\usepackage{algorithm}
\usepackage{subfig}
\usepackage{amsmath}
-
+\usepackage{multirow}
\usepackage{url}
\DeclareUrlCommand\email{\urlstyle{same}}
\newcommand{\Told}{\Xsub{T}{Old}}
\begin{document}
-\title{Energy Consumption Reduction in a Heterogeneous Architecture Using DVFS}
+\title{Energy Consumption Reduction for Message Passing Iterative Applications in Heterogeneous Architecture Using DVFS}
\author{%
\IEEEauthorblockN{%
application must be selected.
In this paper, a new online frequencies selecting algorithm for heterogeneous platforms is presented.
-It selects the frequency that gives the best tradeoff between energy saving and performance degradation,
+It selects the frequency that try to give the best tradeoff between energy saving and performance degradation,
for each node computing the message passing iterative application. The algorithm has a small overhead and
works without training or profiling. It uses a new energy model for message passing iterative applications
-running on a heterogeneous platform. The proposed algorithm was evaluated on the Simgrid simulator while
+running on a heterogeneous platform. The proposed algorithm is evaluated on the Simgrid simulator while
running the NAS parallel benchmarks. The experiments demonstrated that it reduces the energy consumption
-up to 35\% while limiting the performance degradation as much as possible.
+up to 35\% while limiting the performance degradation as much as possible. \textcolor{red}{Furthermore, we compare the
+proposed algorithm with other method. The comparison’s results show that our algorithm gives better
+energy-time trade-off.}
+
\end{abstract}
\section{Introduction}
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 platform might achieve higher floating
point operations per second (FLOPS), but the energy consumption and the heat dissipation are also increased.
-As an example, the chinese supercomputer Tianhe-2 had the highest FLOPS in November 2014 according to the Top500
+As an example, the Chinese supercomputer Tianhe-2 had the highest FLOPS in November 2014 according to the Top500
list \cite{TOP500_Supercomputers_Sites}. However, it was also the most power hungry platform with its over 3 millions
cores consuming around 17.8 megawatts. Moreover, according to the U.S. annual energy outlook 2014
\cite{U.S_Annual.Energy.Outlook.2014}, the price of energy for 1 megawatt-hour
Therefore, the price of the energy consumed by the
Tianhe-2 platform is approximately more than \$10 millions each year.
The computing platforms must be more energy efficient and offer the highest number of FLOPS per watt possible,
-such as the TSUBAME-KFC at the GSIC center of Tokyo which
-became the top of the Green500 list in June 2014 \cite{Green500_List}.
-This heterogeneous platform executes more than four GFLOPS per watt.
+such as the L-CSC from the GSI Helmholtz Center which
+became the top of the Green500 list in November 2014 \cite{Green500_List}.
+This heterogeneous platform executes more than 5 GFLOPS per watt while consumed 57.15 kilowatts.
Besides hardware improvements, there are many software techniques to lower the energy consumption of these platforms,
such as scheduling, DVFS, ... DVFS is a widely used process to reduce the energy consumption of a processor by lowering
-its frequency \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also the reduces the number of FLOPS
+its frequency \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces the number of FLOPS
executed by the processor which might increase the execution time of the application running over that processor.
Therefore, researchers used different optimization strategies to select the frequency that gives the best tradeoff
between the energy reduction and
-performance degradation ratio. \textbf{In our previous paper \cite{Our_first_paper}, a frequency selecting algorithm
-was proposed for distributed iterative application running over homogeneous platform. While in this paper the algorithm is significantly adapted to run over a heterogeneous platform. This platform is a collection of heterogeneous computing nodes interconnected via a high speed homogeneous network.}
-
-The proposed frequency selecting algorithm selects the vector of frequencies for a heterogeneous platform that runs a message passing iterative application, that gives the maximum energy reduction and minimum
-performance degradation ratio simultaneously. The algorithm has a very small
+performance degradation ratio. In \cite{Our_first_paper}, a frequency selecting algorithm
+was proposed to reduce the energy consumption of message passing iterative applications running over homogeneous platforms. The results of the experiments showed significant energy consumption reductions. In this paper, a new frequency selecting algorithm adapted for heterogeneous platform is presented. It selects the vector of frequencies, for a heterogeneous platform running a message passing iterative application, that simultaneously tries to give the maximum energy reduction and minimum performance degradation ratio. The algorithm has a very small
overhead, works online and does not need any training or profiling.
This paper is organized as follows: Section~\ref{sec.relwork} presents some
consumption while minimizing the degradation of the program's performance.
Section~\ref{sec.optim} details the proposed frequency selecting algorithm then the precision of the proposed algorithm is verified.
Section~\ref{sec.expe} presents the results of applying the algorithm on the NAS parallel benchmarks and executing them
-on a heterogeneous platform. It also shows the results of running three
-different power scenarios and comparing them.
-Finally, we conclude in Section~\ref{sec.concl} with a summary and some future works.
+on a heterogeneous platform. It shows the results of running three
+different power scenarios and comparing them. \textcolor{red}{Moreover, it also shows the comparison results
+between our method and other method.}
+Finally, in Section~\ref{sec.concl} the paper is ended with a summary and some future works.
\section{Related works}
\label{sec.relwork}
-Energy reduction process for high performance clusters recently performed using
-dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique enabled
-in 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.
-This paper presents a method
-to find the optimal set of frequencies for heterogeneous cluster to
-simultaneously optimize both the energy and the execution time without adding big
-overhead. This work is developed from our previous work of 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 GPUs and the others executed
-on CPUs. As an example of these works, Luley et al.
+DVFS is a technique enabled
+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 the GPUs to achieve the same goal. Reducing the frequency of a processor lowers its number of FLOPS and might degrade the performance of the application running on that processor, especially if it is compute bound. Therefore selecting the appropriate frequency for a processor to satisfy some objectives and 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,Dhiman_Online.Learning.Power.Management}. Others used offline methods that might need to run the application and profile it before selecting the new frequency, such as ~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The methods could be heuristics, exact or brute force methods that satisfy varied objectives such as energy reduction or performance. They also could be adapted to the execution's environment and the type of the application such as sequential, parallel or distributed architecture, homogeneous or heterogeneous platform, synchronous or asynchronous application, ...
+
+In this paper, we are interested in reducing energy for message passing iterative synchronous applications running over heterogeneous platforms.
+Some works have already been done for such platforms and they can be classified into two types of heterogeneous platforms:
+\begin{itemize}
+
+\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
+\item the platform is only composed of heterogeneous CPUs.
+
+\end{itemize}
+
+For the first type of platform, the compute 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 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 CPU or 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 these 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 less than 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 frequency selecting algorithm has very small overhead and
-it is works without any previous analysis for the application time complexity. The primary
-contributions of our paper are :
+cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal was to maximize the
+energy efficiency of the platform during computation by maximizing the number of FLOPS per watt generated.
+In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et al. developed a scheduling
+algorithm that distributes workloads proportional to the computing power of the nodes which could be a GPU or a CPU. All the tasks must be completed at the same time.
+In~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, Rong et al. showed that
+a heterogeneous (GPUs and CPUs) cluster that enables DVFS gave better energy and performance
+efficiency than other clusters only composed of CPUs.
+
+The work presented in this paper concerns the second type of platform, with heterogeneous CPUs.
+Many methods were conceived to reduce the energy consumption of this type of platform. Naveen et al.~\cite{Naveen_Power.Efficient.Resource.Scaling}
+developed a method that minimizes the value of $energy*delay^2$ (the delay is the sum of slack times that happen during synchronous communications) by dynamically assigning new frequencies to the CPUs of the heterogeneous cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed
+an algorithm that divides the executed tasks into two types: the critical and
+non critical tasks. The algorithm scales down the frequency of non critical tasks proportionally to their slack and communication times while limiting the performance degradation percentage to less than 10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed
+ a heterogeneous cluster composed of two types
+of Intel and AMD processors. They use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks},
+ the best frequencies for a specified heterogeneous cluster are selected offline using some
+heuristic. Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic programming approach to
+minimize the power consumption of heterogeneous severs while respecting given time constraints. This approach
+had considerable overhead.
+In contrast to the above described papers, this paper presents the following contributions :
\begin{enumerate}
-\item It is presents a new online frequency selecting algorithm which has very small
- overhead and not need for any training and profiling.
-\item It is develops a new energy model for iterative distributed applications running over
- a heterogeneous clusters, taking into account the communication and slack times.
-\item The proposed frequency selecting algorithm predicts both the energy and the execution time
- of the iterative application running over heterogeneous platform.
-\item It demonstrates a new optimization function which maximize the performance and
- minimize the energy consumption simultaneously.
+\item two new energy and performance models for message passing iterative synchronous applications running over
+ a heterogeneous platform. Both models takes into account the communication and slack times. The models can predict the required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous platforms. The algorithm has a very small
+ overhead and does not need for any training or profiling. It uses a new optimization function which simultaneously maximizes the performance and minimizes the energy consumption of a message passing iterative synchronous application.
\end{enumerate}
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 platforms. A heterogeneous platform is defined as a collection of
heterogeneous computing nodes interconnected via a high speed homogeneous
network. Therefore, each node has different characteristics such as computing
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
\begin{figure}[t]
\centering
- \includegraphics[scale=0.6]{fig/commtasks}
+ \includegraphics[scale=0.5]{fig/commtasks}
\caption{Parallel tasks on a heterogeneous platform}
\label{fig:heter}
\end{figure}
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}).
+as in (\ref{eq:s}).
\begin{equation}
\label{eq:s}
S = \frac{F_\textit{max}}{F_\textit{new}}
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.
+ until the message is synchronously sent or received.
Since in a heterogeneous platform, each node has different characteristics,
especially different frequency gears, when applying DVFS operations on these
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}).
+vector of scaling factors can be predicted using (\ref{eq:perf}).
\begin{equation}
\label{eq:perf}
\textit T_\textit{new} =
- \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) + MinTcm
+ \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) + MinTcm
+\end{equation}
+Where:\\
+\begin{equation}
+\label{eq:perf}
+ MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
\end{equation}
where $TcpOld_i$ is the computation time of processor $i$ during the first
iteration and $MinTcm$ is the communication time of the slowest processor from
the first iteration. The model computes the maximum computation time
-with scaling factor from each node added to the communication time of the \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 sections (\ref{sec.res}) and (\ref{sec.compare}).
+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
+Therefore, the execution time of the iterative application is
+equal to the execution time of one iteration as in (\ref{eq:perf}) multiplied
by the number of iterations of that application.
-This prediction model is developed from our model for predicting the execution time of
+This prediction model is developed from the 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
+The execution time prediction model is uSpiliopoulossed in the method for optimizing both
energy consumption and performance of iterative methods, which is presented in the
following sections.
consumed as long as the computing unit is turned on, the latter is only consumed during
computation times. The dynamic power $Pd$ is related to the switching
activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
-operational frequency $F$, as shown in EQ(\ref{eq:pd}).
+operational frequency $F$, as shown in (\ref{eq:pd}).
\begin{equation}
\label{eq:pd}
Pd = \alpha \cdot C_L \cdot V^2 \cdot F
E_\textit{ind} = Pd \cdot Tcp + Ps \cdot T
\end{equation}
where $T$ is the execution time of the program, $Tcp$ is the computation
-time and $Tcp \leq T$. $Tcp$ may be equal to $T$ if there is no
+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
+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}).
+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. we can calculate the new frequency
-$F_{new}$ from EQ(\ref{eq:s}) as follow:
+system's kernel to lower a core's frequency. The new frequency
+$F_{new}$ from (\ref{eq:s}) can be calculated as follows:
\begin{equation}
\label{eq:fnew}
F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
-Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following
+Replacing $F_{new}$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
equation for dynamic power consumption:
\begin{multline}
\label{eq:pdnew}
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
+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
\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
+ 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 EQ(\ref{eq:perf}), the execution time of the program
+According to the execution time model in (\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:
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
+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 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
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})
+increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption for the iterative
+application can be measured by measuring the energy consumption for one iteration as in (\ref{eq:energy})
multiplied by the number of iterations of that application.
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
+Moreover, they 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:
+frequency for all nodes) as follows:
\begin{multline}
\label{eq:pnorm}
P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
\end{multline}
-In the same way, we normalize the energy by computing the ratio between the consumed energy
+In the same way, the energy is normalized by computing the ratio between the consumed energy
while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
\begin{multline}
\label{eq:enorm}
\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}).
+Where $E_\textit{Reduced}$ and $E_\textit{Original}$ are computed using (\ref{eq:energy}) and
+ $T_{New}$ and $T_{Old}$ are computed as in (\ref{eq:pnorm}).
- While the main
+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
+to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the vector of frequency
scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
time simultaneously. But the main objective is to produce maximum energy
reduction with minimum execution time reduction.
-
-
-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:
+This problem can be solved by making the optimization process for energy and
+execution time follow the same direction. Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
\begin{figure}
\centering
\subfloat[Homogeneous platform]{%
- \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
- \qquad%
+ \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+
+
\subfloat[Heterogeneous platform]{%
- \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
+ \includegraphics[width=.33\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
+Then, the objective function can be modeled as finding the maximum distance
+between the energy curve (\ref{eq:enorm}) and the performance
+curve (\ref{eq:pnorm_inv}) over all available sets of scaling factors. This
represents the minimum energy consumption with minimum execution time (maximum
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then our objective
+performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then the objective
function has the following form:
\begin{equation}
\label{eq:max}
(\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
+where $N$ is the number of nodes and $F$ is the number of available frequencies for each node.
+Then, the optimal set of scaling factors that satisfies (\ref{eq:max}) can be selected.
+The objective function can work with any energy model or any power values for each node
(static and dynamic powers). However, the most energy reduction gain can be achieved when
the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
\label{sec.optim}
\subsection{The algorithm details}
-In this section we propose algorithm~(\ref{HSA}) which selects the frequency scaling factors
+In this section algorithm \ref{HSA} is presented. It selects the frequency scaling factors
vector that gives the best trade-off between minimizing the energy consumption and maximizing
the performance of a message passing synchronous iterative application executed on a heterogeneous
platform. It works online during the execution time of the iterative message passing program.
It uses information gathered during the first iteration such as the computation time and the
communication time in one iteration for each node. The algorithm is executed after the first
iteration and returns a vector of optimal frequency scaling factors that satisfies the objective
-function EQ(\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs
+function (\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.
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
+The algorithm takes into account this problem and tries to reduce these slack times when selecting the
frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase
the execution times of fast nodes and minimize the differences between the computation times of
fast and slow nodes. The value of the initial frequency scaling factor for each node is inversely
\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
+Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the algorithm computes
the initial frequencies for all nodes as a ratio between the maximum frequency of node $i$
and the computation scaling factor $Scp_i$ as follows:
\begin{equation}
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
+according to (\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}).
+function (\ref{eq:max}).
The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an
application running on a homogeneous platform and a heterogeneous platform respectively while increasing the
\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. $
+ \State $Dist \gets 0 $
\While {(all nodes not reach their minimum frequency)}
\If{(not the last freq. \textbf{and} not the slowest node)}
\State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
\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 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.
\label{dvfs}
\end{algorithm}
-\subsection{The verifications of the proposed algorithm}
-\label{sec.verif}
+\subsection{The evaluation of the proposed algorithm}
+\label{sec.verif.algo}
The precision of the proposed algorithm mainly depends on the execution time prediction model defined in
-EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}).
+(\ref{eq:perf}) and the energy model computed by (\ref{eq:energy}).
The energy model is also significantly dependent on the execution time model because the static energy is
linearly related 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
+the works presented in this paper is based on the execution time model. To verify this model, the predicted
execution time was compared to the real execution time over SimGrid/SMPI simulator, v3.10~\cite{casanova+giersch+legrand+al.2014.versatile},
for all the NAS parallel benchmarks NPB v3.3
\cite{NAS.Parallel.Benchmarks}, running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise,
the maximum normalized difference between the predicted execution time and the real execution time is equal
to 0.03 for all the NAS benchmarks.
-Since the proposed algorithm is not an exact method and 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
+Since the proposed algorithm is not an exact method and does not test all the possible solutions (vectors of scaling factors)
+in the search space. To prove its efficiency, it was compared on small instances to a brute force search algorithm
that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with
different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical
and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time:
for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in
-table~(\ref{table:platform}), it takes on average \np[ms]{0.04} for 4 nodes and \np[ms]{0.15} on average for 144 nodes
+table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4 nodes and \np[ms]{0.15} on average for 144 nodes
to compute the best scaling factors vector. The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number
of iterations and $N$ is the number of computing nodes. The algorithm needs from 12 to 20 iterations to select the best
vector of frequency scaling factors that gives the results of the next sections.
\section{Experimental results}
\label{sec.expe}
-To evaluate the efficiency and the overall energy consumption reduction of algorithm~(\ref{HSA}),
+To evaluate the efficiency and the overall energy consumption reduction of algorithm~ \ref{HSA},
it was applied to the NAS parallel benchmarks NPB v3.3. The experiments were executed
on the simulator SimGrid/SMPI which offers easy tools to create a heterogeneous platform and run
message passing applications over it. The heterogeneous platform that was used in the experiments,
had one core per node because just one process was executed per node.
The heterogeneous platform was composed of four types of nodes. Each type of nodes had different
characteristics such as the maximum CPU frequency, the number of
-available frequencies and the computational power, see table (\ref{table:platform}). The characteristics
+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
& & 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}
\centering
\begin{tabular}{|*{7}{l|}}
\hline
- Method & Execution & Energy & Energy & Performance & Distance \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
CG & 64.64 & 3560.39 &34.16 &6.72 &27.44 \\
\centering
\begin{tabular}{|*{7}{l|}}
\hline
- Method & Execution & Energy & Energy & Performance & Distance \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
CG &36.11 &3263.49 &31.25 &7.12 &24.13 \\
\centering
\begin{tabular}{|*{7}{l|}}
\hline
- Method & Execution & Energy & Energy & Performance & Distance \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
CG &31.74 &4373.90 &26.29 &9.57 &16.72 \\
\centering
\begin{tabular}{|*{7}{l|}}
\hline
- Method & Execution & Energy & Energy & Performance & Distance \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
CG &32.35 &6704.21 &16.15 &5.30 &10.85 \\
\centering
\begin{tabular}{|*{7}{l|}}
\hline
- Method & Execution & Energy & Energy & Performance & Distance \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
CG &46.65 &17521.83 &8.13 &1.68 &6.45 \\
\centering
\begin{tabular}{|*{7}{l|}}
\hline
- Method & Execution & Energy & Energy & Performance & Distance \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
CG &56.92 &41163.36 &4.00 &1.10 &2.90 \\
\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
+consumption model (\ref{eq:energy}), with and without applying the algorithm. The
execution time was also measured for all these experiments. Then, the energy saving
and performance degradation percentages were computed for each instance.
-The results are presented in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n},
+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
\begin{figure}
\centering
\subfloat[Energy saving]{%
- \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%
- \quad%
+ \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
+
\subfloat[Performance degradation ]{%
- \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}
+ \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
\label{fig:avg}
\caption{The energy and performance for all NAS benchmarks running with difference number of nodes}
\end{figure}
Plots (\ref{fig:energy} and \ref{fig:per_deg}) present the energy saving and performance degradation
respectively for all the benchmarks according to the number of used nodes. As shown in the first plot,
-the energy saving percentages of the benchmarks MG, LU, BT and FT are decreased linearly when the the
+the energy saving percentages of the benchmarks MG, LU, BT and FT are decreased linearly when the
number of nodes is increased. While for the EP and SP benchmarks, the energy saving percentage is not
affected by the increase of the number of computing nodes, because in these benchmarks there are little or
no communications. Finally, the energy saving of the GC benchmark is significantly decreased when the number
\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\%
+The NAS parallel benchmarks were executed again over processors that follow the new power scenarios.
+The class C of each benchmark was run over 8 or 9 nodes and the results are presented in Tables
+\ref{table:res_s1} and \ref{table:res_s2}. These tables show that the energy saving percentage of the 70\%-30\%
scenario is less for all benchmarks compared to the energy saving of the 90\%-10\% scenario. Indeed, in the latter
more dynamic power is consumed when nodes are running on their maximum frequencies, thus, scaling down the frequency
of the nodes results in higher energy savings than in the 70\%-30\% scenario. On the other hand, the performance
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 .
+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
+when applying the 90\%-10\% scenario because at maximum frequency the dynamic energy is the most relevant
in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand,
the energy saving is decreased when the 70\%-30\% scenario is used because the dynamic energy is less relevant in
the overall consumed energy and lowering the frequency do not returns big energy savings.
\centering
\begin{tabular}{|*{6}{l|}}
\hline
- Method & Energy & Energy & Performance & Distance \\
+ Program & Energy & Energy & Performance & Distance \\
name & consumption/J & saving\% & degradation\% & \\
\hline
CG &4144.21 &22.42 &7.72 &14.70 \\
\centering
\begin{tabular}{|*{6}{l|}}
\hline
- Method & Energy & Energy & Performance & Distance \\
+ Program & Energy & Energy & Performance & Distance \\
name & consumption/J & saving\% & degradation\% & \\
\hline
CG &2812.38 &36.36 &6.80 &29.56 \\
\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 of the results on 8 nodes]{%
+ \includegraphics[width=.30\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
+
\subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%
- \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
+ \includegraphics[width=.34\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
\label{fig:comp}
\caption{The comparison of the three power scenarios}
\end{figure}
+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
+
+In this section, we compare our scaling factors selection algorithm
+with Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}.
+They developed an online frequency selecting algorithm running over multicore architecture.
+The algorithm predicted both the energy and performance during the runtime of the program, then
+selecting the frequencies that minimized the energy and delay products (EDP), $EDP=Enegry*Delay$.
+To be able to compare with this algorithm, we used our energy and execution time models in prediction process,
+equations (\ref{eq:energy}) and (\ref{eq:fnew}). Also their algorithm is adapted to taking into account
+the heterogeneous platform to starts selecting the
+initial frequencies using the equation (\ref{eq:Fint}). The algorithm built to test all possible frequencies as
+a brute-force search algorithm.
+
+The comparison results of running NAS benchmarks class C on 8 or 9 nodes are
+presented in table \ref{table:compare_EDP}. The results show that our algorithm has a biggest energy saving percentage,
+on average it has 29.76\% and thier algorithm has 25.75\%,
+while the average of performance degradation percentage is approximately the same, the average for our algorithm is
+equal to 3.89\% and for their algorithm is equal to 4.03\%. In general, our algorithm outperforms
+Spiliopoulos et al. algorithm in term of energy and performance tradeoff see figure (\ref{fig:compare_EDP}).
+This because our algorithm maximized the difference (the distance) between the energy saving and the performance degradation
+comparing to their EDP optimization function. It is also keeps the frequency of the slowest node without change
+that gave some enhancements to the energy and performance tradeoff.
+
+
+\begin{table}[h]
+ \caption{Comparing the proposed algorithm}
+ \centering
+\begin{tabular}{|l|l|l|l|l|l|l|l|}
+\hline
+\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Program \\ name\end{tabular}}} & \multicolumn{2}{l|}{Energy saving \%} & \multicolumn{2}{l|}{Perf. degradation \%} & \multicolumn{2}{l|}{Distance} \\ \cline{3-8}
+\multicolumn{2}{|l|}{} & EDP & MaxDist & EDP & MaxDist & EDP & MaxDist \\ \hline
+\multicolumn{2}{|l|}{CG} & 27.58 & 31.25 & 5.82 & 7.12 & 21.76 & 24.13 \\ \hline
+\multicolumn{2}{|l|}{MG} & 29.49 & 33.78 & 3.74 & 6.41 & 25.75 & 27.37 \\ \hline
+\multicolumn{2}{|l|}{LU} & 19.55 & 28.33 & 0.0 & 0.01 & 19.55 & 28.22 \\ \hline
+\multicolumn{2}{|l|}{EP} & 28.40 & 27.04 & 4.29 & 0.49 & 24.11 & 26.55 \\ \hline
+\multicolumn{2}{|l|}{BT} & 27.68 & 32.32 & 6.45 & 7.87 & 21.23 & 24.43 \\ \hline
+\multicolumn{2}{|l|}{SP} & 20.52 & 24.73 & 5.21 & 2.78 & 15.31 & 21.95 \\ \hline
+\multicolumn{2}{|l|}{FT} & 27.03 & 31.02 & 2.75 & 2.54 & 24.28 & 28.48 \\ \hline
+
+\end{tabular}
+\label{table:compare_EDP}
+\end{table}
+
+
+
+\begin{figure}[t]
+ \centering
+ \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
+ \caption{Tradeoff comparison for NAS benchmarks class C}
+ \label{fig:compare_EDP}
+\end{figure}
+
\section{Conclusion}
\label{sec.concl}
-In this paper, we have presented a new online selecting frequency scaling factors algorithm
-that selects the best possible vector of frequency scaling factors for a heterogeneous platform.
-This vector gives the maximum distance (optimal tradeoff) between the predicted energy and
-the predicted performance curves. In addition, we developed a new energy model for measuring
+In this paper, a new online frequency selecting algorithm has been presented. It selects the best possible vector of frequency scaling factors that gives the maximum distance (optimal tradeoff) between the predicted energy and
+the predicted performance curves for a heterogeneous platform. This algorithm uses a new energy model for measuring
and predicting the energy of distributed iterative applications running over heterogeneous
-cluster. The proposed method evaluated on Simgrid/SMPI simulator to built a heterogeneous
-platform to executes NAS parallel benchmarks. The results of the experiments showed the ability of
-the proposed algorithm to changes its behaviour to selects different scaling factors when
-the number of computing nodes and both of the static and the dynamic powers are changed.
-
-In the future, we plan to improve this method to apply on asynchronous iterative applications
-where each task does not wait the others tasks to finish there works. This leads us to develop a new
-energy model to an asynchronous iterative applications, where the number of iterations is not
+platform. To evaluate the proposed method, it was applied on the NAS parallel benchmarks and executed over a heterogeneous platform simulated by Simgrid. The results of the experiments showed that the algorithm reduces up to 35\% the energy consumption of a message passing iterative method while limiting the degradation of the performance. The algorithm also selects different scaling factors according to the percentage of the computing and communication times, and according to the values of the static and dynamic powers of the CPUs. \textcolor{red}{ We compare our algorithm with Spiliopoulos et al. algorithm, the comparison results showed that our
+algorithm outperforms their algorithm in term of energy-time tradeoff.}
+
+In the near future, this method will be applied to real heterogeneous platforms to evaluate its performance in a real study case. It would also be interesting to evaluate its scalability over large scale heterogeneous platform and measure the energy consumption reduction it can produce. Afterward, we would like to develop a similar method that is adapted to asynchronous iterative applications
+where each task does not wait for others tasks to finish there works. The development of such method might require a new
+energy model because the number of iterations is not
known in advance and depends on the global convergence of the iterative system.
\section*{Acknowledgment}
+This work has been partially supported by the Labex
+ACTION project (contract “ANR-11-LABX-01-01”). As a PhD student,
+Mr. Ahmed Fanfakh, would like to thank the University of
+Babylon (Iraq) for supporting his work.
% trigger a \newpage just before the given reference
%%% End:
% 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: CMOS 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
