Remove math mode around enumerations.

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 50fc783f09869fee9ea5478943a807a4f0593bc8..8b0f8073f86f66146f7a7e10f0fa9fc1d4ccc03f 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -8,7 +8,6 @@
 \usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
 \usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}

-\usepackage{multirow}

 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 


@@ -24,33 +23,40 @@
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 

-\newcommand{\Xsub}[2]{\ensuremath{#1_\textit{#2}}}
+\newcommand{\Xsub}[2]{{\ensuremath{#1_\mathit{#2}}}}

 
 

-\newcommand{\Dist}{\textit{Dist}}
+%% 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{\Eind}{\Xsub{E}{ind}}
 \newcommand{\Enorm}{\Xsub{E}{Norm}}
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
 \newcommand{\Eind}{\Xsub{E}{ind}}
 \newcommand{\Enorm}{\Xsub{E}{Norm}}
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}

-\newcommand{\Fdiff}{\Xsub{F}{diff}}
-\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Es}{\Xsub{E}{S}}
+\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
+\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}

 \newcommand{\Fnew}{\Xsub{F}{new}}
 \newcommand{\Ileak}{\Xsub{I}{leak}}
 \newcommand{\Kdesign}{\Xsub{K}{design}}
 \newcommand{\Fnew}{\Xsub{F}{new}}
 \newcommand{\Ileak}{\Xsub{I}{leak}}
 \newcommand{\Kdesign}{\Xsub{K}{design}}

-\newcommand{\MaxDist}{\textit{Max Dist}}
+\newcommand{\MaxDist}{\mathit{Max}\Dist}
+\newcommand{\MinTcm}{\mathit{Min}\Tcm}

 \newcommand{\Ntrans}{\Xsub{N}{trans}}
 \newcommand{\Ntrans}{\Xsub{N}{trans}}

-\newcommand{\Pdyn}{\Xsub{P}{dyn}}
-\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
+\newcommand{\PdNew}{\Xsub{P}{dNew}}
+\newcommand{\PdOld}{\Xsub{P}{dOld}}

 \newcommand{\Pnorm}{\Xsub{P}{Norm}}
 \newcommand{\Pnorm}{\Xsub{P}{Norm}}

-\newcommand{\Tnorm}{\Xsub{T}{Norm}}
-\newcommand{\Pstates}{\Xsub{P}{states}}
-\newcommand{\Pstatic}{\Xsub{P}{static}}
-\newcommand{\Sopt}{\Xsub{S}{opt}}
-\newcommand{\Tcomp}{\Xsub{T}{comp}}
-\newcommand{\TmaxCommOld}{\Xsub{T}{Max Comm Old}}
-\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
-\newcommand{\Tmax}{\Xsub{T}{max}}
+\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
+\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
+\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
+\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
+\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
+\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}

 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Told}{\Xsub{T}{Old}} 
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Told}{\Xsub{T}{Old}} 

+

 \begin{document} 
 
 \title{Energy Consumption Reduction for Message Passing Iterative  Applications in Heterogeneous Architecture Using DVFS}
 \begin{document} 
 
 \title{Energy Consumption Reduction for Message Passing Iterative  Applications in Heterogeneous Architecture Using DVFS}


@@ -63,8 +69,7 @@
     Arnaud Giersch
   } 
   \IEEEauthorblockA{%
     Arnaud Giersch
   } 
   \IEEEauthorblockA{%

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

     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


@@ -82,19 +87,19 @@ 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
 (DVFS) is  one of them. It  reduces the frequency of  a CPU to  lower its energy
 consumption.  However,  lowering the  frequency  of  a  CPU might  increase  the
 execution  time of  an application  running on  that processor.   Therefore, the

-frequency that  gives the best tradeoff  between the energy  consumption and the
+frequency that  gives the best trade-off  between the energy  consumption and the

 performance of an application must be selected.
 
 performance of an application must be selected.
 

-In this  paper, a new  online frequencies selecting algorithm  for heterogeneous
-platforms is presented.   It selects the frequency which tries  to give the best
-tradeoff  between  energy saving  and  performance  degradation,  for each  node
-computing the message  passing iterative application. The algorithm  has a small
+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
 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
+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}
 comparison results showing that it outperforms the latter.
 
 \end{abstract}


@@ -121,13 +126,13 @@ This heterogeneous platform executes more than 5 GFLOPS per watt while consuming
 
 Besides platform  improvements, there are many software  and hardware techniques
 to lower  the energy consumption of  these platforms, such  as scheduling, DVFS,
 
 Besides platform  improvements, there are many software  and hardware techniques
 to lower  the energy consumption of  these platforms, such  as scheduling, DVFS,

-...   DVFS is  a  widely used  process to  reduce  the energy  consumption of  a
+\dots{}  DVFS is a widely used process to reduce the energy consumption of a

 processor            by             lowering            its            frequency
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
 the number of FLOPS executed by the processor which might increase the execution
 time of the application running over that processor.  Therefore, researchers use
 different optimization  strategies to select  the frequency that gives  the best
 processor            by             lowering            its            frequency
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
 the number of FLOPS executed by the processor which might increase the execution
 time of the application running over that processor.  Therefore, researchers use
 different optimization  strategies to select  the frequency that gives  the best

-tradeoff  between the  energy reduction  and performance  degradation  ratio. In
+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
 \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


@@ -154,24 +159,24 @@ Finally, in Section~\ref{sec.concl} the paper ends with a summary and some futur
 \section{Related works}
 \label{sec.relwork}
 DVFS is a technique used in modern processors to scale down both the voltage and
 \section{Related works}
 \label{sec.relwork}
 DVFS is a technique used in modern processors to scale down both the voltage and

-the  frequency  of the  CPU  while  computing, in  order  to  reduce the  energy
-consumption of  the processor. DVFS is also  allowed in  GPUs  to achieve the
-same goal. Reducing the frequency of  a processor lowers its number of FLOPS and
-might  degrade the  performance of  the application  running on  that processor,
-especially if it is compute bound. Therefore selecting the appropriate frequency
-for a processor to satisfy some objectives while taking into account all the
-constraints,  is  not a  trivial  operation.   Many  researchers used  different
-strategies to  tackle this problem. Some  of them developed  online methods that
-compute   the  new   frequency  while   executing  the   application,   such  as
-~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}. Others
-used  offline methods  that might  need to  run the  application and  profile it
-before       selecting       the        new       frequency,       such       as
-~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The
-methods could  be heuristics, exact or  brute force methods  that satisfy varied
-objectives such as  energy reduction or performance. They  also could be adapted
-to  the  execution's  environment  and  the  type of  the  application  such  as
-sequential, parallel  or distributed architecture,  homogeneous or heterogeneous
-platform, synchronous or asynchronous application, ...
+the frequency of the CPU while computing, in order to reduce the energy
+consumption of the processor. DVFS is also allowed in GPUs to achieve the same
+goal. Reducing the frequency of a processor lowers its number of FLOPS and might
+degrade the performance of the application running on that processor, especially
+if it is compute bound. Therefore selecting the appropriate frequency for a
+processor to satisfy some objectives while taking into account all the
+constraints, is not a trivial operation.  Many researchers used different
+strategies to tackle this problem. Some of them developed online methods that
+compute the new frequency while executing the application, such
+as~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}.
+Others used offline methods that might need to run the application and profile
+it before selecting the new frequency, such
+as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
+The methods could be heuristics, exact or brute force methods that satisfy
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+sequential, parallel or distributed architecture, homogeneous or heterogeneous
+platform, synchronous or asynchronous application, \dots{}

 
 In this paper, we are interested in reducing energy for message passing iterative synchronous applications running over heterogeneous platforms.
 Some works have already been done for such platforms and they can be classified into two types of heterogeneous platforms: 
 
 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: 


@@ -182,36 +187,55 @@ Some works have already been done for such platforms and they can be classified
 
 \end{itemize}
 
 
 \end{itemize}
 

-For the first type of platform, the computing intensive parallel tasks are executed on the  GPUs and the rest are executed 
-on the CPUs.  Luley et al.
-~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed  a heterogeneous 
-cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal was to maximize the 
-energy efficiency of the platform during computation by maximizing the number of FLOPS per watt generated. 
-In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et al. developed a scheduling 
-algorithm that distributes  workloads proportional to the computing power of the nodes which could be a GPU or a CPU. All the tasks must be completed at the same time.
-In~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, Rong et al. showed that 
-a heterogeneous (GPUs and CPUs) cluster that enables DVFS gave better energy and performance 
-efficiency than other clusters only composed of  CPUs.
+For the first type of platform, the computing intensive parallel tasks are
+executed on the GPUs and the rest are executed on the CPUs.  Luley et
+al.~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a
+heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
+goal was to maximize the energy efficiency of the platform during computation by
+maximizing the number of FLOPS per watt generated.
+In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
+al. developed a scheduling algorithm that distributes workloads proportional to
+the computing power of the nodes which could be a GPU or a CPU. All the tasks
+must be completed at the same time.  In~\cite{Rong_Effects.of.DVFS.on.K20.GPU},
+Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
+DVFS gave better energy and performance efficiency than other clusters only
+composed of CPUs.

  
  

-The work presented in this paper concerns the second type of platform, with heterogeneous CPUs.
-Many methods were conceived to reduce the energy consumption of this type of platform.  Naveen et al.~\cite{Naveen_Power.Efficient.Resource.Scaling}  
-developed a method that minimizes the value of $energy*delay^2$ (the delay is the sum of slack times that happen during synchronous communications) by dynamically assigning new frequencies to the CPUs of the heterogeneous cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed
-an algorithm that divides the executed tasks into two types: the critical and 
-non critical tasks. The algorithm scales down the frequency of  non critical tasks proportionally to their  slack and communication times while limiting  the performance degradation percentage to less than 10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed 
-  a heterogeneous cluster composed of two  types 
-of Intel and AMD processors. They use a gradient method to predict the impact of DVFS operations on performance.
-In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, 
- the best frequencies for a specified heterogeneous cluster are selected offline using some 
-heuristic. Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic programming approach to  
-minimize the power consumption of heterogeneous servers  while respecting given time constraints. This approach 
-had considerable overhead.
-In contrast to the above described papers, this paper presents the following contributions :
+The work presented in this paper concerns the second type of platform, with
+heterogeneous CPUs.  Many methods were conceived to reduce the energy
+consumption of this type of platform.  Naveen et
+al.~\cite{Naveen_Power.Efficient.Resource.Scaling} developed a method that
+minimizes the value of $\mathit{energy}\times \mathit{delay}^2$ (the delay is
+the sum of slack times that happen during synchronous communications) by
+dynamically assigning new frequencies to the CPUs of the heterogeneous cluster.
+Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed an
+algorithm that divides the executed tasks into two types: the critical and non
+critical tasks. The algorithm scales down the frequency of non critical tasks
+proportionally to their slack and communication times while limiting the
+performance degradation percentage to less than \np[\%]{10}.
+In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
+heterogeneous cluster composed of two types of Intel and AMD processors. They
+use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
+\cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
+frequencies for a specified heterogeneous cluster are selected offline using
+some heuristic.  Chen et
+al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
+programming approach to minimize the power consumption of heterogeneous servers
+while respecting given time constraints.  This approach had considerable
+overhead.  In contrast to the above described papers, this paper presents the
+following contributions :

 \begin{enumerate}
 \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 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 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}
 
       
 \end{enumerate}
 


@@ -231,16 +255,16 @@ 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.
 

-The overall execution time  of a distributed iterative synchronous application 
-over a heterogeneous platform  consists of the sum of the computation time and 
-the communication time for every iteration on a node. However, due to the 
-heterogeneous computation power of the computing nodes, slack times might occur 
-when fast nodes have to  wait, during synchronous communications, for  the slower 
-nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
-Therefore,  the overall execution time  of the program is the execution time of the slowest
-task which has the highest computation time and no slack time.
+The overall execution time of a distributed iterative synchronous application
+over a heterogeneous platform consists of the sum of the computation time and
+the communication time for every iteration on a node. However, due to the
+heterogeneous computation power of the computing nodes, slack times might occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
+overall execution time of the program is the execution time of the slowest task
+which has the highest computation time and no slack time.

   
   

- \begin{figure}[t]
+ \begin{figure}[!t]

   \centering
    \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \centering
    \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}


@@ -257,7 +281,7 @@ 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{F_\textit{max}}{F_\textit{new}}
+ S = \frac{\Fmax}{\Fnew}

 \end{equation}
  The execution time of a compute bound sequential program is linearly proportional 
  to the frequency scaling factor $S$.  On the other hand,  message passing 
 \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 


@@ -282,16 +306,15 @@ 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}

- \textit  T_\textit{new} = 
- \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}
 \end{equation}

-Where:\\
+Where:

 \begin{equation}
 \label{eq:perf2}
 \begin{equation}
 \label{eq:perf2}

- MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
+ \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
+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
 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


@@ -312,28 +335,28 @@ Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Schedul
 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
 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 $C_L$, the supply voltage $V$ and
+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}
 operational frequency $F$, as shown in (\ref{eq:pd}).
 \begin{equation}
   \label{eq:pd}

-  Pd = \alpha \cdot C_L \cdot V^2 \cdot F
+  \Pd = \alpha \cdot \CL \cdot V^2 \cdot F

 \end{equation}
 \end{equation}

-The static power $Ps$ captures the leakage power as follows:
+The static power $\Ps$ captures the leakage power as follows:

 \begin{equation}
   \label{eq:ps}
 \begin{equation}
   \label{eq:ps}

-   Ps  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
+   \Ps  = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak

 \end{equation}
 \end{equation}

-where V is the supply voltage, $N_{trans}$ is the number of transistors,
-$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
+where V is the supply voltage, $\Ntrans$ is the number of transistors,
+$\Kdesign$ is a design dependent parameter and $\Ileak$ is a

 technology dependent parameter.  The energy consumed by an individual processor
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}
 technology dependent parameter.  The energy consumed by an individual processor
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}

-   E_\textit{ind} =  Pd \cdot Tcp + Ps \cdot T
+  \Eind =  \Pd \cdot \Tcp + \Ps \cdot T

 \end{equation}
 \end{equation}

-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
+where $T$ is the execution time of the program, $\Tcp$ is the computation
+time and $\Tcp \le T$.  $\Tcp$ may be equal to $T$ if there is no

 communication and no slack time.
 
 The main objective of DVFS operation is to reduce the overall energy consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  
 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}.  


@@ -344,19 +367,19 @@ 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 
 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 

-$F_{new}$ from (\ref{eq:s}) can be calculated as follows:
+$\Fnew$ from (\ref{eq:s}) can be calculated as follows:

 \begin{equation}
   \label{eq:fnew}
 \begin{equation}
   \label{eq:fnew}

-   F_\textit{new} = S^{-1} \cdot F_\textit{max}
+   \Fnew = S^{-1} \cdot \Fmax

 \end{equation}
 \end{equation}

-Replacing $F_{new}$ 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}
 equation for dynamic power consumption:
 \begin{multline}
   \label{eq:pdnew}

-   {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
-   {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}
+   \PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 \\
+   {} = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}

 \end{multline}
 \end{multline}

-where $ {P}_\textit{dNew}$  and $P_{dOld}$ 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.
 
 According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 
 new frequency and the maximum frequency respectively.
 
 According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 


@@ -366,7 +389,7 @@ The new dynamic energy is the  dynamic power multiplied by the new time of compu
 and is given by the following equation:
 \begin{equation}
   \label{eq:Edyn}
 and is given by the following equation:
 \begin{equation}
   \label{eq:Edyn}

-   E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{dOld} \cdot  Tcp 
+   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp 

 \end{equation}
 The static power is related to 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}, 
 \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}, 


@@ -379,18 +402,18 @@ to the frequency scaling factor, while this scaling factor does not affect the c
 The static energy of a processor after scaling its frequency is computed as follows: 
 \begin{equation}
   \label{eq:Estatic}
 The static energy of a processor after scaling its frequency is computed as follows: 
 \begin{equation}
   \label{eq:Estatic}

- E_\textit{s} = Ps \cdot (Tcp \cdot S  + Tcm)
+  \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)

 \end{equation}
 
 In  the  considered  heterogeneous  platform,  each  processor  $i$  might  have
 \end{equation}
 
 In  the  considered  heterogeneous  platform,  each  processor  $i$  might  have

-different   dynamic  and  static   powers,  noted   as  $Pd_{i}$   and  $Ps_{i}$
+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
 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
+$\Tcp[i]$ 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
 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
+$\Tcm[i]$  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
 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


@@ -400,9 +423,9 @@ 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}) +
-  {MinTcm))}
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + {} \\
+ \sum_{i=1}^{N} (\Ps[i] \cdot (\max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +
+  {\MinTcm))}

  \end{multline}
 
 Reducing the frequencies of the processors according to the vector of
  \end{multline}
 
 Reducing the frequencies of the processors according to the vector of


@@ -449,9 +472,9 @@ scaling  down the  frequencies of  some processors)  and the  initial  one (with
 maximum frequency for all nodes) as follows:
 \begin{multline}
   \label{eq:pnorm}
 maximum frequency for all nodes) as follows:
 \begin{multline}
   \label{eq:pnorm}

-  P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
-       {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +MinTcm}
-           {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
+  \Pnorm = \frac{\Tnew}{\Told}\\
+       {} = \frac{ \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +\MinTcm}
+           {\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}

 \end{multline}
 
 
 \end{multline}
 
 


@@ -459,13 +482,13 @@ In the same way, the energy is normalized by computing the ratio between the con
 while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
 \begin{multline}
   \label{eq:enorm}
 while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
 \begin{multline}
   \label{eq:enorm}

-  E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
-  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
+  \Enorm = \frac{\Ereduced}{\Eoriginal} \\
+  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}

 \end{multline} 
 \end{multline} 

-Where $E_\textit{Reduced}$ and $E_\textit{Original}$ are computed using (\ref{eq:energy}) and
-  $T_{New}$ and $T_{Old}$ are computed as in (\ref{eq:pnorm}).
+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 
 
 While the main 
 goal is to optimize the energy and execution time at the same time, the normalized 


@@ -480,13 +503,13 @@ 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}
 normalized execution time is inverted which gives the normalized performance equation, as follows:
 \begin{multline}
   \label{eq:pnorm_inv}

-  P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
-          = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
-            { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm} 
+  \Pnorm = \frac{\Told}{\Tnew}\\
+          = \frac{\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
+            { \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm} 

 \end{multline}
 
 
 \end{multline}
 
 

-\begin{figure}
+\begin{figure}[!t]

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


@@ -498,92 +521,103 @@ normalized execution time is inverted which gives the normalized performance equ
   \caption{The energy and performance relation}
 \end{figure}
 
   \caption{The energy and performance relation}
 \end{figure}
 

-Then, the objective function can be modeled in order to find the maximum distance
-between the energy curve (\ref{eq:enorm}) and the  performance
-curve (\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
-represents the minimum energy consumption with minimum execution time (maximum 
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then the objective
-function has the following form:
+Then, the objective function can be modeled in order to find the maximum
+distance between the energy curve (\ref{eq:enorm}) and the performance curve
+(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
+represents the minimum energy consumption with minimum execution time (maximum
+performance) at the same time, see Figure~\ref{fig:r1} or
+Figure~\ref{fig:r2}. Then the objective function has the following form:

 \begin{equation}
   \label{eq:max}
 \begin{equation}
   \label{eq:max}

-  Max Dist = 
-  \max_{i=1,\dots F, j=1,\dots,N}
-      (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
-       \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
+  \MaxDist = 
+  \mathop{\max_{i=1,\dots F}}_{j=1,\dots,N}
+      (\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_{ij})}^{\text{Minimize}} )

 \end{equation}
 where $N$ is the number of nodes and $F$ is the  number of available frequencies for each 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 
 \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 energy reduction gain can be achieved when 
+(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}
 
 \subsection{The algorithm details}
 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}
 
 \subsection{The algorithm details}

-In this section, algorithm \ref{HSA} is presented. It selects the frequency scaling factors 
-vector that gives the best trade-off between minimizing the energy consumption  and maximizing 
-the performance of a message passing synchronous iterative application executed on a heterogeneous 
-platform. It works online during the execution time of the iterative message passing program.  
-It uses information gathered during the first iteration such as the computation time and the 
-communication time in one iteration for each node. The algorithm is executed  after the first 
-iteration and returns a vector of optimal frequency scaling factors   that satisfies the objective 
-function (\ref{eq:max}). The program applies DVFS operations to change the frequencies of the CPUs 
-according to the computed scaling factors.  This algorithm is called just once during the execution 
-of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called 
-in the iterative MPI program.
-
-The nodes in a heterogeneous platform have different computing powers, thus while executing message 
-passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their 
-computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). 
-These periods are called idle or slack times. 
-The algorithm takes into account this problem and tries to reduce these slack times when selecting the 
-frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase 
-the execution times of fast nodes and  minimize the  differences between  the  computation times of 
-fast and slow nodes. The value of the initial frequency scaling factor  for each node is inversely 
-proportional to its computation time that was gathered from the first iteration. These initial frequency 
-scaling factors are computed as   a ratio between the computation time of the slowest node and the 
-computation time of the node $i$ as follows:
+In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
+scaling factors vector that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+synchronous iterative application executed on a heterogeneous platform. It works
+online during the execution time of the iterative message passing program.  It
+uses information gathered during the first iteration such as the computation
+time and the communication time in one iteration for each node. The algorithm is
+executed after the first iteration and returns a vector of optimal frequency
+scaling factors that satisfies the objective function (\ref{eq:max}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors.  This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
+scaling algorithm is called in the iterative MPI program.
+
+The nodes in a heterogeneous platform have different computing powers, thus
+while executing message passing iterative synchronous applications, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
+periods are called idle or slack times.  The algorithm takes into account this
+problem and tries to reduce these slack times when selecting the frequency
+scaling factors vector. At first, it selects initial frequency scaling factors
+that increase the execution times of fast nodes and minimize the differences
+between the computation times of fast and slow nodes. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:

 \begin{equation}
   \label{eq:Scp}
 \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}
 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 $i$  
 \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:
+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,\cdots,N}
+ F_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}

 \end{equation}
 \end{equation}

-If the computed initial frequency for a node is not available in the gears of that node, the computed 
-initial frequency is replaced by the nearest available frequency. In  figure (\ref{fig:st_freq}), 
-the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster 
-nodes are scaled down according to the computed initial frequency scaling factors. The resulting new 
-frequencies are colored in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered 
-as a higher bound for the search space of the optimal vector of frequencies because selecting frequency 
-scaling factors higher than the higher bound will not improve the performance of the application and 
-it will increase its overall energy consumption. Therefore the algorithm that selects the frequency 
-scaling factors starts the search method from these initial frequencies and takes a downward search direction 
-toward lower frequencies. The algorithm iterates on all left frequencies, from the higher bound until all 
-nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select 
-the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node 
-according to the equation (\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of  
-all other nodes by one gear.
-The new overall energy consumption and execution time are computed according to the new scaling factors. 
-The optimal set of frequency scaling factors is the set that gives the highest distance according to  the objective 
-function (\ref{eq:max}).
-
-The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an 
-application running on a homogeneous platform and a heterogeneous platform respectively while increasing the 
-scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor 
-should be started from the maximum frequency because the performance and the consumed energy is decreased since  
-the beginning of the plot. On the other hand, in  the heterogeneous platform the performance is  maintained at 
-the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes 
-have computing powers lower than the slowest node. In other words, until they reach the higher bound. It can 
-also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger 
-the maximum distance between the energy curve and the performance curve is while varying the scaling factors 
-which results in bigger energy savings. 
-\begin{figure}[t]
+If the computed initial frequency for a node is not available in the gears of
+that node, it is replaced by the nearest available frequency.  In
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors.  The resulting new
+frequencies are highlighted in Figure~\ref{fig:st_freq}.  This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting frequency scaling factors higher
+than the higher bound will not improve the performance of the application and it
+will increase its overall energy consumption.  Therefore the algorithm that
+selects the frequency scaling factors starts the search method from these
+initial frequencies and takes a downward search direction toward lower
+frequencies. The algorithm iterates on all left frequencies, from the higher
+bound until all nodes reach their minimum frequencies, to compute their overall
+energy consumption and performance, and select the optimal frequency scaling
+factors vector. At each iteration the algorithm determines the slowest node
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+while it lowers the frequency of all other nodes by one gear.  The new overall
+energy consumption and execution time are computed according to the new scaling
+factors.  The optimal set of frequency scaling factors is the set that gives the
+highest distance according to the objective function (\ref{eq:max}).
+
+Figures~\ref{fig:r1} and \ref{fig:r2}  illustrate the normalized performance and
+consumed  energy for  an application  running on  a homogeneous  platform  and a
+heterogeneous platform respectively while increasing the scaling factors. It can
+be noticed  that in a  homogeneous platform the  search for the  optimal scaling
+factor should start  from the maximum frequency because  the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand,
+in the heterogeneous platform the  performance is maintained at the beginning of
+the plot  even if the  frequencies of the  faster nodes decrease  until the
+computing power of scaled down  nodes are lower than the slowest  node. In other
+words, until they reach the higher bound. It can also be noticed that the higher
+the difference between the faster nodes  and the slower nodes is, the bigger the
+maximum distance  between the  energy curve and  the performance curve  is while
+ the scaling factors are varying which results in bigger energy savings.
+\begin{figure}[!t]

   \centering
     \includegraphics[scale=0.5]{fig/start_freq}
   \caption{Selecting the initial frequencies}
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}
   \caption{Selecting the initial frequencies}


@@ -598,41 +632,42 @@ which results in bigger energy savings.
     % \footnotesize
     \Require ~
     \begin{description}
     % \footnotesize
     \Require ~
     \begin{description}

-    \item[$Tcp_i$] array of all computation times for all nodes during one iteration and with highest frequency.
-    \item[$Tcm_i$] array of all communication times for all nodes during one iteration and with highest frequency.
-    \item[$Fmax_i$] array of the maximum frequencies for all nodes.
-    \item[$Pd_i$] array of the dynamic powers for all nodes.
-    \item[$Ps_i$] array of the static powers for all nodes.
-    \item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
+    \item[{$\Tcp[i]$}] array of all computation times for all nodes during one iteration and with highest frequency.
+    \item[{$\Tcm[i]$}] array of all communication times for all nodes during one iteration and with highest frequency.
+    \item[{$\Fmax[i]$}] array of the maximum frequencies for all nodes.
+    \item[{$\Pd[i]$}] array of the dynamic powers for all nodes.
+    \item[{$\Ps[i]$}] array of the static powers for all nodes.
+    \item[{$\Fdiff[i]$}] array of the difference between two successive frequencies for all nodes.

     \end{description}
     \end{description}

-    \Ensure $Sopt_1,Sopt_2 \dots, Sopt_N$ is a vector of optimal scaling factors
+    \Ensure $\Sopt[1],\Sopt[2] \dots, \Sopt[N]$ is a vector of optimal scaling factors

 
 

-    \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
-    \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
+    \State $\Scp[i] \gets \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]} $
+    \State $F_{i} \gets  \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\cdots,N}$

     \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
     \If{(not the first frequency)}
     \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
     \If{(not the first frequency)}

-          \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$

     \EndIf 
     \EndIf 

-    \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
-    \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
-    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
-    \State $Dist \gets 0 $
+    \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)}
     \While {(all nodes not reach their  minimum  frequency)}
         \If{(not the last freq. \textbf{and} not the slowest node)}

-        \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
-        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,\dots,N.$
+        \State $F_i \gets F_i - \Fdiff[i],~i=1,\dots,N.$
+        \State $S_i \gets \frac{\Fmax[i]}{F_i},~i=1,\dots,N.$

         \EndIf
         \EndIf

-       \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm $
-       \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} + $  \hspace*{43 mm} 
-               $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
-       \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
-       \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
+       \State $\Tnew \gets \max_{i=1,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm $
+%       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + \sum_{i=1}^{N} {(\Ps[i] \cdot \rlap{\Tnew)}} $
+       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \rlap{\Tnew)}} $
+       \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+       \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$

       \If{$(\Pnorm - \Enorm > \Dist)$}
       \If{$(\Pnorm - \Enorm > \Dist)$}

-        \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $
+        \State $\Sopt[i] \gets S_{i},~i=1,\dots,N. $

         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile
         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile

-    \State  Return $Sopt_1,Sopt_2,\dots,Sopt_N$
+    \State  Return $\Sopt[1],\Sopt[2],\dots,\Sopt[N]$

   \end{algorithmic}
   \caption{frequency scaling factors selection algorithm}
   \label{HSA}
   \end{algorithmic}
   \caption{frequency scaling factors selection algorithm}
   \label{HSA}


@@ -647,7 +682,7 @@ which results in bigger energy savings.
       \If {$(k=1)$}
         \State Gather all times of computation and\newline\hspace*{3em}%
                communication from each node.
       \If {$(k=1)$}
         \State Gather all times of computation and\newline\hspace*{3em}%
                communication from each node.

-        \State Call algorithm \ref{HSA}.
+        \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.
         \State Compute the new frequencies from the\newline\hspace*{3em}%
                returned optimal scaling factors.
         \State Set the new frequencies to nodes.


@@ -660,68 +695,81 @@ 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 of 
-the works presented in this paper is based on the execution time model. To verify this model, the predicted 
-execution time was compared to  the real execution time over SimGrid/SMPI simulator, v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}, 
-for all  the NAS parallel benchmarks NPB v3.3 
-\cite{NAS.Parallel.Benchmarks}, running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise, 
-the maximum normalized difference between the predicted execution time  and the real execution time is equal 
-to 0.03 for all the NAS benchmarks.
-
-Since  the proposed algorithm is not an exact method and does not test all the possible solutions (vectors of scaling factors) 
-in the search space. To prove its efficiency, it was compared on small instances to a brute force search algorithm 
-that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with 
-different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical 
-and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time: 
-for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in 
-table~\ref{table:platform}, it takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes 
-to compute the best scaling factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number 
-of iterations and $N$ is the number of computing nodes. The algorithm needs  from 12 to 20 iterations to select the best 
-vector of frequency scaling factors that gives the results of the next sections.
+The precision  of the  proposed algorithm mainly  depends on the  execution time
+prediction model  defined in  (\ref{eq:perf}) and the  energy model  computed by
+(\ref{eq:energy}).   The energy  model is  also significantly  dependent  on the
+execution  time model  because  the static  energy  is linearly  related to  the
+execution time  and the dynamic energy  is related to the  computation time. So,
+all the works presented  in this paper are based on the  execution time model. To
+verify  this  model, the  predicted  execution time  was  compared  to the  real
+execution          time           over          SimGrid/SMPI          simulator,
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile},   for   all   the   NAS
+parallel benchmarks NPB v3.3  \cite{NAS.Parallel.Benchmarks}, running class B on
+8 or  9 nodes. The comparison showed  that the proposed execution  time model is
+very precise, the maximum  normalized difference between the predicted execution
+time and the real execution time is equal to 0.03 for all the NAS benchmarks.
+
+Since the proposed algorithm is not an exact method it does not test all the
+possible solutions (vectors of scaling factors) in the search space. To prove
+its efficiency, it was compared on small instances to a brute force search
+algorithm that tests all the possible solutions. The brute force algorithm was
+applied to different NAS benchmarks classes with different number of nodes. The
+solutions returned by the brute force algorithm and the proposed algorithm were
+identical and the proposed algorithm was on average 10 times faster than the
+brute force algorithm. It has a small execution time: for a heterogeneous
+cluster composed of four different types of nodes having the characteristics
+presented in Table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4
+nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling
+factors vector.  The algorithm complexity is $O(F\cdot N)$, where $F$
+is the number of iterations and $N$ is the number of computing nodes. The
+algorithm needs from 12 to 20 iterations to select the best vector of frequency
+scaling factors that gives the results of the next sections.

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

   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering

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

     \hline
     Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
     type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
                   &           & GHz      & GHz          &GHz             &              &       \\
     \hline
     \hline
     Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
     type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
                   &           & GHz      & GHz          &GHz             &              &       \\
     \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}
                   
     \hline
   \end{tabular}


@@ -736,21 +784,24 @@ Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwi
 \label{sec.res}
 
 
 \label{sec.res}
 
 

-The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) 
-and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in 
-this paper, only the results of the biggest class, C, are presented while being run on different number 
-of nodes, ranging  from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the 
-benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes. 
-The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
+MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes:
+A, B and C. However, due to the lack of space in this paper, only the results of
+the biggest class, C, are presented while being run on different number of
+nodes, ranging from 4 to 128 or 144 nodes depending on the benchmark being
+executed. Indeed, the benchmarks CG, MG, LU, EP and FT had to be executed on 1,
+2, 4, 8, 16, 32, 64, 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}[htb]
+\begin{table}[!t]

   \caption{Running NAS benchmarks on 4 nodes }
   % title of Table
   \centering
   \caption{Running NAS benchmarks on 4 nodes }
   % title of Table
   \centering

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

     \hline
     \hline

+    \hspace{-2.2084pt}%

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


@@ -770,14 +821,16 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
 \hline 
   \end{tabular}
   \label{table:res_4n}
 \hline 
   \end{tabular}
   \label{table:res_4n}

-\end{table}
+% \end{table}

 
 

-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]

   \caption{Running NAS benchmarks on 8 and 9 nodes }
   % title of Table
   \centering
   \caption{Running NAS benchmarks on 8 and 9 nodes }
   % title of Table
   \centering

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

     \hline
     \hline

+     \hspace{-2.2084pt}%

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


@@ -797,14 +850,16 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
 \hline 
   \end{tabular}
   \label{table:res_8n}
 \hline 
   \end{tabular}
   \label{table:res_8n}

-\end{table}
+% \end{table}

 
 

-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]

   \caption{Running NAS benchmarks on 16 nodes }
   % title of Table
   \centering
   \caption{Running NAS benchmarks on 16 nodes }
   % title of Table
   \centering

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

     \hline
     \hline

+    \hspace{-2.2084pt}%

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


@@ -824,14 +879,16 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
 \hline 
   \end{tabular}
   \label{table:res_16n}
 \hline 
   \end{tabular}
   \label{table:res_16n}

-\end{table}
+% \end{table}

 
 

-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]

   \caption{Running NAS benchmarks on 32 and 36 nodes }
   % title of Table
   \centering
   \caption{Running NAS benchmarks on 32 and 36 nodes }
   % title of Table
   \centering

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

     \hline
     \hline

+    \hspace{-2.2084pt}%

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


@@ -851,14 +908,16 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
 \hline 
   \end{tabular}
   \label{table:res_32n}
 \hline 
   \end{tabular}
   \label{table:res_32n}

-\end{table}
+% \end{table}

 
 

-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]

   \caption{Running NAS benchmarks on 64 nodes }
   % title of Table
   \centering
   \caption{Running NAS benchmarks on 64 nodes }
   % title of Table
   \centering

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

     \hline
     \hline

+    \hspace{-2.2084pt}%

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


@@ -878,15 +937,16 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
 \hline 
   \end{tabular}
   \label{table:res_64n}
 \hline 
   \end{tabular}
   \label{table:res_64n}

-\end{table}
-
+% \end{table}

 
 

-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]

   \caption{Running NAS benchmarks on 128 and 144 nodes }
   % title of Table
   \centering
   \caption{Running NAS benchmarks on 128 and 144 nodes }
   % title of Table
   \centering

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

     \hline
     \hline

+    \hspace{-2.2084pt}%

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


@@ -907,105 +967,129 @@ The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 6
   \end{tabular}
   \label{table:res_128n}
 \end{table}
   \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 reduce significantly the energy 
-consumption (up to 35\%) and tries to limit the performance degradation. They also show that 
-the  energy saving percentage is decreased  when the number of the computing nodes is increased. 
-This reduction is due to the increase of the communication times compared to the execution times 
-when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C, 
-are executed on different number of nodes, so the computation required for each iteration is divided 
-by the number of computing nodes.   On the other hand, more communications are required when increasing 
-the number of nodes so the static energy is increased linearly according to the communication time and 
-the dynamic power is less relevant in the overall energy consumption. Therefore, reducing the frequency 
-with algorithm~(\ref{HSA}) have less effect in reducing the overall energy savings. It can also be 
-noticed that for the benchmarks EP and SP that contain little or no communications,  the energy savings 
-are not significantly affected with the high number of nodes. No experiments were conducted using bigger 
-classes such as D, because they require a lot of memory(more than 64GB) when being executed by the simulator 
-on one machine. The maximum distance between the normalized energy curve and the normalized performance 
-for each instance is also shown in the result tables. It is decreased in the same way as the energy 
-saving percentage. The tables also show that the performance degradation percentage is not significantly 
-increased when the number of computing nodes is increased because the computation times are small when 
-compared to the communication times.  
+The overall energy consumption was computed for each instance according to the
+energy consumption model (\ref{eq:energy}), with and without applying the
+algorithm. The execution time was also measured for all these experiments. Then,
+the energy saving and performance degradation percentages were computed for each
+instance.  The results are presented in Tables~\ref{table:res_4n},
+\ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n},
+\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
+values from many experiments for energy savings and performance degradation.
+The tables show the experimental results for running the NAS parallel benchmarks
+on different number of nodes.  The experiments show that the algorithm
+significantly reduces the energy consumption (up to \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 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}
+\begin{figure}[!t]

   \centering
   \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}
     \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}

-  \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
+  \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}

 \end{figure}
 
 \end{figure}
 

-Plots (\ref{fig:energy} and \ref{fig:per_deg}) present the energy saving and performance degradation 
-respectively for all the benchmarks according to the number of used nodes. As shown in the first plot, 
-the energy saving percentages of the benchmarks MG, LU, BT and FT are decreased linearly  when the  
-number of nodes is increased. While for the  EP and SP benchmarks, the energy saving percentage is not 
-affected by the increase of the number of computing nodes, because in these benchmarks there are little or 
-no communications. Finally, the energy saving of the GC benchmark  is significantly decreased when the number 
-of nodes is increased because  this benchmark has more communications than the others. The second plot 
-shows that the performance degradation percentages of most of the benchmarks are decreased when they 
-run on a big number of nodes because they spend more time communicating than computing, thus, scaling 
-down the frequencies of some nodes have less effect on the performance. 
+Figures~\ref{fig:energy} and  \ref{fig:per_deg} present  the energy  saving and
+performance  degradation respectively for  all the  benchmarks according  to the
+number of used nodes. As shown  in the first plot, the energy saving percentages
+of the benchmarks MG,  LU, BT and FT decrease linearly when  the number of nodes
+increase. While  for the EP and  SP benchmarks, the energy  saving percentage is
+not affected by the increase of  the number of computing nodes, because in these
+benchmarks there are little or  no communications. Finally, the energy saving of
+the  GC benchmark  significantly  decrease  when the  number  of nodes  increase
+because this benchmark has more  communications than the others. The second plot
+shows that  the performance  degradation percentages of  most of  the benchmarks
+decrease when  they run on a  big number of  nodes because they spend  more time
+communicating than computing,  thus, scaling down the frequencies  of some nodes
+has less effect on the performance.

 
 
 
 
 \subsection{The results for different power consumption scenarios}
 \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 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\% dynamic power  and 30\% static power
-\item 90\% dynamic power  and 10\% 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 less for all benchmarks compared to the energy saving of the 90\%-10\% scenario. Indeed, in the latter 
-more dynamic power is consumed when nodes are running on their maximum frequencies, thus, scaling down the frequency 
-of the nodes results in higher energy savings than in the 70\%-30\% scenario. On the other hand,  the performance 
-degradation percentage is less in the 70\%-30\% scenario  compared to the 90\%-10\%  scenario. This is due to the 
-higher static power percentage in the first scenario which makes it more relevant in the overall consumed energy. 
-Indeed, the static energy is related to the execution time and if the performance is  degraded the total consumed 
-static energy is directly increased. Therefore, the proposed algorithm do not scales down much the frequencies of the 
-nodes  in order to limit the increase of the execution time and thus limiting the effect of the consumed static energy.
-
-The two new power scenarios are compared to the old one in figure (\ref{fig:sen_comp}). It shows the average of 
-the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. 
-The comparison shows that  the energy saving ratio is proportional to the dynamic power ratio: it is increased 
-when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the  most relevant 
-in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, 
-the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in 
-the overall consumed energy and lowering the frequency do not returns big energy savings.
-Moreover, the average of the performance degradation is decreased when using a higher ratio for static power 
-(e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed algorithm optimizes the energy consumption 
-when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in 
-more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens 
-when using a higher ratio for  static  power, the algorithm proportionally  selects  smaller scaling values which 
-results in less energy saving but less performance degradation. 
-
-
- \begin{table}[htb]
-  \caption{The results of 70\%-30\% powers scenario}
+The NAS parallel benchmarks were executed again over processors that follow the
+new power scenarios.  The class C of each benchmark was run over 8 or 9 nodes
+and the results are presented in Tables~\ref{table:res_s1} and
+\ref{table:res_s2}. These tables show that the energy saving percentage of the
+\np[\%]{70}-\np[\%]{30} scenario is smaller for all benchmarks compared to the
+energy saving of the \np[\%]{90}-\np[\%]{10} scenario.  Indeed, in the latter
+more dynamic power is consumed when nodes are running on their maximum
+frequencies, thus, scaling down the frequency of the nodes results in higher
+energy savings than in the \np[\%]{70}-\np[\%]{30} scenario. On the other hand,
+the performance degradation percentage is smaller in the \np[\%]{70}-\np[\%]{30}
+scenario compared to the \np[\%]{90}-\np[\%]{10} scenario. This is due to the
+higher static power percentage in the first scenario which makes it more
+relevant in the overall consumed energy.  Indeed, the static energy is related
+to the execution time and if the performance is degraded the amount of consumed
+static energy directly increases.  Therefore, the proposed algorithm does not
+really significantly scale down much the frequencies of the nodes in order to
+limit the increase of the execution time and thus limiting the effect of the
+consumed static energy.
+
+Both new power scenarios are compared to the old one in
+Figure~\ref{fig:sen_comp}. It shows the average of the performance degradation,
+the energy saving and the distances for all NAS benchmarks of class C running on
+8 or 9 nodes.  The comparison shows that the energy saving ratio is proportional
+to the dynamic power ratio: it is increased when applying the
+\np[\%]{90}-\np[\%]{10} scenario because at maximum frequency the dynamic energy
+is the most relevant in the overall consumed energy and can be reduced by
+lowering the frequency of some processors. On the other hand, the energy saving
+decreases when the \np[\%]{70}-\np[\%]{30} scenario is used because the dynamic
+energy is less relevant in the overall consumed energy and lowering the
+frequency does not return big energy savings.  Moreover, the average of the
+performance degradation is decreased when using a higher ratio for static power
+(e.g.  \np[\%]{70}-\np[\%]{30} scenario and \np[\%]{80}-\np[\%]{20}
+scenario). Since the proposed algorithm optimizes the energy consumption when
+using a higher ratio for dynamic power the algorithm selects bigger frequency
+scaling factors that result in more energy saving but less performance, for
+example see Figure~\ref{fig:scales_comp}. The opposite happens when using a
+higher ratio for static power, the algorithm proportionally selects smaller
+scaling values which result in less energy saving but also less performance
+degradation.
+
+
+ \begin{table}[!t]
+  \caption{The results of the \np[\%]{70}-\np[\%]{30} power scenario}

   % title of Table
   \centering
   % title of Table
   \centering

-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}

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


@@ -1030,11 +1114,11 @@ results in less energy saving but less performance degradation.
 
 
 
 
 
 

-\begin{table}[htb]
-  \caption{The results of 90\%-10\% powers scenario}
+\begin{table}[!t]
+  \caption{The results of the \np[\%]{90}-\np[\%]{10} power scenario}

   % title of Table
   \centering
   % title of Table
   \centering

-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}

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


@@ -1057,82 +1141,106 @@ results in less energy saving but less performance degradation.
   \label{table:res_s2}
 \end{table}
 
   \label{table:res_s2}
 \end{table}
 

-
-\begin{figure}
-  \centering
-  \subfloat[Comparison  of the results on 8 nodes]{%
-    \includegraphics[width=.33\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
-
-  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
-    \includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
-  \label{fig:comp}
-  \caption{The comparison of the three power scenarios}
-\end{figure}  
-
-
-
-
-\subsection{The comparison of the proposed scaling algorithm }
-\label{sec.compare_EDP}
-In this section, the scaling  factors selection algorithm, called MaxDist,
-is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP. 
-They developed a green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Enegry*Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
-To fairly compare both algorithms, the same energy and execution time models, equations (\ref{eq:energy}) and  (\ref{eq:fnew}), were used for both algorithms to predict the energy consumption and the execution times. Also Spiliopoulos et al. algorithm was adapted to  start the search from the 
-initial frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm is an exhaustive search algorithm that minimizes the EDP and has the initial frequencies values as an upper bound.
-
-Both algorithms were applied to the parallel NAS benchmarks to compare their efficiency. Table \ref{table:compare_EDP}  presents the results of comparing the execution times and the energy consumptions for both versions of the NAS benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous nodes. The results show that our algorithm gives better energy savings than Spiliopoulos et al. algorithm, 
-on average it results in 29.76\% energy saving while their algorithm returns just 25.75\%. The average of performance degradation percentage is approximately the same for both algorithms, about 4\%. 
-
-
-For all benchmarks, our algorithm outperforms 
-Spiliopoulos et al. algorithm in term of energy and performance tradeoff, see figure (\ref{fig:compare_EDP}), because it maximizes the distance between the energy saving and the performance degradation values while giving the same weight for both metrics. 
-
-
-
-
-\begin{table}[h]
+\begin{table}[!t]

  \caption{Comparing the proposed algorithm}
  \centering
  \caption{Comparing the proposed algorithm}
  \centering

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

 \hline
 \hline

-\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Program \\ name\end{tabular}}} & \multicolumn{2}{l|}{Energy saving \%} & \multicolumn{2}{l|}{Perf.  degradation \%} & \multicolumn{2}{l|}{Distance} \\ \cline{3-8} 
-\multicolumn{2}{|l|}{}                                                                         & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
-\multicolumn{2}{|l|}{CG}                                                                       & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
-\multicolumn{2}{|l|}{MG}                                                                       & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
-\multicolumn{2}{|l|}{LU}                                                                       & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
-\multicolumn{2}{|l|}{EP}                                                                       & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
-\multicolumn{2}{|l|}{BT}                                                                       & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
-\multicolumn{2}{|l|}{SP}                                                                       & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
-\multicolumn{2}{|l|}{FT}                                                                       & 27.03           & 31.02            & 2.75           & 2.54              & 24.28        & 28.48           \\ \hline
+Program & \multicolumn{2}{c|}{Energy saving \%} & \multicolumn{2}{c|}{Perf.  degradation \%} & \multicolumn{2}{c|}{Distance} \\ \cline{2-7} 
+name    & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
+CG      & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
+MG      & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
+LU      & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
+EP      & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
+BT      & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
+SP      & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
+FT      & 27.03           & 31.02            & 2.75           & 2.54              & 24.28        & 28.48           \\ \hline

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

+\begin{figure}[!t]
+  \centering
+  \subfloat[Comparison  between the results on 8 nodes]{%
+    \includegraphics[width=.33\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%

 
 

+  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
+    \includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
+  \label{fig:comp}
+  \caption{The comparison of the three power scenarios}
+\end{figure}  

 
 

-
-
-\begin{figure}[t]
+\begin{figure}[!t]

   \centering
    \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
   \centering
    \includegraphics[scale=0.5]{fig/compare_EDP.pdf}

-  \caption{Tradeoff comparison for NAS benchmarks class C}
+  \caption{Trade-off comparison for NAS benchmarks class C}

   \label{fig:compare_EDP}
 \end{figure}
 
 
   \label{fig:compare_EDP}
 \end{figure}
 
 

+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
+In this section, the scaling factors selection algorithm, called MaxDist, is
+compared to Spiliopoulos et al. algorithm
+\cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP.  They developed a
+green governor that regularly applies an online frequency selecting algorithm to
+reduce the energy consumed by a multicore architecture without degrading much
+its performance. The algorithm selects the frequencies that minimize the energy
+and delay products, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$ using
+the predicted overall energy consumption and execution time delay for each
+frequency.  To fairly compare both algorithms, the same energy and execution
+time models, equations (\ref{eq:energy}) and (\ref{eq:fnew}), were used for both
+algorithms to predict the energy consumption and the execution times. Also
+Spiliopoulos et al. algorithm was adapted to start the search from the initial
+frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm
+is an exhaustive search algorithm that minimizes the EDP and has the initial
+frequencies values as an upper bound.
+
+Both algorithms were applied to the parallel NAS benchmarks to compare their
+efficiency. Table~\ref{table:compare_EDP} presents the results of comparing the
+execution times and the energy consumption for both versions of the NAS
+benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
+nodes. The results show that our algorithm provides better energy savings than
+Spiliopoulos et al. algorithm, on average it results in \np[\%]{29.76} energy
+saving while their algorithm returns just \np[\%]{25.75}. The average of
+performance degradation percentage is approximately the same for both
+algorithms, about \np[\%]{4}.
+
+
+For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
+terms of  energy and  performance trade-off, see  Figure~\ref{fig:compare_EDP},
+because it maximizes the distance  between the energy saving and the performance
+degradation values while giving the same weight for both metrics.
+
+

 \section{Conclusion}
 \label{sec.concl} 
 \section{Conclusion}
 \label{sec.concl} 

-In this paper, a new online frequency selecting algorithm has been presented. It selects the best possible vector of frequency scaling factors that gives the maximum distance (optimal tradeoff) between the predicted energy and 
-the predicted performance curves for a heterogeneous platform. This algorithm uses a new energy model for measuring  
-and predicting the energy of distributed iterative applications running over heterogeneous 
-platform. To evaluate the proposed method, it  was  applied on the NAS parallel benchmarks and executed over a heterogeneous platform simulated by  Simgrid. The results of the experiments showed that the algorithm reduces up to 35\% the energy consumption of a message passing iterative method while limiting the degradation of the performance. The algorithm also  selects different scaling factors   according to the percentage of the computing and communication times, and according to the values of  the static and  dynamic powers of the CPUs. Finally, the algorithm was compared to Spiliopoulos et al. algorithm and the results showed that it 
- outperforms their algorithm in term of energy-time tradeoff.
-
-In the near future, this method will be applied to real heterogeneous platforms to evaluate its performance in a real study case. It would also be interesting to evaluate its scalability over large scale heterogeneous platform and measure the energy consumption reduction it can produce. Afterward, we would like  to develop a similar method that is adapted to asynchronous  iterative applications 
-where each task does not wait for others tasks to finish their works. The development of such method might require a new 
-energy model because the number of iterations is not 
-known in advance and depends on the global convergence of the iterative system.
+In this paper, a new online frequency selecting algorithm has been presented. It
+selects the best possible vector of frequency scaling factors that gives the
+maximum distance (optimal trade-off) between the predicted energy and the
+predicted performance curves for a heterogeneous platform. This algorithm uses a
+new energy model for measuring and predicting the energy of distributed
+iterative applications running over heterogeneous platforms. To evaluate the
+proposed method, it was applied on the NAS parallel benchmarks and executed over
+a heterogeneous platform simulated by SimGrid. The results of the experiments
+showed that the algorithm reduces up to \np[\%]{35} the energy consumption of a
+message passing iterative method while limiting the degradation of the
+performance. The algorithm also selects different scaling factors according to
+the percentage of the computing and communication times, and according to the
+values of the static and dynamic powers of the CPUs.  Finally, the algorithm was
+compared to Spiliopoulos et al.  algorithm and the results showed that it
+outperforms their algorithm in terms of energy-time trade-off.
+
+In the near future, this method  will be applied to real heterogeneous platforms
+to evaluate its  performance in a real study case. It  would also be interesting
+to evaluate its scalability over large scale heterogeneous platforms and measure
+the energy  consumption reduction it can  produce.  Afterward, we  would like to
+develop a similar method that  is adapted to asynchronous iterative applications
+where  each task  does not  wait for  other tasks  to finish  their  works.  The
+development of such a method might require a new energy model because the number
+of iterations is  not known in advance and depends on  the global convergence of
+the iterative system.

 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}
 


@@ -1160,6 +1268,6 @@ Babylon (Iraq) for supporting his work.
 %%% End:
 
 % LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
 %%% End:
 
 % LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber

-% LocalWords:  CMOS EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
-% LocalWords:  de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
-% LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
+% LocalWords:  CMOS EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex GPU
+% LocalWords:  de badri muslim MPI SimGrid GFlops Xeon EP BT GPUs CPUs AMD
+%  LocalWords:  Spiliopoulos scalability