From b2a65ca728f5564e5df441385c2069715aad9368 Mon Sep 17 00:00:00 2001 From: afanfakh Date: Fri, 21 Mar 2014 17:07:12 +0100 Subject: [PATCH 1/1] objective function changes --- paper.tex | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/paper.tex b/paper.tex index bec8af6..613e83b 100644 --- a/paper.tex +++ b/paper.tex @@ -263,14 +263,14 @@ from the set of scales values $S_i$. Each of these scales are proportional to the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}). \begin{equation} \label{eq:s1} - S_1 = \max_{i=1,2,\dots,F} S_i + S_1 = \max_{i=1,2,\dots,N} S_i \end{equation} \begin{equation} \label{eq:si} S_i = S \cdot \frac{T_1}{T_i} = \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i} \end{equation} -where $F$ is the number of available frequencies. In this paper we depend on +where $N$ is the number of nodes. In this paper we depend on Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this model is used for homogeneous platform that we work on in this paper, and (2) we compare our algorithm with Rauber and Rünger scaling model. Rauber and Rünger @@ -339,9 +339,9 @@ without scaled frequency: \begin{multline} \label{eq:enorm} E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\ - {} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot + {} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + - P_\textit{static} \cdot T_1 \cdot S_i \cdot N }{ + P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{ P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + P_\textit{static} \cdot T_1 \cdot N } \end{multline} @@ -385,16 +385,16 @@ performance as follows: \end{figure*} Then, we can modelize our objective function as finding the maximum distance between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance -curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent -the minimum energy consumption with minimum execution time (better performance) +curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors $S_j$. This represent +the minimum energy consumption with minimum execution time (better performwhere F is the number of available frequenciesance) 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}_\textit{Norm}}^{\text{Maximize}} - - \overbrace{E_\textit{Norm}}^{\text{Minimize}} ) + S_\textit{optimal} = \max_{j=1,2,\dots,F} (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} - + \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} ) \end{equation} -Then we can select the optimal scaling factor that satisfy the +where F is the number of available frequencies. 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 static power values stored in a data file. Moreover, this function works in optimal way when the energy function has a convex form with frequency scaling @@ -419,7 +419,7 @@ scaling factor for both energy and performance at the same time. \State Set $P_{states}$ to the number of available frequencies. \State Set the variable $F_{new}$ to max. frequency, $F_{new} = F_{max} $ \State Set the variable $F_{diff}$ to the scale value between each two frequencies. - \For {$i=1$ to $P_{states} $} + \For {$J:=1$ to $P_{states} $} \State - Calculate the new frequency as $F_{new}=F_{new} - F_{diff} $ \State - Calculate the scale factor $S$ as in EQ~(\ref{eq:s}). \State - Calculate all available scales $S_i$ depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}). @@ -454,10 +454,10 @@ in the MPI program. \caption{DVFS} \label{dvfs} \begin{algorithmic}[1] - \For {$J=1$ to $Some-Iterations \; $} + \For {$K:=1$ to $Some-Iterations \; $} \State -Computations Section. \State -Communications Section. - \If {$(J=1)$} + \If {$(K=1)$} \State -Gather all times of computation and\par\hspace{13 pt} communication from each node. \State -Call EPSA with these times. \State -Calculate the new frequency from optimal scale. -- 2.39.5