X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/08b2589e5657136a0f3a605dbd076b559df777ae..7c598f9d3894ffc00d894cfcbd7684d51fb7c14a:/Heter_paper.tex?ds=inline

diff --git a/Heter_paper.tex b/Heter_paper.tex
index a31c0fb..6be3c9f 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -25,6 +25,9 @@
 
 \newcommand{\Xsub}[2]{{\ensuremath{#1_\mathit{#2}}}}
 
+%% used to put some subscripts lower, and make them more legible
+\newcommand{\fxheight}[1]{\ifx#1\relax\relax\else\rule{0pt}{1.52ex}#1\fi}
+
 \newcommand{\CL}{\Xsub{C}{L}}
 \newcommand{\Dist}{\mathit{Dist}}
 \newcommand{\EdNew}{\Xsub{E}{dNew}}
@@ -33,24 +36,24 @@
 \newcommand{\Eoriginal}{\Xsub{E}{Original}}
 \newcommand{\Ereduced}{\Xsub{E}{Reduced}}
 \newcommand{\Es}{\Xsub{E}{S}}
-\newcommand{\Fdiff}{\Xsub{F}{diff}}
-\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
+\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}
 \newcommand{\Fnew}{\Xsub{F}{new}}
 \newcommand{\Ileak}{\Xsub{I}{leak}}
 \newcommand{\Kdesign}{\Xsub{K}{design}}
 \newcommand{\MaxDist}{\mathit{Max}\Dist}
 \newcommand{\MinTcm}{\mathit{Min}\Tcm}
 \newcommand{\Ntrans}{\Xsub{N}{trans}}
-\newcommand{\Pd}{\Xsub{P}{d}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
 \newcommand{\PdNew}{\Xsub{P}{dNew}}
 \newcommand{\PdOld}{\Xsub{P}{dOld}}
 \newcommand{\Pnorm}{\Xsub{P}{Norm}}
-\newcommand{\Ps}{\Xsub{P}{s}}
-\newcommand{\Scp}{\Xsub{S}{cp}}
-\newcommand{\Sopt}{\Xsub{S}{opt}}
-\newcommand{\Tcm}{\Xsub{T}{cm}}
-\newcommand{\Tcp}{\Xsub{T}{cp}}
-\newcommand{\TcpOld}{\Xsub{T}{cpOld}}
+\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
+\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
+\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
+\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
+\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
+\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\Told}{\Xsub{T}{Old}} 
 
@@ -85,17 +88,18 @@ platforms many techniques have been  used. Dynamic voltage and frequency scaling
 consumption.  However,  lowering the  frequency  of  a  CPU might  increase  the
 execution  time of  an application  running on  that processor.   Therefore, the
 frequency that  gives the best trade-off  between the energy  consumption and the
-performance of an application must be selected.\\
-In this  paper, a new  online frequencies selecting algorithm  for heterogeneous
-platforms is presented.   It selects the frequency which tries  to give the best
-trade-off  between  energy saving  and  performance  degradation,  for each  node
-computing the message  passing iterative application. The algorithm  has a small
+performance of an application must be selected.
+
+In this paper, a new online frequency selecting algorithm for heterogeneous
+platforms is presented.  It selects the frequencies and tries to give the best
+trade-off between energy saving and performance degradation, for each node
+computing the message passing iterative application. The algorithm has a small
 overhead and works without training or profiling. It uses a new energy model for
-message passing iterative applications  running on a heterogeneous platform. The
-proposed algorithm is  evaluated on the SimGrid simulator  while running the NAS
-parallel  benchmarks.  The  experiments   show  that  it  reduces  the  energy
-consumption by up to 35\% while  limiting the performance degradation as much as
-possible.   Finally,  the algorithm  is  compared  to  an existing  method,  the
+message passing iterative applications running on a heterogeneous platform. The
+proposed algorithm is evaluated on the SimGrid simulator while running the NAS
+parallel benchmarks.  The experiments show that it reduces the energy
+consumption by up to \np[\%]{35} while limiting the performance degradation as
+much as possible.  Finally, the algorithm is compared to an existing method, the
 comparison results showing that it outperforms the latter.
 
 \end{abstract}
@@ -203,30 +207,35 @@ consumption of this type of platform.  Naveen et
 al.~\cite{Naveen_Power.Efficient.Resource.Scaling} developed a method that
 minimizes the value of $\mathit{energy}\times \mathit{delay}^2$ (the delay is
 the sum of slack times that happen during synchronous communications) by
-dynamically assigning new frequencies to the CPUs of the heterogeneous
-cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling}
-proposed an algorithm that divides the executed tasks into two types: the
-critical and non critical tasks. The algorithm scales down the frequency of non
-critical tasks proportionally to their slack and communication times while
-limiting the performance degradation percentage to less than
-10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
+dynamically assigning new frequencies to the CPUs of the heterogeneous cluster.
+Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed an
+algorithm that divides the executed tasks into two types: the critical and non
+critical tasks. The algorithm scales down the frequency of non critical tasks
+proportionally to their slack and communication times while limiting the
+performance degradation percentage to less than \np[\%]{10}.
+In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
 heterogeneous cluster composed of two types of Intel and AMD processors. They
 use a gradient method to predict the impact of DVFS operations on performance.
 In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
 \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
 frequencies for a specified heterogeneous cluster are selected offline using
-some heuristic. Chen et
+some heuristic.  Chen et
 al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
 programming approach to minimize the power consumption of heterogeneous servers
-while respecting given time constraints. This approach had considerable
+while respecting given time constraints.  This approach had considerable
 overhead.  In contrast to the above described papers, this paper presents the
 following contributions :
 \begin{enumerate}
-\item  two new energy and performance models for message passing iterative synchronous applications running over 
-       a heterogeneous platform. Both models take into account  communication and slack times. The models can predict the required energy and the execution time of the application.
+\item two new energy and performance models for message passing iterative
+  synchronous applications running over a heterogeneous platform. Both models
+  take into account communication and slack times. The models can predict the
+  required energy and the execution time of the application.
        
-\item a new online frequency selecting algorithm for heterogeneous platforms. The algorithm has a very small 
-      overhead and does not need any training or profiling. It uses a new optimization function which simultaneously maximizes the performance and minimizes the energy consumption of a message passing iterative synchronous application.
+\item a new online frequency selecting algorithm for heterogeneous
+  platforms. The algorithm has a very small overhead and does not need any
+  training or profiling. It uses a new optimization function which
+  simultaneously maximizes the performance and minimizes the energy consumption
+  of a message passing iterative synchronous application.
       
 \end{enumerate}
 
@@ -297,14 +306,14 @@ operation. Then the execution time for one iteration of the application with any
 vector of scaling factors can be predicted using (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
-  \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld_{i}} \cdot S_{i}) +  \MinTcm
+  \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) +  \MinTcm
 \end{equation}
 Where:
 \begin{equation}
 \label{eq:perf2}
- \MinTcm = \min_{i=1,2,\dots,N} (\Tcm_i)
+ \MinTcm = \min_{i=1,2,\dots,N} (\Tcm[i])
 \end{equation}
-where  $\TcpOld_i$ is  the computation  time of  processor $i$  during  the first
+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
@@ -397,13 +406,13 @@ The static energy of a processor after scaling its frequency is computed as foll
 \end{equation}
 
 In  the  considered  heterogeneous  platform,  each  processor  $i$  might  have
-different   dynamic  and  static   powers,  noted   as  $\Pd_{i}$   and  $\Ps_{i}$
+different   dynamic  and  static   powers,  noted   as  $\Pd[i]$   and  $\Ps[i]$
 respectively.  Therefore,  even if  the  distributed  message passing  iterative
 application  is  load balanced,  the  computation time  of  each  CPU $i$  noted
-$\Tcp_{i}$ might  be different and  different frequency scaling factors  might be
+$\Tcp[i]$ might  be different and  different frequency scaling factors  might be
 computed in order to decrease  the overall energy consumption of the application
 and reduce slack  times.  The communication time of a processor  $i$ is noted as
-$\Tcm_{i}$  and could  contain slack  times when  communicating  with slower
+$\Tcm[i]$  and could  contain slack  times when  communicating  with slower
 nodes,  see Figure~\ref{fig:heter}.  Therefore,  all nodes  do  not have  equal
 communication  times. While  the dynamic  energy  is computed  according to  the
 frequency  scaling   factor  and   the  dynamic  power   of  each  node   as  in
@@ -414,8 +423,8 @@ heterogeneous platform during one iteration  is the summation of all dynamic and
 static energies for each processor.  It is computed as follows:
 \begin{multline}
   \label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd_{i} \cdot  \Tcp_i)} + {} \\
- \sum_{i=1}^{N} (\Ps_{i} \cdot (\max_{i=1,2,\dots,N} (\Tcp_i \cdot S_{i}) +
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + {} \\
+ \sum_{i=1}^{N} (\Ps[i] \cdot (\max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +
   {\MinTcm))}
  \end{multline}
 
@@ -464,8 +473,8 @@ maximum frequency for all nodes) as follows:
 \begin{multline}
   \label{eq:pnorm}
   \Pnorm = \frac{\Tnew}{\Told}\\
-       {} = \frac{ \max_{i=1,2,\dots,N} (\Tcp_{i} \cdot S_{i}) +\MinTcm}
-           {\max_{i=1,2,\dots,N}{(\Tcp_i+\Tcm_i)}}
+       {} = \frac{ \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +\MinTcm}
+           {\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
 \end{multline}
 
 
@@ -474,9 +483,9 @@ while scaling down the frequency and the consumed energy with maximum frequency
 \begin{multline}
   \label{eq:enorm}
   \Enorm = \frac{\Ereduced}{\Eoriginal} \\
-  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd_i \cdot  \Tcp_i)} +
- \sum_{i=1}^{N} {(\Ps_i \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i)} +
- \sum_{i=1}^{N} {(\Ps_i \cdot \Told)}}
+  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +
+ \sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}
 \end{multline} 
 Where $\Ereduced$ and $\Eoriginal$ are computed using (\ref{eq:energy}) and
   $\Tnew$ and $\Told$ are computed as in (\ref{eq:pnorm}).
@@ -495,8 +504,8 @@ normalized execution time is inverted which gives the normalized performance equ
 \begin{multline}
   \label{eq:pnorm_inv}
   \Pnorm = \frac{\Told}{\Tnew}\\
-          = \frac{\max_{i=1,2,\dots,N}{(\Tcp_i+\Tcm_i)}}
-            { \max_{i=1,2,\dots,N} (\Tcp_{i} \cdot S_{i}) + \MinTcm} 
+          = \frac{\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
+            { \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm} 
 \end{multline}
 
 
@@ -564,21 +573,21 @@ 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}
+ \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}
 \end{equation}
 Using the initial  frequency scaling factors computed in (\ref{eq:Scp}), the algorithm computes 
 the initial frequencies for all nodes as a ratio between the maximum frequency of node $i$  
-and the computation scaling factor $\Scp_i$ as follows:
+and the computation scaling factor $\Scp[i]$ as follows:
 \begin{equation}
   \label{eq:Fint}
- F_{i} = \frac{\Fmax_i}{\Scp_i},~{i=1,2,\dots,N}
+ F_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}
 \end{equation}
 If the computed initial frequency for a node is not available in the gears of
 that node, it is replaced by the nearest available frequency.  In
 Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
 ascending order and the frequencies of the faster nodes are scaled down
 according to the computed initial frequency scaling factors.  The resulting new
-frequencies are colored in blue in Figure~\ref{fig:st_freq}.  This set of
+frequencies are highlighted in Figure~\ref{fig:st_freq}.  This set of
 frequencies can be considered as a higher bound for the search space of the
 optimal vector of frequencies because selecting frequency scaling factors higher
 than the higher bound will not improve the performance of the application and it
@@ -623,42 +632,42 @@ maximum distance  between the  energy curve and  the performance curve  is while
     % \footnotesize
     \Require ~
     \begin{description}
-    \item[$\Tcp_i$] array of all computation times for all nodes during one iteration and with highest frequency.
-    \item[$\Tcm_i$] array of all communication times for all nodes during one iteration and with highest frequency.
-    \item[$\Fmax_i$] array of the maximum frequencies for all nodes.
-    \item[$\Pd_i$] array of the dynamic powers for all nodes.
-    \item[$\Ps_i$] array of the static powers for all nodes.
-    \item[$\Fdiff_i$] array of the difference between two successive frequencies for all nodes.
+    \item[{$\Tcp[i]$}] array of all computation times for all nodes during one iteration and with highest frequency.
+    \item[{$\Tcm[i]$}] array of all communication times for all nodes during one iteration and with highest frequency.
+    \item[{$\Fmax[i]$}] array of the maximum frequencies for all nodes.
+    \item[{$\Pd[i]$}] array of the dynamic powers for all nodes.
+    \item[{$\Ps[i]$}] array of the static powers for all nodes.
+    \item[{$\Fdiff[i]$}] array of the difference between two successive frequencies for all nodes.
     \end{description}
-    \Ensure $\Sopt_1,\Sopt_2 \dots, \Sopt_N$ is a vector of optimal scaling factors
+    \Ensure $\Sopt[1],\Sopt[2] \dots, \Sopt[N]$ is a vector of optimal scaling factors
 
-    \State $\Scp_i \gets \frac{\max_{i=1,2,\dots,N}(\Tcp_i)}{\Tcp_i} $
-    \State $F_{i} \gets  \frac{\Fmax_i}{\Scp_i},~{i=1,2,\cdots,N}$
+    \State $\Scp[i] \gets \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]} $
+    \State $F_{i} \gets  \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\cdots,N}$
     \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
     \If{(not the first frequency)}
-          \State $F_i \gets F_i+\Fdiff_i,~i=1,\dots,N.$
+          \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
     \EndIf 
-    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp_i+\Tcm_i)$
-    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i)} +\sum_{i=1}^{N} {(\Ps_i \cdot \Told)}$
-    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i + \Ps_i \cdot \Told)}$
-    \State $\Sopt_{i} \gets 1,~i=1,\dots,N. $
+    \State $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
+    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i])} +\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}$
+    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \Told)}$
+    \State $\Sopt[i] \gets 1,~i=1,\dots,N. $
     \State $\Dist \gets 0 $
     \While {(all nodes not reach their  minimum  frequency)}
         \If{(not the last freq. \textbf{and} not the slowest node)}
-        \State $F_i \gets F_i - \Fdiff_i,~i=1,\dots,N.$
-        \State $S_i \gets \frac{\Fmax_i}{F_i},~i=1,\dots,N.$
+        \State $F_i \gets F_i - \Fdiff[i],~i=1,\dots,N.$
+        \State $S_i \gets \frac{\Fmax[i]}{F_i},~i=1,\dots,N.$
         \EndIf
-       \State $\Tnew \gets \max_{i=1,\dots,N} (\Tcp_{i} \cdot S_{i}) + \MinTcm $
-%       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd_i \cdot  \Tcp_i)} + \sum_{i=1}^{N} {(\Ps_i \cdot \rlap{\Tnew)}} $
-       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd_i \cdot  \Tcp_i + \Ps_i \cdot \rlap{\Tnew)}} $
+       \State $\Tnew \gets \max_{i=1,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm $
+%       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i])} + \sum_{i=1}^{N} {(\Ps[i] \cdot \rlap{\Tnew)}} $
+       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot  \Tcp[i] + \Ps[i] \cdot \rlap{\Tnew)}} $
        \State $\Pnorm \gets \frac{\Told}{\Tnew}$
        \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
-        \State $\Sopt_{i} \gets S_{i},~i=1,\dots,N. $
+        \State $\Sopt[i] \gets S_{i},~i=1,\dots,N. $
         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile
-    \State  Return $\Sopt_1,\Sopt_2,\dots,\Sopt_N$
+    \State  Return $\Sopt[1],\Sopt[2],\dots,\Sopt[N]$
   \end{algorithmic}
   \caption{frequency scaling factors selection algorithm}
   \label{HSA}
@@ -711,33 +720,33 @@ 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 $F$
+factors vector.  The algorithm complexity is $O(F\cdot N)$, where $F$
 is the number of iterations and $N$ is the number of computing nodes. The
 algorithm needs from 12 to 20 iterations to select the best vector of frequency
 scaling factors that gives the results of the next sections.
 
 \section{Experimental results}
 \label{sec.expe}
-To  evaluate the  efficiency and  the  overall energy  consumption reduction  of
+To evaluate the efficiency and the overall energy consumption reduction of
 Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The
-experiments were executed on the  simulator SimGrid/SMPI which offers easy tools
+experiments were executed on the simulator SimGrid/SMPI which offers easy tools
 to create a heterogeneous platform and run message passing applications over it.
-The heterogeneous  platform that was used  in the experiments, had  one core per
+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
+was composed of four types of nodes. Each type of nodes had different
+characteristics such as the maximum CPU frequency, the number of available
+frequencies and the computational power, see Table~\ref{table:platform}. The
+characteristics of these different types of nodes are inspired from the
+specifications of real Intel processors.  The heterogeneous platform had up to
 144 nodes and had nodes from the four types in equal proportions, for example if
 a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the
-constructors of  CPUs do not specify the  dynamic and the static  power of their
-CPUs, for  each type of  node they were  chosen proportionally to  its computing
-power  (FLOPS).  In  the initial  heterogeneous platform,  while  computing with
-highest frequency,  each node  consumed an amount  of power proportional  to its
-computing  power  (which  corresponds to  80\%  of  its  dynamic power  and  the
-remaining  20\%  to  the  static   power),  the  same  assumption  was  made  in
-\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.    Finally,  These
+constructors of CPUs do not specify the dynamic and the static power of their
+CPUs, for each type of node they were chosen proportionally to its computing
+power (FLOPS).  In the initial heterogeneous platform, while computing with
+highest frequency, each node consumed an amount of power proportional to its
+computing power (which corresponds to \np[\%]{80} of its dynamic power and the
+remaining \np[\%]{20} to the static power), the same assumption was made in
+\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.  Finally, These
 nodes were connected via an Ethernet network with 1 Gbit/s bandwidth.
 
 
@@ -745,22 +754,22 @@ nodes were connected via an Ethernet network with 1 Gbit/s bandwidth.
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
     Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
     type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
                   &           & GHz      & GHz          &GHz             &              &       \\
     \hline
-    1             &40         & 2.5      & 1.2          & 0.1            & 20~W         &4~W    \\
+    1             &40         & 2.50     & 1.20         & 0.100          & \np[W]{20}   &\np[W]{4} \\
          
     \hline
-    2             &50         & 2.66     & 1.6          & 0.133          & 25~W         &5~W    \\
+    2             &50         & 2.66     & 1.60         & 0.133          & \np[W]{25}   &\np[W]{5} \\
                   
     \hline
-    3             &60         & 2.9      & 1.2          & 0.1            & 30~W         &6~W    \\
+    3             &60         & 2.90     & 1.20         & 0.100          & \np[W]{30}   &\np[W]{6} \\
                   
     \hline
-    4             &70         & 3.4      & 1.6          & 0.133          & 35~W         &7~W    \\
+    4             &70         & 3.40     & 1.60         & 0.133          & \np[W]{35}   &\np[W]{7} \\
                   
     \hline
   \end{tabular}
@@ -958,47 +967,47 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
   \end{tabular}
   \label{table:res_128n}
 \end{table}
-The overall energy  consumption was computed for each  instance according to the
-energy  consumption  model  (\ref{eq:energy}),  with and  without  applying  the
+The overall energy consumption was computed for each instance according to the
+energy consumption model (\ref{eq:energy}), with and without applying the
 algorithm. The execution time was also measured for all these experiments. Then,
 the energy saving and performance degradation percentages were computed for each
-instance.    The   results   are   presented  in   Tables~\ref{table:res_4n},
-\ref{table:res_8n},           \ref{table:res_16n},          \ref{table:res_32n},
+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.
+values from many experiments for energy savings and performance degradation.
 The tables show the experimental results for running the NAS parallel benchmarks
-on  different  number  of  nodes.   The  experiments  show  that  the  algorithm
-significantly reduces the energy consumption (up to 35\%) and tries to limit the
-performance  degradation.  They  also  show that  the  energy saving  percentage
-decreases when the  number of computing nodes increases.   This reduction is due
-to the increase of the communication  times compared to the execution times when
-the benchmarks are run over a high number of nodes.  Indeed, the benchmarks with
-the  same  class,  C,  are  executed  on different  numbers  of  nodes,  so  the
-computation required  for each iteration is  divided by the  number of computing
-nodes.  On the other hand,  more communications are required when increasing the
-number  of  nodes so  the  static energy  increases  linearly  according to  the
-communication time and the dynamic power  is less relevant in the overall energy
-consumption.   Therefore, reducing the  frequency with  Algorithm~\ref{HSA} is
-less effective  in reducing the overall  energy savings. It can  also be noticed
-that for the benchmarks EP and  SP that contain little or no communications, the
-energy savings are  not significantly affected by the high  number of nodes.  No
-experiments were conducted  using bigger classes than D,  because they require a
-lot  of memory (more  than 64GB)  when being  executed by  the simulator  on one
-machine.   The maximum  distance between  the  normalized energy  curve and  the
-normalized performance for each instance is  also shown in the result tables. It
-decrease in the same way as  the energy saving percentage.  The tables also show
-that the performance degradation  percentage is not significantly increased when
-the number  of computing  nodes is increased  because the computation  times are
-small when compared to the communication times.
+on different number of nodes.  The experiments show that the algorithm
+significantly reduces the energy consumption (up to \np[\%]{35}) and tries to
+limit the performance degradation.  They also show that the energy saving
+percentage decreases when the number of computing nodes increases.  This
+reduction is due to the increase of the communication times compared to the
+execution times when the benchmarks are run over a higher number of nodes.
+Indeed, the benchmarks with the same class, C, are executed on different numbers
+of nodes, so the computation required for each iteration is divided by the
+number of computing nodes.  On the other hand, more communications are required
+when increasing the number of nodes so the static energy increases linearly
+according to the communication time and the dynamic power is less relevant in
+the overall energy consumption.  Therefore, reducing the frequency with
+Algorithm~\ref{HSA} is less effective in reducing the overall energy savings. It
+can also be noticed that for the benchmarks EP and SP that contain little or no
+communications, the energy savings are not significantly affected by the high
+number of nodes.  No experiments were conducted using bigger classes than D,
+because they require a lot of memory (more than 64GB) when being executed by the
+simulator on one machine.  The maximum distance between the normalized energy
+curve and the normalized performance for each instance is also shown in the
+result tables. It decrease in the same way as the energy saving percentage.  The
+tables also show that the performance degradation percentage is not
+significantly increased when the number of computing nodes is increased because
+the computation times are small when compared to the communication times.
 
 
  
 \begin{figure}[!t]
   \centering
-  \subfloat[Energy saving]{%
+  \subfloat[Energy saving (\%)]{%
     \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
   
-  \subfloat[Performance degradation ]{%
+  \subfloat[Performance degradation (\%)]{%
     \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
   \label{fig:avg}
   \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}
@@ -1023,59 +1032,61 @@ has less effect on the performance.
 
 \subsection{The results for different power consumption scenarios}
 \label{sec.compare}
-The results  of the previous section  were obtained while  using processors that
-consume during  computation an overall power  which is 80\%  composed of dynamic
-power and of 20\% of static power. In this section, these ratios are changed and
-two new  power scenarios are  considered in order  to evaluate how  the proposed
-algorithm adapts itself  according to the static and  dynamic power values.  The
-two new power scenarios are the following:
+The results of the previous section were obtained while using processors that
+consume during computation an overall power which is \np[\%]{80} composed of
+dynamic power and of \np[\%]{20} of static power. In this section, these ratios
+are changed and two new power scenarios are considered in order to evaluate how
+the proposed algorithm adapts itself according to the static and dynamic power
+values.  The two new power scenarios are the following:
 
 \begin{itemize}
-\item 70\% of dynamic power  and 30\% of static power
-\item 90\% of dynamic power  and 10\% of static power
+\item \np[\%]{70} of dynamic power and \np[\%]{30} of static power
+\item \np[\%]{90} of dynamic power and \np[\%]{10} of static power
 \end{itemize}
 
-The NAS parallel benchmarks were  executed again over processors that follow the
-new power scenarios.   The class C of each  benchmark was run over 8  or 9 nodes
-and   the    results   are   presented   in    Tables~\ref{table:res_s1}   and
-\ref{table:res_s2}. These tables  show that the energy saving  percentage of the
-70\%-30\% scenario is  smaller for all benchmarks compared  to the energy saving
-of the 90\%-10\% scenario. Indeed, in  the latter more dynamic power is consumed
-when  nodes are running  on their  maximum frequencies,  thus, scaling  down the
-frequency of  the nodes results in  higher energy savings than  in the 70\%-30\%
-scenario. On the  other hand, the performance degradation  percentage is smaller
-in the 70\%-30\% scenario compared to the 90\%-10\% scenario. This is due to the
-higher  static  power percentage  in  the first  scenario  which  makes it  more
-relevant in the  overall consumed energy.  Indeed, the  static energy is related
-to the execution time and if  the performance is degraded the amount of consumed
-static  energy directly  increases.  Therefore,  the proposed  algorithm  does not
-really significantly  scale down much the  frequencies of the nodes  in order to
-limit the  increase of the  execution time and  thus limiting the effect  of the
+The NAS parallel benchmarks were executed again over processors that follow the
+new power scenarios.  The class C of each benchmark was run over 8 or 9 nodes
+and the results are presented in Tables~\ref{table:res_s1} and
+\ref{table:res_s2}. These tables show that the energy saving percentage of the
+\np[\%]{70}-\np[\%]{30} scenario is smaller for all benchmarks compared to the
+energy saving of the \np[\%]{90}-\np[\%]{10} scenario.  Indeed, in the latter
+more dynamic power is consumed when nodes are running on their maximum
+frequencies, thus, scaling down the frequency of the nodes results in higher
+energy savings than in the \np[\%]{70}-\np[\%]{30} scenario. On the other hand,
+the performance degradation percentage is smaller in the \np[\%]{70}-\np[\%]{30}
+scenario compared to the \np[\%]{90}-\np[\%]{10} scenario. This is due to the
+higher static power percentage in the first scenario which makes it more
+relevant in the overall consumed energy.  Indeed, the static energy is related
+to the execution time and if the performance is degraded the amount of consumed
+static energy directly increases.  Therefore, the proposed algorithm does not
+really significantly scale down much the frequencies of the nodes in order to
+limit the increase of the execution time and thus limiting the effect of the
 consumed static energy.
 
-Both   new  power   scenarios   are  compared   to   the  old   one  in
-Figure~\ref{fig:sen_comp}. It  shows the average of the  performance degradation, the
-energy saving and the  distances for all NAS benchmarks of class  C running on 8
-or 9 nodes.   The comparison shows that the energy  saving ratio is proportional
-to the dynamic power ratio: it is increased when applying the 90\%-10\% scenario
-because at  maximum frequency  the dynamic  energy is the  most relevant  in the
-overall consumed  energy and can  be reduced by  lowering the frequency  of some
-processors. On  the other hand, the  energy saving decreases  when the 70\%-30\%
-scenario is  used because  the dynamic  energy is less  relevant in  the overall
-consumed energy and  lowering the frequency does not  return big energy savings.
-Moreover, the average  of the performance degradation is  decreased when using a
-higher  ratio   for  static  power  (e.g.   70\%-30\%   scenario  and  80\%-20\%
-scenario). Since  the proposed algorithm  optimizes the energy  consumption when
-using a  higher ratio for dynamic  power the algorithm  selects bigger frequency
-scaling  factors that result  in more  energy saving  but less  performance, for
-example see  Figure~\ref{fig:scales_comp}. The  opposite happens when  using a
-higher  ratio for  static power,  the algorithm  proportionally  selects smaller
-scaling  values which result  in less  energy saving  but also  less performance
+Both new power scenarios are compared to the old one in
+Figure~\ref{fig:sen_comp}. It shows the average of the performance degradation,
+the energy saving and the distances for all NAS benchmarks of class C running on
+8 or 9 nodes.  The comparison shows that the energy saving ratio is proportional
+to the dynamic power ratio: it is increased when applying the
+\np[\%]{90}-\np[\%]{10} scenario because at maximum frequency the dynamic energy
+is the most relevant in the overall consumed energy and can be reduced by
+lowering the frequency of some processors. On the other hand, the energy saving
+decreases when the \np[\%]{70}-\np[\%]{30} scenario is used because the dynamic
+energy is less relevant in the overall consumed energy and lowering the
+frequency does not return big energy savings.  Moreover, the average of the
+performance degradation is decreased when using a higher ratio for static power
+(e.g.  \np[\%]{70}-\np[\%]{30} scenario and \np[\%]{80}-\np[\%]{20}
+scenario). Since the proposed algorithm optimizes the energy consumption when
+using a higher ratio for dynamic power the algorithm selects bigger frequency
+scaling factors that result in more energy saving but less performance, for
+example see Figure~\ref{fig:scales_comp}. The opposite happens when using a
+higher ratio for static power, the algorithm proportionally selects smaller
+scaling values which result in less energy saving but also less performance
 degradation.
 
 
  \begin{table}[!t]
-  \caption{The results of the 70\%-30\% power scenario}
+  \caption{The results of the \np[\%]{70}-\np[\%]{30} power scenario}
   % title of Table
   \centering
   \begin{tabular}{|*{6}{r|}}
@@ -1104,7 +1115,7 @@ degradation.
 
 
 \begin{table}[!t]
-  \caption{The results of the 90\%-10\% power scenario}
+  \caption{The results of the \np[\%]{90}-\np[\%]{10} power scenario}
   % title of Table
   \centering
   \begin{tabular}{|*{6}{r|}}
@@ -1191,9 +1202,10 @@ efficiency. Table~\ref{table:compare_EDP} presents the results of comparing the
 execution times and the energy consumption for both versions of the NAS
 benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
 nodes. The results show that our algorithm provides better energy savings than
-Spiliopoulos et al. algorithm, on average it results in 29.76\% energy saving
-while their algorithm returns just 25.75\%. The average of performance
-degradation percentage is approximately the same for both algorithms, about 4\%.
+Spiliopoulos et al. algorithm, on average it results in \np[\%]{29.76} energy
+saving while their algorithm returns just \np[\%]{25.75}. The average of
+performance degradation percentage is approximately the same for both
+algorithms, about \np[\%]{4}.
 
 
 For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
@@ -1205,20 +1217,20 @@ degradation values while giving the same weight for both metrics.
 \section{Conclusion}
 \label{sec.concl} 
 In this paper, a new online frequency selecting algorithm has been presented. It
-selects the  best possible  vector of frequency  scaling factors that  gives the
-maximum  distance  (optimal  trade-off)  between  the predicted  energy  and  the
+selects the best possible vector of frequency scaling factors that gives the
+maximum distance (optimal trade-off) between the predicted energy and the
 predicted performance curves for a heterogeneous platform. This algorithm uses a
-new  energy  model  for  measuring  and predicting  the  energy  of  distributed
-iterative  applications running  over heterogeneous  platforms. To  evaluate the
+new energy model for measuring and predicting the energy of distributed
+iterative applications running over heterogeneous platforms. To evaluate the
 proposed method, it was applied on the NAS parallel benchmarks and executed over
-a heterogeneous  platform simulated by  SimGrid. The results of  the experiments
-showed that the algorithm reduces up to 35\% the energy consumption of a message
-passing iterative method while limiting  the degradation of the performance. The
-algorithm also selects different scaling  factors according to the percentage of
-the computing and communication times, and according to the values of the static
-and  dynamic  powers  of the  CPUs.   Finally,  the  algorithm was  compared  to
-Spiliopoulos et al.  algorithm and  the results showed that it outperforms their
-algorithm in terms of energy-time trade-off.
+a heterogeneous platform simulated by SimGrid. The results of the experiments
+showed that the algorithm reduces up to \np[\%]{35} the energy consumption of a
+message passing iterative method while limiting the degradation of the
+performance. The algorithm also selects different scaling factors according to
+the percentage of the computing and communication times, and according to the
+values of the static and dynamic powers of the CPUs.  Finally, the algorithm was
+compared to Spiliopoulos et al.  algorithm and the results showed that it
+outperforms their algorithm in terms of energy-time trade-off.
 
 In the near future, this method  will be applied to real heterogeneous platforms
 to evaluate its  performance in a real study case. It  would also be interesting
