X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/869119dabc9fdb4d7eea5f1682c9bc07a9f95258..88ca69b0273314e1fb1b061bd1bf75accff40205:/Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index d7e32df..b7e61b8 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -139,14 +139,11 @@ scaling factors, the communication time and the computation time for all the
 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
@@ -196,14 +193,14 @@ system's kernel to lower a core's frequency. we can calculate the new frequency
 $F_{new}$ 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.
 
@@ -218,7 +215,6 @@ The static energy is the static power multiplied by the execution time of the pr
 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})
@@ -227,9 +223,9 @@ of a processor after scaling its frequency is computed as follows:
 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
@@ -263,7 +259,7 @@ scaled execution time) and the old one as follow:
 \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}
 
@@ -272,7 +268,7 @@ By the same way, we are normalize the energy by calculating the ratio between th
 \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} 
@@ -293,7 +289,6 @@ simultaneously without adding a big overhead.  Our solution for this problem is
 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}}\\
@@ -319,13 +314,13 @@ curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
 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.
@@ -684,10 +679,10 @@ The results of applying the proposed scaling algorithm to NAS benchmarks is demo
 
 \begin{figure}
   \centering
-  \subfloat[Balanced nodes type scenario]{%
+  \subfloat[CG, MG, LU and FT Benchmarks]{%
     \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
   \quad%
-  \subfloat[Imbalanced nodes type scenario]{%
+  \subfloat[BT and SP benchmarks]{%
     \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
   \label{fig:avg}
   \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
