X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/c0f811adf4e7fb3bb57c6042f027be6adc4db147..453664b6bc4c3d4cfee00082a2f26ada681f08a5:/Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 6c37115..2f78db5 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -8,7 +8,6 @@
 \usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
-
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
@@ -24,48 +23,57 @@
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 
-\newcommand{\Xsub}[2]{\ensuremath{#1_\textit{#2}}}
+\newcommand{\Xsub}[2]{{\ensuremath{#1_\mathit{#2}}}}
+
+%% used to put some subscripts lower, and make them more legible
+\newcommand{\fxheight}[1]{\ifx#1\relax\relax\else\rule{0pt}{1.52ex}#1\fi}
 
-\newcommand{\Dist}{\textit{Dist}}
+\newcommand{\CL}{\Xsub{C}{L}}
+\newcommand{\Dist}{\mathit{Dist}}
+\newcommand{\EdNew}{\Xsub{E}{dNew}}
 \newcommand{\Eind}{\Xsub{E}{ind}}
 \newcommand{\Enorm}{\Xsub{E}{Norm}}
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
-\newcommand{\Fdiff}{\Xsub{F}{diff}}
-\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Es}{\Xsub{E}{S}}
+\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
+\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}
 \newcommand{\Fnew}{\Xsub{F}{new}}
 \newcommand{\Ileak}{\Xsub{I}{leak}}
 \newcommand{\Kdesign}{\Xsub{K}{design}}
-\newcommand{\MaxDist}{\textit{Max Dist}}
+\newcommand{\MaxDist}{\mathit{Max}\Dist}
+\newcommand{\MinTcm}{\mathit{Min}\Tcm}
 \newcommand{\Ntrans}{\Xsub{N}{trans}}
-\newcommand{\Pdyn}{\Xsub{P}{dyn}}
-\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
+\newcommand{\PdNew}{\Xsub{P}{dNew}}
+\newcommand{\PdOld}{\Xsub{P}{dOld}}
 \newcommand{\Pnorm}{\Xsub{P}{Norm}}
-\newcommand{\Tnorm}{\Xsub{T}{Norm}}
-\newcommand{\Pstates}{\Xsub{P}{states}}
-\newcommand{\Pstatic}{\Xsub{P}{static}}
-\newcommand{\Sopt}{\Xsub{S}{opt}}
-\newcommand{\Tcomp}{\Xsub{T}{comp}}
-\newcommand{\TmaxCommOld}{\Xsub{T}{Max Comm Old}}
-\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
-\newcommand{\Tmax}{\Xsub{T}{max}}
+\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
+\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
+\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
+\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
+\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
+\newcommand{\Ppeak}[1][]{\Xsub{P}{peak}_{#1}}
+\newcommand{\Pidle}[1][]{\Xsub{P}{idle}_{\fxheight{#1}}}
+\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
-\newcommand{\Told}{\Xsub{T}{Old}} 
-\begin{document} 
+\newcommand{\Told}{\Xsub{T}{Old}}
+
+\begin{document}
 
-\title{Energy Consumption Reduction in a Heterogeneous Architecture Using DVFS}
- 
-\author{% 
+\title{Energy Consumption Reduction with DVFS for \\
+  Message Passing Iterative Applications on \\
+  Heterogeneous Architectures}
+
+\author{%
   \IEEEauthorblockN{%
     Jean-Claude Charr,
     RaphaÃ«l Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch
-the normalized performance equation, as follows:
-  } 
+  }
   \IEEEauthorblockA{%
-    FEMTO-ST Institute\\
-    University of Franche-ComtÃ©\\
+    FEMTO-ST Institute, University of Franche-ComtÃ©\\
     IUT de Belfort-MontbÃ©liard,
     19 avenue du MarÃ©chal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % RaphaÃ«l
@@ -78,523 +86,373 @@ the normalized performance equation, as follows:
 
 \begin{abstract}
   
+
 \end{abstract}
 
 \section{Introduction}
 \label{sec.intro}
-Modern processors continue increasing in a performance. 
-The CPUs constructors are competing to achieve maximum number 
-of floating point operations per second (FLOPS). 
-Thus, the energy consumption and the heat dissipation are increased 
-drastically according to this increase. Because the number of FLOPS 
-is linearly related to the power consumption of a CPU
-~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}.  
-As an example of the more power hungry cluster, Tianhe-2 became in 
-the top of the Top500 list in June 2014 \cite{TOP500_Supercomputers_Sites}. 
-It has more than 3 millions of cores and consumed more than 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, we can consider the price of the energy consumption for the 
-Tianhe-2 platform is approximately more than \$10 millions  for 
-one 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, 
-became the top of the Green500 list in June 2014 \cite{Green500_List}. 
-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 maximum energy reduction against minimum 
-performance degradation ratio simultaneously. The algorithm has very small 
-overhead, works online and not needs for any training or profiling.  
-
-This paper is organized as follows: Section~\ref{sec.relwork} presents some
-related works from other authors.  Section~\ref{sec.exe} describes how the
-execution time of 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
-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{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{Hao_Learning.based.DVFS,Dhiman_Online.Learning.Power.Management}, 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{Our_first_paper}. 
-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{Luley_Energy.efficiency.evaluation.and.benchmarking}, 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{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, 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{Rong_Effects.of.DVFS.on.K20.GPU}, 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{Naveen_Power.Efficient.Resource.Scaling} 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{Lizhe_Energy.aware.parallel.task.scheduling}, 
-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{Joshi_Blackbox.prediction.of.impact.of.DVFS} 
-and \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, 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{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, 
-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{Chen_DVFS.under.quality.of.service.requirements},  
-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. 
+
 
 \section{The performance and energy consumption measurements on heterogeneous architecture}
 \label{sec.exe}
 
-% \JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'',
-%   can be deleted if we need space, we can just say we are interested in this
-%   paper in homogeneous clusters}
-
-\subsection{The execution time of message passing distributed 
-                iterative applications on a heterogeneous platform}
+\subsection{The execution time of message passing distributed iterative
+  applications on a heterogeneous platform}
 
 In this paper, we are interested in reducing the energy consumption of message
 passing distributed iterative synchronous applications running over
-heterogeneous platforms. We define a heterogeneous platform as a collection of
-heterogeneous 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]
+heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
+heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
+and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
+
+\begin{figure}[!t]
   \centering
-    \includegraphics[scale=0.6]{fig/commtasks}
+  \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 
-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 grid 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 clusters, slack times may occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
+overall execution time of the program is the execution time of the slowest task 
+which has the highest computation time and no slack time.
+
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
+modern processors, that reduces the energy consumption of a CPU by scaling
+down its voltage and frequency.  Since DVFS lowers the frequency of a CPU
+and consequently its computing power, the execution time of a program running
+over that scaled down processor may increase, especially if the program is
+compute bound.  The frequency reduction process can be  expressed by the scaling
+factor S which is the ratio between  the maximum and the new frequency of a CPU
+as in (\ref{eq:s}).
 \begin{equation}
   \label{eq:s}
- S = \frac{F_\textit{max}}{F_\textit{new}}
+  S = \frac{\Fmax}{\Fnew}
 \end{equation}
- The execution time of a 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 
- till 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
+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 grid each cluster has different characteristics,
+especially different frequency gears, when applying DVFS operations on the nodes 
+of these clusters, they may get different scaling factors represented by a scaling vector:
+$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
 be able to predict the execution time of message passing synchronous iterative
-applications running over a heterogeneous platform, for different vectors of
+applications running over a heterogeneous grid, for different vectors of
 scaling factors, the communication time and the computation time for all the
 tasks must be measured during the first iteration before applying any DVFS
 operation. Then the execution time for one iteration of the application with any
-vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
+vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
- \textit  T_\textit{new} = 
- \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) +  MinTcm
+  \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
 \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{Our_first_paper}. 
-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 $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
+each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
+and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during 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 in the slowest cluster $h$.
+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 and heterogeneous clusters
+~\cite{Our_first_paper,pdsec2015}.  The execution time prediction model is
+used in the method to optimize both the energy consumption and the performance
+of iterative methods, which is presented in the following sections.
 
 
 \subsection{Energy model for heterogeneous platform}
+
 Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
-Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
-Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by a processor into
-two power metrics: the static and the dynamic power.  While the first one is
-consumed as long as the computing unit is turned on, the latter is only consumed during
-computation times.  The dynamic power $P_{d}$ is related to the switching
-activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
-operational frequency $F$, as shown in EQ(\ref{eq:pd}).
+  Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
+  Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
+a processor into two power metrics: the static and the dynamic power.  While the
+first one is consumed as long as the computing unit is turned on, the latter is
+only consumed during computation times.  The dynamic power $\Pd$ is related to
+the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
+and operational frequency $F$, as shown in (\ref{eq:pd}).
 \begin{equation}
   \label{eq:pd}
-  P_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F
+  \Pd = \alpha \cdot \CL \cdot V^2 \cdot F
 \end{equation}
-The static power $P_{s}$ captures the leakage power as follows:
+The static power $\Ps$ captures the leakage power as follows:
 \begin{equation}
   \label{eq:ps}
-   P_\textit{s}  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
+   \Ps  = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak
 \end{equation}
-where V is the supply voltage, $N_{trans}$ is the number of transistors,
-$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
-technology-dependent parameter.  The energy consumed by an individual processor
+where V is the supply voltage, $\Ntrans$ is the number of transistors,
+$\Kdesign$ is a design dependent parameter and $\Ileak$ is a
+technology dependent parameter.  The energy consumed by an individual processor
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}
-   E_\textit{ind} =  P_\textit{d} \cdot Tcp + P_\textit{s} \cdot T
+  \Eind =  \Pd \cdot \Tcp + \Ps \cdot T
 \end{equation}
-where $T$ is the execution time of the program, $T_{cp}$ is the computation
-time and $T_{cp} \leq T$.  $T_{cp}$ may be equal to $T$ if there is no
+where $T$ is the execution time of the program, $\Tcp$ is the computation
+time and $\Tcp \le T$.  $\Tcp$ may be equal to $T$ if there is no
 communication and no slack time.
 
-The main objective of DVFS operation is to reduce the overall energy consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  
-The operational frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot F$ with some
-constant $\beta$.  This equation is used to study the change of the dynamic
-voltage with respect to various frequency values in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction
-process of the frequency can be expressed by the scaling factor $S$ which is the
-ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
-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}
-   F_\textit{new} = S^{-1} \cdot F_\textit{max}
+   \Fnew = S^{-1} \cdot \Fmax
 \end{equation}
-Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following 
+Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
 equation for dynamic power consumption:
 \begin{multline}
   \label{eq:pdnew}
-   {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
-   {} = \alpha \cdot C_L \cdot V^2 \cdot F_{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}
-where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the 
+where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
 new frequency and the maximum frequency respectively.
 
-According to 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{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is proportional 
-to the frequency of a CPU, the computation time is increased proportionally to $S$.  
-The new dynamic energy is the  dynamic power multiplied by the new time of computation 
-and is given by the following equation:
+According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
+$S^{-3}$ when reducing the frequency by a factor of
+$S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is
+proportional to the frequency of a CPU, the computation time is increased
+proportionally to $S$.  The new dynamic energy is the dynamic power multiplied
+by the new time of computation and is given by the following equation:
 \begin{equation}
   \label{eq:Edyn}
-   E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{dOld} \cdot  Tcp 
+   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp
 \end{equation}
-The static power is related to the power leakage of the CPU and is consumed during computation 
-and even when idle. As in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling}, 
-we assume that the static power of a processor is constant 
-during idle and computation periods, and for all its available frequencies. 
-The static energy is the static power multiplied by the execution time of the program. 
-According to the execution time model in EQ(\ref{eq:perf}), the execution time of the program 
-is the summation of the computation and the communication times. The computation time is linearly related  
-to the frequency scaling factor, while this scaling factor does not affect the communication time. 
-The static energy of a processor after scaling its frequency is computed as follows: 
+The static power is related to the power leakage of the CPU and is consumed
+during computation and even when idle. As
+in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
+the static power of a processor is considered as constant during idle and
+computation periods, and for all its available frequencies.  The static energy
+is the static power multiplied by the execution time of the program.  According
+to the execution time model in (\ref{eq:perf}), the execution time of the
+program is the sum of the computation and the communication times. The
+computation time is linearly related to the frequency scaling factor, while this
+scaling factor does not affect the communication time.  The static energy of a
+processor after scaling its frequency is computed as follows:
 \begin{equation}
   \label{eq:Estatic}
- E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)
+  \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
 \end{equation}
 
-In the considered heterogeneous platform, each processor $i$ might have different dynamic and 
-static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed 
-message passing iterative application is load balanced, the computation time of each CPU $i$ 
-noted $Tcp_{i}$ might be different and different frequency  scaling factors might be computed 
-in order to decrease the overall energy consumption of the application and reduce 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 grid platform, each node $j$ in cluster $i$ may have
+different dynamic and static powers from the nodes of the other clusters, 
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Therefore, even if the distributed 
+message passing iterative application is load balanced, the computation time of each CPU $j$ 
+in cluster $i$ noted $\Tcp[ij]$ 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 $j$ in cluster $i$ is noted as
+$\Tcm[ij]$ 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 grid platform during one iteration is the summation of all dynamic and
+static energies for $M$ processors in $N$ clusters.  It is computed as follows:
 \begin{multline}
   \label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot  Tcp_i)} + {} \\
- \sum_{i=1}^{N} (Ps_{i} \cdot (\max_{i=1,2,\dots,N} (Tcp_i \cdot S_{i}) +
-  {MinTcm))}
- \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}. 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} \sum_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
+ \sum_{i=1}^{N} \sum_{j=1}^{M} (\Ps[ij] \cdot {} \\
+  (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij}) 
+  +\mathop{\min_{j=1,\dots M}} (\Tcm[hj]) ))
+\end{multline}
 
+Reducing the frequencies of the processors according to the vector of scaling
+factors $(S_{11}, S_{12},\dots, S_{NM})$ may degrade the performance of the application
+and thus, increase the static energy because the execution time is
+increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
+for the iterative application can be measured by measuring the energy
+consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
+of iterations of that application.
 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 
-Using the lowest frequency for each processor does not necessarily 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{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, 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{Freeh_Exploring.the.Energy.Time.Tradeoff}.  
-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:
-\begin{multline}
+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,pdsec2015}, we proposed a method that selects the optimal
+frequency scaling factor for a homogeneous and heterogeneous clusters 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 grid 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{equation}
   \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)}}
-\end{multline}
+  \Pnorm = \frac{\Tnew}{\Told}                 
+\end{equation}
 
 
-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:
-\begin{multline}
+Where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told})
+\begin{equation}
+  \label{eq:told}
+   \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])             
+\end{equation}
+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{equation}
   \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:
-\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} 
-\end{multline}
+  \Enorm = \frac{\Ereduced}{\Eoriginal} 
+\end{equation}
+
+Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
+computed as in ().
+
+\textcolor{red}{A reference is missing}
+\begin{equation}
+  \label{eq:eorginal}
+    \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot  \Tcp[ij])  + 
+     \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M} (\Ps[ij] \cdot \Told)       
+\end{equation}
 
+While the main goal is to optimize the energy and execution time at the same
+time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. 
+According to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
+vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy
+and the execution time simultaneously.  But the main objective is to produce
+maximum energy reduction with minimum execution time reduction.
+
+This problem can be solved by making the optimization process for energy and
+execution time follow the same evolution according to the vector of scaling factors
+$(S_{11}, S_{12},\dots, S_{NM})$. Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance
+equation, as follows:
+\begin{equation}
+  \label{eq:pnorm_inv}
+  \Pnorm = \frac{\Told}{\Tnew}          
+\end{equation}
 
-\begin{figure}
+\begin{figure}[!t]
   \centering
-  \subfloat[Homogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
-  \qquad%
-  \subfloat[Heterogeneous platform]{%
-    \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
+  \subfloat[Homogeneous cluster]{%
+    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+
+  \subfloat[Heterogeneous grid]{%
+    \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
   \label{fig:rel}
   \caption{The energy and performance relation}
 \end{figure}
 
-Then, we can model our objective function as finding the maximum distance
-between the energy curve EQ~(\ref{eq:enorm}) and the  performance
-curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
-represents the minimum energy consumption with minimum execution time (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}
-  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{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
+      (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
 \end{equation}
-where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
-Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).  
-Our objective function can work with any energy model or 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{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
-
-\section{The scaling factors selection algorithm for heterogeneous platforms }
+where $N$ is the number of clusters, $M$ is the number of nodes in each cluster 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  grids }
 \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}
-    \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 [{$N$}] number of clusters in the grid.
+    \item [{$M$}] number of nodes in each cluster.
+    \item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
+    \item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
+    \item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
+    \item[{$\Pd[ij]$}] array of the dynamic powers for all nodes.
+    \item[{$\Ps[ij]$}] array of the static powers for all nodes.
+    \item[{$\Fdiff[ij]$}] array of the differences between two successive frequencies for all nodes.
     \end{description}
-    \Ensure $Sopt_1,Sopt_2 \dots, Sopt_N$ is a vector of optimal scaling factors
+    \Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$,  a vector of scaling factors that gives the optimal tradeoff between energy consumption and execution time
 
-    \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.
+    \State $\Scp[ij] \gets \frac{\max_{i=1,2,\dots,N}(\max_{j=1,2,\dots,M_i}(\Tcp[ij]))}{\Tcp[ij]} $
+    \State $F_{ij} \gets  \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
+    \State Round the computed initial frequencies $F_i$ to the closest  available frequency for 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. $
-    \While {(all nodes not reach their  minimum  frequency)}
+          \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
+    \EndIf
+    \State $\Told \gets $ computed as in equations (\ref{eq:told}).
+    \State $\Eoriginal \gets $ computed as in equations (\ref{eq:eorginal}) .
+    \State $\Sopt[ij] \gets 1,~i=1,\dots,N,~{j=1,\dots,M_i}. $
+    \State $\Dist \gets 0 $
+    \While {(all nodes have not reached their  minimum   \newline\hspace*{2.5em} frequency \textbf{or}  $\Pnorm - \Enorm < 0 $)}
         \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 $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
+        \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
         \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 $ computed as  in equations (\ref{eq:perf}). 
+       \State $\Ereduced \gets $ computed as  in equations (\ref{eq:energy}). 
+       \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+       \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
-        \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $
+        \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile
-    \State  Return $Sopt_1,Sopt_2,\dots,Sopt_N$
+    \State  Return $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$
   \end{algorithmic}
-  \caption{Heterogeneous scaling algorithm}
+  \caption{Scaling factors selection algorithm}
   \label{HSA}
 \end{algorithm}
 
@@ -607,7 +465,7 @@ which results in bigger energy savings.
       \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.
@@ -618,425 +476,232 @@ which results in bigger energy savings.
   \label{dvfs}
 \end{algorithm}
 
+\subsection{The algorithm details}
+
+\textcolor{red}{Delete the subsection if there's only one.}
+
+In this section, the scaling factors selection algorithm for  grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency
+scaling factors  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  grid. 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=0.45]{fig/init_freq}
+  \caption{Selecting the initial frequencies}
+  \label{fig:st_freq}
+\end{figure}
+
+Nodes from distinct clusters in a grid 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 vector of the frequency
+scaling factors. 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[ij] =  \frac{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]}
+\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_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M}
+\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 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 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 higher frequencies
+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 until reaching the nodes' minimum frequencies or lower bounds. A node's frequency is considered its lower bound if the computed distance between the energy and performance at this frequency is less than zero.
+A negative distance means that the performance degradation ratio is higher than the energy saving ratio.
+In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances. 
+
+Therefore, the algorithm iterates on all remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
+energy consumption and performance and selects the optimal vector of the frequency scaling
+factors. 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 cluster and a
+ grid platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous cluster 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  grid 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, which results in bigger energy savings. 
+
+
 \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{NAS.Parallel.Benchmarks}. 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{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. 
-Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
-
-
-\begin{table}[htb]
-  \caption{Heterogeneous nodes characteristics}
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
-    type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
-                  &           & GHz      & GHz          &GHz             &              &       \\
-    \hline
-    1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
-                  &           &          &              &                &              &  \\
-    \hline
-    2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
-                  &           &          &              &                &              &  \\
-    \hline
-    3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
-                  &           &          &              &                &              &  \\
-    \hline
-    4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
-                  &           &          &              &                &              &  \\
-    \hline
-  \end{tabular}
-  \label{table:platform}
-\end{table}
+While in~\cite{mpi-energy2} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid.org}, 
+in this paper real experiments were conducted over the grid'5000 platform. 
+
+\subsection{Grid'5000 architature and power consumption}
+\label{sec.grid5000}
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed over all metropolitan France and Luxembourg. All the sites are connected together via 	a special long distance network called RENATER,
+which is the French National Telecommunication Network for Technology.
+Each site of the grid is composed of few heterogeneous 
+computing clusters and each cluster contains many homogeneous nodes. In total,
+ grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
+the clusters and their nodes are connected via  high speed local area networks. 
+Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  
+
+Since grid'5000 is dedicated for testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture 
+the power consumption  for each node in those sites. The measured power is the overall consumed power by by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ...  For more details refer to
+\cite{Energy_measurement}. To just measure the CPU power of one core in a node $j$, 
+ firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $P\max[jx]$. The difference between the two measured power consumption represents the 
+dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 
+
+\textcolor{red}{why maximum and minimum, change peak in the equation and the figure}
+
+The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
+\begin{equation}
+  \label{eq:pdyn}
+    \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (P\max[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
+\end{equation}
 
- 
-%\subsection{Performance prediction verification}
+where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured peak power values, 
+$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
+Therefore, the dynamic power of one core is computed as the difference between the maximum 
+measured value in peak powers vector and the minimum measured value in the idle powers vector.
 
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  np[\%]{20} of dynamic power consumption of the core.
 
-\subsection{The experimental results of the scaling algorithm}
-\label{sec.res}
+In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
 
+Four clusters from the two sites were selected in the experiments: one cluster from 
+Lyon's site, Taurus cluster, and three clusters from Nancy's site, Graphene, 
+Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
+frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 
+the details characteristics of these four clusters. Moreover, the dynamic powers were computed  using the equation (\ref{eq:pdyn}) for all the nodes in the 
+selected clusters and are presented in table  \ref{table:grid5000}.
 
-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]
-  \caption{Running NAS benchmarks on 4 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &  64.64        & 3560.39        &34.16        &6.72               &27.44       \\
-    \hline 
-    MG         & 18.89         & 1074.87	    &35.37	      &4.34	              &31.03       \\
-   \hline
-    EP         &79.73	       &5521.04	        &26.83   	  &3.04               &23.79      \\
-   \hline
-    LU         &308.65	       &21126.00	   &34.00	      &6.16	              &27.84      \\
-    \hline
-    BT         &360.12         &21505.55	   &35.36         &8.49               &26.87     \\
-   \hline
-    SP         &234.24	       &13572.16	   &35.22         &5.70	              &29.52    \\
-   \hline
-    FT         &81.58          &4151.48        &35.58         &0.99        	      &34.59    \\
-\hline 
-  \end{tabular}
-  \label{table:res_4n}
-\end{table}
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 8 and 9 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &36.11    	   &3263.49	        &31.25	      &7.12	               &24.13     \\
-    \hline 
-    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 
-  \end{tabular}
-  \label{table:res_8n}
-\end{table}
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 16 nodes }
-  % title of Table
+\begin{figure}[!t]
   \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &31.74	       &4373.90	        &26.29	      &9.57	               &16.72          \\
-    \hline 
-    MG         &5.71	       &1076.19         &32.49	      &6.05	               &26.44         \\
-   \hline
-    EP         &20.11	       &5638.49	        &26.85	      &0.56	               &26.29         \\
-   \hline
-    LU         &144.13	       &42529.06	    &28.80	      &6.56	               &22.24         \\
-    \hline
-    BT         &97.29	       &22813.86	    &34.95   	  &5.80	               &29.15         \\
-   \hline
-    SP         &66.49	       &20821.67 	    &22.49	      &3.82	               &18.67         \\
-   \hline
-    FT     	   &37.01          &5505.60	        &31.59	      &6.48	               &25.11         \\
-\hline 
-  \end{tabular}
-  \label{table:res_16n}
-\end{table}
+  \includegraphics[scale=1]{fig/grid5000}
+  \caption{The selected two sites of grid'5000}
+  \label{fig:grid5000}
+\end{figure}
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 32 and 36 nodes }
-  % title of Table
+
+The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 
+
+
+
+
+\begin{figure}[!t]
   \centering
-  \begin{tabular}{|*{7}{l|}}
-    \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
-    \hline
-    CG         &32.35	       &6704.21	        &16.15	      &5.30	               &10.85           \\
-    \hline 
-    MG         &4.30	       &1355.58	        &28.93	      &8.85	               &20.08          \\
-   \hline
-    EP         &9.96           &5519.68	        &26.98	      &0.02	               &26.96          \\
-   \hline
-    LU         &99.93	       &67463.43	    &23.60	      &2.45	               &21.15          \\
-    \hline
-    BT         &48.61	       &23796.97	    &34.62	      &5.83	               &28.79          \\
-   \hline
-    SP         &46.01	       &27007.43	    &22.72	      &3.45	               &19.27           \\
-   \hline
-    FT     	   &28.06     	   &7142.69	        &23.09	      &2.90	               &20.19           \\
-\hline 
-  \end{tabular}
-  \label{table:res_32n}
-\end{table}
+  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
+  \caption{The power consumption by one core from Taurus cluster}
+  \label{fig:power_cons}
+\end{figure}
+
 
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 64 nodes }
+
+  
+\begin{table}[!t]
+  \caption{CPUs characteristics of the selected clusters}
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{c|}}
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
+    Cluster     & CPU         & Max   & Min   & Diff. & no. of cores    & dynamic power   \\
+    Name        & model       & Freq. & Freq. & Freq. & per CPU         & of one core     \\
+                &             & GHz   & GHz   & GHz   &                 &           \\
     \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         \\
+    Taurus      & Intel       & 2.3  & 1.2  & 0.1     & 6               & \np[W]{35} \\
+                & Xeon        &       &       &       &                 &            \\
+                & E5-2630     &       &       &       &                 &            \\         
     \hline
-    BT         &39.99 	       &27166.71	    &32.02	      &12.28	           &19.74         \\
-   \hline
-    SP         &52.00	       &49099.28	    &24.84	      &0.03	               &24.81         \\
-   \hline
-    FT         &25.97	       &10416.82        &20.15	      &4.87	               &15.28         \\
-\hline 
-  \end{tabular}
-  \label{table:res_64n}
-\end{table}
-
-
-\begin{table}[htb]
-  \caption{Running NAS benchmarks on 128 and 144 nodes }
-  % title of Table
-  \centering
-  \begin{tabular}{|*{7}{l|}}
+    Graphene    & Intel       & 2.53  & 1.2   & 0.133 & 4               & \np[W]{23} \\
+                & Xeon        &       &       &       &                 &            \\
+                & X3440       &       &       &       &                 &            \\    
     \hline
-    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
-    name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
+    Griffon     & Intel       & 2.5   & 2     & 0.5   & 4               & \np[W]{46} \\
+                & Xeon        &       &       &       &                 &            \\
+                & L5420       &       &       &       &                 &            \\  
     \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         \\
+    Graphite    & Intel       & 2     & 1.2   & 0.1   & 8               & \np[W]{35} \\
+                & Xeon        &       &       &       &                 &            \\
+                & E5-2650     &       &       &       &                 &            \\  
     \hline
-    BT         &37.96          &44243.82	    &23.18	      &1.28	               &21.90         \\
-   \hline
-    SP         &64.53	       &115409.71 	    &26.72	      &0.05	               &26.67         \\
-   \hline
-    FT         &25.51	       &18808.72	    &12.85	      &2.84	               &10.01         \\
-\hline 
   \end{tabular}
-  \label{table:res_128n}
-\end{table}
-The 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}
-  \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}}
-  \label{fig:avg}
-  \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
-\end{figure}
+  \label{table:grid5000}
+\end{table} 
+
 
-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. 
 
 
+\subsection{The experimental results of the scaling algorithm}
+\label{sec.res}
 
+\subsection{The experimental results of multi-cores clusters}
+\label{sec.res}
 
 \subsection{The results for different power consumption scenarios}
+\label{sec.compare}
 
-The results of the previous section were obtained while using processors that consume during computation 
-an overall power which is 80\% composed of  dynamic power and 20\% of static power. In this section, 
-these ratios are changed and two new power scenarios are considered in order to evaluate how the proposed  
-algorithm adapts itself according to the static and dynamic power values.  The two new power scenarios 
-are the following: 
-
-\begin{itemize}
-\item 70\% dynamic power  and 30\% static power
-\item 90\% dynamic power  and 10\% static power
-\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}
-  % title of Table
-  \centering
-  \begin{tabular}{|*{6}{l|}}
-    \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
-    \hline
-    CG         &4144.21          &22.42        &7.72                &14.70         \\
-    \hline 
-    MG         &1133.23          &24.50        &5.34                &19.16          \\
-   \hline
-    EP         &6170.30	        &16.19	       &0.02	            &16.17          \\
-   \hline
-    LU         &39477.28        &20.43	       &0.07	            &20.36          \\
-    \hline
-    BT         &26169.55	    &25.34	       &6.62	            &18.71          \\
-   \hline
-    SP         &19620.09	    &19.32	       &3.66	            &15.66          \\
-   \hline
-    FT         &6094.07	        &23.17	       &0.36	            &22.81          \\
-\hline 
-  \end{tabular}
-  \label{table:res_s1}
-\end{table}
 
 
 
-\begin{table}[htb]
-  \caption{The results of 90\%-10\% powers scenario}
-  % title of Table
-  \centering
-  \begin{tabular}{|*{6}{l|}}
-    \hline
-    Method     & Energy          & Energy      & Performance        & Distance     \\
-    name       & consumption/J   & saving\%    & degradation\%      &              \\
-    \hline
-    CG         &2812.38	         &36.36	       &6.80                &29.56         \\
-    \hline 
-    MG         &825.427	         &38.35	       &6.41	            &31.94         \\
-   \hline
-    EP         &5281.62	         &35.02	       &2.68	            &32.34         \\
-   \hline
-    LU         &31611.28	     &39.15        &3.51	            &35.64        \\
-    \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}
+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
 
 
-\begin{figure}
-  \centering
-  \subfloat[Comparison the average of the results on 8 nodes]{%
-    \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
-  \quad%
-  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
-    \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
-  \label{fig:comp}
-  \caption{The comparison of the three power scenarios}
-\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}).
 
 \section{Conclusion}
 \label{sec.concl}
 
 
+
 \section*{Acknowledgment}
 
+This work  has been  partially supported by  the Labex ACTION  project (contract
+``ANR-11-LABX-01-01'').  Computations  have been performed  on the supercomputer
+facilities  of the  MÃ©socentre de  calcul de  Franche-ComtÃ©. As  a  PhD student,
+Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
+supporting his work.
 
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-% LocalWords:  CMOS EQ EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex
-% LocalWords:  de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
-% LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
+% LocalWords:  CMOS EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex GPU
+% LocalWords:  de badri muslim MPI SimGrid GFlops Xeon EP BT GPUs CPUs AMD
+%  LocalWords:  Spiliopoulos scalability
