From: afanfakh Date: Sat, 15 Mar 2014 13:28:15 +0000 (+0100) Subject: Last version X-Git-Tag: ispa14_submission~38 X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy.git/commitdiff_plain/b950ac52c83730085568b6bf91ac498a5fed2030 Last version --- diff --git a/paper.tex b/paper.tex index 1ce2f60..962a22b 100644 --- a/paper.tex +++ b/paper.tex @@ -201,13 +201,13 @@ $P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$, 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 @@ -216,7 +216,7 @@ of the dynamic and the static power multiply by the execution time for example 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 @@ -229,7 +229,7 @@ expressed by scaling factor \emph S. The scale \emph S is the ratio between the 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. @@ -274,7 +274,7 @@ derivation for this equation (to be minimized) and set it to zero to produce the 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} @@ -307,7 +307,7 @@ any MPI program as a function of the new scaling factor as in the 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 @@ -364,8 +364,8 @@ between the consumed energy with scaled frequency and the consumed energy 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) + @@ -381,9 +381,9 @@ without scaled frequency : 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 @@ -402,8 +402,8 @@ have the same direction. Therefore, we inverse the equation of normalize 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*} @@ -424,8 +424,8 @@ in the same time, see figure~(\ref{fig:r1}). Then our objective function has the 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 @@ -489,9 +489,8 @@ in the MPI program. \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. @@ -537,7 +536,7 @@ frequencies. \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} @@ -590,7 +589,7 @@ same time over all available scales. 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}