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

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\begin{document}

\begin{frontmatter}



\title{Optimizing the Energy Consumption \\ 
of Message Passing Applications with Iterations \\ 
Executed over Grids} 
  



\author{Ahmed Fanfakh,
        Jean-Claude Charr,
        Raphaël Couturier,
        and Arnaud Giersch}

\address{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{{ahmed.fanfakh_badri_muslim,jean-claude.charr,raphael.couturier,arnaud.giersch}@univ-fcomte.fr}
   }

\begin{abstract}

  In recent years, green computing   has  become an important topic 
  in the supercomputing research domain. However, the 
  computing platforms are still  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 can be used to reduce the power consumption of the CPU 
  while computing, by lowering its frequency. 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 grids, composed of heterogeneous clusters, 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  application with iterations. 
  The algorithm has a small
  overhead and works without training or profiling. It uses a new energy model
  for message passing  applications with iterations running on a  grid. 
  The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
  running the NAS parallel benchmarks.  The experiments on 16 nodes, distributed on three clusters, show that it reduces  on average the
  energy consumption  by \np[\%]{30} while  the performance  is on average only degraded
   by \np[\%]{3.2}. Finally, the algorithm is 
  compared to an existing method. The comparison results show that it outperforms the
  latter in terms of energy consumption reduction and performance.
\end{abstract}


\begin{keyword}

Dynamic voltage and frequency scaling \sep Grid computing\sep Green computing and  frequency scaling online algorithm.

%% keywords here, in the form: keyword \sep keyword

%% MSC codes here, in the form: \MSC code \sep code
%% or \MSC[2008] code \sep code (2000 is the default)

\end{keyword}

\end{frontmatter}



\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 June 2015  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 2015
\cite{U.S_Annual.Energy.Outlook.2015}, 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 Shoubu-ExaScaler from RIKEN
which became the top of the Green500 list in June 2015 \cite{Green500_List}.
This heterogeneous platform executes more than 7 GFlops per watt while consuming
50.32 kilowatts.

Besides platform improvements, there are many software and hardware techniques
to lower the energy consumption of these platforms, such as DVFS, scheduling and other techniques.
 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} and \cite{pdsec2015}, a frequency selecting algorithm
was proposed to reduce the energy consumption of message passing 
applications with iterations running over homogeneous and heterogeneous clusters respectively.
The results of the experiments showed significant energy consumption
reductions. All the experimental results were conducted over the SimGrid
simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous  platforms, and to simulate the execution of message passing parallel
applications over them. 


This paper presents the following contributions :
\begin{enumerate}
\item two new energy and performance models for message passing 
  synchronous applications with iterations running over a heterogeneous grid platform. Both models
  take into account communications 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 grid
  platforms. The algorithm has a very small overhead and does not need any
  training nor profiling. It uses a new optimization function which
  simultaneously maximizes the performance and minimizes the energy consumption
  of a message passing  synchronous application with iterations.  The algorithm  was applied to the NAS
parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
\cite{grid5000}.

\end{enumerate}



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 grid 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 frequencies selecting algorithm.
Section~\ref{sec.expe} presents the results of applying the algorithm on the 
NAS parallel benchmarks and executing them on the Grid'5000 testbed. 
It also evaluates the algorithm over multi-core per node architectures and over three different power scenarios. Moreover, it 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.

In this paper, we are interested in reducing the energy consumption of message passing
 synchronous applications with iterations running over heterogeneous grid 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 workload 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.



\section{The performance and energy consumption measurements on heterogeneous grid architecture}
\label{sec.exe}

\subsection{The execution time of message passing distributed 
  applications with iterations on a heterogeneous platform}

In this paper, we are interested in reducing the energy consumption of message
passing distributed  synchronous applications with iterations running over
heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth 
and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.

The overall execution time of a distributed  synchronous application  with iterations running 
over a heterogeneous grid consists of the sum of the computation time and 
the communication time for every iteration on a node. 
However, nodes from distinct clusters in a grid have different computing powers, thus
while the application, 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. 
Therefore, the
overall execution time of the program is the execution time of the slowest task 
which has the highest computation time and almost no slack time. For example, in Figure \ref{fig:heter}, task 1 is the slower task and it does not have to wait for the other nodes to communicate with them because they all finish their computations before it.  

\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 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}
where $\Fmax$ is the maximum frequency before applying any DVFS and $\Fnew$  is the new frequency after applying DVFS.

The execution time of a compute bound sequential program is linearly
proportional to the frequency scaling factor $S$.  On the other hand, message
passing distributed applications consist of two parts: computation and
communication.  The execution time of the computation part is linearly
proportional to the frequency scaling factor $S$ but the communication time is
not affected by the scaling factor because the processors involved remain idle
during the communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  The
communication time for a task is the summation of periods of time that begin
with an MPI call for sending or receiving a message until the message is
synchronously sent or received.

Since in a heterogeneous grid each cluster has different characteristics,
especially different frequency gears, when applying DVFS operations on the nodes 
of these clusters, they may get different scaling factors represented by a scaling vector:
$(S_{11}, S_{12},\dots, S_{NM_i})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
be able to predict the execution time of message passing synchronous 
applications with iterations running over a heterogeneous grid, for different vectors of
scaling factors, the communication time and the computation time for all the
tasks must be measured during the first iteration before applying any DVFS
operation. Then the execution time for one iteration of the application with any
vector of scaling factors can be predicted using Equation (\ref{eq:perf}).
%
\begin{equation}
  \label{eq:perf}
  \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij}) 
  +\mathop{\min_{j=1,\dots,M_h}}  (\Tcm[hj])
\end{equation}
%
where $N$ is the number of  clusters in the grid, $M_i$ is the number of  nodes in
 cluster $i$, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$ 
and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the 
first  iteration.  The execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
and the  communication time of the slowest node without slack time during one iteration.
 The slowest node $h$ is the node which takes the  maximum execution time to execute an iteration  before scaling down its  frequency.
It means that only the communication time without any slack time is taken into account.
Therefore, the execution time of the  application is equal to
the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the
number of iterations of that application.

This model is an adaptation of the one developed in ~\cite{Our_first_paper}  which predicts the execution time
of message passing  applications with iterations running on homogeneous clusters. 
In a  homogeneous cluster only one scaling factor denoted as $S$ was used because all the nodes in the cluster have the same computing power.
In a heterogeneous cluster, each node may have a different scaling factor denoted as  $(S_i)$ where $i$ is the index of the node. In a grid, each node in each cluster may have a scaling factor. The whole set of scaling factors of all the computing nodes in the grid  is denoted by a two dimensional array of scales  
$(S_{11}, S_{12},\dots, S_{NM_i})$ where $N$ is the number of used clusters and $M_i$ is the number of nodes in cluster $i$.  

The execution time model, Equation \ref{eq:perf}, is used in the algorithm presented in section \ref{sec.optim}. The latter selects the scaling factors that optimize both the energy consumption and the performance of message passing applications with iterations running on grids.


\subsection{Energy model for heterogeneous grid platform}

Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
  Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
  Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
a processor into two power metrics: the static and the dynamic power.  While the
first one is consumed as long as the computing unit is turned on, the latter is
only consumed during computation times.  The dynamic power $\Pd$ is related to
the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
and operational frequency $F$, as shown in (\ref{eq:pd}).
\begin{equation}
  \label{eq:pd}
  \Pd = \alpha \cdot \CL \cdot V^2 \cdot F
\end{equation}
The static power $\Ps$ captures the leakage power as follows:
\begin{equation}
  \label{eq:ps}
   \Ps  = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak
\end{equation}
where V is the supply voltage, $\Ntrans$ is the number of transistors,
$\Kdesign$ is a design dependent parameter and $\Ileak$ is a
technology dependent parameter.  The energy consumed by an individual processor
to execute a given program can be computed as:
\begin{equation}
  \label{eq:eind}
  \Eind =  \Pd \cdot \Tcp + \Ps \cdot T
\end{equation}
where $T$ is the execution time of the program, $\Tcp$ is the computation
time and $\Tcp \le T$.  $\Tcp$ may be equal to $T$ if there is no
communication and no slack time.

The main objective of DVFS operation is to reduce the overall energy
consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.  The operational
frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot
F$ with some constant $\beta$. This equation is used to study the change of the
dynamic voltage with respect to various frequency values
in~\cite{Rauber_Analytical.Modeling.for.Energy}.  The reduction process of the
frequency can be expressed by the scaling factor $S$ which is the ratio between
the maximum and the new frequency as in (\ref{eq:s}).  The CPU governors are
power schemes supplied by the operating system's kernel to lower a core's
frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
follows:
\begin{equation}
  \label{eq:fnew}
   \Fnew = S^{-1} \cdot \Fmax
\end{equation}
Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
equation for dynamic power consumption:
\begin{multline}
  \label{eq:pdnew}
   \PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 \\
   {} = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}
\end{multline}
where $\PdNew$  and $\PdOld$ are the  dynamic power consumed with the
new frequency and the maximum frequency respectively.

According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
$S^{-3}$ when reducing the frequency by a factor of
$S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is
proportional to the frequency of a CPU, the computation time is increased
proportionally to $S$.  The new dynamic energy is the dynamic power multiplied
by the new time of computation and is given by the following equation:
\begin{equation}
  \label{eq:Edyn}
   \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot  \Tcp
\end{equation}
The static power is related to the power leakage of the CPU and is consumed
during computation and even when idle. As
in~\cite{Rauber_Analytical.Modeling.for.Energy, Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
the static power of a processor is considered as constant during idle and
computation periods, and for all its available frequencies.  The static energy
is the static power multiplied by the execution time of the program.  According
to the execution time model in (\ref{eq:perf}), the execution time of the
program is the sum of the computation and the communication times. The
computation time is linearly related to the frequency scaling factor, while this
scaling factor does not affect the communication time.  The static energy of a
processor after scaling its frequency is computed as follows:
\begin{equation}
  \label{eq:Estatic}
  \Es = \Ps \cdot (\Tcp \cdot S  + \Tcm)
\end{equation}

In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
different dynamic and static powers from the nodes of the other clusters, 
noted as $\Pd[ij]$ and $\Ps[ij]$ respectively.  Moreover, even if the distributed 
message passing  application with iterations is load balanced, the computation time of each CPU $j$ in cluster $i$
 noted $\Tcp[ij]$ may be slightly different due to the delay caused by the scheduler of the operating system. Therefore, different frequency scaling factors may be
computed in order to decrease the overall energy consumption of the application
and reduce the slack times.  The communication time of a processor $j$ in cluster $i$ is noted as
$\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
see Figure~\ref{fig:heter}.  Therefore, all nodes do not have equal
communication times. While the dynamic energy is computed according to the
frequency scaling factor and the dynamic power of each node as in
(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
of one iteration multiplied by the static power of each processor. 
 The CPU during the communication times consumes only the static power. While 
in the computation times, it consumes both the dynamic and the static powers, for more information refer to \cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.
The overall energy consumption of a message passing distributed application executed over a
heterogeneous grid platform during one iteration is the summation of all dynamic and
static energies for $M_i$ processors in $N$ clusters.  It is computed as follows:
\begin{multline}
  \label{eq:energy}
 E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot  \Tcp[ij])} +  
 \sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot {} \\
  (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij}) 
  +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))
\end{multline}


Reducing the frequencies of the processors according to the vector of scaling
factors $(S_{11}, S_{12},\dots, S_{NM_i})$ 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 a synchronous  application with iterations 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
works, \cite{Our_first_paper} and \cite{pdsec2015}, two methods that select the optimal
frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed. 
Both methods selects the frequencies that gives the best trade-off between 
energy consumption reduction and performance for  message passing
 synchronous applications with iterations.   In this work we
are interested in grids that are composed of heterogeneous clusters. The nodes from distinct clusters may have 
 different characteristics such  as  dynamic power, static power, computation power, 
frequencies range, network latency and bandwidth. 
Due to the heterogeneity of the processors, a vector of scaling factors should be selected
and it must give the best trade-off between energy consumption and performance.

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

\begin{equation}
  \label{eq:told}
   \Told = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} ) 
  +\mathop{\min_{j=1,\dots,M_h}}  (\Tcm[hj])    
\end{equation}

In the same way, the energy is normalized by computing the ratio between the
consumed energy while scaling down the frequency and the consumed energy with
maximum frequency for all  nodes:
%
\begin{equation}
  \label{eq:enorm}
  \Enorm = \frac{\Ereduced}{\Eoriginal} 
\end{equation}
%
where $\Ereduced$  is computed using (\ref{eq:energy}) and $\Eoriginal$ is 
computed as in (\ref{eq:eorginal}).
%
\begin{equation}
  \label{eq:eorginal}
    \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M_i} ( \Pd[ij] \cdot  \Tcp[ij])  + 
     \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M_i} (\Ps[ij] \cdot \Told)       
\end{equation}

While the main goal is to optimize the energy and execution time at the same
time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. 
According to (\ref{eq:pnorm}) and (\ref{eq:enorm}), the
vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduces both the energy
and the execution time,  but the main objective is to produce
maximum energy reduction with minimum execution time reduction.

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

\begin{figure}
  \centering
  \subfloat[Homogeneous cluster]{%
    \includegraphics[width=.48\textwidth]{fig/homo}\label{fig:r1}} \hspace{0.4cm}%
  \subfloat[Heterogeneous grid]{%
    \includegraphics[width=.48\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} and
Figure~\ref{fig:r2}. Then the objective function has the following form:
\begin{equation}
  \label{eq:max}
  \MaxDist =
\mathop{  \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}}_{k=1,\dots,F_j}
      (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
       \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
\end{equation}
where $N$ is the number of clusters, $M_i$ is the number of nodes in the cluster $i$ and
$F_j$ is the number of available frequencies in the node $j$.  Then, the optimal set 
of scaling factors that satisfies (\ref{eq:max}) can be selected.  
The objective function can work with any energy model or any power 
values for each node (static and dynamic powers). However, the most important 
energy reduction gain can be achieved when the energy curve has a convex form as shown 
in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.

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

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

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

\begin{algorithm}
  \begin{algorithmic}[1]
    % \footnotesize
    \For {$k=1$ to \textit{some iterations}}
      \State Computations section.
      \State Communications section.
      \If {$(k=1)$}
        \State Gather all times of computation and communication from each node.
        \State Call Algorithm \ref{HSA}.
        \State Compute the new frequencies from the\newline\hspace*{3em}%
               returned optimal scaling factors.
        \State Set the new frequencies to nodes.
      \EndIf
    \EndFor
  \end{algorithmic}
  \caption{DVFS algorithm}
  \label{dvfs}
\end{algorithm}


In this section, the scaling factors selection algorithm for  grids, Algorithm~\ref{HSA}, 
is presented. It selects the vector of  frequency
scaling factors  that gives the best trade-off between minimizing the
energy consumption and maximizing the performance of a message passing
synchronous  application with iterations executed on a  grid. It works
online during the execution time of the  application.  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 application.

\begin{figure}[!t]
  \centering
  \includegraphics[scale=0.6]{fig/init_freq}
  \caption{Selecting the initial frequencies in a grid platform}
  \label{fig:st_freq}
\end{figure}



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

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

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


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

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

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


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

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

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

In the experiments presented in the following sections, two sites of Grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as shown on Figure~\ref{fig:grid5000}.

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


\begin{figure}[!t]
  \centering
  \includegraphics[scale=1]{fig/grid5000}
  \caption{The selected two sites of Grid'5000}
  \label{fig:grid5000}
\end{figure}
\begin{figure}[!t]
  \centering
  \includegraphics[scale=0.6]{fig/power_consumption.pdf}
  \caption{The power consumption by one core from the Taurus cluster}
  \label{fig:power_cons}
\end{figure}


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


  
\begin{table}[!t]
  \caption{The characteristics of the CPUs in the selected clusters}
  % title of Table
  \centering
  \begin{tabular}{|*{7}{c|}}
    \hline
                &             & Max   & Min   & Diff. &                 &               \\
    Cluster     & CPU         & Freq. & Freq. & Freq. & Cores           & Dynamic power \\
    Name        & model       & GHz   & GHz   & GHz   & per CPU         & of one core   \\
    \hline
                & Intel       &       &       &         &           &              \\
    Taurus      & Xeon        & 2.3   & 1.2   & 0.1     & 6         & \np[W]{35}    \\
                & E5-2630     &       &       &         &           &            \\         
    \hline
                & Intel       &       &       &         &           &             \\
    Graphene    & Xeon        & 2.53  & 1.2   & 0.133   & 4         & \np[W]{23}  \\
                & X3440       &       &       &         &           &             \\    
    \hline
                & Intel       &       &       &         &           &            \\
    Griffon     & Xeon        & 2.5   & 2     & 0.5     & 4         & \np[W]{46}  \\
                & L5420       &       &       &         &           &            \\  
    \hline
                & Intel       &       &       &         &           &            \\
     Graphite   & Xeon        & 2     & 1.2   & 0.1     & 8         & \np[W]{35} \\
                & E5-2650     &       &       &         &           &            \\  
    \hline
  \end{tabular}
  \label{table:grid5000}
\end{table} 



\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
In this section, the results of the application of the scaling factors selection algorithm \ref{HSA} 
to the NAS parallel benchmarks are presented. Each experiment  has been executed many times and the results presented in the figures are the average values of many executions. As mentioned previously, the experiments 
were conducted over two sites of Grid'5000,  Lyon and Nancy sites. 
Two scenarios were considered while selecting the clusters from these two sites :
\begin{itemize}
\item In the first scenario, nodes from two sites and three heterogeneous clusters were selected. The two sites are connected 
 via a long distance network.
\item In the second scenario nodes from three clusters located in one site, Nancy site, were selected.  
\end{itemize}

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

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

\begin{table}[h]

\caption{The different grid scenarios}
\centering
\begin{tabular}{|*{4}{c|}}
\hline
\multirow{2}{*}{Scenario name}        & \multicolumn{3}{c|} {The participating clusters} \\ \cline{2-4} 
                                      & Cluster & Site           & Nodes per cluster     \\ 
\hline
\multirow{3}{*}{Two sites / 16 nodes} & Taurus & Lyon                & 5                      \\ \cline{2-4} 
                                      & Graphene  & Nancy             & 5                      \\ \cline{2-4} 
                                      & Griffon       & Nancy        & 6                      \\ 
\hline
\multirow{3}{*}{Two sites / 32 nodes} & Taurus  & Lyon               & 10                     \\ \cline{2-4} 
                                      & Graphene  & Nancy             & 10                     \\ \cline{2-4} 
                                      & Griffon     &Nancy           & 12                     \\ 
\hline
\multirow{3}{*}{One site / 16 nodes}  & Graphite    & Nancy            & 4                      \\ \cline{2-4} 
                                      & Graphene     & Nancy           & 6                      \\ \cline{2-4} 
                                      & Griffon         & Nancy        & 6                      \\ 
\hline
\multirow{3}{*}{One site / 32 nodes}  & Graphite   & Nancy             & 4                      \\ \cline{2-4} 
                                      & Graphene      & Nancy          & 14                     \\ \cline{2-4} 
                                      & Griffon          & Nancy       & 14                       \\ 
\hline
\end{tabular}
 \label{tab:sc}
\end{table}

\begin{figure}
  \centering
  \subfloat[The energy consumption by the nodes wile executing the NAS benchmarks over different scenarios    
           ]{%
    \includegraphics[width=.48\textwidth]{fig/eng_con_scenarios.eps}\label{fig:eng_sen}} \hspace{0.4cm}%
  \subfloat[The execution times of the NAS benchmarks over different scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/time_scenarios.eps}\label{fig:time_sen}}
  \label{fig:exp-time-energy}
  \caption{The  energy consumption and execution time of NAS  Benchmarks over different scenarios}
\end{figure}

The NAS parallel benchmarks are executed over these two platforms
 with different number of nodes, as in Table~\ref{tab:sc}. 
The overall energy consumption of all the benchmarks solving the class D instance and
using the proposed frequency selection algorithm is measured 
using the equation of the reduced energy consumption, Equation~\ref{eq:energy}. This model uses the measured dynamic power showed in Table~\ref{table:grid5000}
and the static 
power is assumed to be equal to 20\% of the dynamic power as in \cite{Rauber_Analytical.Modeling.for.Energy}. The execution
time is measured for all the benchmarks over these different scenarios.  

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

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

The execution times of these benchmarks 
over one site with 16 and 32 nodes are also lower than those of the  two sites 
scenario. Moreover, most of the benchmarks running over the one site scenario have their execution times  approximately halved  when the number of computing nodes is doubled from 16 to 32 nodes (linear speed up according to the number of the nodes).

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


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

\begin{figure*}[!h]
  \centering
  \subfloat[The energy reduction while executing the NAS benchmarks over different scenarios ]{%
    \includegraphics[width=.48\textwidth]{fig/eng_s.eps}\label{fig:eng_s}} \hspace{0.4cm}%
  \subfloat[The performance degradation of the NAS benchmarks over different scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/per_d.eps}\label{fig:per_d}}\hspace{0.4cm}%
    \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks  
      over different scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/dist.eps}\label{fig:dist}}
  \label{fig:exp-res}
  \caption{The experimental results of different scenarios}
\end{figure*}



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

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


Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
The performance degradation percentage for the benchmarks running on two sites  with
16 or 32  nodes is on average equal to 8.3\% or 4.7\% respectively. 
For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes  without significantly degrading their performance because the communication times are high with 32 nodes which results in smaller  computations to communications ratio.  On the other hand, the performance degradation percentage  for the benchmarks running  on one site  with
16 or 32  nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing 
nodes when the communications occur in high speed network does not decrease the computations to 
communication ratio. 

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

Figure \ref{fig:dist} presents the  distance percentage between the energy saving  and the performance degradation for each benchmark  over both  scenarios. The trade-off distance percentage can be 
computed as in Equation~\ref{eq:max}. The one site scenario with 16 nodes gives the best energy and performance 
trade-off, on average it is equal to  26.8\%. The one site scenario using both 16 and 32 nodes had better energy and performance 
trade-off comparing to the two sites scenario  because the former has high speed local communications 
which increase the computations to communications ratio  and the latter uses long distance communications which decrease this ratio. 

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




\subsection{The experimental results over multi-core clusters}
\label{sec.res-mc}

The  clusters of Grid'5000 have different number of cores embedded in their nodes
as shown in Table~\ref{table:grid5000}. In 
this section, the proposed scaling algorithm is evaluated over the  Grid'5000 platform  while using multi-cores nodes selected according to the one site scenario described in  Section~\ref{sec.res}.
The one site scenario uses  32 cores from multi-core nodes instead of 32 distinct nodes. For example if 
the participating number of cores from a certain cluster is equal to 14, 
in the multi-core scenario  4 nodes are selected and
3 or 4 cores from each node are used. The platforms with one  
core per node and  multi-core nodes are  shown in Table~\ref{table:sen-mc}. 
The energy consumptions and execution times of running the class D of the NAS parallel 
benchmarks over these two different platforms are presented 
in Figures \ref{fig:eng-cons-mc} and \ref{fig:time-mc} respectively.


\begin{table}[!h]
\centering
\caption{The multi-core scenarios}
\begin{tabular}{|*{4}{c|}}
\hline
Scenario name                          & Cluster name & Nodes per cluster & 
                                       Cores per node  \\ \hline
\multirow{3}{*}{One core per node}    & Graphite     & 4               & 1                   \\  \cline{2-4}
                                       & Graphene     & 14              & 1                   \\  \cline{2-4}
                                       & Griffon      & 14              & 1                   \\ \hline
\multirow{3}{*}{Multi-core per node}  & Graphite     & 1               &  4              \\  \cline{2-4}
                                       & Graphene     & 4               & 3 or 4              \\  \cline{2-4}
                                       & Griffon      & 4               & 3 or 4                   \\ \hline
\end{tabular}
\label{table:sen-mc}
\end{table}


\begin{figure}[!h]
  \centering
  \subfloat[Comparing the  execution times of running the NAS benchmarks over one core and multi-core scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/time.eps}\label{fig:time-mc}} \hspace{0.4cm}%
  \subfloat[Comparing the  energy consumptions of running the NAS benchmarks over one core and multi-core scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/eng_con.eps}\label{fig:eng-cons-mc}}
    \label{fig:eng-cons}
  \caption{The energy consumptions and execution times of the  NAS benchmarks running over one core and multi-core per node architectures}
\end{figure}



The execution times for most of  the NAS  benchmarks are higher over the multi-core per node scenario 
than over the single core per node  scenario. Indeed,  
 the communication times  are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node  share  the same node network link which can be  saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.    
Moreover, the energy consumptions of the NAS benchmarks are lower over the 
 one core scenario  than over the multi-core scenario because 
the first scenario had less execution time than the latter which results in less static energy being consumed.
The computations to communications ratios of the NAS benchmarks are higher over 
the one site one core scenario  when compared to the ratio of the multi-core scenario. 
More energy reduction can be gained when this ratio is big because it pushes the proposed scaling algorithm to select smaller frequencies that decrease the dynamic power consumption. These experiments also showed that the energy 
consumption and the execution times of the EP and MG benchmarks do not change significantly over these two
scenarios  because there are no or small communications. Contrary to EP and MG, the  energy consumptions and the execution times of the rest of the  benchmarks  vary according to the  communication times that are different from one scenario to the other.
\begin{figure*}[t]
  \centering
    \subfloat[The energy saving of running NAS benchmarks over one core and multicore scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/eng_s_mc.eps}\label{fig:eng-s-mc}} \hspace{0.4cm}%
    \subfloat[The performance degradation of running NAS benchmarks over one core and multi-core scenarios
      ]{%
    \includegraphics[width=.48\textwidth]{fig/per_d_mc.eps}\label{fig:per-d-mc}}\hspace{0.4cm}%
    \subfloat[The trade-off distance of running NAS benchmarks over one core and multicore scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/dist_mc.eps}\label{fig:dist-mc}}
  \label{fig:exp-res2}
  \caption{The experimental results of one core and multi-core scenarios}
\end{figure*}  
  
The energy saving percentages of all the NAS benchmarks running over these two scenarios are presented in Figure~\ref{fig:eng-s-mc}. 
The figure shows that  the energy saving percentages in the one 
core and the multi-core scenarios
are approximately equivalent, on average they are equal to  25.9\% and 25.1\% respectively.
The energy consumption is reduced at the same rate in the two scenarios when compared to the energy consumption of the executions without DVFS. 


The performance degradation percentages of the NAS benchmarks are presented in
Figure~\ref{fig:per-d-mc}. It shows that the performance degradation percentages are higher for the NAS benchmarks executed over the  one core per node scenario  (on average equal to 10.6\%)  than over the  multi-core scenario (on average equal to 7.5\%). The performance degradation percentages over the multi-core scenario are lower because  the computations to communications ratios are smaller than the ratios of the other scenario. 

The trade-off distances percentages of the NAS benchmarks over both scenarios are presented 
in ~Figure~\ref{fig:dist-mc}. These  trade-off distances between energy consumption reduction and performance  are used to verify which scenario is the best in both terms  at the same time. The figure shows that  the  trade-off distance percentages are on average   bigger over the multi-core scenario  (17.6\%) than over the  one core per node scenario  (15.3\%).







\subsection{Experiments with different static power scenarios}
\label{sec.pow_sen}

In Section~\ref{sec.grid5000}, since it was not possible to measure the static power consumed by a CPU,   the static power was assumed to be equal to 20\% of the measured dynamic power. This power is consumed during the whole execution time, during computation and communication times. Therefore, when the DVFS operations are applied by the scaling algorithm and the CPUs' frequencies lowered, the execution time might increase and consequently the consumed static energy will be increased too. 

The aim of  this section is to evaluate the scaling algorithm while assuming different values of static powers. 
In addition to the previously used  percentage of static power, two new static power ratios,  10\% and 30\% of the measured dynamic power of the core, are used in this section.
The experiments have been executed with these two new static power scenarios  over the one site one core per node scenario.
In these experiments, the class D of the NAS parallel benchmarks were executed over the Nancy site. 16 computing nodes from the three clusters, Graphite, Graphene and Griffon, were used in this experiment. 


\begin{figure*}[t]
  \centering
  \subfloat[The energy saving percentages for the nodes executing the NAS benchmarks over the three power scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/eng_pow.eps}\label{fig:eng-pow}} \hspace{0.4cm}%
  \subfloat[The performance degradation percentages for the NAS benchmarks over the three power scenarios]{%
    \includegraphics[width=.48\textwidth]{fig/per_pow.eps}\label{fig:per-pow}}\hspace{0.4cm}%
    \subfloat[The trade-off distance between the energy reduction and the performance of the NAS benchmarks over the three power scenarios]{%
      
    \includegraphics[width=.48\textwidth]{fig/dist_pow.eps}\label{fig:dist-pow}}
  \label{fig:exp-pow}
  \caption{The experimental results of different static power scenarios}
\end{figure*}



\begin{figure}
  \centering
  \includegraphics[scale=0.5]{fig/three_scenarios.pdf}
  \caption{Comparing the selected frequency scaling factors for the MG benchmark over the three static power scenarios}
  \label{fig:fre-pow}
\end{figure}

The energy saving percentages of the NAS benchmarks with the three static power scenarios are presented 
in Figure~\ref{fig:eng-pow}. This figure shows that the  10\% of static power scenario 
gives the biggest energy saving percentages in comparison to the 20\% and 30\% static power 
scenarios. The small value of the static power consumption makes the proposed 
scaling algorithm  select smaller frequencies for the CPUs. 
These smaller frequencies reduce the dynamic energy consumption more than increasing the consumed static energy which gives             less overall energy consumption. 
The energy saving percentages of the 30\% static power scenario is the smallest between the other scenarios, because the scaling algorithm selects bigger frequencies for the CPUs which increases the energy consumption. Figure \ref{fig:fre-pow} demonstrates that the proposed scaling algorithm selects   the best frequency scaling factors  according to the static power consumption ratio being used.

The performance degradation percentages are presented in Figure~\ref{fig:per-pow}.
The 30\% static power scenario had less performance degradation percentage  because the scaling algorithm
had  selected big frequencies for the CPUs. While, 
the inverse happens in the 10\% and 20\% scenarios because the scaling algorithm had selected  CPUs' frequencies smaller than those of the 30\% scenario. The trade-off distance percentage for the NAS benchmarks with these three static power scenarios 
are presented in Figure~\ref{fig:dist-pow}. 
It shows that the best  trade-off
distance percentage is obtained with  the  10\% static power scenario  and this percentage 
is decreased for the other two scenarios because the scaling algorithm had selected different frequencies according to the static power values.

In the EP benchmark, the energy saving, performance degradation and trade-off 
distance percentages for these static power scenarios are not significantly different because there is no communication in this benchmark. Therefore, the static power is only consumed during computation and   the proposed scaling algorithm selects similar frequencies for the three scenarios.  On the other hand,  for the rest of the benchmarks,  the scaling algorithm  selects  the values of the frequencies according to the communication times of each benchmark because the static energy consumption increases  proportionally to the  communication times.


 
\subsection{Comparison of the proposed frequencies selecting algorithm }
\label{sec.compare_EDP}

Finding the frequencies that give the best trade-off between the energy consumption and the performance for a parallel 
application is not a trivial task.  Many algorithms have been proposed to tackle this problem.  
In this section, the proposed frequencies selecting algorithm is compared to a method that uses the well known  energy and delay product objective function, $EDP=energy \times delay$, that has been used by many researchers  \cite{EDP_for_multi_processors,Energy_aware_application_scheduling,Exploring_Energy_Performance_TradeOffs}. 
This objective function  was also used by Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS} where they select the frequencies that minimize the EDP product and apply them with DVFS operations to  the multi-core 
architecture. Their online algorithm predicts the energy consumption and execution time of a processor before using the EDP method.
\begin{figure*}[t]
  \centering
  \subfloat[The energy reduction induced by the Maxdist method and the EDP method]{%
    \includegraphics[width=.48\textwidth]{fig/edp_eng}\label{fig:edp-eng}} \hspace{0.4cm}%
    \subfloat[The performance degradation induced by  the Maxdist method and the EDP method]{%
    \includegraphics[width=.48\textwidth]{fig/edp_per}\label{fig:edp-perf}}\hspace{0.4cm}%
    \subfloat[The trade-off distance between the energy consumption reduction and the performance for the Maxdist method and the  EDP method]{%
    \includegraphics[width=.48\textwidth]{fig/edp_dist}\label{fig:edp-dist}}
  \label{fig:edp-comparison}
  \caption{The comparison results}
\end{figure*}
To fairly compare the proposed frequencies scaling algorithm to  Spiliopoulos et al. algorithm, called Maxdist and EDP respectively, both algorithms use the same energy model,  Equation~\ref{eq:energy} and
execution time model, Equation~\ref{eq:perf}, to predict the energy consumption and the execution time for each computing node.
Moreover, both algorithms start the search space from the upper bound computed as in Equation~\ref{eq:Fint}.
Finally, the resulting EDP algorithm is an exhaustive search algorithm that tests all the possible frequencies, starting from the initial frequencies (upper bound), 
and selects the vector of frequencies that minimize the EDP product.

Both algorithms were applied to the class D of the NAS benchmarks running over 16 nodes.
The participating computing nodes are distributed  according to the two scenarios described in  Section~\ref{sec.res}. 
The experimental results, the energy saving, performance degradation and trade-off distance percentages, are 
presented in  Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.

As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios. 
The proposed algorithm gives better results than the EDP method because the former selects the set of frequencies that  
gives the best tradeoff between energy saving and performance. 
Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
The high positive values of the trade-off distance percentage mean that the  energy saving percentage is much higher than the performance degradation percentage. 
The  complexity of both algorithms, Maxdist and EDP,  are of order $O(N \cdot M_i \cdot F_j)$ and 
$O(N \cdot M_i \cdot F_j^2)$ respectively, where $N$ is the number of the clusters, $M_i$ is the number of nodes and $F_j$ is the 
maximum number of available frequencies. When Maxdist is applied to a benchmark that is being executed over 32 nodes distributed between Nancy and Lyon sites, it takes on average  $0.01$  $ms$  to compute the best frequencies while the EDP method is on average ten times slower over the same architecture.  


\section{Conclusion}
\label{sec.concl}
This paper presents a new online frequencies selection algorithm.
 The algorithm selects the best vector of 
frequencies that maximizes  the trade-off distance 
between the predicted energy consumption and the predicted execution time of the distributed 
 applications with iterations running over a heterogeneous grid. A new energy model 
is used by the proposed algorithm to predict the energy consumption 
of the application.
To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the  
NAS parallel benchmarks   and the  class D instance was executed over the  Grid'5000 testbed platform. 
The experiments executed on 16 nodes, distributed over three clusters, showed that the algorithm   on average reduces by 30\% the energy consumption
for all the NAS benchmarks   while  on average only degrading by 3.2\%   the performance. 
The Maxdist algorithm was also evaluated in different scenarios that vary in the distribution of the computing nodes between different clusters' sites or  use multi-core per node architecture or consume different static power values. The algorithm selects different vectors of frequencies according to the 
computations and communication times ratios, and  the values of the static and measured dynamic powers of the CPUs. 
Finally, the proposed algorithm was compared to another method that uses
the well known energy and delay product as an objective function. The comparison results showed 
that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better trade-off  between energy consumption reduction and performance. 

In the near future, we will adapt the proposed algorithm to take into consideration the variability between some iterations.  For example, the proposed algorithm can be executed twice: after the first iteration the frequencies are scaled down according to the execution times measured in the first iteration, then after a fixed number of iterations, the frequencies are adjusted according to the execution times measured during the fixed number of iterations. If the  computing power of the system is constantly changing, it would be interesting to implement a mechanism that detects this change and adjusts the frequencies according to the variability of the system. We would like also to develop a similar method that is adapted to
asynchronous  applications with iterations where iterations are not synchronized and communications are overlapped with computations. 
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. Finally, it would be interesting to evaluate the scalability of the proposed algorithm by running it on large platforms composed of many thousands of cores. The scalability of the algorithm can be improved by distributing it  in a hierarchical manner where a leader is chosen for each cluster or a group of nodes  to compute their scaled frequencies and by using asynchronous messages to exchange the the data measured at the first iteration. 



\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 Grid'5000 platform. As  a  PhD student,
Mr. Ahmed  Fanfakh, would  like to  thank the University  of Babylon  (Iraq) for
supporting his work.

%\section*{References}
\bibliography{my_reference}

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