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+\usepackage{url}
+\DeclareUrlCommand\email{\urlstyle{same}}
+
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-\renewcommand*\npunitcommand[1]{\text{#1}}
+\AtBeginDocument{%
+ \renewcommand*\npunitcommand[1]{\text{#1}}
+ \npthousandthpartsep{}}
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-\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}
+\newcommand{\AG}[2][inline]{%
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+\newcommand{\JC}[2][inline]{%
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\begin{document}
-\title{Dynamic Frequency Scaling for Energy Consumption Reduction in Distributed MPI Programs}
+\title{Dynamic Frequency Scaling for Energy Consumption
+ Reduction in Distributed MPI Programs}
\author{%
\IEEEauthorblockN{%
\IEEEauthorblockA{%
FEMTO-ST Institute\\
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,raphael.couturier,ahmed.fanfakh\_badri\_muslim,arnaud.giersch\}@univ-fcomte.fr
+ IUT de Belfort-Montbéliard,
+ 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
+ % Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël
+ % Fax: \mbox{+33 3 84 58 77 81}\\ % Dept Info
+ Email: \email{{jean-claude.charr,raphael.couturier,ahmed.fanfakh_badri_muslim,arnaud.giersch}@univ-fcomte.fr}
}
}
\maketitle
-\AG{Is the fax number correct? Shall we add a telephone number?}
\begin{abstract}
- 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 trade-off
-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.
+ 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 if 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 trade-off 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 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 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
-kW). This large increase in number of computing cores has led to large energy
-consumption by these architectures. Moreover, the price of energy is expected to
-continue its ascent according to the demand. For all these reasons energy
-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. 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
-they use heuristics to find the best scaling factor during the computation. If
-they aim for performance they choose the best scaling factor that reduces the
-consumed energy while affecting as little as possible the performance. On the
-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 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 present an
-algorithm that selects a frequency scaling factor that simultaneously takes into
-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
-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{44}. Our experiments are executed
-using the simulator SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183}
-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 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 trade-offs
-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}.
+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 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 \np[Tflop/s]{33} while consuming
+\np[kW]{17808}). This large increase in number of computing cores has led to
+large energy consumption by these architectures. Moreover, the price of energy
+is expected to continue its ascent according to the demand. For all these
+reasons energy 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. 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 they use heuristics to find the best
+scaling factor during the computation. If they aim for performance they choose
+the best scaling factor that reduces the consumed energy while affecting as
+little as possible the performance. On the 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 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 present an algorithm that selects a
+frequency scaling factor that simultaneously takes into 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 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{44}. Our experiments are executed using the simulator
+SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} 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 some
+related works from other authors. Section~\ref{sec.exe} explains the execution
+of parallel tasks and the sources of slack times. It also presents an energy
+model for homogeneous platforms. Section~\ref{sec.mpip} describes how the
+performance of MPI programs can be predicted. Section~\ref{sec.compet} presents
+the energy-performance objective function that maximizes the reduction of energy
+consumption while minimizing the degradation of the program's performance.
+Section~\ref{sec.optim} details the proposed energy-performance algorithm.
+Section~\ref{sec.expe} 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 methods. Finally, we
+conclude in Section~\ref{sec.concl} with a summary and some future works.
+
\section{Related works}
\label{sec.relwork}
-\AG{Consider introducing the models (sec.~\ref{sec.exe}) before related works}
-In this section, some heuristics to compute the scaling factor are
-presented and classified into two categories: offline and online methods.
+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.
+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 multi-task 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 and Rünger 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 of 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 :
+
+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, \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
+much as needed if the iterations are not regular. Kimura, Peraza, Yu-Liang et
+al.~\cite{11,2,31} used varied heuristics 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 and
+Rünger used an analytical model that can predict the consumed energy and the
+execution time for every frequency gear after measuring the consumed energy and
+the execution time with the highest 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 of 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 our paper is presenting a new online scaling factor
+selection method which has the following characteristics:
\begin{enumerate}
-\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
+\item It is based on Rauber and Rünger analytical 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
+ account the communication time.
+\item It is well adapted to distributed applications with imbalanced tasks.
+\item it 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{Execution and energy of parallel tasks on homogeneous platform}
+\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}
+
+% \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
-executed on this model can be either synchronous or asynchronous. In this paper
+
+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
+executed on this model can be either synchronous or asynchronous. In this paper
we consider execution of the synchronous tasks on distributed homogeneous
-platform. These tasks can exchange the data via synchronous message passing.
+platform. These tasks can exchange the data via synchronous message passing.
\begin{figure*}[t]
\centering
- \subfloat[Sync. imbalanced communications]{\includegraphics[scale=0.67]{commtasks}\label{fig:h1}}
- \subfloat[Sync. imbalanced computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
+ \subfloat[Sync. imbalanced communications]{%
+ \includegraphics[scale=0.67]{fig/commtasks}\label{fig:h1}}
+ \subfloat[Sync. imbalanced computations]{%
+ \includegraphics[scale=0.67]{fig/compt}\label{fig:h2}}
\caption{Parallel tasks on homogeneous platform}
\label{fig:homo}
\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 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 EQ~(\ref{eq:T1}).
+communication time. Moreover, the synchronous communications between tasks can
+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 slack
+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 EQ~(\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 $i$ 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}
-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 EQ~(\ref{eq:pd}).
+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 EQ~(\ref{eq:pd}).
\begin{equation}
\label{eq:pd}
P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
\label{eq:ps}
P_\textit{static} = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
\end{equation}
-where V is the supply voltage, $N_{trans}$ is the number of transistors, $K_{design}$ is a
-design dependent parameter and $I_{leak}$ is a technology-dependent
-parameter. Energy consumed by an individual processor $E_{ind}$ is the summation
-of the dynamic and the static powers multiplied by the execution time~\cite{36,15}.
+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. The energy consumed by an individual processor
+to execute a given program can be computed as:
\begin{equation}
\label{eq:eind}
- E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T
+ E_\textit{ind} = P_\textit{dyn} \cdot T_{Comp} + P_\textit{static} \cdot T
\end{equation}
-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 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}).
+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
+communications, no slack times and no synchronizations.
+
+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 $f$
+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 can be expressed by the scaling factor $S$ which is the
+ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
\begin{equation}
\label{eq:s}
S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
-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}).
+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 and Rünger~\cite{3}, can be written as a
+function of the scaling factor $S$, as in EQ~(\ref{eq:energy}).
\begin{equation}
\label{eq:energy}
P_\textit{static} \cdot T_1 \cdot S_1 \cdot N
\hfill
\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}).
+where $N$ is the number of parallel nodes, $T_i$ and $S_i$ for $i=1,\dots,N$ 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}).
\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}
-In this paper we depend on
-Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this
-model is used for any number of concurrent tasks, 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}).
+In this paper we depend on Rauber and Rünger energy model EQ~(\ref{eq:energy})
+for two reasons: (1) this model is used for any number of concurrent tasks, 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}).
\begin{equation}
\label{eq:sopt}
\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 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~\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}).
+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~\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}
- In this paper, this prediction method is used to select the best scaling factor for each processor as presented in the next section.
+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 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}. 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:
+This section demonstrates our approach for choosing the optimal scaling factor.
+This factor gives maximum energy reduction taking into account 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}. 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:
\begin{multline}
\label{eq:enorm}
E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} +
T_\textit{Max Comm Old}}
\end{equation}
-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
-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 present a method to find the optimal scaling factor \emph S for
+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 scaling factor $S$ 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 present a method to find the optimal scaling factor $S$ 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
+have the same direction. Therefore, we inverse the equation of normalize
performance as follows:
\begin{equation}
\label{eq:pnorm_en}
P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
- = \frac{ T_\textit{Old}}{T_\textit{Max Comp Old} \cdot S +
+ = \frac{T_\textit{Max Comp Old} +
+ T_\textit{Max Comm Old}}{T_\textit{Max Comp Old} \cdot S +
T_\textit{Max Comm Old}}
\end{equation}
\begin{figure*}
\centering
\subfloat[Converted relation.]{%
- \includegraphics[width=.4\textwidth]{file.eps}\label{fig:r1}}%
+ \includegraphics[width=.4\textwidth]{fig/file}\label{fig:r1}}%
\qquad%
\subfloat[Real relation.]{%
- \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}}
+ \includegraphics[width=.4\textwidth]{fig/file3}\label{fig:r2}}
\label{fig:rel}
\caption{The energy and performance relation}
\end{figure*}
Then, we can modelize our objective function as finding the maximum distance
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)
-at the same time, see figure~(\ref{fig:r1}). Then our objective function has the
-following form:
+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:r1}). Then our objective
+function has the following form:
\begin{equation}
\label{eq:max}
- Max Dist = \max_{j=1,2,\dots,F} (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
- \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
+ Max Dist = \max_{j=1,2,\dots,F}
+ (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
+ \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
\end{equation}
-where F is the number of available frequencies. 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 curve has a convex form over the available frequency scaling
-factors as shown in~\cite{15,3,19}.
+where $F$ is the number of available frequencies. Then we can select the optimal
+scaling factor that satisfies 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 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}
- Algorithm~\ref{EPSA} compute the optimal scaling factor according to the objective function described above.
+
+Algorithm~\ref{EPSA} computes the optimal scaling factor according to the
+objective function described above.
\begin{algorithm}[tp]
\caption{Scaling factor selection algorithm}
\label{EPSA}
\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 difference between two successive frequencies.
+ \State Set the variable $F_{diff}$ to the difference between two successive
+ frequencies.
\For {$j:=1$ to $P_{states} $}
- \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 - $E_\textit{Norm} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
- \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{
- P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_\textit{static} \cdot T_1 \cdot N }$
- \State - $P_{NormInv}=T_{old}/T_{new}$
- \If{ $(P_{NormInv}-E_{Norm} > Dist$) }
+ \State $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 $i=1,\dots,N$
+ \State $E_\textit{Norm} =
+ \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
+ \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+ P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{
+ P_\textit{dyn} \cdot
+ \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
+ P_\textit{static} \cdot T_1 \cdot N }$
+ \State $P_{NormInv}=T_{old}/T_{new}$
+ \If{$(P_{NormInv}-E_{Norm} > Dist)$}
\State $S_{opt} = S$
\State $Dist = P_{NormInv} - E_{Norm}$
\EndIf
\end{algorithm}
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}
+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 a small execution time: for a homogeneous cluster
+composed of nodes having the characteristics presented in
+table~\ref{table:platform}, it takes \np[ms]{0.00152} on average for 4 nodes and
+\np[ms]{0.00665} on average 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{table}[htb]
+ \caption{Platform file parameters}
+ % title of Table
+ \centering
+ \begin{tabular}{|*{7}{l|}}
+ \hline
+ Max & Min & Backbone & Backbone & Link & Link & Sharing \\
+ Freq. & Freq. & Bandwidth & Latency & Bandwidth & Latency & Policy \\
+ \hline
+ \np{2.5} & \np{800} & \np[GBps]{2.25} & \np[$\mu$s]{0.5} & \np[GBps]{1} & \np[$\mu$s]{50} & Full \\
+ GHz & MHz & & & & & Duplex \\
+ \hline
+ \end{tabular}
+ \label{table:platform}
+\end{table}
\begin{algorithm}[tp]
\caption{DVFS}
\label{dvfs}
\begin{algorithmic}[1]
- \For {$k:=1$ to $Some-Iterations \; $}
- \State -Computations Section.
- \State -Communications Section.
- \If {$(k=1)$}
- \State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
- \State -Call algorithm~\ref{EPSA} with these times.
- \State -Compute the new frequency from the \par\hspace{13 pt} returned optimal scaling factor.
- \State -Set the new frequency to the CPU.
- \EndIf
-\EndFor
-\end{algorithmic}
+ \For {$k:=1$ to \textit{some iterations}}
+ \State Computations section.
+ \State Communications section.
+ \If {$(k=1)$}
+ \State Gather all times of computation and\newline\hspace*{3em}%
+ communication from each node.
+ \State Call algorithm~\ref{EPSA} with these times.
+ \State Compute the new frequency from the\newline\hspace*{3em}%
+ returned optimal scaling factor.
+ \State Set the new frequency to the CPU.
+ \EndIf
+ \EndFor
+ \end{algorithmic}
\end{algorithm}
-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 EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we
-can calculate the new frequency $F_i$ as follows:
+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 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
-having imbalanced workloads. Thus, EQ~(\ref{eq:fi}) adapts the frequency of the CPU to the nodes' workloads to maintain performance.
+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}
-Our experiments are executed on the simulator SimGrid/SMPI
-v3.10. We configure the simulator to use 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
-from 2.5 GHz to 800 MHz with 100 MHz difference between each two successive
-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
- \centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Max & Min & Backbone & Backbone&Link &Link& Sharing \\
- Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy \\ \hline
- \np{2.5} & \np{800} & \np[GBps]{2.25} &\np[$\mu$s]{0.5}& \np[GBps]{1} & \np[$\mu$s]{50} &Full \\
- GHz& MHz& & & & &Duplex \\\hline
- \end{tabular}
- \label{table:platform}
-\end{table}
+Our experiments are executed on the simulator SimGrid/SMPI v3.10. We configure
+the simulator to use 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
+from \np[GHz]{2.5} to \np[MHz]{800} with \np[MHz]{100} difference between each
+two successive frequencies. The simulated network link is \np[GB]{1} Ethernet
+(TCP/IP). The backbone of the cluster simulates a high performance switch.
+
\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}).
+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=.328\textwidth]{cg_per.eps}\hfill%
- \includegraphics[width=.328\textwidth]{mg_pre.eps}\hfill%
- % \includegraphics[width=.4\textwidth]{bt_pre.eps}\qquad%
- \includegraphics[width=.328\textwidth]{lu_pre.eps}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/cg_per}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/mg_pre}\hfill%
+ % \includegraphics[width=.4\textwidth]{fig/bt_pre}\qquad%
+ \includegraphics[width=.328\textwidth]{fig/lu_pre}\hfill%
\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 maximum normalized error between the predicted execution time and the real time (SimGrid time) for all programs is between 0.0073 to 0.031. The better case is for CG and the worse case is for LU.
+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 and the simulated ones. The maximum
+normalized error between these two execution times varies between \np{0.0073} to
+\np{0.031} dependent on the executed benchmark. The smallest prediction error
+was for CG and the worst one was for LU.
+
\subsection{The experimental results for the scaling algorithm }
-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
+
+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 plots~(\ref{fig:nas}). These plots illustrate that
-there are different distances between the normalized energy and the normalized
-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 leads to small energy savings. The opposite
+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 plots~(\ref{fig:nas}). These plots illustrate that there are
+different distances between the normalized energy and the normalized 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 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 factors give the maximum
-energy saving percent and the minimum performance degradation percent at the
-same time from all available scaling factors.
+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
+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=.328\textwidth]{ep.eps}\hfill%
- \includegraphics[width=.328\textwidth]{cg.eps}\hfill%
- \includegraphics[width=.328\textwidth]{sp.eps}
- \includegraphics[width=.328\textwidth]{lu.eps}\hfill%
- \includegraphics[width=.328\textwidth]{bt.eps}\hfill%
- \includegraphics[width=.328\textwidth]{ft.eps}
+ \includegraphics[width=.328\textwidth]{fig/ep}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/cg}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/sp}
+ \includegraphics[width=.328\textwidth]{fig/lu}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/bt}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/ft}
\caption{Optimal scaling factors for the predicted energy and performance of NAS benchmarks}
\label{fig:nas}
\end{figure*}
\centering
\begin{tabular}{|l|*{4}{r|}}
\hline
- Program & Optimal & Energy & Performance&Energy-Perf.\\
- Name & Scaling Factor& Saving \%&Degradation \% &Distance \\ \hline
- CG & 1.56 &39.23&14.88 &24.35\\ \hline
- MG & 1.47 &34.97&21.70 &13.27 \\ \hline
- EP & 1.04 &22.14&20.73 &1.41\\ \hline
- LU & 1.38 &35.83&22.49 &13.34\\ \hline
- BT & 1.31 &29.60&21.28 &8.32\\ \hline
- SP & 1.38 &33.48&21.36 &12.12\\ \hline
- FT & 1.47 &34.72&19.00 &15.72\\ \hline
- \end{tabular}
+ Program & Optimal & Energy & Performance & Energy-Perf. \\
+ Name & Scaling Factor & Saving \% & Degradation \% & Distance \\
+ \hline
+ CG & 1.56 & 39.23 & 14.88 & 24.35 \\
+ \hline
+ MG & 1.47 & 34.97 & 21.70 & 13.27 \\
+ \hline
+ EP & 1.04 & 22.14 & 20.73 & 1.41 \\
+ \hline
+ LU & 1.38 & 35.83 & 22.49 & 13.34 \\
+ \hline
+ BT & 1.31 & 29.60 & 21.28 & 8.32 \\
+ \hline
+ SP & 1.38 & 33.48 & 21.36 & 12.12 \\
+ \hline
+ FT & 1.47 & 34.72 & 19.00 & 15.72 \\
+ \hline
+ \end{tabular}
\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
-example BT and EP. Our algorithm selects big scaling factor value when the
+FT. The opposite happens when the optimal scaling factor is small value as
+example BT and EP. Our algorithm selects big scaling factor value when the
communication and the other slacks times are big and smaller ones in opposite
-cases. In EP there are no communications inside the iterations. This make our
-algorithm to selects smaller scaling factor values (inducing smaller energy savings).
+cases. In EP there are no communications inside the iterations. This make our
+algorithm to selects smaller scaling factor values (inducing smaller energy
+savings).
\subsection{Results comparison}
-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 $R_{E-P}$. While we refer to our algorithm as EPSA. 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.
+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 $R_{E-P}$. While we refer to our
+algorithm as EPSA (Energy to Performance Scaling Algorithm). 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
\centering
\begin{tabular}{|l|l|*{4}{r|}}
\hline
- Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- Name &Name&Value& Saving \%&Degradation \% &Distance
- \\ \hline
+ Method & Program & Factor & Energy & Performance & Energy-Perf. \\
+ Name & Name & Value & Saving \% & Degradation \% & Distance \\
+ \hline
% \rowcolor[gray]{0.85}
- $EPSA$&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline
- $R_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline
- $R_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
+ $EPSA$ & CG & 1.56 & 37.02 & 13.88 & 23.14 \\ \hline
+ $R_{E-P}$ & CG & 2.14 & 42.77 & 25.27 & 17.50 \\ \hline
+ $R_{E}$ & CG & 2.14 & 42.77 & 26.46 & 16.31 \\ \hline
- $EPSA$&MG & 1.47 &27.66&16.82&10.84\\ \hline
- $R_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
- $R_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline
+ $EPSA$ & MG & 1.47 & 27.66 & 16.82 & 10.84 \\ \hline
+ $R_{E-P}$ & MG & 2.14 & 34.45 & 31.84 & 2.61 \\ \hline
+ $R_{E}$ & MG & 2.14 & 34.48 & 33.65 & 0.80 \\ \hline
- $EPSA$&EP &1.19 &25.32&20.79&4.53\\ \hline
- $R_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
- $R_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline
+ $EPSA$ & EP & 1.19 & 25.32 & 20.79 & 4.53 \\ \hline
+ $R_{E-P}$ & EP & 2.05 & 41.45 & 55.67 & -14.22 \\ \hline
+ $R_{E}$ & EP & 2.05 & 42.09 & 57.59 & -15.50 \\ \hline
- $EPSA$&LU&1.56& 39.55 &19.38& 20.17\\ \hline
- $R_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline
- $R_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
+ $EPSA$ & LU & 1.56 & 39.55 & 19.38 & 20.17 \\ \hline
+ $R_{E-P}$ & LU & 2.14 & 45.62 & 27.00 & 18.62 \\ \hline
+ $R_{E}$ & LU & 2.14 & 45.66 & 33.01 & 12.65 \\ \hline
- $EPSA$&BT&1.31& 29.60&20.53&9.07 \\ \hline
- $R_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline
- $R_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline
+ $EPSA$ & BT & 1.31 & 29.60 & 20.53 & 9.07 \\ \hline
+ $R_{E-P}$ & BT & 2.10 & 45.53 & 49.63 & -4.10 \\ \hline
+ $R_{E}$ & BT & 2.10 & 43.93 & 52.86 & -8.93 \\ \hline
- $EPSA$&SP&1.38& 33.51&15.65&17.86 \\ \hline
- $R_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline
- $R_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
+ $EPSA$ & SP & 1.38 & 33.51 & 15.65 & 17.86 \\ \hline
+ $R_{E-P}$ & SP & 2.11 & 45.62 & 42.52 & 3.10 \\ \hline
+ $R_{E}$ & SP & 2.11 & 45.78 & 43.09 & 2.69 \\ \hline
- $EPSA$&FT&1.25&25.00&10.80&14.20 \\ \hline
- $R_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline
- $R_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline
+ $EPSA$ & FT & 1.25 & 25.00 & 10.80 & 14.20 \\ \hline
+ $R_{E-P}$ & 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:compareA}
% is used to refer this table in the text
\centering
\begin{tabular}{|l|l|*{4}{r|}}
\hline
- Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- Name &Name&Value& Saving \%&Degradation \% &Distance
- \\ \hline
+ Method & Program & Factor & Energy & Performance & Energy-Perf. \\
+ Name & Name & Value & Saving \% & Degradation \% & Distance \\
+ \hline
% \rowcolor[gray]{0.85}
- $EPSA$&CG & 1.66 &39.23&16.63&22.60 \\ \hline
- $R_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline
- $R_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
+ $EPSA$ & CG & 1.66 & 39.23 & 16.63 & 22.60 \\ \hline
+ $R_{E-P}$ & CG & 2.15 & 45.34 & 27.60 & 17.74 \\ \hline
+ $R_{E}$ & CG & 2.15 & 45.34 & 28.88 & 16.46 \\ \hline
- $EPSA$ &MG & 1.47 &34.98&18.35&16.63\\ \hline
- $R_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
- $R_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
+ $EPSA$ & MG & 1.47 & 34.98 & 18.35 & 16.63 \\ \hline
+ $R_{E-P}$ & MG & 2.14 & 43.55 & 36.42 & 7.13 \\ \hline
+ $R_{E}$ & MG & 2.14 & 43.56 & 37.07 & 6.49 \\ \hline
- $EPSA$&EP &1.08 &20.29&17.15&3.14 \\ \hline
- $R_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline
- $R_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline
+ $EPSA$ & EP & 1.08 & 20.29 & 17.15 & 3.14 \\ \hline
+ $R_{E-P}$ & EP & 2.00 & 42.38 & 56.88 & -14.50 \\ \hline
+ $R_{E}$ & EP & 2.00 & 39.73 & 59.94 & -20.21 \\ \hline
- $EPSA$&LU&1.47&38.57&21.34&17.23 \\ \hline
- $R_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline
- $R_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline
+ $EPSA$ & LU & 1.47 & 38.57 & 21.34 & 17.23 \\ \hline
+ $R_{E-P}$ & LU & 2.10 & 43.62 & 36.51 & 7.11 \\ \hline
+ $R_{E}$ & LU & 2.10 & 43.61 & 38.54 & 5.07 \\ \hline
- $EPSA$&BT&1.31& 29.59&20.88&8.71\\ \hline
- $R_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline
- $R_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline
+ $EPSA$ & BT & 1.31 & 29.59 & 20.88 & 8.71 \\ \hline
+ $R_{E-P}$ & BT & 2.10 & 44.53 & 53.05 & -8.52 \\ \hline
+ $R_{E}$ & BT & 2.10 & 42.93 & 52.80 & -9.87 \\ \hline
- $EPSA$&SP&1.38&33.44&19.24&14.20 \\ \hline
- $R_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline
- $R_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
+ $EPSA$ & SP & 1.38 & 33.44 & 19.24 & 14.20 \\ \hline
+ $R_{E-P}$ & SP & 2.15 & 45.69 & 43.20 & 2.49 \\ \hline
+ $R_{E}$ & SP & 2.15 & 45.41 & 44.47 & 0.94 \\ \hline
- $EPSA$&FT&1.38&34.40&14.57&19.83 \\ \hline
- $R_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
- $R_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline
+ $EPSA$ & FT & 1.38 & 34.40 & 14.57 & 19.83 \\ \hline
+ $R_{E-P}$ & 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:compareB}
% is used to refer this table in the text
\centering
\begin{tabular}{|l|l|*{4}{r|}}
\hline
- Method&Program&Factor& Energy& Performance &Energy-Perf.\\
- Name &Name&Value& Saving \%&Degradation \% &Distance
- \\ \hline
+ Method & Program & Factor & Energy & Performance & Energy-Perf. \\
+ Name & Name & Value & Saving \% & Degradation \% & Distance \\
+ \hline
% \rowcolor[gray]{0.85}
- $EPSA$&CG & 1.56 &39.23&14.88&24.35 \\ \hline
- $R_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline
- $R_{E}$&CG &2.15 &45.36&26.70&18.66\\ \hline
+ $EPSA$ & CG & 1.56 & 39.23 & 14.88 & 24.35 \\ \hline
+ $R_{E-P}$ & CG & 2.15 & 45.36 & 25.89 & 19.47 \\ \hline
+ $R_{E}$ & CG & 2.15 & 45.36 & 26.70 & 18.66 \\ \hline
- $EPSA$&MG & 1.47 &34.97&21.69&13.27\\ \hline
- $R_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline
- $R_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
+ $EPSA$ & MG & 1.47 & 34.97 & 21.69 & 13.27 \\ \hline
+ $R_{E-P}$ & MG & 2.15 & 43.65 & 40.45 & 3.20 \\ \hline
+ $R_{E}$ & MG & 2.15 & 43.64 & 41.38 & 2.26 \\ \hline
- $EPSA$&EP &1.04 &22.14&20.73&1.41 \\ \hline
- $R_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline
- $R_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline
+ $EPSA$ & EP & 1.04 & 22.14 & 20.73 & 1.41 \\ \hline
+ $R_{E-P}$ & EP & 1.92 & 39.40 & 56.33 & -16.93 \\ \hline
+ $R_{E}$ & EP & 1.92 & 38.10 & 56.35 & -18.25 \\ \hline
- $EPSA$&LU&1.38&35.83&22.49&13.34 \\ \hline
- $R_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline
- $R_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline
+ $EPSA$ & LU & 1.38 & 35.83 & 22.49 & 13.34 \\ \hline
+ $R_{E-P}$ & LU & 2.15 & 44.97 & 41.00 & 3.97 \\ \hline
+ $R_{E}$ & LU & 2.15 & 44.97 & 41.80 & 3.17 \\ \hline
- $EPSA$&BT&1.31& 29.60&21.28&8.32\\ \hline
- $R_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline
- $R_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline
+ $EPSA$ & BT & 1.31 & 29.60 & 21.28 & 8.32 \\ \hline
+ $R_{E-P}$ & BT & 2.13 & 45.60 & 49.84 & -4.24 \\ \hline
+ $R_{E}$ & BT & 2.13 & 44.90 & 55.16 & -10.26 \\ \hline
- $EPSA$&SP&1.38&33.48&21.35&12.12\\ \hline
- $R_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline
- $R_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline
+ $EPSA$ & SP & 1.38 & 33.48 & 21.35 & 12.12 \\ \hline
+ $R_{E-P}$ & SP & 2.10 & 45.69 & 43.60 & 2.09 \\ \hline
+ $R_{E}$ & SP & 2.10 & 45.75 & 44.10 & 1.65 \\ \hline
- $EPSA$&FT&1.47&34.72&19.00&15.72 \\ \hline
- $R_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline
- $R_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline
+ $EPSA$ & FT & 1.47 & 34.72 & 19.00 & 15.72 \\ \hline
+ $R_{E-P}$ & 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:compareC}
-% is used to refer this table in the text
+ \label{table:compareC}
+ % is used to refer this table in the text
\end{table}
-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.
-Negative values mean that one of the two objectives (energy or performance) have 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 while Rauber and Rünger method
-($R_{E-P}$) gives in some time negative trade-offs such as in BT and
-EP.
-\begin{figure*}[t]
- \centering
- \includegraphics[width=.328\textwidth]{compare_class_A.pdf}
- \includegraphics[width=.328\textwidth]{compare_class_B.pdf}
- \includegraphics[width=.328\textwidth]{compare_class_c.pdf}
- \caption{Comparing our method to Rauber and Rünger methods}
- \label{fig:compare}
-\end{figure*}
+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. Negative values in the energy-performance column mean that one of
+the two objectives (energy or performance) have 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
+while Rauber and Rünger method ($R_{E-P}$) gives in some time negative
+trade-offs such as in BT and EP.
+%\begin{figure*}[t]
+% \centering
+% \includegraphics[width=.328\textwidth]{fig/compare_class_A}
+% \includegraphics[width=.328\textwidth]{fig/compare_class_B}
+% \includegraphics[width=.328\textwidth]{fig/compare_class_C}
+% \caption{Comparing our method to Rauber and Rünger methods}
+% \label{fig:compare}
+%\end{figure*}
+
\section{Conclusion}
\label{sec.concl}
-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 trade-off 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 and Rünger methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber and Rünger 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.
+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 trade-off
+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 and Rünger
+methods while being executed on the simulator SimGrid. The results showed that
+our method, outperforms Rauber and Rünger 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{Jean-Claude, why did you remove the Mésocentre here?}
-As a PhD student, M. Ahmed Fanfakh, would like to thank the University of
-Babylon (Iraq) for supporting his work.
+
+This work has been supported by the Labex ACTION project (contract
+``ANR-11-LABX-01-01''). Computations have been performed on the supercomputer
+facilities of the Mésocentre de calcul de Franche-Comté. As a PhD student,
+Mr. Ahmed Fanfakh, would like to thank the University of Babylon (Iraq) for
+supporting his work.
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