Introduction

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 66a8ac21e7e34e49716405dc4dcea1c0d0e3022b..690c6a18fdb7d1a8830b2d611e164d985649c1e6 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -5,6 +5,7 @@
 \usepackage[english]{babel}
 \usepackage{algpseudocode}
 \usepackage{graphicx}
 \usepackage[english]{babel}
 \usepackage{algpseudocode}
 \usepackage{graphicx}

+\usepackage{algorithm}

 \usepackage{subfig}
 \usepackage{amsmath}
 
 \usepackage{subfig}
 \usepackage{amsmath}
 


@@ -49,11 +50,10 @@
 \newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
 \newcommand{\Tmax}{\Xsub{T}{max}}
 \newcommand{\Tnew}{\Xsub{T}{New}}
 \newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
 \newcommand{\Tmax}{\Xsub{T}{max}}
 \newcommand{\Tnew}{\Xsub{T}{New}}

-\newcommand{\Told}{\Xsub{T}{Old}}
+\newcommand{\Told}{\Xsub{T}{Old}} 
+\begin{document} 

 
 

-\begin{document}
-
-\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
+\title{Energy Consumption Reduction In a Heterogeneous Architecture Using DVFS}

 
 \author{%
   \IEEEauthorblockN{%
 
 \author{%
   \IEEEauthorblockN{%


@@ -61,7 +61,7 @@
     Raphaël Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch
     Raphaël Couturier,
     Ahmed Fanfakh and
     Arnaud Giersch

-  }
+  } 

   \IEEEauthorblockA{%
     FEMTO-ST Institute\\
     University of Franche-Comté\\
   \IEEEauthorblockA{%
     FEMTO-ST Institute\\
     University of Franche-Comté\\


@@ -81,13 +81,30 @@
 
 \section{Introduction}
 \label{sec.intro}
 
 \section{Introduction}
 \label{sec.intro}

-
+Modern processors continue to increased in a performance, achieved  maximum number of floating point operations per second (FLOPS), thus the energy consumption and the heat dissipation are increased drastically  according to this increase. The number of FLOPS is linearly related to power consumption of a CPU~\cite{51}.  
+As an example of more power hungry cluster, according to the Top500 list in June 2014 \cite{43}, Tianhe-2 has more than 3 millions of cores and consumed more  than 17.8 megawatt per second. Moreover, according to the U.S. annual energy outlook 2014 \cite{60}, the price of energy for 1 megawatt per hour is approximately equal to 70\$ (1.16\$ for megawatt per second). Therefore, we can consider the price of the energy consumption for the Tianhe-2 platform is approximately more than 390 millions dollars of megawatt per 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 are require nowadays.  For example, the GSIC center of Tokyo  heterogeneous cluster 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 lead to a disadvantage due to the performance degradation increase. Therefore,  researchers used different optimization strategies to overcame this problem. The best tradeoff relation between the energy reduction and performance degradation ratio is become 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 minimum energy saving against minimum performance degradation ratio simultaneously. The algorithm has very small overhead, works online and not needs for any training or profiling.  
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents some
+related works from other authors.  Section~\ref{sec.exe} describes how the
+execution time of MPI programs can be predicted.  It also presents an energy
+model for heterogeneous platforms. Section~\ref{sec.compet} presents
+the energy-performance objective function that maximizes the reduction of energy
+consumption while minimizing the degradation of the program's performance.
+Section~\ref{sec.optim} details the proposed heterogeneous scaling algorithm.
+Section~\ref{sec.expe} presents the results of running  the NAS benchmarks on 
+the proposed heterogeneous platform. It also shows the comparison of three different power
+scenarios and it verifies the precision of the proposed algorithm.  Finally, we conclude 
+in Section~\ref{sec.concl} with a summary and some future works.

 
 \section{Related works}
 \label{sec.relwork}
 
 \section{Related works}
 \label{sec.relwork}

-
-
-
+Energy reduction process for a high performance clusters recently performed using dynamic voltage and frequency scaling (DVFS) technique. DVFS is a technique  enabled in a modern processors to scaled down both of the  voltage and the frequency of the CPU while it is in the computing mode to reduce the energy consumption. DVFS is also  allowed in the graphical processors GPUs, to achieved the same goal. Applying DVFS has a dramatical side effect if it is applied to minimum levels to gain more energy reduction, producing a high percentage of performance degradations for the parallel applications.  Many researchers used different strategies to solve this nonlinear problem for example in~\cite{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}. 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 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 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 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 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 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 had considerable overhead. In our proposed scaling algorithm has very small overhead and it is works without any previous analysis for the application time complexity. 

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


@@ -107,18 +124,19 @@ power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
 have the same network bandwidth and latency.
 
 
 have the same network bandwidth and latency.
 
 

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

   \centering
     \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 
   \centering
     \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
   \label{fig:heter}
 \end{figure}
 

- The  overall execution time  of a distributed iterative synchronous application over a heterogeneous platform  consists of the sum of the computation time and the communication time for every iteration on a node. However, due to the heterogeneous computation power of the computing nodes, slack times might occur when fast nodes have to
- wait, during synchronous communications, for  the slower nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
- Therefore,  the overall execution time  of the program is the execution time of the slowest
- task which have the highest computation time and no slack time.
- 

 Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
 of a CPU by scaling down its voltage and frequency.  Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound.  The frequency reduction process can be  expressed by the scaling factor S which is the ratio between  the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
 \begin{equation}
 Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
 of a CPU by scaling down its voltage and frequency.  Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound.  The frequency reduction process can be  expressed by the scaling factor S which is the ratio between  the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
 \begin{equation}


@@ -141,17 +159,16 @@ vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
 \begin{equation}
   \label{eq:perf}
  \textit  T_\textit{new} = 
 \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}
 \end{equation}

-where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $MinT_{c}m$ 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 slowest node, it means  only the
  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 the execution time of one iteration as in EQ(\ref{eq:perf}) multiply by the number of iterations of the application.
+ communication time without any slack time. Therefore, we can consider the execution time of the iterative application is equal to the execution time of one iteration as in EQ(\ref{eq:perf}) multiplied by the number of iterations of that application.

 
 This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. 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}
 
 This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. 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
 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
 Many researchers~\cite{9,3,15,26} 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


@@ -198,7 +215,7 @@ power consumption:
 \begin{multline}
   \label{eq:pdnew}
    {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
 \begin{multline}
   \label{eq:pdnew}
    {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\

-   {} = \alpha \cdot C_L \cdot V^2 \cdot F \cdot S^{-3} = P_{dOld} \cdot S^{-3}
+   {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}

 \end{multline}
 where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the new frequency and the maximum frequency respectively.
 
 \end{multline}
 where $ {P}_\textit{dNew}$  and $P_{dOld}$ are the  dynamic power consumed with the new frequency and the maximum frequency respectively.
 


@@ -206,19 +223,18 @@ According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{
 reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$.  The new dynamic energy is the  dynamic power multiplied by the new time of computation and is given by the following equation:
 \begin{equation}
   \label{eq:Edyn}
 reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$.  The new dynamic energy is the  dynamic power multiplied by the new time of computation and is given by the following equation:
 \begin{equation}
   \label{eq:Edyn}

-   E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (T_{cp} \cdot S)= S^{-2}\cdot P_{dOld} \cdot  Tcp 
+   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 during idle and computation periods, and for all its available frequencies. 
 The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}), 
 the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related  
 \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 during idle and computation periods, and for all its available frequencies. 
 The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}), 
 the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related  

-to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy 
-of a processor after scaling its frequency is computed as follows: 
+to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy of a processor after scaling its frequency is computed as follows: 

 \begin{equation}
   \label{eq:Estatic}
  E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)
 \end{equation}
 
 \begin{equation}
   \label{eq:Estatic}
  E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)
 \end{equation}
 

-In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $P_{di}$ and $P_{si}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $T_{cpi}$ might be different and different frequency  scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $T_{cmi}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the frequency scaling factor and the dynamic power of each node as in EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing  distributed application executed over a heterogeneous platform during one iteration is the summation of all dynamic and static energies for each  processor.  It is computed as follows:
+In the considered heterogeneous platform, each processor $i$ might have different dynamic and static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $Tcp_{i}$ might be different and different frequency  scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $Tcm_{i}$ and could contain slack times if it is communicating with slower nodes, see figure(\ref{fig:heter}). Therefore, all nodes do not have equal communication times. While the dynamic energy is computed according to the frequency scaling factor and the dynamic power of each node as in EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing  distributed application executed over a heterogeneous platform during one iteration is the summation of all dynamic and static energies for each  processor.  It is computed as follows:

 \begin{multline}
   \label{eq:energy}
  E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot  Tcp_i)} + {} \\
 \begin{multline}
   \label{eq:energy}
  E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot  Tcp_i)} + {} \\


@@ -230,8 +246,8 @@ 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 
 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 

-application by measuring  the energy consumption from one iteration as in EQ(\ref{eq:energy}) multiply by 
-the number of iterations of the iterative application.
+application by measuring  the energy consumption for one iteration as in EQ(\ref{eq:energy}) multiplied by 
+the number of iterations of that application.

 
 
 \section{Optimization of both energy consumption and performance}
 
 
 \section{Optimization of both energy consumption and performance}


@@ -273,11 +289,6 @@ 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.  
 
 time simultaneously.  But the main objective is to produce maximum energy
 reduction with minimum execution time reduction.  
 

-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 to simultaneously optimize both energy and execution time
- without adding a big overhead. \textbf{put the last two phrases in the related work section}

  
   
 Our solution for this problem is to make the optimization process for energy and execution time follow the same
  
   
 Our solution for this problem is to make the optimization process for energy and execution time follow the same


@@ -320,50 +331,42 @@ Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq
 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}.
 
 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}.
 

-\section{The heterogeneous scaling algorithm }
+\section{The scaling factors selection algorithm for heterogeneous platforms }

 \label{sec.optim}
 
 \label{sec.optim}
 

-In this section we are proposed a heterogeneous scaling algorithm,
-(figure~\ref{HSA}), that selects the optimal vector of the frequency scaling factors from each
-node.  The algorithm is numerates the suitable range of available frequency scaling
-factors for each node in a heterogeneous cluster, returns a vector of optimal
-frequency scaling factors for all node define as $Sopt_1,Sopt_2,\dots,Sopt_N$. Using heterogeneous cluster 
-has different computing powers is produces different workloads for each node. Therefore, the fastest nodes waiting at the
-synchronous barrier for the slowest nodes to finish there work as in figure 
-(\ref{fig:heter}). Our algorithm is takes into account these imbalanced workloads
-when is starts to search for selecting the best vector of the frequency scaling factors. So, the
-algorithm is selects the initial frequencies values for each node proportional
-to the times of computations that gathered from the first iteration. As an
-example in figure (\ref{fig:st_freq}), the algorithm don't tests the first
-frequencies of the computing nodes until it is converge their frequencies to the
-frequency of the slowest node. If the algorithm is starts to test changing the
-frequency of the slowest node from the first gear, we are loosing the performance and
-then the best trade-off relation (the maximum distance) be not reachable. This case will be similar
-to a homogeneous cluster when all nodes scales their frequencies together from
-the first gear. Therefore, there is a small distance between the energy and
-the performance curves in a homogeneous cluster compare to heterogeneous one, for example see the figure(\ref{fig:r1}).  Then the
-algorithm starts to search for the optimal vector of the frequency scaling factors from the selected initial 
-frequencies until all node reach their minimum frequencies.
-\begin{figure}[t]
-  \centering
-    \includegraphics[scale=0.5]{fig/start_freq}
-  \caption{Selecting the initial frequencies}
-  \label{fig:st_freq}
-\end{figure}
+In this section we  propose algorithm~(\ref{HSA}) which selects the frequency scaling factors vector that gives the best trade-off between minimizing the energy consumption  and maximizing the performance of a message passing synchronous iterative application executed on a heterogeneous platform.  
+It works online during the execution time of the iterative message passing program.  It uses information gathered during the first iteration such as the computation time and the communication time in one iteration for each node. The algorithm is executed  after the first iteration and returns a vector of optimal frequency scaling factors   that satisfies the objective function EQ(\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs according to the computed scaling factors.  This algorithm is called just once during the execution of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called in the iterative MPI program.

 
 
 
 

-To compute the initial frequencies in each node, the algorithm firstly needs to compute the computation scaling factors $Scp_i$ of the node $i$. Each one of these factors is represents a ratio between the computation time of the slowest node and the computation time of the node $i$ as follow:
+The nodes in a heterogeneous platform have different computing powers, thus while executing message passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). These periods are called idle or slack times.
+Our algorithm takes into account this problem and tries to reduce these slack times when selecting the frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase the execution times of fast nodes and  minimize the  differences between  the  computation times of fast and slow nodes. The value of the initial frequency scaling factor  for each node is inversely proportional to its computation time that was gathered from the first iteration. These initial frequency scaling factors are computed as   a ratio between the computation time of the slowest node and the computation time of the node $i$ as follows:

 \begin{equation}
   \label{eq:Scp}
  Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
 \end{equation}
 \begin{equation}
   \label{eq:Scp}
  Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
 \end{equation}

-Depending on the initial computation scaling factors EQ(\ref{eq:Scp}), the algorithm computes the initial frequencies for all nodes as a ratio between the 
-maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as follow:
+Using the initial  frequency scaling factors computed in EQ(\ref{eq:Scp}), the algorithm computes the initial frequencies for all nodes as a ratio between the 
+maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as follows:

 \begin{equation}
   \label{eq:Fint}
  F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
 \end{equation}
 \begin{equation}
   \label{eq:Fint}
  F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
 \end{equation}

-\begin{figure}[tp]
+If the computed initial frequency for a node is not available in the gears of that node, the computed initial frequency is replaced by the nearest available frequency.
+In  figure (\ref{fig:st_freq}), the nodes are  sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are colored in blue in  figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal vector of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the algorithm that selects the frequency scaling factors starts the search method from these initial frequencies and takes a downward search direction toward lower frequencies. The algorithm iterates on all left frequencies, from the higher bound until all nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node according to EQ(\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of  all other nodes by one gear.
+The new overall energy consumption and execution time are computed according to the new scaling factors. The optimal set of frequency scaling factors is the set that gives the highest distance according to  the objective function EQ(\ref{eq:max}).
+
+The plots~(\ref{fig:r1} and \ref{fig:r2}) illustrate the normalized performance and consumed energy for an application running on a homogeneous platform and a heterogeneous platform respectively while increasing the scaling factors. It can be noticed that in a homogeneous platform the search for the optimal scaling factor should be started from the maximum frequency because the performance and the consumed energy is decreased since  the beginning of the plot. On the other hand, in  the heterogeneous platform the performance is  maintained at the beginning of the plot even if the frequencies of the faster nodes are decreased until the scaled down nodes have computing powers lower than the slowest node. In other words, until they reach the higher bound. It can also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger the maximum distance between the energy curve and the performance curve is while varying the scaling factors which results in bigger energy savings. 
+\begin{figure}[t]
+  \centering
+    \includegraphics[scale=0.5]{fig/start_freq}
+  \caption{Selecting the initial frequencies}
+  \label{fig:st_freq}
+\end{figure}
+
+
+
+
+
+\begin{algorithm}

   \begin{algorithmic}[1]
     % \footnotesize
     \Require ~
   \begin{algorithmic}[1]
     % \footnotesize
     \Require ~


@@ -375,7 +378,7 @@ maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as fol
     \item[$Ps_i$] array of the static powers for all nodes.
     \item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
     \end{description}
     \item[$Ps_i$] array of the static powers for all nodes.
     \item[$Fdiff_i$] array of the difference between two successive frequencies for all nodes.
     \end{description}

-    \Ensure $Sopt_1, \dots, Sopt_N$ is a set of optimal scaling factors
+    \Ensure $Sopt_1,Sopt_2 \dots, Sopt_N$ is a vector of optimal scaling factors

 
     \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
     \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
 
     \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
     \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$


@@ -406,24 +409,9 @@ maximum frequency of node $i$  and the computation scaling factor $Scp_i$ as fol
   \end{algorithmic}
   \caption{Heterogeneous scaling algorithm}
   \label{HSA}
   \end{algorithmic}
   \caption{Heterogeneous scaling algorithm}
   \label{HSA}

-\end{figure}
-When the initial frequencies are computed, the algorithm numerates all available
-frequency scaling factors starting from the initial frequencies until all nodes reach their
-minimum frequencies. At each iteration the algorithm determine the slowest node according to EQ(\ref{eq:perf}). 
-It is remains the frequency of the slowest node without change, while it is scale down the frequency of the other
-nodes. This is improved the execution time degradation and energy saving in the same time.  
-The proposed algorithm works online during the execution time of the iterative MPI program.  It is 
-returns a vector of optimal frequency scaling factors  depending on the
-objective function EQ(\ref{eq:max}). The program changes the new frequencies of
-the CPUs according to the computed scaling factors.  This algorithm 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 is 
-takes \np[ms]{0.04} on average for 4 nodes and \np[ms]{0.15} on average for 144
-nodes.  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 on average from 12 to 20 iterations to selects the best vector of frequency scaling factors that give the results of the next section. It is called just once during the execution of the program.  The DVFS algorithm in figure~(\ref{dvfs}) shows where
-and when the proposed scaling algorithm is called in the iterative MPI program.
-\begin{figure}[tp]
+\end{algorithm}
+
+\begin{algorithm}

   \begin{algorithmic}[1]
     % \footnotesize
     \For {$k=1$ to \textit{some iterations}}
   \begin{algorithmic}[1]
     % \footnotesize
     \For {$k=1$ to \textit{some iterations}}


@@ -441,49 +429,38 @@ and when the proposed scaling algorithm is called in the iterative MPI program.
   \end{algorithmic}
   \caption{DVFS algorithm}
   \label{dvfs}
   \end{algorithmic}
   \caption{DVFS algorithm}
   \label{dvfs}

-\end{figure}
+\end{algorithm}

 
 \section{Experimental results}
 \label{sec.expe}
 
 \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 on the simulator SimGrid/SMPI
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile} which offers easy tools to create a heterogeneous platform and run message passing applications over it. The heterogeneous platform that was used in the experiments, had one core per node because just one process was executed per node. The heterogeneous platform  was composed of four types of nodes. Each type of nodes had different characteristics such as the maximum CPU frequency, the number of
+available frequencies and the computational power, see table
+(\ref{table:platform}). The characteristics of these different types of  nodes are inspired   from the specifications of real Intel processors. The heterogeneous platform had up to 144 nodes and had nodes from the four types in equal proportions, for example if  a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the constructors of CPUs do not specify the dynamic and the static power of their CPUs, for each type of node they were chosen proportionally to  its computing power (FLOPS).  In the initial heterogeneous platform,  while computing with highest frequency, each node  consumed power proportional to its computing power which 80\% of it was dynamic power and the rest was 20\% for the static power, the same assumption  was made in \cite{45,3}. Finally, These nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
+

 
 

-The experiments of this work are executed on the simulator SimGrid/SMPI
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We are configure the
-simulator to use a heterogeneous cluster with one core per node. The proposed
-heterogeneous cluster has four different types of nodes. Each node in the cluster
-has different characteristics such as the maximum frequency speed, the number of
-available frequencies and dynamic and static powers values, see table
-(\ref{table:platform}). These different types of processing nodes are simulate some
-real Intel processors. The maximum number of nodes that supported by the cluster
-is 144 nodes according to characteristics of some MPI programs of the NAS
-benchmarks that used. We are use the same number from each type of nodes when we
-run the iterative MPI programs, for example if  we are execute the program on 8 node, there
-are 2 nodes from each type participating in the computation. The dynamic and
-static power values is different from one type to other. Each node has a dynamic
-and static power values proportional to their computing power (FLOPS), for more
-details see the Intel data sheets in \cite{47}.  Each node has a percentage of
-80\% for dynamic power and 20\% for static power of the total power
-consumption of a CPU, the same assumption is made in \cite{45,3}. These nodes are
-connected via an ethernet network with 1 Gbit/s bandwidth.

 \begin{table}[htb]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline
 \begin{table}[htb]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
   \begin{tabular}{|*{7}{l|}}
     \hline

-    Node     & Similar     & Max        & Min          & Diff.          & Dynamic      & Static \\
-    type     & to          & Freq. GHz  & Freq. GHz    & Freq GHz       & power        & power \\
+    Node          &Simulated  & Max      & Min          & Diff.          & Dynamic      & Static \\
+    type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
+                  &           & GHz      & GHz          &GHz             &              &       \\

     \hline
     \hline

-    1       & core-i3       & 2.5         & 1.2          & 0.1           & 20~w         &4~w    \\
-            &  2100T        &             &              &               &              &  \\
+    1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
+                  &           &          &              &                &              &  \\

     \hline
     \hline

-    2       & Xeon          & 2.66        & 1.6          & 0.133         & 25~w         &5~w    \\
-            & 7542          &             &              &               &              &  \\
+    2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
+                  &           &          &              &                &              &  \\

     \hline
     \hline

-    3       & core-i5       & 2.9         & 1.2          & 0.1           & 30~w         &6~w    \\
-            & 3470s         &             &              &               &              &  \\
+    3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
+                  &           &          &              &                &              &  \\

     \hline
     \hline

-    4       & core-i7       & 3.4         & 1.6          & 0.133         & 35~w         &7~w    \\
-            & 2600s         &             &              &               &              &  \\
+    4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
+                  &           &          &              &                &              &  \\

     \hline
   \end{tabular}
   \label{table:platform}
     \hline
   \end{tabular}
   \label{table:platform}


@@ -496,11 +473,10 @@ connected via an ethernet network with 1 Gbit/s bandwidth.
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 
 \subsection{The experimental results of the scaling algorithm}
 \label{sec.res}
 

-The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.3 
-\cite{44},  which were run with three classes (A, B and C).
-In this experiments we are interested to run the class C, the biggest class compared to A and B, on different number of 
-nodes, from 4 to 128 or 144 nodes according to the type of the iterative MPI program.  Depending on the proposed energy consumption model EQ(\ref{eq:energy}),
- we are measure the energy consumption for all NAS MPI programs. The dynamic and static power values are used under the same assumption used by \cite{45,3}. We are used a percentage of 80\% for dynamic power and 20\% for static of the total power consumption of a CPU. The heterogeneous nodes in table (\ref{table:platform}) have different  simulated computing power (FLOPS), ranked from the node of type 1 with smaller computing power (FLOPS) to the highest computing power (FLOPS) for node of type 4. Therefore, the power values are used proportionally increased from nodes of type 1 to nodes of type 4 that with highest computing power. Then, we are used an assumption that the power consumption is increased linearly when the computing power (FLOPS) of the processor is increased, see \cite{48}.   
+
+The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes: A,B and C. However, due to the lack of space in this paper, only the results of the biggest class, C, are presented while being run on different number of nodes, ranging  from 4 to 128 or 144 nodes depending on the benchmark being executed. Indeed, the benchmarks CG, MG, LU, EP and FT should be executed on $1, 2, 4, 8, 16, 32, 64, 128$ nodes. The other benchmarks such as BT and SP should be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
+
+ 

  
 \begin{table}[htb]
   \caption{Running NAS benchmarks on 4 nodes }
  
 \begin{table}[htb]
   \caption{Running NAS benchmarks on 4 nodes }


@@ -664,49 +640,43 @@ nodes, from 4 to 128 or 144 nodes according to the type of the iterative MPI pro
   \end{tabular}
   \label{table:res_128n}
 \end{table}
   \end{tabular}
   \label{table:res_128n}
 \end{table}

+The overall energy consumption was computed for each instance according to the energy consumption  model EQ(\ref{eq:energy}), with and without applying the algorithm. The execution time was also measured for all these experiments. Then, the energy saving and performance degradation percentages were computed for each instance.  
+The results are presented in tables (\ref{table:res_4n}, \ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n}, \ref{table:res_64n} and \ref{table:res_128n}). All these results are the average values from many experiments for  energy savings and performance degradation.

 
 

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

 
 

+
+ 

 \begin{figure}
   \centering
 \begin{figure}
   \centering

-  \subfloat[CG, MG, LU and FT benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
+  \subfloat[Energy saving]{%
+    \includegraphics[width=.2315\textwidth]{fig/energy}\label{fig:energy}}%

   \quad%
   \quad%

-  \subfloat[BT and SP benchmarks]{%
-    \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
+  \subfloat[Performance degradation ]{%
+    \includegraphics[width=.2315\textwidth]{fig/per_deg}\label{fig:per_deg}}

   \label{fig:avg}
   \label{fig:avg}

-  \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
+  \caption{The energy and performance for all NAS benchmarks running with difference number of nodes}

 \end{figure}
 
 \end{figure}
 

-In the NAS benchmarks there are some programs executed on different number of
-nodes. The benchmarks CG, MG, LU and FT executed on 2 to a power of (1, 2, 4, 8,
-\dots{}) of nodes. The other benchmarks such as BT and SP executed on 2 to a
-power of (1, 2, 4, 9, \dots{}) of nodes. We are take the average of energy
-saving, performance degradation and distances for all results of NAS
-benchmarks. The average of values of these three objectives are plotted to the number of
-nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}).  In CG, MG, LU, and
-FT benchmarks the average of energy saving is decreased when the number of nodes
-is increased because the communication times is increased as mentioned
-before. Thus, the average of distances (our objective function) is decreased
-linearly with energy saving while keeping the average of performance degradation approximately is 
-the same. In BT and SP benchmarks, the average of the  energy saving is not decreased
-significantly compare to other benchmarks when the number of nodes is
-increased. Nevertheless, the average of performance degradation approximately
-still the same ratio. This difference is depends on the characteristics of the
-benchmarks such as the computation to communication ratio that has.
+ 
+  \textbf{ The energy saving and performance degradation of all benchmarks are plotted to the number of
+nodes as in plots (\ref{fig:energy} and \ref{fig:per_deg}). As shown in the plots, the energy saving percentage of the benchmarks MG, LU, BT and FT is decreased linearly  when the the number of nodes increased. While in EP benchmark the energy saving percentage is approximately the same percentage when the number of computing nodes is increased, because in this benchmark there is no communications. In the SP benchmark the energy saving percentage is decreased when it runs on a small number of nodes, while this percentage is increased when it runs on a big number of nodes. The energy saving of the GC benchmarks  is significantly decreased when the number of nodes is increased, because  this benchmark has more communications compared to other benchmarks. The performance degradation percentage of the benchmarks CG, EP, LU and BT is decreased when they run on a big number of nodes. While in MG benchmark has a higher percentage of performance degradation when it runs on a big number of nodes. The inverse happen in SP benchmark has smaller performance degradation percentage when it runs on a big number of nodes.} 
+

 
 \subsection{The results for different power consumption scenarios}
 
 
 \subsection{The results for different power consumption scenarios}
 

-The results of the previous section are obtained using a percentage of 80\% for
-dynamic power and 20\% for static power of the total power consumption of a CPU. In this
-section we are change these ratio by using two others power scenarios. Because is
-interested to measure the ability of the proposed algorithm when these power ratios are changed. 
-In fact, we are used two different scenarios for dynamic and static power ratios in addition to the previous
-scenario in section (\ref{sec.res}). Therefore, we have three different
-scenarios for three different dynamic and static power ratios refer to these as: 
-70\%-20\%, 80\%-20\% and 90\%-10\% scenario respectively. The results of these scenarios
-running the NAS benchmarks class C on 8 or 9 nodes are place in the tables
-(\ref{table:res_s1} and \ref{table:res_s2}).
+The results of the previous section were obtained while using processors that consume during computation an overall power which is 80\% composed of  dynamic power and 20\% of static power. In this
+section, these ratios are changed and two new power scenarios are considered in order to evaluate how the proposed  algorithm adapts itself according to the static and dynamic power values.  The two new power scenarios are the following: 
+\begin{itemize}
+\item 70\% dynamic power  and 30\% static power
+\item 90\% dynamic power  and 10\% static power
+\end{itemize}
+The NAS parallel benchmarks were executed again over processors that follow the the new power scenarios. The class C of each benchmark was run over 8 or 9 nodes and the results are presented in  tables (\ref{table:res_s1} and \ref{table:res_s2}). \textbf{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, because this scenario uses higher percentage of dynamic dynamic power that is quadratically related to scaling factors. While the performance degradation percentage is less in 70\%-30\% scenario  compared to 90\%-10\%  scenario, because the first scenario used higher percentage for static power consumption that is linearly related to scaling factors and thus the execution time. }
+
+The two new power scenarios are compared to the old one in figure (\ref{fig:sen_comp}). It shows the average of the performance degradation, the energy saving and the distances for all NAS benchmarks of class C running on 8 or 9 nodes. The comparison shows that  the energy saving ratio is proportional to the dynamic power ratio: it is increased when applying the  90\%-10\% scenario because at maximum frequency the dynamic  energy is the the most relevant in the overall consumed energy and can be reduced by lowering the frequency of some processors. On the other hand, the energy saving is decreased when  the 70\%-30\% scenario is used because the dynamic  energy is less relevant in the overall consumed energy and lowering the frequency do not returns big energy savings.
+Moreover, the average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). Since the proposed algorithm optimizes the energy consumption when using a higher ratio for dynamic power the algorithm selects bigger frequency scaling factors that result in more energy saving but less performance, for example see the figure (\ref{fig:scales_comp}). The opposite happens when using a higher ratio for  static  power, the algorithm proportionally  selects  smaller scaling values which results in less energy saving but less performance degradation. 
+

 
  \begin{table}[htb]
   \caption{The results of 70\%-30\% powers scenario}
 
  \begin{table}[htb]
   \caption{The results of 70\%-30\% powers scenario}


@@ -770,18 +740,23 @@ running the NAS benchmarks class C on 8 or 9 nodes are place in the tables
   \subfloat[Comparison the average of the results on 8 nodes]{%
     \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
   \quad%
   \subfloat[Comparison the average of the results on 8 nodes]{%
     \includegraphics[width=.22\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
   \quad%

-  \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
+  \subfloat[Comparison the selected frequency scaling factors of MG benchmark class C running on 8 nodes]{%

     \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}
     \includegraphics[width=.24\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
   \label{fig:comp}
   \caption{The comparison of the three power scenarios}

-\end{figure}
+\end{figure}  
+

 
 

-To compare the results of these three powers scenarios, we are take the average of the performance degradation, the energy saving and the distances for all NAS benchmarks running on 8 or 9 nodes of class C, as in figure (\ref{fig:sen_comp}). Thus, according to the average of these results, the energy saving ratio is increased when using a higher percentage for dynamic power (e.g. 90\%-10\% scenario), due to increase in dynamic energy. While the average of energy saving is decreased in 70\%-30\% scenario. Because the static energy consumption is increase. Moreover, the average of distances is more related to energy saving changes. The average of the performance degradation is decreased when using a higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\% scenario). The raison behind these relations, that the proposed algorithm optimize both energy consumption and performance in the same time. Therefore, when using a higher ratio for dynamic power the algorithm selecting bigger frequency scaling factors values, more energy saving versus more performance degradation, for example see the figure (\ref{fig:scales_comp}). The inverse happen when using a higher ratio for  static  power, the algorithm proportionally  selects a smaller scaling values, less energy saving versus less performance degradation. This is because the  
-algorithm is optimizes the  static energy consumption that is always related to the execution time. 

 
 \subsection{The verifications of the proposed method}
 \label{sec.verif}
 
 \subsection{The verifications of the proposed method}
 \label{sec.verif}

-The precision of the proposed algorithm mainly depends on the execution time prediction model EQ(\ref{eq:perf}) and the energy model EQ(\ref{eq:energy}). The energy model is significantly depends on the execution time model, that the static energy is related linearly. So, our work is depends mainly on execution time model. To verifying thid model, we are compare the predicted execution time with the real execution time (Simgrid time) values that gathered  offline from the NAS benchmarks class B executed on 8 or 9 nodes. The execution time model can predicts the real execution time by maximum normalized error equal to 0.03 for all the NAS benchmarks. The second verification that we are made is for the proposed scaling algorithm to prove its ability to selects the best vector of the frequency scaling factors. Therefore, we are expand the algorithm to test at each iteration the frequency scaling factor of the slowest node with the all available scaling factors of the other nodes, all possible solutions. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the expanded algorithms and the proposed algorithm are identical. While the proposed algorithm is runs  by 10 times faster on average compare to the expanded algorithm.
+The precision of the proposed algorithm mainly depends on the execution time prediction model defined in EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}). 
+The energy model is also significantly dependent  on the execution time model because the static energy is linearly related the execution time and the dynamic energy is related to the computation time. So, all of the work presented in this paper is based on the execution time model. To verify this model, the predicted execution time was compared to  the real execution time over Simgrid for all  the NAS parallel benchmarks running class B on 8 or 9 nodes. The comparison showed that the proposed execution time model is very precise, the maximum normalized difference between  the predicted execution time  and the real execution time is equal to 0.03 for all the NAS benchmarks.
+
+Since  the proposed algorithm is not an exact method and do not test all the possible solutions (vectors of scaling factors) in the search space and to prove its efficiency, it was compared on small instances to a brute force search algorithm that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small
+execution time: for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in table~(\ref{table:platform}), it  
+takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number of iterations and $N$ is the number of computing nodes. The algorithm
+needs  from 12 to 20 iterations to select the best vector of frequency scaling factors that gives the results of the section (\ref{sec.res}).

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