new corrections

[ThesisAhmed.git] / CHAPITRE_02.tex

diff --git a/CHAPITRE_02.tex b/CHAPITRE_02.tex
index 2443000ba475ceab8c0e1e1929de2cfe374edd17..c6a9a2998be6487400b0bb6e952ee883c3fb3524 100644 (file)
--- a/CHAPITRE_02.tex
+++ b/CHAPITRE_02.tex
@@ -26,40 +26,39 @@
 
 Dynamic Voltage and Frequency Scaling (DVFS) can be applied to modern CPUs. This technique is 
 usually used to reduce the energy consumed by a CPU while computing. Indeed,
 
 Dynamic Voltage and Frequency Scaling (DVFS) can be applied to modern CPUs. This technique is 
 usually used to reduce the energy consumed by a CPU while computing. Indeed,

-power consumption by a processor at a given time is exponentially related to its frequency. 
+power consumption by a processor  is exponentially related to its frequency. 

 Thus, decreasing the frequency reduces the power consumed by the CPU. However, it can also 
 Thus, decreasing the frequency reduces the power consumed by the CPU. However, it can also 

-significantly affect the performance of the executed program if it is compute bound and 
-if a low CPU frequency is selected. The performance degradation ratio can even be higher 
+significantly affect the performance of the executed program if it is compute bound. The performance degradation ratio can even be higher 

 than the saved energy ratio. Therefore, the chosen frequency scaling factor must give the best possible
 trade-off between energy reduction and performance. This chapter presents an algorithm 
 that predicts the energy consumed with each frequency gear and selects the one that gives 
 the best ratio between energy consumption reduction and performance. Furthermore, the main
 objective of HPC systems is to execute as fast as possible the application. Therefore, our 
 than the saved energy ratio. Therefore, the chosen frequency scaling factor must give the best possible
 trade-off between energy reduction and performance. This chapter presents an algorithm 
 that predicts the energy consumed with each frequency gear and selects the one that gives 
 the best ratio between energy consumption reduction and performance. Furthermore, the main
 objective of HPC systems is to execute as fast as possible the application. Therefore, our 

-algorithm selects the scaling factor online with very small overhead.  The proposed algorithm 
-takes into account both the computation and communication times of the Message passing 
-programs (MPI) to choose the frequency scaling factor. 
+algorithm selects the scaling factor online with a very small overhead.  The proposed algorithm 
+takes into account both the computation and communication times of the Message Passing 
+Interface  (MPI) programs to choose the frequency scaling factor. 

 This algorithm has the ability to predict both energy consumption and execution time over 
 all available scaling factors.  The prediction achieved depends on some computing time information,
 This algorithm has the ability to predict both energy consumption and execution time over 
 all available scaling factors.  The prediction achieved depends on some computing time information,

-gathered at the beginning of the runtime.  We apply this algorithm to seven MPI
-benchmarks.  These MPI programs are the NAS parallel benchmarks (NPB v3.3)
-developed by NASA~\cite{ref65}.  Our experiments are executed using the simulator
+gathered at the beginning of the runtime.  We have applied this algorithm to the NAS parallel 
+benchmarks (NPB v3.3) developed by the  NASA~\cite{ref65}.  Our experiments are executed using the simulator

 SimGrid/SMPI v3.10~\cite{ref66} over an homogeneous
 distributed memory architecture. 
 
 SimGrid/SMPI v3.10~\cite{ref66} over an homogeneous
 distributed memory architecture. 
 

-Mainly there are two tasks in this chapter, the first one is showed the results of the proposed frequency scaling selection algorithm  using the energy model of the Rauber and Rünger \cite{ref47}. Furthermore, the proposed algorithm  is compared with Rauber and Rünger's method.  The comparison results show that our algorithm gives better energy-time trade-off. In the second task,  a new energy model  that takes into account both the communication and computation times of the MPI programs running over homogeneous cluster is developed.
-It  also shows the new results obtained using the new energy model and comparing them with the ones use Rauber and Rünger energy model.  
+This chapter is composed of two parts. The first part, the proposed frequency scaling selection algorithm  uses the energy model of the Rauber and Rünger \cite{ref47} and  is compared to Rauber and Rünger's method.  The comparison results show that our algorithm gives better energy-time trade-off. In the second part,  a new energy model  that takes into account both the communication and computation times of the MPI programs running over a homogeneous cluster is developed.
+It  also shows the new results obtained using the new energy model. The results are compared to the  ones given by Rauber and Rünger's energy model.  

 
 
 This chapter is organized as follows:  Section~\ref{ch2:2} explains the execution
 of parallel tasks and the sources of slack times.  It also presents an energy
 
 
 This chapter is organized as follows:  Section~\ref{ch2:2} explains the execution
 of parallel tasks and the sources of slack times.  It also presents an energy

-model for homogeneous platforms from other authors.  Section~\ref{ch2:3} describes how the
+model for homogeneous platforms from other researchers.  Section~\ref{ch2:3} describes how the

 performance of MPI programs can be predicted.  Section~\ref{ch2:4} presents
 the energy-performance objective function that maximizes the reduction of energy
 consumption while minimizing the degradation of the program's performance.
 performance of MPI programs can be predicted.  Section~\ref{ch2:4} presents
 the energy-performance objective function that maximizes the reduction of energy
 consumption while minimizing the degradation of the program's performance.

-Section~\ref{ch2:5} details the proposed energy-performance algorithm.
+Section~\ref{ch2:5} details the algorithm that returns the scaling factor that gives the best energy-performance 
+trade-off for a parallel  application with iterations

 Section~\ref{ch2:6} verifies the accuracy of the performance prediction model
 and presents the results of the proposed algorithm.  It also shows the
 Section~\ref{ch2:6} verifies the accuracy of the performance prediction model
 and presents the results of the proposed algorithm.  It also shows the

-comparison results between our method and other existing method.  
+comparison results between our method and other existing methods.  

 Section~\ref{ch2:7} describes the new proposed energy consumption model for 
 homogeneous platforms. Section~\ref{ch2:8} presents the experimental results 
 of using the new energy model. Finally,  section~\ref{ch2:9} summarizes this chapter.
 Section~\ref{ch2:7} describes the new proposed energy consumption model for 
 homogeneous platforms. Section~\ref{ch2:8} presents the experimental results 
 of using the new energy model. Finally,  section~\ref{ch2:9} summarizes this chapter.


@@ -90,14 +89,14 @@ executed on other processors.  Freeh et al. showed in~\cite{ref53} that the
 communication times of MPI programs do not change when the frequency is scaled
 down.  On the other hand, some offline scaling factor selection methods use the
 information gathered from previous full or partial executions of the program. 
 communication times of MPI programs do not change when the frequency is scaled
 down.  On the other hand, some offline scaling factor selection methods use the
 information gathered from previous full or partial executions of the program. 

-The whole program or, a part of it,  is usually executed over all the available frequency
+The whole program or a part of it  is usually executed over all the available frequency

 gears and the execution time and the energy consumed with each frequency
 gear are measured.  Then a heuristic or an exact method uses the retrieved
 information to compute the values of the scaling factor for the processors.
 In~\cite{ref57}, Xie et al. use an exact exponential breadth-first search algorithm
 to compute the scaling factor values that give the optimal energy reduction
 while respecting a deadline for a sequential program.  They also present a
 gears and the execution time and the energy consumed with each frequency
 gear are measured.  Then a heuristic or an exact method uses the retrieved
 information to compute the values of the scaling factor for the processors.
 In~\cite{ref57}, Xie et al. use an exact exponential breadth-first search algorithm
 to compute the scaling factor values that give the optimal energy reduction
 while respecting a deadline for a sequential program.  They also present a

-linear heuristic that approximates the optimal solution.  In~\cite{ref58} , Rountree
+linear heuristic that approximates the optimal solution.  In~\cite{ref58}, Rountree

 et al. use a linear programming algorithm, while in~\cite{ref59,ref60}, Cochran et
 al. use a multi-logistic regression algorithm for the same goal.  The main
 drawback of these methods is that they all require executing the
 et al. use a linear programming algorithm, while in~\cite{ref59,ref60}, Cochran et
 al. use a multi-logistic regression algorithm for the same goal.  The main
 drawback of these methods is that they all require executing the


@@ -109,12 +108,12 @@ the same program.
 The online scaling factor selection methods are executed during the runtime of
 the program.  They are usually integrated into iterative programs where the same
 block of instructions is executed many times.  During the first few iterations,
 The online scaling factor selection methods are executed during the runtime of
 the program.  They are usually integrated into iterative programs where the same
 block of instructions is executed many times.  During the first few iterations,

-a lot of information is measured such as the execution time, the energy consumed
+a lot of information are measured such as the execution time, the energy consumed

 using a multimeter, the slack times, \dots{} Then a method will exploit these
 measurements to compute the scaling factor values for each processor.  This
 using a multimeter, the slack times, \dots{} Then a method will exploit these
 measurements to compute the scaling factor values for each processor.  This

-operation, measurements and computing new scaling factor, can be repeated as
+operation, measurements and computing new scaling factors, can be repeated as

 much as needed if the iterations are not regular.  Kimura, Peraza, Yu-Liang et
 much as needed if the iterations are not regular.  Kimura, Peraza, Yu-Liang et

-al.~\cite{ref61,ref55,ref62} used varied heuristics to select the appropriate scaling
+al.~\cite{ref61,ref55,ref62} used many heuristics to select the appropriate scaling

 factor values to eliminate the slack times during runtime.  However, as seen
 in~\cite{ref63,ref64}, machine learning methods can take a lot of time to converge
 when the number of available gears is big.  To reduce the impact of slack times,
 factor values to eliminate the slack times during runtime.  However, as seen
 in~\cite{ref63,ref64}, machine learning methods can take a lot of time to converge
 when the number of available gears is big.  To reduce the impact of slack times,


@@ -134,7 +133,7 @@ distributed architectures by taking into account the communication times.  The
 primary contribution of this chapter is to present a new online scaling factor
 selection method which has the following characteristics:
 \begin{enumerate}
 primary contribution of this chapter is to present a new online scaling factor
 selection method which has the following characteristics:
 \begin{enumerate}

-\item It is based on both  Rauber and Rünger and the new developed energy models to predict the energy consumption of the application with different frequency gears. 
+\item It is based on both  Rauber and Rünger and the new  energy model to predict the energy consumption of the application with different frequency gears. 

 \item It selects the frequency scaling factor for simultaneously optimizing
   energy reduction and maintaining performance.
 \item It is well adapted to distributed architectures because it takes into
 \item It selects the frequency scaling factor for simultaneously optimizing
   energy reduction and maintaining performance.
 \item It is well adapted to distributed architectures because it takes into


@@ -146,17 +145,17 @@ selection method which has the following characteristics:
 \end{enumerate}
 
 
 \end{enumerate}
 
 

-\section{Execution and energy of parallel tasks on homogeneous platform}
+\section{Execution time and energy consumption of parallel tasks running on a homogeneous platform}

 \label{ch2:2}
 
 \label{ch2:2}
 

-\subsection{Parallel tasks execution on homogeneous platform}
+\subsection{Parallel tasks execution on a homogeneous platform}

 \label{ch2:2:1}
 A homogeneous cluster consists in identical nodes in terms of hardware and
 software.  Each node has its own memory and at least one processor which can be
 a multi-core.  The nodes are connected via a high bandwidth network.  Tasks
 executed on this model can be either synchronous or asynchronous.  In this chapter
 \label{ch2:2:1}
 A homogeneous cluster consists in identical nodes in terms of hardware and
 software.  Each node has its own memory and at least one processor which can be
 a multi-core.  The nodes are connected via a high bandwidth network.  Tasks
 executed on this model can be either synchronous or asynchronous.  In this chapter

-we consider execution of the synchronous tasks on distributed homogeneous
-platform.  These tasks can exchange the data via synchronous message passing.
+we consider the execution of  synchronous tasks on distributed homogeneous
+platform.  These tasks can synchronously exchange  data via message passing.

 
 
 
 
 
 


@@ -165,21 +164,21 @@ platform.  These tasks can exchange the data via synchronous message passing.
 \centering
 \includegraphics[scale=0.73]{fig/ch2/commtasks} 
 \includegraphics[scale=0.73]{fig/ch2/compt}\\ ~ ~ ~ ~ ~  ~(a) ~ ~  ~ ~ ~  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~(b)
 \centering
 \includegraphics[scale=0.73]{fig/ch2/commtasks} 
 \includegraphics[scale=0.73]{fig/ch2/compt}\\ ~ ~ ~ ~ ~  ~(a) ~ ~  ~ ~ ~  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~(b)

-\caption{Parallel tasks execution on homogeneous platform (a) imbalanced communications and (b) imbalanced  
+\caption{Parallel tasks execution on a homogeneous platform (a) imbalanced communications and (b) imbalanced  

 computations}
 \label{fig:homo}
 \end{figure}
 
 computations}
 \label{fig:homo}
 \end{figure}
 

-Therefore, the execution time of a task consists in the computation time and the
+The execution time of a task consists in the computation time and the

 communication time.  Moreover, the synchronous communications between tasks can
 communication time.  Moreover, the synchronous communications between tasks can

-lead to slack times while tasks wait at the synchronization barrier for other
+lead to slack times while tasks wait at a synchronization barrier for other

 tasks to finish their tasks (see figure~\ref{fig:homo}(a)).  The imbalanced
 communications happen when nodes have to send/receive different amounts of data
 or they communicate with different numbers of nodes.  Other sources of slack
 times are imbalanced computations.  This happens when processing different
 amounts of data on each processor (see figure~\ref{fig:homo}(b)).  In this case the
 fastest tasks have to wait at the synchronization barrier for the slowest ones
 tasks to finish their tasks (see figure~\ref{fig:homo}(a)).  The imbalanced
 communications happen when nodes have to send/receive different amounts of data
 or they communicate with different numbers of nodes.  Other sources of slack
 times are imbalanced computations.  This happens when processing different
 amounts of data on each processor (see figure~\ref{fig:homo}(b)).  In this case the
 fastest tasks have to wait at the synchronization barrier for the slowest ones

-to begin the next task.  In both cases the overall execution time of the program
+to continue their computations.  In both cases the overall execution time of the program

 is the execution time of the slowest task as in EQ~(\ref{eq:T1}).
 \begin{equation}
   \label{eq:T1}
 is the execution time of the slowest task as in EQ~(\ref{eq:T1}).
 \begin{equation}
   \label{eq:T1}


@@ -188,10 +187,10 @@ is the execution time of the slowest task as in EQ~(\ref{eq:T1}).
 where $T_i$ is the execution time of task $i$ and all the tasks are executed
 concurrently on different processors.
 
 where $T_i$ is the execution time of task $i$ and all the tasks are executed
 concurrently on different processors.
 

-\subsection{Energy model for homogeneous platform}
+\subsection{Energy consumption model for a homogeneous platform}

 \label{ch2:2:2}
 
 \label{ch2:2:2}
 

-The total energy consumption model for a parallel homogeneous
+The total energy  for a parallel homogeneous

 platform, as presented by Rauber and Rünger~\cite{ref47}, can be written as a
 function of the scaling factor $S$, as in EQ~\ref{eq:energy}.
 Moreover, the scaling factor $S_1$ is the scaling factor which should be the
 platform, as presented by Rauber and Rünger~\cite{ref47}, can be written as a
 function of the scaling factor $S$, as in EQ~\ref{eq:energy}.
 Moreover, the scaling factor $S_1$ is the scaling factor which should be the


@@ -201,11 +200,11 @@ factors of the others tasks $S_i$ are computed as in EQ~\ref{eq:si}.
 \begin{equation}
   \label{eq:si}
   S_i = S \cdot \frac{T_1}{T_i}
 \begin{equation}
   \label{eq:si}
   S_i = S \cdot \frac{T_1}{T_i}

-      = \frac{F_{max}}{F_{new}} \cdot \frac{T_1}{T_i}
+      = \frac{F_{max}}{F_{new}} \cdot \frac{T_1}{T_i},~{i=1,2,\dots,N}

 \end{equation}
 
 \end{equation}
 

-We compare our algorithm with Rauber and Rünger's scaling factor selection
-method which uses the same energy model.  In their method, the optimal scaling factor is
+Rauber and Rünger's scaling factor selection
+method  uses the same energy model.  In their method, the optimal scaling factor is

 computed by minimizing the derivation of EQ~(\ref{eq:energy}) which produces
 EQ~(\ref{eq:sopt}).
 
 computed by minimizing the derivation of EQ~(\ref{eq:energy}) which produces
 EQ~(\ref{eq:sopt}).
 


@@ -215,24 +214,22 @@ EQ~(\ref{eq:sopt}).
     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 
     \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
 \end{equation}
 

-This model is computed the frequency scaling factor which optimizes the energy consumption of the parallel
-program into minimal level.
+This model  computes the frequency scaling factor which minimizes the energy consumption of the parallel program.

 
 \section{Performance evaluation of MPI programs}
 \label{ch2:3}
 
 
 \section{Performance evaluation of MPI programs}
 \label{ch2:3}
 

-The execution time of a parallel synchronous iterative application is equal
-to the execution time of the slowest task as in figure~(\ref{fig:homo}).  
-If there is no communication and the application is not data bounded, the execution time of a
-parallel program is linearly proportional to the operational frequency and any
+The execution time of a parallel synchronous  application with iteration is equal
+to the execution time of its slowest task as in figure~(\ref{fig:homo}).  
+If there is no communication in the application and it is not data bounded, the execution time of this 
+parallel application is linearly proportional to the operational frequency. Any 

 DVFS operation for energy reduction increases the execution time of the parallel
 program.  Therefore, the scaling factor $S$ is linearly proportional to the
 DVFS operation for energy reduction increases the execution time of the parallel
 program.  Therefore, the scaling factor $S$ is linearly proportional to the

-execution time.  However, in most MPI applications the processes exchange
-data.  During these communications the processors involved remain idle until the
-communications are finished.  For that reason, any change in the frequency has no
+execution time of the application.  However, in most MPI applications the processes exchange
+data.  During these communications the processors involved remain idle during a synchronous communication.  For that reason, any change in the frequency has no

 impact on the time of communication~\cite{ref53}.  The communication time for a
 task is the summation of periods of time that begin with an MPI call for sending
 impact on the time of communication~\cite{ref53}.  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.  To
+or receiving a message until the message is synchronously sent or received.  To

 be able to predict the execution time of MPI program, the communication time and
 the computation time for the slowest task must be measured before scaling.  These
 times are used to predict the execution time for any MPI program as a function
 be able to predict the execution time of MPI program, the communication time and
 the computation time for the slowest task must be measured before scaling.  These
 times are used to predict the execution time for any MPI program as a function


@@ -248,7 +245,7 @@ for each processor as presented in the next section.
 
 \section{Performance and energy reduction trade-off}
 \label{ch2:4}
 
 \section{Performance and energy reduction trade-off}
 \label{ch2:4}

-This section presents our method for choosing the optimal scaling factor that
+This section presents our method for choosing the  scaling factor that

 gives the best tradeoff between energy reduction and performance. This method
 takes into account the execution times for both computation and communication to
 compute the scaling factor.  Since the energy consumption and the performance
 gives the best tradeoff between energy reduction and performance. This method
 takes into account the execution times for both computation and communication to
 compute the scaling factor.  Since the energy consumption and the performance


@@ -275,7 +272,7 @@ In the same way we can normalize the performance as follows:
            T_\textit{Min Comm Old}}
 \end{equation}
 
            T_\textit{Min Comm Old}}
 \end{equation}
 

-The relation between the execution time and the consumed energy of a program is nonlinear and complex. In consequences,  the relation between the consumed energy and the scaling factor is also nonlinear, for more details refer to~\cite{ref53}. Therefore, the resulting normalized energy consumption curve and execution time curve, for different scaling factors, do not have the same direction see Figure~\ref{fig:rel}(b).  To tackle this problem and optimize both terms,  we inverse the equation of the normalized execution time (normalized performance) as follows:
+The relation between the execution time and the consumed energy of a program is nonlinear and complex. In consequences,  the relation between the consumed energy and the scaling factor is also nonlinear, for more details refer to~\cite{ref53}. The resulting normalized energy consumption curve and execution time curve, for different scaling factors, do not have the same direction see Figure~\ref{fig:rel}(b).  To tackle this problem and optimize both terms,  we inverse the equation of the normalized execution time which gives the normalized performance and is computed as follows:

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


@@ -289,15 +286,13 @@ The relation between the execution time and the consumed energy of a program is
 \centering
 \centering
 \includegraphics[scale=1]{fig/ch2/file}\\~ ~ ~ ~ ~(a) \\
 \centering
 \centering
 \includegraphics[scale=1]{fig/ch2/file}\\~ ~ ~ ~ ~(a) \\

-

 \includegraphics[scale=1]{fig/ch2/file3}\\~ ~ ~ ~ ~(b)
 \includegraphics[scale=1]{fig/ch2/file3}\\~ ~ ~ ~ ~(b)

-

 \caption{The energy and performance relation (a) Converted relation and (b) Real relation}
 \label{fig:rel}
 \end{figure}
 
 Then, we can model our objective function as finding the maximum distance
 \caption{The energy and performance relation (a) Converted relation and (b) Real relation}
 \label{fig:rel}
 \end{figure}
 
 Then, we can model our objective function as finding the maximum distance

-between the energy curve EQ~\ref{eq:enorm} and the inverse of the execution time (performance)
+between the energy curve EQ~\ref{eq:enorm} and the performance

 curve EQ~\ref{eq:pnorm_en} over all available scaling factors.  This
 represents the minimum energy consumption with minimum execution time (better
 performance) at the same time, see Figure~\ref{fig:rel}(a).  Then
 curve EQ~\ref{eq:pnorm_en} over all available scaling factors.  This
 represents the minimum energy consumption with minimum execution time (better
 performance) at the same time, see Figure~\ref{fig:rel}(a).  Then


@@ -318,11 +313,9 @@ form over the available frequency scaling factors as shown in~\cite{ref48,ref47,
 \section{Optimal scaling factor for performance and energy}
 \label{ch2:5}
 
 \section{Optimal scaling factor for performance and energy}
 \label{ch2:5}
 

-Algorithm~\ref{EPSA} computes the optimal scaling factor according to the
-objective function described above.

 
 \begin{algorithm}[!t]
 
 \begin{algorithm}[!t]

-  \caption{Scaling factor selection algorithm for homogeneous cluster}
+  \caption{Scaling factor selection algorithm for a homogeneous cluster}

   \label{EPSA}
   \begin{algorithmic}[1]
     \State  Initialize the variable $Dist=0$
   \label{EPSA}
   \begin{algorithmic}[1]
     \State  Initialize the variable $Dist=0$


@@ -355,18 +348,18 @@ objective function described above.
 \end{algorithm}
 
 
 \end{algorithm}
 
 

-
+Algorithm~\ref{EPSA} computes the optimal scaling factor according to the
+objective function described above.

 The proposed algorithm works online during the execution time of the MPI
 program.  It selects the optimal scaling factor after gathering the computation
 and communication times from the program after one iteration.  Then the program
 changes the new frequencies of the CPUs according to the computed scaling
 The proposed algorithm works online during the execution time of the MPI
 program.  It selects the optimal scaling factor after gathering the computation
 and communication times from the program after one iteration.  Then the program
 changes the new frequencies of the CPUs according to the computed scaling

-factors. In our experiments over a homogeneous cluster described in
-Section~\ref{ch2:6}, this algorithm has a small execution time. It takes
-\np[$\mu$s]{1.52} on average for 4 nodes and \np[$\mu$s]{6.65} on average for 32
+factors. The  experiments conducted over a homogeneous cluster and described in
+Section~\ref{ch2:6}, showed that this algorithm has a small execution time. It takes on average \np[$\mu$s]{1.52}  for 4 nodes and \np[$\mu$s]{6.65} for 32

 nodes.  The algorithm complexity is $O(F\cdot N)$, where $F$ is the number of
 available frequencies and $N$ is the number of computing nodes.  The algorithm
 is called just once during the execution of the program.  The DVFS algorithm
 nodes.  The algorithm complexity is $O(F\cdot N)$, where $F$ is the number of
 available frequencies and $N$ is the number of computing nodes.  The algorithm
 is called just once during the execution of the program.  The DVFS algorithm

-~\ref{dvfs} shows where and when the algorithm is called in the MPI
+~\ref{dvfs} shows where and when the algorithm \ref{EPSA} is called in the MPI

 program.
 
 
 program.
 
 


@@ -390,7 +383,7 @@ program.
 
 
 After obtaining the optimal scaling factor, the program calculates the new
 
 
 After obtaining the optimal scaling factor, the program calculates the new

-frequency $F_i$ for each task proportionally to its time value $T_i$.  By
+frequency $F_i$ for each task proportionally to its execution time, $T_i$.  By

 substitution of EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we can calculate the new
 frequency $F_i$ as follows:
 \begin{equation}
 substitution of EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we can calculate the new
 frequency $F_i$ as follows:
 \begin{equation}


@@ -427,6 +420,8 @@ two successive frequencies.  The simulated network link is 1 GB Ethernet
   \end{tabular}
   \label{table:platform-homo}
 \end{table}
   \end{tabular}
   \label{table:platform-homo}
 \end{table}

+
+

 \subsection{Performance prediction verification}
 \label{ch2:6:1}
 In this section, the precision of the proposed performance prediction method 
 \subsection{Performance prediction verification}
 \label{ch2:6:1}
 In this section, the precision of the proposed performance prediction method 


@@ -449,26 +444,48 @@ maximum frequency by the new one see EQ~(\ref{eq:s}).
 \end{figure*}
 %see Figure~\ref{fig:pred}
 In our cluster there are 18 available frequency states for each processor.  This
 \end{figure*}
 %see Figure~\ref{fig:pred}
 In our cluster there are 18 available frequency states for each processor.  This

-leads to 18 run states for each program.  We use seven MPI programs of the NAS
-parallel benchmarks: CG, MG, EP, FT, BT, LU and SP.~Figure~(\ref{fig:pred})
-presents plots of the real execution times compared to the simulated ones.  The maximum
+leads to 18 run states for each program.  Seven MPI programs of the NAS
+parallel benchmarks were used: CG, MG, EP, FT, BT, LU and SP. Table \ref{table:NAS-dec} shows 
+the description of these seven benchmarks. Some of these benchmarks are considered  MPI parallel applications with synchronous iterations  or iterative applications that repeat the same block of instructions until convergence. 
+However,  the proposed method can be applied to any application that executes the same block
+of instructions many times and it is not limited to iterative methods. 
+Figure~(\ref{fig:pred}) presents plots of the real execution times compared to the simulated ones.  The maximum

 normalized error between these two execution times varies between 0.0073 to
 normalized error between these two execution times varies between 0.0073 to

-0.031 dependent on the executed benchmark.  The smallest prediction error
+0.031 depending on the executed benchmark.  The smallest prediction error

 was for CG and the worst one was for LU.
 
 was for CG and the worst one was for LU.
 

+\begin{table}[!t]
+\centering
+\caption{NAS Benchmarks description}
+\label{table:NAS-dec}
+\begin{tabular}{|l|l|l|}
+\hline
+Benchmark & Full Name                                                                     & Description                                                                                                                                                                                                                                                                     \\ \hline
+CG        & Conjugate Gradiant                                                            & \begin{tabular}[c]{@{}l@{}}
+It solves a system of linear equations by estimating\\ the smallest eigenvalue of a large sparse  matrix \end{tabular}\\ \hline
+
+MG        & MultiGrid                                                                     & \begin{tabular}[c]{@{}l@{}}It uses the multigrid method to approximate the solution \\of a three-dimensional discrete Poisson equation\end{tabular}                                                                                                                     
+ \\ \hline
+EP       & Embarrassingly Parallel                                                       & \begin{tabular}[c]{@{}l@{}} It applies the Marsaglia polar method to randomly \\generates independent Gaussian variates
+\end{tabular}                                                                                                                                                       \\ \hline
+FT        & Fast Fourier Transform                                                        & \begin{tabular}[c]{@{}l@{}}It uses the fast Fourier transform to solve a \\ three-dimensional partial differential equation
+
+\end{tabular}                                                                                                                               \\ \hline
+BT        & Block Tridiagonal                                                             & \multirow{3}{*}{\begin{tabular}[c]{@{}l@{}} \\They solve nonlinear partial differential equations \end{tabular}} \\ \cline{1-2}
+LU        & \begin{tabular}[c]{@{}l@{}}Lower-Upper symmetric \\ Gauss-Seidel\end{tabular} &                                                                                                                                                                                                                                                                                 \\ \cline{1-2}
+SP        & \begin{tabular}[c]{@{}l@{}}Scalar Pentadiagonal\end{tabular}               &                                                                                                                                                                                                                                                                                 \\ \hline
+\end{tabular}
+\end{table}

 
 \subsection{The experimental results for the scaling algorithm }
 \label{ch2:6:2}
 The proposed algorithm was applied to seven MPI programs of the NAS benchmarks
 (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and C).
 
 \subsection{The experimental results for the scaling algorithm }
 \label{ch2:6:2}
 The proposed algorithm was applied to seven MPI programs of the NAS benchmarks
 (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and C).

-For each instance the benchmarks were executed on a number of processors
-proportional to the size of the class.  Each class represents the problem size
-ascending from  class A to C.  Additionally, depending on some speed up
-points for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes
-respectively.  Depending on EQ~(\ref{eq:energy}), we measure the energy
-consumption for all the NAS MPI programs while assuming that the dynamic power with
-the highest frequency is equal to 20 W and the power static is equal to
-4 W for all experiments.  These power values were also used by Rauber and
+For each instance, the benchmarks were executed on a number of processors
+proportional to the size of the class.  Each class represents the problem size in 
+ascending order from  class A to C.  The classes A, B and C were executed on 4, 8 or 9 and 16 nodes respectively.  The energy consumption for all the NAS MPI programs was measured while assuming that the dynamic power with
+the highest frequency is equal to 20 W and the static power  is equal to
+4 W for all the experiments.  These power values were also used by Rauber and

 Rünger in~\cite{ref47}.  The results showed that the algorithm selected different
 scaling factors for each program depending on the communication features of the
 program as in the plots~(\ref{fig:nas}).  These plots illustrate that there are
 Rünger in~\cite{ref47}.  The results showed that the algorithm selected different
 scaling factors for each program depending on the communication features of the
 program as in the plots~(\ref{fig:nas}).  These plots illustrate that there are


@@ -479,10 +496,10 @@ performance curve is very close to the energy curve.  Then the distance between
 the two curves is very small.  This leads to small energy savings.  The opposite
 happens when there are a lot of communication, the distance between the two
 curves is big.  This leads to more energy savings (e.g. CG and FT), see
 the two curves is very small.  This leads to small energy savings.  The opposite
 happens when there are a lot of communication, the distance between the two
 curves is big.  This leads to more energy savings (e.g. CG and FT), see

-table~(\ref{table:factors results}).  All discovered frequency scaling factors
-optimize both the energy and the performance simultaneously for all NAS
-benchmarks.  In table~(\ref{table:factors results}), we record all optimal
-scaling factors results for each benchmark running class C.  These scaling
+table~(\ref{table:factors results}).  All the discovered frequency scaling factors
+optimize both the energy and the performance simultaneously for all the NAS
+benchmarks.  In table~(\ref{table:factors results}), the optimal
+scaling factors results for each benchmark running class C are presented.  These scaling

 factors give the maximum energy saving percentage and the minimum performance
 degradation percentage at the same time from all available scaling factors.
 \begin{figure*}[h!]
 factors give the maximum energy saving percentage and the minimum performance
 degradation percentage at the same time from all available scaling factors.
 \begin{figure*}[h!]


@@ -527,24 +544,21 @@ degradation percentage at the same time from all available scaling factors.
 As shown in table~(\ref{table:factors results}), when the optimal scaling
 factor has a big value we can gain more energy savings  as in CG and
 FT benchmarks.  The opposite happens when the optimal scaling factor has a  small value as in BT and EP benchmarks.  Our algorithm selects a big scaling factor value when the
 As shown in table~(\ref{table:factors results}), when the optimal scaling
 factor has a big value we can gain more energy savings  as in CG and
 FT benchmarks.  The opposite happens when the optimal scaling factor has a  small value as in BT and EP benchmarks.  Our algorithm selects a big scaling factor value when the

-communication and  other slacks times are big and smaller ones in opposite
-cases.  In EP there are no communication inside the iterations.  This leads our
-algorithm to select smaller scaling factor values (inducing smaller energy
-savings).
+communication and  other slacks times are big.  In EP there are no communication inside the iterations, which leads our algorithm to select smaller scaling factors (inducing smaller energy savings).

 
 
 \subsection{Results comparison}
 \label{ch2:6:3}
 
 
 \subsection{Results comparison}
 \label{ch2:6:3}

-In this section, we compare our scaling factor selection method with Rauber and
-Rünger methods~\cite{ref47}.  They had two scenarios, the first is to reduce energy
+In this section, we compare our scaling factor selection method with the Rauber and
+Rünger's method~\cite{ref47}.  They had two scenarios, the first is to reduce energy

 to the optimal level without considering the performance as in
 EQ~(\ref{eq:sopt}).  We refer to this scenario as $R_{E}$.  The second scenario
 is similar to the first except setting the slower task to the maximum frequency
 to the optimal level without considering the performance as in
 EQ~(\ref{eq:sopt}).  We refer to this scenario as $R_{E}$.  The second scenario
 is similar to the first except setting the slower task to the maximum frequency

-(when the scale $S=1$) to keep the performance from degradation as mush as
-possible.  We refer to this scenario as $R_{E-P}$.  While we refer to our
+(the scale $S=1$) to keep the performance from degradation as mush as
+possible.  We refer to this scenario as $R_{E-P}$ and to our

 algorithm as EPSA (Energy to Performance Scaling Algorithm).  The comparison is
 made in table ~\ref{table:compareC}.  This table shows the results of our method and
 algorithm as EPSA (Energy to Performance Scaling Algorithm).  The comparison is
 made in table ~\ref{table:compareC}.  This table shows the results of our method and

-Rauber and Rünger scenarios for all the NAS benchmarks programs for class C.
+the Rauber and Rünger's scenarios for all the NAS benchmarks programs for class C.

 
 \begin{table}[h!]
   \caption{Comparing results for the NAS class C}
 
 \begin{table}[h!]
   \caption{Comparing results for the NAS class C}


@@ -607,20 +621,19 @@ factor is not optimal for energy reduction, the results in these tables prove
 that our algorithm returns the best scaling factor that satisfy our objective
 method: the largest distance between energy reduction and performance
 degradation. Figure~\ref{fig:compare}  illustrates even better the distance between the energy reduction and performance degradation. The negative values mean that one of
 that our algorithm returns the best scaling factor that satisfy our objective
 method: the largest distance between energy reduction and performance
 degradation. Figure~\ref{fig:compare}  illustrates even better the distance between the energy reduction and performance degradation. The negative values mean that one of

-the two objectives (energy or performance) have been degraded more than 
+the two objectives (energy or performance) has been degraded more than the 

 other.  The positive trade-offs with the highest values lead to maximum energy
 savings while keeping the performance degradation as low as possible.  Our
 algorithm always gives the highest positive energy to performance trade-offs
 other.  The positive trade-offs with the highest values lead to maximum energy
 savings while keeping the performance degradation as low as possible.  Our
 algorithm always gives the highest positive energy to performance trade-offs

-while Rauber and Rünger's method, ($R_{E-P}$), gives sometimes negative
-trade-offs such as in BT and EP.
+while the Rauber and Rünger's method, ($R_{E-P}$), gives sometimes negative
+trade-offs such as for BT and EP.

 
 
 
 

-\section{The new energy model for homogeneous cluster}
+\section{The new energy model for a homogeneous cluster}

 \label{ch2:7}
 As mentioned in chapter \ref{ch1} section \ref{ch1:3}, the power consumed by a processor is divided 
 \label{ch2:7}
 As mentioned in chapter \ref{ch1} section \ref{ch1:3}, the power consumed by a processor is divided 

-into two power metrics: the static and the dynamic power. The  first power metrics  is
-consumed as long as the computing unit is on, while the other one is consumed when the processor is 
-doing the computations. Consequentially, the energy consumed by an individual processor
+into two power metrics: the static and the dynamic power. The  first power metric  is
+consumed as long as the computing unit is on, while the other one is consumed when the processor is doing the computations. Consequentially, the energy consumed by an individual processor

 to execute a given program can be computed as follows:
 
 
 to execute a given program can be computed as follows:
 
 


@@ -633,7 +646,7 @@ where $T$ is the execution time of the program, $T_{Comp}$ is the computation
 time and $T_{Comp} \leq T$.  $T_{Comp}$ may be equal to $T$ if there is no
 communication, no slack time and no synchronization.
 
 time and $T_{Comp} \leq T$.  $T_{Comp}$ may be equal to $T$ if there is no
 communication, no slack time and no synchronization.
 

-Applying DVFS operation leads to a new frequency state, which is represented by the frequency scaling factor $S$ computed as in the equation \ref{eq:s}. According to the Rauber and Rünger energy model \ref{eq:energy}, the dynamic energy is consumed during the overall program's execution time. This assumption is not precise because the CPU is only consumed the dynamic power during computation time. Moreover, the CPU involved remain idle during the communication times and only consumed the static power, see \cite{ref53}. Also, we have conducted some experiments over a real homogeneous cluster by running some MPI programs of the NAS  benchmarks. The results prove that  there is no effect of changing the frequency on the communication times of these programs.  Therefore, the frequency scaling factor $S$ can be increased the computation times  propositionally, while not effecting the communication times. This assumption is acceptable according the used performance prediction model \ref{eq:tnew}. This model is evaluated and its prediction accuracy is showed in section \ref{ch2:6:1}. Therefore, the new dynamic energy is the dynamic power multiplied by the new time of computation and is given by the following equation:
+Applying a DVFS operation leads to a new frequency state which is represented by the frequency scaling factor $S$, computed as in the equation \ref{eq:s}. According to  Rauber and Rünger's energy model \ref{eq:energy}, the dynamic energy is consumed during the overall program's execution time. This assumption is not precise because the CPU  only consumes the dynamic power during computation time. Moreover, the CPU involved remains idle during the communication times and only consumes the static power, see \cite{ref53}. We have also conducted some experiments over a real homogeneous cluster where some MPI programs of the NAS  benchmarks were executed while varying the CPUs frequencies at each execution. The results prove that  changing the frequency does not effect on the communication times of these programs.  Therefore, the frequency scaling factor $S$ can increase the computation times  proportionally to its value,  and does not  effect the communication times. This assumption consort with the used performance prediction model \ref{eq:tnew}. This model is evaluated and its prediction accuracy is showed in section \ref{ch2:6:1}. Therefore, 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_new}
 
 \begin{equation}
   \label{eq:Edyn_new}


@@ -655,9 +668,9 @@ processor after scaling its frequency is computed as follows:
   Es = P_{static} \cdot (T_{comp} \cdot S  + T_{comm})
 \end{equation}
 
   Es = P_{static} \cdot (T_{comp} \cdot S  + T_{comm})
 \end{equation}
 

-In particular, in the homogeneous cluster all the computing nodes have the same computing power and thus they have similar frequencies gears. The execution time of the MPI application is the execution time of the slowest task as shown in section \ref{ch2:2:1}. Therefore, the frequency scaling factor $S$ of the slowest task can be used to modelize the  energy consumption of the parallel tasks execution.
-The dynamic energy consumed by $N$  parallel tasks  is the summation of all the dynamic energies of  all tasks during the computation time $\Tcomp[i]$  of  each task. The static energy of each task is the static power consumed during the execution time of the slower task because they are synchronised together.
-Therefore, the energy consumption model of $N$ parallel task executed  synchronously over a homogeneous platforms is represented in \ref{eq:e-new}. 
+In particular, in a homogeneous cluster all the computing nodes have the same specification and thus their CPUs have similar frequencies gears. The execution time of the MPI application is the execution time of the slowest task as shown in section \ref{ch2:2:1}. Therefore, the frequency scaling factor $S$ of the slowest task can be used to modelize the  energy consumption of the parallel application.
+The dynamic energy consumed by $N$  parallel tasks  is the summation of all the dynamic energies of  all tasks during the computation time $\Tcomp[i]$  of  each task. The static energy of each task is the static power consumed during the execution time of the slower task because all the tasks are synchronised and have the same execution time.
+Therefore, the energy consumption model of $N$ parallel task executed  synchronously over a homogeneous platforms can be represented as in \ref{eq:e-new}. 

 
 \begin{equation}
   \label{eq:e-new}
 
 \begin{equation}
   \label{eq:e-new}


@@ -665,30 +678,28 @@ Therefore, the energy consumption model of $N$ parallel task executed  synchrono
   ( Ps \cdot  ( T_\textit{Max Comp Old} \cdot S + T_{\textit{Min Comm Old}} ) )  \cdot N
 \end{equation}
  
   ( Ps \cdot  ( T_\textit{Max Comp Old} \cdot S + T_{\textit{Min Comm Old}} ) )  \cdot N
 \end{equation}
  

-According to this model, the frequency scaling factor $S$ reducing the energy consumption of the homogeneous architecture by a factor of $S^{-2}$ and increases the execution time by a factor of  $S$.
-This model can be used to predict the energy consumption of the message passing synchronous iterative applications depending on the gathered computation and communication times from the first iteration.
-Furthermore, it can be used to measured the energy consumption of the iterative application by multiplying the energy consumed of all tasks in  one iteration  by the number of the iterations. 
+According to this model, the frequency scaling factor $S$ reduces the energy consumption of the homogeneous architecture by a factor of $S^{-2}$ and increases the execution time by a factor of  $S$.
+This model can be used to predict the energy consumption of the message passing   applications with synchronous iterations after gathering the computation and communication times of the first iteration.
+Furthermore, it can be used to measure the energy consumption of the parallel  application with iterations  by multiplying the energy consumed of all tasks in  one iteration  by the number of the iterations. 

 
 

-Consequently, this model is used in the prediction process of the energy consumption by the 
-algorithm \ref{EPSA} to selects the optimal frequency scaling factor. By the same way in the last section, the new frequency $F_i$ can be computed as in \ref{eq:fi} depending on the new selected frequency scaling factor. 
-In the next section, the algorithm \ref{EPSA} is reimplemented using this new energy model \ref{eq:e-new}  to select new frequency scaling factors and thus a new results are obtained. 
+This model is used by the algorithm \ref{EPSA}  to predict the energy consumption and to select the optimal frequency scaling factor. The new frequency $F_i$ can be computed as in \ref{eq:fi} while using the new selected frequency scaling factor.         
+In the next section,  algorithm \ref{EPSA} is re-evaluated while using this new energy model and the new results are presented.

 
 \section{The experimental results using the new energy model}
 \label{ch2:8}
 
 
 \section{The experimental results using the new energy model}
 \label{ch2:8}
 

-This section presents results of applying the frequency selection algorithm \ref{EPSA} using 
-the new proposed energy model \ref{eq:e-new}. The algorithm is applied to NAS parallel benchmarks class 
-C running on 16  computing nodes using the SimGrid simulator. Same values are used for the static and dynamic powers values as in section \ref{ch2:6:2}. Two measured energy consumptions of the NAS benchmarks class C using the new energy model and Rauber and Rünger's model are presented in the figure \ref{fig:energy_con}. The energy consumptions of both models are computed using similar parameters:  frequency scaling factors, dynamic and static powers values. As shown in this figure,  the majority  of the benchmarks  have smaller computed energies values using the new model compare to those use  Rauber and Rünger's model.
-Indeed, there are two reasons explaining these differences in the energy consumptions. The first one is related to the dynamic power consumption, where the new energy model ensures that this power metric is consumed only during the computation time, while  the other model assumes to consume the dynamic power during both computation and communication times and thus more dynamic energy consumption is given.
-The second one is related to the execution time,  that is only its computation times is increased with the 
-scaling factor value in the new energy model, while other energy model indicates that both  the 
-computation and communication times are increased with scaling factor and hence more static energy consumption is given.  Therefore, the MPI programs that have big communication times, they  have bigger measured energy consumption values using  Rauber and Rünger's model  compare to the new model as in CG, SP, LU and FT benchmarks. Whereas, if the MPI programs have very small communication times, their computed energy values have very small differences  such as in  MG and BT benchmarks, or they are identical such as in EP benchmark where there is no communication and no idle times. 
+This section presents the results of applying the frequency selection algorithm \ref{EPSA} using the new proposed energy model \ref{eq:e-new} to NAS parallel benchmarks.
+The class C of the benchmarks was executed on a homogeneous architecture composed of 16 nodes and simulated by SimGrid. The same  static and dynamic power values were used as in section \ref{ch2:6:2}. Figure \ref{fig:energy_con} presents the energy consumption of the NAS benchmarks class C using the new energy model and the Rauber and Rünger's model. The energy consumptions of both models are computed using similar parameters:  frequency scaling factors, dynamic and static powers values. As shown in this figure,  the majority  of the benchmarks  consumes less energy using the new model than when using the Rauber and Rünger's model.
+Two reasons explain these differences in the energy consumptions: the first one is related to the dynamic power consumption, where the new energy model ensures that this power metric is only consumed  during the computation time, while  the other model assumes that the dynamic power is consumed during both computation and communication times and thus increasing the dynamic energy consumption.
+The second reason is related to the execution time. In the new model only the computation times are increased when the frequency of a processor is scaled down, while
+Rauber and Rünger's model indicates that both the computation and communication times 
+are  increased according to the scaling factor and hence more static energy  is consumed.  Therefore, the MPI programs that have big communication times,  have bigger  energy consumption values using  Rauber and Rünger's model  when compared to the new model as for the CG, SP, LU and FT benchmarks. Whereas, if the MPI programs have very small communication times, their computed energy values have very small differences  using both models such as for the MG and BT benchmarks, or they are identical such as for the  EP benchmark where there is no communication and no idle times. 

 
 
 \begin{figure*}[h!]
   \centering
   \includegraphics[width=.7\textwidth]{fig/ch2/energy_con.eps}
 
 
 \begin{figure*}[h!]
   \centering
   \includegraphics[width=.7\textwidth]{fig/ch2/energy_con.eps}

-  \caption{Comparing the energy consumptions measured using the new and Rauber energy  models}
+  \caption{Comparing the energy consumptions estimated using Rauber energy  model and our own}

   \label{fig:energy_con}
 \end{figure*}
 \begin{table}[h!]
   \label{fig:energy_con}
 \end{figure*}
 \begin{table}[h!]


@@ -710,14 +721,14 @@ FT      & 1.47      & 34.72       & 19.00       & 1.38       & 29.94       & 17.
 \label{table:new-res}
 \end{table}
 
 \label{table:new-res}
 \end{table}
 

-Table \ref{table:new-res} shows  results of energy saving and performance degradation percentages of applying the frequency selecting algorithm  using the new propose energy model. It also presents the new selected frequency scaling factor values and comparing them to ones use Rauber and Rünger's model. It indicates  that the new selected frequency scaling factors are smaller compared to those selected using  other model because the predicted energies by the new energy model are smaller.
-Consequentially, less energy savings and performance degradation percentages are produced according to 
-these smaller frequency scaling factors selected using the new energy model, such as in  CG, MG, LU, SP and FT benchmarks. While in BT and EP benchmarks where a very small or no communication times, similar scaling factors are selected because the predicted energies by the two models are approximately equivalent. 
-On the other hand, the scaling factor selection algorithm can work with any energy model and  it selects proportionally the scaling factor values depending on the predicted energies values. 
+Table \ref{table:new-res} shows  the energy saving and performance degradation percentages when applying the frequency selecting algorithm  using the new proposed  energy model. It also presents the new selected frequency scaling factors and compares them to the ones used by the Rauber and Rünger's model. It shows  that the new selected frequency scaling factors are smaller than  those selected using the other model because the predicted energies by the new energy model are smaller.
+Consequently, less energy savings and performance degradation percentages are produced according to 
+these smaller frequency scaling factors such as for the  CG, MG, LU, SP and FT benchmarks. While in the BT and EP benchmarks where there are very small or no communication times, similar scaling factors are selected because the predicted energies by the two models are approximately equivalent. 

 
 

-As a results,  the new proposed energy model is more accurate than Rauber and Rünger's energy
+Therefore, the new proposed energy model is more accurate than Rauber and Rünger's energy

 model, because  it takes into consideration both the communication and idle times in addition to
 model, because  it takes into consideration both the communication and idle times in addition to

-the computation times of  MPI programs running over homogeneous clusters. 
+the computation times of  message passing programs running over homogeneous clusters. 
+The scaling factor selection algorithm can work with any energy model and  it selects  the scaling factor values according to the predicted energy values. 

 
 \section{Conclusion}
 \label{ch2:9}
 
 \section{Conclusion}
 \label{ch2:9}


@@ -729,10 +740,11 @@ consumption and the performance of the parallel application at every available
 frequency.  Then, it selects the scaling factor that gives the best trade-off
 between energy reduction and performance which is the maximum distance between
 the energy and the performance curves.  To evaluate this method, we
 frequency.  Then, it selects the scaling factor that gives the best trade-off
 between energy reduction and performance which is the maximum distance between
 the energy and the performance curves.  To evaluate this method, we

-have applied it to the NAS benchmarks and it was compared to Rauber and Rünger
-methods while being executed on the SimGrid simulator.  The results showed that
-our method, outperforms Rauber and Rünger's methods in terms of energy-performance
-ratio. Finally, this chapter presents a new energy consumption model for the parallel
-synchronous iterative methods running on homogeneous clusters. This model takes into consideration 
-both  the computation and communication times and them relation with the frequency scaling 
-factor. The results obtained using the new energy model have shown selecting different frequency scaling factors than using other energy model and thus different experimental results have been produced. 
+have applied it to the NAS benchmarks and it was compared to the Rauber and Rünger's 
+method while being executed on the SimGrid simulator.  The results showed that
+our method, outperforms the Rauber and Rünger's method in terms of energy-performance
+ratio. Finally, this chapter presents a new energy consumption model for  parallel
+  applications with synchronous iterations running on homogeneous clusters. This model takes into consideration 
+both  the computation and communication times and their relation with the frequency scaling 
+factor. The results obtained using the new energy model have shown that different frequency scaling factors 
+were  selected which gave new experimental results that are more accurate and realistic.
\ No newline at end of file