objective function changes

diff --git a/paper.tex b/paper.tex
index bec8af60d3db4c8d810429d6bd5aa69b6080738d..613e83bc78a5840d3726c960879c543466ae0ed1 100644 (file)
--- 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.