dNew}}
 \newcommand{\PdOld}{\Xsub{P}{dOld}}
 \newcommand{\Pnorm}{\Xsub{P}{Norm}}

-\newcommand{\Ps}{\Xsub{P}{s}}
-\newcommand{\Scp}{\Xsub{S}{cp}}
-\newcommand{\Sopt}{\Xsub{S}{opt}}
-\newcommand{\Tcm}{\Xsub{T}{cm}}
-\newcommand{\Tcp}{\Xsub{T}{cp}}
-\newcommand{\TcpOld}{\Xsub{T}{cpOld}}
+\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{\Tnew}{\Xsub{T}{New}}

-\newcommand{\Told}{\Xsub{T}{Old}} 
+\newcommand{\Told}{\Xsub{T}{Old}}
+
+\begin{document}

 
 

-\begin{document} 
+\title{Energy Consumption Reduction with DVFS for \\
+  Message Passing Iterative Applications on \\
+  Heterogeneous Architectures}

 
 

-\title{Energy Consumption Reduction for Message Passing Iterative  Applications in Heterogeneous Architecture Using DVFS}
- 
-\author{% 
+\author{%

   \IEEEauthorblockN{%
     Jean-Claude Charr,
     Raphaël Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch
   \IEEEauthorblockN{%
     Jean-Claude Charr,
     Raphaël Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch

-  } 
+  }

   \IEEEauthorblockA{%
   \IEEEauthorblockA{%

-    FEMTO-ST Institute, University of Franche-Comte\\
+    FEMTO-ST Institute, University of Franche-Comté\\

     IUT de Belfort-Montbéliard,
     19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël
     IUT de Belfort-Montbéliard,
     19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël


@@ -78,86 +83,92 @@
 \maketitle
 
 \begin{abstract}
 \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 frequencies selecting algorithm  for heterogeneous
-platforms is presented.   It selects the frequency which 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.
+  Computing platforms are consuming more and more energy due to the increasing
+  number of nodes composing them.  To minimize the operating costs of these
+  platforms many techniques have been used. Dynamic voltage and frequency
+  scaling (DVFS) is one of them. It reduces the frequency of a CPU to lower its
+  energy consumption.  However, lowering the frequency of a CPU may increase
+  the execution time of an application running on that processor.  Therefore,
+  the frequency that gives the best trade-off between the energy consumption and
+  the performance of an application must be selected.
+
+  In this paper, a new online frequency selecting algorithm for heterogeneous
+  platforms is presented.  It selects the frequencies and tries to give the best
+  trade-off between energy saving and performance degradation, for each node
+  computing the message passing iterative application. The algorithm has a small
+  overhead and works without training or profiling. It uses a new energy model
+  for message passing iterative applications running on a heterogeneous
+  platform. The proposed algorithm is evaluated on the SimGrid simulator while
+  running the NAS parallel benchmarks.  The experiments show that it reduces the
+  energy consumption by up to \np[\%]{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}
 
 \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
+
+The need for more computing power is continually increasing. To partially
+satisfy this need, most supercomputers constructors just put more computing
+nodes in their platform. The resulting platforms may achieve higher floating
+point operations per second (FLOPS), but the energy consumption and the heat
+dissipation are also increased.  As an example, the Chinese supercomputer
+Tianhe-2 had the highest FLOPS in November 2014 according to the Top500 list
+\cite{TOP500_Supercomputers_Sites}.  However, it was also the most power hungry
+platform with its over 3 million cores consuming around 17.8 megawatts.
+Moreover, according to the U.S.  annual energy outlook 2014
+\cite{U.S_Annual.Energy.Outlook.2014}, the price of energy for 1 megawatt-hour

 was approximately equal to \$70.  Therefore, the price of the energy consumed by
 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
+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.
 
 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
+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
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces

-the number of FLOPS executed by the processor which might increase the execution
+the number of FLOPS executed by the processor which may increase the execution

 time of the application running over that processor.  Therefore, researchers use
 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
+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
 application, that simultaneously tries to offer the maximum energy reduction and

-minimum performance degradation ratio. The  algorithm has a very small overhead,
+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
 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
+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.
 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~\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}
 
 \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
 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
+goal. Reducing the frequency of a processor lowers its number of FLOPS and may

 degrade the performance of the application running on that processor, especially
 if it is compute bound. Therefore selecting the appropriate frequency for a
 processor to satisfy some objectives while taking into account all the
 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


@@ -165,7 +176,7 @@ 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}.
 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
+Others used offline methods that may need to run the application and profile

 it before selecting the new frequency, such
 as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
 The methods could be heuristics, exact or brute force methods that satisfy
 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


@@ -174,13 +185,13 @@ 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{}
 
 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: 
+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}
 \begin{itemize}

-

 \item the platform is composed of homogeneous GPUs and homogeneous CPUs.
 \item the platform is only composed of heterogeneous CPUs.
 \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
 \end{itemize}
 
 For the first type of platform, the computing intensive parallel tasks are


@@ -196,47 +207,50 @@ 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.
 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
 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
+dynamically assigning new frequencies to the CPUs of the heterogeneous cluster.
+Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed an
+algorithm that divides the executed tasks into two types: the critical and non
+critical tasks. The algorithm scales down the frequency of non critical tasks
+proportionally to their slack and communication times while limiting the
+performance degradation percentage to less than \np[\%]{10}.
+In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a

 heterogeneous cluster composed of two types of Intel and AMD processors. They
 use a gradient method to predict the impact of DVFS operations on performance.
 In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
 \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
 frequencies for a specified heterogeneous cluster are selected offline using
 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
+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
 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
+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}
 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.
-      
+\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}
 
 \end{enumerate}
 
 \section{The performance and energy consumption measurements on heterogeneous architecture}
 \label{sec.exe}
 

-
-
-\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
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over


@@ -246,44 +260,44 @@ 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.
 
 power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
 have the same network bandwidth and latency.
 

+\begin{figure}[!t]
+  \centering
+  \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
 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
+heterogeneous computation power of the computing nodes, slack times may occur

 when fast nodes have to wait, during synchronous communications, for the slower
 nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
 overall execution time of the program is the execution time of the slowest task
 which has the highest computation time and no slack time.
 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}
-  \caption{Parallel tasks on a heterogeneous platform}
-  \label{fig:heter}
-\end{figure}

 
 

-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 
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
+modern processors, that reduces the energy consumption of a CPU by scaling
+down its voltage and frequency.  Since DVFS lowers the frequency of a CPU
+and consequently its computing power, the execution time of a program running
+over that scaled down processor may increase, especially if the program is
+compute bound.  The frequency reduction process can be  expressed by the scaling
+factor S which is the ratio between  the maximum and the new frequency of a CPU

 as in (\ref{eq:s}).
 \begin{equation}
   \label{eq:s}
 as in (\ref{eq:s}).
 \begin{equation}
   \label{eq:s}

- S = \frac{\Fmax}{\Fnew}
+  S = \frac{\Fmax}{\Fnew}

 \end{equation}
 \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{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.
+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
 
 Since in a heterogeneous platform each node has different characteristics,
 especially different frequency gears, when applying DVFS operations on these


@@ -297,38 +311,38 @@ operation. Then the execution time for one iteration of the application with any
 vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
 vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}

-  \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld_{i}} \cdot S_{i}) +  \MinTcm
+  \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) +  \MinTcm

 \end{equation}
 Where:
 \begin{equation}
 \end{equation}
 Where:
 \begin{equation}

-\label{eq:perf2}
- \MinTcm = \min_{i=1,2,\dots,N} (\Tcm_i)
+  \label{eq:perf2}
+  \MinTcm = \min_{i=1,2,\dots,N} (\Tcm[i])

 \end{equation}
 \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
+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
 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
+the execution time of one iteration as in (\ref{eq:perf}) multiplied by the

 number of iterations of that application.
 
 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.
-
+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}
 
 \subsection{Energy model for heterogeneous platform}

+

 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,

-Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
-Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by a processor into
-two power metrics: the static and the dynamic power.  While the first one is
-consumed as long as the computing unit is turned on, the latter is only consumed during
-computation times.  The dynamic power $\Pd$ is related to the switching
-activity $\alpha$, load capacitance $\CL$, the supply voltage $V$ and
-operational frequency $F$, as shown in (\ref{eq:pd}).
+  Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
+  Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
+a processor into two power metrics: the static and the dynamic power.  While the
+first one is consumed as long as the computing unit is turned on, the latter is
+only consumed during computation times.  The dynamic power $\Pd$ is related to
+the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
+and operational frequency $F$, as shown in (\ref{eq:pd}).

 \begin{equation}
   \label{eq:pd}
   \Pd = \alpha \cdot \CL \cdot V^2 \cdot F
 \begin{equation}
   \label{eq:pd}
   \Pd = \alpha \cdot \CL \cdot V^2 \cdot F


@@ -350,162 +364,164 @@ 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.
 
 time and $\Tcp \le T$.  $\Tcp$ may be equal to $T$ if there is no
 communication and no slack time.
 

-The main objective of DVFS operation is to reduce the overall energy consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  
-The operational frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot F$ with some
-constant $\beta$.~This equation is used to study the change of the dynamic
-voltage with respect to various frequency values in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction
-process of the frequency can be expressed by the scaling factor $S$ which is the
-ratio between the maximum and the new frequency as in (\ref{eq:s}).
-The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. The new frequency 
-$\Fnew$ from (\ref{eq:s}) can be calculated as follows:
+The main objective of DVFS operation is to reduce the overall energy
+consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  The operational
+frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot
+F$ with some constant $\beta$.~This equation is used to study the change of the
+dynamic voltage with respect to various frequency values
+in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction process of the
+frequency can be expressed by the scaling factor $S$ which is the ratio between
+the maximum and the new frequency as in (\ref{eq:s}).  The CPU governors are
+power schemes supplied by the operating system's kernel to lower a core's
+frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
+follows:

 \begin{equation}
   \label{eq:fnew}
    \Fnew = S^{-1} \cdot \Fmax
 \end{equation}
 \begin{equation}
   \label{eq:fnew}
    \Fnew = S^{-1} \cdot \Fmax
 \end{equation}

-Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following 
+Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following

 equation for dynamic power consumption:
 \begin{multline}
   \label{eq:pdnew}
    \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}
 equation for dynamic power consumption:
 \begin{multline}
   \label{eq:pdnew}
    \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 $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the 
+where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the

 new frequency and the maximum frequency respectively.
 
 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:
+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}
 \begin{equation}
   \label{eq:Edyn}

-   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp 
+   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp

 \end{equation}
 \end{equation}

-The static power is related to the power leakage of the CPU and is consumed during computation 
-and even when idle. As in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling}, 
- 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: 
+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}
   \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
 \end{equation}
 
 \begin{equation}
   \label{eq:Estatic}
   \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  $\Pd_{i}$   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
+In the considered heterogeneous platform, each processor $i$ may have
+different dynamic and static powers, noted as $\Pd[i]$ and $\Ps[i]$
+respectively.  Therefore, even if the distributed message passing iterative
+application is load balanced, the computation time of each CPU $i$ noted
+$\Tcp[i]$ may be different and different frequency scaling factors may be
+computed in order to decrease the overall energy consumption of the application
+and reduce slack times.  The communication time of a processor $i$ is noted as
+$\Tcm[i]$ and could contain slack times when communicating with slower nodes,
+see Figure~\ref{fig:heter}.  Therefore, all nodes do not have equal
+communication times. While the dynamic energy is computed according to the
+frequency scaling factor and the dynamic power of each node as in
+(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
+of one iteration multiplied by the static power of each processor.  The overall
+energy consumption of a message passing distributed application executed over a
+heterogeneous platform during one iteration is the summation of all dynamic and

 static energies for each processor.  It is computed as follows:
 \begin{multline}
   \label{eq:energy}
 static energies for each processor.  It is computed as follows:
 \begin{multline}
   \label{eq:energy}

- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd_{i} \cdot  \Tcp_i)} + {} \\
- \sum_{i=1}^{N} (\Ps_{i} \cdot (\max_{i=1,2,\dots,N} (\Tcp_i \cdot S_{i}) +
+ 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))}
   {\MinTcm))}

- \end{multline}
-
-Reducing the frequencies of the processors according to the vector of
-scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
-application and thus, increase the static energy because the execution time is
-increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption for the iterative 
-application can be measured by measuring  the energy consumption for one iteration as in (\ref{eq:energy}) 
-multiplied by the number of iterations of that application.
+\end{multline}

 
 

+Reducing the frequencies of the processors according to the vector of scaling
+factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the application
+and thus, increase the static energy because the execution time is
+increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
+for the iterative application can be measured by measuring the energy
+consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
+of iterations of that application.

 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 
 Using the lowest frequency for each processor does not necessarily give the most
 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 
 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
+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}
   \Pnorm = \frac{\Tnew}{\Told}\\
 maximum frequency for all nodes) as follows:
 \begin{multline}
   \label{eq:pnorm}
   \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)}}
+       {} = \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}
 
 \end{multline}
 

-
-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:
+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}
   \Enorm = \frac{\Ereduced}{\Eoriginal} \\
 \begin{multline}
   \label{eq:enorm}
   \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} 
+  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}
+\end{multline}

 Where $\Ereduced$ and $\Eoriginal$ are computed using (\ref{eq:energy}) and
 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.  
-  
-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:
+$\Tnew$ 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.
+
+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}
   \Pnorm = \frac{\Told}{\Tnew}\\
 \begin{multline}
   \label{eq:pnorm_inv}
   \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} 
+          = \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}
 
 \end{multline}
 

-

 \begin{figure}[!t]
   \centering
   \subfloat[Homogeneous platform]{%
     \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
 \begin{figure}[!t]
   \centering
   \subfloat[Homogeneous platform]{%
     \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%

-  
-  
+

   \subfloat[Heterogeneous platform]{%
     \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}
   \subfloat[Heterogeneous platform]{%
     \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}


@@ -520,21 +536,90 @@ 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}
 Figure~\ref{fig:r2}. Then the objective function has the following form:
 \begin{equation}
   \label{eq:max}

-  \MaxDist = 
+  \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}
   \mathop{\max_{i=1,\dots F}}_{j=1,\dots,N}
       (\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
        \overbrace{\Enorm(S_{ij})}^{\text{Minimize}} )
 \end{equation}

-where $N$ is the number of nodes and $F$ is the  number of available frequencies for each node. 
-Then, the optimal set of scaling factors that satisfies (\ref{eq:max}) can be selected.  
-The objective function can work with any energy model or any power values for each node 
-(static and dynamic powers). However, the most important energy reduction gain can be achieved when 
-the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
+where $N$ is the number of nodes and $F$ is the number of available frequencies
+for each node.  Then, the optimal set of scaling factors that satisfies
+(\ref{eq:max}) can be selected.  The objective function can work with any energy
+model or any power values for each node (static and dynamic powers). However,
+the most important energy reduction gain can be achieved when the energy curve
+has a convex form as shown
+in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.

 
 \section{The scaling factors selection algorithm for heterogeneous platforms }
 \label{sec.optim}
 
 
 \section{The scaling factors selection algorithm for heterogeneous platforms }
 \label{sec.optim}
 

+\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.
+    \end{description}
+    \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 Round the computed initial frequencies $F_i$ to the closest one available in each node.
+    \If{(not the first frequency)}
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
+    \EndIf
+    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
+    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}$
+    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \Told)}$
+    \State $\Sopt[i] \gets 1,~i=1,\dots,N. $
+    \State $\Dist \gets 0 $
+    \While {(all nodes not reach their  minimum  frequency)}
+        \If{(not the last freq. \textbf{and} not the slowest node)}
+        \State $F_i \gets F_i - \Fdiff[i],~i=1,\dots,N.$
+        \State $S_i \gets \frac{\Fmax[i]}{F_i},~i=1,\dots,N.$
+        \EndIf
+       \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 $\Dist \gets \Pnorm - \Enorm$
+      \EndIf
+    \EndWhile
+    \State  Return $\Sopt[1],\Sopt[2],\dots,\Sopt[N]$
+  \end{algorithmic}
+  \caption{frequency scaling factors selection algorithm}
+  \label{HSA}
+\end{algorithm}
+
+\begin{algorithm}
+  \begin{algorithmic}[1]
+    % \footnotesize
+    \For {$k=1$ to \textit{some iterations}}
+      \State Computations section.
+      \State Communications section.
+      \If {$(k=1)$}
+        \State Gather all times of computation and\newline\hspace*{3em}%
+               communication from each node.
+        \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.
+      \EndIf
+    \EndFor
+  \end{algorithmic}
+  \caption{DVFS algorithm}
+  \label{dvfs}
+\end{algorithm}
+

 \subsection{The algorithm details}
 \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
 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


@@ -549,6 +634,13 @@ 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.
 
 execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
 scaling algorithm is called in the iterative MPI program.
 

+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.5]{fig/start_freq}
+  \caption{Selecting the initial frequencies}
+  \label{fig:st_freq}
+\end{figure}
+

 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
 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


@@ -564,21 +656,22 @@ 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}
 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}
+  \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}

 \end{equation}
 \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:
+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}
 \begin{equation}
   \label{eq:Fint}

- F_{i} = \frac{\Fmax_i}{\Scp_i},~{i=1,2,\dots,N}
+  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
 \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 are highlighted in Figure~\ref{fig:st_freq}.  This set of

 frequencies can be considered as a higher bound for the search space of the
 optimal vector of frequencies because selecting frequency scaling factors higher
 than the higher bound will not improve the performance of the application and it
 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


@@ -595,109 +688,35 @@ 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}).
 
 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
+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
 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}
-  \label{fig:st_freq}
-\end{figure}
-
-
-
-
-\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.
-    \end{description}
-    \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 Round the computed initial frequencies $F_i$ to the closest one available in each node.
-    \If{(not the first frequency)}
-          \State $F_i \gets F_i+\Fdiff_i,~i=1,\dots,N.$
-    \EndIf 
-    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp_i+\Tcm_i)$
-    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i)} +\sum_{i=1}^{N} {(\Ps_i \cdot \Told)}$
-    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i + \Ps_i \cdot \Told)}$
-    \State $\Sopt_{i} \gets 1,~i=1,\dots,N. $
-    \State $\Dist \gets 0 $
-    \While {(all nodes not reach their  minimum  frequency)}
-        \If{(not the last freq. \textbf{and} not the slowest node)}
-        \State $F_i \gets F_i - \Fdiff_i,~i=1,\dots,N.$
-        \State $S_i \gets \frac{\Fmax_i}{F_i},~i=1,\dots,N.$
-        \EndIf
-       \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 $\Dist \gets \Pnorm - \Enorm$
-      \EndIf
-    \EndWhile
-    \State  Return $\Sopt_1,\Sopt_2,\dots,\Sopt_N$
-  \end{algorithmic}
-  \caption{frequency scaling factors selection algorithm}
-  \label{HSA}
-\end{algorithm}
-
-\begin{algorithm}
-  \begin{algorithmic}[1]
-    % \footnotesize
-    \For {$k=1$ to \textit{some iterations}}
-      \State Computations section.
-      \State Communications section.
-      \If {$(k=1)$}
-        \State Gather all times of computation and\newline\hspace*{3em}%
-               communication from each node.
-        \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.
-      \EndIf
-    \EndFor
-  \end{algorithmic}
-  \caption{DVFS algorithm}
-  \label{dvfs}
-\end{algorithm}
+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.

 
 \subsection{The evaluation of the proposed algorithm}
 \label{sec.verif.algo}
 
 \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
+
+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
 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


@@ -711,81 +730,71 @@ 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
 cluster composed of four different types of nodes having the characteristics
 presented in Table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4
 nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling

-factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$
-is the number of iterations and $N$ is the number of computing nodes. The
-algorithm needs from 12 to 20 iterations to select the best vector of frequency
-scaling factors that gives the results of the next sections.
-
-\section{Experimental results}
-\label{sec.expe}
-To  evaluate the  efficiency and  the  overall energy  consumption reduction  of
-Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. 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.
-
+factors vector.  The algorithm complexity is $O(F\cdot N)$, where $F$ is the
+maximum number of available frequencies, and $N$ is the number of computing
+nodes. The algorithm needs from 12 to 20 iterations to select the best vector of
+frequency scaling factors that gives the results of the next sections.

 
 \begin{table}[!t]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
 
 \begin{table}[!t]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering

-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}

     \hline
     \hline

-    Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
-    type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
-                  &           & GHz      & GHz          &GHz             &              &       \\
+    Node   & Simulated & Max   & Min   & Diff. & Dynamic    & Static    \\
+    type   & GFLOPS    & Freq. & Freq. & Freq. & power      & power     \\
+           &           & GHz   & GHz   & GHz   &            &           \\

     \hline
     \hline

-    1             &40         & 2.5      & 1.2          & 0.1            & 20~W         &4~W    \\
-         
+    1      & 40        & 2.50  & 1.20  & 0.100 & \np[W]{20} & \np[W]{4} \\

     \hline
     \hline

-    2             &50         & 2.66     & 1.6          & 0.133          & 25~W         &5~W    \\
-                  
+    2      & 50        & 2.66  & 1.60  & 0.133 & \np[W]{25} & \np[W]{5} \\

     \hline
     \hline

-    3             &60         & 2.9      & 1.2          & 0.1            & 30~W         &6~W    \\
-                  
+    3      & 60        & 2.90  & 1.20  & 0.100 & \np[W]{30} & \np[W]{6} \\

     \hline
     \hline

-    4             &70         & 3.4      & 1.6          & 0.133          & 35~W         &7~W    \\
-                  
+    4      & 70        & 3.40  & 1.60  & 0.133 & \np[W]{35} & \np[W]{7} \\

     \hline
   \end{tabular}
   \label{table:platform}
 \end{table}
 
     \hline
   \end{tabular}
   \label{table:platform}
 \end{table}
 

- 
-%\subsection{Performance prediction verification}
+\section{Experimental results}
+\label{sec.expe}
+
+To evaluate the efficiency and the overall energy consumption reduction of
+Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The
+experiments were executed on the simulator SimGrid/SMPI which offers easy tools
+to create a heterogeneous platform and run message passing applications over it.
+The heterogeneous platform that was used in the experiments, had one core per
+node because just one process was executed per node.  The heterogeneous platform
+was composed of four types of nodes. Each type of nodes had different
+characteristics such as the maximum CPU frequency, the number of available
+frequencies and the computational power, see Table~\ref{table:platform}. The
+characteristics of these different types of nodes are inspired from the
+specifications of real Intel processors.  The heterogeneous platform had up to
+144 nodes and had nodes from the four types in equal proportions, for example if
+a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the
+constructors of CPUs do not specify the dynamic and the static power of their
+CPUs, for each type of node they were chosen proportionally to its computing
+power (FLOPS).  In the initial heterogeneous platform, while computing with
+highest frequency, each node consumed an amount of power proportional to its
+computing power (which corresponds to \np[\%]{80} of its dynamic power and the
+remaining \np[\%]{20} to the static power), the same assumption was made in
+\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.  Finally, These
+nodes were connected via an Ethernet network with \np[Gbit/s]{1} bandwidth.

 
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 
 
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 

-

 The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
 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:
+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
 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.
-
- 
- 
+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, or 128 nodes.  The other benchmarks such as BT and SP had
+to be executed on 1, 4, 9, 16, 36, 64, or 144 nodes.
+

 \begin{table}[!t]
   \caption{Running NAS benchmarks on 4 nodes }
   % title of Table
 \begin{table}[!t]
   \caption{Running NAS benchmarks on 4 nodes }
   % title of Table


@@ -793,57 +802,57 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%

-    Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

-    CG         &  64.64        & 3560.39        &34.16        &6.72               &27.44       \\
-    \hline 
-    MG         & 18.89         & 1074.87           &35.37            &4.34                   &31.03       \\
-   \hline
-    EP         &79.73         &5521.04         &26.83            &3.04               &23.79      \\
-   \hline
-    LU         &308.65        &21126.00           &34.00             &6.16                   &27.84      \\
+    CG      &  64.64    &  3560.39      & 34.16    & 6.72          & 27.44    \\
+    \hline
+    MG      &  18.89    &  1074.87      & 35.37    & 4.34          & 31.03    \\
+    \hline
+    EP      &  79.73    &  5521.04      & 26.83    & 3.04          & 23.79    \\
+    \hline
+    LU      & 308.65    & 21126.00      & 34.00    & 6.16          & 27.84    \\
+    \hline
+    BT      & 360.12    & 21505.55      & 35.36    & 8.49          & 26.87    \\
+    \hline
+    SP      & 234.24    & 13572.16      & 35.22    & 5.70          & 29.52    \\
+    \hline
+    FT      &  81.58    &  4151.48      & 35.58    & 0.99          & 34.59    \\

     \hline
     \hline

-    BT         &360.12         &21505.55          &35.36         &8.49               &26.87     \\
-   \hline
-    SP         &234.24        &13572.16           &35.22         &5.70               &29.52    \\
-   \hline
-    FT         &81.58          &4151.48        &35.58         &0.99                  &34.59    \\
-\hline 

   \end{tabular}
   \label{table:res_4n}
 % \end{table}
 
   \end{tabular}
   \label{table:res_4n}
 % \end{table}
 

-\medskip
+  \medskip

 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 8 and 9 nodes }
   % title of Table
   \centering
   \begin{tabular}{|*{7}{r|}}
     \hline
 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 8 and 9 nodes }
   % title of Table
   \centering
   \begin{tabular}{|*{7}{r|}}
     \hline

-     \hspace{-2.2084pt}%
-    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\
+    \hline
+    CG      &  36.11    &  3263.49      & 31.25    & 7.12          & 24.13    \\

     \hline
     \hline

-    CG         &36.11             &3263.49             &31.25        &7.12                    &24.13     \\
-    \hline 
-    MG         &8.99          &953.39          &33.78        &6.41                    &27.37     \\
-   \hline
-    EP         &40.39         &5652.81         &27.04        &0.49                    &26.55     \\
-   \hline
-    LU         &218.79            &36149.77        &28.23        &0.01                    &28.22      \\
+    MG      &   8.99    &   953.39      & 33.78    & 6.41          & 27.37    \\
+    \hline
+    EP      &  40.39    &  5652.81      & 27.04    & 0.49          & 26.55    \\
+    \hline
+    LU      & 218.79    & 36149.77      & 28.23    & 0.01          & 28.22    \\
+    \hline
+    BT      & 166.89    & 23207.42      & 32.32    & 7.89          & 24.43    \\
+    \hline
+    SP      & 104.73    & 18414.62      & 24.73    & 2.78          & 21.95    \\
+    \hline
+    FT      &  51.10    &  4913.26      & 31.02    & 2.54          & 28.48    \\

     \hline
     \hline

-    BT         &166.89                &23207.42            &32.32            &7.89                    &24.43      \\
-   \hline
-    SP         &104.73        &18414.62            &24.73            &2.78                    &21.95      \\
-   \hline
-    FT         &51.10         &4913.26         &31.02        &2.54                    &28.48      \\
-\hline 

   \end{tabular}
   \label{table:res_8n}
 % \end{table}
 
   \end{tabular}
   \label{table:res_8n}
 % \end{table}
 

-\medskip
+  \medskip

 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 16 nodes }
   % title of Table
 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 16 nodes }
   % title of Table


@@ -851,28 +860,28 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%

-    Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

-    CG         &31.74         &4373.90         &26.29        &9.57                    &16.72          \\
-    \hline 
-    MG         &5.71          &1076.19         &32.49        &6.05                    &26.44         \\
-   \hline
-    EP         &20.11         &5638.49         &26.85        &0.56                    &26.29         \\
-   \hline
-    LU         &144.13        &42529.06            &28.80            &6.56                    &22.24         \\
+    CG      &  31.74    &  4373.90      & 26.29    & 9.57          & 16.72    \\
+    \hline
+    MG      &   5.71    &  1076.19      & 32.49    & 6.05          & 26.44    \\
+    \hline
+    EP      &  20.11    &  5638.49      & 26.85    & 0.56          & 26.29    \\
+    \hline
+    LU      & 144.13    & 42529.06      & 28.80    & 6.56          & 22.24    \\
+    \hline
+    BT      &  97.29    & 22813.86      & 34.95    & 5.80          & 29.15    \\
+    \hline
+    SP      &  66.49    & 20821.67      & 22.49    & 3.82          & 18.67    \\
+    \hline
+    FT      &  37.01    &  5505.60      & 31.59    & 6.48          & 25.11    \\

     \hline
     \hline

-    BT         &97.29         &22813.86            &34.95        &5.80                &29.15         \\
-   \hline
-    SP         &66.49         &20821.67            &22.49            &3.82                    &18.67         \\
-   \hline
-    FT            &37.01          &5505.60             &31.59        &6.48                    &25.11         \\
-\hline 

   \end{tabular}
   \label{table:res_16n}
 % \end{table}
 
   \end{tabular}
   \label{table:res_16n}
 % \end{table}
 

-\medskip
+  \medskip

 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 32 and 36 nodes }
   % title of Table
 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 32 and 36 nodes }
   % title of Table


@@ -880,28 +889,28 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%

-    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

-    CG         &32.35         &6704.21         &16.15        &5.30                    &10.85           \\
-    \hline 
-    MG         &4.30          &1355.58         &28.93        &8.85                    &20.08          \\
-   \hline
-    EP         &9.96           &5519.68                &26.98        &0.02                    &26.96          \\
-   \hline
-    LU         &99.93         &67463.43            &23.60            &2.45                    &21.15          \\
+    CG      & 32.35     &  6704.21      & 16.15    & 5.30          & 10.85    \\
+    \hline
+    MG      &  4.30     &  1355.58      & 28.93    & 8.85          & 20.08    \\
+    \hline
+    EP      &  9.96     &  5519.68      & 26.98    & 0.02          & 26.96    \\
+    \hline
+    LU      & 99.93     & 67463.43      & 23.60    & 2.45          & 21.15    \\
+    \hline
+    BT      & 48.61     & 23796.97      & 34.62    & 5.83          & 28.79    \\
+    \hline
+    SP      & 46.01     & 27007.43      & 22.72    & 3.45          & 19.27    \\
+    \hline
+    FT      & 28.06     &  7142.69      & 23.09    & 2.90          & 20.19    \\

     \hline
     \hline

-    BT         &48.61         &23796.97            &34.62            &5.83                    &28.79          \\
-   \hline
-    SP         &46.01         &27007.43            &22.72            &3.45                    &19.27           \\
-   \hline
-    FT            &28.06          &7142.69             &23.09        &2.90                    &20.19           \\
-\hline 

   \end{tabular}
   \label{table:res_32n}
 % \end{table}
 
   \end{tabular}
   \label{table:res_32n}
 % \end{table}
 

-\medskip
+  \medskip

 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 64 nodes }
   % title of Table
 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 64 nodes }
   % title of Table


@@ -909,28 +918,28 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%

-    Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

-    CG         &46.65         &17521.83            &8.13             &1.68                    &6.45           \\
-    \hline 
-    MG         &3.27          &1534.70         &29.27        &14.35               &14.92          \\
-   \hline
-    EP         &5.05           &5471.1084          &27.12            &3.11                &24.01         \\
-   \hline
-    LU         &73.92         &101339.16           &21.96            &3.67                    &18.29         \\
+    CG      & 46.65     &  17521.83     &  8.13    &  1.68         &  6.45    \\
+    \hline
+    MG      &  3.27     &   1534.70     & 29.27    & 14.35         & 14.92    \\
+    \hline
+    EP      &  5.05     &   5471.11     & 27.12    &  3.11         & 24.01    \\
+    \hline
+    LU      & 73.92     & 101339.16     & 21.96    &  3.67         & 18.29    \\
+    \hline
+    BT      & 39.99     &  27166.71     & 32.02    & 12.28         & 19.74    \\
+    \hline
+    SP      & 52.00     &  49099.28     & 24.84    &  0.03         & 24.81    \\
+    \hline
+    FT      & 25.97     &  10416.82     & 20.15    &  4.87         & 15.28    \\

     \hline
     \hline

-    BT         &39.99         &27166.71            &32.02            &12.28               &19.74         \\
-   \hline
-    SP         &52.00         &49099.28            &24.84            &0.03                    &24.81         \\
-   \hline
-    FT         &25.97         &10416.82        &20.15        &4.87                    &15.28         \\
-\hline 

   \end{tabular}
   \label{table:res_64n}
 % \end{table}
 
   \end{tabular}
   \label{table:res_64n}
 % \end{table}
 

-\medskip
+  \medskip

 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 128 and 144 nodes }
   % title of Table
 % \begin{table}[!t]
   \caption{Running NAS benchmarks on 128 and 144 nodes }
   % title of Table


@@ -938,215 +947,221 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%
   \begin{tabular}{|*{7}{r|}}
     \hline
     \hspace{-2.2084pt}%

-    Program    & Execution     & Energy         & Energy      & Performance        & Distance     \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

-    CG         &56.92         &41163.36        &4.00         &1.10                    &2.90          \\
-    \hline 
-    MG         &3.55           &2843.33         &18.77       &10.38               &8.39          \\
-   \hline
-    EP         &2.67           &5669.66                &27.09        &0.03                    &27.06         \\
-   \hline
-    LU         &51.23         &144471.90       &16.67        &2.36                    &14.31         \\
+    CG      & 56.92     &  41163.36     &  4.00    &  1.10         &  2.90    \\
+    \hline
+    MG      &  3.55     &   2843.33     & 18.77    & 10.38         &  8.39    \\
+    \hline
+    EP      &  2.67     &   5669.66     & 27.09    &  0.03         & 27.06    \\
+    \hline
+    LU      & 51.23     & 144471.90     & 16.67    &  2.36         & 14.31    \\
+    \hline
+    BT      & 37.96     &  44243.82     & 23.18    &  1.28         & 21.90    \\
+    \hline
+    SP      & 64.53     & 115409.71     & 26.72    &  0.05         & 26.67    \\
+    \hline
+    FT      & 25.51     &  18808.72     & 12.85    &  2.84         & 10.01    \\

     \hline
     \hline

-    BT         &37.96          &44243.82           &23.18            &1.28                    &21.90         \\
-   \hline
-    SP         &64.53         &115409.71           &26.72            &0.05                    &26.67         \\
-   \hline
-    FT         &25.51         &18808.72            &12.85            &2.84                    &10.01         \\
-\hline 

   \end{tabular}
   \label{table:res_128n}
 \end{table}
   \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.
-
-
- 
+

 \begin{figure}[!t]
   \centering
 \begin{figure}[!t]
   \centering

-  \subfloat[Energy saving]{%
+  \subfloat[Energy saving (\%)]{%

     \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
     \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%

-  
-  \subfloat[Performance degradation ]{%
+
+  \subfloat[Performance degradation (\%)]{%

     \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}
   \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}
 \end{figure}
 
     \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}
   \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}
 \end{figure}
 

-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
+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 \np[\%]{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 higher number of nodes.
+Indeed, the benchmarks with the same class, C, are executed on different numbers
+of nodes, so the computation required for each iteration is divided by the
+number of computing nodes.  On the other hand, more communications are required
+when increasing the number of nodes so the static energy increases linearly
+according to the communication time and the dynamic power is less relevant in
+the overall energy consumption.  Therefore, reducing the frequency with
+Algorithm~\ref{HSA} is less effective in reducing the overall energy savings. It
+can also be noticed that for the benchmarks EP and SP that contain little or no
+communications, the energy savings are not significantly affected by the high
+number of nodes.  No experiments were conducted using bigger classes than D,
+because they require a lot of memory (more than \np[GB]{64}) when being executed
+by the simulator on one machine.  The maximum distance between the normalized
+energy curve and the normalized performance for each instance is also shown in
+the result tables. It decrease in the same way as the energy saving percentage.
+The tables also show that the performance degradation percentage is not
+significantly increased when the number of computing nodes is increased because
+the computation times are small when compared to the communication times.
+
+Figures~\ref{fig:energy} and \ref{fig:per_deg} present the energy saving and
+performance degradation respectively for all the benchmarks according to the
+number of used nodes. As shown in the first plot, the energy saving percentages
+of the benchmarks MG, LU, BT and FT decrease linearly when the number of nodes
+increase. While for the EP and SP benchmarks, the energy saving percentage is
+not affected by the increase of the number of computing nodes, because in these
+benchmarks there are little or no communications. Finally, the energy saving of
+the 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.
 
 has less effect on the performance.
 

-
-
-

 \subsection{The results for different power consumption scenarios}
 \label{sec.compare}
 \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:
+
+The results of the previous section were obtained while using processors that
+consume during computation an overall power which is \np[\%]{80} composed of
+dynamic power and of \np[\%]{20} of static power. In this section, these ratios
+are changed and two new power scenarios are considered in order to evaluate how
+the proposed algorithm adapts itself according to the static and dynamic power
+values.  The two new power scenarios are the following:

 
 \begin{itemize}
 
 \begin{itemize}

-\item 70\% of dynamic power  and 30\% of static power
-\item 90\% of dynamic power  and 10\% of static power
+\item \np[\%]{70} of dynamic power and \np[\%]{30} of static power
+\item \np[\%]{90} of dynamic power and \np[\%]{10} of static power

 \end{itemize}
 
 \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
+The NAS parallel benchmarks were executed again over processors that follow the
+new power scenarios.  The class C of each benchmark was run over 8 or 9 nodes
+and the results are presented in Tables~\ref{table:res_s1} and
+\ref{table:res_s2}. These tables show that the energy saving percentage of the
+\np[\%]{70}-\np[\%]{30} scenario is smaller for all benchmarks compared to the
+energy saving of the \np[\%]{90}-\np[\%]{10} scenario.  Indeed, in the latter
+more dynamic power is consumed when nodes are running on their maximum
+frequencies, thus, scaling down the frequency of the nodes results in higher
+energy savings than in the \np[\%]{70}-\np[\%]{30} scenario. On the other hand,
+the performance degradation percentage is smaller in the \np[\%]{70}-\np[\%]{30}
+scenario compared to the \np[\%]{90}-\np[\%]{10} scenario. This is due to the
+higher static power percentage in the first scenario which makes it more
+relevant in the overall consumed energy.  Indeed, the static energy is related
+to the execution time and if the performance is degraded the amount of consumed
+static energy directly increases.  Therefore, the proposed algorithm does not
+really significantly scale down much the frequencies of the nodes in order to
+limit the increase of the execution time and thus limiting the effect of the

 consumed static energy.
 
 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
+Both new power scenarios are compared to the old one in
+Figure~\ref{fig:sen_comp}. It shows the average of the performance degradation,
+the energy saving and the distances for all NAS benchmarks of class C running on
+8 or 9 nodes.  The comparison shows that the energy saving ratio is proportional
+to the dynamic power ratio: it is increased when applying the
+\np[\%]{90}-\np[\%]{10} scenario because at maximum frequency the dynamic energy
+is the most relevant in the overall consumed energy and can be reduced by
+lowering the frequency of some processors. On the other hand, the energy saving
+decreases when the \np[\%]{70}-\np[\%]{30} scenario is used because the dynamic
+energy is less relevant in the overall consumed energy and lowering the
+frequency does not return big energy savings.  Moreover, the average of the
+performance degradation is decreased when using a higher ratio for static power
+(e.g.  \np[\%]{70}-\np[\%]{30} scenario and \np[\%]{80}-\np[\%]{20}
+scenario). Since the proposed algorithm optimizes the energy consumption when
+using a higher ratio for dynamic power the algorithm selects bigger frequency
+scaling factors that result in more energy saving but less performance, for
+example see Figure~\ref{fig:scales_comp}. The opposite happens when using a
+higher ratio for static power, the algorithm proportionally selects smaller
+scaling values which result in less energy saving but also less performance

 degradation.
 
 degradation.
 

-
- \begin{table}[!t]
-  \caption{The results of the 70\%-30\% power scenario}
+\begin{table}[!t]
+  \caption{The results of the \np[\%]{70}-\np[\%]{30} power scenario}

   % title of Table
   \centering
   \begin{tabular}{|*{6}{r|}}
     \hline
   % title of Table
   \centering
   \begin{tabular}{|*{6}{r|}}
     \hline

-    Program    & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
+    Program & Energy        & Energy   & Performance   & Distance \\
+    name    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

-    CG         &4144.21          &22.42        &7.72                &14.70         \\
-    \hline 
-    MG         &1133.23          &24.50        &5.34                &19.16          \\
+    CG      &  4144.21      & 22.42    & 7.72          & 14.70    \\
+    \hline
+    MG      &  1133.23      & 24.50    & 5.34          & 19.16    \\

    \hline
    \hline

-    EP         &6170.30                &16.19         &0.02                &16.17          \\
+    EP      &  6170.30      & 16.19    & 0.02          & 16.17    \\

    \hline
    \hline

-    LU         &39477.28        &20.43        &0.07                &20.36          \\
+    LU      & 39477.28      & 20.43    & 0.07          & 20.36    \\

     \hline
     \hline

-    BT         &26169.55           &25.34             &6.62                &18.71          \\
+    BT      & 26169.55      & 25.34    & 6.62          & 18.71    \\

    \hline
    \hline

-    SP         &19620.09           &19.32             &3.66                &15.66          \\
+    SP      & 19620.09      & 19.32    & 3.66          & 15.66    \\

    \hline
    \hline

-    FT         &6094.07                &23.17         &0.36                &22.81          \\
-\hline 
+    FT      &  6094.07      & 23.17    & 0.36          & 22.81    \\
+    \hline

   \end{tabular}
   \label{table:res_s1}
 \end{table}
 
   \end{tabular}
   \label{table:res_s1}
 \end{table}
 

-
-

 \begin{table}[!t]
 \begin{table}[!t]

-  \caption{The results of the 90\%-10\% power scenario}
+  \caption{The results of the \np[\%]{90}-\np[\%]{10} power scenario}

   % title of Table
   \centering
   \begin{tabular}{|*{6}{r|}}
     \hline
   % title of Table
   \centering
   \begin{tabular}{|*{6}{r|}}
     \hline

-    Program    & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
+    Program & Energy          & Energy      & Performance   & Distance \\
+    name    & consumption/J   & saving\%    & degradation\% &          \\

     \hline
     \hline

-    CG         &2812.38                 &36.36        &6.80                &29.56         \\
-    \hline 
-    MG         &825.427                 &38.35        &6.41                &31.94         \\
-   \hline
-    EP         &5281.62                 &35.02        &2.68                &32.34         \\
-   \hline
-    LU         &31611.28            &39.15        &3.51                    &35.64        \\
+    CG      &  2812.38        & 36.36       &  6.80         & 29.56    \\
+    \hline
+    MG      &   825.43        & 38.35       &  6.41         & 31.94    \\
+    \hline
+    EP      &  5281.62        & 35.02       &  2.68         & 32.34    \\
+    \hline
+    LU      & 31611.28        & 39.15       &  3.51         & 35.64    \\
+    \hline
+    BT      & 21296.46        & 36.70       &  6.60         & 30.10    \\
+    \hline
+    SP      & 15183.42        & 35.19       & 11.76         & 23.43    \\
+    \hline
+    FT      &  3856.54        & 40.80       &  5.67         & 35.13    \\

     \hline
     \hline

-    BT         &21296.46            &36.70            &6.60                &30.10       \\
-   \hline
-    SP         &15183.42            &35.19            &11.76               &23.43        \\
-   \hline
-    FT         &3856.54                 &40.80        &5.67                &35.13        \\
-\hline 

   \end{tabular}
   \label{table:res_s2}
 \end{table}
 
 \begin{table}[!t]
   \end{tabular}
   \label{table:res_s2}
 \end{table}
 
 \begin{table}[!t]

- \caption{Comparing the proposed algorithm}
- \centering
-\begin{tabular}{|*{7}{r|}}
-\hline
-Program & \multicolumn{2}{c|}{Energy saving \%} & \multicolumn{2}{c|}{Perf.  degradation \%} & \multicolumn{2}{c|}{Distance} \\ \cline{2-7} 
-name    & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
-CG      & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
-MG      & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
-LU      & 19.55           & 28.33            & 0.0            & 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}
+  \caption{Comparing the proposed algorithm}
+  \centering
+  \begin{tabular}{|*{7}{r|}}
+    \hline
+    Program & \multicolumn{2}{c|}{Energy saving \%}
+            & \multicolumn{2}{c|}{Perf.  degradation \%}
+            & \multicolumn{2}{c|}{Distance} \\
+    \cline{2-7}
+    name    & EDP   & MaxDist & EDP  & MaxDist & EDP   & MaxDist \\
+    \hline
+    CG      & 27.58 & 31.25   & 5.82 & 7.12    & 21.76 & 24.13   \\
+    \hline
+    MG      & 29.49 & 33.78   & 3.74 & 6.41    & 25.75 & 27.37   \\
+    \hline
+    LU      & 19.55 & 28.33   & 0.00 & 0.01    & 19.55 & 28.22   \\
+    \hline
+    EP      & 28.40 & 27.04   & 4.29 & 0.49    & 24.11 & 26.55   \\
+    \hline
+    BT      & 27.68 & 32.32   & 6.45 & 7.87    & 21.23 & 24.43   \\
+    \hline
+    SP      & 20.52 & 24.73   & 5.21 & 2.78    & 15.31 & 21.95   \\
+    \hline
+    FT      & 27.03 & 31.02   & 2.75 & 2.54    & 24.28 & 28.48   \\
+    \hline
+  \end{tabular}
+  \label{table:compare_EDP}

 \end{table}
 
 \begin{figure}[!t]
 \end{table}
 
 \begin{figure}[!t]


@@ -1158,7 +1173,7 @@ FT      & 27.03           & 31.02            & 2.75           & 2.54
     \includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
     \includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}

-\end{figure}  
+\end{figure}

 
 \begin{figure}[!t]
   \centering
 
 \begin{figure}[!t]
   \centering


@@ -1170,6 +1185,7 @@ FT      & 27.03           & 31.02            & 2.75           & 2.54
 
 \subsection{The comparison of the proposed scaling algorithm }
 \label{sec.compare_EDP}
 
 \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
 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


@@ -1191,34 +1207,34 @@ 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
 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\%.
-
+Spiliopoulos et al. algorithm, on average it results in \np[\%]{29.76} energy
+saving while their algorithm returns just \np[\%]{25.75}. The average of
+performance degradation percentage is approximately the same for both
+algorithms, about \np[\%]{4}.

 
 For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
 terms of  energy and  performance trade-off, see  Figure~\ref{fig:compare_EDP},
 because it maximizes the distance  between the energy saving and the performance
 degradation values while giving the same weight for both metrics.
 
 
 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}
 \section{Conclusion}

-\label{sec.concl} 
+\label{sec.concl}
+

 In this paper, a new online frequency selecting algorithm has been presented. It
 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
+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
 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
+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
 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.
+a heterogeneous platform simulated by SimGrid. The results of the experiments
+showed that the algorithm reduces up to \np[\%]{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
 
 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


@@ -1233,9 +1249,9 @@ the iterative system.
 \section*{Acknowledgment}
 
 This work has been partially supported by the Labex
 \section*{Acknowledgment}
 
 This work has been partially supported by the Labex

-ACTION project (contract “ANR-11-LABX-01-01”). As a PhD student, 
+ACTION project (contract ``ANR-11-LABX-01-01''). As a PhD student,

 Mr. Ahmed Fanfakh, would like to thank the University of
 Mr. Ahmed Fanfakh, would like to thank the University of

-Babylon (Iraq) for supporting his work. 
+Babylon (Iraq) for supporting his work.

 
 
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@@ -1247,7 +1263,7 @@ Babylon (Iraq) for supporting his work.
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+

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