% \usepackage{secdot}
%\usepackage[font={footnotesize,bt}]{caption}
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-\usepackage{todonotes}
-\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\small\textbf{AG:} #2}}
+\usepackage{xspace}
+\usepackage[textsize=footnotesize]{todonotes}
+\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
\begin{document}
\author{%
\IEEEauthorblockN{%
- Ahmed Badri,
Jean-Claude Charr,
- Raphaël Couturier and
+ Raphaël Couturier,
+ Ahmed Fanfakh and
Arnaud Giersch
}
\IEEEauthorblockA{%
\maketitle
-\AG{``Optimal'' is a bit pretentious in the title}
+\AG{``Optimal'' is a bit pretentious in the title.\\
+ Complete affiliation, add an email address, etc.}
\begin{abstract}
- \AG{complete the abstract\dots}
+The important technique for energy reduction of parallel systems is CPU frequency
+scaling. This operation used by many researchers to reduce energy consumption in many
+ways. Frequency scaling operation also has big impact on the performance. In some cases,
+the performance degradation ratio is bigger than energy saving ratio when the frequency scaled
+to down level. Therefore, the trade offs between the energy and performance becomes more
+important topic when using this technique. In this paper we developed an algorithm that
+select the frequency scaling factor for both energy and performance simultaneously.
+This algorithm takes into account the communication times when selecting the frequency scaling
+factor. It is works online without training or profiling to have very small overhead.
+The algorithm has better energy-performance trade offs compared to other methods.
\end{abstract}
\section{Introduction}
+\label{sec.intro}
The need for computing power is still increasing and it is not expected to slow
down in the coming years. To satisfy this demand, researchers and supercomputers
constructors have been regularly increasing the number of computing cores in
-supercomputers (for example in November 2013, according to the top 500
+supercomputers (for example in November 2013, according to the TOP500
list~\cite{43}, the Tianhe-2 was the fastest supercomputer. It has more than 3
millions of cores and delivers more than 33 Tflop/s while consuming 17808
kW). This large increase in number of computing cores has led to large energy
minimum value does not always give the most efficient execution due to energy
leakage. The best scaling factor might be chosen during execution (online) or
during a pre-execution phase. In this paper we emphasize to develop an
-algorithm that selects the optimal frequency scaling factor that takes into
-consideration simultaneously the energy consumption and the performance. The
+algorithm that selects a frequency scaling factor that simultaneously takes into
+consideration the energy consumption and the performance. The
main objective of HPC systems is to run the application with less execution
-time. Therefore, our algorithm selects the optimal scaling factor online with
+time. Therefore, our algorithm selects the scaling factor online with
very small footprint. The proposed algorithm takes into account the
-communication times of the MPI programs to choose the scaling factor. This
+communication times of the MPI program to choose the scaling factor. This
algorithm has ability to 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
benchmarks (NPB v3.3) developed by NASA~\cite{44}. Our experiments are executed
using the simulator SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183}
over an homogeneous distributed memory architecture. Furthermore, we compare the
-proposed algorithm with Rauber's methods. The comparison's results show that our
+proposed algorithm with Rauber and R\"{u}nger methods~\cite{3}.
+The comparison's results show that our
algorithm gives better energy-time trade off.
+This paper is organized as follows: Section~\ref{sec.relwork} presents the works from other authors.
+Section~\ref{sec.ptasks} shows the execution of parallel tasks and sources of idle times. Section~\ref{sec.energy} resumes the
+energy model of homogenous platform. Section~\ref{sec.mpip} evaluates the performance of MPI program.
+Section~\ref{sec.verif} verifies the performance prediction model. Section~\ref{sec.compet} presents
+the energy-performance trade offs objective function. Section~\ref{sec.optim} demonstrates the proposed
+energy-performance algorithm. Section~\ref{sec.expe} presents the results of our experiments.
+Section~\ref{sec.compare} shows the comparison results. Finally, we conclude in Section~\ref{sec.concl}.
+
\section{Related Works}
+\label{sec.relwork}
+
+\AG{Consider introducing the models (sec.~\ref{sec.ptasks},
+ maybe~\ref{sec.energy}) before related works}
-In the this section some heuristics, to compute the scaling factor, are
-presented and classified in two parts : offline and online methods.
+In the this section some heuristics to compute the scaling factor are
+presented and classified in two parts: offline and online methods.
\subsection{The offline DVFS orientations}
The DVFS offline methods are static and are not executed during the runtime of
the program. Some approaches used heuristics to select the best DVFS state
-during the compilation phases as an example in Azevedo et al.~\cite{40}. He used
-intra-task algorithm to choose the DVFS setting when there are dependency points
+during the compilation phases as for example in Azevedo et al.~\cite{40}. They use
+dynamic voltage scaling (DVS) algorithm to choose the DVS setting when there are dependency points
between tasks. While in~\cite{29}, Xie et al. used breadth-first search
-algorithm to do that. Their goal is saving energy with time limits. Another
-approaches gathers and stores the runtime information for each DVFS state, then
-used their methods offline to select the suitable DVFS that optimize energy-time
-trade offs. As an example~\cite{8}, Rountree et al. used liner programming
-algorithm, while in~\cite{38,34}, Cochran et al. used multi logistic regression
-algorithm for the same goal. The offline study that shown the DVFS impact on the
-communication time of the MPI program is~\cite{17}, Freeh et al. show that these
-times not changed when the frequency is scaled down.
+algorithm to do that. Their goal is to save energy with time limits. Another
+approach gathers and stores the runtime information for each DVFS state, then
+selects the suitable DVFS offline to optimize energy-time
+trade offs. As an example Rountree et al.~\cite{8}, use liner programming
+algorithm, while in~\cite{38,34}, Cochran et al. use multi logistic regression
+algorithm for the same goal. The offline study that shows the DVFS impact on the
+communication time of the MPI program is~\cite{17}, where Freeh et al. show that these
+times do not change when the frequency is scaled down.
\subsection{The online DVFS orientations}
-The objective of these works is to dynamically compute and set the frequency of
+The objective of online DVFS orientations works is to dynamically compute and set the frequency of
the CPU during the runtime of the program for saving energy. Estimating and
-predicting approaches for the energy-time trade offs developed by
+predicting approaches for the energy-time trade offs are developed by Kimura, Peraza, Yu-Liang et al.
~\cite{11,2,31}. These works select the best DVFS setting depending on the slack
times. These times happen when the processors have to wait for data from other
processors to compute their task. For example, during the synchronous
-communication time that take place in the MPI programs, the processors are
-idle. The optimal DVFS can be selected using the learning methods. Therefore, in
+communications that take place in MPI programs, some processors are
+idle. The optimal DVFS can be selected using learning methods. Therefore, in Dhiman, Hao Shen et al.
~\cite{39,19} used machine learning to converge to the suitable DVFS
-configuration. Their learning algorithms have big time to converge when the
-number of available frequencies is high. Also, the communication time of the MPI
-program used online for saving energy as in~\cite{1}, Lim et al. developed an
+configuration. Their learning algorithms take big time to converge when the
+number of available frequencies is high. Also, the communication sections of the MPI
+program can be used to save energy. In~\cite{1}, Lim et al. developed an
algorithm that detects the communication sections and changes the frequency
during these sections only. This approach changes the frequency many times
because an iteration may contain more than one communication section. The domain
-of analytical modeling used for choosing the optimal frequency as in~\cite{3},
-Rauber et al. developed an analytical mathematical model for determining the
-optimal frequency scaling factor for any number of concurrent tasks, without
-considering communication times. They set the slowest task to maximum frequency
-for maintaining performance. In this paper we compare our algorithm with
-Rauber's model~\cite{3}, because his model can be used for any number of
-concurrent tasks for homogeneous platform and this is the same direction of this
-paper. However, the primary contributions of this paper are:
+of analytical modeling used for choosing the optimal frequency as inRauber and R\"{u}nger~\cite{3}. they
+developed an analytical mathematical model to determine the
+optimal frequency scaling factor for any number of concurrent tasks. They set the slowest task to maximum frequency for maintaining performance. In this paper we compare our algorithm with
+Rauber and R\"{u}nger model~\cite{3}, because their model can be used for any number of
+concurrent tasks for homogeneous platforms. The primary contributions of this paper are:
\begin{enumerate}
-\item Selecting the optimal frequency scaling factor for energy and performance
- simultaneously. While taking into account the communication time.
-\item Adapting our scale factor to taking into account the imbalanced tasks.
+\item Selecting the frequency scaling factor for simultaneously optimizing energy and performance,
+ while taking into account the communication time.
+\item Adapting our scaling factor to take into account the imbalanced tasks.
\item The execution time of our algorithm is very small when compared to other
methods (e.g.,~\cite{19}).
\item The proposed algorithm works online without profiling or training as
\end{enumerate}
\section{Parallel Tasks Execution on Homogeneous Platform}
+\label{sec.ptasks}
-A homogeneous cluster consists of identical nodes in terms of the hardware and
-the software. Each node has its own memory and at least one processor which can
+A homogeneous cluster consists of identical nodes in terms of hardware and software.
+Each node has its own memory and at least one processor which can
be a multi-core. The nodes are connected via a high bandwidth network. Tasks
executed on this model can be either synchronous or asynchronous. In this paper
we consider execution of the synchronous tasks on distributed homogeneous
-platform. These tasks can exchange the data via synchronous memory passing.
+platform. These tasks can exchange the data via synchronous message passing.
\begin{figure*}[t]
\centering
- \subfloat[Synch. Imbalanced Communications]{\includegraphics[scale=0.67]{synch_tasks}\label{fig:h1}}
- \subfloat[Synch. Imbalanced Computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
+ \subfloat[Sync. Imbalanced Communications]{\includegraphics[scale=0.67]{commtasks}\label{fig:h1}}
+ \subfloat[Sync. Imbalanced Computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
\caption{Parallel Tasks on Homogeneous Platform}
\label{fig:homo}
\end{figure*}
Therefore, the execution time of a task consists of the computation time and the
communication time. Moreover, the synchronous communications between tasks can
-lead to idle time while tasks wait at the synchronous point for others tasks to
-finish their communications see figure~(\ref{fig:h1}). Another source for idle
-times is the imbalanced computations. This happen when processing different
-amounts of data on each processor as an example see figure~(\ref{fig:h2}). In
-this case the fastest tasks have to wait at the synchronous barrier for the
-slowest tasks to finish their job. In both two cases the overall execution time
-of the program is the execution time of the slowest task as :
+lead to idle time while tasks wait at the synchronization barrier for other tasks to
+finish their communications (see figure~(\ref{fig:h1})). The imbalanced communications happen when nodes have to send/receive different amount of data or each node is communicates with different number of nodes. Another source for idle times is the imbalanced computations. This happen when processing different
+amounts of data on each processor (see figure~(\ref{fig:h2})). In
+this case the fastest tasks have to wait at the synchronization barrier for the
+slowest tasks to finish their job. In both cases the overall execution time
+of the program is the execution time of the slowest task as:
\begin{equation}
\label{eq:T1}
\textit{Program Time} = \max_{i=1,2,\dots,N} T_i
\end{equation}
-where $T_i$ is the execution time of process $i$.
+where $T_i$ is the execution time of task $i$.
\section{Energy Model for Homogeneous Platform}
+\label{sec.energy}
-The energy consumption by the processor consists of two powers metric: the
+The energy consumption by the processor consists of two power metrics: the
dynamic and the static power. This general power formulation is used by many
-researchers see~\cite{9,3,15,26}. The dynamic power of the CMOS processors
+researchers~\cite{9,3,15,26}. The dynamic power of the CMOS processors
$P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
the supply voltage $V$ and operational frequency $f$ respectively as follow :
\begin{equation}
\label{eq:pd}
- P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
+ P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
\end{equation}
The static power $P_{static}$ captures the leakage power consumption as well as
the power consumption of peripheral devices like the I/O subsystem.
\begin{equation}
\label{eq:ps}
- P_{static} = V \cdot N \cdot K_{design} \cdot I_{leak}
+ P_\textit{static} = V \cdot N \cdot K_{design} \cdot I_{leak}
\end{equation}
where V is the supply voltage, N is the number of transistors, $K_{design}$ is a
design dependent parameter and $I_{leak}$ is a technology-dependent
parameter. Energy consumed by an individual processor $E_{ind}$ is the summation
-of the dynamic and the static power multiply by the execution time for example
-see~\cite{36,15} .
+of the dynamic and the static power multiplied by the execution time for example
+see~\cite{36,15}.
\begin{equation}
\label{eq:eind}
- E_{ind} = ( P_{dyn} + P_{static} ) \cdot T
+ E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T
\end{equation}
-The dynamic voltage and frequency scaling (DVFS) is a process that allowed in
+The dynamic voltage and frequency scaling (DVFS) is a process that is allowed in
modern processors to reduce the dynamic power by scaling down the voltage and
frequency. Its main objective is to reduce the overall energy
consumption~\cite{37}. The operational frequency \emph f depends linearly on the
-supply voltage $V$, i.e., $V = \beta . f$ with some constant $\beta$. This
+supply voltage $V$, i.e., $V = \beta \cdot f$ with some constant $\beta$. This
equation is used to study the change of the dynamic voltage with respect to
various frequency values in~\cite{3}. The reduction process of the frequency are
expressed by scaling factor \emph S. The scale \emph S is the ratio between the
maximum and the new frequency as in EQ~(\ref{eq:s}).
\begin{equation}
\label{eq:s}
- S = \frac{F_{max}}{F_{new}}
+ S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
-The value of the scale \emph S is grater than 1 when changing the frequency to
-any new frequency value(\emph {P-state}) in governor. It is equal to 1 when the
-frequency are set to the maximum frequency. The energy consumption model for
-parallel homogeneous platform is depending on the scaling factor \emph S. This
-factor reduces quadratically the dynamic power. Also, this factor increases the
+The value of the scale $S$ is greater than 1 when changing the frequency to
+any new frequency value~(\emph {P-state}) in governor, the CPU governor is an interface
+driver supplied by the operating system kernel (e.g. Linux) to lowering core's frequency.
+The scaling factor is equal to 1 when the frequency set is to the maximum frequency.
+The energy consumption model for parallel homogeneous platform depends on the scaling factor \emph S. This factor reduces quadratically the dynamic power. Also, this factor increases the
static energy linearly because the execution time is increased~\cite{36}. The
-energy model, depending on the frequency scaling factor, of homogeneous platform
-for any number of concurrent tasks develops by Rauber~\cite{3}. This model
-consider the two powers metric for measuring the energy of the parallel tasks as
-in EQ~(\ref{eq:energy}).
+energy model depending on the frequency scaling factor for homogeneous platform
+for any number of concurrent tasks was developed by Rauber and R\"{u}nger~\cite{3}. This model
+considers the two power metrics for measuring the energy of the parallel tasks as
+in EQ~(\ref{eq:energy}):
\begin{equation}
\label{eq:energy}
- E = P_{dyn} \cdot S_1^{-2} \cdot
+ E = P_\textit{dyn} \cdot S_1^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
- P_{static} \cdot T_1 \cdot S_1 \cdot N
+ P_\textit{static} \cdot T_1 \cdot S_1 \cdot N
\hfill
\end{equation}
-Where \emph N is the number of parallel nodes, $T_1 $ is the time of the slower
+where \emph N is the number of parallel nodes, $T_1 $ is the time of the slowest
task, $T_i$ is the time of the task $i$ and $S_1$ is the maximum scaling factor
for the slower task. The scaling factor $S_1$, as in EQ~(\ref{eq:s1}), selects
from the set of scales values $S_i$. Each of these scales are proportional to
\begin{equation}
\label{eq:si}
S_i = S \cdot \frac{T_1}{T_i}
- = \frac{F_{max}}{F_{new}} \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
-Rauber's energy model EQ~(\ref{eq:energy}) for two reasons : 1-this model used
-for homogeneous platform that we work on in this paper. 2-we are compare our
-algorithm with Rauber's scaling model. Rauber's optimal scaling factor for
-optimal energy reduction derived from the EQ~(\ref{eq:energy}). He takes the
+where $F$ is the number of available frequencies. In this paper we depend on
+Rauber and R\"{u}nger energy model EQ~(\ref{eq:energy}) for two reasons: (1)-this model is used
+for homogeneous platform that we work on in this paper. 2-we compare our
+algorithm with Rauber and R\"{u}nger scaling model. Rauber and R\"{u}nger scaling factor that reduce
+ energy consumption derived from the EQ~(\ref{eq:energy}). They take the
derivation for this equation (to be minimized) and set it to zero to produce the
scaling factor as in EQ~(\ref{eq:sopt}).
\begin{equation}
\label{eq:sopt}
- S_{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_{dyn}}{P_{static}} \cdot
+ S_\textit{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \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 the parallel MPI applications are depends on
+The performance (execution time) of parallel MPI applications depend on
the time of the slowest task as in figure~(\ref{fig:homo}). Normally the
execution time of the parallel programs are proportional to the operational
-frequency. Therefore, any DVFS operation for the energy reduction increase the
+frequency. Therefore, any DVFS operation for the energy reduction increases the
execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) the
-energy affected by the scaling factor $S$. This factor also has a great impact
+energy is affected by the scaling factor $S$. This factor also has a great impact
on the performance. When scaling down the frequency to the new value according
-to EQ~(\ref{eq:s}) lead to the value of the scale $S$ has inverse relation with
-new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decrease the
-frequency value, the execution time increase. Then the new frequency value has
-inverse relation with time ($F_{new} \propto \frac{1}{T}$). This lead to the
+to EQ~(\ref{eq:s}), the value of the scale $S$ has inverse relation with
+new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decreasing the
+frequency value, the execution time increases. Then the new frequency value has
+inverse relation with time ($F_{new} \propto \frac{1}{T}$). This leads to the
frequency scaling factor $S$ proportional linearly with execution time ($S
\propto T$). Large scale MPI applications such as NAS benchmarks have
considerable amount of communications embedded in these programs. During the
-communication process the processor remain idle until the communication has
+communication process the processors remain idle until the communication has
finished. For that reason any change in the frequency has no impact on the time
of communication but it has obvious impact on the time of
-computation~\cite{17}. We are made many tests on real cluster to prove that the
+computation~\cite{17}. We have made many tests on a real cluster to prove that the
frequency scaling factor \emph S has a linear relation with computation time
-only also see~\cite{41}. To predict the execution time of MPI program, firstly
-must be precisely specifying communication time and the computation time for the
-slower task. Secondly, we use these times for predicting the execution time for
-any MPI program as a function of the new scaling factor as in the
-EQ~(\ref{eq:tnew}).
+only. To predict the execution time of MPI program, the communication time and
+the computation time for the slower task must be first precisely specified. Secondly,
+these times are used to predict the execution time for any MPI program as a function of
+the new scaling factor as in the EQ~(\ref{eq:tnew}).
\begin{equation}
\label{eq:tnew}
- T_{new} = T_{\textit{Max Comp Old}} \cdot S + T_{\textit{Max Comm Old}}
+ \textit T_\textit{new} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
\end{equation}
The above equation shows that the scaling factor \emph S has linear relation
with the computation time without affecting the communication time. The
investigation study for the EQ~(\ref{eq:tnew}).
\section{Performance Prediction Verification}
+\label{sec.verif}
In this section we evaluate the precision of our performance prediction methods
on the NAS benchmark. We use the EQ~(\ref{eq:tnew}) that predicts the execution
figure~(\ref{fig:pred}).
\section{Performance to Energy Competition}
+\label{sec.compet}
+
This section demonstrates our approach for choosing the optimal scaling
factor. This factor gives maximum energy reduction taking into account the
-execution time for both computation and communication times . The relation
+execution time for both computation and communication times. The relation
between the energy and the performance are nonlinear and complex, because the
relation of the energy with scaling factor is nonlinear and with the performance
it is linear see~\cite{17}. The relation between the energy and the performance
without scaled frequency :
\begin{multline}
\label{eq:enorm}
- E_\textit{Norm} = \frac{E_{Reduced}}{E_{Original}}\\
- {} = \frac{ P_{dyn} \cdot S_i^{-2} \cdot
+ E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
+ {} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_{static} \cdot T_1 \cdot S_i \cdot N }{
- P_{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_{static} \cdot T_1 \cdot N }
+ P_\textit{static} \cdot T_1 \cdot S_i \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}
-\AG{Use \texttt{\textbackslash{}text\{xxx\}} or
- \texttt{\textbackslash{}textit\{xxx\}} for all subscripted words in equations
- (e.g. \mbox{\texttt{E\_\{\textbackslash{}text\{Norm\}\}}}).
-
- Don't hesitate to define new commands :
- \mbox{\texttt{\textbackslash{}newcommand\{\textbackslash{}ENorm\}\{E\_\{\textbackslash{}text\{Norm\}\}\}}}
-}
By the same way we can normalize the performance as follows :
\begin{equation}
\label{eq:pnorm}
- P_{Norm} = \frac{T_{New}}{T_{Old}}
- = \frac{T_{\textit{Max Comp Old}} \cdot S +
- T_{\textit{Max Comm Old}}}{T_{Old}}
+ 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{Old}}
\end{equation}
The second problem is the optimization operation for both energy and performance
is not in the same direction. In other words, the normalized energy and the
performance as follows :
\begin{equation}
\label{eq:pnorm_en}
- P^{-1}_{Norm} = \frac{T_{Old}}{T_{New}}
- = \frac{T_{Old}}{T_{\textit{Max Comp Old}} \cdot S +
- T_{\textit{Max Comm Old}}}
+ P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
+ = \frac{ T_\textit{Old}}{T_\textit{Max Comp Old} \cdot S +
+ T_\textit{Max Comm Old}}
\end{equation}
\begin{figure*}
\centering
following form:
\begin{equation}
\label{eq:max}
- \textit{MaxDist} = \max (\overbrace{P^{-1}_{Norm}}^{\text{Maximize}} -
- \overbrace{E_{Norm}}^{\text{Minimize}} )
+ \textit{MaxDist} = \max (\overbrace{P^{-1}_\textit{Norm}}^{\text{Maximize}} -
+ \overbrace{E_\textit{Norm}}^{\text{Minimize}} )
\end{equation}
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
factor as shown in~\cite{15,3,19}. Energy measurement model is not the
-objective of this paper and we choose Rauber's model as an example with two
+objective of this paper and we choose Rauber and R\"{u}nger model as an example with two
reasons that mentioned before.
\section{Optimal Scaling Factor for Performance and Energy}
+\label{sec.optim}
In the previous section we described the objective function that satisfy our
goal in discovering optimal scaling factor for both performance and energy at
the same time. Therefore, we develop an energy to performance scaling algorithm
-(EPSA). This algorithm is simple and has a direct way to calculate the optimal
+($EPSA$). This algorithm is simple and has a direct way to calculate the optimal
scaling factor for both energy and performance at the same time.
\begin{algorithm}[tp]
\caption{EPSA}
\For {$i=1$ 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 in EQ~(\ref{eq:si}).
- \State - Select the maximum scale factor $S_1$ from the set of scales $S_i$.
- \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ as in EQ~(\ref{eq:enorm}).
- \State - Calculate the normalize inverse of performance\par
+ \State - Calculate all available scales $S_i$ depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}).
+ \State - Select the maximum scale factor $S_1$ from the set\par\hspace{1 pt} of scales $S_i$.
+ \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$
+ \par\hspace{1 pt} as in EQ~(\ref{eq:enorm}).
+ \State - Calculate the normalize inverse of performance\par\hspace{1 pt}
$P_{NormInv}=T_{old}/T_{new}$ as in EQ~(\ref{eq:pnorm_en}).
\If{ $(P_{NormInv}-E_{Norm} > Dist$) }
\State $S_{optimal} = S$
program. The DVFS algorithm~(\ref{dvfs}) shows where and when the EPSA algorithm is called
in the MPI program.
%\begin{minipage}{\textwidth}
-%\AG{Use the same format as for Algorithm~\ref{EPSA}}
+%\AG{Use the same format as for Algorithm~\ref{$EPSA$}}
\begin{algorithm}[tp]
\caption{DVFS}
\label{dvfs}
- \begin{algorithmic}
+ \begin{algorithmic}[1]
\For {$J:=1$ to $Some-Iterations \; $}
\State -Computations Section.
\State -Communications Section.
\If {$(J==1)$}
- \State -Gather all times of computation and\par
- \State communication from each node.
+ \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 -Set the new frequency to the system.
can calculate the new frequency $F_i$ as follows :
\begin{equation}
\label{eq:fi}
- F_i = \frac{F_{max} \cdot T_i}{S_{optimal} \cdot T_{max}}
+ F_i = \frac{F_\textit{max} \cdot T_i}{S_\textit{optimal} \cdot T_\textit{max}}
\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
the frequency according to the nodes workloads.
\section{Experimental Results}
+\label{sec.expe}
The proposed EPSA algorithm was applied to seven MPI programs of the NAS
benchmarks (EP, CG, MG, FT, BT, LU and SP). We work on three classes (A, B and
table~(\ref{table:platform}). Each node in the cluster has 18 frequency values
from 2.5 GHz to 800 MHz with 100 MHz difference between each two successive
frequencies.
-\begin{table}[ht]
+\begin{table}[htb]
\caption{Platform File Parameters}
% title of Table
\centering
\hline
Max & Min & Backbone & Backbone&Link &Link& Sharing \\
Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy \\ \hline
- 2.5 &800 & 2.25 GBps &$5\times 10^{-7} s$& 1 GBps & $5\times 10^{-5}$ s&Full \\
+ 2.5 &800 & 2.25 GBps &$5\times 10^{-7} s$& 1 GBps & $5\times 10^{-5} s$ &Full \\
GHz& MHz& & & & &Duplex \\\hline
\end{tabular}
\label{table:platform}
factors results for each program on class C. These factors give the maximum
energy saving percent and the minimum performance degradation percent in the
same time over all available scales.
-\begin{figure*}
+\begin{figure*}[t]
\centering
\includegraphics[width=.33\textwidth]{ep.eps}\hfill%
\includegraphics[width=.33\textwidth]{cg.eps}\hfill%
\caption{Optimal scaling factors for The NAS MPI Programs}
\label{fig:nas}
\end{figure*}
-\begin{table}
+\begin{table}[htb]
\caption{Optimal Scaling Factors Results}
% title of Table
\centering
\AG{Use the same number of decimals for all numbers in a column,
and vertically align the numbers along the decimal points.
The same for all the following tables.}
- \begin{tabular}{ | l | l | l |l | l | p{2cm} |}
+ \begin{tabular}{ | l | l | l |l | r |}
\hline
Program & Optimal & Energy & Performance&Energy-Perf.\\
Name & Scaling Factor& Saving \%&Degradation \% &Distance \\ \hline
BT & 1.31 &29.60&21.28 &8.32\\ \hline
SP & 1.38 &33.48&21.36 &12.12\\ \hline
FT & 1.47 &34.72&19.00 &15.72\\ \hline
- \end{tabular}
+ \end{tabular}
\label{table:factors results}
% is used to refer this table in the text
\end{table}
EPSA to selects smaller scaling factor values (inducing smaller energy savings).
\section{Comparing Results}
+\label{sec.compare}
-In this section, we compare our EPSA algorithm results with Rauber's
+In this section, we compare our EPSA algorithm results with Rauber and R\"{u}nger
methods~\cite{3}. He had two scenarios, the first is to reduce energy to optimal
level without considering the performance as in EQ~(\ref{eq:sopt}). We refer to
-this scenario as $Rauber_{E}$. The second scenario is similar to the first
+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 possible. We refer to this
-scenario as $Rauber_{E-P}$. The comparison is made in tables~(\ref{table:compare
+scenario as $R_{E-P}$. The comparison is made in tables~(\ref{table:compare
Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
-tables show the results of our EPSA and Rauber's two scenarios for all the NAS
+tables show the results of our EPSA and Rauber and R\"{u}nger scenarios for all the NAS
benchmarks programs for classes A,B and C.
-\begin{table}[ht]
+\begin{table}[p]
\caption{Comparing Results for The NAS Class A}
% title of Table
\centering
- \begin{tabular}{ | l | l | l |l | l | l| }
+ \begin{tabular}{ | l | l | l |l | l | r| }
\hline
Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- name &name&value& Saving \%&Degradation \% &Distance
+ Name &Name&Value& Saving \%&Degradation \% &Distance
\\ \hline
% \rowcolor[gray]{0.85}
- EPSA&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
- $Rauber_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline
- $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
+ $EPSA$&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
+ $R_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline
+ $R_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
- EPSA&MG & 1.47 &27.66&16.82&10.84\\ \hline
- $Rauber_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
- $Rauber_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline
+ $EPSA$&MG & 1.47 &27.66&16.82&10.84\\ \hline
+ $R_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
+ $R_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline
- EPSA&EP &1.19 &25.32&20.79&4.53\\ \hline
- $Rauber_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
- $Rauber_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline
+ $EPSA$&EP &1.19 &25.32&20.79&4.53\\ \hline
+ $R_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
+ $R_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline
- EPSA&LU&1.56& 39.55 &19.38& 20.17\\ \hline
- $Rauber_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline
- $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
+ $EPSA$&LU&1.56& 39.55 &19.38& 20.17\\ \hline
+ $R_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline
+ $R_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
- EPSA&BT&1.31& 29.60&20.53&9.07 \\ \hline
- $Rauber_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline
- $Rauber_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline
+ $EPSA$&BT&1.31& 29.60&20.53&9.07 \\ \hline
+ $R_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline
+ $R_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline
- EPSA&SP&1.38& 33.51&15.65&17.86 \\ \hline
- $Rauber_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline
- $Rauber_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
+ $EPSA$&SP&1.38& 33.51&15.65&17.86 \\ \hline
+ $R_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline
+ $R_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
- EPSA&FT&1.25&25.00&10.80&14.20 \\ \hline
- $Rauber_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
- $Rauber_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline
+ $EPSA$&FT&1.25&25.00&10.80&14.20 \\ \hline
+ $R_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
+ $R_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline
\end{tabular}
\label{table:compare Class A}
% is used to refer this table in the text
\end{table}
-\begin{table}[ht]
+\begin{table}[p]
\caption{Comparing Results for The NAS Class B}
% title of Table
\centering
- \begin{tabular}{ | l | l | l |l | l |l| }
+ \begin{tabular}{ | l | l | l |l | l |r| }
\hline
Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- name &name&value& Saving \%&Degradation \% &Distance
+ Name &Name&Value& Saving \%&Degradation \% &Distance
\\ \hline
% \rowcolor[gray]{0.85}
- EPSA&CG & 1.66 &39.23&16.63&22.60 \\ \hline
- $Rauber_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline
- $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
+ $EPSA$&CG & 1.66 &39.23&16.63&22.60 \\ \hline
+ $R_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline
+ $R_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
- EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline
- $Rauber_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
- $Rauber_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
+ $EPSA$ &MG & 1.47 &34.98&18.35&16.63\\ \hline
+ $R_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
+ $R_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
- EPSA&EP &1.08 &20.29&17.15&3.14 \\ \hline
- $Rauber_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline
- $Rauber_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline
+ $EPSA$&EP &1.08 &20.29&17.15&3.14 \\ \hline
+ $R_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline
+ $R_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline
- EPSA&LU&1.47&38.57&21.34&17.23 \\ \hline
- $Rauber_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline
- $Rauber_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline
+ $EPSA$&LU&1.47&38.57&21.34&17.23 \\ \hline
+ $R_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline
+ $R_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline
- EPSA&BT&1.31& 29.59&20.88&8.71\\ \hline
- $Rauber_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline
- $Rauber_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline
+ $EPSA$&BT&1.31& 29.59&20.88&8.71\\ \hline
+ $R_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline
+ $R_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline
- EPSA&SP&1.38&33.44&19.24&14.20 \\ \hline
- $Rauber_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline
- $Rauber_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
+ $EPSA$&SP&1.38&33.44&19.24&14.20 \\ \hline
+ $R_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline
+ $R_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
- EPSA&FT&1.38&34.40&14.57&19.83 \\ \hline
- $Rauber_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
- $Rauber_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline
+ $EPSA$&FT&1.38&34.40&14.57&19.83 \\ \hline
+ $R_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
+ $R_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline
\end{tabular}
\label{table:compare Class B}
% is used to refer this table in the text
\end{table}
-\begin{table}[ht]
+\begin{table}[p]
\caption{Comparing Results for The NAS Class C}
% title of Table
\centering
- \begin{tabular}{ | l | l | l |l | l |l| }
+ \begin{tabular}{ | l | l | l |l | l |r| }
\hline
Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- name &name&value& Saving \%&Degradation \% &Distance
+ Name &Name&Value& Saving \%&Degradation \% &Distance
\\ \hline
% \rowcolor[gray]{0.85}
- EPSA&CG & 1.56 &39.23&14.88&24.35 \\ \hline
- $Rauber_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline
- $Rauber_{E}$&CG &2.15 &45.36&26.70&18.66\\ \hline
+ $EPSA$&CG & 1.56 &39.23&14.88&24.35 \\ \hline
+ $R_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline
+ $R_{E}$&CG &2.15 &45.36&26.70&18.66\\ \hline
- EPSA&MG & 1.47 &34.97&21.69&13.27\\ \hline
- $Rauber_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline
- $Rauber_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
+ $EPSA$&MG & 1.47 &34.97&21.69&13.27\\ \hline
+ $R_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline
+ $R_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
- EPSA&EP &1.04 &22.14&20.73&1.41 \\ \hline
- $Rauber_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline
- $Rauber_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline
+ $EPSA$&EP &1.04 &22.14&20.73&1.41 \\ \hline
+ $R_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline
+ $R_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline
- EPSA&LU&1.38&35.83&22.49&13.34 \\ \hline
- $Rauber_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline
- $Rauber_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline
+ $EPSA$&LU&1.38&35.83&22.49&13.34 \\ \hline
+ $R_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline
+ $R_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline
- EPSA&BT&1.31& 29.60&21.28&8.32\\ \hline
- $Rauber_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline
- $Rauber_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline
+ $EPSA$&BT&1.31& 29.60&21.28&8.32\\ \hline
+ $R_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline
+ $R_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline
- EPSA&SP&1.38&33.48&21.35&12.12\\ \hline
- $Rauber_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline
- $Rauber_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline
+ $EPSA$&SP&1.38&33.48&21.35&12.12\\ \hline
+ $R_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline
+ $R_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline
- EPSA&FT&1.47&34.72&19.00&15.72 \\ \hline
- $Rauber_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
- $Rauber_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline
+ $EPSA$&FT&1.47&34.72&19.00&15.72 \\ \hline
+ $R_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
+ $R_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline
\end{tabular}
\label{table:compare Class C}
% is used to refer this table in the text
\end{table}
As shown in these tables our scaling factor is not optimal for energy saving
such as Rauber's scaling factor EQ~(\ref{eq:sopt}), but it is optimal for both
-the energy and the performance simultaneously. Our EPSA optimal scaling factors
+the energy and the performance simultaneously. Our $EPSA$ optimal scaling factors
has better simultaneous optimization for both the energy and the performance
-compared to Rauber's energy-performance method ($Rauber_{E-P}$). Also, in
-($Rauber_{E-P}$) method when setting the frequency to maximum value for the
+compared to Rauber and R\"{u}nger energy-performance method ($R_{E-P}$). Also, in
+($R_{E-P}$) method when setting the frequency to maximum value for the
slower task lead to a small improvement of the performance. Also the results
show that this method keep or improve energy saving. Because of the energy
consumption decrease when the execution time decreased while the frequency value
Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving
percent and the performance degradation percent. Therefore, this means it is the
same resultant of our objective function EQ~(\ref{eq:max}). Our algorithm always
-gives positive energy to performance trade offs while Rauber's method
-($Rauber_{E-P}$) gives in some time negative trade offs such as in BT and
+gives positive energy to performance trade offs while Rauber and R\"{u}nger method
+($R_{E-P}$) gives in some time negative trade offs such as in BT and
EP. The positive trade offs with highest values lead to maximum energy savings
concatenating with less performance degradation and this the objective of this
paper. While the negative trade offs refers to improving energy saving (or may
be the performance) while degrading the performance (or may be the energy) more
than the first.
-\begin{figure*}
+\begin{figure}[t]
\centering
- \includegraphics[width=.33\textwidth]{compare_class_A.pdf}\hfill%
- \includegraphics[width=.33\textwidth]{compare_class_B.pdf}\hfill%
+ \includegraphics[width=.33\textwidth]{compare_class_A.pdf}
+ \includegraphics[width=.33\textwidth]{compare_class_B.pdf}
\includegraphics[width=.33\textwidth]{compare_class_c.pdf}
- \caption{Comparing Our EPSA with Rauber's Methods}
+ \caption{Comparing Our EPSA with Rauber and R\"{u}nger Methods}
\label{fig:compare}
-\end{figure*}
+\end{figure}
\section{Conclusion}
-\label{sec.conc}
+\label{sec.concl}
\AG{the conclusion needs to be written\dots{} one day}
\section*{Acknowledgment}
+\AG{Right?}
Computations have been performed on the supercomputer facilities of the
Mésocentre de calcul de Franche-Comté.
+% trigger a \newpage just before the given reference
+% number - used to balance the columns on the last page
+% adjust value as needed - may need to be readjusted if
+% the document is modified later
+%\IEEEtriggeratref{15}
+
\bibliographystyle{IEEEtran}
\bibliography{IEEEabrv,my_reference}
\end{document}
%%% mode: latex
%%% TeX-master: t
%%% fill-column: 80
-%%%ispell-local-dictionary: "american"
+%%% ispell-local-dictionary: "american"
%%% End:
% LocalWords: Badri Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords: CMOS EQ EPSA
+% LocalWords: CMOS EQ $$EPSA$$ Franche Comté Tflop
