[mpi-energy2.git] / Heter_paper.tex

\documentclass[conference]{IEEEtran}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{algpseudocode}
\usepackage{graphicx}
\usepackage{algorithm}
\usepackage{subfig}
\usepackage{amsmath}
\usepackage{url}
\DeclareUrlCommand\email{\urlstyle{same}}

\usepackage[autolanguage,np]{numprint}
\AtBeginDocument{%
  \renewcommand*\npunitcommand[1]{\text{#1}}
  \npthousandthpartsep{}}

\usepackage{xspace}
\usepackage[textsize=footnotesize]{todonotes}
\newcommand{\AG}[2][inline]{%
  \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
\newcommand{\JC}[2][inline]{%
  \todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}

\newcommand{\Xsub}[2]{{\ensuremath{#1_\mathit{#2}}}}

%% used to put some subscripts lower, and make them more legible
\newcommand{\fxheight}[1]{\ifx#1\relax\relax\else\rule{0pt}{1.52ex}#1\fi}

\newcommand{\CL}{\Xsub{C}{L}}
\newcommand{\Dist}{\mathit{Dist}}
\newcommand{\EdNew}{\Xsub{E}{dNew}}
\newcommand{\Eind}{\Xsub{E}{ind}}
\newcommand{\Enorm}{\Xsub{E}{Norm}}
\newcommand{\Eoriginal}{\Xsub{E}{Original}}
\newcommand{\Ereduced}{\Xsub{E}{Reduced}}
\newcommand{\Es}{\Xsub{E}{S}}
\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}
\newcommand{\Fnew}{\Xsub{F}{new}}
\newcommand{\Ileak}{\Xsub{I}{leak}}
\newcommand{\Kdesign}{\Xsub{K}{design}}
\newcommand{\MaxDist}{\mathit{Max}\Dist}
\newcommand{\MinTcm}{\mathit{Min}\Tcm}
\newcommand{\Ntrans}{\Xsub{N}{trans}}
\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
\newcommand{\PdNew}{\Xsub{P}{dNew}}
\newcommand{\PdOld}{\Xsub{P}{dOld}}
\newcommand{\Pnorm}{\Xsub{P}{Norm}}
\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
\newcommand{\Tnew}{\Xsub{T}{New}}
\newcommand{\Told}{\Xsub{T}{Old}}

\begin{document}

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

\author{%
  \IEEEauthorblockN{%
    Jean-Claude Charr,
    Raphaël Couturier,
    Ahmed Fanfakh and
    Arnaud Giersch
  }
  \IEEEauthorblockA{%
    FEMTO-ST Institute, University of Franche-Comté\\
    IUT de Belfort-Montbéliard,
    19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
    % Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël
    % Fax: \mbox{+33 3 84 58 77 81}\\      % Dept Info
    Email: \email{{jean-claude.charr,raphael.couturier,ahmed.fanfakh_badri_muslim,arnaud.giersch}@univ-fcomte.fr}
   }
  }

\maketitle

\begin{abstract}
  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 may 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 frequency selecting algorithm for heterogeneous
  platforms (heterogeneous CPUs) is presented.  It selects the frequencies and 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 \np[\%]{34} while limiting the performance
  degradation as much as possible.  Finally, the algorithm is compared to an
  existing method, the comparison results show that it outperforms the
  latter, on average it saves \np[\%]{4} more energy while keeping the same performance. 

\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 may 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 may 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 may
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 may 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 $\mathit{energy}\times \mathit{delay}^2$ (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 \np[\%]{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.

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

The overall execution time of a distributed iterative synchronous application
over a heterogeneous platform consists of the sum of the computation time and
the communication time for every iteration on a node. However, due to the
heterogeneous computation power of the computing nodes, slack times may occur
when fast nodes have to wait, during synchronous communications, for the slower
nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
overall execution time of the program is the execution time of the slowest task
which has the highest computation time and no slack time.

Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
modern processors, that reduces the energy consumption of a CPU by scaling
down its voltage and frequency.  Since DVFS lowers the frequency of a CPU
and consequently its computing power, the execution time of a program running
over that scaled down processor may increase, especially if the program is
compute bound.  The frequency reduction process can be  expressed by the scaling
factor S which is the ratio between  the maximum and the new frequency of a CPU
as in (\ref{eq:s}).
\begin{equation}
  \label{eq:s}
  S = \frac{\Fmax}{\Fnew}
\end{equation}
The execution time of a compute bound sequential program is linearly
proportional to the frequency scaling factor $S$.  On the other hand, message
passing distributed applications consist of two parts: computation and
communication.  The execution time of the computation part is linearly
proportional to the frequency scaling factor $S$ but the communication time is
not affected by the scaling factor because the processors involved remain idle
during the communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  The
communication time for a task is the summation of periods of time that begin
with an MPI call for sending or receiving a message until the message is
synchronously sent or received.

Since in a heterogeneous 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 $i$ may 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
$\Tcp[i]$ may be different and different frequency scaling factors may be
computed in order to decrease the overall energy consumption of the application
and reduce slack times.  The communication time of a processor $i$ 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 do not evolve (increase/decrease) in the same way. According to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy
and the execution time simultaneously.  But the main objective is to produce
maximum energy reduction with minimum execution time reduction.

This problem can be solved by making the optimization process for energy and
execution time follow the same evolution according to the vector of scaling factors. 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}

\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 differences 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_{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 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.

\begin{figure}[!t]
  \centering
  \includegraphics[scale=0.5]{fig/start_freq}
  \caption{Selecting the initial frequencies}
  \label{fig:st_freq}
\end{figure}

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 highlighted in Figure~\ref{fig:st_freq}.  This set of
frequencies can be considered as a higher bound for the search space of the
optimal vector of frequencies because selecting 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 remaining 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. 
Finally, in a homogeneous platform the energy consumption is increased when the scaling factor is very high. 
Indeed, the dynamic energy  saved  by reducing the frequency of the processor is compensated by the significant increase of the execution time and thus the increased of the static energy. On the other hand, in a heterogeneous platform this is not the case.

\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 $O(F\cdot N)$, where $F$ is the
maximum number of available frequencies, 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.

\begin{table}[!t]
  \caption{Heterogeneous nodes characteristics}
  % title of Table
  \centering
  \begin{tabular}{|*{7}{r|}}
    \hline
    Node   & Simulated & Max   & Min   & Diff. & Dynamic    & Static    \\
    type   & GFLOPS    & Freq. & Freq. & Freq. & power      & power     \\
           &           & GHz   & GHz   & GHz   &            &           \\
    \hline
    1      & 40        & 2.50  & 1.20  & 0.100 & \np[W]{20} & \np[W]{4} \\
    \hline
    2      & 50        & 2.66  & 1.60  & 0.133 & \np[W]{25} & \np[W]{5} \\
    \hline
    3      & 60        & 2.90  & 1.20  & 0.100 & \np[W]{30} & \np[W]{6} \\
    \hline
    4      & 70        & 3.40  & 1.60  & 0.133 & \np[W]{35} & \np[W]{7} \\
    \hline
  \end{tabular}
  \label{table:platform}
\end{table}

\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 which 
is composed of synchronous message passing applications. 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 \np[\%]{80} of its dynamic power and the
remaining \np[\%]{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 \np[Gbit/s]{1} bandwidth.


\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 8 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, or 128 nodes.  The other benchmarks such as BT and SP had
to be executed on 1, 4, 9, 16, 36, 64, or 144 nodes.

\begin{table}[!t]
  
% \end{table}

  
% \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     &   5471.11     & 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}

\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}

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_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 numbers of nodes.  The experiments show that the algorithm
significantly reduces the energy consumption (up to \np[\%]{34}) 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 higher 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 \np[GB]{64}) 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.

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 CG benchmark significantly decreases 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 \np[\%]{80} composed of
dynamic power and of \np[\%]{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 \np[\%]{70} of dynamic power and \np[\%]{30} of static power
\item \np[\%]{90} of dynamic power and \np[\%]{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
\np[\%]{70}-\np[\%]{30} scenario is smaller for all benchmarks compared to the
energy saving of the \np[\%]{90}-\np[\%]{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 \np[\%]{70}-\np[\%]{30} scenario. On the other hand,
the performance degradation percentage is smaller in the \np[\%]{70}-\np[\%]{30}
scenario compared to the \np[\%]{90}-\np[\%]{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
\np[\%]{90}-\np[\%]{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 \np[\%]{70}-\np[\%]{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.  \np[\%]{70}-\np[\%]{30} scenario and \np[\%]{80}-\np[\%]{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 \np[\%]{70}-\np[\%]{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 \np[\%]{90}-\np[\%]{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      &   825.43        & 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{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   & 0.00 & 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
  \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}

\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}


\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, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$ 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 \np[\%]{29.76} energy
saving while their algorithm returns just \np[\%]{25.75}. The average of
performance degradation percentage is approximately the same for both
algorithms, about \np[\%]{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.

\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 \np[\%]{34} 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'').  Computations  have been performed  on the supercomputer
facilities  of the  Mésocentre de  calcul de  Franche-Comté. As  a  PhD student,
Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
supporting his work.

% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{15}

\bibliographystyle{IEEEtran}
\bibliography{IEEEabrv,my_reference}
\end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% fill-column: 80
%%% ispell-local-dictionary: "american"
%%% End:

% LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
% LocalWords:  CMOS EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex GPU
% LocalWords:  de badri muslim MPI SimGrid GFlops Xeon EP BT GPUs CPUs AMD
%  LocalWords:  Spiliopoulos scalability