the supply voltage $V$ and operational frequency $f$ respectively as follow :
\begin{equation}
\label{eq:pd}
- P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
+ \textit P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
\end{equation}
The static power $P_{static}$ captures the leakage power consumption as well as
the power consumption of peripheral devices like the I/O subsystem.
\begin{equation}
\label{eq:ps}
- P_{static} = V \cdot N \cdot K_{design} \cdot I_{leak}
+ \textit P_{static} = V \cdot N \cdot K_{design} \cdot I_{leak}
\end{equation}
where V is the supply voltage, N is the number of transistors, $K_{design}$ is a
design dependent parameter and $I_{leak}$ is a technology-dependent
see~\cite{36,15}.
\begin{equation}
\label{eq:eind}
- E_{ind} = ( P_{dyn} + P_{static} ) \cdot T
+ \textit E_{ind} = ( P_{dyn} + P_{static} ) \cdot T
\end{equation}
The dynamic voltage and frequency scaling (DVFS) is a process that allowed in
modern processors to reduce the dynamic power by scaling down the voltage and
maximum and the new frequency as in EQ~(\ref{eq:s}).
\begin{equation}
\label{eq:s}
- S = \frac{F_{max}}{F_{new}}
+ S = \frac{F_{max}}{F_{new}}
\end{equation}
The value of the scale $S$ is greater than 1 when changing the frequency to
any new frequency value (\emph {P-state}) in governor.
scaling factor as in EQ~(\ref{eq:sopt}).
\begin{equation}
\label{eq:sopt}
- S_{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_{dyn}}{P_{static}} \cdot
+ \textit S_{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_{dyn}}{P_{static}} \cdot
\left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
\end{equation}
EQ~(\ref{eq:tnew}).
\begin{equation}
\label{eq:tnew}
- T_{new} = T_{\textit{Max Comp Old}} \cdot S + T_{\textit{Max Comm Old}}
+ \textit T_{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
without scaled frequency :
\begin{multline}
\label{eq:enorm}
- E_\textit{Norm} = \frac{E_{Reduced}}{E_{Original}}\\
- {} = \frac{ P_{dyn} \cdot S_i^{-2} \cdot
+ \textit E_{Norm} = \frac{\textit E_{Reduced}}{\textit E_{Original}} \\
+ {} = \frac{ P_{dyn} \cdot S_i^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
P_{static} \cdot T_1 \cdot S_i \cdot N }{
P_{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
By the same way we can normalize the performance as follows :
\begin{equation}
\label{eq:pnorm}
- P_{Norm} = \frac{T_{New}}{T_{Old}}
+\textit P_{Norm} = \frac{\textit T_{New}}{\textit T_{Old}}
= \frac{T_{\textit{Max Comp Old}} \cdot S +
- T_{\textit{Max Comm Old}}}{T_{Old}}
+ T_{\textit{Max Comm Old}}}{\textit T_{Old}}
\end{equation}
The second problem is the optimization operation for both energy and performance
is not in the same direction. In other words, the normalized energy and the
performance as follows :
\begin{equation}
\label{eq:pnorm_en}
- P^{-1}_{Norm} = \frac{T_{Old}}{T_{New}}
- = \frac{T_{Old}}{T_{\textit{Max Comp Old}} \cdot S +
+\textit P^{-1}_{Norm} = \frac{\textit T_{Old}}{\textit T_{New}}
+ = \frac{\textit T_{Old}}{T_{\textit{Max Comp Old}} \cdot S +
T_{\textit{Max Comm Old}}}
\end{equation}
\begin{figure*}
following form:
\begin{equation}
\label{eq:max}
- \textit{MaxDist} = \max (\overbrace{P^{-1}_{Norm}}^{\text{Maximize}} -
- \overbrace{E_{Norm}}^{\text{Minimize}} )
+ \textit{MaxDist} = \max (\overbrace{\textit P^{-1}_{Norm}}^{\text{Maximize}} -
+ \overbrace{\textit E_{Norm}}^{\text{Minimize}} )
\end{equation}
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
\For {$J:=1$ to $Some-Iterations \; $}
\State -Computations Section.
\State -Communications Section.
- \If {$(J==1)$}
- \State -Gather all times of computation and\par
- \State communication from each node.
+ \If {$(J==1)$}
+ \State -Gather all times of computation and communication from\par each node.
\State -Call EPSA with these times.
\State -Calculate the new frequency from optimal scale.
\State -Set the new frequency to the system.
\hline
Max & Min & Backbone & Backbone&Link &Link& Sharing \\
Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy \\ \hline
- 2.5 &800 & 2.25 GBps &$5\times 10^{-7} s$& 1 GBps & $5\times 10^{-5}$ s&Full \\
+ 2.5 &800 & 2.25 GBps &$5\times 10^{-7} s$& 1 GBps & $5\times 10^{-5} s$ &Full \\
GHz& MHz& & & & &Duplex \\\hline
\end{tabular}
\label{table:platform}
BT & 1.31 &29.60&21.28 &8.32\\ \hline
SP & 1.38 &33.48&21.36 &12.12\\ \hline
FT & 1.47 &34.72&19.00 &15.72\\ \hline
- \end{tabular}
+ \end{tabular}
\label{table:factors results}
% is used to refer this table in the text
\end{table}
