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\begin{document}
-\begin{center}
-\Large
-\title*\textbf{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs}
- \end{center}
-\parskip 0pt
-\linespread{1.18}
-\normalsize
-\makeatletter
-\renewcommand*{\@seccntformat}[1]{\csname the#1\endcsname\hspace{0.01cm}}
-\makeatother
-\sectionfont{\large}
-\section{.~Introduction }
-The need for computing power is still increasing 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 supercomputers (for example in November 2013, according to the top 500 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. However, this operation 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 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 algorithm that selects the optimal frequency scaling factor that takes into consideration simultaneously 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 optimal scaling factor online with very small footprint. The proposed algorithm takes into account the communication times of the MPI programs 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 penchmarks (NPB v3.3) developed by NASA~\cite{44}. Our experiments are executed using the simulator Simgrid/SMPI v3.10~\cite{45} over an homogeneous distributed memory architecture. Furthermore, we compare the proposed algorithm with Rauber's methods. The comparison's results show that our algorithm gives better energy-time trade off.
-\sectionfont{\large}
-\section{.~Related Works }
-In the this section some heuristics, to compute the scaling factor, are presented and classified in two parts : offline and online methods.
- \sectionfont{\large}
-\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 an example in Azevedo et al.~\cite{40}. He used intra-task algorithm to choose the DVFS 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 saving energy with time limits. Another approaches gathers and stores the runtime information for each DVFS state , then used their methods offline to select the suitable DVFS that optimize energy-time trade offs. As an example~\cite{8}, Rountree et al. used liner programming algorithm, while in~\cite{38,34}, Cochran et al. used multi logistic regression algorithm for the same goal. The offline study that shown the DVFS impact on the communication time of the MPI program is~\cite{17}, Freeh et al. show that these times not changed when the frequency is scaled down.
-\sectionfont{\large}
-\subsection{~The online DVFS orientations}
-The objective of these 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 developed by ~\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 communication time that take place in the MPI programs, the processors are idle. The optimal DVFS can be selected using the learning methods. Therefore, in ~\cite{39,19} used machine learning to converge to the suitable DVFS configuration. Their learning algorithms have big time to converge when the number of available frequencies is high. Also, the communication time of the MPI program used online for saving energy as 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 ~\cite{3}, Rauber et al. developed an analytical mathematical model for determining the optimal frequency scaling factor for any number of concurrent tasks, without considering communication times. They set the slowest task to maximum frequency for maintaining performance.
-In this paper we compare our algorithm with Rauber's model~\cite{3}, because his model can be used for any number of concurrent tasks for homogeneous platform and this is the same direction of this paper.
-However, the primary contributions of this paper are:
-\\1-Selecting the optimal frequency scaling factor for energy and performance
- simultaneously. While taking into account the communication time.
-\\2-Adapting our scale factor to taking into account the imbalanced tasks.
-\\3-The execution time of our algorithm is very small when compared to other methods (e.g.,~\cite{19}).
-\\4-The proposed algorithm works online without profiling or training as in~\cite{38,34}.
-\sectionfont{\large}
-\section{.~Parallel Tasks Execution on Homogeneous Platform}
-A homogeneous cluster consists of identical nodes in terms of the hardware and the 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 memory passing.
-\begin{figure}[h]
-\centering
-\subfloat[Synch. Imbalanced Communications]{\includegraphics[scale=0.67]{synch_tasks}\label{fig:h1}}
-\subfloat[Synch. Imbalanced Computations]{\includegraphics[scale=0.67]{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 idle time while tasks wait at the synchronous point for others tasks to finish their communications see figure~(\ref{fig:h1}).
-Another source for idle times is the imbalanced computations. This happen when processing different amounts of data on each processor as an example see figure~(\ref{fig:h2}). In this case the fastest tasks have to wait at the synchronous barrier for the slowest tasks to finish their job. In both two cases the overall execution time of the program is the execution time of the slowest task as :
-\begin{equation} \label{eq:T1}
- Program Time=MAX_{i=1,2,..,N} (T_i) \hfill
+
+\title{Dynamic Frequency Scaling for Energy Consumption
+ Reduction in Distributed MPI Programs}
+
+\author{%
+ \IEEEauthorblockN{%
+ Jean-Claude Charr,
+ Raphaël Couturier,
+ Ahmed Fanfakh and
+ Arnaud Giersch
+ }
+ \IEEEauthorblockA{%
+ FEMTO-ST Institute\\
+ University of Franche-Comté\\
+ 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
+
+\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 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 \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}
+
+
+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 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, \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 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}
+\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
+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.
+\begin{figure*}[t]
+ \centering
+ \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 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 process $i$.
-\sectionfont{\large}
-\section{.~Energy Model for Homogeneous Platform}
-The energy consumption by the processor consists of two powers metric: the dynamic and the static power. This general power formulation is used by many researchers see ~\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 :
-\begin{equation} \label{eq:pd}
-\displaystyle P_{dyn} = \alpha . C_L . V^2 . f
+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}).
+\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.
-\begin{equation} \label{eq:ps}
-\displaystyle P_{static} = V . N . K_{design} . I_{leak}
+The static power $P_{static}$ captures the leakage power as follows:
+\begin{equation}
+ \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 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 multiply by the execution time for example see~\cite{36,15} .
-\begin{equation} \label{eq:eind}
-\displaystyle E_{ind} = (P_{dyn} + P_{static} ) . T
+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} \cdot T_{Comp} + P_\textit{static} \cdot T
\end{equation}
-The dynamic voltage and frequency scaling (DVFS) is a process that 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., $V = \beta . f$ 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 maximum and the new frequency as in EQ~(\ref{eq:s}).
-\begin{equation} \label{eq:s}
- S=\:\frac{F_{max}}{F_{new}} \hfill \newline
+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 scale \emph S is grater than 1 when changing the frequency to any new frequency value(\emph {P-state}) in governor.~It is equal to 1 when the frequency are set to the maximum frequency.
-The energy consumption model for parallel homogeneous platform is depending 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, of homogeneous platform for any number of concurrent tasks develops by Rauber~\cite{3}. This model consider the two powers metric for measuring the energy of the parallel tasks as in EQ~(\ref{eq:energy}).
-
-\begin{equation} \label{eq:energy}
-E= \displaystyle \;P_{dyn}\,.\,S_1^{-2}\;.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_1\;\,.\,N
+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}
+ E = 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
\hfill
\end{equation}
-Where \emph N is the number of parallel nodes, $T_1 $ is the time of the slower 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,..,F} (S_i) \hfill
+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}
-\begin{equation} \label{eq:si}
- S_i=\:S\: .\:(\frac{T_1}{T_i})=\: (\frac{F_{max}}{F_{new}}).(\frac{T_1}{T_i}) \hfill
+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}
+ 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}
-Where $F$ is the number of available frequencies. In this paper we depend on Rauber's energy model EQ~(\ref{eq:energy}) for two reasons : 1-this model used for homogeneous platform that we work on in this paper. 2-we are compare our algorithm with Rauber's scaling model.
-Rauber's optimal scaling factor for optimal energy reduction derived from the EQ~(\ref{eq:energy}). He takes the derivation for this equation (to be minimized) and set it to zero to produce the scaling factor as in EQ~(\ref{eq:sopt}).
- \begin{equation} \label{eq:sopt}
- S_{opt}= {\sqrt [3~]{\frac{2}{n} \frac{P_{dyn}}{P_{static}} \Big(1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^3}\Big) }} \hfill
-\end{equation}
-%[\Big 3]
-\sectionfont{\large}
-\section{.~Performance Evaluation of MPI Programs}
-The performance (execution time) of the parallel MPI applications are depends 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 increase the execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) the energy 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}) lead to the value of the scale $S$ has inverse relation with new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decrease the frequency value, the execution time increase. Then the new frequency value has inverse relation with time ($F_{new} \propto \frac{1}{T}$). This lead 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 processor remain idle until the communication has 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 are made many tests on real cluster to prove that the frequency scaling factor \emph S has a linear relation with computation time only also see~\cite{41}. To predict the execution time of MPI program, firstly must be precisely specifying communication time and the computation time for the slower task. Secondly, we use these times for predicting the execution time for any MPI program as a function of the new scaling factor as in the EQ~(\ref{eq:tnew}).
- \begin{equation} \label{eq:tnew}
- \displaystyle T_{new}= T_{Max \:Comp \:Old} \; . \:S \;+ \;T_{Max\: Comm\: Old}
- \hfill
- \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-performace scaling method as we will show in the coming sections. In the next section we make an investigation study for the EQ~(\ref{eq:tnew}).
-\sectionfont{\large}
-\section{.~Performance Prediction Verification }
-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}[width=\textwidth,height=\textheight,keepaspectratio]
+
+
+\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}).
+\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.
+
+\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:
+\begin{multline}
+ \label{eq:enorm}
+ E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
+ {} = \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 }
+\end{multline}
+By the same way we can normalize the performance as follows:
+\begin{equation}
+ \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{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 $S$ 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
+performance as follows:
+\begin{equation}
+ \label{eq:pnorm_en}
+ P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
+ = \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
- \includegraphics[scale=0.60]{cg_per.eps}
- \includegraphics[scale=0.60]{mg_pre.eps}
- \includegraphics[scale=0.60]{bt_pre.eps}
- \includegraphics[scale=0.60]{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}).
-\sectionfont{\large}
-\section{.~Performance to Energy Competition}
-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 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. 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{equation} \label{eq:enorm}
- E_{Norm}=\displaystyle\frac{E_{Reduced}}{E_{Orginal}}= \frac{\displaystyle \;P_{dyn}\,.\,S_i^{-2}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_i\;\,.\,N }{\displaystyle \;P_{dyn}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,\,.\,N }
-\end{equation}
-By the same way we can normalize the performance as follows :
-\begin{equation} \label{eq:pnorm}
- P_{Norm}=\displaystyle \frac{T_{New}}{T_{Old}}=\frac{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}}{T_{Old}} \;\;
- \end{equation}
-The second problem is 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 are present a method to find the optimal scaling factor \emph S for optimize 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 :
-\begin{equation} \label{eq:pnorm_en}
-\displaystyle P^{-1}_{Norm}= \frac{T_{Old}}{T_{New}}=\frac{T_{Old}}{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}}
-\end{equation}
-\begin{figure}
-\centering
-\subfloat[Converted Relation.]{\includegraphics[scale=0.70]{file.eps}\label{fig:r1}}
-\subfloat[Real Relation.]{\includegraphics[scale=0.70]{file3.eps}\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) in the same time, see figure~(\ref{fig:r1}). Then our objective function has the following form:
-\begin{equation} \label{eq:max}
-\displaystyle MaxDist = Max \;(\;\overbrace{P^{-1}_{Norm}}^{Maximize}\; -\; \overbrace{E_{Norm}}^{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 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's model as an example with two reasons that mentioned before.
-\sectionfont{\large}
-\section{.~Optimal Scaling Factor for Performance and Energy }
-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.
-\clearpage
-\linespread{1}
-\begin{algorithm}[width=\textwidth,height=\textheight,keepaspectratio]
-\caption{EPSA}
-\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.
-\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 in EQ~(\ref{eq:si}).
-\State - Select the maximum scale factor $S_1$ from the set of scales $S_i$.
-\State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ as in EQ~(\ref{eq:enorm}).
-\State - Calculate the normalize inverse of performance $P_{NormInv}=T_{old}/T_{new}$
-
- as in EQ~(\ref{eq:pnorm_en}).
- \If{ $(P_{NormInv}-E_{Norm}$ $>$ $Dist$) }
- \State $S_{optimal}=S$
- \State $Dist = P_{NormInv} - E_{Norm}$
- \EndIf
-\EndFor
-\State $ Return \; \; (S_{optimal})$
-\end{algorithmic}
-\end{algorithm}
-\linespread{1.2}
-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 following example shows where and when the EPSA algorithm is called in the MPI program :
-\clearpage
-\begin{lstlisting}
-FOR J:=1 to Some_iterations Do
- -Computations Section.
- -Communications Section.
- IF (J==1) THEN
- -Gather all times of computation and communication
- from each node.
- -Call EPSA with these times.
- -Calculate the new frequency from optimal scale.
- -Set the new frequency to the system.
- ENDIF
-ENDFOR
-\end{lstlisting}
-After obtaining the optimal scale factor from the EPSA algorithm. 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 can calculate the new frequency $F_i$ as follows :
-\begin{equation} \label{eq:fi}
- F_i=\frac{F_{max} \; . \;T_i}{S_{optimal} \; . \;T_{max}} \hfill
+ \subfloat[Converted relation.]{%
+ \includegraphics[width=.4\textwidth]{fig/file}\label{fig:r1}}%
+ \qquad%
+ \subfloat[Real relation.]{%
+ \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
+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}} )
\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 freguency according to the nodes workloads.
-\sectionfont{\large}
-\section{.~Experimental Results}
-The proposed ESPA 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 detailed characteristics of our platform file are shown in thetable~(\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.
-\begin{table}[ht]
-\caption{Platform File Parameters}
-% title of Table
-\centering
- \begin{tabular}{ | l | l | l |l | l |l |l | p{2cm} |}
+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} computes the optimal scaling factor according to the
+objective function described above.
+\begin{algorithm}[tp]
+ \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 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 $i=1,\dots,N$
+ \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
+ \EndFor
+ \State Return $S_{opt}$
+ \end{algorithmic}
+\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 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
- Max & Min & Backbone & Backbone&Link &Link& Sharing \\
- Freq. & Freq. & Bandwidth & Latency & Bandwidth& Latency&Policy \\ \hline
- 2.5 &800 & 2.25 GBps &5E-7 s & 1 GBps & 5E-5 s&Full \\
- GHz& MHz& & & & &Duplex \\\hline
- \end{tabular}
-\label{table:platform}
+ \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}
-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 ESPA algorithm for all these programs. 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 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 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, the distance between the two curves is big. This lead 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.
-\begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
-\centering
- \includegraphics[scale=0.47]{ep.eps}
- \includegraphics[scale=0.47]{cg.eps}
- \includegraphics[scale=0.47]{sp.eps}
- \includegraphics[scale=0.47]{lu.eps}
- \includegraphics[scale=0.47]{bt.eps}
- \includegraphics[scale=0.47]{ft.eps}
- \caption{Optimal scaling factors for The NAS MPI Programs}
- \label{fig:nas}
-\end{figure}
-\linespread{1.1}
-\begin{table}[width=\textwidth,height=\textheight,keepaspectratio]
-\caption{Optimal Scaling Factors Results}
-% title of Table
-\centering
- \begin{tabular}{ | l | l | l |l | l | p{2cm} |}
+
+\begin{algorithm}[tp]
+ \caption{DVFS}
+ \label{dvfs}
+ \begin{algorithmic}[1]
+ \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:
+\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.
+
+\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 \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}).
+\begin{figure*}[t]
+ \centering
+ \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
+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
+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
+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.
+\begin{figure*}[t]
+ \centering
+ \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*}
+\begin{table}[htb]
+ \caption{The scaling factors results}
+ % title of Table
+ \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
- 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&21.7& 13.27 \\ \hline
- EP & 1.04 &22.14&20.73 &1.41\\ \hline
- LU & 1.388 &35.83&22.49 &13.34\\ \hline
- BT & 1.315 &29.6&21.28 &8.32\\ \hline
- SP & 1.388 &33.48 &21.36&12.12\\ \hline
- FT & 1.47 &34.72 &19&15.72\\ \hline
- \end{tabular}
-\label{table:factors results}
-% is used to refer this table in the text
+ 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}
-\linespread{1.2}
-
-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 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 EPSA to selects smaller scaling factor values (inducing smaller energy savings).
-
-% \clearpage
-\sectionfont{\large}
-\section{.~Comparing Results}
-In this section, we compare our EPSA algorithm results with Rauber's 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 $Rauber_{E}$. 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 $Rauber_{E-P}$. 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's two scenarios for all the NAS benchmarks programs for classes A,B and C.
-%\linespread{1}
-\begin{table}[ht]
-\caption{Comparing Results for The NAS Class A}
-% title of Table
-\centering
- \begin{tabular}{ | l | l | l |l | l |l| }
+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
+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).
+
+\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 (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
- $Rauber_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.5\\ \hline
- $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline
-
- EPSA&MG & 1.47 &27.66&16.82&10.84\\ \hline
- $Rauber_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline
- $Rauber_{E}$&MG &2.14&34.48&33.65&0.8 \\ \hline
-
- EPSA&EP &1.19 &25.32&20.79&4.53\\ \hline
- $Rauber_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline
- $Rauber_{E}$&EP&2.05&42.09&57.59&-15.5\\ \hline
-
- EPSA&LU&1.56& 39.55 &19.38& 20.17\\ \hline
- $Rauber_{E-P}$&LU&2.14&45.62&27&18.62 \\ \hline
- $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline
-
- EPSA&BT&1.315& 29.6&20.53&9.07 \\ \hline
- $Rauber_{E-P}$&BT&2.1&45.53&49.63&-4.1\\ \hline
- $Rauber_{E}$&BT&2.1&43.93&52.86&-8.93\\ \hline
-
- EPSA&SP&1.388& 33.51&15.65&17.86 \\ \hline
- $Rauber_{E-P}$&SP&2.11&45.62&42.52&3.1\\ \hline
- $Rauber_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline
-
- EPSA&FT&1.25& 25&10.8&14.2 \\ \hline
- $Rauber_{E-P}$&FT&2.1&39.29&34.3&4.99 \\ \hline
- $Rauber_{E}$&FT&2.1&37.56&38.21&-0.65\\ \hline
- \end{tabular}
-\label{table:compare Class A}
-% is used to refer this table in the text
+ % \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
+
+ $EPSA$ & 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
+
+ $EPSA$ & 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
+
+ $EPSA$ & 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
+ \end{tabular}
+ \label{table:compareA}
+ % is used to refer this table in the text
\end{table}
-\begin{table}[ht]
-\caption{Comparing Results for The NAS Class B}
-% title of Table
-\centering
- \begin{tabular}{ | l | l | l |l | l |l| }
+\begin{table}[p]
+ \caption{Comparing results for the NAS class B}
+ % 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
- % \rowcolor[gray]{0.85}
- EPSA&CG & 1.66 &39.23&16.63&22.6 \\ \hline
- $Rauber_{E-P}$&CG &2.15 &45.34&27.6&17.74\\ \hline
- $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline
-
- EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline
- $Rauber_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline
- $Rauber_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline
-
- EPSA&EP &1.08 &20.29&17.15&3.14 \\ \hline
- $Rauber_{E-P}$&EP&2&42.38&56.88&-14.5\\ \hline
- $Rauber_{E}$&EP&2&39.73&59.94&-20.21\\ \hline
-
- EPSA&LU&1.47&38.57&21.34&17.23 \\ \hline
- $Rauber_{E-P}$&LU&2.1&43.62&36.51&7.11 \\ \hline
- $Rauber_{E}$&LU&2.1&43.61&38.54&5.07 \\ \hline
-
- EPSA&BT&1.315& 29.59&20.88&8.71\\ \hline
- $Rauber_{E-P}$&BT&2.1&44.53&53.05&-8.52\\ \hline
- $Rauber_{E}$&BT&2.1&42.93&52.806&-9.876\\ \hline
-
- EPSA&SP&1.388&33.44&19.24&14.2 \\ \hline
- $Rauber_{E-P}$&SP&2.15&45.69&43.2&2.49\\ \hline
- $Rauber_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline
-
- EPSA&FT&1.388&34.4&14.57&19.83 \\ \hline
- $Rauber_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline
- $Rauber_{E}$&FT&2.13&43.04&37.9&5.14\\ \hline
- \end{tabular}
-\label{table:compare Class B}
-% is used to refer this table in the text
+ 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
+
+ $EPSA$ & 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
+
+ $EPSA$ & 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
+
+ $EPSA$ & 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
+ \end{tabular}
+ \label{table:compareB}
+ % is used to refer this table in the text
\end{table}
-
- \begin{table}[ht]
-\caption{Comparing Results for The NAS Class C}
-% title of Table
-\centering
- \begin{tabular}{ | l | l | l |l | l |l| }
+
+\begin{table}[p]
+ \caption{Comparing results for the NAS class C}
+ % 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 &39.23&14.88&24.35 \\ \hline
- $Rauber_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline
- $Rauber_{E}$&CG &2.15 &45.36&26.7&18.66\\ \hline
-
- EPSA&MG & 1.47 &34.97&21.697&13.273\\ \hline
- $Rauber_{E-P}$&MG &2.15&43.65&40.45&3.2 \\ \hline
- $Rauber_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline
-
- EPSA&EP &1.04 &22.14&20.73&1.41 \\ \hline
- $Rauber_{E-P}$&EP&1.92&39.4&56.33&-16.93\\ \hline
- $Rauber_{E}$&EP&1.92&38.1&56.35&-18.25\\ \hline
-
- EPSA&LU&1.388&35.83&22.49&13.34 \\ \hline
- $Rauber_{E-P}$&LU&2.15&44.97&41&3.97 \\ \hline
- $Rauber_{E}$&LU&2.15&44.97&41.8&3.17 \\ \hline
-
- EPSA&BT&1.315& 29.6&21.28&8.32\\ \hline
- $Rauber_{E-P}$&BT&2.13&45.6&49.84&-4.24\\ \hline
- $Rauber_{E}$&BT&2.13&44.9&55.16&-10.26\\ \hline
-
- EPSA&SP&1.388&33.48&21.35&12.12\\ \hline
- $Rauber_{E-P}$&SP&2.1&45.69&43.6&2.09\\ \hline
- $Rauber_{E}$&SP&2.1&45.75&44.1&1.65\\ \hline
-
- EPSA&FT&1.47&34.72&19&15.72 \\ \hline
- $Rauber_{E-P}$&FT&2.04&39.4&37.1&2.3\\ \hline
- $Rauber_{E}$&FT&2.04&39.35&37.7&1.65\\ \hline
- \end{tabular}
-\label{table:compare Class C}
-% is used to refer this table in the text
+ % \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
+
+ $EPSA$ & 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
+
+ $EPSA$ & 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
+
+ $EPSA$ & 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
+ \end{tabular}
+ \label{table:compareC}
+ % is used to refer this table in the text
\end{table}
-%\linespread{1.2}
-\clearpage
-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's energy-performance method ($Rauber_{E-P}$). Also, in ($Rauber_{E-P}$) 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.
-
-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's method ($Rauber_{E-P}$) 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.
- \begin{figure}[width=\textwidth,height=\textheight,keepaspectratio]
-\centering
- \includegraphics[scale=0.60]{compare_class_A.pdf}
- \includegraphics[scale=0.60]{compare_class_B.pdf}
- \includegraphics[scale=0.60]{compare_class_c.pdf}
- % use scale 35 for all to be in the same line
- \caption{Comparing Our EPSA with Rauber's Methods}
- \label{fig:compare}
-
-\end{figure}
- \clearpage
-\bibliographystyle{plain}
-\bibliography{my_reference}
+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.
+
+\section*{Acknowledgment}
+
+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.
+
+% trigger a \newpage just before the given reference
+% number - used to balance the columns on the last page
+% adjust value as needed - may need to be readjusted if
+% the document is modified later
+%\IEEEtriggeratref{15}
+
+\bibliographystyle{IEEEtran}
+\bibliography{IEEEabrv,my_reference}
\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% fill-column: 80
+%%% ispell-local-dictionary: "american"
+%%% End:
+
+% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
+% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
