Typo.

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 06dbb3052ebd438e55f8abeaab475998268171f8..832d62f3c30f56a0106c34b2ea823cb16be5036a 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 +23,33 @@
 \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{\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


@@ -208,28 +233,28 @@ our paper is to present a new online scaling factor selection method which has t
 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
 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 +264,10 @@ 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 +277,32 @@ 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 and scaling factors 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 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}
 


@@ -301,10 +325,10 @@ 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.


@@ -321,27 +345,26 @@ the consumed energy with scaled frequency and the consumed energy without scaled
 frequency:
 \begin{multline}
   \label{eq:enorm}
 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}
 In the same way we can normalize the performance as follows:
 \begin{equation}
   \label{eq:pnorm}
 \end{multline}
 In the same way we can normalize the performance as follows:
 \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}}
+  \Pnorm = \frac{\Tnew}{\Told}
+         = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{
+             \TmaxCompOld + \TmaxCommOld}

 \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
 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
 \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
 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
+equations~\eqref{eq:enorm} and~\eqref{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
 researchers used different strategies to solve this nonlinear problem for
 both the energy and the performance simultaneously.  But the main objective is
 to produce maximum energy reduction with minimum performance reduction.  Many
 researchers used different strategies to solve this nonlinear problem for


@@ -354,10 +377,10 @@ direction.  Therefore, we inverse the equation of the normalized performance 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^{-1} = \frac{ \Told}{ \Tnew}
+               = \frac{\TmaxCompOld +
+                 \TmaxCommOld}{\TmaxCompOld \cdot S +
+                 \TmaxCommOld}

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


@@ -369,19 +392,19 @@ 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 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^{-1}(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 +417,32 @@ 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  Initialize the variable $\Dist=0$

     \State Set dynamic and static power values.
     \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
+    \State Set $\Pstates$ to the number of available frequencies.
+    \State Set the variable $\Fnew$ to max. frequency,  $\Fnew = \Fmax $
+    \State Set the variable $\Fdiff$ to the difference between two successive

            frequencies.
            frequencies.

-    \For {$j:=1$   to   $P_{states} $}
-      \State $F_{new}=F_{new} - F_{diff} $
-      \State $S = \frac{F_\textit{max}}{F_\textit{new}}$
+    \For {$j := 1$ to $\Pstates $}
+      \State $\Fnew = \Fnew - \Fdiff $
+      \State $S = \frac{\Fmax}{\Fnew}$

       \State $S_i = S \cdot \frac{T_1}{T_i}
       \State $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}$

              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 =
+          \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 $\PnormInv = \Told / \Tnew$
+      \If{$(\PnormInv - \Enorm > \Dist)$}
+        \State $\Sopt = S$
+        \State $\Dist = \PnormInv - \Enorm$

       \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}


@@ -474,22 +497,22 @@ 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. 


@@ -497,12 +520,12 @@ two successive frequencies. The nodes are connected via an ethernet network with
 \subsection{Performance prediction verification}
 
 In this section we evaluate the precision of our performance prediction method
 \subsection{Performance 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
+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 +551,7 @@ 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


@@ -575,7 +598,7 @@ 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
 to the optimal level without considering the performance as in
 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
+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