adding ref. keys

[mpi-energy2.git] / Heter_paper.tex

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\begin{document} 

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

\maketitle

\begin{abstract}
  
\end{abstract}

\section{Introduction}
\label{sec.intro}
Modern processors continue increasing in a performance. 
The CPUs constructors are competing to achieve maximum number 
of floating point operations per second (FLOPS). 
Thus, the energy consumption and the heat dissipation are increased 
drastically according to this increase. Because the number of FLOPS 
is linearly related to the power consumption of a CPU
~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}.  
As an example of the more power hungry cluster, Tianhe-2 became in 
the top of the Top500 list in June 2014 \cite{TOP500_Supercomputers_Sites}. 
It has more than 3 millions of cores and consumed more than 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, we can consider the price of the energy consumption for the 
Tianhe-2 platform is approximately more than \$10 millions  for 
one year. For this reason, the heterogeneous clusters must be offer more 
energy efficiency due to the increase in the energy cost and the environment 
influences. Therefore, a green computing clusters with maximum number of 
FLOPS per watt are required nowadays. For example, the GSIC center of Tokyo, 
became the top of the Green500 list in June 2014 \cite{Green500_List}. 
This platform has more than four thousand of  MFLOPS per watt. Dynamic 
voltage and frequency scaling (DVFS) is a process used widely to reduce the energy 
consumption of the processor. In a heterogeneous clusters enabled DVFS, many researchers 
used DVFS  in a different ways. DVFS can be minimized the energy consumption 
but it leads to a disadvantage due to increase in performance degradation. 
Therefore,  researchers used different optimization strategies to overcame 
this problem. The best tradeoff relation between the energy reduction and 
performance degradation ratio is became a key challenges in a heterogeneous 
platforms. In this paper we are propose a heterogeneous scaling algorithm  
that selects the optimal vector of the frequency scaling factors for distributed 
iterative application, producing maximum energy reduction against minimum 
performance degradation ratio simultaneously. The algorithm has very small 
overhead, works online and not needs for 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 MPI programs can be predicted.  It also presents an energy
model for heterogeneous platforms. 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 heterogeneous scaling algorithm.
Section~\ref{sec.expe} presents the results of running  the NAS benchmarks on 
the proposed heterogeneous platform. It also shows the comparison of three 
different power scenarios and it verifies the precision of the proposed algorithm.  
Finally, we conclude in Section~\ref{sec.concl} with a summary and some future works.

\section{Related works}
\label{sec.relwork}
Energy reduction process for a high performance clusters recently performed using 
dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique enabled 
in a modern processors to scaled down both of the  voltage and the frequency of 
the CPU while it is in the computing mode to reduce the energy consumption. DVFS is 
also  allowed in the graphical processors GPUs, to achieved the same goal. Applying 
DVFS has a dramatical side effect if it is applied to minimum levels to gain more 
energy reduction, producing a high percentage of performance degradations for the 
parallel applications.  Many researchers used different strategies to solve this 
nonlinear problem for example in
~\cite{Hao_Learning.based.DVFS,Dhiman_Online.Learning.Power.Management}, their methods 
add big overheads to the algorithm to select the suitable frequency.  
In this paper we  present a method 
to find the optimal set of frequency scaling factors for a heterogeneous cluster to 
simultaneously optimize both the energy and the execution time  without adding a big 
overhead. This work is developed from our previous work of a homogeneous cluster~\cite{Our_first_paper}. 
Therefore we are interested to present some works that concerned the heterogeneous clusters 
enabled DVFS. In general, the heterogeneous cluster works fall into two categorizes: 
GPUs-CPUs heterogeneous clusters and CPUs-CPUs heterogeneous clusters. In GPUs-CPUs 
heterogeneous clusters some parallel tasks executed on a GPUs and the others executed 
on a CPUs. As an example of this works, 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 is to determined the 
energy efficiency as a function of performance per watt, the best tradeoff is done when the 
performance per watt function is maximized. In the work of Kia Ma et al.
~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, They developed a scheduling 
algorithm to distributed different workloads proportional to the computing power of the node 
to be executed on a CPU or a GPU, emphasize all tasks must be finished in the same time. 
Recently, Rong et al.~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, Their study explain that 
a heterogeneous clusters enabled DVFS using GPUs and CPUs gave better energy and performance 
efficiency than other clusters composed of only CPUs. 
The CPUs-CPUs heterogeneous clusters consist of number of computing nodes  all of the type CPU. 
Our work in this paper can be classified to this type of the clusters. 
As an example of this works see  Naveen et al.~\cite{Naveen_Power.Efficient.Resource.Scaling} work, 
They developed a policy to dynamically assigned the frequency to a heterogeneous cluster. 
The goal is to minimizing a fixed metric of $energy*delay^2$. Where our proposed method is automatically 
optimized  the relation between the energy and the delay of the iterative applications. 
Other works such as Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling}, 
their algorithm divided the executed tasks into two types: the critical and 
non critical tasks. The algorithm scaled down the frequency of the non critical tasks 
as function to the  amount of the slack and communication times that 
have with maximum of performance degradation percentage of 10\%. In our method there is no 
fixed bounds for performance degradation percentage and the bound is dynamically computed 
according to the energy and the performance tradeoff relation of the executed application. 
There are some approaches used a heterogeneous cluster composed from two different types 
of Intel and AMD processors such as~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS} 
and \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, they predicated  both the energy 
and the performance for each frequency gear, then the algorithm selected the best gear that gave 
the best tradeoff. In contrast our algorithm works over a heterogeneous  platform composed of 
four different types of processors. Others approaches such as 
\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, 
they are selected the best frequencies for a specified heterogeneous clusters offline using some 
heuristic methods. While our proposed algorithm works online during the execution time of 
iterative application. Greedy dynamic approach used by Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements},  
minimized the power consumption of a heterogeneous severs  with time/space complexity, this approach 
had considerable overhead. In our proposed scaling algorithm has very small overhead and 
it is works without any previous analysis for the application time complexity. 

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

% \JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'',
%   can be deleted if we need space, we can just say we are interested in this
%   paper in homogeneous clusters}

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

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

Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in 
modern processors, that reduces the energy consumption of a CPU by scaling 
down its voltage and frequency.  Since DVFS lowers the frequency of a CPU 
and consequently its computing power, the execution time of a program running 
over that scaled down processor might increase, especially if the program is 
compute bound.  The frequency reduction process can be  expressed by the scaling 
factor S which is the ratio between  the maximum and the new frequency of a CPU 
as in EQ (\ref{eq:s}).
\begin{equation}
  \label{eq:s}
 S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
 The execution time of a compute bound sequential program is linearly proportional 
 to the frequency scaling factor $S$.  On the other hand,  message passing 
 distributed applications consist of two parts: computation and communication. 
 The execution time of the computation part is linearly proportional to the 
 frequency scaling factor $S$ but  the communication time is not affected by the 
 scaling factor because  the processors involved remain idle during the  
 communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}. 
 The communication time for a task is the summation of  periods of 
 time that begin with an MPI call for sending or receiving   a message 
 till 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 EQ (\ref{eq:perf}).
\begin{equation}
  \label{eq:perf}
 \textit  T_\textit{new} = 
 \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) +  MinTcm
\end{equation}
where $TcpOld_i$ 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. 
Therefore, we can consider the execution time of the iterative application is 
equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied 
by the number of iterations of that application.

This prediction model is based on our model for predicting the execution time of 
message passing distributed applications for homogeneous architectures~\cite{Our_first_paper}. 
The execution time prediction model is used in our method for optimizing both 
energy consumption and 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 $P_{d}$ is related to the switching
activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
operational frequency $F$, as shown in EQ(\ref{eq:pd}).
\begin{equation}
  \label{eq:pd}
  P_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F
\end{equation}
The static power $P_{s}$ captures the leakage power as follows:
\begin{equation}
  \label{eq:ps}
   P_\textit{s}  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
\end{equation}
where V is the supply voltage, $N_{trans}$ is the number of transistors,
$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
technology-dependent parameter.  The energy consumed by an individual processor
to execute a given program can be computed as:
\begin{equation}
  \label{eq:eind}
   E_\textit{ind} =  P_\textit{d} \cdot Tcp + P_\textit{s} \cdot T
\end{equation}
where $T$ is the execution time of the program, $T_{cp}$ is the computation
time and $T_{cp} \leq T$.  $T_{cp}$ 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 EQ~(\ref{eq:s}).
The CPU governors are power schemes supplied by the operating
system's kernel to lower a core's frequency. we can calculate the new frequency 
$F_{new}$ from EQ(\ref{eq:s}) as follow:
\begin{equation}
  \label{eq:fnew}
   F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following 
equation for dynamic power consumption:
\begin{multline}
  \label{eq:pdnew}
   {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
   {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}
\end{multline}
where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the 
new frequency and the maximum frequency respectively.

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

In the considered heterogeneous platform, each processor $i$ might have different dynamic and 
static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed 
message passing iterative application is load balanced, the computation time of each CPU $i$ 
noted $Tcp_{i}$ might be different and different frequency  scaling factors might be computed 
in order to decrease the overall energy consumption of the application and reduce the slack times. 
The communication time of a processor $i$ is noted as $Tcm_{i}$ and could contain slack times 
if it is 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 EQ(\ref{eq:Edyn}), the static energy is 
computed as the sum of the execution time of each processor multiplied by its static power. 
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}. We can measure the overall energy consumption for the iterative 
application by measuring  the energy consumption for one iteration as in EQ(\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 gives 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 and thus the execution time might be drastically increased during which 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) in order to reduce the overall energy consumption and not 
significantly increase 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, not 
one but 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 of the energy with the frequency scaling
factors is nonlinear, for more details refer to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.  
Moreover, they are not measured using the same metric.  To solve this problem, we normalize the
execution time by computing the ratio between the new execution time (after 
scaling down the frequencies of some processors) and the initial one (with maximum 
frequency for all nodes,) as follows:
\begin{multline}
  \label{eq:pnorm}
  P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
       {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +MinTcm}
           {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
\end{multline}


In the same way, we normalize the energy by computing the ratio between the consumed energy 
while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
\begin{multline}
  \label{eq:enorm}
  E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
 \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
 \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
\end{multline} 
Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).

 While the main 
goal is to optimize the energy and execution time at the same time, the normalized 
energy and execution time curves are not in the same direction. According 
to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), 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.  

 
  
Our solution for this problem is to make the optimization process for energy and 
execution time follow the same direction.  Therefore, we inverse the equation of the 
normalized execution time which gives the normalized performance equation, as follows:
\begin{multline}
  \label{eq:pnorm_inv}
  P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
          = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
            { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm} 
\end{multline}


\begin{figure}
  \centering
  \subfloat[Homogeneous platform]{%
    \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
  \qquad%
  \subfloat[Heterogeneous platform]{%
    \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
  \label{fig:rel}
  \caption{The energy and performance relation}
\end{figure}

Then, we can model our objective function as finding the maximum distance
between the energy curve EQ~(\ref{eq:enorm}) and the  performance
curve EQ~(\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 our objective
function has the following form:
\begin{equation}
  \label{eq:max}
  Max Dist = 
  \max_{i=1,\dots F, j=1,\dots,N}
      (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
       \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
\end{equation}
where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).  
Our objective function can work with any energy model or any power values for each node 
(static and dynamic powers). However, the most energy reduction gain can be achieved when 
the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.

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

In this section we  propose algorithm~(\ref{HSA}) which 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 EQ(\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs 
according to the computed scaling factors.  This algorithm is called just once during the execution 
of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called 
in the iterative MPI program.

The nodes in a heterogeneous platform have different computing powers, thus while executing message 
passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their 
computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). 
These periods are called idle or slack times. 
Our 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 EQ(\ref{eq:Scp}), the algorithm computes 
the initial frequencies for all nodes as a ratio between the maximum frequency of node $i$  
and the computation scaling factor $Scp_i$ as follows:
\begin{equation}
  \label{eq:Fint}
 F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
\end{equation}
If the computed initial frequency for a node is not available in the gears of that node, the computed 
initial frequency is replaced by the nearest available frequency. In  figure (\ref{fig:st_freq}), 
the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster 
nodes are scaled down according to the computed initial frequency scaling factors. The resulting new 
frequencies are colored in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered 
as a higher bound for the search space of the optimal vector of frequencies because selecting frequency 
scaling factors higher than the higher bound will not improve the performance of the application and 
it will increase its overall energy consumption. Therefore the algorithm that selects the frequency 
scaling factors starts the search method from these initial frequencies and takes a downward search direction 
toward lower frequencies. The algorithm iterates on all left frequencies, from the higher bound until all 
nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select 
the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node 
according to EQ(\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 EQ(\ref{eq:max}).

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




\begin{algorithm}
  \begin{algorithmic}[1]
    % \footnotesize
    \Require ~
    \begin{description}
    \item[$Tcp_i$] array of all computation times for all nodes during one iteration and with highest frequency.
    \item[$Tcm_i$] array of all communication times for all nodes during one iteration and with highest frequency.
    \item[$Fmax_i$] array of the maximum frequencies for all nodes.
    \item[$Pd_i$] array of the dynamic powers for all nodes.
    \item[$Ps_i$] array of the static powers for all nodes.
    \item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
    \end{description}
    \Ensure $Sopt_1,Sopt_2 \dots, Sopt_N$ is a vector of optimal scaling factors

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

\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  \cite{NAS.Parallel.Benchmarks}. The experiments were executed 
on the simulator SimGrid/SMPI v3.10~\cite{casanova+giersch+legrand+al.2014.versatile} which offers 
easy tools to create a heterogeneous platform and run message passing applications over it. The 
heterogeneous platform that was used in the experiments, had one core per node because just one 
process was executed per node. The heterogeneous platform  was composed of four types of nodes. 
Each type of nodes had different characteristics such as the maximum CPU frequency, the number of
available frequencies and the computational power, see table (\ref{table:platform}). The characteristics 
of these different types of  nodes are inspired   from the specifications of real Intel processors. 
The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, 
for example if  a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors 
of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were 
chosen proportionally to  its computing power (FLOPS).  In the initial heterogeneous platform,  while computing 
with highest frequency, each node  consumed power proportional to its computing power which 80\% of it was 
dynamic power and the rest was 20\% for the static power, the same assumption  was made in \cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. 
Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.


\begin{table}[htb]
  \caption{Heterogeneous nodes characteristics}
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
    type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
                  &           & GHz      & GHz          &GHz             &              &       \\
    \hline
    1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
                  &           &          &              &                &              &  \\
    \hline
    2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
                  &           &          &              &                &              &  \\
    \hline
    3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
                  &           &          &              &                &              &  \\
    \hline
    4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
                  &           &          &              &                &              &  \\
    \hline
  \end{tabular}
  \label{table:platform}
\end{table}

 
%\subsection{Performance prediction verification}


\subsection{The experimental results of the scaling algorithm}
\label{sec.res}


The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) 
and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in 
this paper, only the results of the biggest class, C, are presented while being run on different number 
of nodes, ranging  from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the 
benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes. 
The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.

 
 
\begin{table}[htb]
  \caption{Running NAS benchmarks on 4 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &  64.64        & 3560.39        &34.16        &6.72               &27.44       \\
    \hline 
    MG         & 18.89         & 1074.87            &35.37            &4.34                   &31.03       \\
   \hline
    EP         &79.73          &5521.04         &26.83            &3.04               &23.79      \\
   \hline
    LU         &308.65         &21126.00           &34.00             &6.16                   &27.84      \\
    \hline
    BT         &360.12         &21505.55           &35.36         &8.49               &26.87     \\
   \hline
    SP         &234.24         &13572.16           &35.22         &5.70               &29.52    \\
   \hline
    FT         &81.58          &4151.48        &35.58         &0.99                   &34.59    \\
\hline 
  \end{tabular}
  \label{table:res_4n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 8 and 9 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &36.11              &3263.49             &31.25        &7.12                    &24.13     \\
    \hline 
    MG         &8.99           &953.39          &33.78        &6.41                    &27.37     \\
   \hline
    EP         &40.39          &5652.81         &27.04        &0.49                    &26.55     \\
   \hline
    LU         &218.79             &36149.77        &28.23        &0.01                    &28.22      \\
    \hline
    BT         &166.89         &23207.42            &32.32            &7.89                    &24.43      \\
   \hline
    SP         &104.73         &18414.62            &24.73            &2.78                    &21.95      \\
   \hline
    FT         &51.10          &4913.26         &31.02        &2.54                    &28.48      \\
\hline 
  \end{tabular}
  \label{table:res_8n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 16 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &31.74          &4373.90         &26.29        &9.57                    &16.72          \\
    \hline 
    MG         &5.71           &1076.19         &32.49        &6.05                    &26.44         \\
   \hline
    EP         &20.11          &5638.49         &26.85        &0.56                    &26.29         \\
   \hline
    LU         &144.13         &42529.06            &28.80            &6.56                    &22.24         \\
    \hline
    BT         &97.29          &22813.86            &34.95        &5.80                &29.15         \\
   \hline
    SP         &66.49          &20821.67            &22.49            &3.82                    &18.67         \\
   \hline
    FT             &37.01          &5505.60             &31.59        &6.48                    &25.11         \\
\hline 
  \end{tabular}
  \label{table:res_16n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 32 and 36 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &32.35          &6704.21         &16.15        &5.30                    &10.85           \\
    \hline 
    MG         &4.30           &1355.58         &28.93        &8.85                    &20.08          \\
   \hline
    EP         &9.96           &5519.68         &26.98        &0.02                    &26.96          \\
   \hline
    LU         &99.93          &67463.43            &23.60            &2.45                    &21.15          \\
    \hline
    BT         &48.61          &23796.97            &34.62            &5.83                    &28.79          \\
   \hline
    SP         &46.01          &27007.43            &22.72            &3.45                    &19.27           \\
   \hline
    FT             &28.06          &7142.69             &23.09        &2.90                    &20.19           \\
\hline 
  \end{tabular}
  \label{table:res_32n}
\end{table}

\begin{table}[htb]
  \caption{Running NAS benchmarks on 64 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance      \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
    \hline
    CG         &46.65          &17521.83            &8.13             &1.68                    &6.45           \\
    \hline 
    MG         &3.27           &1534.70         &29.27        &14.35               &14.92          \\
   \hline
    EP         &5.05           &5471.1084           &27.12            &3.11                &24.01         \\
   \hline
    LU         &73.92          &101339.16           &21.96            &3.67                    &18.29         \\
    \hline
    BT         &39.99          &27166.71            &32.02            &12.28               &19.74         \\
   \hline
    SP         &52.00          &49099.28            &24.84            &0.03                    &24.81         \\
   \hline
    FT         &25.97          &10416.82        &20.15        &4.87                    &15.28         \\
\hline 
  \end{tabular}
  \label{table:res_64n}
\end{table}


\begin{table}[htb]
  \caption{Running NAS benchmarks on 128 and 144 nodes }
  % title of Table
  \centering
  \begin{tabular}{|*{7}{l|}}
    \hline
    Method     & Execution     & Energy         & Energy      & Performance        & Distance     \\
    name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
    \hline
    CG         &56.92          &41163.36        &4.00         &1.10                    &2.90          \\
    \hline 
    MG         &3.55           &2843.33         &18.77        &10.38               &8.39          \\
   \hline
    EP         &2.67           &5669.66         &27.09        &0.03                    &27.06         \\
   \hline
    LU         &51.23          &144471.90       &16.67        &2.36                    &14.31         \\
    \hline
    BT         &37.96          &44243.82            &23.18            &1.28                    &21.90         \\
   \hline
    SP         &64.53          &115409.71           &26.72            &0.05                    &26.67         \\
   \hline
    FT         &25.51          &18808.72            &12.85            &2.84                    &10.01         \\
\hline 
  \end{tabular}
  \label{table:res_128n}
\end{table}
The overall energy consumption was computed for each instance according to the energy 
consumption  model EQ(\ref{eq:energy}), with and without applying the algorithm. The 
execution time was also measured for all these experiments. Then, the energy saving 
and performance degradation percentages were computed for each instance.  
The results are presented in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, 
\ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}). All these results are the 
average values from many experiments for  energy savings and performance degradation.

The tables  show the experimental results for running the NAS parallel benchmarks on different 
number of nodes. The experiments show that the algorithm reduce significantly the energy 
consumption (up to 35\%) and tries to limit the performance degradation. They also show that 
the  energy saving percentage is decreased  when the number of the computing nodes is increased. 
This reduction is due to the increase of the communication times compared to the execution times 
when the benchmarks are run over a high number of nodes. Indeed, the benchmarks with the same class, C, 
are executed on different number of nodes, so the computation required for each iteration is divided 
by the number of computing nodes.   On the other hand, more communications are required when increasing 
the number of nodes so the static energy is increased linearly according to the communication time and 
the dynamic power is less relevant in the overall energy consumption. Therefore, reducing the frequency 
with algorithm~(\ref{HSA}) have less effect in reducing the overall energy savings. It can also be 
noticed that for the benchmarks EP and SP that contain little or no communications,  the energy savings 
are not significantly affected with the high number of nodes. No experiments were conducted using bigger 
classes such as D, because they require a lot of memory(more than 64GB) when being executed by the simulator 
on one machine. The maximum distance between the normalized energy curve and the normalized performance 
for each instance is also shown in the result tables. It is decreased in the same way as the energy 
saving percentage. The tables also show that the performance degradation percentage is not significantly 
increased when the number of computing nodes is increased because the computation times are small when 
compared to the communication times.  


 
\begin{figure}
  \centering
  \subfloat[Energy saving]{%
    \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%
  \quad%
  \subfloat[Performance degradation ]{%
    \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}
  \label{fig:avg}
  \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
\end{figure}

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




\subsection{The results for different power consumption scenarios}

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

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

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

The two new power scenarios are compared to the old one in figure (\ref{fig:sen_comp}). It shows the average of 
the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. 
The comparison shows that  the energy saving ratio is proportional to the dynamic power ratio: it is increased 
when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the the most relevant 
in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, 
the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in 
the overall consumed energy and lowering the frequency do not returns big energy savings.
Moreover, the average of the performance degradation is decreased when using a higher ratio for static power 
(e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed algorithm optimizes the energy consumption 
when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in 
more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens 
when using a higher ratio for  static  power, the algorithm proportionally  selects  smaller scaling values which 
results in less energy saving but less performance degradation. 


 \begin{table}[htb]
  \caption{The results of 70\%-30\% powers scenario}
  % title of Table
  \centering
  \begin{tabular}{|*{6}{l|}}
    \hline
    Method     & Energy          & Energy      & Performance        & Distance     \\
    name       & consumption/J   & saving\%    & degradation\%      &              \\
    \hline
    CG         &4144.21          &22.42        &7.72                &14.70         \\
    \hline 
    MG         &1133.23          &24.50        &5.34                &19.16          \\
   \hline
    EP         &6170.30         &16.19         &0.02                &16.17          \\
   \hline
    LU         &39477.28        &20.43         &0.07                &20.36          \\
    \hline
    BT         &26169.55            &25.34             &6.62                &18.71          \\
   \hline
    SP         &19620.09            &19.32             &3.66                &15.66          \\
   \hline
    FT         &6094.07         &23.17         &0.36                &22.81          \\
\hline 
  \end{tabular}
  \label{table:res_s1}
\end{table}



\begin{table}[htb]
  \caption{The results of 90\%-10\% powers scenario}
  % title of Table
  \centering
  \begin{tabular}{|*{6}{l|}}
    \hline
    Method     & Energy          & Energy      & Performance        & Distance     \\
    name       & consumption/J   & saving\%    & degradation\%      &              \\
    \hline
    CG         &2812.38          &36.36        &6.80                &29.56         \\
    \hline 
    MG         &825.427          &38.35        &6.41                &31.94         \\
   \hline
    EP         &5281.62          &35.02        &2.68                &32.34         \\
   \hline
    LU         &31611.28             &39.15        &3.51                    &35.64        \\
    \hline
    BT         &21296.46             &36.70            &6.60                &30.10       \\
   \hline
    SP         &15183.42             &35.19            &11.76               &23.43        \\
   \hline
    FT         &3856.54          &40.80        &5.67                &35.13        \\
\hline 
  \end{tabular}
  \label{table:res_s2}
\end{table}


\begin{figure}
  \centering
  \subfloat[Comparison the average of the results on 8 nodes]{%
    \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
  \quad%
  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
    \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
  \label{fig:comp}
  \caption{The comparison of the three power scenarios}
\end{figure}  



\subsection{The verifications of the proposed method}
\label{sec.verif}
The precision of the proposed algorithm mainly depends on the execution time prediction model defined in 
EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}). 
The energy model is also significantly dependent  on the execution time model because the static energy is 
linearly related the execution time and the dynamic energy is related to the computation time. So, all of 
the work presented in this paper is based on the execution time model. To verify this model, the predicted 
execution time was compared to  the real execution time over Simgrid for all  the NAS parallel benchmarks 
running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise, 
the maximum normalized difference between  the predicted execution time  and the real execution time is equal 
to 0.03 for all the NAS benchmarks.

Since  the proposed algorithm is not an exact method and do not test all the possible solutions (vectors of scaling factors) 
in the search space and to prove its efficiency, it was compared on small instances to a brute force search algorithm 
that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with 
different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical 
and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time: 
for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in 
table~(\ref{table:platform}), it takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes 
to compute the best scaling factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number 
of iterations and $N$ is the number of computing nodes. The algorithm needs  from 12 to 20 iterations to select the best 
vector of frequency scaling factors that gives the results of the section (\ref{sec.res}).

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


\section*{Acknowledgment}


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% LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
% LocalWords:  CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
% LocalWords:  de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
% LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT