and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the
first iteration. The execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
and the communication time of the slowest node without slack time during one iteration.
- The slowest node $h$ is the node which takes the maximum execution time to execute an iteration before scaling down its frequency.
+The slowest node in cluster $h$ is the node which takes the maximum execution time to execute an iteration before scaling down its frequency.
It means that only the communication time without any slack time is taken into account.
Therefore, the execution time of the application is equal to
the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the
E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} +
\sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot {} \\
(\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij})
- +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))
+ +\mathop{\min_{j=1,\dots M_h}} (\Tcm[hj]) ))
\end{multline}
While the main goal is to optimize the energy and execution time at the same
time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way.
According to (\ref{eq:pnorm}) and (\ref{eq:enorm}), the
-vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduces both the energy
+vector of frequency scaling factors $S_{11},S_{12},\dots,S_{NM_i}$ reduces both the energy
and the execution time, but the main objective is to produce
maximum energy reduction with minimum execution time reduction.
This problem can be solved by making the optimization process for energy and
execution time follow the same evolution according to the vector of scaling factors
-$(S_{11}, S_{12},\dots, S_{NM})$. Therefore, the equation of the
+$(S_{11}, S_{12},\dots, S_{NM_i})$. Therefore, the equation of the
normalized execution time is inverted which gives the normalized performance
equation, as follows:
\begin{equation}
communication ratio.
The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to
-the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its
+the other benchmarks. Indeed, in the EP benchmark, there are no communication and no slack times and its
performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
The rest of the benchmarks showed different performance degradation percentages which decrease
when the communication times increase and vice versa.
The execution times for most of the NAS benchmarks are higher over the multi-core per node scenario
than over the single core per node scenario. Indeed,
- the communication times are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.
+ the communication times are higher in the multi-core scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.
Moreover, the energy consumptions of the NAS benchmarks are lower over the
one core scenario than over the multi-core scenario because
the first scenario had less execution time than the latter which results in less static energy being consumed.
\section*{Acknowledgment}
This work has been partially supported by the Labex ACTION project (contract
-``ANR-11-LABX-01-01''). Computations have been performed on the Grid'5000 platform. As a PhD student,
+``ANR-11-LABX-01-01''). Computations have been performed on the Grid'5000
+platform and on the mésocentre of Franche-Comté. As a PhD student,
Mr. Ahmed Fanfakh, would like to thank the University of Babylon (Iraq) for
supporting his work.
