X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy.git/blobdiff_plain/449ff37f88ef932d63e318c50e4130759c59be99..43905e2e8c50ec820f6060f3d119c1cb8461c596:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index a7860cd..832d62f 100644
--- a/paper.tex
+++ b/paper.tex
@@ -23,6 +23,33 @@
 \newcommand{\JC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
 
+\newcommand{\Xsub}[2]{\ensuremath{#1_\textit{#2}}}
+
+\newcommand{\Dist}{\textit{Dist}}
+\newcommand{\Eind}{\Xsub{E}{ind}}
+\newcommand{\Enorm}{\Xsub{E}{Norm}}
+\newcommand{\Eoriginal}{\Xsub{E}{Original}}
+\newcommand{\Ereduced}{\Xsub{E}{Reduced}}
+\newcommand{\Fdiff}{\Xsub{F}{diff}}
+\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Fnew}{\Xsub{F}{new}}
+\newcommand{\Ileak}{\Xsub{I}{leak}}
+\newcommand{\Kdesign}{\Xsub{K}{design}}
+\newcommand{\MaxDist}{\textit{Max Dist}}
+\newcommand{\Ntrans}{\Xsub{N}{trans}}
+\newcommand{\Pdyn}{\Xsub{P}{dyn}}
+\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
+\newcommand{\Pnorm}{\Xsub{P}{Norm}}
+\newcommand{\Pstates}{\Xsub{P}{states}}
+\newcommand{\Pstatic}{\Xsub{P}{static}}
+\newcommand{\Sopt}{\Xsub{S}{opt}}
+\newcommand{\Tcomp}{\Xsub{T}{comp}}
+\newcommand{\TmaxCommOld}{\Xsub{T}{Max Comm Old}}
+\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
+\newcommand{\Tmax}{\Xsub{T}{max}}
+\newcommand{\Tnew}{\Xsub{T}{New}}
+\newcommand{\Told}{\Xsub{T}{Old}}
+
 \begin{document}
 
 \title{Dynamic Frequency Scaling for Energy Consumption
@@ -206,28 +233,28 @@ our paper is to present a new online scaling factor selection method which has t
 Many researchers~\cite{9,3,15,26} divide the power consumed by a processor into
 two power metrics: the static and the dynamic power.  While the first one is
 consumed as long as the computing unit is on, the latter is only consumed during
-computation times.  The dynamic power $P_{dyn}$ is related to the switching
+computation times.  The dynamic power $\Pdyn$ is related to the switching
 activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
 operational frequency $f$, as shown in EQ~\eqref{eq:pd}.
 \begin{equation}
   \label{eq:pd}
-  P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
+  \Pdyn = \alpha \cdot C_L \cdot V^2 \cdot f
 \end{equation}
-The static power $P_{static}$ captures the leakage power as follows:
+The static power $\Pstatic$ captures the leakage power as follows:
 \begin{equation}
   \label{eq:ps}
-   P_\textit{static}  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
+   \Pstatic  = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak
 \end{equation}
-where V is the supply voltage, $N_{trans}$ is the number of transistors,
-$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
+where V is the supply voltage, $\Ntrans$ is the number of transistors,
+$\Kdesign$ is a design dependent parameter and $\Ileak$ is a
 technology-dependent parameter.  The energy consumed by an individual processor
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}
-   E_\textit{ind} =  P_\textit{dyn} \cdot T_{Comp} + P_\textit{static} \cdot T
+   \Eind =  \Pdyn \cdot \Tcomp + \Pstatic \cdot T
 \end{equation}
-where $T$ is the execution time of the program, $T_{Comp}$ is the computation
-time and $T_{Comp} \leq T$.  $T_{Comp}$ may be equal to $T$ if there is no
+where $T$ is the execution time of the program, $\Tcomp$ is the computation
+time and $\Tcomp \leq T$. $\Tcomp$ may be equal to $T$ if there is no
 communication, no slack time and no synchronization.
 
 DVFS is a process that is allowed in modern processors to reduce the dynamic
@@ -240,7 +267,7 @@ process of the frequency can be expressed by the scaling factor $S$ which is the
 ratio between the maximum and the new frequency as in EQ~\eqref{eq:s}.
 \begin{equation}
   \label{eq:s}
- S = \frac{F_\textit{max}}{F_\textit{new}}
+  S = \frac{\Fmax}{\Fnew}
 \end{equation}
 The value of the scaling factor $S$ is greater than 1 when changing the
 frequency of the CPU to any new frequency value~(\emph{P-state}) in the
@@ -254,20 +281,19 @@ function of the scaling factor $S$, as in EQ~\eqref{eq:energy}.
 
 \begin{equation}
   \label{eq:energy}
-  E = P_\textit{dyn} \cdot S_1^{-2} \cdot
+  E = \Pdyn \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_1 \cdot N
- \hfill
+      \Pstatic \cdot T_1 \cdot S_1 \cdot N
 \end{equation}
 where $N$ is the number of parallel nodes, $T_i$ for $i=1,\dots,N$ are
-the execution times and scaling factors of the sorted tasks.  Therefore, $T1$ is
+the execution times and scaling factors of the sorted tasks.  Therefore, $T_1$ is
 the time of the slowest task, and $S_1$ its scaling factor which should be the
 highest because they are proportional to the time values $T_i$.  The scaling
 factors are computed as in EQ~\eqref{eq:si}.
 \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}
+      = \frac{\Fmax}{\Fnew} \cdot \frac{T_1}{T_i}
 \end{equation}
 In this paper we use Rauber and Rünger's energy model, EQ~\eqref{eq:energy}, because it can be applied to homogeneous clusters if the communication time is taken in consideration. Moreover, we compare our algorithm with Rauber and Rünger's scaling factor selection
 method which uses the same energy model.  In their method, the optimal scaling factor is
@@ -276,7 +302,7 @@ EQ~\eqref{eq:sopt}.
 
 \begin{equation}
   \label{eq:sopt}
-  S_\textit{opt} = \sqrt[3]{\frac{2}{N} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot
+  \Sopt = \sqrt[3]{\frac{2}{N} \cdot \frac{\Pdyn}{\Pstatic} \cdot
     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 
@@ -302,7 +328,7 @@ times are used to predict the execution time for any MPI program as a function
 of the new scaling factor as in EQ~\eqref{eq:tnew}.
 \begin{equation}
   \label{eq:tnew}
- \textit  T_\textit{new} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
+  \Tnew = \TmaxCompOld \cdot S + \TmaxCommOld
 \end{equation}
 In this paper, this prediction method is used to select the best scaling factor
 for each processor as presented in the next section.
@@ -319,20 +345,19 @@ the consumed energy with scaled frequency and the consumed energy without scaled
 frequency:
 \begin{multline}
   \label{eq:enorm}
-  E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
-        {} = \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_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 }
+  \Enorm = \frac{ \Ereduced}{\Eoriginal} \\
+      {} = \frac{\Pdyn \cdot S_1^{-2} \cdot
+             \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+               \Pstatic \cdot T_1 \cdot S_1 \cdot N}{
+             \Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+               \Pstatic \cdot T_1 \cdot N }
 \end{multline}
 In the same way we can normalize the performance as follows:
 \begin{equation}
   \label{eq:pnorm}
-  P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}
-          = \frac{T_\textit{Max Comp Old} \cdot S +
-           T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} +
-           T_\textit{Max Comm Old}}
+  \Pnorm = \frac{\Tnew}{\Told}
+         = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{
+             \TmaxCompOld + \TmaxCommOld}
 \end{equation}
 The second problem is that the optimization operation for both energy and
 performance is not in the same direction.  In other words, the normalized energy
@@ -352,10 +377,10 @@ direction.  Therefore, we inverse the equation of the normalized performance as
 follows:
 \begin{equation}
   \label{eq:pnorm_en}
-  P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
-               = \frac{T_\textit{Max Comp Old} +
-                 T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} \cdot S +
-                 T_\textit{Max Comm Old}}
+  \Pnorm^{-1} = \frac{ \Told}{ \Tnew}
+               = \frac{\TmaxCompOld +
+                 \TmaxCommOld}{\TmaxCompOld \cdot S +
+                 \TmaxCommOld}
 \end{equation}
 \begin{figure}
   \centering
@@ -374,9 +399,9 @@ performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}.  Then
 our objective function has the following form:
 \begin{equation}
   \label{eq:max}
-  \textit{Max Dist} = \max_{j=1,2,\dots,F}
-      (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
-       \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
+  \MaxDist = \max_{j=1,2,\dots,F}
+      (\overbrace{\Pnorm^{-1}(S_j)}^{\text{Maximize}} -
+       \overbrace{\Enorm(S_j)}^{\text{Minimize}} )
 \end{equation}
 where $F$ is the number of available frequencies. Then we can select the optimal
 scaling factor that satisfies EQ~\eqref{eq:max}.  Our objective function can
@@ -392,32 +417,32 @@ the objective function described above.
 \begin{figure}[tp]
   \begin{algorithmic}[1]
     % \footnotesize
-    \State  Initialize the variable $Dist=0$
+    \State  Initialize the variable $\Dist=0$
     \State Set dynamic and static power values.
-    \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 difference between two successive
+    \State Set $\Pstates$ to the number of available frequencies.
+    \State Set the variable $\Fnew$ to max. frequency,  $\Fnew = \Fmax $
+    \State Set the variable $\Fdiff$ to the difference between two successive
            frequencies.
-    \For {$j:=1$   to   $P_{states} $}
-      \State $F_{new}=F_{new} - F_{diff} $
-      \State $S = \frac{F_\textit{max}}{F_\textit{new}}$
+    \For {$j := 1$ to $\Pstates $}
+      \State $\Fnew = \Fnew - \Fdiff $
+      \State $S = \frac{\Fmax}{\Fnew}$
       \State $S_i = S \cdot \frac{T_1}{T_i}
-                  = \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}$
+                  = \frac{\Fmax}{\Fnew} \cdot \frac{T_1}{T_i}$
              for $i=1,\dots,N$
-      \State $E_\textit{Norm} =
-          \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
+      \State $\Enorm =
+          \frac{\Pdyn \cdot S_1^{-2} \cdot
                   \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
-                  P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{
-                P_\textit{dyn} \cdot
+                  \Pstatic \cdot T_1 \cdot S_1 \cdot N }{
+                \Pdyn \cdot
                   \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
-                  P_\textit{static} \cdot T_1 \cdot N }$
-      \State $P_{NormInv}=T_{old}/T_{new}$
-      \If{$(P_{NormInv}-E_{Norm} > Dist)$}
-        \State $S_{opt} = S$
-        \State $Dist = P_{NormInv} - E_{Norm}$
+                  \Pstatic \cdot T_1 \cdot N }$
+      \State $\PnormInv = \Told / \Tnew$
+      \If{$(\PnormInv - \Enorm > \Dist)$}
+        \State $\Sopt = S$
+        \State $\Dist = \PnormInv - \Enorm$
       \EndIf
     \EndFor
-    \State  Return $S_{opt}$
+    \State  Return $\Sopt$
   \end{algorithmic}
   \caption{Scaling factor selection algorithm}
   \label{EPSA}
@@ -476,7 +501,7 @@ substitution of EQ~\eqref{eq:s} in EQ~\eqref{eq:si}, we can calculate the new
 frequency $F_i$ as follows:
 \begin{equation}
   \label{eq:fi}
-  F_i = \frac{F_\textit{max} \cdot T_i}{S_\textit{optimal} \cdot T_\textit{max}}
+  F_i = \frac{\Fmax \cdot T_i}{\Sopt \cdot \Tmax}
 \end{equation}
 According to this equation all the nodes may have the same frequency value if
 they have balanced workloads, otherwise, they take different frequencies when