Raphaël Couturier,
Ahmed Fanfakh and
Arnaud Giersch
-the normalized performance equation, as follows:
}
\IEEEauthorblockA{%
FEMTO-ST Institute\\
\maketitle
\begin{abstract}
-
+Computing platforms are consuming more and more energy due to the increase of the number of nodes composing them. To minimize the operating costs of these platforms many techniques have been used. Dynamic voltage and frequency scaling (DVFS) is one of them, it reduces the frequency of a CPU to lower its energy consumption. However, lowering the frequency of a CPU might increase the execution time of an application running on that processor. Therefore, the frequency that gives the best tradeoff between the energy consumption and the performance of an application must be selected.
+
+In this paper, a new online frequencies selecting algorithm for heterogeneous platforms is presented. It selects the frequency that gives 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 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.
\end{abstract}
\section{Introduction}
\label{sec.intro}
-Modern processors continue to increased 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{51}.
-As an example of the more power hungry cluster, Tianhe-2 became in
-the top of the Top500 list in June 2014 \cite{43}. 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{60},
-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{59}. 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.
+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 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
+was approximately equal to \$70.
+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.
+
+ 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. \textbf{put a reference to DVFS} However, it also the 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{you should talk about the first paper here and say that the algorithm was applied to a homogeneous platform then define what is a heterogeneous platform, you can take it from the firdt paragraph in section 3 }
+
+In this paper, a frequency selecting algorithm is proposed. It 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
+overhead, works online and does not need any training or profiling.
This paper is organized as follows: Section~\ref{sec.relwork} presents some
related works from other authors. Section~\ref{sec.exe} describes how the
-execution time of MPI programs can be predicted. It also presents an energy
-model for heterogeneous platforms. Section~\ref{sec.compet} presents
+execution time of message passing programs can be predicted. It also presents an energy
+model that predicts the energy consumption of an application running over a heterogeneous platform. Section~\ref{sec.compet} presents
the energy-performance objective function that maximizes the reduction of energy
consumption while minimizing the degradation of the program's performance.
-Section~\ref{sec.optim} details the proposed 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.
+Section~\ref{sec.optim} details the proposed frequency selecting algorithm then the precision of the proposed algorithm is verified.\textbf{the verification should be put here}
+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.
+\textbf{never use we in an article and the algorithm is not heterogeneous! you cannot use scaling factors before defining what they are.}
\section{Related works}
\label{sec.relwork}
-Energy reduction process for a high performance clusters recently performed using
+Energy reduction process for 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
+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
+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{19,42}, 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{45}.
+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 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 a GPUs and the others executed
-on a CPUs. As an example of this works, Luley et al.~\cite{51}, proposed a heterogeneous
+heterogeneous clusters some parallel tasks executed on GPUs and the others executed
+on 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{49},
-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{50}, 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{52} 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
+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{53}, 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
+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{54} and \cite{55}, they predicated both the energy
+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{56} and \cite{57}, they
-are selected the best frequencies for a specified heterogeneous clusters offline using some
+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{58}, minimized
-the power consumption of a heterogeneous severs with time/space complexity, this approach
+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.
+it is works without any previous analysis for the application time complexity. The primary
+contributions of our paper are :
+\begin{enumerate}
+\item It is presents a new online heterogeneous scaling 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 scaling algorithm predicts both the energy and the execution time
+ of the iterative application.
+\item It demonstrates a new optimization function which maximize the performance and
+ minimize the energy consumption simultaneously.
+
+\end{enumerate}
\section{The performance and energy consumption measurements on heterogeneous architecture}
\label{sec.exe}
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
+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}).
+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.
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{17}. 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
+ 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,
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{45}.
+This prediction model is developed from 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{9,3,15,26} divide the power consumed by a processor into
+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
+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}).
\begin{equation}
\label{eq:pd}
- P_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F
+ Pd = \alpha \cdot C_L \cdot V^2 \cdot F
\end{equation}
-The static power $P_{s}$ captures the leakage power as follows:
+The static power $Ps$ captures the leakage power as follows:
\begin{equation}
\label{eq:ps}
- P_\textit{s} = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
+ Ps = 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
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
+ E_\textit{ind} = Pd \cdot Tcp + Ps \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
+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
communication and no slack time.
-The main objective of DVFS operation is to
-reduce the overall energy consumption~\cite{37}. The operational frequency $F$
-depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot F$ with some
+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{3}. The reduction
+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 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:
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{3}. Since the FLOPS of a CPU is proportional
+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:
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{3,46}, we assume that the static power of a processor is constant
+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
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)
+ E_\textit{s} = Ps \cdot (Tcp \cdot S + Tcm)
\end{equation}
In the considered heterogeneous platform, each processor $i$ might have different dynamic and
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{36}. We can measure the overall energy consumption for the iterative
+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.
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{45}, we proposed a method
+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
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{17}. Moreover, they are
-not measured using the same metric. To solve this problem, we normalize the
+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:
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{15,3,19}.
+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}
\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{44}. The experiments were executed
+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
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
+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{45,3}.
+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.
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
+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
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
\subsection{The results for different power consumption scenarios}
-
+\label{sec.compare}
The results of the previous section were obtained while using processors that consume during computation
an overall power which is 80\% composed of dynamic power and 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
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
+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)
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}).
+vector of frequency scaling factors that gives the results of the sections (\ref{sec.res}) and (\ref{sec.compare}).
\section{Conclusion}
\label{sec.concl}
-
+In this paper, we have presented a new online heterogeneous scaling algorithm
+that selects the best possible vector of frequency scaling factors. 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
+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
+known in advance and depends on the global convergence of the iterative system.
\section*{Acknowledgment}
+
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