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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 performance of parallel tasks on heterogeneous cluster}

The  heterogeneous cluster is a collection of non identical computing nodes. Each node in 
a cluster is connected via a high speed network. The communication capabilities between nodes  
are  identical or different. In this work we are interested in identical communications. While each 
node has different processing capabilities such as CPU speeds and memory. Tasks executed
on this model can be either synchronous or asynchronous. In this paper we are consider execution of 
the synchronous tasks on distributed heterogeneous platform. These tasks can exchange
the data via synchronous message passing. 

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

 Therefore, the  execution time of a task consists of the computation time and
 the communication time. Due to  heterogeneous computations can lead to slack times while the tasks 
 wait at the synchronization barrier for other tasks to finish  their jobs (see Figure~(\ref{fig:heter})). 
 In this case the fastest tasks  have to wait at the synchronization barrier for the slowest ones  to begin
 the next task. Therefore,  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 the task $i$ and all the tasks are executed concurrently on different processors. DVFS is a process that is allowed in modern processors to reduce the dynamic
power by scaling down the voltage and frequency. Then any DVFS operation used to reduce energy of the processor has direct affect on the execution time of the MPI program. 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}).
\begin{equation}
  \label{eq:s}
 S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
 The execution time of a parallel program is linearly proportional to the frequency scaling factor $S$. 
 However, in most  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.
 Each node has different DVFS features such as frequency values and the number of available frequencies 
 (Pstates) for each node. By contrast there are different frequency scaling factors for each node $S_1, S_2,..., S_N$. To be able to predict the execution time of MPI program, the  communication time and the computation time for the slowest
 task must be measured before scaling. These times are used to  predict the execution time for any MPI program running on heterogeneous cluster as a function
 of the new scaling factors as in EQ (\ref{eq:perf}). The model is computes the maximum production of  computation time 
 with scaling factor from each node  added to the minimum communication time of the slowest node, it means  only the
 communication time without slack times, because in MPI the slack times is measured with communication times. 
\begin{multline}
  \label{eq:perf}
 \textit  T_\textit{new} = \\ 
 {} \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  
 \min_{i=1,2,\dots,N} Tcm Old_{i}
\end{multline}
This prediction modal is developed from our model for predicting the execution time of parallel task on homogeneous architecture~\cite{45}. The execution time predicting model is useful to used in our method for optimizing both energy and performance of iterative methods as in the coming 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 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, no slack time and no synchronization.

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 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. 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} .  F_\textit{max}
\end{equation}
By substituting this equation in EQ(\ref{eq:pd}) results the following equation for dynamic 
power consumption:
\begin{multline}
  \label{eq:pdnew}
   {P}_\textit{d} = \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_{d} \cdot S^{-3}
\end{multline}
According to EQ(\ref{eq:pdnew}) the dynamic power is reduce by a factor of $S^{-3}$ when 
reducing the frequency by a factor of $S$~\cite{3}. The dynamic energy is the energy consumed by a CPU when its in the computation mode. 
So, the dynamic energy is the  dynamic power multiply by the time of computations. While the time of computation is decreased by a factor of $S$. Therefore the 
the dynamic energy  is decreased by a factor of $S^{-2}$ as follow:
\begin{equation}
  \label{eq:Edyn}
   E_\textit{d} = P_{d} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{d} \cdot  Tcp 
\end{equation}
The static power is related to leakage power consumption, its mean the CPU continue consumes energy 
whereas in idle state. Therefore, we are make an assumption that the static power is constant as in~\cite{3,46}.
The static energy is the static power multiply by the execution time of the program. Moreover, the CPU consumes static
energy in all times of the program such as computation, communication and slacks times. 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 related  
to frequency scaling factor linearly, while this scaling factor not affecting on the time of communication~\cite{17}, then  the static energy 
of individual processor is as follow: 

\begin{equation}
  \label{eq:Estatic}
 E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)
\end{equation}
In heterogeneous architecture there is a  number of different processors $P_1, P_2,...,P_N$, where $N$ is the number of nodes . Moreover, each processor perhaps has different frequency  scaling factor, so there are a set of frequency scaling factors for such platform  $S_1,S_2,...,S_N$. According  to these  different 
scaling factors producing different computation time, $Tcp_1,Tcp_2,...,Tcp_N$, because these times linearly related to these scaling factors. In MPI program the communication times is measured with slacks times. The slack times also has linear relation with the scaling factors. So, there are different mesured communication times $Tcm_1,Tcm_2,...,Tcm_N$ even if its identical communications, e.g. see figure(\ref{fig:heter}). The energy modal of an  heterogeneous architecture represents the summation of all dynamic and static energies from each  processors, each processor has its dynamic and static powers, for example, in the hole architecture their are: $Pd_1,Pd_2,...,Pd_N$ and $Ps_1,Ps_2,...,Ps_N$.  The dynamic energy is computes as in EQ(\ref{eq:Edyn}) with regarding to the frequency scaling factor and the dynamic power of each node. While the static energy is computes using EQ(\ref{eq:perf}) multiplied  by the static power of each processor. So, the energy modal of an heterogeneous platform has the following form:
\begin{multline}
  \label{eq:energy}
 E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +\\ 
 {}\sum_{i=1}^{N} {(Ps_i \cdot (\max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +} 
 {}\min_{i=1,2,\dots,N} {Tcm_{i}))} 
 \end{multline}
 
These set of frequency scaling factors $S_i$ reduce 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}.

\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, $Sopt_1,Sopt_2,...,Sopt_N$, 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 $S_1,S_2,...,S_N$ 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{multline}
  \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{multline}
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 tradeoff (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 $F_i \gets F_i+Fdiff_i,~i=1,...,N.$
    \EndIf 
    \State $T_\textit{Old} \gets max_{~i=1,...,N } (Tcp_i+Tcm_i)$
    \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,...,N. $
    \While {(all nodes not reach their  minimum  frequency)}
        \If{(not the last freq. \textbf{and} not the slowest node)}
        \State $F_i \gets F_i - Fdiff_i,~i=1,...,N.$ 
        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,...,N.$ 
        \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,...,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 tradeoff. 
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. 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, ...) of nodes. The other benchmarks such as BT and SP executed on 2 to a power of (1, 2, 4, 9, ...) 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   behaviour 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:avg}
  \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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