\usepackage[utf8]{inputenc}
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\usepackage{algpseudocode}
-\usepackage{graphicx,graphics}
+\usepackage{graphicx}
\usepackage{subfig}
\usepackage{amsmath}
\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{\Tnorm}{\Xsub{T}{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
predict both energy consumption and execution time over all available scaling
factors. The prediction achieved depends on some computing time information,
gathered at the beginning of the runtime. We apply this algorithm to the NAS parallel benchmarks (NPB v3.3)~\cite{44}. Our experiments are executed using the simulator
-SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over an homogeneous
+SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} over a homogeneous
distributed memory architecture. Furthermore, we compare the proposed algorithm
with Rauber and Rünger methods~\cite{3}. The comparison's results show that our
algorithm gives better energy-time trade-off.
% paper in homogeneous clusters}
-\section{Energy model for homogeneous platform}
+\section{Energy model for a homogeneous platform}
\label{sec.exe}
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
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
\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 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}.
+factors $S_i$ 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
\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}
\section{Performance evaluation of MPI programs}
\label{sec.mpip}
-The performance (execution time) of parallel synchronous MPI applications depends
-on the time of the slowest task. If there is no
+The execution time of a parallel synchronous iterative application is
+equal to the execution time of the slowest task. If there is no
communication and the application is not data bounded, the execution time of a
parallel program is linearly proportional to the operational frequency and any
DVFS operation for energy reduction increases the execution time of the parallel
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.
\section{Performance and energy reduction trade-off}
\label{sec.compet}
-This section presents our approach for choosing the optimal scaling factor.
-This factor gives maximum energy reduction while taking into account the execution
-times for both computation and communication. The relation between the performance
-and the energy is nonlinear and complex. Thus, unlike the relation between the performance and the scaling factor, the relation of energy with the scaling factor is nonlinear, for more details refer to~\cite{17}. Moreover, they are not measured using the same metric. To
-solve this problem, we normalize the energy by calculating the ratio between
-the consumed energy with scaled frequency and the consumed energy without scaled
+This section presents our method for choosing the optimal scaling factor that
+gives the best tradeoff between energy reduction and performance. This method
+takes into account the execution times for both computation and communication to
+compute the scaling factor. Since the energy consumption and the performance
+are not measured using the same metric, a normalized value of both measurements
+can be used to compare them. The normalized energy is the ratio between 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:
+In the same way, the normalized execution time of a program is computed 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}}
+ \Tnorm = \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
-and the performance curves are not at the same direction see
-Figure~\ref{fig:rel}\subref{fig:r2}. While the main goal is to optimize the
-energy and performance in the same time. According to the
-equations~\eqref{eq:enorm} and~\eqref{eq:pnorm}, the scaling factor $S$ reduce
-both the energy and the performance simultaneously. But the main objective is
-to produce maximum energy reduction with minimum performance reduction. Many
-researchers used different strategies to solve this nonlinear problem for
-example see~\cite{19,42}, their methods add big overheads to the algorithm to
-select the suitable frequency. In this paper we present a method to find the
-optimal scaling factor $S$ to optimize both energy and performance
-simultaneously without adding a big overhead. Our solution for this problem is
-to make the optimization process for energy and performance follow the same
-direction. Therefore, we inverse the equation of the normalized performance as
-follows:
+The relation between the execution time and the consumed energy of a program is nonlinear and complex. In consequences, the relation between the consumed energy and the scaling factor is also nonlinear, for more details refer to~\cite{17}. Therefore, the resulting normalized energy consumption curve and execution time curve, for different scaling factors, do not have the same direction see Figure~\ref{fig:rel}\subref{fig:r2}. To tackle this problem and optimize both terms, we inverse the equation of the normalized execution time 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 = \frac{ \Told}{ \Tnew}
+ = \frac{\TmaxCompOld +
+ \TmaxCommOld}{\TmaxCompOld \cdot S +
+ \TmaxCommOld}
\end{equation}
\begin{figure}
\centering
\label{fig:rel}
\end{figure}
Then, we can model our objective function as finding the maximum distance
-between the energy curve EQ~\eqref{eq:enorm} and the inverse of performance
+between the energy curve EQ~\eqref{eq:enorm} and the inverse of the execution time (performance)
curve EQ~\eqref{eq:pnorm_en} over all available scaling factors. This
represents the minimum energy consumption with minimum execution time (better
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(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
\begin{figure}[tp]
\begin{algorithmic}[1]
% \footnotesize
- \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
- frequencies.
- \For {$j:=1$ to $P_{states} $}
- \State $F_{new}=F_{new} - F_{diff} $
- \State $S = \frac{F_\textit{max}}{F_\textit{new}}$
- \State $S_i = S \cdot \frac{T_1}{T_i}
- = \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}$
+ \Require ~
+ \begin{description}
+ \item[$\Pstatic$] static power value
+ \item[$\Pdyn$] dynamic power value
+ \item[$\Pstates$] number of available frequencies
+ \item[$\Fmax$] maximum frequency
+ \item[$\Fdiff$] difference between two successive freq.
+ \end{description}
+ \Ensure $\Sopt$ is the optimal scaling factor
+
+ \State $\Sopt \gets 1$
+ \State $\Dist \gets 0$
+ \State $\Fnew \gets \Fmax$
+ \For {$j = 2$ to $\Pstates$}
+ \State $\Fnew \gets \Fnew - \Fdiff$
+ \State $S \gets \Fmax / \Fnew$
+ \State $S_i \gets S \cdot \frac{T_1}{T_i}
+ = \frac{\Fmax}{\Fnew} \cdot \frac{T_1}{T_i}$
for $i=1,\dots,N$
- \State $E_\textit{Norm} =
- \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
+ \State $\Enorm \gets
+ \frac{\Pdyn \cdot S_1^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{
- P_\textit{dyn} \cdot
+ \Pstatic \cdot T_1 \cdot S_1 \cdot N }{
+ \Pdyn \cdot
\left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_\textit{static} \cdot T_1 \cdot N }$
- \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 $\Pnorm \gets \Told / \Tnew$
+ \If{$(\Pnorm - \Enorm > \Dist)$}
+ \State $\Sopt \gets S$
+ \State $\Dist \gets \Pnorm - \Enorm$
\EndIf
\EndFor
- \State Return $S_{opt}$
+ \State Return $\Sopt$
\end{algorithmic}
\caption{Scaling factor selection algorithm}
\label{EPSA}
\begin{figure}[tp]
\begin{algorithmic}[1]
% \footnotesize
- \For {$k:=1$ to \textit{some iterations}}
+ \For {$k=1$ to \textit{some iterations}}
\State Computations section.
\State Communications section.
\If {$(k=1)$}
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
from \np[GHz]{2.5} to \np[MHz]{800} with \np[MHz]{100} difference between each
two successive frequencies. The nodes are connected via an ethernet network with 1Gbit/s bandwidth.
-\subsection{Performance prediction verification}
+\subsection{Execution time prediction verification}
-In this section we evaluate the precision of our performance prediction method
+In this section we evaluate the precision of our execution time prediction method
based on EQ~\eqref{eq:tnew} by applying it to the NAS benchmarks. The NAS programs
are executed with the class B option to compare the real execution time with
the predicted execution time. Each program runs offline with all available
different scaling factors for each program depending on the communication
features of the program as in the plots from Figure~\ref{fig:nas}. These plots
illustrate that there are different distances between the normalized energy and
-the normalized inverted performance curves, because there are different
+the normalized inverted execution time curves, because there are different
communication features for each benchmark. When there are little or no
-communications, the inverted performance curve is very close to the energy
+communications, the inverted execution time curve is very close to the energy
curve. Then the distance between the two curves is very small. This leads to
small energy savings. The opposite happens when there are a lot of
communication, the distance between the two curves is big. This leads to more
energy savings (e.g. CG and FT), see Table~\ref{table:compareC}. All discovered
-frequency scaling factors optimize both the energy and the performance
+frequency scaling factors optimize both the energy and the execution time
simultaneously for all NAS benchmarks. In Table~\ref{table:compareC}, we record
all optimal scaling factors results for each benchmark running class C. These
scaling factors give the maximum energy saving percentage and the minimum
In this section, we compare our scaling factor selection method with Rauber and
Rünger methods~\cite{3}. They had two scenarios, the first is to reduce energy
-to the optimal level without considering the performance as in
+to the optimal level without considering the execution time as in
EQ~\eqref{eq:sopt}. We refer to this scenario as $R_{E}$. The second scenario
is similar to the first except setting the slower task to the maximum frequency
(when the scale $S=1$) to keep the performance from degradation as mush as
In this paper, we have presented a new online scaling factor selection method
that optimizes simultaneously the energy and performance of a distributed
-application running on an homogeneous cluster. It uses the computation and
+application running on a homogeneous cluster. It uses the computation and
communication times measured at the first iteration to predict energy
-consumption and the performance of the parallel application at every available
+consumption and the execution time of the parallel application at every available
frequency. Then, it selects the scaling factor that gives the best trade-off
between energy reduction and performance which is the maximum distance between
-the energy and the inverted performance curves. To evaluate this method, we
+the energy and the inverted execution time curves. To evaluate this method, we
have applied it to the NAS benchmarks and it was compared to Rauber and Rünger
methods while being executed on the simulator SimGrid. The results showed that
our method, outperforms Rauber and Rünger's methods in terms of energy-performance
% the document is modified later
%\IEEEtriggeratref{15}
+\newpage
\bibliographystyle{IEEEtran}
\bibliography{IEEEabrv,my_reference}
\end{document}
