\usepackage{xspace}
\usepackage[textsize=footnotesize]{todonotes}
\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
+\newcommand{\JC}[2][inline]{\todo[color=red!10,#1]{\sffamily\textbf{JC:} #2}\xspace}
\begin{document}
-\title{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
+\title{Dynamic Frequency Scaling for Energy Consumption Reduction in Distributed MPI Programs}
\author{%
\IEEEauthorblockN{%
}
\IEEEauthorblockA{%
FEMTO-ST Institute\\
- University of Franche-Comté
+ University of Franche-Comté\\
+ IUT de Belfort-Montbéliard, 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
+ Fax : +33~3~84~58~77~32\\
+ Email: \{jean-claude.charr, ahmed.fanfakh\_badri\_muslim, raphael.couturier, arnaud.giersch\}@univ-fcomte.fr
+ }
}
-}
\maketitle
-\AG{``Optimal'' is a bit pretentious in the title.\\
- Complete affiliation, add an email address, etc.}
-
+\AG{Is the fax number correct? Shall we add a telephone number?}
+\JC{Use Capital letters for only the first letter in the title of a section, table, figure, ...}
\begin{abstract}
-The important technique for energy reduction of parallel systems is CPU frequency
-scaling. This operation used by many researchers to reduce energy consumption in many
-ways. Frequency scaling operation also has big impact on the performance. In some cases,
-the performance degradation ratio is bigger than energy saving ratio when the frequency scaled
-to down level. Therefore, the trade offs between the energy and performance becomes more
-important topic when using this technique. In this paper we developed an algorithm that
-select the frequency scaling factor for both energy and performance simultaneously.
-This algorithm takes into account the communication times when selecting the frequency scaling
-factor. It is works online without training or profiling to have very small overhead.
-The algorithm has better energy-performance trade offs compared to other methods.
+ Dynamic Voltage 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 instant is
+exponentially related to its frequency. 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 a low CPU
+frequency is selected. The performance degradation ratio can even be higher than
+the saved energy ratio.Therefore, the chosen scaling factor must give the best possible tradeoff
+between energy reduction and performance.
+
+In this paper we present 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.
+This algorithm works online without training or profiling and
+has a very small overhead. It also takes into account synchronous communications between the nodes
+that are executing the distributed algorithm. The algorithm has been evaluated over the SimGrid simulator
+while being applied to the NAS parallel benchmark programs. The results of the experiments show that it outperforms other existing scaling factor selection algorithms.
\end{abstract}
\section{Introduction}
\label{sec.intro}
-The need for computing power is still increasing and it is not expected to slow
+The need and demand for more computing power have been increasing since the birth of the first computing unit and it is not expected to slow
down in the coming years. To satisfy this demand, researchers and supercomputers
-constructors have been regularly increasing the number of computing cores in
+constructors have been regularly increasing the number of computing cores and processors in
supercomputers (for example in November 2013, according to the TOP500
list~\cite{43}, the Tianhe-2 was the fastest supercomputer. It has more than 3
millions of cores and delivers more than 33 Tflop/s while consuming 17808
reduction became an important topic in the high performance computing field. To
tackle this problem, many researchers used DVFS (Dynamic Voltage Frequency
Scaling) operations which reduce dynamically the frequency and voltage of cores
-and thus their energy consumption. However, this operation also degrades the
+and thus their energy consumption. Indeed, modern CPUs offer a set of acceptable frequencies which are usually called gears, and the user or the operating system can modify the frequency of the processor according to its needs. However, DVFS also degrades the
performance of computation. Therefore researchers try to reduce the frequency to
minimum when processors are idle (waiting for data from other processors or
communicating with other processors). Moreover, depending on their objectives
other hand, if they aim for energy reduction, the chosen scaling factor must
produce the most energy efficient execution without considering the degradation
of the performance. It is important to notice that lowering the frequency to
-minimum value does not always give the most efficient execution due to energy
+minimum value does not always give the most energy efficient execution due to energy
leakage. The best scaling factor might be chosen during execution (online) or
-during a pre-execution phase. In this paper we emphasize to develop an
+during a pre-execution phase. In this paper, we present an
algorithm that selects a frequency scaling factor that simultaneously takes into
-consideration the energy consumption and the performance. The
-main objective of HPC systems is to run the application with less execution
-time. Therefore, our algorithm selects the scaling factor online with
+consideration the energy consumption by the CPU and the performance of the application. 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 footprint. The proposed algorithm takes into account the
communication times of the MPI program to choose the scaling factor. This
algorithm has ability to predict both energy consumption and execution time over
over an homogeneous distributed memory architecture. Furthermore, we compare the
proposed algorithm with Rauber and Rünger methods~\cite{3}.
The comparison's results show that our
-algorithm gives better energy-time trade off.
-
-This paper is organized as follows: Section~\ref{sec.relwork} presents the works from other authors.
-Section~\ref{sec.ptasks} shows the execution of parallel tasks and sources of idle times. Section~\ref{sec.energy} resumes the
-energy model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance of MPI program.
-Section~\ref{sec.verif} verifies the performance prediction model. Section~\ref{sec.compet} presents
-the energy-performance trade offs objective function. Section~\ref{sec.optim} demonstrates the proposed
-energy-performance algorithm. Section~\ref{sec.expe} presents the results of our experiments.
-Section~\ref{sec.compare} shows the comparison results. Finally, we conclude in Section~\ref{sec.concl}.
-
+algorithm gives better energy-time tradeoff.
+
+This paper is organized as follows: Section~\ref{sec.relwork} presents related works
+from other authors. Section~\ref{sec.exe} shows the execution of parallel
+tasks and sources of idle times. It resumes the energy
+model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance
+of MPI program. Section~\ref{sec.compet} presents the energy-performance tradeoffs
+objective function. Section~\ref{sec.optim} demonstrates the proposed
+energy-performance algorithm. Section~\ref{sec.expe} verifies the performance prediction
+model and presents the results of the proposed algorithm. Also, It shows the comparison results. Finally,
+we conclude in Section~\ref{sec.concl}.
\section{Related Works}
\label{sec.relwork}
-\AG{Consider introducing the models (sec.~\ref{sec.ptasks},
- maybe~\ref{sec.energy}) before related works}
-
-In the this section some heuristics to compute the scaling factor are
-presented and classified in two parts: offline and online methods.
-
-\subsection{The offline DVFS orientations}
-
-The DVFS offline methods are static and are not executed during the runtime of
-the program. Some approaches used heuristics to select the best DVFS state
-during the compilation phases as for example in Azevedo et al.~\cite{40}. They use
-dynamic voltage scaling (DVS) algorithm to choose the DVS setting when there are dependency points
-between tasks. While in~\cite{29}, Xie et al. used breadth-first search
-algorithm to do that. Their goal is to save energy with time limits. Another
-approach gathers and stores the runtime information for each DVFS state, then
-selects the suitable DVFS offline to optimize energy-time
-trade offs. As an example Rountree et al.~\cite{8}, use liner programming
-algorithm, while in~\cite{38,34}, Cochran et al. use multi logistic regression
-algorithm for the same goal. The offline study that shows the DVFS impact on the
-communication time of the MPI program is~\cite{17}, where Freeh et al. show that these
-times do not change when the frequency is scaled down.
-
-\subsection{The online DVFS orientations}
-
-The objective of online DVFS orientations works is to dynamically compute and set the frequency of
-the CPU during the runtime of the program for saving energy. Estimating and
-predicting approaches for the energy-time trade offs are developed by Kimura, Peraza, Yu-Liang et al.
-~\cite{11,2,31}. These works select the best DVFS setting depending on the slack
-times. These times happen when the processors have to wait for data from other
-processors to compute their task. For example, during the synchronous
-communications that take place in MPI programs, some processors are
-idle. The optimal DVFS can be selected using learning methods. Therefore, in Dhiman, Hao Shen et al.
-~\cite{39,19} used machine learning to converge to the suitable DVFS
-configuration. Their learning algorithms take big time to converge when the
-number of available frequencies is high. Also, the communication sections of the MPI
-program can be used to save energy. In~\cite{1}, Lim et al. developed an
-algorithm that detects the communication sections and changes the frequency
-during these sections only. This approach changes the frequency many times
-because an iteration may contain more than one communication section. The domain
-of analytical modeling used for choosing the optimal frequency as in Rauber and Rünger~\cite{3}. they
-developed an analytical mathematical model to determine the
-optimal frequency scaling factor for any number of concurrent tasks. They set the slowest task to maximum frequency for maintaining performance. In this paper we compare our algorithm with
-Rauber and Rünger model~\cite{3}, because their model can be used for any number of
-concurrent tasks for homogeneous platforms. The primary contributions of this paper are:
+\AG{Consider introducing the models sec.~\ref{sec.exe} maybe before related works}
+
+In this section, some heuristics to compute the scaling factor are
+presented and classified into two categories: offline and online methods.
+
+\subsection{Offline scaling factor selection methods}
+
+The offline scaling factor selection methods are executed before the runtime of
+the program. They return static scaling factor values to the processors
+participating in the execution of the parallel program. On one hand, the scaling
+factor
+values could be computed based on information retrieved by analyzing the code of
+the program and the computing system that will execute it. In ~\cite{40},
+Azevedo et
+al. detect during compilation the dependency points between
+tasks in a parallel program. This information is then used to lower the frequency of
+some processors in order to eliminate slack times. A slack time is the period of time during which a processor that have already finished its computation, have to wait
+for a set of processors to finish their computations and send their results to the
+waiting processor in order to continue its task that is
+dependent on the results of computations being executed on other processors.
+Freeh et al. showed in ~\cite{17} 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. A part or the whole program is usually executed over all the available frequency gears and the the execution time and the energy consumed with each frequency gear are measured. Then an heuristic or an exact method uses the retrieved information to compute the values of the scaling factor for the processors.
+ In~\cite{29}, 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{8} , Rountree et al. use a linear programming
+algorithm, while in~\cite{38,34}, Cochran et al. use multi logistic regression algorithm for the same goal.
+The main drawback for these methods is that they all require executing a part or the whole program on all frequency gears for each new instance of the same program.
+
+\subsection{Online scaling factor selection methods}
+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, many informations are measured such as the execution time, the energy consumed using a multimeter, the slack times, ... 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 much as needed if the iterations are not regular. Kimura, Peraza, Yu-Liang et al. ~\cite{11,2,31} used learning methods to select the appropriate scaling factor values to eliminate the slack times during runtime. However, as seen in ~\cite{39,19}, 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, in~\cite{1}, Lim et al. developed an algorithm that detects the
+communication sections and changes the frequency during these sections
+only. This approach might change the frequency of each processor many times per iteration if an iteration
+contains more than one communication section. In ~\cite{3}, Rauber et al. used an analytical model that after measuring the energy consumed and the execution time with the highest frequency gear, it can predict the energy consumed and the execution time for every frequency gear . These predictions may be used to choose the optimal gear for each processor executing the parallel program to reduce energy consumption.
+To maintain the performance of the parallel program , they
+set the processor with the biggest load to the highest gear and then compute the scaling factor values for the rest pf the processors. Although this model was built for parallel architectures, it can be adapted to distributed architectures by taking into account the communications.
+The primary contribution of this paper is presenting a new online scaling factor selection method which has the following characteristics :
\begin{enumerate}
-\item Selecting the frequency scaling factor for simultaneously optimizing energy and performance,
- while taking into account the communication time.
-\item Adapting our scaling factor to take into account the imbalanced tasks.
-\item The execution time of our algorithm is very small when compared to other
- methods (e.g.,~\cite{19}).
-\item The proposed algorithm works online without profiling or training as
+\item Based on Rauber's analytical model to predict the energy consumption and the execution time of the application with different frequency gears.
+\item Selects the frequency scaling factor for simultaneously optimizing energy reduction and maintaining performance.
+\item Well adapted to distributed architectures because it takes into account the communication time.
+\item Well adapted to distributed applications with imbalanced tasks.
+\item Has very small footprint when compared to other
+ methods (e.g.,~\cite{19}) and does not require profiling or training as
in~\cite{38,34}.
\end{enumerate}
-\section{Parallel Tasks Execution on Homogeneous Platform}
-\label{sec.ptasks}
+\section{Execution and Energy of Parallel Tasks on Homogeneous Platform}
+\label{sec.exe}
+\JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'', can be deleted if we need space, we can just say we are interested in this paper in homogeneous clusters}
+\subsection{Parallel Tasks Execution on Homogeneous Platform}
A homogeneous cluster consists of 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
\end{figure*}
Therefore, the execution time of a task consists of the computation time and the
communication time. Moreover, the synchronous communications between tasks can
-lead to idle time while tasks wait at the synchronization barrier for other tasks to
-finish their communications (see figure~(\ref{fig:h1})). The imbalanced communications happen when nodes have to send/receive different amount of data or each node is communicates with different number of nodes. Another source for idle times is the imbalanced computations. This happen when processing different
-amounts of data on each processor (see figure~(\ref{fig:h2})). In
-this case the fastest tasks have to wait at the synchronization barrier for the
-slowest tasks to finish their job. In both cases the overall execution time
-of the program is the execution time of the slowest task as:
+lead to slack times while tasks wait at the synchronization barrier for other tasks to
+finish their tasks (see figure~(\ref{fig:h1})). The imbalanced communications
+happen when nodes have to send/receive different amount of data or they communicate
+with different number of nodes. Another source of idle times is the imbalanced computations.
+This happens when processing different amounts of data on each processor (see figure~(\ref{fig:h2})).
+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 is the execution time of the slowest task as in equation \ref{eq:T1}.
\begin{equation}
\label{eq:T1}
\textit{Program Time} = \max_{i=1,2,\dots,N} T_i
\end{equation}
-where $T_i$ is the execution time of task $i$.
+where $T_i$ is the execution time of task $i$ and all the tasks are executed concurrently on different processors.
-\section{Energy Model for Homogeneous Platform}
-\label{sec.energy}
+\subsection{Energy Model for Homogeneous Platform}
-The energy consumption by the processor consists of two power metrics: the
-dynamic and the static power. This general power formulation is used by many
-researchers~\cite{9,3,15,26}. The dynamic power of the CMOS processors
-$P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
-the supply voltage $V$ and operational frequency $f$ respectively as follow:
+Many researchers~\cite{9,3,15,26} divide the power consumed by a processor to two power metrics: the
+ static and the dynamic power. While the first one is consumed as long as the computing unit is on, the latter is only consumed during computation times. The dynamic power
+$P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
+the supply voltage $V$ and operational frequency $f$, as shown in equation ~\ref{eq:pd}.
\begin{equation}
\label{eq:pd}
P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
\end{equation}
-The static power $P_{static}$ captures the leakage power consumption as well as
-the power consumption of peripheral devices like the I/O subsystem.
+The static power $P_{static}$ captures the leakage power as follows:
\begin{equation}
\label{eq:ps}
- P_\textit{static} = V \cdot N \cdot K_{design} \cdot I_{leak}
+ P_\textit{static} = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
\end{equation}
-where V is the supply voltage, N is the number of transistors, $K_{design}$ is a
+where V is the supply voltage, $N_{trans}$ is the number of transistors, $K_{design}$ is a
design dependent parameter and $I_{leak}$ is a technology-dependent
parameter. Energy consumed by an individual processor $E_{ind}$ is the summation
-of the dynamic and the static power multiplied by the execution time for example
-see~\cite{36,15}.
+of the dynamic and the static powers multiplied by the execution time~\cite{36,15}.
\begin{equation}
\label{eq:eind}
E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T
\end{equation}
-The dynamic voltage and frequency scaling (DVFS) is a process that is allowed in
+DVFS is a process that is allowed in
modern processors to reduce the dynamic power by scaling down the voltage and
frequency. Its main objective is to reduce the overall energy
consumption~\cite{37}. The operational frequency \emph f depends linearly on the
supply voltage $V$, i.e., $V = \beta \cdot f$ with some constant $\beta$. This
equation is used to study the change of the dynamic voltage with respect to
-various frequency values in~\cite{3}. The reduction process of the frequency are
-expressed by scaling factor \emph S. The scale \emph S is the ratio between the
+various frequency values in~\cite{3}. The reduction process of the frequency can be
+expressed by the scaling factor \emph S which is the ratio between the
maximum and the new frequency as in EQ~(\ref{eq:s}).
\begin{equation}
\label{eq:s}
S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
-The value of the scale $S$ is greater than 1 when changing the frequency to
-any new frequency value~(\emph {P-state}) in governor, the CPU governor is an interface
-driver supplied by the operating system kernel (e.g. Linux) to lowering core's frequency.
-The scaling factor is equal to 1 when the frequency set is to the maximum frequency.
-The energy consumption model for parallel homogeneous platform depends on the scaling factor \emph S. This factor reduces quadratically the dynamic power. Also, this factor increases the
-static energy linearly because the execution time is increased~\cite{36}. The
-energy model depending on the frequency scaling factor for homogeneous platform
-for any number of concurrent tasks was developed by Rauber and Rünger~\cite{3}. This model
-considers the two power metrics for measuring the energy of the parallel tasks as
-in EQ~(\ref{eq:energy}):
-
+The value of the scaling factor $S$ is greater than 1 when changing the frequency of the CPU to any
+new frequency value~(\emph {P-state}) in the governor. The CPU governor is an
+interface driver supplied by the operating system's kernel to
+lower a core's frequency. This factor reduces
+quadratically the dynamic power which may cause degradation in performance and thus, the increase of the static energy because the execution time is increased~\cite{36}. If the tasks are sorted according to their execution times before scaling in a descending order, the total energy consumption model for a parallel
+homogeneous platform, as presented by Rauber et al.~\cite{3}, can be written as a function of the scaling factor \emph S, as in EQ~(\ref{eq:energy}).
+
+\JC{Are you sure of the following equation}
\begin{equation}
\label{eq:energy}
E = P_\textit{dyn} \cdot S_1^{-2} \cdot
P_\textit{static} \cdot T_1 \cdot S_1 \cdot N
\hfill
\end{equation}
-where \emph N is the number of parallel nodes, $T_1 $ is the time of the slowest
-task, $T_i$ is the time of the task $i$ and $S_1$ is the maximum scaling factor
-for the slower task. The scaling factor $S_1$, as in EQ~(\ref{eq:s1}), selects
-from the set of scales values $S_i$. Each of these scales are proportional to
-the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}).
-\begin{equation}
- \label{eq:s1}
- S_1 = \max_{i=1,2,\dots,F} S_i
-\end{equation}
+where \emph N is the number of parallel nodes, $T_i \ and \ S_i \ for \ i=1,...,N$ are the execution times and scaling factors of the sorted tasks. Therefore, $T1$ is the time of the slowest task, and $S_1$ its scaling factor which should be the highest because they are proportional to
+the time values $T_i$. The scaling factors are computed as in EQ~(\ref{eq:si}).
+\JC{This equation does not make sense either, what's S? there is no F}
\begin{equation}
\label{eq:si}
S_i = S \cdot \frac{T_1}{T_i}
= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}
\end{equation}
+\JC{The Rauber model was used for a parallel machine not a homogeneous platform}
where $F$ is the number of available frequencies. In this paper we depend on
-Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1)-this model is used
-for homogeneous platform that we work on in this paper. 2-we compare our
-algorithm with Rauber and Rünger scaling model. Rauber and Rünger scaling factor that reduce
- energy consumption derived from the EQ~(\ref{eq:energy}). They take the
-derivation for this equation (to be minimized) and set it to zero to produce the
-scaling factor as in EQ~(\ref{eq:sopt}).
+Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this
+model is used for homogeneous platform that we work on in this paper, and (2) we
+compare our algorithm with Rauber and Rünger scaling factor selection method which is based on
+EQ~(\ref{eq:energy}). The optimal scaling factor is computed by minimizing the derivation for this equation which produces EQ~(\ref{eq:sopt}).
+\JC{what's the small n in the equation}
+
\begin{equation}
\label{eq:sopt}
S_\textit{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot
\left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
\end{equation}
+\JC{The following 2 sections can be merged easily}
+
\section{Performance Evaluation of MPI Programs}
\label{sec.mpip}
-The performance (execution time) of parallel MPI applications depend on
-the time of the slowest task as in figure~(\ref{fig:homo}). Normally the
-execution time of the parallel programs are proportional to the operational
-frequency. Therefore, any DVFS operation for the energy reduction increases the
-execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) the
-energy is affected by the scaling factor $S$. This factor also has a great impact
-on the performance. When scaling down the frequency to the new value according
-to EQ~(\ref{eq:s}), the value of the scale $S$ has inverse relation with
-new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decreasing the
-frequency value, the execution time increases. Then the new frequency value has
-inverse relation with time ($F_{new} \propto \frac{1}{T}$). This leads to the
-frequency scaling factor $S$ proportional linearly with execution time ($S
-\propto T$). Large scale MPI applications such as NAS benchmarks have
-considerable amount of communications embedded in these programs. During the
-communication process the processors remain idle until the communication has
+The performance (execution time) of parallel synchronous MPI applications depend on
+the 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 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 of 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 impact on the time
-of communication but it has obvious impact on the time of
-computation~\cite{17}. We have made many tests on a real cluster to prove that the
-frequency scaling factor \emph S has a linear relation with computation time
-only. To predict the execution time of MPI program, the communication time and
-the computation time for the slower task must be first precisely specified. Secondly,
-these times are used to predict the execution time for any MPI program as a function of
-the new scaling factor as in the EQ~(\ref{eq:tnew}).
+of communication~\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. To be able to predict the execution time of MPI program, the communication time and
+the computation time for the slower task must be measured before scaling. These times are used to predict the execution time for any MPI program as a function of
+the new scaling factor as in EQ~(\ref{eq:tnew}).
\begin{equation}
\label{eq:tnew}
\textit T_\textit{new} = T_\textit{Max Comp Old} \cdot S + T_{\textit{Max Comm Old}}
\end{equation}
-The above equation shows that the scaling factor \emph S has linear relation
-with the computation time without affecting the communication time. The
-communication time consists of the beginning times which an MPI calls for
-sending or receiving till the message is synchronously sent or received. In this
-paper we predict the execution time of the program for any new scaling factor
-value. Depending on this prediction we can produce our energy-performance scaling
-method as we will show in the coming sections. In the next section we make to finishan
-investigation study for the EQ~(\ref{eq:tnew}).
-
-\section{Performance Prediction Verification}
-\label{sec.verif}
-
-In this section we evaluate the precision of our performance prediction methods
-on the NAS benchmark. We use the EQ~(\ref{eq:tnew}) that predicts the execution
-time for any scale value. The NAS programs run the class B for comparing the
-real execution time with the predicted execution time. Each program runs offline
-with all available scaling factors on 8 or 9 nodes to produce real execution
-time values. These scaling factors are computed by dividing the maximum
-frequency by the new one see EQ~(\ref{eq:s}). In all tests, we use the simulator
-SimGrid/SMPI v3.10 to run the NAS programs.
-\begin{figure*}[t]
- \centering
- \includegraphics[width=.4\textwidth]{cg_per.eps}\qquad%
- \includegraphics[width=.4\textwidth]{mg_pre.eps}
- \includegraphics[width=.4\textwidth]{bt_pre.eps}\qquad%
- \includegraphics[width=.4\textwidth]{lu_pre.eps}
- \caption{Fitting Predicted to Real Execution Time}
- \label{fig:pred}
-\end{figure*}
-%see Figure~\ref{fig:pred}
-In our cluster there are 18 available frequency states for each processor from
-2.5 GHz to 800 MHz, there is 100 MHz difference between two successive
-frequencies. For more details on the characteristics of the platform refer to
-table~(\ref{table:platform}). This lead 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. The average normalized errors between the predicted execution time and
-the real time (SimGrid time) for all programs is between 0.0032 to 0.0133. AS an
-example, we are present the execution times of the NAS benchmarks as in the
-figure~(\ref{fig:pred}).
+ In this paper, this prediction method is used to select the best scaling factor for each processor as presented in the next section.
+
\section{Performance to Energy Competition}
\label{sec.compet}
This section demonstrates our approach for choosing the optimal scaling
factor. This factor gives maximum energy reduction taking into account the
-execution time for both computation and communication times. The relation
-between the energy and the performance are nonlinear and complex, because the
+execution times for both computation and communication. The relation
+between the energy and the performance is nonlinear and complex, because the
relation of the energy with scaling factor is nonlinear and with the performance
-it is linear see~\cite{17}. The relation between the energy and the performance
-is not straightforward. Moreover, they are not measured using the same metric.
+it is linear see~\cite{17}. Moreover, they are not measured using the same metric.
For solving this problem, we normalize the energy by calculating the ratio
between the consumed energy with scaled frequency and the consumed energy
without scaled frequency:
\label{eq:pnorm}
P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}
= \frac{T_\textit{Max Comp Old} \cdot S +
- T_\textit{Max Comm Old}}{ T_\textit{Old}}
+ T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} +
+ T_\textit{Max Comm Old}}
\end{equation}
-The second problem is the optimization operation for both energy and performance
+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
performance curves are not in the same direction see figure~(\ref{fig:r2}).
While the main goal is to optimize the energy and performance in the same
-time. According to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}) the
+time. According to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the
scaling factor \emph S reduce both the energy and the performance
simultaneously. But the main objective is to produce maximum energy reduction
with minimum performance reduction. Many researchers used different strategies
to solve this nonlinear problem for example see~\cite{19,42}, their methods add
big overhead to the algorithm for selecting the suitable frequency. In this
-paper we are present a method to find the optimal scaling factor \emph S for
-optimize both energy and performance simultaneously without adding big
+paper we present a method to find the optimal scaling factor \emph S for
+optimizing both energy and performance simultaneously without adding big
overheads. Our solution for this problem is to make the optimization process
have the same direction. Therefore, we inverse the equation of normalize
performance as follows:
between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance
curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent
the minimum energy consumption with minimum execution time (better performance)
-in the same time, see figure~(\ref{fig:r1}). Then our objective function has the
+at the same time, see figure~(\ref{fig:r1}). Then our objective function has the
following form:
\begin{equation}
\label{eq:max}
\textit{MaxDist} = \max (\overbrace{P^{-1}_\textit{Norm}}^{\text{Maximize}} -
\overbrace{E_\textit{Norm}}^{\text{Minimize}} )
\end{equation}
-Then we can select the optimal scaling factor that satisfy the
-EQ~(\ref{eq:max}). Our objective function can works with any energy model or
+Then we can select the optimal scaling factor that satisfy
+EQ~(\ref{eq:max}). Our objective function can work with any energy model or
static power values stored in a data file. Moreover, this function works in
-optimal way when the energy function has a convex form with frequency scaling
-factor as shown in~\cite{15,3,19}. Energy measurement model is not the
-objective of this paper and we choose Rauber and Rünger model as an example with two
-reasons that mentioned before.
+optimal way when the energy curve has a convex form over the available frequency scaling
+factors as shown in~\cite{15,3,19}.
\section{Optimal Scaling Factor for Performance and Energy}
\label{sec.optim}
-
-In the previous section we described the objective function that satisfy our
-goal in discovering optimal scaling factor for both performance and energy at
-the same time. Therefore, we develop an energy to performance scaling algorithm
-($EPSA$). This algorithm is simple and has a direct way to calculate the optimal
-scaling factor for both energy and performance at the same time.
+ Algorithm~\ref{EPSA} compute the optimal scaling factor according to the objective function described above.
\begin{algorithm}[tp]
- \caption{EPSA}
+ \caption{Scaling factor selection algorithm}
\label{EPSA}
\begin{algorithmic}[1]
\State Initialize the variable $Dist=0$
\State Set dynamic and static power values.
\State Set $P_{states}$ to the number of available frequencies.
\State Set the variable $F_{new}$ to max. frequency, $F_{new} = F_{max} $
- \State Set the variable $F_{diff}$ to the scale value between each two frequencies.
+ \State Set the variable $F_{diff}$ to the difference between two successive frequencies.
\For {$i=1$ to $P_{states} $}
- \State - Calculate the new frequency as $F_{new}=F_{new} - F_{diff} $
- \State - Calculate the scale factor $S$ as in EQ~(\ref{eq:s}).
- \State - Calculate all available scales $S_i$ depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}).
- \State - Select the maximum scale factor $S_1$ from the set\par\hspace{1 pt} of scales $S_i$.
- \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$
- \par\hspace{1 pt} as in EQ~(\ref{eq:enorm}).
- \State - Calculate the normalize inverse of performance\par\hspace{1 pt}
- $P_{NormInv}=T_{old}/T_{new}$ as in EQ~(\ref{eq:pnorm_en}).
+ \State - $F_{new}=F_{new} - F_{diff} $
+ \State - $S = \frac{F_\textit{max}}{F_\textit{new}}$
+ \State - $S_i = S \cdot \frac{T_1}{T_i}= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i} \ for \ i=1,...,N$
+ \State - $E_\textit{Norm} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot
+ \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+ P_\textit{static} \cdot T_1 \cdot S_i \cdot N }{
+ P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+ P_\textit{static} \cdot T_1 \cdot N }$
+ \State - $P_{NormInv}=T_{old}/T_{new}$
\If{ $(P_{NormInv}-E_{Norm} > Dist$) }
\State $S_{optimal} = S$
\State $Dist = P_{NormInv} - E_{Norm}$
\State Return $S_{optimal}$
\end{algorithmic}
\end{algorithm}
-The proposed EPSA algorithm works online during the execution time of the MPI
-program. It selects the optimal scaling factor by gathering some information
-from the program after one iteration. This algorithm has small execution time
-(between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ for 32 nodes). The data
-required by this algorithm is the computation time and the communication time
-for each task from the first iteration only. When these times are measured, the
-MPI program calls the EPSA algorithm to choose the new frequency using the
-optimal scaling factor. Then the program set the new frequency to the
-system. The algorithm is called just one time during the execution of the
-program. The DVFS algorithm~(\ref{dvfs}) shows where and when the EPSA algorithm is called
+
+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. This algorithm has small execution time
+(between 0.00152 $ms$ for 4 nodes to 0.00665 $ms$ 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~(\ref{dvfs}) shows where and when the algorithm is called
in the MPI program.
%\begin{minipage}{\textwidth}
-%\AG{Use the same format as for Algorithm~\ref{$EPSA$}}
\begin{algorithm}[tp]
\caption{DVFS}
\For {$J:=1$ to $Some-Iterations \; $}
\State -Computations Section.
\State -Communications Section.
- \If {$(J==1)$}
+ \If {$(J=1)$}
\State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
- \State -Call EPSA with these times.
- \State -Calculate the new frequency from optimal scale.
- \State -Set the new frequency to the system.
+ \State -Call algorithm~\ref{EPSA} with these times.
+ \State -Compute the new frequency from the returned optimal scaling factor.
+ \State -Set the new frequency to the CPU.
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
-
-After obtaining the optimal scale factor from the EPSA algorithm. The program
+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 substitution of the EQ~(\ref{eq:s}) in the EQ~(\ref{eq:si}), we
+value $T_i$. By substitution of EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we
can calculate the new frequency $F_i$ as follows:
\begin{equation}
\label{eq:fi}
F_i = \frac{F_\textit{max} \cdot T_i}{S_\textit{optimal} \cdot T_\textit{max}}
\end{equation}
According to this equation all the nodes may have the same frequency value if
-they have balanced workloads. Otherwise, they take different frequencies when
-have imbalanced workloads. Then EQ~(\ref{eq:fi}) works in adaptive way to change
-the frequency according to the nodes workloads.
+they have balanced workloads, otherwise, they take different frequencies when
+having imbalanced workloads. Thus, EQ~(\ref{eq:fi}) adapts the frequency of the CPU to the nodes' workloads to maintain performance.
\section{Experimental Results}
\label{sec.expe}
-
-The proposed EPSA algorithm was applied to seven MPI programs of the NAS
-benchmarks (EP, CG, MG, FT, BT, LU and SP). We work on three classes (A, B and
-C) for each program. Each program runs on specific number of processors
-proportional to the size of the class. Each class represents the problem size
-ascending from the 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. Our experiments are executed on the simulator SimGrid/SMPI
-v3.10. We design a platform file that simulates a cluster with one core per
-node. This cluster is a homogeneous architecture with distributed memory. The
+Our experiments are executed on the simulator SimGrid/SMPI
+v3.10. We configure the simulator to use a a homogeneous cluster with one core per
+node. The
detailed characteristics of our platform file are shown in the
-table~(\ref{table:platform}). Each node in the cluster has 18 frequency values
+table~(\ref{table:platform}).
+Each node in the cluster has 18 frequency values
from 2.5 GHz to 800 MHz with 100 MHz difference between each two successive
-frequencies.
+frequencies. The simulated network link is 1 GB Ethernet (TCP/IP).
+The backbone of the cluster simulates a high performance switch.
\begin{table}[htb]
\caption{Platform File Parameters}
% title of Table
\end{tabular}
\label{table:platform}
\end{table}
-Depending on the EQ~(\ref{eq:energy}), we measure the energy consumption for all
-the NAS MPI programs while assuming the power dynamic is equal to 20W and the
-power static is equal to 4W for all experiments. We run the proposed EPSA
-algorithm for all these programs. The results showed that the algorithm selected
+\subsection{Performance Prediction Verification}
+
+In this section we evaluate the precision of our performance prediction method based on EQ~(\ref{eq:tnew}) by applying it the NAS benchmarks. The NAS programs are executed with the class B option for comparing the
+real execution time with the predicted execution time. Each program runs offline
+with all available scaling factors on 8 or 9 nodes (depending on the benchmark) to produce real execution
+time values. These scaling factors are computed by dividing the maximum
+frequency by the new one see EQ~(\ref{eq:s}).
+\begin{figure*}[t]
+ \centering
+ \includegraphics[width=.4\textwidth]{cg_per.eps}\qquad%
+ \includegraphics[width=.4\textwidth]{mg_pre.eps}
+ \includegraphics[width=.4\textwidth]{bt_pre.eps}\qquad%
+ \includegraphics[width=.4\textwidth]{lu_pre.eps}
+ \caption{Comparing predicted to real execution time}
+ \label{fig:pred}
+\end{figure*}
+%see Figure~\ref{fig:pred}
+In our cluster there are 18 available frequency states for each processor.
+This lead 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 and the simulated ones. The average normalized errors between the predicted execution time and
+the real time (SimGrid time) for all programs is between 0.0032 to 0.0133.
+\JC{why compute the average error not the max}
+\subsection{The experimental results}
+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 the 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 the power dynamic with the highest frequency is equal to \np[W]{20} and
+the power static is equal to \np[W]{4} for all experiments. These power values were also
+used by Rauber and Rünger in~\cite{3}. The results showed that the algorithm selected
different scaling factors for each program depending on the communication
-features of the program as in the figure~(\ref{fig:nas}). This figure shows that
+features of the program as in the plots~(\ref{fig:nas}). These plots illustrate that
there are different distances between the normalized energy and the normalized
-inversed performance curves, because there are different communication features
-for each MPI program. When there are little or not communications, the inversed
+inverted performance curves, because there are different communication features
+for each benchmark. When there are little or not communications, the inverted
performance curve is very close to the energy curve. Then the distance between
-the two curves is very small. This lead to small energy savings. The opposite
-happens when there are a lot of communication, theto finish distance between the two
-curves is big. This lead 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 the NAS
-programs. In table~(\ref{table:factors results}), we record all optimal scaling
-factors results for each program on class C. These factors give the maximum
-energy saving percent and the minimum performance degradation percent in the
-same time over all available scales.
+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 factors give the maximum
+energy saving percent and the minimum performance degradation percent at the
+same time from all available scaling factors.
\begin{figure*}[t]
\centering
\includegraphics[width=.33\textwidth]{ep.eps}\hfill%
\includegraphics[width=.33\textwidth]{lu.eps}\hfill%
\includegraphics[width=.33\textwidth]{bt.eps}\hfill%
\includegraphics[width=.33\textwidth]{ft.eps}
- \caption{Optimal scaling factors for The NAS MPI Programs}
+ \caption{Optimal scaling factors for The parallel NAS benchmarks}
\label{fig:nas}
\end{figure*}
\begin{table}[htb]
- \caption{The EPSA Scaling Factors Results}
+ \caption{The Scaling Factors Results}
% title of Table
\centering
\begin{tabular}{|l|*{4}{r|}}
\label{table:factors results}
% is used to refer this table in the text
\end{table}
-
As shown in the table~(\ref{table:factors results}), when the optimal scaling
factor has big value we can gain more energy savings for example as in CG and
FT. The opposite happens when the optimal scaling factor is small value as
cases. In EP there are no communications inside the iterations. This make our
EPSA to selects smaller scaling factor values (inducing smaller energy savings).
-\section{Comparing Results}
-\label{sec.compare}
+\subsection{Comparing Results}
-In this section, we compare our EPSA algorithm results with Rauber and Rünger
-methods~\cite{3}. He had two scenarios, the first is to reduce energy to 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
+In this section, we compare our scaling factor selection method with Rauber and Rünger
+methods~\cite{3}. 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 (when the scale $S=1$)
to keep the performance from degradation as mush as possible. We refer to this
-scenario as $R_{E-P}$. The comparison is made in tables~(\ref{table:compare
- Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
-tables show the results of our EPSA and Rauber and Rünger scenarios for all the NAS
-benchmarks programs for classes A,B and C.
+scenario as $R_{E-P}$. The comparison is made in tables~(\ref{table:compareA},
+\ref{table:compareB}, and \ref{table:compareC}). These
+tables show the results of our method and Rauber and Rünger scenarios for all the
+NAS benchmarks programs for classes A,B and C.
\begin{table}[p]
\caption{Comparing Results for The NAS Class A}
% title of Table
$R_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
$R_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline
\end{tabular}
- \label{table:compare Class A}
+ \label{table:compareA}
% is used to refer this table in the text
\end{table}
\begin{table}[p]
$R_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
$R_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline
\end{tabular}
- \label{table:compare Class B}
+ \label{table:compareB}
% is used to refer this table in the text
\end{table}
$R_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
$R_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline
\end{tabular}
-\label{table:compare Class C}
+\label{table:compareC}
% is used to refer this table in the text
\end{table}
-As shown in these tables our scaling factor is not optimal for energy saving
-such as Rauber's scaling factor EQ~(\ref{eq:sopt}), but it is optimal for both
-the energy and the performance simultaneously. Our $EPSA$ optimal scaling factors
-has better simultaneous optimization for both the energy and the performance
-compared to Rauber and Rünger energy-performance method ($R_{E-P}$). Also, in
-($R_{E-P}$) method when setting the frequency to maximum value for the
-slower task lead to a small improvement of the performance. Also the results
-show that this method keep or improve energy saving. Because of the energy
-consumption decrease when the execution time decreased while the frequency value
-increased.
+As shown in tables~\ref{table:compareA},~\ref{table:compareB} and~\ref{table:compareC}, the ($R_{E-P}$) method outperforms the ($R_{E}$) method in terms of performance and energy reduction. The ($R_{E-P}$) method also gives better energy savings than our method. However, although our scaling 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}) shows the maximum distance between the energy saving
-percent and the performance degradation percent. Therefore, this means it is the
-same resultant of our objective function EQ~(\ref{eq:max}). Our algorithm always
-gives positive energy to performance trade offs while Rauber and Rünger method
-($R_{E-P}$) gives in some time negative trade offs such as in BT and
-EP. The positive trade offs with highest values lead to maximum energy savings
-concatenating with less performance degradation and this the objective of this
-paper. While the negative trade offs refers to improving energy saving (or may
-be the performance) while degrading the performance (or may be the energy) more
-than the first.
+percent and the performance degradation percent.
+Negative values mean that one of the two objectives (energy or performance) have been degraded more than the other. The positive tradeoffs 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 tradeoffs while Rauber and Rünger method
+($R_{E-P}$) gives in some time negative tradeoffs such as in BT and
+EP.
\begin{figure}[t]
\centering
\includegraphics[width=.33\textwidth]{compare_class_A.pdf}
\includegraphics[width=.33\textwidth]{compare_class_B.pdf}
\includegraphics[width=.33\textwidth]{compare_class_c.pdf}
- \caption{Comparing Our EPSA with Rauber and Rünger Methods}
+ \caption{Comparing our method to Rauber and Rünger Methods}
\label{fig:compare}
\end{figure}
\section{Conclusion}
\label{sec.concl}
-In this paper we develop the simultaneous energy-performance algorithm. It is works based on the trade off relation between the energy and performance. The results showed that when the scaling factor is big value leads to more energy saving. Also, it show that when the the scaling factor is small value leads to the fact that the scaling factor has bigger impact on performance than energy. Then the algorithm optimize the energy saving and performance in the same time to have positive trade off. The optimal trade off refer to maximum distance between the energy and the inversed performance curves. Also, the results explained when setting the slowest task to maximum frequency usually not have a big improvement on performance.
+In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed application running on an homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy consumption and the performance of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best tradeoff between energy reduction and performance which is the maximum distance between the energy and the inverted performance curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber's methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber's methods in terms of energy-performance ratio.
+
+In the near future, we would like to adapt this scaling factor selection method to heterogeneous platforms where each node has different characteristics. In particular, each CPU has different available frequencies, energy consumption and performance. It would be also interesting to develop a new energy model for asynchronous parallel iterative methods where the number of iterations is not known in advance and depends on the global convergence of the iterative system.
-\section*{Acknowledgment}
-\AG{Right?}
-Computations have been performed on the supercomputer facilities of the
-Mésocentre de calcul de Franche-Comté. As a PhD student , M. Ahmed Fanfakh , would
-likes to thank the University of Babylon /Iraq for supporting my scholarship program that allows me
-working on this paper.
+\section*{Acknowledgment}
+As a PhD student, M. Ahmed Fanfakh, would like to thank the University of
+Babylon (Iraq) for supporting his work.
+\JC{delete the online paths for each reference}
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