\left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
\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
-the execution times and scaling factors of the sorted tasks. Therefore, $T1$ is
+where $N$ is the number of parallel nodes, $T_i$ and $S_i$ for $i=1,\dots,N$ are
+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
factors are computed as in EQ~\eqref{eq:si}.
frequency:
\begin{multline}
\label{eq:enorm}
- \Enorm = \frac{ \Ereduced}{\Eoriginal} \\
- {} = \frac{\Pdyn \cdot S_1^{-2} \cdot
+ \Enorm(S) = \frac{ \Ereduced}{\Eoriginal} \\
+ {} = \frac{\Pdyn \cdot S^{-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}{
+ \Pstatic \cdot T_1 \cdot S \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}
- \Pnorm = \frac{\Tnew}{\Told}
- = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{
+ \Pnorm(S) = \frac{\Tnew}{\Told}
+ = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{
\TmaxCompOld + \TmaxCommOld}
\end{equation}
The second problem is that the optimization operation for both energy and
follows:
\begin{equation}
\label{eq:pnorm_en}
- \Pnorm^{-1} = \frac{ \Told}{ \Tnew}
+ \Pnorm^{-1}(S) = \frac{ \Told}{ \Tnew}
= \frac{\TmaxCompOld +
\TmaxCommOld}{\TmaxCompOld \cdot S +
\TmaxCommOld}
\begin{figure}[tp]
\begin{algorithmic}[1]
% \footnotesize
- \State Initialize the variable $\Dist=0$
- \State Set dynamic and static power values.
- \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.
- \For {$j := 1$ to $\Pstates $}
- \State $\Fnew = \Fnew - \Fdiff $
- \State $S = \frac{\Fmax}{\Fnew}$
- \State $S_i = S \cdot \frac{T_1}{T_i}
- = \frac{\Fmax}{\Fnew} \cdot \frac{T_1}{T_i}$
- for $i=1,\dots,N$
- \State $\Enorm =
- \frac{\Pdyn \cdot S_1^{-2} \cdot
+ \Require ~
+ \begin{description}
+ \item[$\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 $\Enorm \gets
+ \frac{\Pdyn \cdot S^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- \Pstatic \cdot T_1 \cdot S_1 \cdot N }{
+ \Pstatic \cdot T_1 \cdot S \cdot N }{
\Pdyn \cdot
\left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
\Pstatic \cdot T_1 \cdot N }$
- \State $\PnormInv = \Told / \Tnew$
+ \State $\PnormInv \gets \Told / \Tnew$
\If{$(\PnormInv - \Enorm > \Dist)$}
- \State $\Sopt = S$
- \State $\Dist = \PnormInv - \Enorm$
+ \State $\Sopt \gets S$
+ \State $\Dist \gets \PnormInv - \Enorm$
\EndIf
\EndFor
\State Return $\Sopt$
\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 {$(k=1)$}
