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
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}
P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
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}
S = \frac{F_\textit{max}}{F_\textit{new}}
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}
the execution times and scaling factors of the sorted tasks. Therefore, $T1$ 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
-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}
= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}
\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
-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}
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}
\textit T_\textit{new} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
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
\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:
\overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
\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}.
\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}
\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.
-%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.
\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
-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%
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
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
