some corrections

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 3ba32046ce6d91f8eefba744bb466a3c787c05e9..babdfc2373f30e70758c5d0a7a777cbcd4283a7d 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -76,92 +76,99 @@
 \maketitle
 
 \begin{abstract}
 \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}
 \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 
+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. 
 \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, 
+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}. 
 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 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 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. 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{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 
+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
 
 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.
 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. 
+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.
 
 \section{Related works}
 \label{sec.relwork}
 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 
+Energy reduction process for high performance clusters recently performed using 

 dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique enabled 
 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 
 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{Hao_Learning.based.DVFS,Dhiman_Online.Learning.Power.Management}, their methods 
 add big overheads to the algorithm to select the suitable frequency.  
 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}. 
+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 
 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.
+heterogeneous clusters some parallel tasks executed on  GPUs and the others executed 
+on  CPUs. As an example of these 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{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 
+~\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 
 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. 
+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. 
 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, 
+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. 
 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. 


@@ -169,7 +176,7 @@ Other works such as Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduli
 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 
 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 
+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 
 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 


@@ -183,15 +190,25 @@ they are selected the best frequencies for a specified heterogeneous clusters of
 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 
 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. 
+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 :
+\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.
+      
+\end{enumerate}

 
 \section{The performance and energy consumption measurements on heterogeneous architecture}
 \label{sec.exe}
 
 
 \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}
 
 \subsection{The execution time of message passing distributed 
                 iterative applications on a heterogeneous platform}


@@ -204,12 +221,12 @@ 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.
 
 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 
 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.
   
 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.
   


@@ -261,13 +278,34 @@ vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
 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 
 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 
+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}).

 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.
 
 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 
+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 
 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 


@@ -280,17 +318,17 @@ Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Schedul
 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
 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}
 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}
 \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}
 \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
 \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


@@ -298,10 +336,10 @@ technology-dependent parameter.  The energy consumed by an individual processor
 to execute a given program can be computed as:
 \begin{equation}
   \label{eq:eind}
 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}
 \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{Le_DVFS.Laws.of.Diminishing.Returns}.  
 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}.  


@@ -309,7 +347,7 @@ The operational frequency $F$ depends linearly on the supply voltage $V$, i.e.,
 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
 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 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:
 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:


@@ -347,7 +385,7 @@ to the frequency scaling factor, while this scaling factor does not affect the c
 The static energy of a processor after scaling its frequency is computed as follows: 
 \begin{equation}
   \label{eq:Estatic}
 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 
 \end{equation}
 
 In the considered heterogeneous platform, each processor $i$ might have different dynamic and 


@@ -480,6 +518,7 @@ the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynam
 \section{The scaling factors selection algorithm for heterogeneous platforms }
 \label{sec.optim}
 
 \section{The scaling factors selection algorithm for heterogeneous platforms }
 \label{sec.optim}
 

+\subsection{The algorithm details}

 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 
 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 


@@ -617,10 +656,33 @@ which results in bigger energy savings.
   \label{dvfs}
 \end{algorithm}
 
   \label{dvfs}
 \end{algorithm}
 

+\subsection{The verifications of the proposed algorithm}
+\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 
+\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 
+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 next sections.
+

 \section{Experimental results}
 \label{sec.expe}
 To evaluate the efficiency and the overall energy consumption reduction of algorithm~(\ref{HSA}), 
 \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 
+it was applied to the NAS parallel benchmarks NPB v3.3. 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 
 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 


@@ -631,7 +693,7 @@ of these different types of  nodes are inspired   from the specifications of rea
 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 
 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{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. 
 Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
 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.


@@ -887,8 +949,8 @@ Plots (\ref{fig:energy} and \ref{fig:per_deg}) present the energy saving and per
 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 
 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 
 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 


@@ -898,7 +960,7 @@ down the frequencies of some nodes have less effect on the performance.
 
 
 \subsection{The results for different power consumption scenarios}
 
 
 \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 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  


@@ -1007,36 +1069,29 @@ results in less energy saving but less performance degradation.
 
 
 
 
 
 

-\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}
 
 \section{Conclusion}

-\label{sec.concl}
-
+\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  
+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}
 
 
 
 \section*{Acknowledgment}
 
 

+

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