Rewrite formulae to be shorter.

[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-Comte\\
    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 trade-off  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
trade-off  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,
\dots{}  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
trade-off  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, \dots{}

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 $energy\cdot delay^2$ (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{\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 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}
  \Tnew = \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 $\CL$, 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  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}
  \Pnorm = \frac{\Tnew}{\Told}\\
       {} = \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}
  \Enorm = \frac{\Ereduced}{\Eoriginal} \\
  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd_i \cdot  \Tcp_i)} +
 \sum_{i=1}^{N} {(\Ps_i \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i)} +
 \sum_{i=1}^{N} {(\Ps_i \cdot \Told)}}
\end{multline} 
Where $\Ereduced$ and $\Eoriginal$ are computed using (\ref{eq:energy}) and
  $\Tnew$ and $\Told$ 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}
  \Pnorm = \frac{\Told}{\Tnew}\\
          = \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}[!t]
  \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}
  \MaxDist = 
  \mathop{\max_{i=1,\dots F}}_{j=1,\dots,N}
      (\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
       \overbrace{\Enorm(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 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 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,\dots,N}
\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 power 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}).

Figures~\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 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 heterogeneous 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 while
 the scaling factors are varying 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 $\Told \gets max_{~i=1,\dots,N } (\Tcp_i+\Tcm_i)$
    % \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i)} +\sum_{i=1}^{N} {(\Ps_i \cdot \Told)}$
    \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd_i \cdot  \Tcp_i + \Ps_i \cdot \Told)}$
    \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 $\Tnew \gets max_\textit{~i=1,\dots,N} (\Tcp_{i} \cdot S_{i}) + \MinTcm $
%       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd_i \cdot  \Tcp_i)} + \sum_{i=1}^{N} {(\Ps_i \cdot \rlap{\Tnew)}} $
       \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd_i \cdot  \Tcp_i + \Ps_i \cdot \rlap{\Tnew)}} $
       \State $\Pnorm \gets \frac{\Told}{\Tnew}$
       \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
      \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 the works presented  in this paper are 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 it 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 an amount  of power proportional  to its
computing  power  (which  corresponds to  80\%  of  its  dynamic power  and  the
remaining  20\%  to  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}[!t]
  \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 had to be executed on $1,
2, 4, 8, 16, 32, 64, 128$ nodes.   The other benchmarks such as BT and SP had to
be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.

 
 
\begin{table}[!t]
  \caption{Running NAS benchmarks on 4 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{r|}}
    \hline
    \hspace{-2.2084pt}%
    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}

\medskip
% \begin{table}[!t]
  \caption{Running NAS benchmarks on 8 and 9 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{r|}}
    \hline
     \hspace{-2.2084pt}%
    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}

\medskip
% \begin{table}[!t]
  \caption{Running NAS benchmarks on 16 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{r|}}
    \hline
    \hspace{-2.2084pt}%
    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}

\medskip
% \begin{table}[!t]
  \caption{Running NAS benchmarks on 32 and 36 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{r|}}
    \hline
    \hspace{-2.2084pt}%
    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}

\medskip
% \begin{table}[!t]
  \caption{Running NAS benchmarks on 64 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{r|}}
    \hline
    \hspace{-2.2084pt}%
    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}

\medskip
% \begin{table}[!t]
  \caption{Running NAS benchmarks on 128 and 144 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{r|}}
    \hline
    \hspace{-2.2084pt}%
    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
significantly reduces the energy consumption (up to 35\%) and tries to limit the
performance  degradation.  They  also  show that  the  energy saving  percentage
decreases when the  number of computing nodes increases.   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  numbers  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  increases  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} is
less effective  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 by the high  number of nodes.  No
experiments were conducted  using bigger classes than 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
decrease 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}[!t]
  \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 a different number of nodes}
\end{figure}

Figures~\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 decrease linearly when  the number of nodes
increase. 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  significantly  decrease  when the  number  of nodes  increase
because this benchmark has more  communications than the others. The second plot
shows that  the performance  degradation percentages of  most of  the benchmarks
decrease 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
has 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 of 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\% of dynamic power  and 30\% of static power
\item 90\% of dynamic power  and 10\% of 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  smaller 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 smaller
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 amount of consumed
static  energy directly  increases.  Therefore,  the proposed  algorithm  does not
really significantly  scale 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.

Both   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 decreases  when the 70\%-30\%
scenario is  used because  the dynamic  energy is less  relevant in  the overall
consumed energy and  lowering the frequency does not  return 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  Figure~\ref{fig:scales_comp}. The  opposite happens when  using a
higher  ratio for  static power,  the algorithm  proportionally  selects smaller
scaling  values which result  in less  energy saving  but also  less performance
degradation.


 \begin{table}[!t]
  \caption{The results of the 70\%-30\% power scenario}
  % title of Table
  \centering
  \begin{tabular}{|*{6}{r|}}
    \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}[!t]
  \caption{The results of the 90\%-10\% power scenario}
  % title of Table
  \centering
  \begin{tabular}{|*{6}{r|}}
    \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}[!t]
  \centering
  \subfloat[Comparison  between 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, $EDP=Energy\cdot Delay$ 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 consumption 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 provides 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
terms of  energy and  performance trade-off, 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}[!t]
 \caption{Comparing the proposed algorithm}
 \centering
\begin{tabular}{|*{7}{r|}}
\hline
Program & \multicolumn{2}{c|}{Energy saving \%} & \multicolumn{2}{c|}{Perf.  degradation \%} & \multicolumn{2}{c|}{Distance} \\ \cline{2-7} 
name    & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
CG      & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
MG      & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
LU      & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
EP      & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
BT      & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
SP      & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
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{Trade-off 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  trade-off)  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  platforms. 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 terms of energy-time trade-off.

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 platforms 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  other tasks  to finish  their  works.  The
development of such a 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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% LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
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