some corr.

[mpi-energy2.git] / Heter_paper.tex

diff --git a/Heter_paper.tex b/Heter_paper.tex
index c5f885adc134a9dec61972503f2d68b4945ace20..66a8ac21e7e34e49716405dc4dcea1c0d0e3022b 100644 (file)
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -89,7 +89,6 @@
 
 
 
 
 
 

-

 \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}
 


@@ -104,8 +103,8 @@ passing distributed iterative synchronous applications running over
 heterogeneous platforms. We define a heterogeneous platform as a collection of
 heterogeneous computing nodes interconnected via a high speed homogeneous
 network. Therefore, each node has different characteristics such as computing
 heterogeneous platforms. We define a heterogeneous platform as a collection of
 heterogeneous computing nodes interconnected via a high speed homogeneous
 network. Therefore, each node has different characteristics such as computing

-power (FLOPS), energy consumption, CPU's frequency range, ... 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.

 
 
 \begin{figure}[t]
 
 
 \begin{figure}[t]


@@ -129,20 +128,24 @@ of a CPU by scaling down its voltage and frequency.  Since DVFS lowers the frequ
  The execution time of a compute bound sequential program is linearly proportional to the frequency scaling factor $S$. 
  On the other hand,  message passing distributed applications consist of two parts: computation and communication. The execution time of the computation part is linearly proportional to the frequency scaling factor $S$ but  the communication time is not affected by the scaling factor because  the processors involved remain idle during the  communications~\cite{17}. The communication time for a task is the summation of periods of time that begin with an MPI call for sending or receiving   a message till the message is synchronously sent or received.
 
  The execution time of a compute bound sequential program is linearly proportional to the frequency scaling factor $S$. 
  On the other hand,  message passing distributed applications consist of two parts: computation and communication. The execution time of the computation part is linearly proportional to the frequency scaling factor $S$ but  the communication time is not affected by the scaling factor because  the processors involved remain idle during the  communications~\cite{17}. The communication time for a task is the summation of periods of time that begin with an MPI call for sending or receiving   a message till the message is synchronously sent or received.
 

-Since in a heterogeneous platform, each node has different characteristics, especially different frequency gears, when applying DVFS operations on these nodes, they may get different scaling factors represented by a scaling vector: $(S_1, S_2,..., S_N)$ where $S_i$ is the scaling factor of processor $i$. To be able to predict the execution time of message passing synchronous iterative applications running over a heterogeneous platform, for different vectors of scaling factors, the  communication time and the computation time  for all the 
- tasks must be measured during the first iteration before applying any DVFS operation. Then the execution time for one iteration of the application with any vector of scaling factors  can be  predicted using EQ (\ref{eq:perf}).
- 
-  
-  
-\begin{multline}
+Since in a heterogeneous platform, each node has different characteristics,
+especially different frequency gears, when applying DVFS operations on these
+nodes, they may get different scaling factors represented by a scaling vector:
+$(S_1, S_2,\dots, S_N)$ where $S_i$ is the scaling factor of processor $i$. To
+be able to predict the execution time of message passing synchronous iterative
+applications running over a heterogeneous platform, for different vectors of
+scaling factors, the communication time and the computation time for all the
+tasks must be measured during the first iteration before applying any DVFS
+operation. Then the execution time for one iteration of the application with any
+vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
+\begin{equation}

   \label{eq:perf}
  \textit  T_\textit{new} = 
   \label{eq:perf}
  \textit  T_\textit{new} = 

- {} \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  TcmOld_{j}
-\end{multline}
-where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $TcmOld_j$ is the communication time of the slowest processor $j$.
-  The model computes the maximum computation time 
+ \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) +  MinTcm
+\end{equation}
+where $TcpOld_i$ is the computation time  of processor $i$ during the first iteration and $MinT_{c}m$ 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. 
+ 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.

 
 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.
 
 
 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.
 


@@ -170,7 +173,7 @@ 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 T_{cp} + P_\textit{s} \cdot T
+   E_\textit{ind} =  P_\textit{d} \cdot Tcp + P_\textit{s} \cdot T

 \end{equation}
 where $T$ is the execution time of the program, $T_{cp}$ is the computation
 time and $T_{cp} \leq T$.  $T_{cp}$ may be equal to $T$ if there is no
 \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


@@ -188,14 +191,14 @@ system's kernel to lower a core's frequency. we can calculate the new frequency
 $F_{new}$ from EQ(\ref{eq:s}) as follow:
 \begin{equation}
   \label{eq:fnew}
 $F_{new}$ from EQ(\ref{eq:s}) as follow:
 \begin{equation}
   \label{eq:fnew}

-   F_\textit{new} = S^{-1} .  F_\textit{max}
+   F_\textit{new} = S^{-1} \cdot F_\textit{max}

 \end{equation}
 Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic 
 power consumption:
 \begin{multline}
   \label{eq:pdnew}
    {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
 \end{equation}
 Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic 
 power consumption:
 \begin{multline}
   \label{eq:pdnew}
    {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\

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


@@ -203,69 +206,88 @@ 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  T_{cp} 
+   E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (T_{cp} \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  
 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: 
 \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: 

-

 \begin{equation}
   \label{eq:Estatic}
 \begin{equation}
   \label{eq:Estatic}

- E_\textit{s} = P_\textit{s} \cdot (T_{cp} \cdot S  + T_{cm})
+ E_\textit{s} = P_\textit{s} \cdot (Tcp \cdot S  + Tcm)

 \end{equation}
 
 \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 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 $P_{di}$ 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:

 \begin{multline}
   \label{eq:energy}
 \begin{multline}
   \label{eq:energy}

- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot P_{di} \cdot  T_{cpi})} +\\ 
- {}\sum_{i=1}^{N} {(P_{si} \cdot (\max_{i=1,2,\dots,N} (T_{cpi} \cdot S_{i}) +} 
- {}\min_{i=1,2,\dots,N} {T_{cmi}))} 
+ E = \sum_{i=1}^{N} {(S_i^{-2} \cdot Pd_{i} \cdot  Tcp_i)} + {} \\
+ \sum_{i=1}^{N} (Ps_{i} \cdot (\max_{i=1,2,\dots,N} (Tcp_i \cdot S_{i}) +
+  {MinTcm))}

  \end{multline}
  \end{multline}

- 
-Reducing the the frequencies of the processors according to the vector of scaling factors $(S_1, S_2,..., S_N)$  may degrade the  performance of the application and thus,
-increase the static energy because the execution time is increased~\cite{36}.
+
+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 
+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.
+

 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}
 
 \section{Optimization of both energy consumption and performance}
 \label{sec.compet}

-Applying DVFS to lower level  not surly reducing the energy consumption to minimum level. Also, a big scaling for the frequency produces high performance degradation percent. Moreover, by considering the  drastically  increase in execution time of parallel program, the static energy is related to this time 
-and it also increased by the same ratio. Thus, the opportunity for gaining more energy reduction is restricted. For that choosing frequency scaling factors is very important process to taking into account both energy and performance. In our previous work~\cite{45}, we are proposed a method that selects the optimal frequency scaling factor for an homogeneous cluster, depending on the trade-off relation between the energy and performance. In this work we have an  heterogeneous cluster, at each node there is different scaling factors, so our goal is to selects the optimal set of frequency scaling factors, $Sopt_1,Sopt_2,...,Sopt_N$, that gives the best trade-off  between energy consumption and performance. The relation between the energy and the execution time is complex and nonlinear, Thus, unlike the relation between the performance and  the scaling factor,  the relation of the energy with the frequency scaling factors is nonlinear, for more details refer to~\cite{17}.  Moreover, they are not measured using the same metric.  To solve this problem, we normalize the execution time by calculating the ratio between the new execution time (the scaled execution time) and the old one as follow:
+
+Using the lowest frequency for each processor does not necessarily gives the most energy efficient execution of an application. Indeed, even though the dynamic power is reduced while scaling down the frequency of a processor, its computation power is proportionally decreased and thus the execution time might be drastically increased during which dynamic and static powers are being consumed. Therefore,  it might cancel any gains achieved by scaling down the frequency of all nodes to the minimum  and the overall energy consumption of the application might not be the optimal one. It is not trivial to select the appropriate frequency scaling factor for each processor while considering the characteristics of each processor (computation power, range of frequencies, dynamic and static powers) and the task executed (computation/communication ratio) in order to reduce the overall energy consumption and not significantly increase the execution time. In our previous work~\cite{45}, we  proposed a method that selects the optimal 
+frequency scaling factor for a homogeneous cluster executing a message passing iterative synchronous application while giving the best trade-off
+ between the energy consumption and the performance for such applications. In this work we are interested in 
+heterogeneous clusters as described above. Due to the heterogeneity of the processors, not one but a  vector of scaling factors should be selected and it must  give the best trade-off between energy consumption and performance. 
+
+The relation between the energy consumption and the execution
+time for an application is complex and nonlinear, Thus, unlike the relation between the execution time 
+and the scaling factor, the relation of the energy with the frequency scaling
+factors is nonlinear, for more details refer to~\cite{17}.  Moreover, they are
+not measured using the same metric.  To solve this problem, we normalize the
+execution time by computing the ratio between the new execution time (after scaling down the frequencies of some processors) and the initial one (with maximum frequency for all nodes,) as follows:

 \begin{multline}
   \label{eq:pnorm}
   P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
 \begin{multline}
   \label{eq:pnorm}
   P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\

-          = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}} 
+       {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +MinTcm}

            {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
 \end{multline}
 
 
            {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
 \end{multline}
 
 

-By the same way, we are normalize the energy by calculating the ratio between the consumed energy with scaled frequency and the consumed energy without scaled frequency:
+In the same way, we normalize the energy by computing the ratio between the consumed energy while scaling down the frequency and the consumed energy with maximum frequency for all nodes:

 \begin{multline}
   \label{eq:enorm}
   E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
 \begin{multline}
   \label{eq:enorm}
   E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\

-   = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +
+  {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} +

  \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
  \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
 \end{multline} 
  \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +
  \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
 \end{multline} 

-Where $T_{New}$ and $T_{Old}$ is computed as in EQ(\ref{eq:pnorm}). The second problem 
-is that the optimization operation for both energy and performance is not in the same direction.  
-In other words, the normalized energy and the normalized execution time curves are not at the same direction.  
-While the main goal is to optimize the energy and execution time in the same time.  According to the 
-equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the set of frequency scaling factors $S_1,S_2,...,S_N$ reduce both the energy and the
-execution time simultaneously.  But the main objective is to produce maximum energy
-reduction with minimum execution time reduction.  Many researchers used different
-strategies to solve this nonlinear problem for example see~\cite{19,42}, their
-methods add big overheads to the algorithm to select the suitable frequency.
-In this paper we are present a method to find the optimal set of frequency scaling factors to optimize both energy and execution time simultaneously
-without adding a big overhead.  Our solution for this problem is to make the optimization process
-for energy and execution time follow the same direction.  Therefore, we inverse the equation of the normalized
-execution time, the normalized performance,  as follows:
-
+Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
+
+ While the main 
+goal is to optimize the energy and execution time at the same time, the normalized energy and execution time curves are not in the same direction. According 
+to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the vector  of frequency
+scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
+time simultaneously.  But the main objective is to produce maximum energy
+reduction with minimum execution time reduction.  
+
+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
+direction.  Therefore, we inverse the equation of the normalized execution time which gives 
+the normalized performance equation, as follows:

 \begin{multline}
   \label{eq:pnorm_inv}
   P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
           = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
 \begin{multline}
   \label{eq:pnorm_inv}
   P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
           = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}

-            { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +\min_{i=1,2,\dots,N} {Tcm_{i}}} 
+            { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm} 

 \end{multline}
 
 
 \end{multline}
 
 


@@ -283,46 +305,45 @@ execution time, the normalized performance,  as follows:
 Then, we can model our objective function as finding the maximum distance
 between the energy curve EQ~(\ref{eq:enorm}) and the  performance
 curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
 Then, we can model our objective function as finding the maximum distance
 between the energy curve EQ~(\ref{eq:enorm}) and the  performance
 curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This

-represents the minimum energy consumption with minimum execution time (better
+represents the minimum energy consumption with minimum execution time (maximum 

 performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}) .  Then our objective
 function has the following form:
 performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}) .  Then our objective
 function has the following form:

-\begin{multline}
+\begin{equation}

   \label{eq:max}
   Max Dist = 
   \max_{i=1,\dots F, j=1,\dots,N}
       (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
        \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
   \label{eq:max}
   Max Dist = 
   \max_{i=1,\dots F, j=1,\dots,N}
       (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
        \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )

-\end{multline}
+\end{equation}

 where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
 Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).  Our objective function can
 where $N$ is the number of nodes and $F$ is the  number of available frequencies for each nodes. 
 Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}).  Our objective function can

-work with any energy model or energy values stored in a data file.
-Moreover, this function works in optimal way when the energy curve has a convex
-form over the available frequency scaling factors 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 }
 \label{sec.optim}
 
 
 \section{The heterogeneous scaling algorithm }
 \label{sec.optim}
 

-In this section we proposed an heterogeneous scaling algorithm,
-(figure~\ref{HSA}), that selects the optimal set of scaling factors from each
-node.  The algorithm is numerates the suitable range of available scaling
-factors for each node in the heterogeneous cluster, returns a set of optimal
-frequency scaling factors for each node. Using heterogeneous cluster is produces
-different workloads for each node. Therefore, the fastest nodes waiting at the
-barrier for the slowest nodes to finish there work as in figure
-(\ref{fig:heter}). Our algorithm takes into account these imbalanced workloads
-when is starts to search for selecting the best scaling factors. So, the
-algorithm is selecting the initial frequencies values for each node proportional
+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
 to the times of computations that gathered from the first iteration. As an

-example in figure (\ref{fig:st_freq}), the algorithm don't test the first
-frequencies of the fastest nodes until it converge their frequencies to the
-frequency of the slowest node. If the algorithm is starts test changing the
-frequency of the slowest nodes from beginning, we are loosing performance and
-then not selecting the best trade-off (the distance). This case will be similar
-to the homogeneous cluster when all nodes scales their frequencies together from
-the beginning. In this case there is a small distance between energy and
-performance curves, for example see the figure(\ref{fig:r1}).  Then the
-algorithm searching for optimal frequency scaling factor from the selected
-frequencies until the last available ones.
+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}
 \begin{figure}[t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}


@@ -331,7 +352,7 @@ frequencies until the last available ones.
 \end{figure}
 
 
 \end{figure}
 
 

-To compute the initial frequencies, the algorithm firstly needs to compute the computation scaling factors $Scp_i$ for each node. Each one of these factors represent a ratio between the computation time of the slowest node and the computation time of the node $i$ as follow:
+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:

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


@@ -354,56 +375,54 @@ 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 $Sopt_1, \dots, Sopt_N$ is a set of optimal scaling factors

 
     \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
     \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
     \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
     \If{(not the first frequency)}
 
     \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
     \State $F_{i} \gets  \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
     \State Round the computed initial frequencies $F_i$ to the closest one available in each node.
     \If{(not the first frequency)}

-          \State $F_i \gets F_i+Fdiff_i,~i=1,...,N.$
+          \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$

     \EndIf 
     \EndIf 

-    \State $T_\textit{Old} \gets max_{~i=1,...,N } (Tcp_i+Tcm_i)$
+    \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$

     \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
     \State $Dist \gets 0$
     \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot  Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
     \State $Dist \gets 0$

-    \State  $Sopt_{i} \gets 1,~i=1,...,N. $
+    \State  $Sopt_{i} \gets 1,~i=1,\dots,N. $

     \While {(all nodes not reach their  minimum  frequency)}
         \If{(not the last freq. \textbf{and} not the slowest node)}
     \While {(all nodes not reach their  minimum  frequency)}
         \If{(not the last freq. \textbf{and} not the slowest node)}

-        \State $F_i \gets F_i - Fdiff_i,~i=1,...,N.$ 
-        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,...,N.$ 
+        \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
+        \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,\dots,N.$

         \EndIf
         \EndIf

-       \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + min_\textit{~i=1,\dots,N}(Tcm_i) $
+       \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm $

        \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} + $  \hspace*{43 mm} 
                $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
        \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
        \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
       \If{$(\Pnorm - \Enorm > \Dist)$}
        \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot  Tcp_i)} + $  \hspace*{43 mm} 
                $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
        \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
        \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
       \If{$(\Pnorm - \Enorm > \Dist)$}

-        \State $Sopt_{i} \gets S_{i},~i=1,...,N. $
+        \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $

         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile
         \State $\Dist \gets \Pnorm - \Enorm$
       \EndIf
     \EndWhile

-    \State  Return $Sopt_1,Sopt_2, \dots ,Sopt_N$
+    \State  Return $Sopt_1,Sopt_2,\dots,Sopt_N$

   \end{algorithmic}
   \caption{Heterogeneous scaling algorithm}
   \label{HSA}
 \end{figure}
   \end{algorithmic}
   \caption{Heterogeneous scaling algorithm}
   \label{HSA}
 \end{figure}

-When the initial frequencies are computed the algorithm numerates all available
-scaling factors starting from these frequencies until all nodes reach their
-minimum frequencies. At each iteration the algorithm remains the frequency of
-the slowest node without change and scaling the frequency of the other
-nodes. This is gives better performance and energy trade-off.  The proposed
-algorithm works online during the execution time of the MPI program.  Its
-returns a set of optimal frequency scaling factors $Sopt_i$ depending on the
+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
 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 an heterogeneous cluster composed of four different types of
-nodes having the characteristics presented in table~(\ref{table:platform}), it
-takes \np[ms]{0.04} on average for 4 nodes and \np[ms]{0.1} on average for 128
+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
 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 for all the NAS benchmark on class C
-to selects the best set of frequency scaling factors. Its called just once
-during the execution of the program.  The DVFS figure~(\ref{dvfs}) shows where
-and when the algorithm is called in the MPI program.
+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]
   \begin{algorithmic}[1]
     % \footnotesize
 \begin{figure}[tp]
   \begin{algorithmic}[1]
     % \footnotesize


@@ -428,23 +447,23 @@ and when the algorithm is called in the MPI program.
 \label{sec.expe}
 
 The experiments of this work are executed on the simulator SimGrid/SMPI
 \label{sec.expe}
 
 The experiments of this work are executed on the simulator SimGrid/SMPI

-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We configure the
+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
 simulator to use a heterogeneous cluster with one core per node. The proposed

-heterogeneous cluster has four different types of nodes. Each node in cluster
+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
 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 simulate some
+(\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
 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
-running the MPI programs, for example if we execute the program on 8 node, there
-are 2 nodes from each type participating in the computing. The dynamic and
+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
 static power values is different from one type to other. Each node has a dynamic

-and static power values proportional to their performance/GFlops, for more
+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
 details see the Intel data sheets in \cite{47}.  Each node has a percentage of

-80\% for dynamic power and 20\% for static power from the hole power
-consumption, the same assumption is made in \cite{45,3}. These nodes are
-connected via an Ethernet network with 1 Gbit/s bandwidth.
+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
 \begin{table}[htb]
   \caption{Heterogeneous nodes characteristics}
   % title of Table


@@ -477,11 +496,11 @@ 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.
+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).
 \cite{44},  which were run with three classes (A, B and C).

-In this experiments we are focus on running of 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 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 used under the same assumption used by \cite{45,3}. We used a percentage of 80\% for dynamic power from all power consumption of the CPU and 20\% for static power. The heterogeneous nodes in table (\ref{table:platform}) have different  simulated performance, ranked from the node of type 1 with smaller performance/GFlops to the highest performance/GFlops for node 4. Therefore, the power values used proportionally increased from node 1 to node 4 that with highest performance. Then we used an assumption that the power consumption increases linearly with the performance/GFlops of the processor see \cite{48}.   
+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}.   

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


@@ -646,34 +665,47 @@ nodes, from 4 to 128 or 144 nodes according to the type of the MPI program.  Dep
   \label{table:res_128n}
 \end{table}
 
   \label{table:res_128n}
 \end{table}
 

-The results of applying the proposed scaling algorithm to 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 show the 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. While the distance is decreased by the same direction. This because when we are run the MPI programs on a big number of nodes the communications is biggest than the computations, so the static energy is increased linearly to these times. The tables also show that performance degradation percent still approximately the same or decreased a little when the number of computing nodes is increased. This  gives good a prove that the proposed algorithm keeping as mush as possible the 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. 

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

-  \subfloat[Balanced nodes type scenario]{%
+  \subfloat[CG, MG, LU and FT benchmarks]{%

     \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
   \quad%
     \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
   \quad%

-  \subfloat[Imbalanced nodes type scenario]{%
+  \subfloat[BT and SP benchmarks]{%

     \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
   \label{fig:avg}
   \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
 \end{figure}
 
     \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
   \label{fig:avg}
   \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
 \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, ...) of nodes. The other benchmarks such as BT and SP executed on 2 to a power of (1, 2, 4, 9, ...) of nodes. We are take the average of energy saving, performance degradation and distances for all results of NAS benchmarks. The average 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 due to the increasing in the communication times as mentioned before. Thus, the average of distances (our objective function) is decreased linearly with energy saving while keeping the average of performance degradation the same. In BT and SP benchmarks, the average of 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. 
-
-\subsection{The results for different powers scenarios}
+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.
+
+\subsection{The results for different power consumption scenarios}

 
 The results of the previous section are obtained using a percentage of 80\% for
 
 The results of the previous section are obtained using a percentage of 80\% for

-dynamic power and 20\% for static power of total power consumption. In this
-section we are change these ratio by using two others scenarios. Because is
-interested to measure the ability of the proposed algorithm to changes it
-behavior when these power ratios are changed. In fact, we are use two different
-scenarios for dynamic and static power ratios in addition to the previous
+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
 scenario in section (\ref{sec.res}). Therefore, we have three different

-scenarios for three different dynamic and static power ratios refer to as:
-70\%-20\%, 80\%-20\% and 90\%-10\% scenario. The results of these scenarios
-running NAS benchmarks class C on 8 or 9 nodes are place in the tables
+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}).
 
  \begin{table}[htb]
 (\ref{table:res_s1} and \ref{table:res_s2}).
 
  \begin{table}[htb]


@@ -744,14 +776,12 @@ running NAS benchmarks class C on 8 or 9 nodes are place in the tables
   \caption{The comparison of the three power scenarios}
 \end{figure}
 
   \caption{The comparison of the three power scenarios}
 \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 the results, the energy saving ratio is increased when using the a higher percentage for dynamic power (e.g. 90\%-10\% scenario). 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 also keeps as much as possible the static energy consumption that is always related to execution time. 
+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 and the energy model. The energy model is significantly depends on the execution time model EQ(\ref{eq:perf}), that the energy static related linearly. So, we are compare the predicted execution time with the real execution time (Simgrid time)  values that gathered  offline for 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 0.03 of all NAS benchmarks. The second verification that we are make is for the scaling algorithm to prove its ability to selects the best set of 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 scaling factors available of the other nodes. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the two algorithms is identical. While the proposed algorithm is runs faster, 10 times faster on average than the expanded algorithm.
+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.

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