corrections

[mpi-energy2.git] / mpi-energy2-extension / Heter_paper.tex

diff --git a/mpi-energy2-extension/Heter_paper.tex b/mpi-energy2-extension/Heter_paper.tex
index ba53471aef2b2a5b5d4490e19cb267e330566f41..9108cb1280e4f0c9757839a22b1761a96d5bc9e5 100644 (file)
--- a/mpi-energy2-extension/Heter_paper.tex
+++ b/mpi-energy2-extension/Heter_paper.tex
@@ -107,9 +107,9 @@
 
 
 
 
 
 

-\title{Optimizing Energy Consumption with DVFS for Message \\
-         Passing Applications \textcolor{blue}{with iterations} on \\
-                    Grid Architectures} 
+\title{Optimizing the Energy Consumption \\ 
+of Message Passing Applications with Iterations \\ 
+Executed over Grids} 

   
 
 
   
 
 


@@ -143,10 +143,10 @@ scaling (DVFS) is one of them. It can be used to reduce the power consumption of
   In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.  
   It selects the frequencies and tries to give the best
   trade-off between energy saving and performance degradation, for each node
   In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.  
   It selects the frequencies and tries to give the best
   trade-off between energy saving and performance degradation, for each node

-  computing the message passing  application \textcolor{blue}{with iterations}. 
+  computing the message passing  application with iterations. 

   The algorithm has a small
   overhead and works without training or profiling. It uses a new energy model
   The algorithm has a small
   overhead and works without training or profiling. It uses a new energy model

-  for message passing  applications \textcolor{blue}{with iterations} running on a  grid. 
+  for message passing  applications with iterations running on a  grid. 

   The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
   running the NAS parallel benchmarks.  The experiments on 16 nodes, distributed on three clusters, show that it reduces  on average the
   energy consumption  by \np[\%]{30} while  the performance  is on average only degraded
   The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
   running the NAS parallel benchmarks.  The experiments on 16 nodes, distributed on three clusters, show that it reduces  on average the
   energy consumption  by \np[\%]{30} while  the performance  is on average only degraded


@@ -192,7 +192,7 @@ This heterogeneous platform executes more than 7 GFlops per watt while consuming
 50.32 kilowatts.
 
 Besides platform improvements, there are many software and hardware techniques
 50.32 kilowatts.
 
 Besides platform improvements, there are many software and hardware techniques

-to lower the energy consumption of these platforms, such as DVFS, scheduling \textcolor{blue}{and other techniques}.
+to lower the energy consumption of these platforms, such as DVFS, scheduling and other techniques.

  DVFS is a widely used process to reduce the energy consumption of a
 processor by lowering its frequency
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
  DVFS is a widely used process to reduce the energy consumption of a
 processor by lowering its frequency
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces


@@ -202,20 +202,30 @@ different optimization strategies to select the frequency that gives the best
 trade-off between the energy reduction and performance degradation ratio. In
 \cite{Our_first_paper} and \cite{pdsec2015}, a frequency selecting algorithm
 was proposed to reduce the energy consumption of message passing 
 trade-off between the energy reduction and performance degradation ratio. In
 \cite{Our_first_paper} and \cite{pdsec2015}, a frequency selecting algorithm
 was proposed to reduce the energy consumption of message passing 

-applications \textcolor{blue}{with iterations} running over homogeneous and heterogeneous clusters respectively.
+applications with iterations running over homogeneous and heterogeneous clusters respectively.

 The results of the experiments showed significant energy consumption
 reductions. All the experimental results were conducted over the SimGrid
 simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous  platforms, and to simulate the execution of message passing parallel
 applications over them. 
 
 The results of the experiments showed significant energy consumption
 reductions. All the experimental results were conducted over the SimGrid
 simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous  platforms, and to simulate the execution of message passing parallel
 applications over them. 
 

-In this paper, a new frequency selecting algorithm, adapted to grid platforms
-composed of heterogeneous clusters, is presented. It is applied to the NAS
+
+This paper presents the following contributions :
+\begin{enumerate}
+\item two new energy and performance models for message passing 
+  synchronous applications with iterations running over a heterogeneous grid platform. Both models
+  take into account communications and slack times. The models can predict the
+  required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous grid
+  platforms. The algorithm has a very small overhead and does not need any
+  training nor profiling. It uses a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing  synchronous application with iterations.  The algorithm  was applied to the NAS

 parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
 parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform

-\cite{grid5000}. It selects for a grid platform running a message passing
- application \textcolor{blue}{with iterations} the vector of frequencies that simultaneously tries to
-offer the maximum energy reduction and minimum performance degradation
-ratios. The algorithm has a very small overhead, works online and does not need
-any training or profiling.
+\cite{grid5000}.
+
+\end{enumerate}
+

 
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some
 
 
 This paper is organized as follows: Section~\ref{sec.relwork} presents some


@@ -255,8 +265,8 @@ adapted to the execution's environment and the type of the application such as
 sequential, parallel or distributed architecture, homogeneous or heterogeneous
 platform, synchronous or asynchronous application.
 
 sequential, parallel or distributed architecture, homogeneous or heterogeneous
 platform, synchronous or asynchronous application.
 

-In this paper, we are interested in reducing energy for message passing
- synchronous applications \textcolor{blue}{with iterations} running over heterogeneous grid platforms.  Some
+In this paper, we are interested in reducing the energy consumption of message passing
+ synchronous applications with iterations running over heterogeneous grid platforms.  Some

 works have already been done for such platforms and they can be classified into
 two types of heterogeneous platforms:
 \begin{itemize}
 works have already been done for such platforms and they can be classified into
 two types of heterogeneous platforms:
 \begin{itemize}


@@ -300,21 +310,7 @@ some heuristic.  Chen et
 al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
 programming approach to minimize the power consumption of heterogeneous servers
 while respecting given time constraints.  This approach had considerable
 al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
 programming approach to minimize the power consumption of heterogeneous servers
 while respecting given time constraints.  This approach had considerable

-overhead.  In contrast to the above described papers, this paper presents the
-following contributions :
-\begin{enumerate}
-\item two new energy and performance models for message passing 
-  synchronous applications \textcolor{blue}{with iterations} running over a heterogeneous grid platform. Both models
-  take into account communication and slack times. The models can predict the
-  required energy and the execution time of the application.
-
-\item a new online frequency selecting algorithm for heterogeneous grid
-  platforms. The algorithm has a very small overhead and does not need any
-  training nor profiling. It uses a new optimization function which
-  simultaneously maximizes the performance and minimizes the energy consumption
-  of a message passing  synchronous application \textcolor{blue}{with iterations}.
-
-\end{enumerate}
+overhead.

 
 
 
 
 
 


@@ -322,25 +318,25 @@ following contributions :
 \label{sec.exe}
 
 \subsection{The execution time of message passing distributed 
 \label{sec.exe}
 
 \subsection{The execution time of message passing distributed 

-  applications \textcolor{blue}{with iterations} on a heterogeneous platform}
+  applications with iterations on a heterogeneous platform}

 
 In this paper, we are interested in reducing the energy consumption of message
 
 In this paper, we are interested in reducing the energy consumption of message

-passing distributed  synchronous applications \textcolor{blue}{with iterations} running over
+passing distributed  synchronous applications with iterations running over

 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.
 
 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.
 

-The overall execution time of a distributed  synchronous application  \textcolor{blue}{with iterations} 
+The overall execution time of a distributed  synchronous application  with iterations running 

 over a heterogeneous grid consists of the sum of the computation time and 
 the communication time for every iteration on a node. 
 over a heterogeneous grid consists of the sum of the computation time and 
 the communication time for every iteration on a node. 

-\textcolor{blue}{However, nodes from distinct clusters in a grid have different computing powers, thus
-while executing message passing \textcolor{blue}{with iterations} synchronous applications, fast nodes
+However, nodes from distinct clusters in a grid have different computing powers, thus
+while the application, 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
 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. } 
+periods are called idle or slack times. 

 Therefore, the
 overall execution time of the program is the execution time of the slowest task 
 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. \textcolor{blue}{For example, in Figure \ref{fig:heter}  the task 1 is the slower task which has no slack time (not waits for the other nodes) and it is only has the communication times.}
+which has the highest computation time and almost no slack time. For example, in Figure \ref{fig:heter}, task 1 is the slower task and it does not have to wait for the other nodes to communicate with them because they all finish their computations before it.  

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


@@ -361,7 +357,8 @@ as in (\ref{eq:s}).
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}
   \label{eq:s}
   S = \frac{\Fmax}{\Fnew}
 \end{equation}

-\textcolor{blue}{Where $\Fmax$ is the maximum frequency before applying DVFS and $\Fnew$  is the new frequency after applying DVFS.}
+where $\Fmax$ is the maximum frequency before applying any DVFS and $\Fnew$  is the new frequency after applying DVFS.
+

 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
 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


@@ -378,7 +375,7 @@ 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:
 $(S_{11}, S_{12},\dots, S_{NM_i})$ 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 
 of these clusters, they may get different scaling factors represented by a scaling vector:
 $(S_{11}, S_{12},\dots, S_{NM_i})$ 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 

-applications \textcolor{blue}{with iterations} running over a heterogeneous grid, for different vectors of
+applications with iterations 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
 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


@@ -387,24 +384,27 @@ vector of scaling factors can be predicted using Equation (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
   \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij}) 
 \begin{equation}
   \label{eq:perf}
   \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij}) 

-  +\mathop{\min_{j=1,\dots,M_i}}  (\Tcm[hj])
+  +\mathop{\min_{j=1,\dots,M_h}}  (\Tcm[hj])

 \end{equation}
 %
 where $N$ is the number of  clusters in the grid, $M_i$ is the number of  nodes in
  cluster $i$, $\TcpOld[ij]$ 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 execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
 \end{equation}
 %
 where $N$ is the number of  clusters in the grid, $M_i$ is the number of  nodes in
  cluster $i$, $\TcpOld[ij]$ 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 execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors

-and \textcolor{blue}{the  communication time of the slower node without slack time during one iteration.
-The slower node $h$ is the node that gives maximum execution time in all clusters befor scaling its frequency.}
+and the  communication time of the slowest node without slack time during one iteration.
+The slowest node in cluster $h$ is the node which takes the  maximum execution time to execute an iteration  before scaling down its  frequency.

 It means that only the communication time without any slack time is taken into account.
 It means that only the communication time without any slack time is taken into account.

-Therefore, the execution time of the  application \textcolor{blue}{with iterations} is equal to
+Therefore, the execution time of the  application is equal to

 the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the
 number of iterations of that application.
 
 the execution time of one iteration as in Equation (\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}. \textcolor{blue}{where the homogeneous cluster predication model was used one scaling factor denoted as $S$, because all the nodes in the cluster have the same computing powers. Whereas, in heterogeneous cluster prediction model all the nodes have different scales and the scaling factors have denoted as one dimensional vector $(S_1, S_2, \dots, S_N)$. The execution time prediction model for a grid Equation \ref{eq:perf} defines a two dimensional array of scales  
-$(S_{11}, S_{12},\dots, S_{NM_i})$}. This model is used in the method to optimize both the energy consumption and the performance of iterative methods, which is presented in the following sections.
+This model is an adaptation of the one developed in ~\cite{Our_first_paper}  which predicts the execution time
+of message passing  applications with iterations running on homogeneous clusters. 
+In a  homogeneous cluster only one scaling factor denoted as $S$ was used because all the nodes in the cluster have the same computing power.
+In a heterogeneous cluster, each node may have a different scaling factor denoted as  $(S_i)$ where $i$ is the index of the node. In a grid, each node in each cluster may have a scaling factor. The whole set of scaling factors of all the computing nodes in the grid  is denoted by a two dimensional array of scales  
+$(S_{11}, S_{12},\dots, S_{NM_i})$ where $N$ is the number of used clusters and $M_i$ is the number of nodes in cluster $i$.  
+
+The execution time model, Equation \ref{eq:perf}, is used in the algorithm presented in section \ref{sec.optim}. The latter selects the scaling factors that optimize both the energy consumption and the performance of message passing applications with iterations running on grids.

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


@@ -491,9 +491,9 @@ processor after scaling its frequency 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, 
 
 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.  \textcolor{blue}{Therefore, even if the distributed 
-message passing  application \textcolor{blue}{with iterations} is load balanced, the computation time of each CPU $j$ 
-in cluster $i$ noted $\Tcp[ij]$ may be slightly different due to the delay caused by the scheduler of the operating system}. Therefore, different frequency scaling factors may be
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Moreover, even if the distributed 
+message passing  application with iterations is load balanced, the computation time of each CPU $j$ in cluster $i$
+ noted $\Tcp[ij]$ may be slightly different due to the delay caused by the scheduler of the operating system. Therefore, different frequency scaling factors may be

 computed in order to decrease the overall energy consumption of the application
 and reduce the 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,
 computed in order to decrease the overall energy consumption of the application
 and reduce the 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,


@@ -502,8 +502,8 @@ 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. 
 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. 

-\textcolor{blue}{ The CPU during the communication times consumes only the static power. While 
-in the computation times, it consumes both the dynamic and the static power refer to \cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.} 
+ The CPU during the communication times consumes only the static power. While 
+in the computation times, it consumes both the dynamic and the static powers, for more information refer to \cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.

 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 $M_i$ processors in $N$ clusters.  It is computed as follows:
 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 $M_i$ processors in $N$ clusters.  It is computed as follows:


@@ -512,7 +512,7 @@ static energies for $M_i$ processors in $N$ clusters.  It is computed as follows
  E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
  \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot {} \\
   (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij}) 
  E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
  \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot {} \\
   (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij}) 

-  +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))
+  +\mathop{\min_{j=1,\dots M_h}} (\Tcm[hj]) ))

 \end{multline}
 
 
 \end{multline}
 
 


@@ -520,7 +520,7 @@ Reducing the frequencies of the processors according to the vector of scaling
 factors $(S_{11}, S_{12},\dots, S_{NM_i})$ 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
 factors $(S_{11}, S_{12},\dots, S_{NM_i})$ 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  application \textcolor{blue}{with iterations} can be measured by measuring the energy
+for a synchronous  application with iterations 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.
 
 consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
 of iterations of that application.
 


@@ -545,9 +545,9 @@ works, \cite{Our_first_paper} and \cite{pdsec2015}, two methods that select the
 frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed. 
 Both methods selects the frequencies that gives the best trade-off between 
 energy consumption reduction and performance for  message passing
 frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed. 
 Both methods selects the frequencies that gives the best trade-off between 
 energy consumption reduction and performance for  message passing

- synchronous applications \textcolor{blue}{with iterations}.   In this work we
-are interested in grids that are composed of heterogeneous clusters, \textcolor{blue}{where} the nodes 
-have different characteristics such  as  dynamic power, static power, computation power, 
+ synchronous applications with iterations.   In this work we
+are interested in grids that are composed of heterogeneous clusters. The nodes from distinct clusters may have 
+ different characteristics such  as  dynamic power, static power, computation power, 

 frequencies range, network latency and bandwidth. 
 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.
 frequencies range, network latency and bandwidth. 
 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.


@@ -568,13 +568,13 @@ maximum frequency for all nodes) as follows:
 \end{equation}
 %
 where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).
 \end{equation}
 %
 where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).

-\textcolor{blue}{
+

 \begin{equation}
   \label{eq:told}
    \Told = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} ) 
 \begin{equation}
   \label{eq:told}
    \Told = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} ) 

-  +\mathop{\min_{j=1,\dots,M_i}}  (\Tcm[hj])    
+  +\mathop{\min_{j=1,\dots,M_h}}  (\Tcm[hj])    

 \end{equation}
 \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:
 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:


@@ -596,13 +596,13 @@ computed as in (\ref{eq:eorginal}).
 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 (\ref{eq:pnorm}) and (\ref{eq:enorm}), the
 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 (\ref{eq:pnorm}) and (\ref{eq:enorm}), the

-vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduces both the energy
+vector of frequency scaling factors $S_{11},S_{12},\dots,S_{NM_i}$ reduces both the energy

 and the execution time,  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
 and the execution time,  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
+$(S_{11}, S_{12},\dots, S_{NM_i})$. Therefore, the equation of the

 normalized execution time is inverted which gives the normalized performance
 equation, as follows:
 \begin{equation}
 normalized execution time is inverted which gives the normalized performance
 equation, as follows:
 \begin{equation}


@@ -652,7 +652,7 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
     \Require ~
     \begin{description}
     \item [{$N$}] number of clusters in the grid. 
     \Require ~
     \begin{description}
     \item [{$N$}] number of clusters in the grid. 

-    \item [{$M$}] number of nodes in each cluster.
+    \item [{$M_i$}] 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[{$\Tcp[ij]$}] 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.


@@ -712,11 +712,11 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin
 
 
 In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 
 
 
 In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 

-is presented. It selects the vector of the frequency
+is presented. It selects the vector of  frequency

 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing
 scaling factors  that gives the best trade-off between minimizing the
 energy consumption and maximizing the performance of a message passing

-synchronous  application \textcolor{blue}{with iterations} executed on a  grid. It works
-online during the execution time of the  message passing program \textcolor{blue}{with iterations}.  It
+synchronous  application with iterations executed on a  grid. It works
+online during the execution time of the  application.  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
 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


@@ -724,7 +724,7 @@ 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
 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  MPI program \textcolor{blue}{with iterations}.
+scaling algorithm is called in the application.

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


@@ -750,7 +750,7 @@ of the slowest node and the computation time of the node $i$ as follows:
 \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
 \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
+maximum frequency of node and its computed scaling factor as

 follows:
 \begin{equation}
   \label{eq:Fint}
 follows:
 \begin{equation}
   \label{eq:Fint}


@@ -800,7 +800,7 @@ in this paper real experiments were conducted over the Grid'5000 platform.
 
 \subsection{Grid'5000 architecture and power consumption}
 \label{sec.grid5000}
 
 \subsection{Grid'5000 architecture and power consumption}
 \label{sec.grid5000}

-Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed all over  metropolitan France and Luxembourg. All the sites are connected together via       a special long distance network called RENATER,
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed all over  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 a few heterogeneous 
 computing clusters and each cluster contains many homogeneous nodes. In total,
 which is the French National Telecommunication Network for Technology.
 Each site of the grid is composed of a few heterogeneous 
 computing clusters and each cluster contains many homogeneous nodes. In total,


@@ -809,7 +809,7 @@ 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 to  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 
 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 to  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 all the components of a node at a given instant, such as CPU, hard drive, main-board and memory. For more details refer to
+the power consumption  for each node in those sites. The measured power is the overall consumed power  by all the components of a node at a given instant. For more details refer to

 \cite{Energy_measurement}. In order to correctly 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 $\Pmax[jx]$. The difference between the two measured power consumptions represents the 
 dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 
 \cite{Energy_measurement}. In order to correctly 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 $\Pmax[jx]$. The difference between the two measured power consumptions represents the 
 dynamic power consumption of that core with the maximum frequency, see  Figure~\ref{fig:power_cons}. 


@@ -854,11 +854,9 @@ selected clusters and are presented in Table~\ref{table:grid5000}.
 
 
 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 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. \textcolor{blue}{These benchmarks are message passing  applications with iterations compute
-the same block of operations several times, starting from the initial solution until reaching
-the acceptable approximation of the exact solution.}
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These benchmarks are considered as message passing  applications with iterations because the same block of operations is executed many times.

  These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
  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,  class D was used for all benchmarks in all the experiments presented in the next sections. 
+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 the next sections, the  class D was used for all the benchmarks in all the experiments. 

 
 
   
 
 
   


@@ -897,9 +895,7 @@ The benchmarks have seven different classes, S, W, A, B, C, D and E, that repres
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 

-to the NAS parallel benchmarks are presented. \textcolor{blue}{Each experiment of this section and next sections has been executed many times and the results presented in the figures are the average values of many execution.}
-
-As mentioned previously, the experiments 
+to the NAS parallel benchmarks are presented. Each experiment  has been executed many times and the results presented in the figures are the average values of many executions. As mentioned previously, the experiments 

 were conducted over two sites of Grid'5000,  Lyon and Nancy sites. 
 Two scenarios were considered while selecting the clusters from these two sites :
 \begin{itemize}
 were conducted over two sites of Grid'5000,  Lyon and Nancy sites. 
 Two scenarios were considered while selecting the clusters from these two sites :
 \begin{itemize}


@@ -911,7 +907,7 @@ Two scenarios were considered while selecting the clusters from these two sites
 The main reason 
 for using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
 scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 
 The main reason 
 for using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
 scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 

-is very low due to the higher communication times which reduce the effect of DVFS operations.
+is very low due to the higher communication times which reduces the effect of the DVFS operations.

 
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 
 
 The NAS parallel benchmarks are executed over 
 16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster 


@@ -931,7 +927,7 @@ Table~\ref{tab:sc} shows the number of nodes used from each cluster for each sce
                                       & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                       & Griffon       & Nancy        & 6                      \\ 
 \hline
                                       & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                       & Griffon       & Nancy        & 6                      \\ 
 \hline

-\multirow{3}{*}{Tow sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
+\multirow{3}{*}{Two sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 

                                       & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                       & Griffon     &Nancy           & 12                     \\ 
 \hline
                                       & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                       & Griffon     &Nancy           & 12                     \\ 
 \hline


@@ -964,7 +960,7 @@ The overall energy consumption of all the benchmarks solving the class D instanc
 using the proposed frequency selection algorithm is measured 
 using the equation of the reduced energy consumption, Equation~\ref{eq:energy}. This model uses the measured dynamic power showed in Table~\ref{table:grid5000}
 and the static 
 using the proposed frequency selection algorithm is measured 
 using the equation of the reduced energy consumption, Equation~\ref{eq:energy}. This model uses the measured dynamic power showed in Table~\ref{table:grid5000}
 and the static 

-power is assumed to be equal to 20\% of the dynamic power \textcolor{blue}{as in \cite{Rauber_Analytical.Modeling.for.Energy}}. The execution
+power is assumed to be equal to 20\% of the dynamic power as in \cite{Rauber_Analytical.Modeling.for.Energy}. The execution

 time is measured for all the benchmarks over these different scenarios.  
 
 The energy consumptions  and the execution times for all the benchmarks are 
 time is measured for all the benchmarks over these different scenarios.  
 
 The energy consumptions  and the execution times for all the benchmarks are 


@@ -997,7 +993,7 @@ results in  a lower energy consumption. Indeed, the dynamic  consumed power
 is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can 
 increase the communication times and thus produces less energy saving depending on the 
 benchmarks being executed. The results of benchmarks CG, MG, BT and FT show more 
 is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can 
 increase the communication times and thus produces less energy saving depending on the 
 benchmarks being executed. The results of benchmarks CG, MG, BT and FT show more 

-energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes.  LU and SP consume more energy with 16 nodes than 32 in one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 
+energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes.  LU and SP consume more energy with 16 nodes than 32 nodes on one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 

 
 \begin{figure*}[!h]
   \centering
 
 \begin{figure*}[!h]
   \centering


@@ -1031,13 +1027,13 @@ The best energy saving percentage was obtained in the one site scenario with 16
 Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
 The performance degradation percentage for the benchmarks running on two sites  with
 16 or 32  nodes is on average equal to 8.3\% or 4.7\% respectively. 
 Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
 The performance degradation percentage for the benchmarks running on two sites  with
 16 or 32  nodes is on average equal to 8.3\% or 4.7\% respectively. 

-For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are high with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with

 16 or 32  nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
 nodes when the communications occur in high speed network does not decrease the computations to 
 communication ratio. 
 
 The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to 
 16 or 32  nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
 nodes when the communications occur in high speed network does not decrease the computations to 
 communication ratio. 
 
 The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to 

-the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its 
+the other benchmarks. Indeed, in the EP benchmark, there are no communication and no slack times and its 

 performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
 The rest of the benchmarks showed different performance degradation percentages which decrease
 when the communication times increase and vice versa.
 performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
 The rest of the benchmarks showed different performance degradation percentages which decrease
 when the communication times increase and vice versa.


@@ -1102,7 +1098,7 @@ Scenario name                          & Cluster name & Nodes per cluster &
 
 The execution times for most of  the NAS  benchmarks are higher over the multi-core per node scenario 
 than over the single core per node  scenario. Indeed,  
 
 The execution times for most of  the NAS  benchmarks are higher over the multi-core per node scenario 
 than over the single core per node  scenario. Indeed,  

- the communication times  are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.    
+ the communication times  are higher in the  multi-core scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.    

 Moreover, the energy consumptions of the NAS benchmarks are lower over the 
  one core scenario  than over the multi-core scenario because 
 the first scenario had less execution time than the latter which results in less static energy being consumed.
 Moreover, the energy consumptions of the NAS benchmarks are lower over the 
  one core scenario  than over the multi-core scenario because 
 the first scenario had less execution time than the latter which results in less static energy being consumed.


@@ -1229,8 +1225,8 @@ The experimental results, the energy saving, performance degradation and trade-o
 presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
 
 As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
 presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
 
 As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 

-The proposed algorithm gives better results than the EDP method because it 
-maximizes the energy saving and the performance at the same time. 
+The proposed algorithm gives better results than the EDP method because the former selects the set of frequencies that  
+gives the best tradeoff between energy saving and performance. 

 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
 These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
 Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
 Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
 These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.


@@ -1246,9 +1242,9 @@ This paper presents a new online frequencies selection algorithm.
  The algorithm selects the best vector of 
 frequencies that maximizes  the trade-off distance 
 between the predicted energy consumption and the predicted execution time of the distributed 
  The algorithm selects the best vector of 
 frequencies that maximizes  the trade-off distance 
 between the predicted energy consumption and the predicted execution time of the distributed 

- applications \textcolor{blue}{with iterations} running over a heterogeneous grid. A new energy model 
+ applications with iterations running over a heterogeneous grid. A new energy model 

 is used by the proposed algorithm to predict the energy consumption 
 is used by the proposed algorithm to predict the energy consumption 

-of the distributed  message passing application \textcolor{blue}{with iterations} running over a grid architecture.
+of the application.

 To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
 NAS parallel benchmarks   and the  class D instance was executed over the  Grid'5000 testbed platform. 
 The experiments executed on 16 nodes, distributed over three clusters, showed that the algorithm   on average reduces by 30\% the energy consumption
 To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
 NAS parallel benchmarks   and the  class D instance was executed over the  Grid'5000 testbed platform. 
 The experiments executed on 16 nodes, distributed over three clusters, showed that the algorithm   on average reduces by 30\% the energy consumption


@@ -1259,19 +1255,19 @@ Finally, the proposed algorithm was compared to another method that uses
 the well known energy and delay product as an objective function. The comparison results showed 
 that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better trade-off  between energy consumption reduction and performance. 
 
 the well known energy and delay product as an objective function. The comparison results showed 
 that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better trade-off  between energy consumption reduction and performance. 
 

-In the near future, \textcolor{blue}{we will adapt the proposed algorithm to take the variability between some iterations in two steps. In the first step,  the algorithm selects the best frequencies at the end of the first iterations and apply them to the system. In the second step, after some iterations (e.g. 5 iterations) the algorithm recomputes the frequencies depending on the average of the communication and computation times  for all  previous iterations. It will change the frequency of each node if the new frequency is different from the old one. Otherwise, it keeps the old frequency.} 
-Also, we would like to develop a similar method that is adapted to
-asynchronous  applications \textcolor{blue}{with iterations} where iterations are not synchronized and communications are overlapped with computations. 
+In the near future, we will adapt the proposed algorithm to take into consideration the variability between some iterations.  For example, the proposed algorithm can be executed twice: after the first iteration the frequencies are scaled down according to the execution times measured in the first iteration, then after a fixed number of iterations, the frequencies are adjusted according to the execution times measured during the fixed number of iterations. If the  computing power of the system is constantly changing, it would be interesting to implement a mechanism that detects this change and adjusts the frequencies according to the variability of the system. We would like also to develop a similar method that is adapted to
+asynchronous  applications with iterations where iterations are not synchronized and communications are overlapped with computations. 

 The development of such a method might require a new energy model because the
 number of iterations is not known in advance and depends on
 The development of such a method might require a new energy model because the
 number of iterations is not known in advance and depends on

-the global convergence of the iterative system.
+the global convergence of the iterative system. Finally, it would be interesting to evaluate the scalability of the proposed algorithm by running it on large platforms composed of many thousands of cores. The scalability of the algorithm can be improved by distributing it  in a hierarchical manner where a leader is chosen for each cluster or a group of nodes  to compute their scaled frequencies and by using asynchronous messages to exchange the the data measured at the first iteration. 

 
 
 
 \section*{Acknowledgment}
 
 This work  has been  partially supported by  the Labex ACTION  project (contract
 
 
 
 \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 Grid'5000 platform. As  a  PhD student,
+``ANR-11-LABX-01-01'').  Computations  have been performed  on the Grid'5000
+platform and on the mésocentre of Franche-Comté. As  a  PhD student,

 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.
 
 Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
 supporting his work.