1st english corrections

[mpi-energy2.git] / Heter_paper.tex

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\begin{document} 

\title{Energy Consumption Reduction for Message Passing Iterative  Applications 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}
Computing platforms  are consuming  more and more  energy due to  the increasing
number  of nodes  composing  them.  To  minimize  the operating  costs of  these
platforms many techniques have been  used. Dynamic voltage and frequency scaling
(DVFS) is  one of them. It  reduces the frequency of  a CPU to  lower its energy
consumption.  However,  lowering the  frequency  of  a  CPU might  increase  the
execution  time of  an application  running on  that processor.   Therefore, the
frequency that  gives the best tradeoff  between the energy  consumption and the
performance of an application must be selected.

In this  paper, a new  online frequencies selecting algorithm  for heterogeneous
platforms is presented.   It selects the frequency which tries  to give the best
tradeoff  between  energy saving  and  performance  degradation,  for each  node
computing the message  passing iterative application. The algorithm  has a small
overhead and works without training or profiling. It uses a new energy model for
message passing iterative applications  running on a heterogeneous platform. The
proposed algorithm is  evaluated on the Simgrid simulator  while running the NAS
parallel  benchmarks.  The  experiments   show  that  it  reduces  the  energy
consumption by up to 35\% while  limiting the performance degradation as much as
possible.   Finally,  the algorithm  is  compared  to  an existing  method,  the
comparison results showing that it outperforms the latter.

\end{abstract}

\section{Introduction}
\label{sec.intro}
The  need for  more  computing  power is  continually  increasing. To  partially
satisfy  this need,  most supercomputers  constructors just  put  more computing
nodes in their  platform. The resulting platforms might  achieve higher floating
point operations  per second  (FLOPS), but the  energy consumption and  the heat
dissipation  are  also increased.   As  an  example,  the Chinese  supercomputer
Tianhe-2 had  the highest FLOPS  in November 2014  according to the  Top500 list
\cite{TOP500_Supercomputers_Sites}.  However, it was  also the most power hungry
platform  with  its  over  3  million cores  consuming  around  17.8  megawatts.
Moreover,    according   to    the    U.S.    annual    energy   outlook    2014
\cite{U.S_Annual.Energy.Outlook.2014}, the  price of energy  for 1 megawatt-hour
was approximately equal to \$70.  Therefore, the price of the energy consumed by
the Tianhe-2  platform is approximately more  than \$10 million  each year.  The
computing platforms must  be more energy efficient and  offer the highest number
of FLOPS  per watt  possible, such as  the L-CSC  from the GSI  Helmholtz Center
which became the top of the Green500 list in November 2014 \cite{Green500_List}.
This heterogeneous platform executes more than 5 GFLOPS per watt while consuming
57.15 kilowatts.

Besides platform  improvements, there are many software  and hardware techniques
to lower  the energy consumption of  these platforms, such  as scheduling, DVFS,
...   DVFS is  a  widely used  process to  reduce  the energy  consumption of  a
processor            by             lowering            its            frequency
\cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
the number of FLOPS executed by the processor which might increase the execution
time of the application running over that processor.  Therefore, researchers use
different optimization  strategies to select  the frequency that gives  the best
tradeoff  between the  energy reduction  and performance  degradation  ratio. In
\cite{Our_first_paper}, a  frequency selecting algorithm was  proposed to reduce
the energy  consumption of message  passing iterative applications  running over
homogeneous platforms.  The results of  the experiments show  significant energy
consumption  reductions. In  this  paper, a  new  frequency selecting  algorithm
adapted  for heterogeneous  platform  is  presented. It  selects  the vector  of
frequencies, for  a heterogeneous platform  running a message  passing iterative
application, that simultaneously tries to offer the maximum energy reduction and
minimum performance degradation ratio. The  algorithm has a very small overhead,
works online and does not need any training or profiling.

This paper is organized as follows: Section~\ref{sec.relwork} presents some
related works from other authors.  Section~\ref{sec.exe} describes how the
execution time of message passing programs can be predicted.  It also presents an energy
model that predicts the energy consumption of an application running over a heterogeneous platform. Section~\ref{sec.compet} presents
the energy-performance objective function that maximizes the reduction of energy
consumption while minimizing the degradation of the program's performance.
Section~\ref{sec.optim} details the proposed frequency selecting algorithm then the precision of the proposed algorithm is verified. 
Section~\ref{sec.expe} presents the results of applying the algorithm on  the NAS parallel benchmarks and executing them 
on a heterogeneous platform. It shows the results of running three 
different power scenarios and comparing them. Moreover, it also shows the comparison results
between the proposed method and an existing method.
Finally, in Section~\ref{sec.concl} the paper ends with a summary and some future works.

\section{Related works}
\label{sec.relwork}
DVFS is a technique used in modern processors to scale down both the voltage and
the  frequency  of the  CPU  while  computing, in  order  to  reduce the  energy
consumption of  the processor. DVFS is also  allowed in  GPUs  to achieve the
same goal. Reducing the frequency of  a processor lowers its number of FLOPS and
might  degrade the  performance of  the application  running on  that processor,
especially if it is compute bound. Therefore selecting the appropriate frequency
for a processor to satisfy some objectives while taking into account all the
constraints,  is  not a  trivial  operation.   Many  researchers used  different
strategies to  tackle this problem. Some  of them developed  online methods that
compute   the  new   frequency  while   executing  the   application,   such  as
~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}. Others
used  offline methods  that might  need to  run the  application and  profile it
before       selecting       the        new       frequency,       such       as
~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The
methods could  be heuristics, exact or  brute force methods  that satisfy varied
objectives such as  energy reduction or performance. They  also could be adapted
to  the  execution's  environment  and  the  type of  the  application  such  as
sequential, parallel  or distributed architecture,  homogeneous or heterogeneous
platform, synchronous or asynchronous application, ...

In this paper, we are interested in reducing energy for message passing iterative synchronous applications running over heterogeneous platforms.
Some works have already been done for such platforms and they can be classified into two types of heterogeneous platforms: 
\begin{itemize}

\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
\item the platform is only composed of heterogeneous CPUs.

\end{itemize}

For the first type of platform, the computing intensive parallel tasks are executed on the  GPUs and the rest are executed 
on the CPUs.  Luley et al.
~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed  a heterogeneous 
cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal was to maximize the 
energy efficiency of the platform during computation by maximizing the number of FLOPS per watt generated. 
In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et al. developed a scheduling 
algorithm that distributes  workloads proportional to the computing power of the nodes which could be a GPU or a CPU. All the tasks must be completed at the same time.
In~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, Rong et al. showed that 
a heterogeneous (GPUs and CPUs) cluster that enables DVFS gave better energy and performance 
efficiency than other clusters only composed of  CPUs.
 
The work presented in this paper concerns the second type of platform, with heterogeneous CPUs.
Many methods were conceived to reduce the energy consumption of this type of platform.  Naveen et al.~\cite{Naveen_Power.Efficient.Resource.Scaling}  
developed a method that minimizes the value of $energy*delay^2$ (the delay is the sum of slack times that happen during synchronous communications) by dynamically assigning new frequencies to the CPUs of the heterogeneous cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed
an algorithm that divides the executed tasks into two types: the critical and 
non critical tasks. The algorithm scales down the frequency of  non critical tasks proportionally to their  slack and communication times while limiting  the performance degradation percentage to less than 10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed 
  a heterogeneous cluster composed of two  types 
of Intel and AMD processors. They use a gradient method to predict the impact of DVFS operations on performance.
In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, 
 the best frequencies for a specified heterogeneous cluster are selected offline using some 
heuristic. Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic programming approach to  
minimize the power consumption of heterogeneous servers  while respecting given time constraints. This approach 
had considerable overhead.
In contrast to the above described papers, this paper presents the following contributions :
\begin{enumerate}
\item  two new energy and performance models for message passing iterative synchronous applications running over 
       a heterogeneous platform. Both models take into account  communication and slack times. The models can predict the required energy and the execution time of the application.
       
\item a new online frequency selecting algorithm for heterogeneous platforms. The algorithm has a very small 
      overhead and does not need any training or profiling. It uses a new optimization function which simultaneously maximizes the performance and minimizes the energy consumption of a message passing iterative synchronous application.
      
\end{enumerate}

\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 platforms. A heterogeneous platform is defined as a collection of
heterogeneous computing nodes interconnected via a high speed homogeneous
network. Therefore, each node has different characteristics such as computing
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
have the same network bandwidth and latency.

The overall execution time  of a distributed iterative synchronous application 
over a heterogeneous platform  consists of the sum of the computation time and 
the communication time for every iteration on a node. However, due to the 
heterogeneous computation power of the computing nodes, slack times might occur 
when fast nodes have to  wait, during synchronous communications, for  the slower 
nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
Therefore,  the overall execution time  of the program is the execution time of the slowest
task which has the highest computation time and no slack time.
  
 \begin{figure}[t]
  \centering
   \includegraphics[scale=0.6]{fig/commtasks}
  \caption{Parallel tasks on a heterogeneous platform}
  \label{fig:heter}
\end{figure}

Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in 
modern processors, that reduces the energy consumption of a CPU by scaling 
down its voltage and frequency.  Since DVFS lowers the frequency of a CPU 
and consequently its computing power, the execution time of a program running 
over that scaled down processor might increase, especially if the program is 
compute bound.  The frequency reduction process can be  expressed by the scaling 
factor S which is the ratio between  the maximum and the new frequency of a CPU 
as in (\ref{eq:s}).
\begin{equation}
  \label{eq:s}
 S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
 The execution time of a compute bound sequential program is linearly proportional 
 to the frequency scaling factor $S$.  On the other hand,  message passing 
 distributed applications consist of two parts: computation and communication. 
 The execution time of the computation part is linearly proportional to the 
 frequency scaling factor $S$ but  the communication time is not affected by the 
 scaling factor because  the processors involved remain idle during the  
 communications~\cite{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 platform each node has different characteristics,
especially different frequency gears, when applying DVFS operations on these
nodes, they may get different scaling factors represented by a scaling vector:
$(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
be able to predict the execution time of message passing synchronous iterative
applications running over a heterogeneous platform, for different vectors of
scaling factors, the communication time and the computation time for all the
tasks must be measured during the first iteration before applying any DVFS
operation. Then the execution time for one iteration of the application with any
vector of scaling factors can be predicted using (\ref{eq:perf}).
\begin{equation}
  \label{eq:perf}
 \textit  T_\textit{new} = 
 \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) +  MinTcm 
\end{equation}
Where:\\
\begin{equation}
\label{eq:perf2}
 MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
\end{equation}
where  $TcpOld_i$ is  the computation  time of  processor $i$  during  the first
iteration and $MinTcm$  is the communication time of  the slowest processor from
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. 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
architectures~\cite{Our_first_paper}.   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 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 $C_L$, the supply voltage $V$ and
operational frequency $F$, as shown in (\ref{eq:pd}).
\begin{equation}
  \label{eq:pd}
  Pd = \alpha \cdot C_L \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 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} =  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 
$F_{new}$ from (\ref{eq:s}) can be calculated as follows:
\begin{equation}
  \label{eq:fnew}
   F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
Replacing $F_{new}$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following 
equation for dynamic power consumption:
\begin{multline}
  \label{eq:pdnew}
   {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
   {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}
\end{multline}
where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the 
new frequency and the maximum frequency respectively.

According to (\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}
   E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{dOld} \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}
 E_\textit{s} = Ps \cdot (Tcp \cdot S  + Tcm)
\end{equation}

In  the  considered  heterogeneous  platform,  each  processor  $i$  might  have
different   dynamic  and  static   powers,  noted   as  $Pd_{i}$   and  $Ps_{i}$
respectively.  Therefore,  even if  the  distributed  message passing  iterative
application  is  load balanced,  the  computation time  of  each  CPU $i$  noted
$Tcp_{i}$ might  be different and  different frequency scaling factors  might be
computed in order to decrease  the overall energy consumption of the application
and reduce slack  times.  The communication time of a processor  $i$ is noted as
$Tcm_{i}$  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 platform during one iteration  is the summation of all dynamic and
static energies for each processor.  It is computed as follows:
\begin{multline}
  \label{eq:energy}
 E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot  Tcp_i)} + {} \\
 \sum_{i=1}^{N} (Ps_{i} \cdot (\max_{i=1,2,\dots,N} (Tcp_i \cdot S_{i}) +
  {MinTcm))}
 \end{multline}

Reducing the frequencies of the processors according to the vector of
scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
application and thus, increase the static energy because the execution time is
increased~\cite{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},  we  proposed a  method  that  selects the  optimal
frequency scaling factor  for a homogeneous cluster 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  clusters  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{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}) +MinTcm}
           {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
\end{multline}


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{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 $E_\textit{Reduced}$ and $E_\textit{Original}$ are computed using (\ref{eq:energy}) and
  $T_{New}$ and $T_{Old}$ are computed as in (\ref{eq:pnorm}).

While the main 
goal is to optimize the energy and execution time at the same time, the normalized 
energy and execution time curves are not in the same direction. 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 following the same direction.  Therefore, the equation of the 
normalized execution time is inverted which gives the normalized performance equation, 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}) + MinTcm} 
\end{multline}


\begin{figure}
  \centering
  \subfloat[Homogeneous platform]{%
    \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
  
  
  \subfloat[Heterogeneous platform]{%
    \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}
  Max Dist = 
  \max_{i=1,\dots F, j=1,\dots,N}
      (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
       \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
\end{equation}
where $N$ is the number of nodes and $F$ is the  number of available frequencies for each 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 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 heterogeneous platforms }
\label{sec.optim}

\subsection{The algorithm details}
In this section, algorithm \ref{HSA} is presented. It selects the frequency scaling factors 
vector 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 heterogeneous 
platform. 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.

The nodes in a heterogeneous platform 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 
frequency scaling factors vector. 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_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
\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_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
\end{equation}
If the computed initial frequency for a node is not available in the gears of that node, the computed 
initial frequency 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 colored in blue 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 frequency 
scaling factors higher 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. The algorithm iterates on all left frequencies, from the higher bound until all 
nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select 
the optimal frequency scaling factors vector. 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}).

The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an 
application running on a homogeneous platform and a heterogeneous platform respectively while increasing the 
scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor 
should be started from the maximum frequency because the performance and the consumed energy is decreased since  
the beginning of the plot. On the other hand, in  the heterogeneous platform the performance is  maintained at 
the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes 
have computing powers 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 while varying the scaling factors 
which results in bigger energy savings. 
\begin{figure}[t]
  \centering
    \includegraphics[scale=0.5]{fig/start_freq}
  \caption{Selecting the initial frequencies}
  \label{fig:st_freq}
\end{figure}




\begin{algorithm}
  \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 $Sopt_1,Sopt_2 \dots, Sopt_N$ is a vector of optimal scaling factors

    \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
    \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
    \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
    \If{(not the first frequency)}
          \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
    \EndIf 
    \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
    \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $
    \State $Dist \gets 0 $
    \While {(all nodes not reach their  minimum  frequency)}
        \If{(not the last freq. \textbf{and} not the slowest node)}
        \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,\dots,N.$
        \EndIf
       \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm $
       \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} + $  \hspace*{43 mm} 
               $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
       \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
       \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
      \If{$(\Pnorm - \Enorm > \Dist)$}
        \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $
        \State $\Dist \gets \Pnorm - \Enorm$
      \EndIf
    \EndWhile
    \State  Return $Sopt_1,Sopt_2,\dots,Sopt_N$
  \end{algorithmic}
  \caption{frequency 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}

\subsection{The evaluation of the proposed algorithm}
\label{sec.verif.algo}
The precision of the proposed algorithm mainly depends on the execution time prediction model defined in 
(\ref{eq:perf}) and the energy model computed by (\ref{eq:energy}). 
The energy model is also significantly dependent  on the execution time model because the static energy is 
linearly related to the execution time and the dynamic energy is related to the computation time. So, all of 
the works presented in this paper is based on the execution time model. To verify this model, the predicted 
execution time was compared to  the real execution time over SimGrid/SMPI simulator, v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}, 
for all  the NAS parallel benchmarks NPB v3.3 
\cite{NAS.Parallel.Benchmarks}, running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise, 
the maximum normalized difference between the predicted execution time  and the real execution time is equal 
to 0.03 for all the NAS benchmarks.

Since  the proposed algorithm is not an exact method and does not test all the possible solutions (vectors of scaling factors) 
in the search space. To prove its efficiency, it was compared on small instances to a brute force search algorithm 
that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with 
different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical 
and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time: 
for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in 
table~\ref{table:platform}, it takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes 
to compute the best scaling factors vector.  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  from 12 to 20 iterations to select the best 
vector of frequency scaling factors that gives the results of the next sections.

\section{Experimental results}
\label{sec.expe}
To evaluate the efficiency and the overall energy consumption reduction of algorithm~ \ref{HSA}, 
it was applied to the NAS parallel benchmarks NPB v3.3. The experiments were executed 
on the simulator SimGrid/SMPI which offers easy tools to create a heterogeneous platform and run 
message passing applications over it. The  heterogeneous platform that was used in the experiments, 
had one core per node because just one  process was executed per node. 
The heterogeneous platform  was composed of four types of nodes. Each type of nodes had different 
characteristics such as the maximum CPU frequency, the number of
available frequencies and the computational power, see Table \ref{table:platform}. The characteristics 
of these different types of  nodes are inspired   from the specifications of real Intel processors. 
The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, 
for example if  a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors 
of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were 
chosen proportionally to its computing power (FLOPS).  In the initial heterogeneous platform,  while computing 
with highest frequency, each node  consumed power proportional to its computing power which 80\% of it was 
dynamic power and the rest was 20\% for the static power, the same assumption  was made in \cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. 
Finally, These nodes were 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          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
    type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
                  &           & GHz      & GHz          &GHz             &              &       \\
    \hline
    1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
         
    \hline
    2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
                  
    \hline
    3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
                  
    \hline
    4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
                  
    \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 the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) 
and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in 
this paper, only the results of the biggest class, C, are presented while being run on different number 
of nodes, ranging  from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the 
benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes. 
The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.

 
 
\begin{table}[htb]
  \caption{Running NAS benchmarks on 4 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Program     & 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
    Program    & 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
    Program     & 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
    Program    & 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
    Program    & 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
    Program    & 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 overall energy consumption was computed for each instance according to the energy 
consumption  model (\ref{eq:energy}), with and without applying the algorithm. The 
execution time was also measured for all these experiments. Then, the energy saving 
and performance degradation percentages were computed for each instance.  
The results are presented 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}). All these results are the 
average values from many experiments for  energy savings and performance degradation.
The tables  show the experimental results for running the NAS parallel benchmarks on different 
number of nodes. The experiments show that the algorithm reduce significantly the energy 
consumption (up to 35\%) and tries to limit the performance degradation. They also show that 
the  energy saving percentage is decreased  when the number of the computing nodes is increased. 
This reduction is due to the increase of the communication times compared to the execution times 
when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C, 
are executed on different number of nodes, so the computation required for each iteration is divided 
by the number of computing nodes.   On the other hand, more communications are required when increasing 
the number of nodes so the static energy is increased linearly according to the communication time and 
the dynamic power is less relevant in the overall energy consumption. Therefore, reducing the frequency 
with algorithm~(\ref{HSA}) have less effect in reducing the overall energy savings. It can also be 
noticed that for the benchmarks EP and SP that contain little or no communications,  the energy savings 
are not significantly affected with the high number of nodes. No experiments were conducted using bigger 
classes such as D, because they require a lot of memory(more than 64GB) when being executed by the simulator 
on one machine. The maximum distance between the normalized energy curve and the normalized performance 
for each instance is also shown in the result tables. It is decreased in the same way as the energy 
saving percentage. The tables also show that the performance degradation percentage is not significantly 
increased when the number of computing nodes is increased because the computation times are small when 
compared to the communication times.  


 
\begin{figure}
  \centering
  \subfloat[Energy saving]{%
    \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
  
  \subfloat[Performance degradation ]{%
    \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
  \label{fig:avg}
  \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
\end{figure}

Plots (\ref{fig:energy} and \ref{fig:per_deg}) present the energy saving and performance degradation 
respectively for all the benchmarks according to the number of used nodes. As shown in the first plot, 
the energy saving percentages of the benchmarks MG, LU, BT and FT are decreased linearly  when the  
number of nodes is increased. While for the  EP and SP benchmarks, the energy saving percentage is not 
affected by the increase of the number of computing nodes, because in these benchmarks there are little or 
no communications. Finally, the energy saving of the GC benchmark  is significantly decreased when the number 
of nodes is increased because  this benchmark has more communications than the others. The second plot 
shows that the performance degradation percentages of most of the benchmarks are decreased when they 
run on a big number of nodes because they spend more time communicating than computing, thus, scaling 
down the frequencies of some nodes have less effect on the performance. 




\subsection{The results for different power consumption scenarios}
\label{sec.compare}
The results of the previous section were obtained while using processors that consume during computation 
an overall power which is 80\% composed of  dynamic power and 20\% of static power. In this section, 
these ratios are changed and two new power scenarios are considered in order to evaluate how the proposed  
algorithm adapts itself according to the static and dynamic power values.  The two new power scenarios 
are the following: 

\begin{itemize}
\item 70\% dynamic power  and 30\% static power
\item 90\% dynamic power  and 10\% static power
\end{itemize}

The NAS parallel benchmarks were executed again over processors that follow the new power scenarios. 
The class C of each benchmark was run over 8 or 9 nodes and the results are presented in  Tables 
\ref{table:res_s1} and \ref{table:res_s2}. These tables show that the energy saving percentage of the 70\%-30\% 
scenario is less for all benchmarks compared to the energy saving of the 90\%-10\% scenario. Indeed, in the latter 
more dynamic power is consumed when nodes are running on their maximum frequencies, thus, scaling down the frequency 
of the nodes results in higher energy savings than in the 70\%-30\% scenario. On the other hand,  the performance 
degradation percentage is less in the 70\%-30\% scenario  compared to the 90\%-10\%  scenario. This is due to the 
higher static power percentage in the first scenario which makes it more relevant in the overall consumed energy. 
Indeed, the static energy is related to the execution time and if the performance is  degraded the total consumed 
static energy is directly increased. Therefore, the proposed algorithm do not scales down much the frequencies of the 
nodes  in order to limit the increase of the execution time and thus limiting the effect of the consumed static energy.

The two new power scenarios are compared to the old one in figure (\ref{fig:sen_comp}). It shows the average of 
the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. 
The comparison shows that  the energy saving ratio is proportional to the dynamic power ratio: it is increased 
when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the  most relevant 
in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, 
the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in 
the overall consumed energy and lowering the frequency do not returns big energy savings.
Moreover, 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). Since the proposed algorithm optimizes the energy consumption 
when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in 
more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens 
when using a higher ratio for  static  power, the algorithm proportionally  selects  smaller scaling values which 
results in less energy saving but less performance degradation. 


 \begin{table}[htb]
  \caption{The results of 70\%-30\% powers scenario}
  % title of Table
  \centering
  \begin{tabular}{|*{6}{l|}}
    \hline
    Program    & 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
    Program    & 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  of the results on 8 nodes]{%
    \includegraphics[width=.33\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%

  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
    \includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
  \label{fig:comp}
  \caption{The comparison of the three power scenarios}
\end{figure}  




\subsection{The comparison of the proposed scaling algorithm }
\label{sec.compare_EDP}
In this section, the scaling  factors selection algorithm, called MaxDist,
is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP. 
They developed a green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Enegry*Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
To fairly compare both algorithms, the same energy and execution time models, equations (\ref{eq:energy}) and  (\ref{eq:fnew}), were used for both algorithms to predict the energy consumption and the execution times. Also Spiliopoulos et al. algorithm was adapted to  start the search from the 
initial frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm is an exhaustive search algorithm that minimizes the EDP and has the initial frequencies values as an upper bound.

Both algorithms were applied to the parallel NAS benchmarks to compare their efficiency. Table \ref{table:compare_EDP}  presents the results of comparing the execution times and the energy consumptions for both versions of the NAS benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous nodes. The results show that our algorithm gives better energy savings than Spiliopoulos et al. algorithm, 
on average it results in 29.76\% energy saving while their algorithm returns just 25.75\%. The average of performance degradation percentage is approximately the same for both algorithms, about 4\%. 


For all benchmarks, our algorithm outperforms 
Spiliopoulos et al. algorithm in term of energy and performance tradeoff, see figure (\ref{fig:compare_EDP}), because it maximizes the distance between the energy saving and the performance degradation values while giving the same weight for both metrics. 




\begin{table}[h]
 \caption{Comparing the proposed algorithm}
 \centering
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Program \\ name\end{tabular}}} & \multicolumn{2}{l|}{Energy saving \%} & \multicolumn{2}{l|}{Perf.  degradation \%} & \multicolumn{2}{l|}{Distance} \\ \cline{3-8} 
\multicolumn{2}{|l|}{}                                                                         & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
\multicolumn{2}{|l|}{CG}                                                                       & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
\multicolumn{2}{|l|}{MG}                                                                       & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
\multicolumn{2}{|l|}{LU}                                                                       & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
\multicolumn{2}{|l|}{EP}                                                                       & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
\multicolumn{2}{|l|}{BT}                                                                       & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
\multicolumn{2}{|l|}{SP}                                                                       & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
\multicolumn{2}{|l|}{FT}                                                                       & 27.03           & 31.02            & 2.75           & 2.54              & 24.28        & 28.48           \\ \hline

\end{tabular}
\label{table:compare_EDP}
\end{table}





\begin{figure}[t]
  \centering
   \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
  \caption{Tradeoff comparison for NAS benchmarks class C}
  \label{fig:compare_EDP}
\end{figure}


\section{Conclusion}
\label{sec.concl} 
In this paper, a new online frequency selecting algorithm has been presented. It selects the best possible vector of frequency scaling factors that gives the maximum distance (optimal tradeoff) between the predicted energy and 
the predicted performance curves for a heterogeneous platform. This algorithm uses a new energy model for measuring  
and predicting the energy of distributed iterative applications running over heterogeneous 
platform. To evaluate the proposed method, it  was  applied on the NAS parallel benchmarks and executed over a heterogeneous platform simulated by  Simgrid. The results of the experiments showed that the algorithm reduces up to 35\% the energy consumption of a message passing iterative method while limiting the degradation of the performance. The algorithm also  selects different scaling factors   according to the percentage of the computing and communication times, and according to the values of  the static and  dynamic powers of the CPUs. Finally, the algorithm was compared to Spiliopoulos et al. algorithm and the results showed that it 
 outperforms their algorithm in term of energy-time tradeoff.

In the near future, this method will be applied to real heterogeneous platforms to evaluate its performance in a real study case. It would also be interesting to evaluate its scalability over large scale heterogeneous platform and measure the energy consumption reduction it can produce. Afterward, we would like  to develop a similar method that is adapted to asynchronous  iterative applications 
where each task does not wait for others tasks to finish their works. The development of such method might require a new 
energy model because the number of iterations is not 
known in advance and depends on the global convergence of the iterative system.

\section*{Acknowledgment}

This work has been partially supported by the Labex
ACTION project (contract “ANR-11-LABX-01-01”). As a PhD student, 
Mr. Ahmed Fanfakh, would like to thank the University of
Babylon (Iraq) for supporting his work. 


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