[mpi-energy.git] / paper.tex

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\begin{document}

\title{Dynamic Frequency Scaling for Energy Consumption Reduction in Distributed MPI Programs}

\author{%
  \IEEEauthorblockN{%
    Jean-Claude Charr,
    Raphaël Couturier,
    Ahmed Fanfakh and
    Arnaud Giersch
  }
  \IEEEauthorblockA{%
    FEMTO-ST Institute\\
    University of Franche-Comté\\
    IUT de Belfort-Montbéliard, 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
    % Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël
    % Fax: \mbox{+33 3 84 58 77 81}\\      % Dept Info
    Email: \email{{jean-claude.charr,raphael.couturier,ahmed.fanfakh_badri_muslim,arnaud.giersch}@univ-fcomte.fr}
   }
  }

\maketitle

\begin{abstract}
  Dynamic Voltage Frequency Scaling (DVFS) can be applied to modern CPUs. 
This technique is usually used to reduce the energy consumed by a CPU while
computing . Indeed, power consumption by a processor at a given instant is
exponentially related to its frequency. Thus, decreasing the frequency reduces
the power consumed by the CPU. However, it can also significantly affect the
performance of the executed program if it is compute bound and if a low CPU
frequency is selected. The performance degradation ratio can even be higher than
the saved energy ratio. Therefore, the chosen scaling factor must give the best possible trade-off
between energy reduction and performance. 

In this paper we present an algorithm
that predicts the energy consumed with each frequency gear and selects the one that
gives the best ratio between energy consumption reduction and performance.
This algorithm works online without training or profiling and
has a very small overhead. It also takes into account synchronous communications between the nodes
that are executing the distributed algorithm. The algorithm has been evaluated over the SimGrid simulator
while being applied to the NAS parallel benchmark programs. The results of the experiments show that it outperforms other existing scaling factor selection algorithms.
\end{abstract}

\section{Introduction}
\label{sec.intro}

The need and demand for more computing power have been increasing since the birth of the first computing unit 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 and processors in
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
consumption by these architectures. Moreover, the price of energy is expected to
continue its ascent according to the demand. For all these reasons energy
reduction became an important topic in the high performance computing field. To
tackle this problem, many researchers used DVFS (Dynamic Voltage Frequency
Scaling) operations which reduce dynamically the frequency and voltage of cores
and thus their energy consumption. Indeed, modern CPUs offer a set of acceptable frequencies which are usually called gears, and the user or the operating system can modify the frequency of the processor according to its needs. However, DVFS also degrades the
performance of computation. Therefore researchers try to reduce the frequency to
minimum when processors are idle (waiting for data from other processors or
communicating with other processors). Moreover, depending on their objectives
they use heuristics to find the best scaling factor during the computation. If
they aim for performance they choose the best scaling factor that reduces the
consumed energy while affecting as little as possible the performance. On the
other hand, if they aim for energy reduction, the chosen scaling factor must
produce the most energy efficient execution without considering the degradation
of the performance. It is important to notice that lowering the frequency to
minimum value does not always give the most energy 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 present an
algorithm that selects a frequency scaling factor that simultaneously takes into
consideration  the energy consumption by the CPU and the performance of the application. The
main objective of HPC systems is to execute as fast as possible the application. 
Therefore, our algorithm selects the scaling factor online with
very small footprint. The proposed algorithm takes into account the
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
this algorithm to seven MPI benchmarks. These MPI programs are the NAS parallel
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 and Rünger 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 some related works
from other authors.  Section~\ref{sec.exe} explains the execution of parallel
tasks and the sources of slack times.  It also presents an energy
model for homogeneous platforms. Section~\ref{sec.mpip} describes how the performance
of MPI programs can be predicted .  Section~\ref{sec.compet} presents the energy-performance 
objective function that maximizes the reduction of energy consumption while minimizing the degradation of the program's performance. Section~\ref{sec.optim} details the proposed energy-performance algorithm. Section~\ref{sec.expe} verifies the accuracy of the performance prediction
model and presents the results of the proposed algorithm.  It also shows the comparison results between our method and other existing methods. Finally,
we conclude in Section~\ref{sec.concl} with a summary and some future works.
\section{Related works}
\label{sec.relwork}


In this section, some heuristics to compute the scaling factor are
presented and classified into two categories: offline and online methods.

\subsection{Offline scaling factor selection methods}

The offline scaling factor selection methods are executed before the runtime of
the program. They return static scaling factor values to the processors
participating in the execution of the parallel program. On one hand, the scaling
factor
values could be computed based on information retrieved by analyzing the code of
the program and the computing system that will execute it. In ~\cite{40},
Azevedo et
al. detect during compilation the dependency points between  
tasks in a multi-task program. This information is then used to lower the frequency of
some processors in order to eliminate slack times. A slack time is the period of time during which a processor that have already finished its computation, have to wait
for a set of processors to finish their computations and send their results to the
waiting processor in order to continue its task that is
dependent on the results of computations being executed on other processors.
Freeh et al. showed in ~\cite{17} that the 
communication times of MPI programs do not change when the frequency is scaled down.
On the other hand, some offline scaling factor selection methods use the
information gathered from previous full or
partial executions of the program. A part or the whole program is usually executed over all the available frequency gears and the the execution time and the energy consumed  with each frequency gear are measured. Then an heuristic or an exact method uses the retrieved information to compute the values of the scaling factor for the processors.  
   In~\cite{29}, Xie et al. use an exact exponential breadth-first search algorithm to compute the scaling factor values that give the optimal energy reduction while respecting a deadline for a sequential program. They also present a linear heuristic that approximates the optimal solution. In~\cite{8} , Rountree et al. use a linear programming
algorithm, while in~\cite{38,34}, Cochran et al. use multi logistic regression algorithm for the same goal. 
The main drawback for these methods is that they all require executing a part or the whole program on all frequency gears for each new instance of the same program. 

\subsection{Online scaling factor selection methods}
The online scaling factor selection methods are executed during the runtime of the program. They are usually integrated into iterative programs where the same block of instructions is executed many times.  During the first few iterations, many informations are measured such as the execution time, the energy consumed using a multimeter, the slack times, ... Then a  method will exploit these measurements to compute the scaling factor values for each processor. This operation, measurements and computing new scaling factor, can be repeated as much as needed if the iterations are not regular. Kimura, Peraza, Yu-Liang et al.  ~\cite{11,2,31} used learning methods to select the appropriate scaling factor values  to eliminate the slack times during runtime. However, as seen in ~\cite{39,19}, machine learning methods can take a lot of time to converge when the number of available gears is big. To reduce the impact of slack times,  in~\cite{1}, Lim et al. developed an algorithm that detects the
communication sections and changes the frequency during these sections
only. This approach might change the frequency of each processor many times per iteration if an iteration
contains more than one communication section. In ~\cite{3}, Rauber and Rünger used an analytical model that can predict the consumed energy  and the execution time for every frequency gear after measuring the consumed energy  and the execution time with the highest frequency gear.   These predictions may be used to choose the optimal gear for each processor executing the parallel program to reduce energy consumption. 
To maintain the performance of the parallel program , they
set the  processor with the biggest load to the highest gear and then compute the scaling  factor values for the rest of the processors. Although this model was built for parallel architectures, it can be adapted  to distributed architectures by taking into account the communications. 
The primary contribution of our paper is presenting a new online scaling factor selection method which has the following characteristics :
\begin{enumerate}
\item It is based on Rauber and Rünger analytical model to predict the energy consumption  of the application with different frequency gears. 
\item It selects the frequency scaling factor for simultaneously optimizing energy reduction and maintaining performance.
\item It is well adapted to distributed architectures because it takes into account the communication time.
\item It is well adapted to distributed applications with imbalanced tasks.
\item it has very small footprint when compared to other
  methods (e.g.,~\cite{19}) and does not require profiling or training as
  in~\cite{38,34}.
\end{enumerate}


\section{Execution and energy of parallel tasks on homogeneous platform} 
\label{sec.exe}
%\JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'', can be deleted if we need space, we can just say we are interested in this paper in homogeneous clusters}
\subsection{Parallel tasks execution on homogeneous platform}
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 message passing.
\begin{figure*}[t]
  \centering
  \subfloat[Sync. imbalanced communications]{\includegraphics[scale=0.67]{fig/commtasks}\label{fig:h1}}
  \subfloat[Sync. imbalanced computations]{\includegraphics[scale=0.67]{fig/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 slack times while tasks wait at the synchronization barrier for other tasks to
finish their tasks (see figure~(\ref{fig:h1})). The imbalanced communications 
happen when nodes have to send/receive different amount of data or they communicate
with different number of nodes. Another source of slack times is the imbalanced computations.  
This happens 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 ones to begin the next task. In both cases the overall execution time 
of the program is the execution time of the slowest task as in EQ~(\ref{eq:T1}).
\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 task $i$ and all the tasks are executed concurrently on different processors.

\subsection{Energy model for homogeneous platform}

Many researchers~\cite{9,3,15,26} divide the power consumed by a processor to 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 activity $\alpha$, load capacitance $C_L$,
the supply voltage $V$ and operational frequency $f$, as shown in EQ~(\ref{eq:pd}).
\begin{equation}
  \label{eq:pd}
  P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
\end{equation}
The static power $P_{static}$ captures the leakage power as follows:
\begin{equation}
  \label{eq:ps}
   P_\textit{static}  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
\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 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
\end{equation}
where $T$ is the execution time of the program, $T_{Comp}$ is the computation time and $T_{Comp} \le T$. $T_{Comp}$ may be equal to $T$ if there is no communications, no slack times and no synchronizations. 

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., $V = \beta \cdot f$ 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 can be
expressed by the scaling factor \emph S which is the ratio between the
maximum and the new frequency as in EQ~(\ref{eq:s}).
\begin{equation}
  \label{eq:s}
 S = \frac{F_\textit{max}}{F_\textit{new}}
\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 governor. The CPU governor is an
interface driver supplied by the operating system's kernel to
lower a core's frequency.  This factor reduces
quadratically the dynamic power which may cause degradation in performance and thus, the increase of the static energy because the execution time is increased~\cite{36}. If the tasks are sorted according to their execution times before scaling in a descending order,   the total energy consumption model for a parallel
homogeneous platform, as  presented by Rauber and Rünger~\cite{3}, can be written as a function of   the scaling factor \emph S,   as in EQ~(\ref{eq:energy}).

\begin{equation}
  \label{eq:energy}
  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_\textit{static} \cdot T_1 \cdot S_1 \cdot N
 \hfill
\end{equation}
where \emph N is the number of parallel nodes,  $T_i  \  and  \  S_i \  for \  i=1,...,N$ are the execution times and scaling factors of the sorted tasks. Therefore,  $T1$ 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~(\ref{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}
\end{equation}
In this paper we depend on
Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this
model is used for any number of concurrent tasks, and (2) we
compare our algorithm with Rauber and Rünger scaling factor selection method which is based on
EQ~(\ref{eq:energy}). The optimal scaling factor is computed by minimizing the derivation for this equation which produces EQ~(\ref{eq:sopt}).

\begin{equation}
  \label{eq:sopt}
  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  parallel synchronous MPI applications depend on
the time of the slowest task as in figure~(\ref{fig:homo}).  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 program. Therefore, the scaling factor $S$ is  linearly proportional to the  execution time. However, in most of MPI applications  the processes exchange data. During these
communications the processors involved remain idle until the communications are
finished. For that reason any change in the frequency has no impact on the time
of communication~\cite{17}. The
communication time for a task is the summation of  periods of time that  begin with an MPI call for
sending or receiving a message till the message is synchronously sent or received. To be able to predict the execution time of MPI program, the communication time and 
the computation time for the slower task must be  measured before scaling. These times are used to predict the execution time for any MPI program as a function of 
the new scaling factor as in EQ~(\ref{eq:tnew}).
\begin{equation}
  \label{eq:tnew}
 \textit  T_\textit{new} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
\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 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 times for both computation and communication. The relation
between the energy and the performance is nonlinear and complex, because the
relation of the energy with scaling factor is nonlinear and with the performance
it is linear see~\cite{17}.  Moreover, they are not measured using the same metric.
For solving this problem, we normalize the energy by calculating 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 }
\end{multline}
By 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}}
\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 in the same direction see figure~(\ref{fig:r2}).
While the main goal is to optimize the energy and performance in the same
time. According to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the
scaling factor \emph 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 overhead to the algorithm for selecting the suitable frequency. In this
paper we present a method to find the optimal scaling factor \emph S for
optimizing both energy and performance simultaneously without adding big
overheads.  Our solution for this problem is to make the optimization process
have the same direction. Therefore, we inverse the equation of normalize
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}}
\end{equation}
\begin{figure*}
  \centering
  \subfloat[Converted relation.]{%
    \includegraphics[width=.4\textwidth]{fig/file}\label{fig:r1}}%
  \qquad%
  \subfloat[Real relation.]{%
    \includegraphics[width=.4\textwidth]{fig/file3}\label{fig:r2}}
  \label{fig:rel}
  \caption{The energy and performance relation}
\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 represents
the minimum energy consumption with minimum execution time (better performance)
at the same time, see figure~(\ref{fig:r1}). Then our objective function has the
following form:
\begin{equation}
  \label{eq:max}
 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}} )
\end{equation}
where F is the number of available frequencies. Then we can select the optimal scaling factor that satisfies
EQ~(\ref{eq:max}).  Our objective function can work with any energy model or
static power values stored in a data file. Moreover, this function works in
optimal way when the energy curve has a convex form over the available frequency scaling
factors as shown in~\cite{15,3,19}. 

\section{Optimal scaling factor for performance and energy}
\label{sec.optim}
 Algorithm~\ref{EPSA} computes the optimal scaling factor according to the objective function described above.
\begin{algorithm}[tp]
  \caption{Scaling factor selection algorithm}
  \label{EPSA}
  \begin{algorithmic}[1]
    \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 - $S_i = S \cdot \frac{T_1}{T_i}= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i} \                   
       for \  i=1,...,N$
      \State - $E_\textit{Norm} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
               \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
               P_\textit{static} \cdot T_1 \cdot S_1 \cdot N  }{
              P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
              P_\textit{static} \cdot T_1 \cdot N }$
      \State -  $P_{NormInv}=T_{old}/T_{new}$ 
      \If{  $(P_{NormInv}-E_{Norm} > Dist$) }
        \State $S_{opt} = S$
        \State $Dist = P_{NormInv} - E_{Norm}$
      \EndIf
    \EndFor
    \State  Return $S_{opt}$
  \end{algorithmic}
\end{algorithm}

The proposed algorithm works online during the execution time of the MPI
program. It selects the optimal scaling factor after gathering the computation and communication times 
from the program after one iteration. Then the program changes the new frequencies of the CPUs according to the computed scaling factors. This algorithm has a small execution time: for a homogeneous cluster composed of nodes having the characteristics presented in table~\ref{table:platform}, it takes  0.00152 $ms$ on average for 4 nodes and 0.00665 $ms$ on average for 32 nodes. The algorithm complexity is O(F$\cdot$N), 
where F is the number of available frequencies and N is the number of computing nodes.  The algorithm is called just 
once during the execution of the program. The DVFS algorithm~(\ref{dvfs}) shows where and when the algorithm is called
in the MPI program.
\begin{table}[htb]
  \caption{Platform file parameters}
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Max & Min & Backbone & Backbone&Link &Link& Sharing  \\
    Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy  \\ \hline
    \np{2.5} & \np{800} & \np[GBps]{2.25} &\np[$\mu$s]{0.5}& \np[GBps]{1} & \np[$\mu$s]{50} &Full  \\
    GHz& MHz&  & & &  &Duplex  \\\hline
  \end{tabular}
  \label{table:platform}
\end{table}

%\begin{minipage}{\textwidth}

\begin{algorithm}[tp]
  \caption{DVFS}
  \label{dvfs}
  \begin{algorithmic}[1]
 \For {$k:=1$ to $Some-Iterations \; $}
  \State -Computations section.
   \State -Communications section.
   \If {$(k=1)$} 
     \State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
     \State -Call algorithm~\ref{EPSA} with these times.
     \State -Compute the new frequency from the \par\hspace{13 pt} returned optimal scaling factor.
     \State -Set the new frequency to the CPU.
   \EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
After obtaining the optimal scaling factor, the program
calculates the new frequency $F_i$ for each task proportionally to its time
value $T_i$. By substitution of  EQ~(\ref{eq:s}) in EQ~(\ref{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}}
\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
having imbalanced workloads. Thus, EQ~(\ref{eq:fi}) adapts the frequency of the CPU to the nodes' workloads to maintain performance.

\section{Experimental results}
\label{sec.expe}
Our experiments are executed on the simulator SimGrid/SMPI
v3.10. We configure the simulator to use a homogeneous cluster with one core per
node. The
detailed characteristics of our platform file are shown in the
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. The simulated network link is 1 GB Ethernet (TCP/IP). 
The backbone of the cluster simulates a high performance switch.

\subsection{Performance prediction verification}

In this section we evaluate the precision of our performance prediction method based on EQ~(\ref{eq:tnew}) by applying it the NAS benchmarks. The NAS programs are executed with the class B option for comparing the
real execution time with the predicted execution time. Each program runs offline
with all available scaling factors on 8 or 9 nodes (depending on the benchmark) to produce real execution
time values. These scaling factors are computed by dividing the maximum
frequency by the new one see EQ~(\ref{eq:s}). 
\begin{figure*}[t]
  \centering
  \includegraphics[width=.328\textwidth]{fig/cg_per}\hfill%
  \includegraphics[width=.328\textwidth]{fig/mg_pre}\hfill%
 % \includegraphics[width=.4\textwidth]{fig/bt_pre}\qquad%
  \includegraphics[width=.328\textwidth]{fig/lu_pre}\hfill%
  \caption{Comparing predicted to real execution time}
  \label{fig:pred}
\end{figure*}
%see Figure~\ref{fig:pred}
In our cluster there are 18 available frequency states for each processor. 
This leads to 18 run states for each program. We use seven MPI programs of the
 NAS parallel benchmarks: CG, MG, EP, FT, BT, LU
and SP. Figure~(\ref{fig:pred}) presents plots of the real execution times and the simulated ones. The maximum normalized error between these two execution times varies between 0.0073 to 0.031 dependent on the executed benchmark. The smallest prediction error was for CG and the worst one was for LU. 
\subsection{The experimental results for the scaling algorithm }
The proposed algorithm was applied to seven MPI programs of the NAS
benchmarks (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and
C). For each instance the benchmarks were executed on a number of processors
proportional to the size of the class.  Each class represents the problem size
ascending from the class A to C. Additionally, depending on some speed up points
for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes
respectively. 
Depending on EQ~(\ref{eq:energy}), we measure the energy consumption for all
the NAS MPI programs while assuming the power dynamic with the highest frequency is equal to \np[W]{20} and
the power static is equal to \np[W]{4} for all experiments. These power values were also
used by Rauber and Rünger in~\cite{3}.  The results showed that the algorithm selected
different scaling factors for each program depending on the communication
features of the program as in the plots~(\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 communication features
for each benchmark.  When there are little or not communications, the inverted
performance 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:factors results}). All discovered frequency scaling factors
optimize both the energy and the performance simultaneously for all  NAS
benchmarks. In table~(\ref{table:factors results}), we record all optimal scaling
factors results for each benchmark running class C. These scaling factors give the maximum
energy saving percent and the minimum performance degradation percent at the
same time from all available scaling factors.
\begin{figure*}[t]
  \centering
  \includegraphics[width=.328\textwidth]{fig/ep}\hfill%
  \includegraphics[width=.328\textwidth]{fig/cg}\hfill%
  \includegraphics[width=.328\textwidth]{fig/sp}
  \includegraphics[width=.328\textwidth]{fig/lu}\hfill%
  \includegraphics[width=.328\textwidth]{fig/bt}\hfill%
  \includegraphics[width=.328\textwidth]{fig/ft}
  \caption{Optimal scaling factors for the predicted energy and performance of NAS benchmarks}
  \label{fig:nas}
\end{figure*}
\begin{table}[htb]
  \caption{The scaling factors results}
  % title of Table
  \centering
  \begin{tabular}{|l|*{4}{r|}}
    \hline
    Program & Optimal & Energy  & Performance&Energy-Perf.\\
    Name & Scaling Factor& Saving \%&Degradation \% &Distance  \\ \hline
    CG & 1.56 &39.23&14.88 &24.35\\ \hline
    MG & 1.47 &34.97&21.70 &13.27 \\ \hline
    EP & 1.04 &22.14&20.73 &1.41\\ \hline
    LU & 1.38 &35.83&22.49 &13.34\\ \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} 
  \label{table:factors results}
  % is used to refer this table in the text
\end{table}
As shown in the table~(\ref{table:factors results}), when the optimal scaling
factor has big value we can gain more energy savings for example as in CG and
FT. The opposite happens when the optimal scaling factor is small value as
example BT and EP. Our algorithm selects big scaling factor value when the
communication and the other slacks times are big and smaller ones in opposite
cases. In EP there are no communications inside the iterations. This make our
algorithm to selects smaller scaling factor values (inducing smaller energy savings).

\subsection{Results comparison}

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 EQ~(\ref{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 possible. We refer to this
scenario as $R_{E-P}$. While we refer to our algorithm as EPSA (Energy to Performance Scaling Algorithm). The comparison
is made in tables \ref{table:compareA}, \ref{table:compareB},
and~\ref{table:compareC}. These
tables show the results of our method and Rauber and Rünger scenarios for all the
NAS benchmarks programs for classes A, B and C.
\begin{table}[p]
  \caption{Comparing results for  the NAS class A}
  % title of Table
  \centering
  \begin{tabular}{|l|l|*{4}{r|}}
    \hline
    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
    Name &Name&Value& Saving \%&Degradation \% &Distance
    \\ \hline
    % \rowcolor[gray]{0.85}
    $EPSA$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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:compareA}
  % is used to refer this table in the text
\end{table}
\begin{table}[p]
  \caption{Comparing results for the NAS class B}
  % title of Table
  \centering
  \begin{tabular}{|l|l|*{4}{r|}}
    \hline
    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
    Name &Name&Value& Saving \%&Degradation \% &Distance
    \\ \hline
    % \rowcolor[gray]{0.85}
    $EPSA$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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:compareB}
  % is used to refer this table in the text
\end{table}

\begin{table}[p]
  \caption{Comparing results for the NAS class C}
  % title of Table
  \centering
  \begin{tabular}{|l|l|*{4}{r|}}
    \hline
    Method&Program&Factor& Energy& Performance &Energy-Perf.\\
    Name &Name&Value& Saving \%&Degradation \% &Distance
    \\ \hline
    % \rowcolor[gray]{0.85}
    $EPSA$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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
    $R_{E-P}$&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:compareC}
% is used to refer this table in the text
\end{table}
As shown in tables~\ref{table:compareA},~\ref{table:compareB} and~\ref{table:compareC}, the ($R_{E-P}$) method outperforms the ($R_{E}$) method in terms of performance and energy reduction. The ($R_{E-P}$) method also gives better energy savings than our method. However, although our scaling factor is not optimal for energy reduction, the results in these tables prove that our algorithm returns the best scaling factor that satisfy our objective method : the largest distance between energy reduction and performance degradation. Negative values in the energy-performance column mean that one of the two objectives (energy or performance) have been degraded more than the other. The positive trade-offs with the highest values lead to maximum energy savings
while keeping the performance degradation as low as possible. Our algorithm always
gives the highest positive energy to performance trade-offs while Rauber and Rünger method
($R_{E-P}$) gives in some time negative trade-offs such as in BT and
EP.
%\begin{figure*}[t]
%  \centering
%  \includegraphics[width=.328\textwidth]{fig/compare_class_A}
%  \includegraphics[width=.328\textwidth]{fig/compare_class_B}
%  \includegraphics[width=.328\textwidth]{fig/compare_class_C}
%  \caption{Comparing our method to Rauber and Rünger methods}
%  \label{fig:compare}
%\end{figure*}
\section{Conclusion}
\label{sec.concl}
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 communication times measured at the first iteration to predict energy consumption and the performance 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 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 methods in terms of energy-performance ratio. 

In the near future, we would like to adapt this scaling factor selection method to heterogeneous platforms where each node has different characteristics. In particular, each CPU has different available frequencies, energy consumption and performance. It would be also interesting to develop a new energy model for asynchronous parallel iterative methods where the number of iterations is not known in advance and depends on the global convergence of the iterative system.


\section*{Acknowledgment}
This work has been supported by the Labex ACTION project (contract ``ANR-11-LABX-01-01'').Computations have been performed on the supercomputer facilities of the
Mésocentre de calcul de Franche-Comté. As a PhD student, M. Ahmed Fanfakh, would like to thank the University of
Babylon (Iraq) for supporting his work.

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%  LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
%  LocalWords:  CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex