tasks must be measured during the first iteration before applying any DVFS
operation. Then the execution time for one iteration of the application with any
vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
-
-
-
-\begin{multline}
+\begin{equation}
\label{eq:perf}
\textit T_\textit{new} =
- {} \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) + TcmOld_{j}
-\end{multline}
+ \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) + TcmOld_{j}
+\end{equation}
where $TcpOld_i$ is the computation time of processor $i$ during the first iteration and $TcmOld_j$ is the communication time of the slowest processor $j$.
The model computes the maximum computation time
with scaling factor from each node added to the communication time of the slowest node, it means only the
$F_{new}$ from EQ(\ref{eq:s}) as follow:
\begin{equation}
\label{eq:fnew}
- F_\textit{new} = S^{-1} . F_\textit{max}
+ F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic
power consumption:
\begin{multline}
\label{eq:pdnew}
{P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
- = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{dOld} \cdot S^{-3}
+ {} = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{dOld} \cdot S^{-3}
\end{multline}
where $ {P}_\textit{dNew}$ and $P_{dOld}$ are the dynamic power consumed with the new frequency and the maximum frequency respectively.
the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related
to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy
of a processor after scaling its frequency is computed as follows:
-
\begin{equation}
\label{eq:Estatic}
E_\textit{s} = P_\textit{s} \cdot (T_{cp} \cdot S + T_{cm})
In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $P_{di}$ and $P_{si}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $T_{cpi}$ might be different and different frequency scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $T_{cmi}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the frequency scaling factor and the dynamic power of each node as in EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing distributed application executed over a heterogeneous platform is the summation of all dynamic and static energies for each processor. It is computed as follows:
\begin{multline}
\label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot P_{di} \cdot T_{cpi})} +\\
- {}\sum_{i=1}^{N} {(P_{si} \cdot (\max_{i=1,2,\dots,N} (T_{cpi} \cdot S_{i}) +}
- {}\min_{i=1,2,\dots,N} {T_{cmi}))}
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot P_{di} \cdot T_{cpi})} + {} \\
+ \sum_{i=1}^{N} (P_{si} \cdot (\max_{i=1,2,\dots,N} (T_{cpi} \cdot S_{i}) +
+ \min_{i=1,2,\dots,N} {T_{cmi}))}
\end{multline}
Reducing the the frequencies of the processors according to the vector of
\begin{multline}
\label{eq:pnorm}
P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
- = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}}
+ {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}}
{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
\end{multline}
\begin{multline}
\label{eq:enorm}
E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
- = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} +
+ {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} +
\sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot Tcp_i)} +
\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
\end{multline}
to make the optimization process for energy and execution time follow the same
direction. Therefore, we inverse the equation of the normalized execution time,
the normalized performance, as follows:
-
\begin{multline}
\label{eq:pnorm_inv}
P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
represents the minimum energy consumption with minimum execution time (better
performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}) . Then our objective
function has the following form:
-\begin{multline}
+\begin{equation}
\label{eq:max}
Max Dist =
\max_{i=1,\dots F, j=1,\dots,N}
(\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
\overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
-\end{multline}
+\end{equation}
where $N$ is the number of nodes and $F$ is the number of available frequencies for each nodes.
Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}). Our objective function can
work with any energy model or energy values stored in a data file.
