Add review from PDSEC 2015.

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index e8c58607d067ba5ea2a0ef85c64748523e4fa43d..a9a1e6da497ddbd13099d0c00c389ec2b6a52213 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{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 


@@ -24,36 +23,45 @@
 \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}}}}
+
+%% used to put some subscripts lower, and make them more legible
+\newcommand{\fxheight}[1]{\ifx#1\relax\relax\else\rule{0pt}{1.52ex}#1\fi}

 
 

-\newcommand{\Dist}{\textit{Dist}}
+\newcommand{\CL}{\Xsub{C}{L}}
+\newcommand{\Dist}{\mathit{Dist}}
+\newcommand{\EdNew}{\Xsub{E}{dNew}}

 \newcommand{\Eind}{\Xsub{E}{ind}}
 \newcommand{\Enorm}{\Xsub{E}{Norm}}
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
 \newcommand{\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{\Tnew}{\Xsub{T}{New}}

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

 
 

-\title{Energy Consumption Reduction In a Heterogeneous Architecture Using DVFS}
+\title{Energy Consumption Reduction with DVFS for \\
+  Message Passing Iterative Applications on \\
+  Heterogeneous Architectures}

 
 \author{%
   \IEEEauthorblockN{%
 
 \author{%
   \IEEEauthorblockN{%


@@ -61,10 +69,9 @@
     Raphaël Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch
     Raphaël Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch

-  } 
+  }

   \IEEEauthorblockA{%
   \IEEEauthorblockA{%

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

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


@@ -76,77 +83,223 @@
 \maketitle
 
 \begin{abstract}
 \maketitle
 
 \begin{abstract}

-  
+  Computing platforms are consuming more and more energy due to the increasing
+  number of nodes composing them.  To minimize the operating costs of these
+  platforms many techniques have been used. Dynamic voltage and frequency
+  scaling (DVFS) is one of them. It reduces the frequency of a CPU to lower its
+  energy consumption.  However, lowering the frequency of a CPU may increase
+  the execution time of an application running on that processor.  Therefore,
+  the frequency that gives the best trade-off between the energy consumption and
+  the performance of an application must be selected.
+
+  In this paper, a new online frequency selecting algorithm for heterogeneous
+  platforms is presented.  It selects the frequencies and tries to give the best
+  trade-off between energy saving and performance degradation, for each node
+  computing the message passing iterative application. The algorithm has a small
+  overhead and works without training or profiling. It uses a new energy model
+  for message passing iterative applications running on a heterogeneous
+  platform. The proposed algorithm is evaluated on the SimGrid simulator while
+  running the NAS parallel benchmarks.  The experiments show that it reduces the
+  energy consumption by up to \np[\%]{35} while limiting the performance
+  degradation as much as possible.  Finally, the algorithm is compared to an
+  existing method, the comparison results showing that it outperforms the
+  latter.
+

 \end{abstract}
 
 \section{Introduction}
 \label{sec.intro}
 \end{abstract}
 
 \section{Introduction}
 \label{sec.intro}

-Modern processors continue to increased in a performance, achieved  maximum number of floating point operations per second (FLOPS), thus the energy consumption and the heat dissipation are increased drastically  according to this increase. The number of FLOPS is linearly related to power consumption of a CPU~\cite{51}.  
-As an example of more power hungry cluster, according to the Top500 list in June 2014 \cite{43}, Tianhe-2 has more than 3 millions of cores and consumed more  than 17.8 megawatt per second. Moreover, according to the U.S. annual energy outlook 2014 \cite{60}, the price of energy for 1 megawatt per hour is approximately equal to 70\$ (1.16\$ for megawatt per second). Therefore, we can consider the price of the energy consumption for the Tianhe-2 platform is approximately more than 390 millions dollars of megawatt per year. For this reason, the heterogeneous clusters must be offer more energy efficiency due to the increase in the energy cost and the environment influences. Therefore, a green computing clusters with maximum number of FLOPS per watt are required nowadays.  For example, the GSIC center of Tokyo  heterogeneous cluster became the top of the Green500 list in June 2014 \cite{59}. This platform  has more than four thousand of  MFLOPS per watt. Dynamic voltage and frequency scaling (DVFS) is a process used widely to reduce the energy consumption of the processor. In a heterogeneous clusters enabled DVFS, many researchers used DVFS  in a different ways. DVFS can be minimized the energy consumption but it leads to a disadvantage due to increase in performance degradation. Therefore,  researchers used different optimization strategies to overcame this problem. The best tradeoff relation between the energy reduction and performance degradation ratio is became a key challenges in a heterogeneous platforms. In this paper we are propose a heterogeneous scaling algorithm  that selects the optimal vector of the frequency scaling factors for distributed iterative application, producing minimum energy saving against minimum performance degradation ratio simultaneously. The algorithm has very small overhead, works online and not needs for any training or profiling.  
+
+The need for more computing power is continually increasing. To partially
+satisfy this need, most supercomputers constructors just put more computing
+nodes in their platform. The resulting platforms may achieve higher floating
+point operations per second (FLOPS), but the energy consumption and the heat
+dissipation are also increased.  As an example, the Chinese supercomputer
+Tianhe-2 had the highest FLOPS in November 2014 according to the Top500 list
+\cite{TOP500_Supercomputers_Sites}.  However, it was also the most power hungry
+platform with its over 3 million cores consuming around 17.8 megawatts.
+Moreover, according to the U.S.  annual energy outlook 2014
+\cite{U.S_Annual.Energy.Outlook.2014}, the price of energy for 1 megawatt-hour
+was approximately equal to \$70.  Therefore, the price of the energy consumed by
+the Tianhe-2 platform is approximately more than \$10 million each year.  The
+computing platforms must be more energy efficient and offer the highest number
+of FLOPS per watt possible, such as the L-CSC from the GSI Helmholtz Center
+which became the top of the Green500 list in November 2014 \cite{Green500_List}.
+This heterogeneous platform executes more than 5 GFLOPS per watt while consuming
+57.15 kilowatts.
+
+Besides platform improvements, there are many software and hardware techniques
+to lower the energy consumption of these platforms, such as scheduling, DVFS,
+\dots{} DVFS is a widely used process to reduce the energy consumption of a
+processor by lowering its frequency
+\cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
+the number of FLOPS executed by the processor which may increase the execution
+time of the application running over that processor.  Therefore, researchers use
+different optimization strategies to select the frequency that gives the best
+trade-off between the energy reduction and performance degradation ratio. In
+\cite{Our_first_paper}, a frequency selecting algorithm was proposed to reduce
+the energy consumption of message passing iterative applications running over
+homogeneous platforms.  The results of the experiments show significant energy
+consumption reductions. In this paper, a new frequency selecting algorithm
+adapted for heterogeneous platform is presented. It selects the vector of
+frequencies, for a heterogeneous platform running a message passing iterative
+application, that simultaneously tries to offer the maximum energy reduction and
+minimum performance degradation ratio. The algorithm has a very small overhead,
+works online and does not need any training or profiling.

 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some
 related works from other authors.  Section~\ref{sec.exe} describes how the
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some
 related works from other authors.  Section~\ref{sec.exe} describes how the

-execution time of MPI programs can be predicted.  It also presents an energy
-model for heterogeneous platforms. Section~\ref{sec.compet} presents
-the energy-performance objective function that maximizes the reduction of energy
+execution time of message passing programs can be predicted.  It also presents
+an energy model that predicts the energy consumption of an application running
+over a heterogeneous platform. Section~\ref{sec.compet} presents the
+energy-performance objective function that maximizes the reduction of energy

 consumption while minimizing the degradation of the program's performance.
 consumption while minimizing the degradation of the program's performance.

-Section~\ref{sec.optim} details the proposed heterogeneous scaling algorithm.
-Section~\ref{sec.expe} presents the results of running  the NAS benchmarks on 
-the proposed heterogeneous platform. It also shows the comparison of three different power
-scenarios and it verifies the precision of the proposed algorithm.  Finally, we conclude 
-in Section~\ref{sec.concl} with a summary and some future works.
+Section~\ref{sec.optim} details the proposed frequency selecting algorithm then
+the precision of the proposed algorithm is verified.  Section~\ref{sec.expe}
+presents the results of applying the algorithm on the NAS parallel benchmarks
+and executing them on a heterogeneous platform. It shows the results of running
+three different power scenarios and comparing them. Moreover, it also shows the
+comparison results between the proposed method and an existing method.  Finally,
+in Section~\ref{sec.concl} the paper ends with a summary and some future works.

 
 \section{Related works}
 \label{sec.relwork}
 
 \section{Related works}
 \label{sec.relwork}

-Energy reduction process for a high performance clusters recently performed using dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique  enabled in a modern processors to scaled down both of the  voltage and the frequency of the CPU while it is in the computing mode to reduce the energy consumption. DVFS is also  allowed in the graphical processors GPUs, to achieved the same goal. Applying DVFS has a dramatical side effect if it is applied to minimum levels to gain more energy reduction, producing a high percentage of performance degradations for the parallel applications.  Many researchers used different strategies to solve this nonlinear problem for example in~\cite{19,42}, their methods add big overheads to the algorithm to select the
-suitable frequency.  In this paper we  present a method to find the optimal
-set of frequency scaling factors for a heterogeneous cluster to simultaneously optimize both the energy and the execution time  without adding a big overhead.
-This work is developed from our previous work of a homogeneous cluster~\cite{45}. Therefore we are interested to present some works that concerned the heterogeneous clusters enabled DVFS. In general, the heterogeneous cluster works fall into two categorizes: GPUs-CPUs heterogeneous clusters and CPUs-CPUs heterogeneous clusters. In GPUs-CPUs heterogeneous clusters some parallel tasks executed on a GPUs and the others executed on a CPUs. As an example of this works, Luley et al.~\cite{51}, proposed  a heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal is to determined the energy efficiency as a function of performance per watt, the best tradeoff is done when the performance per watt function is maximized. In the work of Kia Ma et al.~\cite{49}, They developed a scheduling algorithm to distributed different workloads proportional to the computing power of the node to be executed on a CPU or a GPU, emphasize all tasks must be finished in the same time. 
-Recently, Rong et al.~\cite{50}, Their study explain that a heterogeneous clusters enabled DVFS using GPUs and CPUs gave better energy and performance efficiency 
-than other clusters composed of only CPUs. The CPUs-CPUs heterogeneous clusters consist of number of computing nodes  all of the type CPU. Our work in this paper can be classified to this type of the clusters. As an example of this works see  Naveen et al.~\cite{52} work, They developed a policy to dynamically assigned the frequency to a heterogeneous cluster. The goal is to minimizing a fixed metric of $energy*delay^2$. Where our proposed method is automatically optimized  the relation between the energy and the delay of the iterative applications. Other works such as Lizhe et al.~\cite{53}, their algorithm divided the executed tasks into two types: the critical and non critical tasks. The algorithm scaled down the frequency of the non critical tasks as function to the  amount of the slack and communication times that have with maximum of performance degradation percentage of 10\%. In our method there is no fixed bounds for performance degradation percentage and the bound is dynamically computed according to the energy and the performance tradeoff relation of the executed application. 
-There are some approaches used a heterogeneous cluster composed from two different types of Intel and AMD processors such as~\cite{54} and \cite{55}, they predicated  both the energy and the performance for each frequency gear, then the algorithm selected the best gear that gave the best tradeoff. In contrast our algorithm works over a heterogeneous  platform composed of four different types of processors. Others approaches such as \cite{56} and \cite{57}, they are selected the best frequencies for a specified heterogeneous clusters offline using some heuristic methods. While our proposed algorithm works online during the execution time of iterative application. Greedy dynamic approach used by Chen et al.~\cite{58},  minimized the power consumption of a heterogeneous severs  with time/space complexity, this approach had considerable overhead. In our proposed scaling algorithm has very small overhead and it is works without any previous analysis for the application time complexity. 
+
+DVFS is a technique used in modern processors to scale down both the voltage and
+the frequency of the CPU while computing, in order to reduce the energy
+consumption of the processor. DVFS is also allowed in GPUs to achieve the same
+goal. Reducing the frequency of a processor lowers its number of FLOPS and may
+degrade the performance of the application running on that processor, especially
+if it is compute bound. Therefore selecting the appropriate frequency for a
+processor to satisfy some objectives while taking into account all the
+constraints, is not a trivial operation.  Many researchers used different
+strategies to tackle this problem. Some of them developed online methods that
+compute the new frequency while executing the application, such
+as~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}.
+Others used offline methods that may need to run the application and profile
+it before selecting the new frequency, such
+as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
+The methods could be heuristics, exact or brute force methods that satisfy
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+sequential, parallel or distributed architecture, homogeneous or heterogeneous
+platform, synchronous or asynchronous application, \dots{}
+
+In this paper, we are interested in reducing energy for message passing
+iterative synchronous applications running over heterogeneous platforms.  Some
+works have already been done for such platforms and they can be classified into
+two types of heterogeneous platforms:
+\begin{itemize}
+\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
+\item the platform is only composed of heterogeneous CPUs.
+\end{itemize}
+
+For the first type of platform, the computing intensive parallel tasks are
+executed on the GPUs and the rest are executed on the CPUs.  Luley et
+al.~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a
+heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
+goal was to maximize the energy efficiency of the platform during computation by
+maximizing the number of FLOPS per watt generated.
+In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
+al. developed a scheduling algorithm that distributes workloads proportional to
+the computing power of the nodes which could be a GPU or a CPU. All the tasks
+must be completed at the same time.  In~\cite{Rong_Effects.of.DVFS.on.K20.GPU},
+Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
+DVFS gave better energy and performance efficiency than other clusters only
+composed of CPUs.
+
+The work presented in this paper concerns the second type of platform, with
+heterogeneous CPUs.  Many methods were conceived to reduce the energy
+consumption of this type of platform.  Naveen et
+al.~\cite{Naveen_Power.Efficient.Resource.Scaling} developed a method that
+minimizes the value of $\mathit{energy}\times \mathit{delay}^2$ (the delay is
+the sum of slack times that happen during synchronous communications) by
+dynamically assigning new frequencies to the CPUs of the heterogeneous cluster.
+Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed an
+algorithm that divides the executed tasks into two types: the critical and non
+critical tasks. The algorithm scales down the frequency of non critical tasks
+proportionally to their slack and communication times while limiting the
+performance degradation percentage to less than \np[\%]{10}.
+In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
+heterogeneous cluster composed of two types of Intel and AMD processors. They
+use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
+\cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
+frequencies for a specified heterogeneous cluster are selected offline using
+some heuristic.  Chen et
+al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
+programming approach to minimize the power consumption of heterogeneous servers
+while respecting given time constraints.  This approach had considerable
+overhead.  In contrast to the above described papers, this paper presents the
+following contributions :
+\begin{enumerate}
+\item two new energy and performance models for message passing iterative
+  synchronous applications running over a heterogeneous platform. Both models
+  take into account communication and slack times. The models can predict the
+  required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous
+  platforms. The algorithm has a very small overhead and does not need any
+  training or profiling. It uses a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing iterative synchronous application.
+
+\end{enumerate}

 
 \section{The performance and energy consumption measurements on heterogeneous architecture}
 \label{sec.exe}
 
 
 \section{The performance and energy consumption measurements on heterogeneous architecture}
 \label{sec.exe}
 

-% \JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'',
-%   can be deleted if we need space, we can just say we are interested in this
-%   paper in homogeneous clusters}
-
-\subsection{The execution time of message passing distributed iterative applications on a heterogeneous platform}
+\subsection{The execution time of message passing distributed iterative
+  applications on a heterogeneous platform}

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

-heterogeneous platforms. We define a heterogeneous platform as a collection of
+heterogeneous platforms. A heterogeneous platform is defined as a collection of

 heterogeneous computing nodes interconnected via a high speed homogeneous
 network. Therefore, each node has different characteristics such as computing
 power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
 have the same network bandwidth and latency.
 
 heterogeneous computing nodes interconnected via a high speed homogeneous
 network. Therefore, each node has different characteristics such as computing
 power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
 have the same network bandwidth and latency.
 

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

   \centering
   \centering

-    \includegraphics[scale=0.6]{fig/commtasks}
+  \includegraphics[scale=0.6]{fig/commtasks}

   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 

-Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
-of a CPU by scaling down its voltage and frequency.  Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound.  The frequency reduction process can be  expressed by the scaling factor S which is the ratio between  the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
+The overall execution time of a distributed iterative synchronous application
+over a heterogeneous platform consists of the sum of the computation time and
+the communication time for every iteration on a node. However, due to the
+heterogeneous computation power of the computing nodes, slack times may occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
+overall execution time of the program is the execution time of the slowest task
+which has the highest computation time and no slack time.
+
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
+modern processors, that reduces the energy consumption of a CPU by scaling
+down its voltage and frequency.  Since DVFS lowers the frequency of a CPU
+and consequently its computing power, the execution time of a program running
+over that scaled down processor may increase, especially if the program is
+compute bound.  The frequency reduction process can be  expressed by the scaling
+factor S which is the ratio between  the maximum and the new frequency of a CPU
+as in (\ref{eq:s}).

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

- S = \frac{F_\textit{max}}{F_\textit{new}}
+  S = \frac{\Fmax}{\Fnew}

 \end{equation}
 \end{equation}

- The execution time of a compute bound sequential program is linearly proportional to the frequency scaling factor $S$. 
- On the other hand,  message passing distributed applications consist of two parts: computation and communication. The execution time of the computation part is linearly proportional to the frequency scaling factor $S$ but  the communication time is not affected by the scaling factor because  the processors involved remain idle during the  communications~\cite{17}. The communication time for a task is the summation of periods of time that begin with an MPI call for sending or receiving   a message till the message is synchronously sent or received.
-
-Since in a heterogeneous platform, each node has different characteristics,
+The execution time of a compute bound sequential program is linearly
+proportional to the frequency scaling factor $S$.  On the other hand, message
+passing distributed applications consist of two parts: computation and
+communication.  The execution time of the computation part is linearly
+proportional to the frequency scaling factor $S$ but the communication time is
+not affected by the scaling factor because the processors involved remain idle
+during the communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  The
+communication time for a task is the summation of periods of time that begin
+with an MPI call for sending or receiving a message until the message is
+synchronously sent or received.
+
+Since in a heterogeneous platform each node has different characteristics,

 especially different frequency gears, when applying DVFS operations on these
 nodes, they may get different scaling factors represented by a scaling vector:
 $(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
 especially different frequency gears, when applying DVFS operations on these
 nodes, they may get different scaling factors represented by a scaling vector:
 $(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To


@@ -155,259 +308,293 @@ applications running over a heterogeneous platform, for different vectors of
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any

-vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
+vector of scaling factors can be predicted using (\ref{eq:perf}).

 \begin{equation}
   \label{eq:perf}
 \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 $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $MinTcm$ is the communication time of the slowest processor from the first iteration.  The model computes the maximum computation time 
- with scaling factor from each node  added to the communication time of the slowest node, it means  only the
- communication time without any slack time. Therefore, we can consider the execution time of the iterative application is equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied by the number of iterations of that application.
-
-This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. The execution time prediction model is used in our method for optimizing both energy consumption and performance of iterative methods, which is presented in the following sections.
-
+Where:
+\begin{equation}
+  \label{eq:perf2}
+  \MinTcm = \min_{i=1,2,\dots,N} (\Tcm[i])
+\end{equation}
+where $\TcpOld[i]$ is the computation time of processor $i$ during the first
+iteration and $\MinTcm$ is the communication time of the slowest processor from
+the first iteration.  The model computes the maximum computation time with
+scaling factor from each node added to the communication time of the slowest
+node. It means only the communication time without any slack time is taken into
+account.  Therefore, the execution time of the iterative application is equal to
+the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
+number of iterations of that application.
+
+This prediction model is developed from the model to predict the execution time
+of message passing distributed applications for homogeneous
+architectures~\cite{Our_first_paper}.  The execution time prediction model is
+used in the method to optimize both the energy consumption and the performance
+of iterative methods, which is presented in the following sections.

 
 \subsection{Energy model for heterogeneous platform}
 
 \subsection{Energy model for heterogeneous platform}

-Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
-two power metrics: the static and the dynamic power.  While the first one is
-consumed as long as the computing unit is turned on, the latter is only consumed during
-computation times.  The dynamic power $P_{d}$ is related to the switching
-activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
-operational frequency $F$, as shown in EQ(\ref{eq:pd}).
+
+Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
+  Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
+  Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
+a processor into two power metrics: the static and the dynamic power.  While the
+first one is consumed as long as the computing unit is turned on, the latter is
+only consumed during computation times.  The dynamic power $\Pd$ is related to
+the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
+and operational frequency $F$, as shown in (\ref{eq:pd}).

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

-  P_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F
+  \Pd = \alpha \cdot \CL \cdot V^2 \cdot F

 \end{equation}
 \end{equation}

-The static power $P_{s}$ captures the leakage power as follows:
+The static power $\Ps$ captures the leakage power as follows:

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

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

 \end{equation}
 \end{equation}

-where V is the supply voltage, $N_{trans}$ is the number of transistors,
-$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
-technology-dependent parameter.  The energy consumed by an individual processor
+where V is the supply voltage, $\Ntrans$ is the number of transistors,
+$\Kdesign$ is a design dependent parameter and $\Ileak$ is a
+technology dependent parameter.  The energy consumed by an individual processor

 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}

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

 \end{equation}
 \end{equation}

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

 communication and no slack time.
 
 communication and no slack time.
 

-The main objective of DVFS operation is to
-reduce the overall energy consumption~\cite{37}.  The operational frequency $F$
-depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot F$ with some
-constant $\beta$.  This equation is used to study the change of the dynamic
-voltage with respect to various frequency values in~\cite{3}.  The reduction
-process of the frequency can be expressed by the scaling factor $S$ which is the
-ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
-The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. we can calculate the new frequency 
-$F_{new}$ from EQ(\ref{eq:s}) as follow:
+The main objective of DVFS operation is to reduce the overall energy
+consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  The operational
+frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot
+F$ with some constant $\beta$.~This equation is used to study the change of the
+dynamic voltage with respect to various frequency values
+in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction process of the
+frequency can be expressed by the scaling factor $S$ which is the ratio between
+the maximum and the new frequency as in (\ref{eq:s}).  The CPU governors are
+power schemes supplied by the operating system's kernel to lower a core's
+frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
+follows:

 \begin{equation}
   \label{eq:fnew}
 \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 EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic 
-power consumption:
+Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
+equation for dynamic power consumption:

 \begin{multline}
   \label{eq:pdnew}
 \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 new frequency and the maximum frequency respectively.
-
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 
-reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$.  The new dynamic energy is the  dynamic power multiplied by the new time of computation and is given by the following equation:
+where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
+new frequency and the maximum frequency respectively.
+
+According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
+$S^{-3}$ when reducing the frequency by a factor of
+$S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is
+proportional to the frequency of a CPU, the computation time is increased
+proportionally to $S$.  The new dynamic energy is the dynamic power multiplied
+by the new time of computation and is given by the following equation:

 \begin{equation}
   \label{eq:Edyn}
 \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}
 \end{equation}

-The static power is related to the power leakage of the CPU and is consumed during computation and even when idle. As in~\cite{3,46}, we assume that the static power of a processor is constant during idle and computation periods, and for all its available frequencies. 
-The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}), 
-the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related  
-to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy of a processor after scaling its frequency is computed as follows: 
+The static power is related to the power leakage of the CPU and is consumed
+during computation and even when idle. As
+in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
+the static power of a processor is considered as constant during idle and
+computation periods, and for all its available frequencies.  The static energy
+is the static power multiplied by the execution time of the program.  According
+to the execution time model in (\ref{eq:perf}), the execution time of the
+program is the sum of the computation and the communication times. The
+computation time is linearly related to the frequency scaling factor, while this
+scaling factor does not affect the communication time.  The static energy of a
+processor after scaling its frequency is computed as follows:

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

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

 \end{equation}
 
 \end{equation}
 

-In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $Tcp_{i}$ might be different and different frequency  scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $Tcm_{i}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the frequency scaling factor and the dynamic power of each node as in EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing  distributed application executed over a heterogeneous platform during one iteration is the summation of all dynamic and static energies for each  processor.  It is computed as follows:
+In the considered heterogeneous platform, each processor $i$ may have
+different dynamic and static powers, noted as $\Pd[i]$ and $\Ps[i]$
+respectively.  Therefore, even if the distributed message passing iterative
+application is load balanced, the computation time of each CPU $i$ noted
+$\Tcp[i]$ may be different and different frequency scaling factors may be
+computed in order to decrease the overall energy consumption of the application
+and reduce slack times.  The communication time of a processor $i$ is noted as
+$\Tcm[i]$ and could contain slack times when communicating with slower nodes,
+see Figure~\ref{fig:heter}.  Therefore, all nodes do not have equal
+communication times. While the dynamic energy is computed according to the
+frequency scaling factor and the dynamic power of each node as in
+(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
+of one iteration multiplied by the static power of each processor.  The overall
+energy consumption of a message passing distributed application executed over a
+heterogeneous platform during one iteration is the summation of all dynamic and
+static energies for each processor.  It is computed as follows:

 \begin{multline}
   \label{eq:energy}
 \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))}
- \end{multline}
-
-Reducing the frequencies of the processors according to the vector of
-scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
-application and thus, increase the static energy because the execution time is
-increased~\cite{36}. We can measure the overall energy consumption for the iterative 
-application by measuring  the energy consumption for one iteration as in EQ(\ref{eq:energy}) multiplied by 
-the number of iterations of that application.
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + {} \\
+ \sum_{i=1}^{N} (\Ps[i] \cdot (\max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +
+  {\MinTcm))}
+\end{multline}

 
 

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

 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 
 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 

-Using the lowest frequency for each processor does not necessarily gives the most energy efficient execution of an application. Indeed, even though the dynamic power is reduced while scaling down the frequency of a processor, its computation power is proportionally decreased and thus the execution time might be drastically increased during which dynamic and static powers are being consumed. Therefore,  it might cancel any gains achieved by scaling down the frequency of all nodes to the minimum  and the overall energy consumption of the application might not be the optimal one. It is not trivial to select the appropriate frequency scaling factor for each processor while considering the characteristics of each processor (computation power, range of frequencies, dynamic and static powers) and the task executed (computation/communication ratio) in order to reduce the overall energy consumption and not significantly increase the execution time. In our previous work~\cite{45}, we  proposed a method that selects the optimal 
-frequency scaling factor for a homogeneous cluster executing a message passing iterative synchronous application while giving the best trade-off
- between the energy consumption and the performance for such applications. In this work we are interested in 
-heterogeneous clusters as described above. Due to the heterogeneity of the processors, not one but a  vector of scaling factors should be selected and it must  give the best trade-off between energy consumption and performance. 
-
-The relation between the energy consumption and the execution
-time for an application is complex and nonlinear, Thus, unlike the relation between the execution time 
-and the scaling factor, the relation of the energy with the frequency scaling
-factors is nonlinear, for more details refer to~\cite{17}.  Moreover, they are
-not measured using the same metric.  To solve this problem, we normalize the
-execution time by computing the ratio between the new execution time (after scaling down the frequencies of some processors) and the initial one (with maximum frequency for all nodes,) as follows:
+Using the lowest frequency for each processor does not necessarily give the most
+energy efficient execution of an application. Indeed, even though the dynamic
+power is reduced while scaling down the frequency of a processor, its
+computation power is proportionally decreased. Hence, the execution time might
+be drastically increased and during that time, dynamic and static powers are
+being consumed.  Therefore, it might cancel any gains achieved by scaling down
+the frequency of all nodes to the minimum and the overall energy consumption of
+the application might not be the optimal one.  It is not trivial to select the
+appropriate frequency scaling factor for each processor while considering the
+characteristics of each processor (computation power, range of frequencies,
+dynamic and static powers) and the task executed (computation/communication
+ratio). The aim being to reduce the overall energy consumption and to avoid
+increasing significantly the execution time.  In our previous
+work~\cite{Our_first_paper}, we proposed a method that selects the optimal
+frequency scaling factor for a homogeneous cluster executing a message passing
+iterative synchronous application while giving the best trade-off between the
+energy consumption and the performance for such applications.  In this work we
+are interested in heterogeneous clusters as described above.  Due to the
+heterogeneity of the processors, a vector of scaling factors should be selected
+and it must give the best trade-off between energy consumption and performance.
+
+The relation between the energy consumption and the execution time for an
+application is complex and nonlinear, Thus, unlike the relation between the
+execution time and the scaling factor, the relation between the energy and the
+frequency scaling factors is nonlinear, for more details refer
+to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  Moreover, these relations
+are not measured using the same metric.  To solve this problem, the execution
+time is normalized by computing the ratio between the new execution time (after
+scaling down the frequencies of some processors) and the initial one (with
+maximum frequency for all nodes) as follows:

 \begin{multline}
   \label{eq:pnorm}
 \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}
 

-
-In the same way, we normalize the energy by computing the ratio between the consumed energy while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
+In the same way, the energy is normalized by computing the ratio between the
+consumed energy while scaling down the frequency and the consumed energy with
+maximum frequency for all nodes:

 \begin{multline}
   \label{eq:enorm}
 \begin{multline}
   \label{eq:enorm}

-  E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
-  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
-\end{multline} 
-Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
-
- While the main 
-goal is to optimize the energy and execution time at the same time, the normalized energy and execution time curves are not in the same direction. According 
-to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the vector  of frequency
-scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
-time simultaneously.  But the main objective is to produce maximum energy
-reduction with minimum execution time reduction.  
-
- 
-  
-Our solution for this problem is to make the optimization process for energy and execution time follow the same
-direction.  Therefore, we inverse the equation of the normalized execution time which gives 
-the normalized performance equation, as follows:
+  \Enorm = \frac{\Ereduced}{\Eoriginal} \\
+  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}
+\end{multline}
+Where $\Ereduced$ and $\Eoriginal$ are computed using (\ref{eq:energy}) and
+$\Tnew$ and $\Told$ are computed as in (\ref{eq:pnorm}).
+
+While the main goal is to optimize the energy and execution time at the same
+time, the normalized energy and execution time curves are not in the same
+direction. According to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
+vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy
+and the execution time simultaneously.  But the main objective is to produce
+maximum energy reduction with minimum execution time reduction.
+
+This problem can be solved by making the optimization process for energy and
+execution time following the same direction.  Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance
+equation, as follows:

 \begin{multline}
   \label{eq:pnorm_inv}
 \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]{%
   \centering
   \subfloat[Homogeneous platform]{%

-    \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
-  \qquad%
+    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+

   \subfloat[Heterogeneous platform]{%
   \subfloat[Heterogeneous platform]{%

-    \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
+    \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}

   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}
 
   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}
 

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

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

-where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
-Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).  Our objective function can
-work with any energy model or any power values for each node (static and dynamic powers).
-However, the most energy reduction gain can be achieved when the energy curve has a convex form as shown in~\cite{15,3,19}.
+where $N$ is the number of nodes and $F$ is the number of available frequencies
+for each node.  Then, the optimal set of scaling factors that satisfies
+(\ref{eq:max}) can be selected.  The objective function can work with any energy
+model or any power values for each node (static and dynamic powers). However,
+the most important energy reduction gain can be achieved when the energy curve
+has a convex form as shown
+in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.

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

-In this section we  propose algorithm~(\ref{HSA}) which selects the frequency scaling factors vector that gives the best trade-off between minimizing the energy consumption  and maximizing the performance of a message passing synchronous iterative application executed on a heterogeneous platform.  
-It works online during the execution time of the iterative message passing program.  It uses information gathered during the first iteration such as the computation time and the communication time in one iteration for each node. The algorithm is executed  after the first iteration and returns a vector of optimal frequency scaling factors   that satisfies the objective function EQ(\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs 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.
-Our algorithm takes into account this problem and tries to reduce these slack times when selecting the frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase the execution times of fast nodes and  minimize the  differences between  the  computation times of fast and slow nodes. The value of the initial frequency scaling factor  for each node is inversely proportional to its computation time that was gathered from the first iteration. These initial frequency scaling factors are computed as   a ratio between the computation time of the slowest node and the computation time of the node $i$ as follows:
-\begin{equation}
-  \label{eq:Scp}
- Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
-\end{equation}
-Using the initial  frequency scaling factors computed in EQ(\ref{eq:Scp}), the algorithm computes the initial frequencies for all nodes as a ratio between the 
-maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as follows:
-\begin{equation}
-  \label{eq:Fint}
- F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
-\end{equation}
-If the computed initial frequency for a node is not available in the gears of that node, the computed initial frequency is replaced by the nearest available frequency.
-In  figure (\ref{fig:st_freq}), the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are colored in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal vector of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the algorithm that selects the frequency scaling factors starts the search method from these initial frequencies and takes a downward search direction toward lower frequencies. The algorithm iterates on all left frequencies, from the higher bound until all nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node according to EQ(\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of  all other nodes by one gear.
-The new overall energy consumption and execution time are computed according to the new scaling factors. The optimal set of frequency scaling factors is the set that gives the highest distance according to  the objective function EQ(\ref{eq:max}).
-
-The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an application running on a homogeneous platform and a heterogeneous platform respectively while increasing the scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor should be started from the maximum frequency because the performance and the consumed energy is decreased since  the beginning of the plot. On the other hand, in  the heterogeneous platform the performance is  maintained at the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes have computing powers lower than the slowest node. In other words, until they reach the higher bound. It can also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger the maximum distance between the energy curve and the performance curve is while varying the scaling factors which results in bigger energy savings. 
-\begin{figure}[t]
-  \centering
-    \includegraphics[scale=0.5]{fig/start_freq}
-  \caption{Selecting the initial frequencies}
-  \label{fig:st_freq}
-\end{figure}
-
-
-
-
-

 \begin{algorithm}
   \begin{algorithmic}[1]
     % \footnotesize
     \Require ~
     \begin{description}
 \begin{algorithm}
   \begin{algorithmic}[1]
     % \footnotesize
     \Require ~
     \begin{description}

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

     \end{description}
     \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 $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
-    \EndIf 
-    \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
-    \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
-    \State $Dist \gets 0$
-    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
+    \EndIf
+    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
+    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}$
+    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \Told)}$
+    \State $\Sopt[i] \gets 1,~i=1,\dots,N. $
+    \State $\Dist \gets 0 $

     \While {(all nodes not reach their  minimum  frequency)}
         \If{(not the last freq. \textbf{and} not the slowest node)}
     \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}
   \end{algorithmic}

-  \caption{Heterogeneous scaling algorithm}
+  \caption{frequency scaling factors selection algorithm}

   \label{HSA}
 \end{algorithm}
 
   \label{HSA}
 \end{algorithm}
 


@@ -420,7 +607,7 @@ The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance
       \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 from Figure~\ref{HSA} with these times.
+        \State Call Algorithm \ref{HSA}.

         \State Compute the new frequencies from the\newline\hspace*{3em}%
                returned optimal scaling factors.
         \State Set the new frequencies to nodes.
         \State Compute the new frequencies from the\newline\hspace*{3em}%
                returned optimal scaling factors.
         \State Set the new frequencies to nodes.


@@ -431,341 +618,641 @@ The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance
   \label{dvfs}
 \end{algorithm}
 
   \label{dvfs}
 \end{algorithm}
 

-\section{Experimental results}
-\label{sec.expe}
-To evaluate the efficiency and the overall energy consumption reduction of algorithm~(\ref{HSA}), it was applied to the NAS parallel benchmarks NPB v3.3 
-\cite{44}. The experiments were executed on the simulator SimGrid/SMPI
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile} which offers easy tools to create a heterogeneous platform and run message passing applications over it. The heterogeneous platform that was used in the experiments, had one core per node because just one process was executed per node. The heterogeneous platform  was composed of four types of nodes. Each type of nodes had different characteristics such as the maximum CPU frequency, the number of
-available frequencies and the computational power, see table
-(\ref{table:platform}). The characteristics of these different types of  nodes are inspired   from the specifications of real Intel processors. The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, for example if  a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were chosen proportionally to  its computing power (FLOPS).  In the initial heterogeneous platform,  while computing with highest frequency, each node  consumed power proportional to its computing power which 80\% of it was dynamic power and the rest was 20\% for the static power, the same assumption  was made in \cite{45,3}. Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
-
+\subsection{The algorithm details}
+
+In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
+scaling factors vector that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+synchronous iterative application executed on a heterogeneous platform. It works
+online during the execution time of the iterative message passing program.  It
+uses information gathered during the first iteration such as the computation
+time and the communication time in one iteration for each node. The algorithm is
+executed after the first iteration and returns a vector of optimal frequency
+scaling factors that satisfies the objective function (\ref{eq:max}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors.  This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
+scaling algorithm is called in the iterative MPI program.
+
+\begin{figure}[!t]
+  \centering
+  \includegraphics[scale=0.5]{fig/start_freq}
+  \caption{Selecting the initial frequencies}
+  \label{fig:st_freq}
+\end{figure}

 
 

-\begin{table}[htb]
+The nodes in a heterogeneous platform have different computing powers, thus
+while executing message passing iterative synchronous applications, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
+periods are called idle or slack times.  The algorithm takes into account this
+problem and tries to reduce these slack times when selecting the frequency
+scaling factors vector. At first, it selects initial frequency scaling factors
+that increase the execution times of fast nodes and minimize the differences
+between the computation times of fast and slow nodes. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
+\begin{equation}
+  \label{eq:Scp}
+  \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}
+\end{equation}
+Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
+algorithm computes the initial frequencies for all nodes as a ratio between the
+maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+follows:
+\begin{equation}
+  \label{eq:Fint}
+  F_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}
+\end{equation}
+If the computed initial frequency for a node is not available in the gears of
+that node, it is replaced by the nearest available frequency.  In
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors.  The resulting new
+frequencies are highlighted in Figure~\ref{fig:st_freq}.  This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting 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.
+
+\subsection{The evaluation of the proposed algorithm}
+\label{sec.verif.algo}
+
+The precision of the proposed algorithm mainly depends on the execution time
+prediction model defined in (\ref{eq:perf}) and the energy model computed by
+(\ref{eq:energy}).  The energy model is also significantly dependent on the
+execution time model because the static energy is linearly related to the
+execution time and the dynamic energy is related to the computation time. So,
+all the works presented in this paper are based on the execution time model. To
+verify this model, the predicted execution time was compared to the real
+execution time over SimGrid/SMPI simulator,
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}, for all the NAS
+parallel benchmarks NPB v3.3 \cite{NAS.Parallel.Benchmarks}, running class B on
+8 or 9 nodes. The comparison showed that the proposed execution time model is
+very precise, the maximum normalized difference between the predicted execution
+time and the real execution time is equal to 0.03 for all the NAS benchmarks.
+
+Since the proposed algorithm is not an exact method it does not test all the
+possible solutions (vectors of scaling factors) in the search space. To prove
+its efficiency, it was compared on small instances to a brute force search
+algorithm that tests all the possible solutions. The brute force algorithm was
+applied to different NAS benchmarks classes with different number of nodes. The
+solutions returned by the brute force algorithm and the proposed algorithm were
+identical and the proposed algorithm was on average 10 times faster than the
+brute force algorithm. It has a small execution time: for a heterogeneous
+cluster composed of four different types of nodes having the characteristics
+presented in Table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4
+nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling
+factors vector.  The algorithm complexity is $O(F\cdot N)$, where $F$ is the
+maximum number of available frequencies, and $N$ is the number of computing
+nodes. The algorithm needs from 12 to 20 iterations to select the best vector of
+frequency scaling factors that gives the results of the next sections.
+
+\begin{table}[!t]

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

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

     \hline
     \hline

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

     \hline
     \hline

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

     \hline
     \hline

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

     \hline
     \hline

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

     \hline
     \hline

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

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

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

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

+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
+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.

 
 

-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.
-
- 
- 
-\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

-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

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

     \hline
     \hline

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

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

-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

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

     \hline
     \hline

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

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

-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

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

     \hline
     \hline

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

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

-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

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

     \hline
     \hline

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

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

-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

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

     \hline
     \hline

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

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

-    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
+    \hspace{-2.2084pt}%
+    Program & Execution & Energy        & Energy   & Performance   & Distance \\
+    name    & time/s    & consumption/J & saving\% & degradation\% &          \\

     \hline
     \hline

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

     \hline
     \hline

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

   \end{tabular}
   \label{table:res_128n}
 \end{table}
   \end{tabular}
   \label{table:res_128n}
 \end{table}

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

 
 

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

   \centering
   \centering

-  \subfloat[Energy saving]{%
-    \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%
-  \quad%
-  \subfloat[Performance degradation ]{%
-    \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}
+  \subfloat[Energy saving (\%)]{%
+    \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
+
+  \subfloat[Performance degradation (\%)]{%
+    \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}

   \label{fig:avg}
   \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 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 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. 
-
-
-
+The overall energy consumption was computed for each instance according to the
+energy consumption model (\ref{eq:energy}), with and without applying the
+algorithm. The execution time was also measured for all these experiments. Then,
+the energy saving and performance degradation percentages were computed for each
+instance.  The results are presented in Tables~\ref{table:res_4n},
+\ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n},
+\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
+values from many experiments for energy savings and performance degradation.
+The tables show the experimental results for running the NAS parallel benchmarks
+on different number of nodes.  The experiments show that the algorithm
+significantly reduces the energy consumption (up to \np[\%]{35}) and tries to
+limit the performance degradation.  They also show that the energy saving
+percentage decreases when the number of computing nodes increases.  This
+reduction is due to the increase of the communication times compared to the
+execution times when the benchmarks are run over a higher number of nodes.
+Indeed, the benchmarks with the same class, C, are executed on different numbers
+of nodes, so the computation required for each iteration is divided by the
+number of computing nodes.  On the other hand, more communications are required
+when increasing the number of nodes so the static energy increases linearly
+according to the communication time and the dynamic power is less relevant in
+the overall energy consumption.  Therefore, reducing the frequency with
+Algorithm~\ref{HSA} is less effective in reducing the overall energy savings. It
+can also be noticed that for the benchmarks EP and SP that contain little or no
+communications, the energy savings are not significantly affected by the high
+number of nodes.  No experiments were conducted using bigger classes than D,
+because they require a lot of memory (more than \np[GB]{64}) when being executed
+by the simulator on one machine.  The maximum distance between the normalized
+energy curve and the normalized performance for each instance is also shown in
+the result tables. It decrease in the same way as the energy saving percentage.
+The tables also show that the performance degradation percentage is not
+significantly increased when the number of computing nodes is increased because
+the computation times are small when compared to the communication times.
+
+Figures~\ref{fig:energy} and \ref{fig:per_deg} present the energy saving and
+performance degradation respectively for all the benchmarks according to the
+number of used nodes. As shown in the first plot, the energy saving percentages
+of the benchmarks MG, LU, BT and FT decrease linearly when the number of nodes
+increase. While for the EP and SP benchmarks, the energy saving percentage is
+not affected by the increase of the number of computing nodes, because in these
+benchmarks there are little or no communications. Finally, the energy saving of
+the GC benchmark significantly decrease when the number of nodes increase
+because this benchmark has more communications than the others. The second plot
+shows that the performance degradation percentages of most of the benchmarks
+decrease when they run on a big number of nodes because they spend more time
+communicating than computing, thus, scaling down the frequencies of some nodes
+has less effect on the performance.

 
 \subsection{The results for different power consumption scenarios}
 
 \subsection{The results for different power consumption scenarios}

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

 
 

-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: 

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

 
 

-
- \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
     \hline

-    Method     & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
+    CG      &  4144.21      & 22.42    & 7.72          & 14.70    \\

     \hline
     \hline

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

    \hline
    \hline

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

    \hline
    \hline

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

     \hline
     \hline

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

    \hline
    \hline

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

    \hline
    \hline

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

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

-
-
-\begin{table}[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
     \hline

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

     \hline
     \hline

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

     \hline
     \hline

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

   \end{tabular}
   \label{table:res_s2}
 \end{table}
 
   \end{tabular}
   \label{table:res_s2}
 \end{table}
 

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

 
 

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

   \centering
   \centering

-  \subfloat[Comparison the average of the results on 8 nodes]{%
-    \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
-  \quad%
+  \subfloat[Comparison  between the results on 8 nodes]{%
+    \includegraphics[width=.33\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
+

   \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
   \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%

-    \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
+    \includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}

   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}

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

 
 

+\begin{figure}[!t]
+  \centering
+   \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
+  \caption{Trade-off comparison for NAS benchmarks class C}
+  \label{fig:compare_EDP}
+\end{figure}

 
 

-\subsection{The verifications of the proposed method}
-\label{sec.verif}
-The precision of the proposed algorithm mainly depends on the execution time prediction model defined in EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}). 
-The energy model is also significantly dependent  on the execution time model because the static energy is linearly related the execution time and the dynamic energy is related to the computation time. So, all of the work presented in this paper is based on the execution time model. To verify this model, the predicted execution time was compared to  the real execution time over Simgrid for all  the NAS parallel benchmarks running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise, the maximum normalized difference between  the predicted execution time  and the real execution time is equal to 0.03 for all the NAS benchmarks.

 
 

-Since  the proposed algorithm is not an exact method and do not test all the possible solutions (vectors of scaling factors) in the search space and to prove its efficiency, it was compared on small instances to a brute force search algorithm that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small
-execution time: for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in table~(\ref{table:platform}), it  
-takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number of iterations and $N$ is the number of computing nodes. The algorithm
-needs  from 12 to 20 iterations to select the best vector of frequency scaling factors that gives the results of the section (\ref{sec.res}).
+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
+
+In this section, the scaling factors selection algorithm, called MaxDist, is
+compared to Spiliopoulos et al. algorithm
+\cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP.  They developed a
+green governor that regularly applies an online frequency selecting algorithm to
+reduce the energy consumed by a multicore architecture without degrading much
+its performance. The algorithm selects the frequencies that minimize the energy
+and delay products, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$ 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 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}
 

+This work has been partially supported by the Labex
+ACTION project (contract ``ANR-11-LABX-01-01''). As a PhD student,
+Mr. Ahmed Fanfakh, would like to thank the University of
+Babylon (Iraq) for supporting his work.
+

 
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@@ -785,6 +1272,6 @@ needs  from 12 to 20 iterations to select the best vector of frequency scaling f
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 % LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber

-% LocalWords:  CMOS EQ 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