Slight changes in equations (typography).

[mpi-energy2.git] / Heter_paper.tex

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\begin{document}

\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}

\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}
  
\end{abstract}

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


\section{Related works}
\label{sec.relwork}





\section{The performance and energy consumption measurements on heterogeneous architecture}
\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{The execution time of message passing distributed iterative applications on a heterogeneous platform}

In this paper, we are interested in reducing the energy consumption of message
passing distributed iterative synchronous applications running over
heterogeneous platforms. We define a heterogeneous platform as a collection of
heterogeneous computing nodes interconnected via a high speed homogeneous
network. Therefore, each node has different characteristics such as computing
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
have the same network bandwidth and latency.


\begin{figure}[t]
  \centering
    \includegraphics[scale=0.6]{fig/commtasks}
  \caption{Parallel tasks on a heterogeneous platform}
  \label{fig:heter}
\end{figure}

 The  overall execution time  of a distributed iterative synchronous application over a heterogeneous platform  consists of the sum of the computation time and the communication time for every iteration on a node. However, due to the heterogeneous computation power of the computing nodes, slack times might occur when fast nodes have to
 wait, during synchronous communications, for  the slower nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
 Therefore,  the overall execution time  of the program is the execution time of the slowest
 task which have the highest computation time and no slack time.
 
Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
of a CPU by scaling down its voltage and frequency.  Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound.  The frequency reduction process can be  expressed by the scaling factor S which is the ratio between  the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
\begin{equation}
  \label{eq:s}
 S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
 The execution time of a compute bound sequential program is linearly proportional to the frequency scaling factor $S$. 
 On the other hand,  message passing distributed applications consist of two parts: computation and communication. The execution time of the computation part is linearly proportional to the frequency scaling factor $S$ but  the communication time is not affected by the scaling factor because  the processors involved remain idle during the  communications~\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.

Since in a heterogeneous platform, each node has different characteristics,
especially different frequency gears, when applying DVFS operations on these
nodes, they may get different scaling factors represented by a scaling vector:
$(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
be able to predict the execution time of message passing synchronous iterative
applications running over a heterogeneous platform, for different vectors of
scaling factors, the communication time and the computation time for all the
tasks must be measured during the first iteration before applying any DVFS
operation. Then the execution time for one iteration of the application with any
vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
\begin{equation}
  \label{eq:perf}
 \textit  T_\textit{new} = 
 \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  TcmOld_{j}
\end{equation}
where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $TcmOld_j$ is the communication time of the slowest processor $j$.
  The model computes the maximum computation time 
 with scaling factor from each node  added to the communication time of the slowest node, it means  only the
 communication time without any slack time. 

This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. The execution time prediction model is used in our method for optimizing both energy consumption and performance of iterative methods, which is presented in the following sections.


\subsection{Energy model for heterogeneous platform}

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 turned on, the latter is only consumed during
computation times.  The dynamic power $P_{d}$ 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{d} = \alpha \cdot C_L \cdot V^2 \cdot F
\end{equation}
The static power $P_{s}$ captures the leakage power as follows:
\begin{equation}
  \label{eq:ps}
   P_\textit{s}  = 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{d} \cdot T_{cp} + P_\textit{s} \cdot T
\end{equation}
where $T$ is the execution time of the program, $T_{cp}$ is the computation
time and $T_{cp} \leq T$.  $T_{cp}$ may be equal to $T$ if there is no
communication and no slack time.

The main objective of DVFS operation is to
reduce the overall energy consumption~\cite{37}.  The operational frequency $F$
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 $S$ which is the
ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
The CPU governors are power schemes supplied by the operating
system's kernel to lower a core's frequency. we can calculate the new frequency 
$F_{new}$ from EQ(\ref{eq:s}) as follow:
\begin{equation}
  \label{eq:fnew}
   F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic 
power consumption:
\begin{multline}
  \label{eq:pdnew}
   {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
   {} = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{dOld} \cdot S^{-3}
\end{multline}
where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the new frequency and the maximum frequency respectively.

According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when 
reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$.  The new dynamic energy is the  dynamic power multiplied by the new time of computation and is given by the following equation:
\begin{equation}
  \label{eq:Edyn}
   E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (T_{cp} \cdot S)= S^{-2}\cdot P_{dOld} \cdot  T_{cp} 
\end{equation}
The static power is related to the power leakage of the CPU and is consumed during computation and even when idle. As in~\cite{3,46}, we assume that the static power of a processor is constant during idle and computation periods, and for all its available frequencies. 
The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}), 
the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related  
to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy 
of a processor after scaling its frequency is computed as follows: 
\begin{equation}
  \label{eq:Estatic}
 E_\textit{s} = P_\textit{s} \cdot (T_{cp} \cdot S  + T_{cm})
\end{equation}

In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $P_{di}$ and $P_{si}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $T_{cpi}$ might be different and different frequency  scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $T_{cmi}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the  frequency scaling factor and the dynamic power of each node as in EQ(\ref{eq:Edyn}),  the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing  distributed application  executed over a heterogeneous platform is the summation of all dynamic and static energies for each  processor.  It is computed as follows:
\begin{multline}
  \label{eq:energy}
 E = \sum_{i=1}^{N} {(S_i^{-2} \cdot P_{di} \cdot  T_{cpi})} + {} \\
 \sum_{i=1}^{N} (P_{si} \cdot (\max_{i=1,2,\dots,N} (T_{cpi} \cdot S_{i}) +
 \min_{i=1,2,\dots,N} {T_{cmi}))}
 \end{multline}
 
Reducing the the frequencies of the processors according to the vector of
scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
application and thus, increase the static energy because the execution time is
increased~\cite{36}.

\section{Optimization of both energy consumption and performance}
\label{sec.compet}

Applying DVFS to lower level not surly reducing the energy consumption to
minimum level. Also, a big scaling for the frequency produces high performance
degradation percent. Moreover, by considering the drastically increase in
execution time of parallel program, the static energy is related to this time
and it also increased by the same ratio. Thus, the opportunity for gaining more
energy reduction is restricted. For that choosing frequency scaling factors is
very important process to taking into account both energy and performance. In
our previous work~\cite{45}, we are proposed a method that selects the optimal
frequency scaling factor for an homogeneous cluster, depending on the trade-off
relation between the energy and performance. In this work we have an
heterogeneous cluster, at each node there is different scaling factors, so our
goal is to selects the optimal set of frequency scaling factors,
$Sopt_1,Sopt_2,\dots,Sopt_N$, that gives the best trade-off between energy
consumption and performance. The relation between the energy and the execution
time is complex and nonlinear, Thus, unlike the relation between the performance
and the scaling factor, the relation of the energy with the frequency scaling
factors 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
execution time by calculating the ratio between the new execution time (the
scaled execution time) and the old one as follow:
\begin{multline}
  \label{eq:pnorm}
  P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
       {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}}
           {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
\end{multline}


By the same way, we are 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{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
 \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
 \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
\end{multline} 
Where $T_{New}$ and $T_{Old}$ is computed as in EQ(\ref{eq:pnorm}). 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
normalized execution time curves are not at the same direction.  While the main
goal is to optimize the energy and execution time in the same time.  According
to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the set of frequency
scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
time simultaneously.  But the main objective is to produce maximum energy
reduction with minimum execution time 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 are present a method to find the optimal
set of frequency scaling factors to optimize both energy and execution time
simultaneously without adding a big overhead.  Our solution for this problem is
to make the optimization process for energy and execution time follow the same
direction.  Therefore, we inverse the equation of the normalized execution time,
the normalized performance, as follows:
\begin{multline}
  \label{eq:pnorm_inv}
  P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
          = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
            { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}} 
\end{multline}


\begin{figure}
  \centering
  \subfloat[Homogeneous platform]{%
    \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
  \qquad%
  \subfloat[Heterogeneous platform]{%
    \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
  \label{fig:rel}
  \caption{The energy and performance relation}
\end{figure}

Then, we can model our objective function as finding the maximum distance
between the energy curve EQ~(\ref{eq:enorm}) and the  performance
curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
represents the minimum energy consumption with minimum execution time (better
performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}) .  Then our objective
function has the following form:
\begin{equation}
  \label{eq:max}
  Max Dist = 
  \max_{i=1,\dots F, j=1,\dots,N}
      (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
       \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
\end{equation}
where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).  Our objective function can
work with any energy model or energy 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{The heterogeneous scaling algorithm }
\label{sec.optim}

In this section we proposed an heterogeneous scaling algorithm,
(figure~\ref{HSA}), that selects the optimal set of scaling factors from each
node.  The algorithm is numerates the suitable range of available scaling
factors for each node in the heterogeneous cluster, returns a set of optimal
frequency scaling factors for each node. Using heterogeneous cluster is produces
different workloads for each node. Therefore, the fastest nodes waiting at the
barrier for the slowest nodes to finish there work as in figure
(\ref{fig:heter}). Our algorithm takes into account these imbalanced workloads
when is starts to search for selecting the best scaling factors. So, the
algorithm is selecting the initial frequencies values for each node proportional
to the times of computations that gathered from the first iteration. As an
example in figure (\ref{fig:st_freq}), the algorithm don't test the first
frequencies of the fastest nodes until it converge their frequencies to the
frequency of the slowest node. If the algorithm is starts test changing the
frequency of the slowest nodes from beginning, we are loosing performance and
then not selecting the best trade-off (the distance). This case will be similar
to the homogeneous cluster when all nodes scales their frequencies together from
the beginning. In this case there is a small distance between energy and
performance curves, for example see the figure(\ref{fig:r1}).  Then the
algorithm searching for optimal frequency scaling factor from the selected
frequencies until the last available ones.
\begin{figure}[t]
  \centering
    \includegraphics[scale=0.5]{fig/start_freq}
  \caption{Selecting the initial frequencies}
  \label{fig:st_freq}
\end{figure}


To compute the initial frequencies, the algorithm firstly needs to compute the computation scaling factors $Scp_i$ for each node. Each one of these factors represent a ratio between the computation time of the slowest node and the computation time of the node $i$ as follow:
\begin{equation}
  \label{eq:Scp}
 Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
\end{equation}
Depending on the initial computation scaling factors EQ(\ref{eq:Scp}), the algorithm computes the initial frequencies for all nodes as a ratio between the 
maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as follow:
\begin{equation}
  \label{eq:Fint}
 F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
\end{equation}
\begin{figure}[tp]
  \begin{algorithmic}[1]
    % \footnotesize
    \Require ~
    \begin{description}
    \item[$Tcp_i$] array of all computation times for all nodes during one iteration and with highest frequency.
    \item[$Tcm_i$] array of all communication times for all nodes during one iteration and with highest frequency.
    \item[$Fmax_i$] array of the maximum frequencies for all nodes.
    \item[$Pd_i$] array of the dynamic powers for all nodes.
    \item[$Ps_i$] array of the static powers for all nodes.
    \item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
    \end{description}
    \Ensure $Sopt_1, \dots, Sopt_N$ is a set of optimal scaling factors

    \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
    \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
    \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
    \If{(not the first frequency)}
          \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
    \EndIf 
    \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
    \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
    \State $Dist \gets 0$
    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
    \While {(all nodes not reach their  minimum  frequency)}
        \If{(not the last freq. \textbf{and} not the slowest node)}
        \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,\dots,N.$
        \EndIf
       \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + min_\textit{~i=1,\dots,N}(Tcm_i) $
       \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} + $  \hspace*{43 mm} 
               $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
       \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
       \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
      \If{$(\Pnorm - \Enorm > \Dist)$}
        \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $
        \State $\Dist \gets \Pnorm - \Enorm$
      \EndIf
    \EndWhile
    \State  Return $Sopt_1,Sopt_2,\dots,Sopt_N$
  \end{algorithmic}
  \caption{Heterogeneous scaling algorithm}
  \label{HSA}
\end{figure}
When the initial frequencies are computed the algorithm numerates all available
scaling factors starting from these frequencies until all nodes reach their
minimum frequencies. At each iteration the algorithm remains the frequency of
the slowest node without change and scaling the frequency of the other
nodes. This is gives better performance and energy trade-off.  The proposed
algorithm works online during the execution time of the MPI program.  Its
returns a set of optimal frequency scaling factors $Sopt_i$ depending on the
objective function EQ(\ref{eq:max}). The program changes the new frequencies of
the CPUs according to the computed scaling factors.  This algorithm has a small
execution time: for an heterogeneous cluster composed of four different types of
nodes having the characteristics presented in table~(\ref{table:platform}), it
takes \np[ms]{0.04} on average for 4 nodes and \np[ms]{0.1} on average for 128
nodes.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the
number of iterations and $N$ is the number of computing nodes. The algorithm
needs on average from 12 to 20 iterations for all the NAS benchmark on class C
to selects the best set of frequency scaling factors. Its called just once
during the execution of the program.  The DVFS figure~(\ref{dvfs}) shows where
and when the algorithm is called in the MPI program.
\begin{figure}[tp]
  \begin{algorithmic}[1]
    % \footnotesize
    \For {$k=1$ to \textit{some iterations}}
      \State Computations section.
      \State Communications section.
      \If {$(k=1)$}
        \State Gather all times of computation and\newline\hspace*{3em}%
               communication from each node.
        \State Call algorithm from Figure~\ref{HSA} with these times.
        \State Compute the new frequencies from the\newline\hspace*{3em}%
               returned optimal scaling factors.
        \State Set the new frequencies to nodes.
      \EndIf
    \EndFor
  \end{algorithmic}
  \caption{DVFS algorithm}
  \label{dvfs}
\end{figure}

\section{Experimental results}
\label{sec.expe}

The experiments of this work are executed on the simulator SimGrid/SMPI
v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We configure the
simulator to use a heterogeneous cluster with one core per node. The proposed
heterogeneous cluster has four different types of nodes. Each node in cluster
has different characteristics such as the maximum frequency speed, the number of
available frequencies and dynamic and static powers values, see table
(\ref{table:platform}). These different types of processing nodes simulate some
real Intel processors. The maximum number of nodes that supported by the cluster
is 144 nodes according to characteristics of some MPI programs of the NAS
benchmarks that used. We are use the same number from each type of nodes when
running the MPI programs, for example if we execute the program on 8 node, there
are 2 nodes from each type participating in the computing. The dynamic and
static power values is different from one type to other. Each node has a dynamic
and static power values proportional to their performance/GFlops, for more
details see the Intel data sheets in \cite{47}.  Each node has a percentage of
80\% for dynamic power and 20\% for static power from the hole power
consumption, the same assumption is made in \cite{45,3}. These nodes are
connected via an Ethernet network with 1 Gbit/s bandwidth.
\begin{table}[htb]
  \caption{Heterogeneous nodes characteristics}
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\
    type     & to          & Freq. GHz  & Freq. GHz    & Freq GHz       & power        & power \\
    \hline
    1       & core-i3       & 2.5         & 1.2          & 0.1           & 20~w         &4~w    \\
            &  2100T        &             &              &               &              &  \\
    \hline
    2       & Xeon          & 2.66        & 1.6          & 0.133         & 25~w         &5~w    \\
            & 7542          &             &              &               &              &  \\
    \hline
    3       & core-i5       & 2.9         & 1.2          & 0.1           & 30~w         &6~w    \\
            & 3470s         &             &              &               &              &  \\
    \hline
    4       & core-i7       & 3.4         & 1.6          & 0.133         & 35~w         &7~w    \\
            & 2600s         &             &              &               &              &  \\
    \hline
  \end{tabular}
  \label{table:platform}
\end{table}

 
%\subsection{Performance prediction verification}


\subsection{The experimental results of the scaling algorithm}
\label{sec.res}

The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.
\cite{44},  which were run with three classes (A, B and C).
In this experiments we are focus on running of the class C, the biggest class compared to A and B, on different number of 
nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Depending on the proposed energy consumption model EQ(\ref{eq:energy}),
 we are measure the energy consumption for all NAS MPI programs. The dynamic and static power values used under the same assumption used by \cite{45,3}. We used a percentage of 80\% for dynamic power from all power consumption of the CPU and 20\% for static power. The heterogeneous nodes in table (\ref{table:platform}) have different  simulated performance, ranked from the node of type 1 with smaller performance/GFlops to the highest performance/GFlops for node 4. Therefore, the power values used proportionally increased from node 1 to node 4 that with highest performance. Then we used an assumption that the power consumption increases linearly with the performance/GFlops of the processor see \cite{48}.   
 
\begin{table}[htb]
  \caption{Running NAS benchmarks on 4 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &  64.64        & 3560.39        &34.16        &6.72               &27.44       \\
    \hline 
    MG         & 18.89         & 1074.87            &35.37            &4.34                   &31.03       \\
   \hline
    EP         &79.73          &5521.04         &26.83            &3.04               &23.79      \\
   \hline
    LU         &308.65         &21126.00           &34.00             &6.16                   &27.84      \\
    \hline
    BT         &360.12         &21505.55           &35.36         &8.49               &26.87     \\
   \hline
    SP         &234.24         &13572.16           &35.22         &5.70               &29.52    \\
   \hline
    FT         &81.58          &4151.48        &35.58         &0.99                   &34.59    \\
\hline 
  \end{tabular}
  \label{table:res_4n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 8 and 9 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &36.11              &3263.49             &31.25        &7.12                    &24.13     \\
    \hline 
    MG         &8.99           &953.39          &33.78        &6.41                    &27.37     \\
   \hline
    EP         &40.39          &5652.81         &27.04        &0.49                    &26.55     \\
   \hline
    LU         &218.79             &36149.77        &28.23        &0.01                    &28.22      \\
    \hline
    BT         &166.89         &23207.42            &32.32            &7.89                    &24.43      \\
   \hline
    SP         &104.73         &18414.62            &24.73            &2.78                    &21.95      \\
   \hline
    FT         &51.10          &4913.26         &31.02        &2.54                    &28.48      \\
\hline 
  \end{tabular}
  \label{table:res_8n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 16 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &31.74          &4373.90         &26.29        &9.57                    &16.72          \\
    \hline 
    MG         &5.71           &1076.19         &32.49        &6.05                    &26.44         \\
   \hline
    EP         &20.11          &5638.49         &26.85        &0.56                    &26.29         \\
   \hline
    LU         &144.13         &42529.06            &28.80            &6.56                    &22.24         \\
    \hline
    BT         &97.29          &22813.86            &34.95        &5.80                &29.15         \\
   \hline
    SP         &66.49          &20821.67            &22.49            &3.82                    &18.67         \\
   \hline
    FT             &37.01          &5505.60             &31.59        &6.48                    &25.11         \\
\hline 
  \end{tabular}
  \label{table:res_16n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 32 and 36 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &32.35          &6704.21         &16.15        &5.30                    &10.85           \\
    \hline 
    MG         &4.30           &1355.58         &28.93        &8.85                    &20.08          \\
   \hline
    EP         &9.96           &5519.68         &26.98        &0.02                    &26.96          \\
   \hline
    LU         &99.93          &67463.43            &23.60            &2.45                    &21.15          \\
    \hline
    BT         &48.61          &23796.97            &34.62            &5.83                    &28.79          \\
   \hline
    SP         &46.01          &27007.43            &22.72            &3.45                    &19.27           \\
   \hline
    FT             &28.06          &7142.69             &23.09        &2.90                    &20.19           \\
\hline 
  \end{tabular}
  \label{table:res_32n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 64 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &46.65          &17521.83            &8.13             &1.68                    &6.45           \\
    \hline 
    MG         &3.27           &1534.70         &29.27        &14.35               &14.92          \\
   \hline
    EP         &5.05           &5471.1084           &27.12            &3.11                &24.01         \\
   \hline
    LU         &73.92          &101339.16           &21.96            &3.67                    &18.29         \\
    \hline
    BT         &39.99          &27166.71            &32.02            &12.28               &19.74         \\
   \hline
    SP         &52.00          &49099.28            &24.84            &0.03                    &24.81         \\
   \hline
    FT         &25.97          &10416.82        &20.15        &4.87                    &15.28         \\
\hline 
  \end{tabular}
  \label{table:res_64n}
\end{table}


\begin{table}[htb]
  \caption{Running NAS benchmarks on 128 and 144 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
    \hline
    CG         &56.92          &41163.36        &4.00         &1.10                    &2.90          \\
    \hline 
    MG         &3.55           &2843.33         &18.77        &10.38               &8.39          \\
   \hline
    EP         &2.67           &5669.66         &27.09        &0.03                    &27.06         \\
   \hline
    LU         &51.23          &144471.90       &16.67        &2.36                    &14.31         \\
    \hline
    BT         &37.96          &44243.82            &23.18            &1.28                    &21.90         \\
   \hline
    SP         &64.53          &115409.71           &26.72            &0.05                    &26.67         \\
   \hline
    FT         &25.51          &18808.72            &12.85            &2.84                    &10.01         \\
\hline 
  \end{tabular}
  \label{table:res_128n}
\end{table}

The results of applying the proposed scaling algorithm to NAS benchmarks is demonstrated in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, 
\ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}). These tables show the results for running the NAS benchmarks on different number of nodes. In general the energy saving percent is decreased  when the number of the computing nodes is increased. While the distance is decreased by the same direction. This because when we are run the MPI programs on a big number of nodes the communications is biggest than the computations, so the static energy is increased linearly to these times. The tables also show that performance degradation percent still approximately the same or decreased a little when the number of computing nodes is increased. This  gives good a prove that the proposed algorithm keeping as mush as possible the performance degradation. 

\begin{figure}
  \centering
  \subfloat[Balanced nodes type scenario]{%
    \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
  \quad%
  \subfloat[Imbalanced nodes type scenario]{%
    \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
  \label{fig:avg}
  \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
\end{figure}

In the NAS benchmarks there are some programs executed on different number of
nodes. The benchmarks CG, MG, LU and FT executed on 2 to a power of (1, 2, 4, 8,
\dots{}) of nodes. The other benchmarks such as BT and SP executed on 2 to a
power of (1, 2, 4, 9, \dots{}) of nodes. We are take the average of energy
saving, performance degradation and distances for all results of NAS
benchmarks. The average of these three objectives are plotted to the number of
nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}).  In CG, MG, LU, and
FT benchmarks the average of energy saving is decreased when the number of nodes
is increased due to the increasing in the communication times as mentioned
before. Thus, the average of distances (our objective function) is decreased
linearly with energy saving while keeping the average of performance degradation
the same. In BT and SP benchmarks, the average of energy saving is not decreased
significantly compare to other benchmarks when the number of nodes is
increased. Nevertheless, the average of performance degradation approximately
still the same ratio. This difference is depends on the characteristics of the
benchmarks such as the computation to communication ratio that has.

\subsection{The results for different powers scenarios}

The results of the previous section are obtained using a percentage of 80\% for
dynamic power and 20\% for static power of total power consumption. In this
section we are change these ratio by using two others scenarios. Because is
interested to measure the ability of the proposed algorithm to changes it
behavior when these power ratios are changed. In fact, we are use two different
scenarios for dynamic and static power ratios in addition to the previous
scenario in section (\ref{sec.res}). Therefore, we have three different
scenarios for three different dynamic and static power ratios refer to as:
70\%-20\%, 80\%-20\% and 90\%-10\% scenario. The results of these scenarios
running NAS benchmarks class C on 8 or 9 nodes are place in the tables
(\ref{table:res_s1} and \ref{table:res_s2}).

 \begin{table}[htb]
  \caption{The results of 70\%-30\% powers scenario}
  % title of Table
  \centering
  \begin{tabular}{|*{6}{l|}}
    \hline
    Method     & Energy          & Energy      & Performance        & Distance     \\
    name       & consumption/J   & saving\%    & degradation\%      &              \\
    \hline
    CG         &4144.21          &22.42        &7.72                &14.70         \\
    \hline 
    MG         &1133.23          &24.50        &5.34                &19.16          \\
   \hline
    EP         &6170.30         &16.19         &0.02                &16.17          \\
   \hline
    LU         &39477.28        &20.43         &0.07                &20.36          \\
    \hline
    BT         &26169.55            &25.34             &6.62                &18.71          \\
   \hline
    SP         &19620.09            &19.32             &3.66                &15.66          \\
   \hline
    FT         &6094.07         &23.17         &0.36                &22.81          \\
\hline 
  \end{tabular}
  \label{table:res_s1}
\end{table}



\begin{table}[htb]
  \caption{The results of 90\%-10\% powers scenario}
  % title of Table
  \centering
  \begin{tabular}{|*{6}{l|}}
    \hline
    Method     & Energy          & Energy      & Performance        & Distance     \\
    name       & consumption/J   & saving\%    & degradation\%      &              \\
    \hline
    CG         &2812.38          &36.36        &6.80                &29.56         \\
    \hline 
    MG         &825.427          &38.35        &6.41                &31.94         \\
   \hline
    EP         &5281.62          &35.02        &2.68                &32.34         \\
   \hline
    LU         &31611.28             &39.15        &3.51                    &35.64        \\
    \hline
    BT         &21296.46             &36.70            &6.60                &30.10       \\
   \hline
    SP         &15183.42             &35.19            &11.76               &23.43        \\
   \hline
    FT         &3856.54          &40.80        &5.67                &35.13        \\
\hline 
  \end{tabular}
  \label{table:res_s2}
\end{table}


\begin{figure}
  \centering
  \subfloat[Comparison the average of the results on 8 nodes]{%
    \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
  \quad%
  \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
    \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
  \label{fig:comp}
  \caption{The comparison of the three power scenarios}
\end{figure}

To compare the results of these three powers scenarios, we are take the average of the performance degradation, the energy saving and the distances for all NAS benchmarks running on 8 or 9 nodes of class C, as in figure (\ref{fig:sen_comp}). Thus, according to the average of the results, the energy saving ratio is increased when using the a higher percentage for dynamic power (e.g. 90\%-10\% scenario). While the average of energy saving is decreased in 70\%-30\% scenario. 
Because the static energy consumption is increase. Moreover, the average of distances is more related to energy saving changes. The average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). The raison behind these relations, that the proposed algorithm optimize both energy consumption and performance in the same time. Therefore, when using a higher ratio for dynamic power the algorithm selecting bigger frequency scaling factors values, more energy saving versus more performance degradation, for example see the figure (\ref{fig:scales_comp}). The inverse happen when using a higher ratio for  static  power, the algorithm proportionally  selects a smaller scaling values, less energy saving versus less performance degradation. This is because the  
algorithm also keeps as much as possible the static energy consumption that is always related to execution time. 

\subsection{The verifications of the proposed method}
\label{sec.verif}
The precision of the proposed algorithm mainly depends on the execution time prediction model and the energy model. The energy model is significantly depends on the execution time model EQ(\ref{eq:perf}), that the energy static related linearly. So, we are compare the predicted execution time with the real execution time (Simgrid time)  values that gathered  offline for the NAS benchmarks class B executed on 8 or 9 nodes. The execution time model can predicts 
the real execution time by maximum normalized error 0.03 of all NAS benchmarks. The second verification that we are make is for the scaling algorithm to prove its ability to selects the best set of frequency scaling factors. Therefore, we are expand the algorithm to test at each iteration the frequency scaling factor of the slowest node with the all scaling factors available of the other nodes. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the two algorithms is identical. While the proposed algorithm is runs faster, 10 times faster on average than the expanded algorithm.

\section{Conclusion}
\label{sec.concl}


\section*{Acknowledgment}


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% LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
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