consumption by these architectures. Moreover, the price of energy is expected to
continue its ascent according to the demand. For all these reasons energy
reduction became an important topic in the high performance computing field. To
-tackle this problem, many researchers used DVFS (Dynamic Voltage Frequency
+tackle this problem, many researchers used DVFS (Dynamic Voltage and Frequency
Scaling) operations which reduce dynamically the frequency and voltage of cores
and thus their energy consumption. However, this operation also degrades the
performance of computation. Therefore researchers try to reduce the frequency to
This paper is organized as follows: Section~\ref{sec.relwork} presents the works
from other authors. Section~\ref{sec.exe} shows the execution of parallel
-tasks and sources of idle times. It resumes the energy
+tasks and sources of idle times. Also, it resumes the energy
model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance
of MPI program. Section~\ref{sec.compet} presents the energy-performance trade offs
objective function. Section~\ref{sec.optim} demonstrates the proposed
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 are
-expressed by the scaling factor \emph S. The scale \emph S is the ratio between the
+expressed by the scaling factor \emph S. This scaling factor is the ratio between the
maximum and the new frequency as in EQ~(\ref{eq:s}).
\begin{equation}
\label{eq:s}
The value of the scale $S$ is greater than 1 when changing the frequency to any
new frequency value~(\emph {P-state}) in governor, the CPU governor is an
interface driver supplied by the operating system kernel (e.g. Linux) to
-lowering core's frequency. The scaling factor is equal to 1 when the frequency
-set is to the maximum frequency. The energy consumption model for parallel
+lowering core's frequency. The scaling factor is equal to 1 when the new frequency is
+set to the maximum frequency. The energy consumption model for parallel
homogeneous platform depends on the scaling factor \emph S. This factor reduces
quadratically the dynamic power. Also, this factor increases the static energy
linearly because the execution time is increased~\cite{36}. The energy model
the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}).
\begin{equation}
\label{eq:s1}
- S_1 = \max_{i=1,2,\dots,F} S_i
+ S_1 = \max_{i=1,2,\dots,N} S_i
\end{equation}
\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}
-where $F$ is the number of available frequencies. In this paper we depend on
+where $N$ is the number of nodes. In this paper we depend on
Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this
model is used for homogeneous platform that we work on in this paper, and (2) we
compare our algorithm with Rauber and Rünger scaling model. Rauber and Rünger
the new scaling factor as in EQ~(\ref{eq:tnew}).
\begin{equation}
\label{eq:tnew}
- \textit T_\textit{new} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
+ \textit T_\textit{New} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
\end{equation}
The above equation shows that the scaling factor \emph S has linear relation
with the computation time without affecting the communication time. The
\begin{multline}
\label{eq:enorm}
E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
- {} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot
+ {} = \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_i \cdot N }{
+ 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 }
\end{multline}
\end{figure*}
Then, we can modelize 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 represent
-the minimum energy consumption with minimum execution time (better performance)
+curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors $S_j$. This represent
+the minimum energy consumption with minimum execution time (better performwhere F is the number of available frequenciesance)
in the same time, see figure~(\ref{fig:r1}). Then our objective function has the
following form:
\begin{equation}
\label{eq:max}
- \textit{MaxDist} = \max (\overbrace{P^{-1}_\textit{Norm}}^{\text{Maximize}} -
- \overbrace{E_\textit{Norm}}^{\text{Minimize}} )
+ S_\textit{optimal} = \max_{j=1,2,\dots,F} (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
+ \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
\end{equation}
-Then we can select the optimal scaling factor that satisfy the
+where F is the number of available frequencies. Then we can select the optimal scaling factor that satisfy the
EQ~(\ref{eq:max}). Our objective function can works with any energy model or
static power values stored in a data file. Moreover, this function works in
optimal way when the energy function has a convex form with frequency scaling
In the previous section we described the objective function that satisfy our
goal in discovering optimal scaling factor for both performance and energy at
the same time. Therefore, we develop an energy to performance scaling algorithm
-($EPSA$). This algorithm is simple and has a direct way to calculate the optimal
+(EPSA). This algorithm is simple and has a direct way to calculate the optimal
scaling factor for both energy and performance at the same time.
\begin{algorithm}[tp]
\caption{EPSA}
\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 scale value between each two frequencies.
- \For {$i=1$ to $P_{states} $}
+ \For {$J:=1$ to $P_{states} $}
\State - Calculate the new frequency as $F_{new}=F_{new} - F_{diff} $
\State - Calculate the scale factor $S$ as in EQ~(\ref{eq:s}).
\State - Calculate all available scales $S_i$ depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}).
\caption{DVFS}
\label{dvfs}
\begin{algorithmic}[1]
- \For {$J=1$ to $Some-Iterations \; $}
+ \For {$K:=1$ to $Some-Iterations \; $}
\State -Computations Section.
\State -Communications Section.
- \If {$(J=1)$}
+ \If {$(K=1)$}
\State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
\State -Call EPSA with these times.
\State -Calculate the new frequency from optimal scale.
\end{figure}
\section{Conclusion}
\label{sec.concl}
-In this paper we developed the simultaneous energy-performance algorithm. It works based on the trade off relation between the energy and performance. The results showed that when the scaling factor is big value refer to more energy saving. Also, when the scaling factor is smaller value, Then it has bigger impact on performance than energy. The algorithm optimizes the energy saving and performance in the same time to have positive trade off. The optimal trade off represents the maximum distance between the energy and the inversed performance curves. Also, the results explained when setting the slowest task to maximum frequency usually not have a big improvement on performance. In future, we will apply the EPSA algorithm on heterogeneous platform.
+In this paper we developed the simultaneous energy-performance algorithm. It works based on the trade off relation between the energy and performance. The results showed that when the scaling factor is big value refer to more energy saving. Also, when the scaling factor is smaller value, then it has bigger impact on performance than energy. The algorithm optimizes the energy saving and performance in the same time to have positive trade off. The optimal trade off represents the maximum distance between the energy and the inversed performance curves. Also, the results explained when setting the slowest task to maximum frequency usually not have a big improvement on performance. In future, we will apply the EPSA algorithm on heterogeneous platform.
\section*{Acknowledgment}
Computations have been performed on the supercomputer facilities of the
