objective function changes

[mpi-energy.git] / paper.tex

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}
 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}
 \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
 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}} \\
 \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) +
                \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}
               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
 \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}
 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}
 \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
 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.
     \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}).
       \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]
   \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.
   \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.
      \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.