Subsection C have been corrected. There are still a lot of errors!

[mpi-energy2.git] / mpi-energy2-extension / Heter_paper.tex

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\begin{document}

\title{Energy Consumption Reduction with DVFS for \\
  Message Passing Iterative Applications on \\
  Heterogeneous Architectures}

\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}

\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 grid platforms. A heterogeneous grid platform could be defined as a collection of
heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, 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 grid 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 clusters, slack times may 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 has 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 may 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 (\ref{eq:s}).
\begin{equation}
  \label{eq:s}
  S = \frac{\Fmax}{\Fnew}
\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{Freeh_Exploring.the.Energy.Time.Tradeoff}.  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 until the message is
synchronously sent or received.

Since in a heterogeneous grid each cluster has different characteristics,
especially different frequency gears, when applying DVFS operations on the nodes 
of these clusters, they may get different scaling factors represented by a scaling vector:
$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
be able to predict the execution time of message passing synchronous iterative
applications running over a heterogeneous grid, 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 (\ref{eq:perf}).
\begin{equation}
  \label{eq:perf}
  \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij}) 
  +\mathop{\min_{j=1,\dots,M}}  (\Tcm[hj])
\end{equation}

where $N$ is the number of  clusters in the grid, $M$ is the number of  nodes in
each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
first  iteration. The model computes the maximum computation time with scaling factor 
from each node added to the communication time of the slowest node in the slowest cluster $h$.
It means only the communication time without any slack time is taken into account.  
Therefore, the execution time of the iterative application is equal to
the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
number of iterations of that application.

This prediction model is developed from the model to predict the execution time
of message passing distributed applications for homogeneous and heterogeneous clusters
~\cite{Our_first_paper,pdsec2015}.  The execution time prediction model is
used in the method to optimize both the energy consumption and the performance
of iterative methods, which is presented in the following sections.


\subsection{Energy model for heterogeneous grid platform}

Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
  Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
  Rizvandi_Some.Observations.on.Optimal.Frequency} 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 $\Pd$ is related to
the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
and operational frequency $F$, as shown in (\ref{eq:pd}).
\begin{equation}
  \label{eq:pd}
  \Pd = \alpha \cdot \CL \cdot V^2 \cdot F
\end{equation}
The static power $\Ps$ captures the leakage power as follows:
\begin{equation}
  \label{eq:ps}
   \Ps  = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak
\end{equation}
where V is the supply voltage, $\Ntrans$ is the number of transistors,
$\Kdesign$ is a design dependent parameter and $\Ileak$ is a
technology dependent parameter.  The energy consumed by an individual processor
to execute a given program can be computed as:
\begin{equation}
  \label{eq:eind}
  \Eind =  \Pd \cdot \Tcp + \Ps \cdot T
\end{equation}
where $T$ is the execution time of the program, $\Tcp$ is the computation
time and $\Tcp \le T$.  $\Tcp$ 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{Le_DVFS.Laws.of.Diminishing.Returns}.  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{Rauber_Analytical.Modeling.for.Energy}.  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 (\ref{eq:s}).  The CPU governors are
power schemes supplied by the operating system's kernel to lower a core's
frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
follows:
\begin{equation}
  \label{eq:fnew}
   \Fnew = S^{-1} \cdot \Fmax
\end{equation}
Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
equation for dynamic power consumption:
\begin{multline}
  \label{eq:pdnew}
   \PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 \\
   {} = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}
\end{multline}
where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
new frequency and the maximum frequency respectively.

According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
$S^{-3}$ when reducing the frequency by a factor of
$S$~\cite{Rauber_Analytical.Modeling.for.Energy}. 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}
   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp
\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{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
the static power of a processor is considered as 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 (\ref{eq:perf}), the execution time of the
program is the sum 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}
  \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
\end{equation}

In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
different dynamic and static powers from the nodes of the other clusters, 
noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Therefore, even if the distributed 
message passing iterative application is load balanced, the computation time of each CPU $j$ 
in cluster $i$ noted $\Tcp[ij]$ may be different and different frequency scaling factors may be
computed in order to decrease the overall energy consumption of the application
and reduce slack times.  The communication time of a processor $j$ in cluster $i$ is noted as
$\Tcm[ij]$ and could contain slack times when 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
(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
of one iteration multiplied by the static power of each processor.  The overall
energy consumption of a message passing distributed application executed over a
heterogeneous grid platform during one iteration is the summation of all dynamic and
static energies for $M$ processors in $N$ clusters.  It is computed as follows:
\begin{multline}
  \label{eq:energy}
 E = \sum_{i=1}^{N} \sum_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
 \sum_{i=1}^{N} \sum_{j=1}^{M} (\Ps[ij] \cdot {} \\
  (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij}) 
  +\mathop{\min_{j=1,\dots M}} (\Tcm[hj]) ))
\end{multline}

Reducing the frequencies of the processors according to the vector of scaling
factors $(S_{11}, S_{12},\dots, S_{NM})$ may degrade the performance of the application
and thus, increase the static energy because the execution time is
increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
for the iterative application can be measured by measuring the energy
consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
of iterations of that application.

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

Using the lowest frequency for each processor does not necessarily give the most
energy efficient execution of an application. Indeed, even though the dynamic
power is reduced while scaling down the frequency of a processor, its
computation power is proportionally decreased. Hence, the execution time might
be drastically increased and during that time, dynamic and static powers are
being consumed.  Therefore, it might cancel any gains achieved by scaling down
the frequency of all nodes to the minimum and the overall energy consumption of
the application might not be the optimal one.  It is not trivial to select the
appropriate frequency scaling factor for each processor while considering the
characteristics of each processor (computation power, range of frequencies,
dynamic and static powers) and the task executed (computation/communication
ratio). The aim being to reduce the overall energy consumption and to avoid
increasing significantly the execution time.  In our previous
work~\cite{Our_first_paper,pdsec2015}, we proposed a method that selects the optimal
frequency scaling factor for a homogeneous and heterogeneous clusters executing a message passing
iterative synchronous application while giving the best trade-off between the
energy consumption and the performance for such applications.  In this work we
are interested in heterogeneous grid as described above.  Due to the
heterogeneity of the processors, a vector of scaling factors should be selected
and it must give the best trade-off between energy consumption and performance.

The relation between the energy consumption and the execution time for an
application is complex and nonlinear, Thus, unlike the relation between the
execution time and the scaling factor, the relation between the energy and the
frequency scaling factors is nonlinear, for more details refer
to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  Moreover, these relations
are not measured using the same metric.  To solve this problem, the execution
time is normalized by computing the ratio between the new execution time (after
scaling down the frequencies of some processors) and the initial one (with
maximum frequency for all nodes) as follows:
\begin{equation}
  \label{eq:pnorm}
  \Pnorm = \frac{\Tnew}{\Told}                 
\end{equation}


Where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told})
\begin{equation}
  \label{eq:told}
   \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])             
\end{equation}
In the same way, the energy is normalized by computing the ratio between the
consumed energy while scaling down the frequency and the consumed energy with
maximum frequency for all  nodes:
\begin{equation}
  \label{eq:enorm}
  \Enorm = \frac{\Ereduced}{\Eoriginal} 
\end{equation}

Where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
computed as in (\ref{eq:eorginal}).


\begin{equation}
  \label{eq:eorginal}
    \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot  \Tcp[ij])  + 
     \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M} (\Ps[ij] \cdot \Told)       
\end{equation}

While the main goal is to optimize the energy and execution time at the same
time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. 
According to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
vector 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.

This problem can be solved by making the optimization process for energy and
execution time follow the same evolution according to the vector of scaling factors
$(S_{11}, S_{12},\dots, S_{NM})$. Therefore, the equation of the
normalized execution time is inverted which gives the normalized performance
equation, as follows:
\begin{equation}
  \label{eq:pnorm_inv}
  \Pnorm = \frac{\Told}{\Tnew}          
\end{equation}

\begin{figure}[!t]
  \centering
  \subfloat[Homogeneous cluster]{%
    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%

  \subfloat[Heterogeneous grid]{%
    \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
  \label{fig:rel}
  \caption{The energy and performance relation}
\end{figure}

Then, the objective function can be modeled in order to find the maximum
distance between the energy curve (\ref{eq:enorm}) and the performance curve
(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
represents the minimum energy consumption with minimum execution time (maximum
performance) at the same time, see Figure~\ref{fig:r1} or
Figure~\ref{fig:r2}. Then the objective function has the following form:
\begin{equation}
  \label{eq:max}
  \MaxDist =
\mathop{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
      (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
       \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
\end{equation}
where $N$ is the number of clusters, $M$ is the number of nodes in each cluster and
$F$ is the number of available frequencies for each node.  Then, the optimal set 
of scaling factors that satisfies (\ref{eq:max}) can be selected.  
The objective function can work with any energy model or any power 
values for each node (static and dynamic powers). However, the most important 
energy reduction gain can be achieved when the energy curve has a convex form as shown 
in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.

\section{The scaling factors selection algorithm for  grids }
\label{sec.optim}

\begin{algorithm}
  \begin{algorithmic}[1]
    % \footnotesize
    \Require ~
    \begin{description}
    \item [{$N$}] number of clusters in the grid.
    \item [{$M$}] number of nodes in each cluster.
    \item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
    \item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
    \item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
    \item[{$\Pd[ij]$}] array of the dynamic powers for all nodes.
    \item[{$\Ps[ij]$}] array of the static powers for all nodes.
    \item[{$\Fdiff[ij]$}] array of the differences between two successive frequencies for all nodes.
    \end{description}
    \Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$,  a vector of scaling factors that gives the optimal tradeoff between energy consumption and execution time

    \State $\Scp[ij] \gets \frac{\max_{i=1,2,\dots,N}(\max_{j=1,2,\dots,M_i}(\Tcp[ij]))}{\Tcp[ij]} $
    \State $F_{ij} \gets  \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
    \State Round the computed initial frequencies $F_i$ to the closest  available frequency for each node.
    \If{(not the first frequency)}
          \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
    \EndIf
    \State $\Told \gets $ computed as in equations (\ref{eq:told}).
    \State $\Eoriginal \gets $ computed as in equations (\ref{eq:eorginal}) .
    \State $\Sopt[ij] \gets 1,~i=1,\dots,N,~{j=1,\dots,M_i}. $
    \State $\Dist \gets 0 $
    \While {(all nodes have not reached their  minimum   \newline\hspace*{2.5em} frequency \textbf{or}  $\Pnorm - \Enorm < 0 $)}
        \If{(not the last freq. \textbf{and} not the slowest node)}
        \State $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
        \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
        \EndIf
       \State $\Tnew \gets $ computed as  in equations (\ref{eq:perf}). 
       \State $\Ereduced \gets $ computed as  in equations (\ref{eq:energy}). 
       \State $\Pnorm \gets \frac{\Told}{\Tnew}$
       \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
      \If{$(\Pnorm - \Enorm > \Dist)$}
        \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
        \State $\Dist \gets \Pnorm - \Enorm$
      \EndIf
    \EndWhile
    \State  Return $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$
  \end{algorithmic}
  \caption{Scaling factors selection algorithm}
  \label{HSA}
\end{algorithm}

\begin{algorithm}
  \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 \ref{HSA}.
        \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{algorithm}



In this section, the scaling factors selection algorithm for  grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency
scaling factors  that gives the best trade-off between minimizing the
energy consumption and maximizing the performance of a message passing
synchronous iterative application executed on a  grid. It works
online during the execution time of the iterative message passing program.  It
uses information gathered during the first iteration such as the computation
time and the communication time in one iteration for each node. The algorithm is
executed after the first iteration and returns a vector of optimal frequency
scaling factors that satisfies the objective function (\ref{eq:max}). The
program applies DVFS operations to change the frequencies of the CPUs according
to the computed scaling factors.  This algorithm is called just once during the
execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
scaling algorithm is called in the iterative MPI program.

\begin{figure}[!t]
  \centering
  \includegraphics[scale=0.45]{fig/init_freq}
  \caption{Selecting the initial frequencies}
  \label{fig:st_freq}
\end{figure}

Nodes from distinct clusters in a grid have different computing powers, thus
while executing message passing iterative synchronous applications, fast nodes
have to wait for the slower ones to finish their computations before being able
to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
periods are called idle or slack times.  The algorithm takes into account this
problem and tries to reduce these slack times when selecting the vector of the frequency
scaling factors. At first, it selects initial frequency scaling factors
that increase the execution times of fast nodes and minimize the differences
between the computation times of fast and slow nodes. The value of the initial
frequency scaling factor for each node is inversely proportional to its
computation time that was gathered from the first iteration. These initial
frequency scaling factors are computed as a ratio between the computation time
of the slowest node and the computation time of the node $i$ as follows:
\begin{equation}
  \label{eq:Scp}
  \Scp[ij] =  \frac{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]}
\end{equation}
Using the initial frequency scaling factors computed in (\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
follows:
\begin{equation}
  \label{eq:Fint}
  F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M}
\end{equation}
If the computed initial frequency for a node is not available in the gears of
that node, it is replaced by the nearest available frequency.  In
Figure~\ref{fig:st_freq}, the nodes are sorted by their computing powers in
ascending order and the frequencies of the faster nodes are scaled down
according to the computed initial frequency scaling factors.  The resulting new
frequencies are highlighted in Figure~\ref{fig:st_freq}.  This set of
frequencies can be considered as a higher bound for the search space of the
optimal vector of frequencies because selecting higher frequencies
than the higher bound will not improve the performance of the application and it
will increase its overall energy consumption.  Therefore the algorithm that
selects the frequency scaling factors starts the search method from these
initial frequencies and takes a downward search direction toward lower
frequencies until reaching the nodes' minimum frequencies or lower bounds. A node's frequency is considered its lower bound if the computed distance between the energy and performance at this frequency is less than zero.
A negative distance means that the performance degradation ratio is higher than the energy saving ratio.
In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances. 

Therefore, the algorithm iterates on all remaining frequencies, from the higher
bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
energy consumption and performance and selects the optimal vector of the frequency scaling
factors. At each iteration the algorithm determines the slowest node
according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
while it lowers the frequency of all other nodes by one gear.  The new overall
energy consumption and execution time are computed according to the new scaling
factors.  The optimal set of frequency scaling factors is the set that gives the
highest distance according to the objective function (\ref{eq:max}).

Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
consumed energy for an application running on a homogeneous cluster and a
 grid platform respectively while increasing the scaling factors. It can
be noticed that in a homogeneous cluster the search for the optimal scaling
factor should start from the maximum frequency because the performance and the
consumed energy decrease from the beginning of the plot. On the other hand, in
the  grid platform the performance is maintained at the beginning of the
plot even if the frequencies of the faster nodes decrease until the computing
power of scaled down nodes are lower than the slowest node. In other words,
until they reach the higher bound. It can also be noticed that the higher the
difference between the faster nodes and the slower nodes is, the bigger the
maximum distance between the energy curve and the performance curve is, which results in bigger energy savings. 


\section{Experimental results}
\label{sec.expe}
While in~\cite{pdsec2015} the energy  model and the scaling factors selection algorithm were applied to a heterogeneous cluster and  evaluated over the SimGrid simulator~\cite{SimGrid}, 
in this paper real experiments were conducted over the grid'5000 platform. 

\subsection{Grid'5000 architature and power consumption}
\label{sec.grid5000}
Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed over all metropolitan France and Luxembourg. All the sites are connected together via         a special long distance network called RENATER,
which is the French National Telecommunication Network for Technology.
Each site of the grid is composed of few heterogeneous 
computing clusters and each cluster contains many homogeneous nodes. In total,
grid'5000 has about  one thousand heterogeneous nodes and eight thousand cores.  In each site,
the clusters and their nodes are connected via  high speed local area networks. 
Two types of local networks are used, Ethernet or Infiniband networks which have  different characteristics in terms of bandwidth and latency.  

Since grid'5000 is dedicated for testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture 
the power consumption  for each node in those sites. The measured power is the overall consumed power by by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ...  For more details refer to
\cite{Energy_measurement}. To just measure the CPU power of one core in a node $j$, 
 firstly,  the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumption represents the 
dynamic power consumption of that core with the maximum frequency, see  figure(\ref{fig:power_cons}). 


The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
\begin{equation}
  \label{eq:pdyn}
    \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (\Pmax[jx])  -  \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
\end{equation}

where $\Pd[j]$ is the dynamic power consumption for one core of node $j$, 
$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured maximum power values, 
$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured  idle power values.
Therefore, the dynamic power of one core is computed as the difference between the maximum 
measured value in maximum powers vector and the minimum measured value in the idle powers vector.

On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that  the static power  represents a ratio of the dynamic power, the value of the static power is assumed as  20\% of dynamic power consumption of the core.

In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).

Four clusters from the two sites were selected in the experiments: one cluster from 
Lyon's site, Taurus cluster, and three clusters from Nancy's site, Graphene, 
Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available 
frequency ranges and local network features: the bandwidth and the latency.  Table \ref{table:grid5000} shows 
the details characteristics of these four clusters. Moreover, the dynamic powers were computed  using the equation (\ref{eq:pdyn}) for all the nodes in the 
selected clusters and are presented in table  \ref{table:grid5000}.




\begin{figure}[!t]
  \centering
  \includegraphics[scale=1]{fig/grid5000}
  \caption{The selected two sites of grid'5000}
  \label{fig:grid5000}
\end{figure}


The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections. 




\begin{figure}[!t]
  \centering
  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
  \caption{The power consumption by one core from Taurus cluster}
  \label{fig:power_cons}
\end{figure}



  
\begin{table}[!t]
  \caption{CPUs characteristics of the selected clusters}
  % title of Table
  \centering
  \begin{tabular}{|*{7}{c|}}
    \hline
    Cluster     & CPU         & Max   & Min   & Diff. & no. of cores    & dynamic power   \\
    Name        & model       & Freq. & Freq. & Freq. & per CPU         & of one core     \\
                &             & GHz   & GHz   & GHz   &                 &           \\
    \hline
    Taurus      & Intel       & 2.3  & 1.2  & 0.1     & 6               & \np[W]{35} \\
                & Xeon        &       &       &       &                 &            \\
                & E5-2630     &       &       &       &                 &            \\         
    \hline
    Graphene    & Intel       & 2.53  & 1.2   & 0.133 & 4               & \np[W]{23} \\
                & Xeon        &       &       &       &                 &            \\
                & X3440       &       &       &       &                 &            \\    
    \hline
    Griffon     & Intel       & 2.5   & 2     & 0.5   & 4               & \np[W]{46} \\
                & Xeon        &       &       &       &                 &            \\
                & L5420       &       &       &       &                 &            \\  
    \hline
    Graphite    & Intel       & 2     & 1.2   & 0.1   & 8               & \np[W]{35} \\
                & Xeon        &       &       &       &                 &            \\
                & E5-2650     &       &       &       &                 &            \\  
    \hline
  \end{tabular}
  \label{table:grid5000}
\end{table} 



\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 
to the NAS parallel benchmarks are presented. 

As mentioned previously, the experiments 
were conducted over two sites of grid'5000,  Lyon and Nancy sites. 
Two scenarios were considered while selecting the clusters from these two sites :
\begin{itemize}
\item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected 
 via a long distance network.
\item In the second scenario nodes from three clusters that are located in one site, Nancy site.  
\end{itemize}

The main reason 
behind using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the 
scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio 
is very low due to the higher communication times which reduces the effect of DVFS operations.

The NAS parallel benchmarks are executed over 
16 and 32 nodes for each scenario. The number of participating computing nodes form each cluster 
are different because all the selected clusters do not have the same available number of nodes and all benchmarks do not require the same number of computing nodes.
Table \ref{tab:sc} shows the number of nodes used from each cluster for each scenario. 

\begin{table}[h]

\caption{The different clusters scenarios}
\centering
\begin{tabular}{|*{4}{c|}}
\hline
\multirow{2}{*}{Scenario name}        & \multicolumn{3}{c|} {The participating clusters} \\ \cline{2-4} 
                                      & Cluster & Site           & No. of  nodes     \\ 
\hline
\multirow{3}{*}{Two sites / 16 nodes} & Taurus & Lyon                & 5                      \\ \cline{2-4} 
                                      & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                      & Griffon       & Nancy        & 6                      \\ 
\hline
\multirow{3}{*}{Tow sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
                                      & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                      & Griffon     &Nancy           & 12                     \\ 
\hline
\multirow{3}{*}{One site / 16 nodes}  & Graphite    & Nancy            & 4                      \\ \cline{2-4} 
                                      & Graphene     & Nancy           & 6                      \\ \cline{2-4} 
                                      & Griffon         & Nancy        & 6                      \\ 
\hline
\multirow{3}{*}{One site / 32 nodes}  & Graphite   & Nancy             & 4                      \\ \cline{2-4} 
                                      & Graphene      & Nancy          & 12                     \\ \cline{2-4} 
                                      & Griffon          & Nancy       & 12                       \\ 
\hline
\end{tabular}
 \label{tab:sc}
\end{table}

\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/eng_con_scenarios.eps}
  \caption{The energy consumptions of NAS benchmarks over different scenarios }
  \label{fig:eng_sen}
\end{figure}



\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/time_scenarios.eps}
  \caption{The execution times of NAS benchmarks over different scenarios }
  \label{fig:time_sen}
\end{figure}

The NAS parallel benchmarks are executed over these two platforms
 with different number of nodes, as in Table \ref{tab:sc}. 
The overall energy consumption of all the benchmarks solving the class D instance and
using the proposed frequency selection algorithm is measured 
using the equation of the reduced energy consumption, equation 
(\ref{eq:energy}). This model uses the measured dynamic and static 
power values  showed in Table \ref{table:grid5000}. The execution
time is measured for all the benchmarks over these different scenarios.  

The energy consumptions  and the execution times for all the benchmarks are 
presented in the plots \ref{fig:eng_sen} and \ref{fig:time_sen} respectively. 

For the majority of the benchmarks, the energy consumed while executing  the NAS benchmarks over one site scenario 
for  16 and 32 nodes is lower than the energy consumed while using two sites. 
The long distance communications between the two distributed sites increase the idle time, which leads to more static energy consumption. 
The execution times of these benchmarks 
over one site with 16 and 32 nodes are also lower when  compared to those of the  two sites 
scenario.

However, the  execution times and the energy consumptions of EP and MG benchmarks, which have no or small communications, are not significantly affected 
 in both scenarios. Even when the number of nodes is doubled. On the other hand, the communications of the rest of the benchmarks increases when using long distance communications between two sites or increasing the number of computing nodes.

\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/eng_s.eps}
  \caption{The energy saving of NAS benchmarks over different scenarios }
  \label{fig:eng_s}
\end{figure}


\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/per_d.eps}
  \caption{The performance degradation of NAS benchmarks over different scenarios }
  \label{fig:per_d}
\end{figure}


\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/dist.eps}
  \caption{The tradeoff distance of NAS benchmarks over different scenarios }
  \label{fig:dist}
\end{figure}

The energy saving percentage is computed as the ratio between the reduced 
energy consumption, equation (\ref{eq:energy}), and the original energy consumption,
equation (\ref{eq:eorginal}), for all benchmarks as in figure \ref{fig:eng_s}. 
This figure shows that the energy saving percentages of one site scenario for
16 and 32 nodes are bigger than those of the two sites scenario which is due
to the higher  computations to communications ratio in the first scenario   
than in the second one. Moreover, the frequency selecting algorithm selects smaller frequencies when the computations times are bigger than the communication times which 
results in  a lower energy consumption. Indeed, the dynamic  consumed power
is exponentially related to the CPU's frequency value. On the other side, the increase in the number of computing nodes can 
increase the communication times and thus produces less energy saving depending on the 
benchmarks being executed. The results of the benchmarks CG, MG, BT and FT show more 
energy saving percentage in one site scenario when executed over 16 nodes comparing to 32 nodes. While, LU and SP consume more energy with 16 nodes than 32 in one site  because their computations to communications ratio is not affected by the increase of the number of local communications. 


The energy saving percentage is reduced for all the benchmarks because of the long distance communications in the two sites 
scenario, except for the   EP benchmark which has  no communications. Therefore, the energy saving percentage of this benchmark is 
dependent on the maximum difference between the computing powers of the heterogeneous computing nodes, for example 
in the one site scenario, the graphite cluster is selected but in the two sits scenario 
this cluster is replaced with Taurus cluster which is more powerful. 
Therefore, the energy saving of EP benchmarks are bigger in the two sites scenario due 
to the higher maximum difference between the computing powers of the nodes. 

In fact, high differences between the nodes' computing powers make the proposed frequencies selecting  
algorithm  select smaller frequencies for the powerful nodes which 
produces less energy consumption and thus more energy saving.
The best energy saving percentage was obtained in the one site scenario with 16 nodes, the energy consumption was on average reduced up to 30\%.


Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
The performance degradation percentage for the benchmarks running on two sites  with
16 or 32  nodes is on average equal to 8\% or 4\% respectively. 
For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with
16 or 32  nodes is on average equal to 3\% or 10\% respectively. In opposition to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
nodes when the communications occur in high speed network does not decrease the computations to 
communication ratio. 


 Figure \ref{fig:time_sen} presents the execution times for all the benchmarks over the two scenarios. For most of the benchmarks running over the one site scenario, their execution times are approximately divided by two  when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).  
\textcolor{red}{The transition between the execution times to the performance degradation is not clear}


The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to 
the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its 
performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
The rest of the benchmarks showed different performance degradation percentages, which decrease
when the communication times increase and vice versa.

Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The tradeoff distance percentage can be 
computed as in equation \ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
tradeoff, on average it is equal to  26\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
tradeoff comparing to the two sites scenario  because the former has high speed local communications 
which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 


 Finally, the best energy and performance tradeoff depends on all of the following:
1) the computations to communications ratio when there are  communications and slack times, 2) the heterogeneity of the computing powers of the nodes and 3) the heterogeneity of the consumed  static and dynamic powers of the nodes.




\subsection{The experimental results of multi-cores clusters}
\label{sec.res-mc}
The  clusters of grid'5000 have different number of cores embedded in their nodes
as shown in Table \ref{table:grid5000}. The cores of each node can exchange 
data via the shared memory \cite{rauber_book}. In 
this section, the proposed scaling algorithm is evaluated over the grid'5000 grid while using multi-core nodes 
selected according to the two  platform scenarios described in the section \ref{sec.res}.
The two platform scenarios, the two sites and one site scenarios, use  32 
cores from multi-cores nodes instead of 32 distinct nodes. For example if 
the participating number of cores from a certain cluster is equal to 12, 
in the multi-core scenario the selected nodes is equal to 3 nodes while using 
4 cores from each node. The platforms with one  
core per node and  multi-cores nodes are  shown in Table \ref{table:sen-mc}. 
The energy consumptions and execution times of running the NAS parallel 
benchmarks, class D, over these four different scenarios are presented 
in the figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.

The execution times for most of  the NAS  benchmarks are higher over the one site multi-cores per node scenario 
 than the execution time of those running over one site single core per node  scenario. Indeed,  
   the communication times  are higher in the one site multi-cores scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. On the other hand,  the execution times for most of the NAS benchmarks  are lower over 
the two sites  multi-cores scenario than those over the two sites one core scenario. In the two sites multi-cores scenario, There are three types of communications :
\begin{itemize}
\item between cores on the same node via shared memory 
\item between cores from distinct nodes but belonging to the same cluster or site via local network
\item between cores from distinct sites via long distance network
\end{itemize}
The latency of the communications increases from shared memory to LAN to WAN.
Therefore, using multi-cores communicating via shared memory
has reduced the communication times, and thus the overall execution time is also decreased.



The experiments showed that for most of the NAS benchmarks and between the four scenarios,  
the one site one core scenario gives the best execution times because the communication times are the lowest. 
Indeed, in this scenario each core has a dedicated network link and all the communications are local.  
Moreover, the energy consumptions of the NAS benchmarks are lower over the 
one site one core scenario  than over the one site multi-cores scenario because 
the first scenario had less execution time than the latter which results in less static energy being consumed.

The computations to communications ratios of the NAS benchmarks are higher over 
the one site one core scenario  when compared to the ratios of the other scenarios. 
More energy reduction was achieved when this ratio is increased because the proposed scaling algorithm selects smaller frequencies that decrease the dynamic power consumption. 

 
  \textcolor{red}{ The next sentence is completely false! It is impossible to have these results!  Whereas, the energy consumption in the two sites multi-cores scenario is higher than the energy consumption 
of the two sites one core scenario. 
Actually, using multi-cores in this scenario decreased the communication times that decreased the static energy consumption.}


These experiments also showed that the energy 
consumption and the execution times of the EP and MG benchmarks do not change significantly over these four 
scenarios  because there are no or small communications,  
which could increase or decrease the static power consumptions. Contrary to EP and MG, the  energy consumptions 
and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.


The energy saving percentages of all NAS benchmarks running over these four scenarios are presented in the figure \ref{fig:eng-s-mc}. It shows that  the energy saving percentages   over the two sites multi-cores scenario 
and over the two sites one core scenario are on average  equal to 22\% and 18\%
respectively. The energy saving percentages   are higher in the former scenario because  its computations to communications  ratio is higher than the ratio of the latter scenario  as mentioned previously.


In contrast, in the one site one 
core and one site multi-cores scenarios the energy saving percentages 
are approximately equivalent, on average they are up to 25\%. In both scenarios there 
are a small difference  in the computations to communications ratios, which leads 
the proposed scaling algorithm to select similar frequencies for both scenarios.  

The performance degradation percentages of the NAS benchmarks are presented in
figure \ref{fig:per-d-mc}. It shows that the performance degradation percentages for the NAS benchmarks are higher over the two sites 
multi-cores scenario than over the  two sites  one core scenario, equal on average to 7\% and 4\% respectively. 
Moreover, using the two sites multi-cores scenario increased 
the computations to communications ratio, which may increase 
the overall execution time  when the proposed scaling algorithm is applied and the frequencies scaled down.  


When the benchmarks are executed  over the one 
site one core scenario, their performance degradation percentages are equal  on average
to 10\% and are higher than those executed over the one site multi-cores scenario, 
which on average is equal to 7\%. 

\textcolor{red}{the next sentence is completely false! 
The higher performance degradation percentages over the first scenario is due to the use of multi-cores which 
decreases the computations to communications ratio. Therefore, selecting small 
frequencies by the scaling algorithm do not increase the execution time significantly. }


The tradeoff distance percentages of the NAS 
benchmarks over all scenarios are presented in the figure \ref{fig:dist-mc}.
These  tradeoff distance percentages are used to verify which scenario is the best in terms of energy reduction and performance. The figure shows that using muti-cores in both of the one site and two sites scenarios gives bigger  tradeoff distance percentages, on overage equal to 17.6\% and 15.3\% respectively, than using one core per node in both of one site and two sites scenarios,  on average  equal to 14.7\% and 13.3\% respectively. 

\begin{table}[]
\centering
\caption{The multicores scenarios}

\begin{tabular}{|*{4}{c|}}
\hline
Scenario name                          & Cluster name & \begin{tabular}[c]{@{}c@{}}No. of  nodes\\ in each cluster\end{tabular} & 
                                       \begin{tabular}[c]{@{}c@{}}No. of  cores\\ for each node\end{tabular}  \\ \hline
\multirow{3}{*}{Two sites/ one core}   & Taurus       & 10              & 1                   \\ \cline{2-4}
                                       & Graphene     & 10              & 1                   \\ \cline{2-4}
                                       & Griffon      & 12              & 1                   \\ \hline
\multirow{3}{*}{Two sites/ multicores} & Taurus       & 3               & 3 or 4              \\ \cline{2-4}
                                       & Graphene     & 3               & 3 or 4              \\  \cline{2-4}
                                       & Griffon      & 3               & 4                   \\ \hline
\multirow{3}{*}{One site/ one core}    & Graphite     & 4               & 1                   \\  \cline{2-4}
                                       & Graphene     & 12              & 1                   \\  \cline{2-4}
                                       & Griffon      & 12              & 1                   \\ \hline
\multirow{3}{*}{One site/ multicores}  & Graphite     & 3               & 3 or 4              \\  \cline{2-4}
                                       & Graphene     & 3               & 3 or 4              \\  \cline{2-4}
                                       & Griffon      & 3               & 4                   \\ \hline
\end{tabular}
\label{table:sen-mc}
\end{table}

\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/eng_con.eps}
  \caption{Comparing the  energy consumptions of running NAS benchmarks over one core and multicores scenarios }
  \label{fig:eng-cons-mc}
\end{figure}


  \begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/time.eps}
  \caption{Comparing the  execution times of running NAS benchmarks over one core and multicores scenarios }
  \label{fig:time-mc}
\end{figure}

 \begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/eng_s_mc.eps}
  \caption{The energy saving of running NAS benchmarks over one core and multicores scenarios }
  \label{fig:eng-s-mc}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/per_d_mc.eps}
  \caption{The performance degradation of running NAS benchmarks over one core and multicores scenarios }
  \label{fig:per-d-mc}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/dist_mc.eps}
  \caption{The tradeoff distance of running NAS benchmarks over one core and multicores scenarios }
  \label{fig:dist-mc}
\end{figure}

\subsection{The results of using different static power consumption scenarios}
\label{sec.pow_sen}
\textcolor{blue}{
The static power consumption for one core is the leakage power
consumption when it is idle. The measured static  power of the node, 
as in section \ref{sec.grid5000}, had a collection of power values such as 
all cores static powers and the power consumptions of the other devices. Furthermore, the static power for one core is hard to measured precisely. On the other hand,  the core has consumed the static power during 
the communication and computation times.  However, the static power consumption becomes more important  when the execution time is
increased using DVFS. Therefore, the objective of this section is to verify the ability of the proposed 
scaling algorithm to select the best frequencies when the static power consumption is changing. 
All the results obtained in the previous sections depend on the measured dynamic power 
consumptions as in table \ref{table:grid5000}. Moreover, the static power consumption for one core is represented by 20\% of the measured dynamic power consumption. 
This assumption is extended in this section to taking into account other ratios for the static power consumption.
In addition to the previous ratio of the static power consumption, two other static power ratios are used, which are 10\% and 30\% of the measured dynamic power of the core.
As a result, all of these static power scenarios is denoted as follow: 
\begin{itemize}
\item 10\% of static power scenario 
\item 20\% of static power scenario 
\item 30\% of static power scenario 
\end{itemize}
The NAS parallel benchmarks, class D, are executed over Nancy site. 
The number of computing nodes used is 16 nodes distributed between three cluster, which are Graphite, Graphene and Griffon.  The NAS benchmarks rerun
with these two new static power scenarios over one site scenario
using one core per node. }

 \begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/eng_pow.eps}
  \caption{The energy saving percentages for NAS benchmarks of the three power scenario}
  \label{fig:eng-pow}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/per_pow.eps}
  \caption{The performance degradation percentages for NAS benchmarks of the three power scenario}
  \label{fig:per-pow}
\end{figure}


\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/dist_pow.eps}
  \caption{The tradeoff distance for NAS benchmarks of the three power scenario}
  \label{fig:dist-pow}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[scale=0.47]{fig/three_scenarios.pdf}
  \caption{Comparing the selected frequency scaling factors of MG benchmark for three static power scenarios}
  \label{fig:fre-pow}
\end{figure}

\textcolor{blue}{
The energy saving percentages of NAS benchmarks with these three static power scenarios are presented 
in figure \ref{fig:eng_sen}. This figure shows that  10\% of static power scenario 
gives the biggest energy saving percentage comparing to 20\% and 30\% static power 
scenarios. The smaller ratio of the static power consumption makes the proposed 
scaling algorithm to select smaller frequencies, bigger scaling factors. 
These smaller frequencies has reduced the dynamic energy consumption and thus the 
overall energy consumption is decreased. 
The energy saving percentages of 30\% static power scenario is the smallest between the other scenarios, because of the scaling algorithm selects bigger frequencies, smaller scaling factors, that increased the energy consumption. For example, figure \ref{fig:fre-pow}, illustrates that the proposed scaling algorithm is proportionally selected the best frequency scaling factors according to the static power consumption ratio being used.
Furthermore, the proposed scaling algorithm tries to limit selecting smaller frequencies, which increased the execution time. Hence, the increase in the execution time is relatively increased the static energy consumption.
The performance degradation percentages are presented in the figure \ref{fig:per-pow},
the 30\% of static power scenario had less performance degradation percentage. This  because
bigger frequencies was selected due to the big ratio in the static power consumption.
The inverse happens in the 20\% and 30\% scenarios, the scaling algorithm is selecting
smaller frequencies, bigger scaling factors, according to the ratio of the static power.
The tradeoff distance percentage for the NAS benchmarks with these three static power scenarios 
are presented in the figure \ref{fig:dist}. It shows that the tradeoff
distance percentage is the best when the  10\% of static power scenario is used, and this percentage 
is decreased for the other two scenarios propositionally to their static power ratios.
In EP benchmark, the results of energy saving, performance degradation and tradeoff 
distance are showed small differences when the these static power scenarios were used.
The absent of the communications in this benchmark made the proposed scaling algorithm to select equivalent frequencies even if the static power values are different. While, the 
inverse has been shown  for the rest of the benchmarks, which have  different communication times
that increased the static energy consumption proportionally. Therefore, the scaling algorithm relatively selects 
different frequencies for each benchmark when these static power scenarios are used. }

 
\subsection{The comparison of the proposed frequencies selecting algorithm }
\label{sec.compare_EDP}
\textcolor{blue}{
The tradeoff between the energy consumption and the performance of the parallel 
applications had significant importance in the domain of the research. 
Many researchers, \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs},
have optimized the tradeoff between the energy and the performance using the well known  energy and delay product, $EDP=energy \times delay$. 
This model is also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS},
the  objective is to select the frequencies that minimized EDP product for the multi-cores 
architecture when DVFS is used. Moreover, their algorithm is applied online, which synchronously optimized the energy consumption 
and the execution time. Both energy consumption and execution time of a processor are predicted by the their algorithm.
In this section the proposed frequencies selection algorithm, called Maxdist is compared with Spiliopoulos et al. algorithm, called EDP.
To make both of the algorithms follow the same direction and  fairly  comparing them, the same energy model,  equation \ref{eq:energy} and
the execution time model, equation \ref{eq:perf}, are used in the prediction process to select the best vector of the frequencies. 
In contrast, the proposed algorithm starts the search space from the lower bound computed as in equation the  \ref{eq:Fint}. Also, the algorithm
stops  the search process when it is reached to the lower bound as mentioned before. In the same way, the EDP algorithm is developed to start from the 
same upper bound used in Maxdist algorithm, and it stops the search process when  a minimum available frequencies is reached. 
Finally, the resulting EDP algorithm is an exhaustive search algorithm that test all possible frequencies, starting from the initial frequencies, 
and selecting those minimized the EDP product.
Both algorithms were applied to NAS benchmarks, class D, over 16 nodes selected from grid'5000 clusters.
The participating computing nodes are distributed between two sites and one site to have two different scenarios that used in the section \ref{sec.res}. 
The experimental results: the energy saving, performance degradation and tradeoff distance percentages are 
presented in the figures \ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively. 
\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/edp_eng}
  \caption{Comparing of the energy saving for the proposed method with EDP method}
  \label{fig:edp-eng}
\end{figure}
\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/edp_per}
  \caption{Comparing of the performance degradation for the proposed method with EDP method}
  \label{fig:edp-perf}
\end{figure}
\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/edp_dist}
  \caption{Comparing of the tradeoff distance for the proposed method with EDP method}
  \label{fig:edp-dist}
\end{figure}
As shown form these figures, the proposed frequencies selection algorithm, Maxdist, outperform the EDP algorithm in term of energy and performance for all of the benchmarks executed over the two scenarios. 
Generally, the proposed algorithm gives better results for all benchmarks because it is
optimized the distance between the energy saving and the performance degradation in the same time. 
Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
Whereas, the EDP algorithm gives some times negative tradeoff values for some benchmarks in the two sites scenarios.
These negative tradeoff values mean that the performance degradation percentage is higher than energy saving percentage.
The higher positive value of the tradeoff distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
The time complexity of both Maxdist and EDP algorithms are $O(N \cdot M \cdot F)$ and 
$O(N \cdot M \cdot F^2)$ respectively. Where $N$ is the number of the clusters, $M$ is the number of nodes and $F$ is the 
maximum number of available frequencies. The proposed algorithm, Maxdist, has selected the best frequencies in a small execution time, 
on average is equal to  0.01 $ms$, when it is executed over 32 nodes distributed between Nancy and Lyon sites.
While the EDP algorithm was slower than Maxdist algorithm by ten times over the same number of nodes and same distribution, its execution time on average 
is equal to 0.1 $ms$. 
}


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



\section*{Acknowledgment}

This work  has been  partially 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,
Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
supporting his work.

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