change of performance to execution time in a paper and plots

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 06dbb3052ebd438e55f8abeaab475998268171f8..b272684ae8931f84ada9d6f2543cea793822fe50 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -6,8 +6,6 @@
 \usepackage{algpseudocode}
 \usepackage{graphicx,graphics}
 \usepackage{subfig}
 \usepackage{algpseudocode}
 \usepackage{graphicx,graphics}
 \usepackage{subfig}

-\usepackage{listings}
-\usepackage{colortbl}

 \usepackage{amsmath}
 
 \usepackage{url}
 \usepackage{amsmath}
 
 \usepackage{url}


@@ -25,6 +23,34 @@
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 

+\newcommand{\Xsub}[2]{\ensuremath{#1_\textit{#2}}}
+
+\newcommand{\Dist}{\textit{Dist}}
+\newcommand{\Eind}{\Xsub{E}{ind}}
+\newcommand{\Enorm}{\Xsub{E}{Norm}}
+\newcommand{\Eoriginal}{\Xsub{E}{Original}}
+\newcommand{\Ereduced}{\Xsub{E}{Reduced}}
+\newcommand{\Fdiff}{\Xsub{F}{diff}}
+\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Fnew}{\Xsub{F}{new}}
+\newcommand{\Ileak}{\Xsub{I}{leak}}
+\newcommand{\Kdesign}{\Xsub{K}{design}}
+\newcommand{\MaxDist}{\textit{Max Dist}}
+\newcommand{\Ntrans}{\Xsub{N}{trans}}
+\newcommand{\Pdyn}{\Xsub{P}{dyn}}
+\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
+\newcommand{\Pnorm}{\Xsub{P}{Norm}}
+\newcommand{\Tnorm}{\Xsub{T}{Norm}}
+\newcommand{\Pstates}{\Xsub{P}{states}}
+\newcommand{\Pstatic}{\Xsub{P}{static}}
+\newcommand{\Sopt}{\Xsub{S}{opt}}
+\newcommand{\Tcomp}{\Xsub{T}{comp}}
+\newcommand{\TmaxCommOld}{\Xsub{T}{Max Comm Old}}
+\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
+\newcommand{\Tmax}{\Xsub{T}{max}}
+\newcommand{\Tnew}{\Xsub{T}{New}}
+\newcommand{\Told}{\Xsub{T}{Old}}
+

 \begin{document}
 
 \title{Dynamic Frequency Scaling for Energy Consumption
 \begin{document}
 
 \title{Dynamic Frequency Scaling for Energy Consumption


@@ -111,7 +137,7 @@ the MPI program to choose the scaling factor.  This algorithm has the ability to
 predict both energy consumption and execution time over all available scaling
 factors.  The prediction achieved depends on some computing time information,
 gathered at the beginning of the runtime.  We apply this algorithm to the NAS parallel benchmarks (NPB v3.3)~\cite{44}.  Our experiments are executed using the simulator
 predict both energy consumption and execution time over all available scaling
 factors.  The prediction achieved depends on some computing time information,
 gathered at the beginning of the runtime.  We apply this algorithm to the NAS parallel benchmarks (NPB v3.3)~\cite{44}.  Our experiments are executed using the simulator

-SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over an homogeneous
+SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over a homogeneous

 distributed memory architecture.  Furthermore, we compare the proposed algorithm
 with Rauber and Rünger methods~\cite{3}.  The comparison's results show that our
 algorithm gives better energy-time trade-off.
 distributed memory architecture.  Furthermore, we compare the proposed algorithm
 with Rauber and Rünger methods~\cite{3}.  The comparison's results show that our
 algorithm gives better energy-time trade-off.


@@ -203,33 +229,33 @@ our paper is to present a new online scaling factor selection method which has t
 %   paper in homogeneous clusters}
 
 
 %   paper in homogeneous clusters}
 
 

-\section{Energy model for homogeneous platform}
+\section{Energy model for a homogeneous platform}

 \label{sec.exe}
 Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
 two power metrics: the static and the dynamic power.  While the first one is
 consumed as long as the computing unit is on, the latter is only consumed during
 \label{sec.exe}
 Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
 two power metrics: the static and the dynamic power.  While the first one is
 consumed as long as the computing unit is on, the latter is only consumed during

-computation times.  The dynamic power $P_{dyn}$ is related to the switching
+computation times.  The dynamic power $\Pdyn$ is related to the switching

 activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
 activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and

-operational frequency $f$, as shown in EQ~(\ref{eq:pd}).
+operational frequency $f$, as shown in EQ~\eqref{eq:pd}.

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

-  P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
+  \Pdyn = \alpha \cdot C_L \cdot V^2 \cdot f

 \end{equation}
 \end{equation}

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

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

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

 \end{equation}
 \end{equation}

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

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

-   E_\textit{ind} =  P_\textit{dyn} \cdot T_{Comp} + P_\textit{static} \cdot T
+   \Eind =  \Pdyn \cdot \Tcomp + \Pstatic \cdot T

 \end{equation}
 \end{equation}

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

 communication, no slack time and no synchronization.
 
 DVFS is a process that is allowed in modern processors to reduce the dynamic
 communication, no slack time and no synchronization.
 
 DVFS is a process that is allowed in modern processors to reduce the dynamic


@@ -239,10 +265,10 @@ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot f$ with some
 constant $\beta$.  This equation is used to study the change of the dynamic
 voltage with respect to various frequency values in~\cite{3}.  The reduction
 process of the frequency can be expressed by the scaling factor $S$ which is the
 constant $\beta$.  This equation is used to study the change of the dynamic
 voltage with respect to various frequency values in~\cite{3}.  The reduction
 process of the frequency can be expressed by the scaling factor $S$ which is the

-ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
+ratio between the maximum and the new frequency as in EQ~\eqref{eq:s}.

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

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

 \end{equation}
 The value of the scaling factor $S$ is greater than 1 when changing the
 frequency of the CPU to any new frequency value~(\emph{P-state}) in the
 \end{equation}
 The value of the scaling factor $S$ is greater than 1 when changing the
 frequency of the CPU to any new frequency value~(\emph{P-state}) in the


@@ -252,33 +278,32 @@ increase of the static energy because the execution time is increased~\cite{36}.
 If the tasks are sorted according to their execution times before scaling in a
 descending order, the total energy consumption model for a parallel homogeneous
 platform, as presented by Rauber and Rünger~\cite{3}, can be written as a
 If the tasks are sorted according to their execution times before scaling in a
 descending order, the total energy consumption model for a parallel homogeneous
 platform, as presented by Rauber and Rünger~\cite{3}, can be written as a

-function of the scaling factor $S$, as in EQ~(\ref{eq:energy}).
+function of the scaling factor $S$, as in EQ~\eqref{eq:energy}.

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

-  E = P_\textit{dyn} \cdot S_1^{-2} \cdot
+  E = \Pdyn \cdot S_1^{-2} \cdot

     \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
     \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +

-    P_\textit{static} \cdot T_1 \cdot S_1 \cdot N
- \hfill
+      \Pstatic \cdot T_1 \cdot S_1 \cdot N

 \end{equation}
 where $N$ is the number of parallel nodes, $T_i$ for $i=1,\dots,N$ are
 \end{equation}
 where $N$ is the number of parallel nodes, $T_i$ for $i=1,\dots,N$ are

-the execution times and scaling factors of the sorted tasks.  Therefore, $T1$ is
+the execution times of the sorted tasks.  Therefore, $T_1$ is

 the time of the slowest task, and $S_1$ its scaling factor which should be the
 highest because they are proportional to the time values $T_i$.  The scaling
 the time of the slowest task, and $S_1$ its scaling factor which should be the
 highest because they are proportional to the time values $T_i$.  The scaling

-factors are computed as in EQ~(\ref{eq:si}).
+factors $S_i$ are computed as in EQ~\eqref{eq:si}.

 \begin{equation}
   \label{eq:si}
   S_i = S \cdot \frac{T_1}{T_i}
 \begin{equation}
   \label{eq:si}
   S_i = S \cdot \frac{T_1}{T_i}

-      = \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}
+      = \frac{\Fmax}{\Fnew} \cdot \frac{T_1}{T_i}

 \end{equation}
 \end{equation}

-In this paper we use Rauber and Rünger's energy model, EQ~(\ref{eq:energy}), because it can be applied to homogeneous clusters if the communication time is taken in consideration. Moreover, we compare our algorithm with Rauber and Rünger's scaling factor selection
+In this paper we use Rauber and Rünger's energy model, EQ~\eqref{eq:energy}, because it can be applied to homogeneous clusters if the communication time is taken in consideration. Moreover, we compare our algorithm with Rauber and Rünger's scaling factor selection

 method which uses the same energy model.  In their method, the optimal scaling factor is
 method which uses the same energy model.  In their method, the optimal scaling factor is

-computed by minimizing the derivation of EQ~(\ref{eq:energy}) which produces
-EQ~(\ref{eq:sopt}).
+computed by minimizing the derivation of EQ~\eqref{eq:energy} which produces
+EQ~\eqref{eq:sopt}.

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

-  S_\textit{opt} = \sqrt[3]{\frac{2}{N} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot
+  \Sopt = \sqrt[3]{\frac{2}{N} \cdot \frac{\Pdyn}{\Pstatic} \cdot

     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 
     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 


@@ -286,7 +311,7 @@ EQ~(\ref{eq:sopt}).
 \section{Performance evaluation of MPI programs}
 \label{sec.mpip}
 
 \section{Performance evaluation of MPI programs}
 \label{sec.mpip}
 

-The performance (execution time) of parallel synchronous MPI applications depends
+The execution time of parallel synchronous MPI applications depends

 on the time of the slowest task.  If there is no
 communication and the application is not data bounded, the execution time of a
 parallel program is linearly proportional to the operational frequency and any
 on the time of the slowest task.  If there is no
 communication and the application is not data bounded, the execution time of a
 parallel program is linearly proportional to the operational frequency and any


@@ -301,10 +326,10 @@ or receiving a message till the message is synchronously sent or received.  To
 be able to predict the execution time of MPI program, the communication time and
 the computation time for the slowest task must be measured before scaling.  These
 times are used to predict the execution time for any MPI program as a function
 be able to predict the execution time of MPI program, the communication time and
 the computation time for the slowest task must be measured before scaling.  These
 times are used to predict the execution time for any MPI program as a function

-of the new scaling factor as in EQ~(\ref{eq:tnew}).
+of the new scaling factor as in EQ~\eqref{eq:tnew}.

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

- \textit  T_\textit{new} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
+  \Tnew = \TmaxCompOld \cdot S + \TmaxCommOld

 \end{equation}
 In this paper, this prediction method is used to select the best scaling factor
 for each processor as presented in the next section.
 \end{equation}
 In this paper, this prediction method is used to select the best scaling factor
 for each processor as presented in the next section.


@@ -314,50 +339,49 @@ for each processor as presented in the next section.
 
 This section presents our approach for choosing the optimal scaling factor.
 This factor gives maximum energy reduction while taking into account the execution
 
 This section presents our approach for choosing the optimal scaling factor.
 This factor gives maximum energy reduction while taking into account the execution

-times for both computation and communication.  The relation between the performance
-and the energy is nonlinear and complex. Thus, unlike the relation between the performance and  the scaling factor,  the relation of energy with the scaling factor is nonlinear, for more details refer to~\cite{17}.  Moreover, they are not measured using the same metric.  To
+times for both computation and communication.  The relation between the execution time
+and the energy is nonlinear and complex. Thus, unlike the relation between the execution time and  the scaling factor,  the relation of energy with the scaling factor is nonlinear, for more details refer to~\cite{17}.  Moreover, they are not measured using the same metric.  To

 solve this problem, we normalize the energy by calculating the ratio between
 the consumed energy with scaled frequency and the consumed energy without scaled
 frequency:
 \begin{multline}
   \label{eq:enorm}
 solve this problem, we normalize the energy by calculating the ratio between
 the consumed energy with scaled frequency and the consumed energy without scaled
 frequency:
 \begin{multline}
   \label{eq:enorm}

-  E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
-        {} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
-               \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
-               P_\textit{static} \cdot T_1 \cdot S_1 \cdot N  }{
-              P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
-              P_\textit{static} \cdot T_1 \cdot N }
+  \Enorm = \frac{ \Ereduced}{\Eoriginal} \\
+      {} = \frac{\Pdyn \cdot S_1^{-2} \cdot
+             \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+               \Pstatic \cdot T_1 \cdot S_1 \cdot N}{
+             \Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+               \Pstatic \cdot T_1 \cdot N }

 \end{multline}
 \end{multline}

-In the same way we can normalize the performance as follows:
+In the same way we can normalize the time as follows:

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

-  P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}
-          = \frac{T_\textit{Max Comp Old} \cdot S +
-           T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} +
-           T_\textit{Max Comm Old}}
+  \Tnorm = \frac{\Tnew}{\Told}
+         = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{
+             \TmaxCompOld + \TmaxCommOld}

 \end{equation}
 The second problem is that the optimization operation for both energy and
 \end{equation}
 The second problem is that the optimization operation for both energy and

-performance is not in the same direction.  In other words, the normalized energy
-and the performance curves are not at the same direction see
+execution time is not in the same direction.  In other words, the normalized energy
+and the execution time curves are not at the same direction see

 Figure~\ref{fig:rel}\subref{fig:r2}.  While the main goal is to optimize the
 Figure~\ref{fig:rel}\subref{fig:r2}.  While the main goal is to optimize the

-energy and performance in the same time.  According to the
-equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the scaling factor $S$ reduce
-both the energy and the performance simultaneously.  But the main objective is
-to produce maximum energy reduction with minimum performance reduction.  Many
+energy and execution time in the same time.  According to the
+equations~\eqref{eq:enorm} and~\eqref{eq:pnorm}, the scaling factor $S$ reduce
+both the energy and the execution time simultaneously.  But the main objective is
+to produce maximum energy reduction with minimum execution time reduction.  Many

 researchers used different strategies to solve this nonlinear problem for
 example see~\cite{19,42}, their methods add big overheads to the algorithm to
 select the suitable frequency.  In this paper we present a method to find the
 researchers used different strategies to solve this nonlinear problem for
 example see~\cite{19,42}, their methods add big overheads to the algorithm to
 select the suitable frequency.  In this paper we present a method to find the

-optimal scaling factor $S$ to optimize both energy and performance
+optimal scaling factor $S$ to optimize both energy and execution time

 simultaneously without adding a big overhead.  Our solution for this problem is
 simultaneously without adding a big overhead.  Our solution for this problem is

-to make the optimization process for energy and performance follow the same
-direction.  Therefore, we inverse the equation of the normalized performance as
+to make the optimization process for energy and execution time follow the same
+direction.  Therefore, we inverse the equation of the normalized execution time as

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

-  P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
-               = \frac{T_\textit{Max Comp Old} +
-                 T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} \cdot S +
-                 T_\textit{Max Comm Old}}
+  \Pnorm = \frac{ \Told}{ \Tnew}
+               = \frac{\TmaxCompOld +
+                 \TmaxCommOld}{\TmaxCompOld \cdot S +
+                 \TmaxCommOld}

 \end{equation}
 \begin{figure}
   \centering
 \end{equation}
 \begin{figure}
   \centering


@@ -369,19 +393,19 @@ follows:
   \label{fig:rel}
 \end{figure}
 Then, we can model our objective function as finding the maximum distance
   \label{fig:rel}
 \end{figure}
 Then, we can model our objective function as finding the maximum distance

-between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance
-curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors.  This
+between the energy curve EQ~\eqref{eq:enorm} and the inverse of execution time (performance)
+curve EQ~\eqref{eq:pnorm_en} over all available scaling factors.  This

 represents the minimum energy consumption with minimum execution time (better
 performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}.  Then
 our objective function has the following form:
 \begin{equation}
   \label{eq:max}
 represents the minimum energy consumption with minimum execution time (better
 performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}.  Then
 our objective function has the following form:
 \begin{equation}
   \label{eq:max}

-  \textit{Max Dist} = \max_{j=1,2,\dots,F}
-      (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
-       \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
+  \MaxDist = \max_{j=1,2,\dots,F}
+      (\overbrace{\Pnorm(S_j)}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_j)}^{\text{Minimize}} )

 \end{equation}
 where $F$ is the number of available frequencies. Then we can select the optimal
 \end{equation}
 where $F$ is the number of available frequencies. Then we can select the optimal

-scaling factor that satisfies EQ~(\ref{eq:max}).  Our objective function can
+scaling factor that satisfies EQ~\eqref{eq:max}.  Our objective function can

 work with any energy model or static power values stored in a data file.
 Moreover, this function works in optimal way when the energy curve has a convex
 form over the available frequency scaling factors as shown in~\cite{15,3,19}.
 work with any energy model or static power values stored in a data file.
 Moreover, this function works in optimal way when the energy curve has a convex
 form over the available frequency scaling factors as shown in~\cite{15,3,19}.


@@ -394,32 +418,39 @@ the objective function described above.
 \begin{figure}[tp]
   \begin{algorithmic}[1]
     % \footnotesize
 \begin{figure}[tp]
   \begin{algorithmic}[1]
     % \footnotesize

-    \State  Initialize the variable $Dist=0$
-    \State Set dynamic and static power values.
-    \State Set $P_{states}$ to the number of available frequencies.
-    \State Set the variable $F_{new}$ to max. frequency,  $F_{new} = F_{max} $
-    \State Set the variable $F_{diff}$ to the difference between two successive
-           frequencies.
-    \For {$j:=1$   to   $P_{states} $}
-      \State $F_{new}=F_{new} - F_{diff} $
-      \State $S = \frac{F_\textit{max}}{F_\textit{new}}$
-      \State $S_i = S \cdot \frac{T_1}{T_i}
-                  = \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}$
+    \Require ~
+    \begin{description}
+    \item[$\Pstatic$] static power value
+    \item[$\Pdyn$] dynamic power value
+    \item[$\Pstates$] number of available frequencies
+    \item[$\Fmax$] maximum frequency
+    \item[$\Fdiff$] difference between two successive freq.
+    \end{description}
+    \Ensure $\Sopt$ is the optimal scaling factor
+
+    \State $\Sopt \gets 1$
+    \State $\Dist \gets 0$
+    \State $\Fnew \gets \Fmax$
+    \For {$j = 2$ to $\Pstates$}
+      \State $\Fnew \gets \Fnew - \Fdiff$
+      \State $S \gets \Fmax / \Fnew$
+      \State $S_i \gets S \cdot \frac{T_1}{T_i}
+                  = \frac{\Fmax}{\Fnew} \cdot \frac{T_1}{T_i}$

              for $i=1,\dots,N$
              for $i=1,\dots,N$

-      \State $E_\textit{Norm} =
-          \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
+      \State $\Enorm \gets
+          \frac{\Pdyn \cdot S_1^{-2} \cdot

                   \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
                   \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +

-                  P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{
-                P_\textit{dyn} \cdot
+                  \Pstatic \cdot T_1 \cdot S_1 \cdot N }{
+                \Pdyn \cdot

                   \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
                   \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +

-                  P_\textit{static} \cdot T_1 \cdot N }$
-      \State $P_{NormInv}=T_{old}/T_{new}$
-      \If{$(P_{NormInv}-E_{Norm} > Dist)$}
-        \State $S_{opt} = S$
-        \State $Dist = P_{NormInv} - E_{Norm}$
+                  \Pstatic \cdot T_1 \cdot N }$
+      \State $\Pnorm \gets \Told / \Tnew$
+      \If{$(\Pnorm - \Enorm > \Dist)$}
+        \State $\Sopt \gets S$
+        \State $\Dist \gets \Pnorm - \Enorm$

       \EndIf
     \EndFor
       \EndIf
     \EndFor

-    \State  Return $S_{opt}$
+    \State  Return $\Sopt$

   \end{algorithmic}
   \caption{Scaling factor selection algorithm}
   \label{EPSA}
   \end{algorithmic}
   \caption{Scaling factor selection algorithm}
   \label{EPSA}


@@ -456,7 +487,7 @@ program.
 \begin{figure}[tp]
   \begin{algorithmic}[1]
     % \footnotesize
 \begin{figure}[tp]
   \begin{algorithmic}[1]
     % \footnotesize

-    \For {$k:=1$ to \textit{some iterations}}
+    \For {$k=1$ to \textit{some iterations}}

       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}
       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}


@@ -474,35 +505,35 @@ program.
 \end{figure}
 After obtaining the optimal scaling factor, the program calculates the new
 frequency $F_i$ for each task proportionally to its time value $T_i$.  By
 \end{figure}
 After obtaining the optimal scaling factor, the program calculates the new
 frequency $F_i$ for each task proportionally to its time value $T_i$.  By

-substitution of EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we can calculate the new
+substitution of EQ~\eqref{eq:s} in EQ~\eqref{eq:si}, we can calculate the new

 frequency $F_i$ as follows:
 \begin{equation}
   \label{eq:fi}
 frequency $F_i$ as follows:
 \begin{equation}
   \label{eq:fi}

-  F_i = \frac{F_\textit{max} \cdot T_i}{S_\textit{optimal} \cdot T_\textit{max}}
+  F_i = \frac{\Fmax \cdot T_i}{\Sopt \cdot \Tmax}

 \end{equation}
 According to this equation all the nodes may have the same frequency value if
 they have balanced workloads, otherwise, they take different frequencies when
 \end{equation}
 According to this equation all the nodes may have the same frequency value if
 they have balanced workloads, otherwise, they take different frequencies when

-having imbalanced workloads.  Thus, EQ~(\ref{eq:fi}) adapts the frequency of the
+having imbalanced workloads.  Thus, EQ~\eqref{eq:fi} adapts the frequency of the

 CPU to the nodes' workloads to maintain the performance of the program.
 
 \section{Experimental results}
 \label{sec.expe}
 Our experiments are executed on the simulator SimGrid/SMPI v3.10.  We configure
 the simulator to use a homogeneous cluster with one core per node.  
 CPU to the nodes' workloads to maintain the performance of the program.
 
 \section{Experimental results}
 \label{sec.expe}
 Our experiments are executed on the simulator SimGrid/SMPI v3.10.  We configure
 the simulator to use a homogeneous cluster with one core per node.  

-%The detailed characteristics of our platform file are shown in Table~(\ref{table:platform}).  
+%The detailed characteristics of our platform file are shown in Table~\ref{table:platform}.

 Each node in the cluster has 18 frequency values
 from \np[GHz]{2.5} to \np[MHz]{800} with \np[MHz]{100} difference between each
 two successive frequencies. The nodes are connected via an ethernet network with 1Gbit/s  bandwidth. 
 
 Each node in the cluster has 18 frequency values
 from \np[GHz]{2.5} to \np[MHz]{800} with \np[MHz]{100} difference between each
 two successive frequencies. The nodes are connected via an ethernet network with 1Gbit/s  bandwidth. 
 

-\subsection{Performance prediction verification}
+\subsection{Execution time prediction verification}

 
 

-In this section we evaluate the precision of our performance prediction method
-based on EQ~(\ref{eq:tnew}) by applying it to the NAS benchmarks.  The NAS programs
+In this section we evaluate the precision of our execution time prediction method
+based on EQ~\eqref{eq:tnew} by applying it to the NAS benchmarks.  The NAS programs

 are executed with the class B option to compare the real execution time with
 the predicted execution time.  Each program runs offline with all available
 scaling factors on 8 or 9 nodes (depending on the benchmark) to produce real
 execution time values.  These scaling factors are computed by dividing the
 are executed with the class B option to compare the real execution time with
 the predicted execution time.  Each program runs offline with all available
 scaling factors on 8 or 9 nodes (depending on the benchmark) to produce real
 execution time values.  These scaling factors are computed by dividing the

-maximum frequency by the new one see EQ~(\ref{eq:s}).
+maximum frequency by the new one see EQ~\eqref{eq:s}.

 \begin{figure}
   \centering
   \includegraphics[width=.5\linewidth]{fig/cg_per}\hfill%
 \begin{figure}
   \centering
   \includegraphics[width=.5\linewidth]{fig/cg_per}\hfill%


@@ -528,7 +559,7 @@ For each instance the benchmarks were executed on a number of processors
 proportional to the size of the class.  Each class represents the problem size
 ascending from class A to C.  Additionally, depending on some speed up points
 for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes
 proportional to the size of the class.  Each class represents the problem size
 ascending from class A to C.  Additionally, depending on some speed up points
 for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes

-respectively.  Depending on EQ~(\ref{eq:energy}), we measure the energy
+respectively.  Depending on EQ~\eqref{eq:energy}, we measure the energy

 consumption for all the NAS MPI programs while assuming that the dynamic power
 with the highest frequency is equal to \np[W]{20} and the power static is equal
 to \np[W]{4} for all experiments.  These power values were also used by Rauber
 consumption for all the NAS MPI programs while assuming that the dynamic power
 with the highest frequency is equal to \np[W]{20} and the power static is equal
 to \np[W]{4} for all experiments.  These power values were also used by Rauber


@@ -536,14 +567,14 @@ and Rünger in~\cite{3}.  The results showed that the algorithm selected
 different scaling factors for each program depending on the communication
 features of the program as in the plots from Figure~\ref{fig:nas}.  These plots
 illustrate that there are different distances between the normalized energy and
 different scaling factors for each program depending on the communication
 features of the program as in the plots from Figure~\ref{fig:nas}.  These plots
 illustrate that there are different distances between the normalized energy and

-the normalized inverted performance curves, because there are different
+the normalized inverted execution time curves, because there are different

 communication features for each benchmark.  When there are little or no
 communication features for each benchmark.  When there are little or no

-communications, the inverted performance curve is very close to the energy
+communications, the inverted execution time curve is very close to the energy

 curve.  Then the distance between the two curves is very small.  This leads to
 small energy savings.  The opposite happens when there are a lot of
 communication, the distance between the two curves is big.  This leads to more
 energy savings (e.g. CG and FT), see Table~\ref{table:compareC}.  All discovered
 curve.  Then the distance between the two curves is very small.  This leads to
 small energy savings.  The opposite happens when there are a lot of
 communication, the distance between the two curves is big.  This leads to more
 energy savings (e.g. CG and FT), see Table~\ref{table:compareC}.  All discovered

-frequency scaling factors optimize both the energy and the performance
+frequency scaling factors optimize both the energy and the execution time

 simultaneously for all NAS benchmarks.  In Table~\ref{table:compareC}, we record
 all optimal scaling factors results for each benchmark running class C.  These
 scaling factors give the maximum energy saving percentage and the minimum
 simultaneously for all NAS benchmarks.  In Table~\ref{table:compareC}, we record
 all optimal scaling factors results for each benchmark running class C.  These
 scaling factors give the maximum energy saving percentage and the minimum


@@ -574,8 +605,8 @@ savings).
 
 In this section, we compare our scaling factor selection method with Rauber and
 Rünger methods~\cite{3}.  They had two scenarios, the first is to reduce energy
 
 In this section, we compare our scaling factor selection method with Rauber and
 Rünger methods~\cite{3}.  They had two scenarios, the first is to reduce energy

-to the optimal level without considering the performance as in
-EQ~(\ref{eq:sopt}).  We refer to this scenario as $R_{E}$.  The second scenario
+to the optimal level without considering the execution time as in
+EQ~\eqref{eq:sopt}.  We refer to this scenario as $R_{E}$.  The second scenario

 is similar to the first except setting the slower task to the maximum frequency
 (when the scale $S=1$) to keep the performance from degradation as mush as
 possible.  We refer to this scenario as $R_{E-P}$.  While we refer to our
 is similar to the first except setting the slower task to the maximum frequency
 (when the scale $S=1$) to keep the performance from degradation as mush as
 possible.  We refer to this scenario as $R_{E-P}$.  While we refer to our


@@ -652,12 +683,12 @@ trade-offs such as in BT and EP.
 
 In this paper, we have presented a new online scaling factor selection method
 that optimizes simultaneously the energy and performance of a distributed
 
 In this paper, we have presented a new online scaling factor selection method
 that optimizes simultaneously the energy and performance of a distributed

-application running on an homogeneous cluster.  It uses the computation and
+application running on a homogeneous cluster.  It uses the computation and

 communication times measured at the first iteration to predict energy
 communication times measured at the first iteration to predict energy

-consumption and the performance of the parallel application at every available
+consumption and the execution time of the parallel application at every available

 frequency.  Then, it selects the scaling factor that gives the best trade-off
 between energy reduction and performance which is the maximum distance between
 frequency.  Then, it selects the scaling factor that gives the best trade-off
 between energy reduction and performance which is the maximum distance between

-the energy and the inverted performance curves.  To evaluate this method, we
+the energy and the inverted eexecution time curves.  To evaluate this method, we

 have applied it to the NAS benchmarks and it was compared to Rauber and Rünger
 methods while being executed on the simulator SimGrid.  The results showed that
 our method, outperforms Rauber and Rünger's methods in terms of energy-performance
 have applied it to the NAS benchmarks and it was compared to Rauber and Rünger
 methods while being executed on the simulator SimGrid.  The results showed that
 our method, outperforms Rauber and Rünger's methods in terms of energy-performance