\usepackage{subfig}
\usepackage{amsmath}
\usepackage{url}
+
\DeclareUrlCommand\email{\urlstyle{same}}
\usepackage[autolanguage,np]{numprint}
\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
\newcommand{\Tnew}{\Xsub{T}{New}}
-\newcommand{\Told}{\Xsub{T}{Old}}
+\newcommand{\Told}{\Xsub{T}{Old}}
+
+\begin{document}
-\begin{document}
+\title{Energy Consumption Reduction with DVFS for \\
+ Message Passing Iterative Applications on \\
+ Heterogeneous Architectures}
-\title{Energy Consumption Reduction for Message Passing Iterative Applications in Heterogeneous Architecture Using DVFS}
-
-\author{%
+\author{%
\IEEEauthorblockN{%
Jean-Claude Charr,
Raphaël Couturier,
Ahmed Fanfakh and
Arnaud Giersch
- }
+ }
\IEEEauthorblockA{%
- FEMTO-ST Institute, University of Franche-Comte\\
+ 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
\maketitle
\begin{abstract}
-Computing platforms are consuming more and more energy due to the increasing
-number of nodes composing them. To minimize the operating costs of these
-platforms many techniques have been used. Dynamic voltage and frequency scaling
-(DVFS) is one of them. It reduces the frequency of a CPU to lower its energy
-consumption. However, lowering the frequency of a CPU might increase the
-execution time of an application running on that processor. Therefore, the
-frequency that gives the best trade-off between the energy consumption and the
-performance of an application must be selected.
-
-In this paper, a new online frequency selecting algorithm for heterogeneous
-platforms is presented. It selects the frequencies and tries to give the best
-trade-off between energy saving and performance degradation, for each node
-computing the message passing iterative application. The algorithm has a small
-overhead and works without training or profiling. It uses a new energy model for
-message passing iterative applications running on a heterogeneous platform. The
-proposed algorithm is evaluated on the SimGrid simulator while running the NAS
-parallel benchmarks. The experiments show that it reduces the energy
-consumption by up to \np[\%]{35} while limiting the performance degradation as
-much as possible. Finally, the algorithm is compared to an existing method, the
-comparison results showing that it outperforms the latter.
+ Computing platforms are consuming more and more energy due to the increasing
+ number of nodes composing them. To minimize the operating costs of these
+ platforms many techniques have been used. Dynamic voltage and frequency
+ scaling (DVFS) is one of them. It reduces the frequency of a CPU to lower its
+ energy consumption. However, lowering the frequency of a CPU may increase
+ the execution time of an application running on that processor. Therefore,
+ the frequency that gives the best trade-off between the energy consumption and
+ the performance of an application must be selected.
+
+ In this paper, a new online frequency selecting algorithm for heterogeneous
+ platforms (heterogeneous CPUs) is presented. It selects the frequencies and tries to give the best
+ trade-off between energy saving and performance degradation, for each node
+ computing the message passing iterative application. The algorithm has a small
+ overhead and works without training or profiling. It uses a new energy model
+ for message passing iterative applications running on a heterogeneous
+ platform. The proposed algorithm is evaluated on the SimGrid simulator while
+ running the NAS parallel benchmarks. The experiments show that it reduces the
+ energy consumption by up to \np[\%]{34} while limiting the performance
+ degradation as much as possible. Finally, the algorithm is compared to an
+ existing method, the comparison results show that it outperforms the
+ latter, on average it saves \np[\%]{4} more energy while keeping the same performance.
\end{abstract}
\section{Introduction}
\label{sec.intro}
-The need for more computing power is continually increasing. To partially
-satisfy this need, most supercomputers constructors just put more computing
-nodes in their platform. The resulting platforms might achieve higher floating
-point operations per second (FLOPS), but the energy consumption and the heat
-dissipation are also increased. As an example, the Chinese supercomputer
-Tianhe-2 had the highest FLOPS in November 2014 according to the Top500 list
-\cite{TOP500_Supercomputers_Sites}. However, it was also the most power hungry
-platform with its over 3 million cores consuming around 17.8 megawatts.
-Moreover, according to the U.S. annual energy outlook 2014
-\cite{U.S_Annual.Energy.Outlook.2014}, the price of energy for 1 megawatt-hour
+
+The need for more computing power is continually increasing. To partially
+satisfy this need, most supercomputers constructors just put more computing
+nodes in their platform. The resulting platforms may achieve higher floating
+point operations per second (FLOPS), but the energy consumption and the heat
+dissipation are also increased. As an example, the Chinese supercomputer
+Tianhe-2 had the highest FLOPS in November 2014 according to the Top500 list
+\cite{TOP500_Supercomputers_Sites}. However, it was also the most power hungry
+platform with its over 3 million cores consuming around 17.8 megawatts.
+Moreover, according to the U.S. annual energy outlook 2014
+\cite{U.S_Annual.Energy.Outlook.2014}, the price of energy for 1 megawatt-hour
was approximately equal to \$70. Therefore, the price of the energy consumed by
-the Tianhe-2 platform is approximately more than \$10 million each year. The
-computing platforms must be more energy efficient and offer the highest number
-of FLOPS per watt possible, such as the L-CSC from the GSI Helmholtz Center
+the Tianhe-2 platform is approximately more than \$10 million each year. The
+computing platforms must be more energy efficient and offer the highest number
+of FLOPS per watt possible, such as the L-CSC from the GSI Helmholtz Center
which became the top of the Green500 list in November 2014 \cite{Green500_List}.
This heterogeneous platform executes more than 5 GFLOPS per watt while consuming
57.15 kilowatts.
-Besides platform improvements, there are many software and hardware techniques
-to lower the energy consumption of these platforms, such as scheduling, DVFS,
-\dots{} DVFS is a widely used process to reduce the energy consumption of a
-processor by lowering its frequency
+Besides platform improvements, there are many software and hardware techniques
+to lower the energy consumption of these platforms, such as scheduling, DVFS,
+\dots{} DVFS is a widely used process to reduce the energy consumption of a
+processor by lowering its frequency
\cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
-the number of FLOPS executed by the processor which might increase the execution
+the number of FLOPS executed by the processor which may increase the execution
time of the application running over that processor. Therefore, researchers use
-different optimization strategies to select the frequency that gives the best
-trade-off between the energy reduction and performance degradation ratio. In
-\cite{Our_first_paper}, a frequency selecting algorithm was proposed to reduce
-the energy consumption of message passing iterative applications running over
-homogeneous platforms. The results of the experiments show significant energy
-consumption reductions. In this paper, a new frequency selecting algorithm
-adapted for heterogeneous platform is presented. It selects the vector of
-frequencies, for a heterogeneous platform running a message passing iterative
+different optimization strategies to select the frequency that gives the best
+trade-off between the energy reduction and performance degradation ratio. In
+\cite{Our_first_paper}, a frequency selecting algorithm was proposed to reduce
+the energy consumption of message passing iterative applications running over
+homogeneous platforms. The results of the experiments show significant energy
+consumption reductions. In this paper, a new frequency selecting algorithm
+adapted for heterogeneous platform is presented. It selects the vector of
+frequencies, for a heterogeneous platform running a message passing iterative
application, that simultaneously tries to offer the maximum energy reduction and
-minimum performance degradation ratio. The algorithm has a very small overhead,
+minimum performance degradation ratio. The algorithm has a very small overhead,
works online and does not need any training or profiling.
This paper is organized as follows: Section~\ref{sec.relwork} presents some
related works from other authors. Section~\ref{sec.exe} describes how the
-execution time of message passing programs can be predicted. It also presents an energy
-model that predicts the energy consumption of an application running over a heterogeneous platform. Section~\ref{sec.compet} presents
-the energy-performance objective function that maximizes the reduction of energy
+execution time of message passing programs can be predicted. It also presents
+an energy model that predicts the energy consumption of an application running
+over a heterogeneous platform. 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 frequency selecting algorithm then the precision of the proposed algorithm is verified.
-Section~\ref{sec.expe} presents the results of applying the algorithm on the NAS parallel benchmarks and executing them
-on a heterogeneous platform. It shows the results of running three
-different power scenarios and comparing them. Moreover, it also shows the comparison results
-between the proposed method and an existing method.
-Finally, in Section~\ref{sec.concl} the paper ends with a summary and some future works.
+Section~\ref{sec.optim} details the proposed frequency selecting algorithm then
+the precision of the proposed algorithm is verified. Section~\ref{sec.expe}
+presents the results of applying the algorithm on the NAS parallel benchmarks
+and executing them on a heterogeneous platform. It shows the results of running
+three different power scenarios and comparing them. Moreover, it also shows the
+comparison results between the proposed method and an existing method. Finally,
+in Section~\ref{sec.concl} the paper ends with a summary and some future works.
\section{Related works}
\label{sec.relwork}
+
DVFS is a technique used in modern processors to scale down both the voltage and
the frequency of the CPU while computing, in order to reduce the energy
consumption of the processor. DVFS is also allowed in GPUs to achieve the same
-goal. Reducing the frequency of a processor lowers its number of FLOPS and might
+goal. Reducing the frequency of a processor lowers its number of FLOPS and may
degrade the performance of the application running on that processor, especially
if it is compute bound. Therefore selecting the appropriate frequency for a
-processor to satisfy some objectives while taking into account all the
+processor to satisfy some objectives, while taking into account all the
constraints, is not a trivial operation. Many researchers used different
strategies to tackle this problem. Some of them developed online methods that
compute the new frequency while executing the application, such
as~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}.
-Others used offline methods that might need to run the application and profile
+Others used offline methods that may need to run the application and profile
it before selecting the new frequency, such
as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
The methods could be heuristics, exact or brute force methods that satisfy
sequential, parallel or distributed architecture, homogeneous or heterogeneous
platform, synchronous or asynchronous application, \dots{}
-In this paper, we are interested in reducing energy for message passing iterative synchronous applications running over heterogeneous platforms.
-Some works have already been done for such platforms and they can be classified into two types of heterogeneous platforms:
+In this paper, we are interested in reducing energy for message passing
+iterative synchronous applications running over heterogeneous platforms. Some
+works have already been done for such platforms and they can be classified into
+two types of heterogeneous platforms:
\begin{itemize}
-
\item the platform is composed of homogeneous GPUs and homogeneous CPUs.
\item the platform is only composed of heterogeneous CPUs.
-
\end{itemize}
For the first type of platform, the computing intensive parallel tasks are
Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
DVFS gave better energy and performance efficiency than other clusters only
composed of CPUs.
-
+
The work presented in this paper concerns the second type of platform, with
heterogeneous CPUs. Many methods were conceived to reduce the energy
consumption of this type of platform. Naveen et
synchronous applications running over a heterogeneous platform. Both models
take into account communication and slack times. The models can predict the
required energy and the execution time of the application.
-
+
\item a new online frequency selecting algorithm for heterogeneous
platforms. The algorithm has a very small overhead and does not need any
training or profiling. It uses a new optimization function which
simultaneously maximizes the performance and minimizes the energy consumption
of a message passing iterative synchronous application.
-
+
\end{enumerate}
\section{The performance and energy consumption measurements on heterogeneous architecture}
\label{sec.exe}
-
-
-\subsection{The execution time of message passing distributed
- iterative applications on a heterogeneous platform}
+\subsection{The execution time of message passing distributed iterative
+ applications on a heterogeneous platform}
In this paper, we are interested in reducing the energy consumption of message
passing distributed iterative synchronous applications running over
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
have the same network bandwidth and latency.
+\begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.6]{fig/commtasks}
+ \caption{Parallel tasks on a heterogeneous platform}
+ \label{fig:heter}
+\end{figure}
+
The overall execution time of a distributed iterative synchronous application
over a heterogeneous platform consists of the sum of the computation time and
the communication time for every iteration on a node. However, due to the
-heterogeneous computation power of the computing nodes, slack times might occur
+heterogeneous computation power of the computing nodes, slack times may occur
when fast nodes have to wait, during synchronous communications, for the slower
nodes to finish their computations (see Figure~\ref{fig:heter}). Therefore, the
overall execution time of the program is the execution time of the slowest task
which has the highest computation time and no slack time.
-
- \begin{figure}[!t]
- \centering
- \includegraphics[scale=0.6]{fig/commtasks}
- \caption{Parallel tasks on a heterogeneous platform}
- \label{fig:heter}
-\end{figure}
-Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
-modern processors, that reduces the energy consumption of a CPU by scaling
-down its voltage and frequency. Since DVFS lowers the frequency of a CPU
-and consequently its computing power, the execution time of a program running
-over that scaled down processor might increase, especially if the program is
-compute bound. The frequency reduction process can be expressed by the scaling
-factor S which is the ratio between the maximum and the new frequency of a CPU
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
+modern processors, that reduces the energy consumption of a CPU by scaling
+down its voltage and frequency. Since DVFS lowers the frequency of a CPU
+and consequently its computing power, the execution time of a program running
+over that scaled down processor may increase, especially if the program is
+compute bound. The frequency reduction process can be expressed by the scaling
+factor S which is the ratio between the maximum and the new frequency of a CPU
as in (\ref{eq:s}).
\begin{equation}
\label{eq:s}
- S = \frac{\Fmax}{\Fnew}
+ S = \frac{\Fmax}{\Fnew}
\end{equation}
- The execution time of a compute bound sequential program is linearly proportional
- to the frequency scaling factor $S$. On the other hand, message passing
- distributed applications consist of two parts: computation and communication.
- The execution time of the computation part is linearly proportional to the
- frequency scaling factor $S$ but the communication time is not affected by the
- scaling factor because the processors involved remain idle during the
- communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.
- 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
- until the message is synchronously sent or received.
+The execution time of a compute bound sequential program is linearly
+proportional to the frequency scaling factor $S$. On the other hand, message
+passing distributed applications consist of two parts: computation and
+communication. The execution time of the computation part is linearly
+proportional to the frequency scaling factor $S$ but the communication time is
+not affected by the scaling factor because the processors involved remain idle
+during the communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}. 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 until the message is
+synchronously sent or received.
Since in a heterogeneous platform each node has different characteristics,
especially different frequency gears, when applying DVFS operations on these
\end{equation}
Where:
\begin{equation}
-\label{eq:perf2}
- \MinTcm = \min_{i=1,2,\dots,N} (\Tcm[i])
+ \label{eq:perf2}
+ \MinTcm = \min_{i=1,2,\dots,N} (\Tcm[i])
\end{equation}
-where $\TcpOld[i]$ is the computation time of processor $i$ during the first
-iteration and $\MinTcm$ is the communication time of the slowest processor from
-the first iteration. The model computes the maximum computation time with
-scaling factor from each node added to the communication time of the slowest
-node. It means only the communication time without any slack time is taken into
+where $\TcpOld[i]$ is the computation time of processor $i$ during the first
+iteration and $\MinTcm$ is the communication time of the slowest processor from
+the first iteration. The model computes the maximum computation time with
+scaling factor from each node added to the communication time of the slowest
+node. It means only the communication time without any slack time is taken into
account. Therefore, the execution time of the iterative application is equal to
-the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
+the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
number of iterations of that application.
-This prediction model is developed from the model to predict the execution time
-of message passing distributed applications for homogeneous
-architectures~\cite{Our_first_paper}. The execution time prediction model is
-used in the method to optimize both the energy consumption and the performance of
-iterative methods, which is presented in the following sections.
-
+This prediction model is developed from the model to predict the execution time
+of message passing distributed applications for homogeneous
+architectures~\cite{Our_first_paper}. The execution time prediction model is
+used in the method to optimize both the energy consumption and the performance
+of iterative methods, which is presented in the following sections.
\subsection{Energy model for heterogeneous platform}
+
Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
-Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
-Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by a processor into
-two power metrics: the static and the dynamic power. While the first one is
-consumed as long as the computing unit is turned on, the latter is only consumed during
-computation times. The dynamic power $\Pd$ is related to the switching
-activity $\alpha$, load capacitance $\CL$, the supply voltage $V$ and
-operational frequency $F$, as shown in (\ref{eq:pd}).
+ Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
+ Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
+a processor into two power metrics: the static and the dynamic power. While the
+first one is consumed as long as the computing unit is turned on, the latter is
+only consumed during computation times. The dynamic power $\Pd$ is related to
+the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
+and operational frequency $F$, as shown in (\ref{eq:pd}).
\begin{equation}
\label{eq:pd}
\Pd = \alpha \cdot \CL \cdot V^2 \cdot F
time and $\Tcp \le T$. $\Tcp$ may be equal to $T$ if there is no
communication and no slack time.
-The main objective of DVFS operation is to reduce the overall energy consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}.
-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{Rauber_Analytical.Modeling.for.Energy}. 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 (\ref{eq:s}).
-The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. The new frequency
-$\Fnew$ from (\ref{eq:s}) can be calculated as follows:
+The main objective of DVFS operation is to reduce the overall energy
+consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}. 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{Rauber_Analytical.Modeling.for.Energy}. 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 (\ref{eq:s}). The CPU governors are
+power schemes supplied by the operating system's kernel to lower a core's
+frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
+follows:
\begin{equation}
\label{eq:fnew}
\Fnew = S^{-1} \cdot \Fmax
\end{equation}
-Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
+Replacing $\Fnew$ in (\ref{eq:pd}) as in (\ref{eq:fnew}) gives the following
equation for dynamic power consumption:
\begin{multline}
\label{eq:pdnew}
\PdNew = \alpha \cdot \CL \cdot V^2 \cdot \Fnew = \alpha \cdot \CL \cdot \beta^2 \cdot \Fnew^3 \\
{} = \alpha \cdot \CL \cdot V^2 \cdot \Fmax \cdot S^{-3} = \PdOld \cdot S^{-3}
\end{multline}
-where $\PdNew$ and $\PdOld$ are the dynamic power consumed with the
+where $\PdNew$ and $\PdOld$ are the dynamic power consumed with the
new frequency and the maximum frequency respectively.
-According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when
-reducing the frequency by a factor of $S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is proportional
-to the frequency of a CPU, the computation time is increased proportionally to $S$.
-The new dynamic energy is the dynamic power multiplied by the new time of computation
-and is given by the following equation:
+According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
+$S^{-3}$ when reducing the frequency by a factor of
+$S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is
+proportional to the frequency of a CPU, the computation time is increased
+proportionally to $S$. The new dynamic energy is the dynamic power multiplied
+by the new time of computation and is given by the following equation:
\begin{equation}
\label{eq:Edyn}
- \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot \Tcp
+ \EdNew = \PdOld \cdot S^{-3} \cdot (\Tcp \cdot S)= S^{-2}\cdot \PdOld \cdot \Tcp
\end{equation}
-The static power is related to the power leakage of the CPU and is consumed during computation
-and even when idle. As in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
- the static power of a processor is considered as constant
-during idle and computation periods, and for all its available frequencies.
-The static energy is the static power multiplied by the execution time of the program.
-According to the execution time model in (\ref{eq:perf}), the execution time of the program
-is the sum of the computation and the communication times. The computation time is linearly related
-to the frequency scaling factor, while this scaling factor does not affect the communication time.
-The static energy of a processor after scaling its frequency is computed as follows:
+The static power is related to the power leakage of the CPU and is consumed
+during computation and even when idle. As
+in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
+the static power of a processor is considered as constant during idle and
+computation periods, and for all its available frequencies. The static energy
+is the static power multiplied by the execution time of the program. According
+to the execution time model in (\ref{eq:perf}), the execution time of the
+program is the sum of the computation and the communication times. The
+computation time is linearly related to the frequency scaling factor, while this
+scaling factor does not affect the communication time. The static energy of a
+processor after scaling its frequency is computed as follows:
\begin{equation}
\label{eq:Estatic}
\Es = \Ps \cdot (\Tcp \cdot S + \Tcm)
\end{equation}
-In the considered heterogeneous platform, each processor $i$ might have
-different dynamic and static powers, noted as $\Pd[i]$ and $\Ps[i]$
-respectively. Therefore, even if the distributed message passing iterative
-application is load balanced, the computation time of each CPU $i$ noted
-$\Tcp[i]$ might be different and different frequency scaling factors might be
-computed in order to decrease the overall energy consumption of the application
-and reduce slack times. The communication time of a processor $i$ is noted as
-$\Tcm[i]$ and could contain slack times when communicating with slower
-nodes, see Figure~\ref{fig:heter}. Therefore, all nodes do not have equal
-communication times. While the dynamic energy is computed according to the
-frequency scaling factor and the dynamic power of each node as in
-(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
-of one iteration multiplied by the static power of each processor. The overall
-energy consumption of a message passing distributed application executed over a
-heterogeneous platform during one iteration is the summation of all dynamic and
+In the considered heterogeneous platform, each processor $i$ may have
+different dynamic and static powers, noted as $\Pd[i]$ and $\Ps[i]$
+respectively. Therefore, even if the distributed message passing iterative
+application is load balanced, the computation time of each CPU $i$ noted
+$\Tcp[i]$ may be different and different frequency scaling factors may be
+computed in order to decrease the overall energy consumption of the application
+and reduce slack times. The communication time of a processor $i$ is noted as
+$\Tcm[i]$ and could contain slack times when communicating with slower nodes,
+see Figure~\ref{fig:heter}. Therefore, all nodes do not have equal
+communication times. While the dynamic energy is computed according to the
+frequency scaling factor and the dynamic power of each node as in
+(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
+of one iteration multiplied by the static power of each processor. The overall
+energy consumption of a message passing distributed application executed over a
+heterogeneous platform during one iteration is the summation of all dynamic and
static energies for each processor. It is computed as follows:
\begin{multline}
\label{eq:energy}
E = \sum_{i=1}^{N} {(S_i^{-2} \cdot \Pd[i] \cdot \Tcp[i])} + {} \\
\sum_{i=1}^{N} (\Ps[i] \cdot (\max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +
{\MinTcm))}
- \end{multline}
-
-Reducing the frequencies of the processors according to the vector of
-scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
-application and thus, increase the static energy because the execution time is
-increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption for the iterative
-application can be measured by measuring the energy consumption for one iteration as in (\ref{eq:energy})
-multiplied by the number of iterations of that application.
+\end{multline}
+Reducing the frequencies of the processors according to the vector of scaling
+factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the application
+and thus, increase the static energy because the execution time is
+increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
+for the iterative application can be measured by measuring the energy
+consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
+of iterations of that application.
\section{Optimization of both energy consumption and performance}
\label{sec.compet}
Using the lowest frequency for each processor does not necessarily give the most
-energy efficient execution of an application. Indeed, even though the dynamic
-power is reduced while scaling down the frequency of a processor, its
-computation power is proportionally decreased. Hence, the execution time might
-be drastically increased and during that time, dynamic and static powers are
-being consumed. Therefore, it might cancel any gains achieved by scaling down
-the frequency of all nodes to the minimum and the overall energy consumption of
-the application might not be the optimal one. It is not trivial to select the
-appropriate frequency scaling factor for each processor while considering the
-characteristics of each processor (computation power, range of frequencies,
-dynamic and static powers) and the task executed (computation/communication
-ratio). The aim being to reduce the overall energy consumption and to avoid
-increasing significantly the execution time. In our previous
-work~\cite{Our_first_paper}, we proposed a method that selects the optimal
-frequency scaling factor for a homogeneous cluster executing a message passing
-iterative synchronous application while giving the best trade-off between the
-energy consumption and the performance for such applications. In this work we
-are interested in heterogeneous clusters as described above. Due to the
-heterogeneity of the processors, a vector of scaling factors should
-be selected and it must give the best trade-off between energy consumption and
-performance.
-
-The relation between the energy consumption and the execution time for an
-application is complex and nonlinear, Thus, unlike the relation between the
-execution time and the scaling factor, the relation between the energy and the
-frequency scaling factors is nonlinear, for more details refer
-to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}. Moreover, these relations
-are not measured using the same metric. To solve this problem, the execution
-time is normalized by computing the ratio between the new execution time (after
-scaling down the frequencies of some processors) and the initial one (with
+energy efficient execution of an application. Indeed, even though the dynamic
+power is reduced while scaling down the frequency of a processor, its
+computation power is proportionally decreased. Hence, the execution time might
+be drastically increased and during that time, dynamic and static powers are
+being consumed. Therefore, it might cancel any gains achieved by scaling down
+the frequency of all nodes to the minimum and the overall energy consumption of
+the application might not be the optimal one. It is not trivial to select the
+appropriate frequency scaling factor for each processor while considering the
+characteristics of each processor (computation power, range of frequencies,
+dynamic and static powers) and the task executed (computation/communication
+ratio). The aim being to reduce the overall energy consumption and to avoid
+increasing significantly the execution time. In our previous
+work~\cite{Our_first_paper}, we proposed a method that selects the optimal
+frequency scaling factor for a homogeneous cluster executing a message passing
+iterative synchronous application while giving the best trade-off between the
+energy consumption and the performance for such applications. In this work we
+are interested in heterogeneous clusters as described above. Due to the
+heterogeneity of the processors, a vector of scaling factors should be selected
+and it must give the best trade-off between energy consumption and performance.
+
+The relation between the energy consumption and the execution time for an
+application is complex and nonlinear, Thus, unlike the relation between the
+execution time and the scaling factor, the relation between the energy and the
+frequency scaling factors is nonlinear, for more details refer
+to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}. Moreover, these relations
+are not measured using the same metric. To solve this problem, the execution
+time is normalized by computing the ratio between the new execution time (after
+scaling down the frequencies of some processors) and the initial one (with
maximum frequency for all nodes) as follows:
\begin{multline}
\label{eq:pnorm}
{\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
\end{multline}
-
-In the same way, the energy is normalized by computing the ratio between the consumed energy
-while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
+In the same way, the energy is normalized by computing the ratio between the
+consumed energy while scaling down the frequency and the consumed energy with
+maximum frequency for all nodes:
\begin{multline}
\label{eq:enorm}
\Enorm = \frac{\Ereduced}{\Eoriginal} \\
{} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot \Tcp[i])} +
\sum_{i=1}^{N} {(\Ps[i] \cdot \Tnew)}}{\sum_{i=1}^{N}{( \Pd[i] \cdot \Tcp[i])} +
\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}}
-\end{multline}
+\end{multline}
Where $\Ereduced$ and $\Eoriginal$ are computed using (\ref{eq:energy}) and
- $\Tnew$ and $\Told$ are computed as in (\ref{eq:pnorm}).
-
-While the main
-goal is to optimize the energy and execution time at the same time, the normalized
-energy and execution time curves are not in the same direction. According
-to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the vector of frequency
-scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy and the execution
-time simultaneously. But the main objective is to produce maximum energy
-reduction with minimum execution time reduction.
-
-This problem can be solved by making the optimization process for energy and
-execution time following the same direction. Therefore, the equation of the
-normalized execution time is inverted which gives the normalized performance equation, as follows:
+$\Tnew$ and $\Told$ are computed as in (\ref{eq:pnorm}).
+
+While the main goal is to optimize the energy and execution time at the same
+time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way. According to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
+vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy
+and the execution time simultaneously. But the main objective is to produce
+maximum energy reduction with minimum execution time reduction.
+
+This problem can be solved by making the optimization process for energy and
+execution time follow the same evolution according to the vector of scaling factors. Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance
+equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
\Pnorm = \frac{\Told}{\Tnew}\\
= \frac{\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
- { \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm}
+ { \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm}
\end{multline}
-
\begin{figure}[!t]
\centering
\subfloat[Homogeneous platform]{%
\includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
-
-
+
\subfloat[Heterogeneous platform]{%
\includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
\label{fig:rel}
Figure~\ref{fig:r2}. Then the objective function has the following form:
\begin{equation}
\label{eq:max}
- \MaxDist =
+ \MaxDist =
\mathop{\max_{i=1,\dots F}}_{j=1,\dots,N}
(\overbrace{\Pnorm(S_{ij})}^{\text{Maximize}} -
\overbrace{\Enorm(S_{ij})}^{\text{Minimize}} )
\end{equation}
-where $N$ is the number of nodes and $F$ is the number of available frequencies for each node.
-Then, the optimal set of scaling factors that satisfies (\ref{eq:max}) can be selected.
-The objective function can work with any energy model or any power values for each node
-(static and dynamic powers). However, the most important energy reduction gain can be achieved when
-the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
+where $N$ is the number of nodes and $F$ is the number of available frequencies
+for each node. Then, the optimal set of scaling factors that satisfies
+(\ref{eq:max}) can be selected. The objective function can work with any energy
+model or any power values for each node (static and dynamic powers). However,
+the most important energy reduction gain can be achieved when the energy curve
+has a convex form as shown
+in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
\section{The scaling factors selection algorithm for heterogeneous platforms }
\label{sec.optim}
-\subsection{The algorithm details}
-In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
-scaling factors vector that gives the best trade-off between minimizing the
-energy consumption and maximizing the performance of a message passing
-synchronous iterative application executed on a heterogeneous platform. It works
-online during the execution time of the iterative message passing program. It
-uses information gathered during the first iteration such as the computation
-time and the communication time in one iteration for each node. The algorithm is
-executed after the first iteration and returns a vector of optimal frequency
-scaling factors that satisfies the objective function (\ref{eq:max}). The
-program applies DVFS operations to change the frequencies of the CPUs according
-to the computed scaling factors. This algorithm is called just once during the
-execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
-scaling algorithm is called in the iterative MPI program.
-
-The nodes in a heterogeneous platform have different computing powers, thus
-while executing message passing iterative synchronous applications, fast nodes
-have to wait for the slower ones to finish their computations before being able
-to synchronously communicate with them as in Figure~\ref{fig:heter}. These
-periods are called idle or slack times. The algorithm takes into account this
-problem and tries to reduce these slack times when selecting the frequency
-scaling factors vector. At first, it selects initial frequency scaling factors
-that increase the execution times of fast nodes and minimize the differences
-between the computation times of fast and slow nodes. The value of the initial
-frequency scaling factor for each node is inversely proportional to its
-computation time that was gathered from the first iteration. These initial
-frequency scaling factors are computed as a ratio between the computation time
-of the slowest node and the computation time of the node $i$ as follows:
-\begin{equation}
- \label{eq:Scp}
- \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}
-\end{equation}
-Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the algorithm computes
-the initial frequencies for all nodes as a ratio between the maximum frequency of node $i$
-and the computation scaling factor $\Scp[i]$ as follows:
-\begin{equation}
- \label{eq:Fint}
- F_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}
-\end{equation}
-If the computed initial frequency for a node is not available in the gears of
-that node, it is replaced by the nearest available frequency. In
-Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
-ascending order and the frequencies of the faster nodes are scaled down
-according to the computed initial frequency scaling factors. The resulting new
-frequencies are highlighted in Figure~\ref{fig:st_freq}. This set of
-frequencies can be considered as a higher bound for the search space of the
-optimal vector of frequencies because selecting frequency scaling factors higher
-than the higher bound will not improve the performance of the application and it
-will increase its overall energy consumption. Therefore the algorithm that
-selects the frequency scaling factors starts the search method from these
-initial frequencies and takes a downward search direction toward lower
-frequencies. The algorithm iterates on all left frequencies, from the higher
-bound until all nodes reach their minimum frequencies, to compute their overall
-energy consumption and performance, and select the optimal frequency scaling
-factors vector. At each iteration the algorithm determines the slowest node
-according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
-while it lowers the frequency of all other nodes by one gear. The new overall
-energy consumption and execution time are computed according to the new scaling
-factors. The optimal set of frequency scaling factors is the set that gives the
-highest distance according to the objective function (\ref{eq:max}).
-
-Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
-consumed energy for an application running on a homogeneous platform and a
-heterogeneous platform respectively while increasing the scaling factors. It can
-be noticed that in a homogeneous platform the search for the optimal scaling
-factor should start from the maximum frequency because the performance and the
-consumed energy decrease from the beginning of the plot. On the other hand,
-in the heterogeneous platform the performance is maintained at the beginning of
-the plot even if the frequencies of the faster nodes decrease until the
-computing power of scaled down nodes are lower than the slowest node. In other
-words, until they reach the higher bound. It can also be noticed that the higher
-the difference between the faster nodes and the slower nodes is, the bigger the
-maximum distance between the energy curve and the performance curve is while
- the scaling factors are varying which results in bigger energy savings.
-\begin{figure}[!t]
- \centering
- \includegraphics[scale=0.5]{fig/start_freq}
- \caption{Selecting the initial frequencies}
- \label{fig:st_freq}
-\end{figure}
-
-
-
-
\begin{algorithm}
\begin{algorithmic}[1]
% \footnotesize
\item[{$\Fmax[i]$}] array of the maximum frequencies for all nodes.
\item[{$\Pd[i]$}] array of the dynamic powers for all nodes.
\item[{$\Ps[i]$}] array of the static powers for all nodes.
- \item[{$\Fdiff[i]$}] array of the difference between two successive frequencies for all nodes.
+ \item[{$\Fdiff[i]$}] array of the differences between two successive frequencies for all nodes.
\end{description}
\Ensure $\Sopt[1],\Sopt[2] \dots, \Sopt[N]$ is a vector of optimal scaling factors
\State Round the computed initial frequencies $F_i$ to the closest one available in each node.
\If{(not the first frequency)}
\State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
- \EndIf
+ \EndIf
\State $\Told \gets \max_{i=1,\dots,N} (\Tcp[i]+\Tcm[i])$
% \State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot \Tcp[i])} +\sum_{i=1}^{N} {(\Ps[i] \cdot \Told)}$
\State $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot \Tcp[i] + \Ps[i] \cdot \Told)}$
\label{dvfs}
\end{algorithm}
+\subsection{The algorithm details}
+
+In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
+scaling factors vector that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+synchronous iterative application executed on a heterogeneous platform. It works
+online during the execution time of the iterative message passing program. It
+uses information gathered during the first iteration such as the computation
+time and the communication time in one iteration for each node. The algorithm is
+executed after the first iteration and returns a vector of optimal frequency
+scaling factors that satisfies the objective function (\ref{eq:max}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors. This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
+scaling algorithm is called in the iterative MPI program.
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.5]{fig/start_freq}
+ \caption{Selecting the initial frequencies}
+ \label{fig:st_freq}
+\end{figure}
+
+The nodes in a heterogeneous platform have different computing powers, thus
+while executing message passing iterative synchronous applications, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}. These
+periods are called idle or slack times. The algorithm takes into account this
+problem and tries to reduce these slack times when selecting the frequency
+scaling factors vector. At first, it selects initial frequency scaling factors
+that increase the execution times of fast nodes and minimize the differences
+between the computation times of fast and slow nodes. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
+\begin{equation}
+ \label{eq:Scp}
+ \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}
+\end{equation}
+Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
+algorithm computes the initial frequencies for all nodes as a ratio between the
+maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+follows:
+\begin{equation}
+ \label{eq:Fint}
+ F_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}
+\end{equation}
+If the computed initial frequency for a node is not available in the gears of
+that node, it is replaced by the nearest available frequency. In
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors. The resulting new
+frequencies are highlighted in Figure~\ref{fig:st_freq}. This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting scaling factors higher
+than the higher bound will not improve the performance of the application and it
+will increase its overall energy consumption. Therefore the algorithm that
+selects the frequency scaling factors starts the search method from these
+initial frequencies and takes a downward search direction toward lower
+frequencies. The algorithm iterates on all remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies, to compute their overall
+energy consumption and performance, and select the optimal frequency scaling
+factors vector. At each iteration the algorithm determines the slowest node
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+while it lowers the frequency of all other nodes by one gear. The new overall
+energy consumption and execution time are computed according to the new scaling
+factors. The optimal set of frequency scaling factors is the set that gives the
+highest distance according to the objective function (\ref{eq:max}).
+
+Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
+consumed energy for an application running on a homogeneous platform and a
+heterogeneous platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous platform the search for the optimal scaling
+factor should start from the maximum frequency because the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand, in
+the heterogeneous platform the performance is maintained at the beginning of the
+plot even if the frequencies of the faster nodes decrease until the computing
+power of scaled down nodes are lower than the slowest node. In other words,
+until they reach the higher bound. It can also be noticed that the higher the
+difference between the faster nodes and the slower nodes is, the bigger the
+maximum distance between the energy curve and the performance curve is while the
+scaling factors are varying which results in bigger energy savings.
+Finally, in a homogeneous platform the energy consumption is increased when the scaling factor is very high.
+Indeed, the dynamic energy saved by reducing the frequency of the processor is compensated by the significant increase of the execution time and thus the increased of the static energy. On the other hand, in a heterogeneous platform this is not the case.
+
\subsection{The evaluation of the proposed algorithm}
\label{sec.verif.algo}
-The precision of the proposed algorithm mainly depends on the execution time
-prediction model defined in (\ref{eq:perf}) and the energy model computed by
-(\ref{eq:energy}). The energy model is also significantly dependent on the
-execution time model because the static energy is linearly related to the
-execution time and the dynamic energy is related to the computation time. So,
-all the works presented in this paper are based on the execution time model. To
-verify this model, the predicted execution time was compared to the real
-execution time over SimGrid/SMPI simulator,
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}, for all the NAS
-parallel benchmarks NPB v3.3 \cite{NAS.Parallel.Benchmarks}, running class B on
-8 or 9 nodes. The comparison showed that the proposed execution time model is
-very precise, the maximum normalized difference between the predicted execution
+
+The precision of the proposed algorithm mainly depends on the execution time
+prediction model defined in (\ref{eq:perf}) and the energy model computed by
+(\ref{eq:energy}). The energy model is also significantly dependent on the
+execution time model because the static energy is linearly related to the
+execution time and the dynamic energy is related to the computation time. So,
+all the works presented in this paper are based on the execution time model. To
+verify this model, the predicted execution time was compared to the real
+execution time over SimGrid/SMPI simulator,
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}, for all the NAS
+parallel benchmarks NPB v3.3 \cite{NAS.Parallel.Benchmarks}, running class B on
+8 or 9 nodes. The comparison showed that the proposed execution time model is
+very precise, the maximum normalized difference between the predicted execution
time and the real execution time is equal to 0.03 for all the NAS benchmarks.
-Since the proposed algorithm is not an exact method it does not test all the
+Since the proposed algorithm is not an exact method, it does not test all the
possible solutions (vectors of scaling factors) in the search space. To prove
its efficiency, it was compared on small instances to a brute force search
algorithm that tests all the possible solutions. The brute force algorithm was
cluster composed of four different types of nodes having the characteristics
presented in Table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4
nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling
-factors vector. The algorithm complexity is $O(F\cdot N)$, where $F$
-is the number of iterations and $N$ is the number of computing nodes. The
-algorithm needs from 12 to 20 iterations to select the best vector of frequency
-scaling factors that gives the results of the next sections.
+factors vector. The algorithm complexity is $O(F\cdot N)$, where $F$ is the
+maximum number of available frequencies, and $N$ is the number of computing
+nodes. The algorithm needs from 12 to 20 iterations to select the best vector of
+frequency scaling factors that gives the results of the next sections.
+
+\begin{table}[!t]
+ \caption{Heterogeneous nodes characteristics}
+ % title of Table
+ \centering
+ \begin{tabular}{|*{7}{r|}}
+ \hline
+ Node & Simulated & Max & Min & Diff. & Dynamic & Static \\
+ type & GFLOPS & Freq. & Freq. & Freq. & power & power \\
+ & & GHz & GHz & GHz & & \\
+ \hline
+ 1 & 40 & 2.50 & 1.20 & 0.100 & \np[W]{20} & \np[W]{4} \\
+ \hline
+ 2 & 50 & 2.66 & 1.60 & 0.133 & \np[W]{25} & \np[W]{5} \\
+ \hline
+ 3 & 60 & 2.90 & 1.20 & 0.100 & \np[W]{30} & \np[W]{6} \\
+ \hline
+ 4 & 70 & 3.40 & 1.60 & 0.133 & \np[W]{35} & \np[W]{7} \\
+ \hline
+ \end{tabular}
+ \label{table:platform}
+\end{table}
\section{Experimental results}
\label{sec.expe}
+
To evaluate the efficiency and the overall energy consumption reduction of
-Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The
+Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3 which
+is composed of synchronous message passing applications. The
experiments were executed on the simulator SimGrid/SMPI which offers easy tools
to create a heterogeneous platform and run message passing applications over it.
The heterogeneous platform that was used in the experiments, had one core per
computing power (which corresponds to \np[\%]{80} of its dynamic power and the
remaining \np[\%]{20} to the static power), the same assumption was made in
\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. Finally, These
-nodes were connected via an Ethernet network with 1 Gbit/s bandwidth.
-
-
-\begin{table}[!t]
- \caption{Heterogeneous nodes characteristics}
- % title of Table
- \centering
- \begin{tabular}{|*{7}{r|}}
- \hline
- Node &Simulated & Max & Min & Diff. & Dynamic & Static \\
- type &GFLOPS & Freq. & Freq. & Freq. & power & power \\
- & & GHz & GHz &GHz & & \\
- \hline
- 1 &40 & 2.50 & 1.20 & 0.100 & \np[W]{20} &\np[W]{4} \\
-
- \hline
- 2 &50 & 2.66 & 1.60 & 0.133 & \np[W]{25} &\np[W]{5} \\
-
- \hline
- 3 &60 & 2.90 & 1.20 & 0.100 & \np[W]{30} &\np[W]{6} \\
-
- \hline
- 4 &70 & 3.40 & 1.60 & 0.133 & \np[W]{35} &\np[W]{7} \\
-
- \hline
- \end{tabular}
- \label{table:platform}
-\end{table}
-
-
-%\subsection{Performance prediction verification}
+nodes were connected via an Ethernet network with \np[Gbit/s]{1} bandwidth.
\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
-
The proposed algorithm was applied to the seven parallel NAS benchmarks (EP, CG,
-MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes:
+MG, FT, BT, LU and SP) and the benchmarks were executed with the three classes:
A, B and C. However, due to the lack of space in this paper, only the results of
-the biggest class, C, are presented while being run on different number of
-nodes, ranging from 4 to 128 or 144 nodes depending on the benchmark being
-executed. Indeed, the benchmarks CG, MG, LU, EP and FT had to be executed on $1,
-2, 4, 8, 16, 32, 64, 128$ nodes. The other benchmarks such as BT and SP had to
-be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
-
-
-
+the biggest class, C, are presented while being run on different number of
+nodes, ranging from 8 to 128 or 144 nodes depending on the benchmark being
+executed. Indeed, the benchmarks CG, MG, LU, EP and FT had to be executed on 1,
+2, 4, 8, 16, 32, 64, or 128 nodes. The other benchmarks such as BT and SP had
+to be executed on 1, 4, 9, 16, 36, 64, or 144 nodes.
+
\begin{table}[!t]
- \caption{Running NAS benchmarks on 4 nodes }
- % title of Table
- \centering
- \begin{tabular}{|*{7}{r|}}
- \hline
- \hspace{-2.2084pt}%
- Program & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
- \hline
- CG & 64.64 & 3560.39 &34.16 &6.72 &27.44 \\
- \hline
- MG & 18.89 & 1074.87 &35.37 &4.34 &31.03 \\
- \hline
- EP &79.73 &5521.04 &26.83 &3.04 &23.79 \\
- \hline
- LU &308.65 &21126.00 &34.00 &6.16 &27.84 \\
- \hline
- BT &360.12 &21505.55 &35.36 &8.49 &26.87 \\
- \hline
- SP &234.24 &13572.16 &35.22 &5.70 &29.52 \\
- \hline
- FT &81.58 &4151.48 &35.58 &0.99 &34.59 \\
-\hline
- \end{tabular}
- \label{table:res_4n}
+
% \end{table}
-\medskip
+
% \begin{table}[!t]
\caption{Running NAS benchmarks on 8 and 9 nodes }
% title of Table
\centering
\begin{tabular}{|*{7}{r|}}
\hline
- \hspace{-2.2084pt}%
- Program & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
+ \hspace{-2.2084pt}%
+ Program & Execution & Energy & Energy & Performance & Distance \\
+ name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
- CG &36.11 &3263.49 &31.25 &7.12 &24.13 \\
- \hline
- MG &8.99 &953.39 &33.78 &6.41 &27.37 \\
- \hline
- EP &40.39 &5652.81 &27.04 &0.49 &26.55 \\
- \hline
- LU &218.79 &36149.77 &28.23 &0.01 &28.22 \\
+ CG & 36.11 & 3263.49 & 31.25 & 7.12 & 24.13 \\
+ \hline
+ MG & 8.99 & 953.39 & 33.78 & 6.41 & 27.37 \\
+ \hline
+ EP & 40.39 & 5652.81 & 27.04 & 0.49 & 26.55 \\
+ \hline
+ LU & 218.79 & 36149.77 & 28.23 & 0.01 & 28.22 \\
+ \hline
+ BT & 166.89 & 23207.42 & 32.32 & 7.89 & 24.43 \\
+ \hline
+ SP & 104.73 & 18414.62 & 24.73 & 2.78 & 21.95 \\
+ \hline
+ FT & 51.10 & 4913.26 & 31.02 & 2.54 & 28.48 \\
\hline
- BT &166.89 &23207.42 &32.32 &7.89 &24.43 \\
- \hline
- SP &104.73 &18414.62 &24.73 &2.78 &21.95 \\
- \hline
- FT &51.10 &4913.26 &31.02 &2.54 &28.48 \\
-\hline
\end{tabular}
\label{table:res_8n}
% \end{table}
-\medskip
+ \medskip
% \begin{table}[!t]
\caption{Running NAS benchmarks on 16 nodes }
% title of Table
\begin{tabular}{|*{7}{r|}}
\hline
\hspace{-2.2084pt}%
- Program & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
+ name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
- CG &31.74 &4373.90 &26.29 &9.57 &16.72 \\
- \hline
- MG &5.71 &1076.19 &32.49 &6.05 &26.44 \\
- \hline
- EP &20.11 &5638.49 &26.85 &0.56 &26.29 \\
- \hline
- LU &144.13 &42529.06 &28.80 &6.56 &22.24 \\
+ CG & 31.74 & 4373.90 & 26.29 & 9.57 & 16.72 \\
+ \hline
+ MG & 5.71 & 1076.19 & 32.49 & 6.05 & 26.44 \\
+ \hline
+ EP & 20.11 & 5638.49 & 26.85 & 0.56 & 26.29 \\
+ \hline
+ LU & 144.13 & 42529.06 & 28.80 & 6.56 & 22.24 \\
+ \hline
+ BT & 97.29 & 22813.86 & 34.95 & 5.80 & 29.15 \\
+ \hline
+ SP & 66.49 & 20821.67 & 22.49 & 3.82 & 18.67 \\
+ \hline
+ FT & 37.01 & 5505.60 & 31.59 & 6.48 & 25.11 \\
\hline
- BT &97.29 &22813.86 &34.95 &5.80 &29.15 \\
- \hline
- SP &66.49 &20821.67 &22.49 &3.82 &18.67 \\
- \hline
- FT &37.01 &5505.60 &31.59 &6.48 &25.11 \\
-\hline
\end{tabular}
\label{table:res_16n}
% \end{table}
-\medskip
+ \medskip
% \begin{table}[!t]
\caption{Running NAS benchmarks on 32 and 36 nodes }
% title of Table
\begin{tabular}{|*{7}{r|}}
\hline
\hspace{-2.2084pt}%
- Program & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
+ name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
- CG &32.35 &6704.21 &16.15 &5.30 &10.85 \\
- \hline
- MG &4.30 &1355.58 &28.93 &8.85 &20.08 \\
- \hline
- EP &9.96 &5519.68 &26.98 &0.02 &26.96 \\
- \hline
- LU &99.93 &67463.43 &23.60 &2.45 &21.15 \\
+ CG & 32.35 & 6704.21 & 16.15 & 5.30 & 10.85 \\
+ \hline
+ MG & 4.30 & 1355.58 & 28.93 & 8.85 & 20.08 \\
+ \hline
+ EP & 9.96 & 5519.68 & 26.98 & 0.02 & 26.96 \\
+ \hline
+ LU & 99.93 & 67463.43 & 23.60 & 2.45 & 21.15 \\
+ \hline
+ BT & 48.61 & 23796.97 & 34.62 & 5.83 & 28.79 \\
+ \hline
+ SP & 46.01 & 27007.43 & 22.72 & 3.45 & 19.27 \\
+ \hline
+ FT & 28.06 & 7142.69 & 23.09 & 2.90 & 20.19 \\
\hline
- BT &48.61 &23796.97 &34.62 &5.83 &28.79 \\
- \hline
- SP &46.01 &27007.43 &22.72 &3.45 &19.27 \\
- \hline
- FT &28.06 &7142.69 &23.09 &2.90 &20.19 \\
-\hline
\end{tabular}
\label{table:res_32n}
% \end{table}
-\medskip
+ \medskip
% \begin{table}[!t]
\caption{Running NAS benchmarks on 64 nodes }
% title of Table
\begin{tabular}{|*{7}{r|}}
\hline
\hspace{-2.2084pt}%
- Program & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
+ name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
- CG &46.65 &17521.83 &8.13 &1.68 &6.45 \\
- \hline
- MG &3.27 &1534.70 &29.27 &14.35 &14.92 \\
- \hline
- EP &5.05 &5471.1084 &27.12 &3.11 &24.01 \\
- \hline
- LU &73.92 &101339.16 &21.96 &3.67 &18.29 \\
+ CG & 46.65 & 17521.83 & 8.13 & 1.68 & 6.45 \\
+ \hline
+ MG & 3.27 & 1534.70 & 29.27 & 14.35 & 14.92 \\
+ \hline
+ EP & 5.05 & 5471.11 & 27.12 & 3.11 & 24.01 \\
+ \hline
+ LU & 73.92 & 101339.16 & 21.96 & 3.67 & 18.29 \\
+ \hline
+ BT & 39.99 & 27166.71 & 32.02 & 12.28 & 19.74 \\
+ \hline
+ SP & 52.00 & 49099.28 & 24.84 & 0.03 & 24.81 \\
+ \hline
+ FT & 25.97 & 10416.82 & 20.15 & 4.87 & 15.28 \\
\hline
- BT &39.99 &27166.71 &32.02 &12.28 &19.74 \\
- \hline
- SP &52.00 &49099.28 &24.84 &0.03 &24.81 \\
- \hline
- FT &25.97 &10416.82 &20.15 &4.87 &15.28 \\
-\hline
\end{tabular}
\label{table:res_64n}
% \end{table}
-\medskip
+ \medskip
% \begin{table}[!t]
\caption{Running NAS benchmarks on 128 and 144 nodes }
% title of Table
\begin{tabular}{|*{7}{r|}}
\hline
\hspace{-2.2084pt}%
- Program & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
+ Program & Execution & Energy & Energy & Performance & Distance \\
+ name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
- CG &56.92 &41163.36 &4.00 &1.10 &2.90 \\
- \hline
- MG &3.55 &2843.33 &18.77 &10.38 &8.39 \\
- \hline
- EP &2.67 &5669.66 &27.09 &0.03 &27.06 \\
- \hline
- LU &51.23 &144471.90 &16.67 &2.36 &14.31 \\
+ CG & 56.92 & 41163.36 & 4.00 & 1.10 & 2.90 \\
+ \hline
+ MG & 3.55 & 2843.33 & 18.77 & 10.38 & 8.39 \\
+ \hline
+ EP & 2.67 & 5669.66 & 27.09 & 0.03 & 27.06 \\
+ \hline
+ LU & 51.23 & 144471.90 & 16.67 & 2.36 & 14.31 \\
+ \hline
+ BT & 37.96 & 44243.82 & 23.18 & 1.28 & 21.90 \\
+ \hline
+ SP & 64.53 & 115409.71 & 26.72 & 0.05 & 26.67 \\
+ \hline
+ FT & 25.51 & 18808.72 & 12.85 & 2.84 & 10.01 \\
\hline
- BT &37.96 &44243.82 &23.18 &1.28 &21.90 \\
- \hline
- SP &64.53 &115409.71 &26.72 &0.05 &26.67 \\
- \hline
- FT &25.51 &18808.72 &12.85 &2.84 &10.01 \\
-\hline
\end{tabular}
\label{table:res_128n}
\end{table}
+
+\begin{figure}[!t]
+ \centering
+ \subfloat[Energy saving (\%)]{%
+ \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
+
+ \subfloat[Performance degradation (\%)]{%
+ \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
+ \label{fig:avg}
+ \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}
+\end{figure}
+
The overall energy consumption was computed for each instance according to the
energy consumption model (\ref{eq:energy}), with and without applying the
algorithm. The execution time was also measured for all these experiments. Then,
the energy saving and performance degradation percentages were computed for each
-instance. The results are presented in Tables~\ref{table:res_4n},
+instance. The results are presented in Tables
\ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n},
\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
values from many experiments for energy savings and performance degradation.
The tables show the experimental results for running the NAS parallel benchmarks
-on different number of nodes. The experiments show that the algorithm
-significantly reduces the energy consumption (up to \np[\%]{35}) and tries to
+on different numbers of nodes. The experiments show that the algorithm
+significantly reduces the energy consumption (up to \np[\%]{34}) and tries to
limit the performance degradation. They also show that the energy saving
percentage decreases when the number of computing nodes increases. This
reduction is due to the increase of the communication times compared to the
-execution times when the benchmarks are run over a high number of nodes.
+execution times when the benchmarks are run over a higher number of nodes.
Indeed, the benchmarks with the same class, C, are executed on different numbers
of nodes, so the computation required for each iteration is divided by the
number of computing nodes. On the other hand, more communications are required
can also be noticed that for the benchmarks EP and SP that contain little or no
communications, the energy savings are not significantly affected by the high
number of nodes. No experiments were conducted using bigger classes than D,
-because they require a lot of memory (more than 64GB) when being executed by the
-simulator on one machine. The maximum distance between the normalized energy
-curve and the normalized performance for each instance is also shown in the
-result tables. It decrease in the same way as the energy saving percentage. The
-tables also show that the performance degradation percentage is not
+because they require a lot of memory (more than \np[GB]{64}) when being executed
+by the simulator on one machine. The maximum distance between the normalized
+energy curve and the normalized performance for each instance is also shown in
+the result tables. It decrease in the same way as the energy saving percentage.
+The tables also show that the performance degradation percentage is not
significantly increased when the number of computing nodes is increased because
the computation times are small when compared to the communication times.
-
-
-\begin{figure}[!t]
- \centering
- \subfloat[Energy saving (\%)]{%
- \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
-
- \subfloat[Performance degradation (\%)]{%
- \includegraphics[width=.33\textwidth]{fig/per_deg}\label{fig:per_deg}}
- \label{fig:avg}
- \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}
-\end{figure}
-
-Figures~\ref{fig:energy} and \ref{fig:per_deg} present the energy saving and
-performance degradation respectively for all the benchmarks according to the
-number of used nodes. As shown in the first plot, the energy saving percentages
-of the benchmarks MG, LU, BT and FT decrease linearly when the number of nodes
-increase. While for the EP and SP benchmarks, the energy saving percentage is
-not affected by the increase of the number of computing nodes, because in these
-benchmarks there are little or no communications. Finally, the energy saving of
-the GC benchmark significantly decrease when the number of nodes increase
-because this benchmark has more communications than the others. The second plot
-shows that the performance degradation percentages of most of the benchmarks
-decrease when they run on a big number of nodes because they spend more time
-communicating than computing, thus, scaling down the frequencies of some nodes
+Figures~\ref{fig:energy} and \ref{fig:per_deg} present the energy saving and
+performance degradation respectively for all the benchmarks according to the
+number of used nodes. As shown in the first plot, the energy saving percentages
+of the benchmarks MG, LU, BT and FT decrease linearly when the number of nodes
+increase. While for the EP and SP benchmarks, the energy saving percentage is
+not affected by the increase of the number of computing nodes, because in these
+benchmarks there are little or no communications. Finally, the energy saving of
+the CG benchmark significantly decreases when the number of nodes increase
+because this benchmark has more communications than the others. The second plot
+shows that the performance degradation percentages of most of the benchmarks
+decrease when they run on a big number of nodes because they spend more time
+communicating than computing, thus, scaling down the frequencies of some nodes
has less effect on the performance.
-
-
-
\subsection{The results for different power consumption scenarios}
\label{sec.compare}
+
The results of the previous section were obtained while using processors that
consume during computation an overall power which is \np[\%]{80} composed of
dynamic power and of \np[\%]{20} of static power. In this section, these ratios
scaling values which result in less energy saving but also less performance
degradation.
-
- \begin{table}[!t]
+\begin{table}[!t]
\caption{The results of the \np[\%]{70}-\np[\%]{30} power scenario}
% title of Table
\centering
\begin{tabular}{|*{6}{r|}}
\hline
- Program & Energy & Energy & Performance & Distance \\
- name & consumption/J & saving\% & degradation\% & \\
+ Program & Energy & Energy & Performance & Distance \\
+ name & consumption/J & saving\% & degradation\% & \\
+ \hline
+ CG & 4144.21 & 22.42 & 7.72 & 14.70 \\
\hline
- CG &4144.21 &22.42 &7.72 &14.70 \\
- \hline
- MG &1133.23 &24.50 &5.34 &19.16 \\
+ MG & 1133.23 & 24.50 & 5.34 & 19.16 \\
\hline
- EP &6170.30 &16.19 &0.02 &16.17 \\
+ EP & 6170.30 & 16.19 & 0.02 & 16.17 \\
\hline
- LU &39477.28 &20.43 &0.07 &20.36 \\
+ LU & 39477.28 & 20.43 & 0.07 & 20.36 \\
\hline
- BT &26169.55 &25.34 &6.62 &18.71 \\
+ BT & 26169.55 & 25.34 & 6.62 & 18.71 \\
\hline
- SP &19620.09 &19.32 &3.66 &15.66 \\
+ SP & 19620.09 & 19.32 & 3.66 & 15.66 \\
\hline
- FT &6094.07 &23.17 &0.36 &22.81 \\
-\hline
+ FT & 6094.07 & 23.17 & 0.36 & 22.81 \\
+ \hline
\end{tabular}
\label{table:res_s1}
\end{table}
-
-
\begin{table}[!t]
\caption{The results of the \np[\%]{90}-\np[\%]{10} power scenario}
% title of Table
\centering
\begin{tabular}{|*{6}{r|}}
\hline
- Program & Energy & Energy & Performance & Distance \\
- name & consumption/J & saving\% & degradation\% & \\
+ Program & Energy & Energy & Performance & Distance \\
+ name & consumption/J & saving\% & degradation\% & \\
\hline
- CG &2812.38 &36.36 &6.80 &29.56 \\
- \hline
- MG &825.427 &38.35 &6.41 &31.94 \\
- \hline
- EP &5281.62 &35.02 &2.68 &32.34 \\
- \hline
- LU &31611.28 &39.15 &3.51 &35.64 \\
+ CG & 2812.38 & 36.36 & 6.80 & 29.56 \\
+ \hline
+ MG & 825.43 & 38.35 & 6.41 & 31.94 \\
+ \hline
+ EP & 5281.62 & 35.02 & 2.68 & 32.34 \\
+ \hline
+ LU & 31611.28 & 39.15 & 3.51 & 35.64 \\
+ \hline
+ BT & 21296.46 & 36.70 & 6.60 & 30.10 \\
+ \hline
+ SP & 15183.42 & 35.19 & 11.76 & 23.43 \\
+ \hline
+ FT & 3856.54 & 40.80 & 5.67 & 35.13 \\
\hline
- BT &21296.46 &36.70 &6.60 &30.10 \\
- \hline
- SP &15183.42 &35.19 &11.76 &23.43 \\
- \hline
- FT &3856.54 &40.80 &5.67 &35.13 \\
-\hline
\end{tabular}
\label{table:res_s2}
\end{table}
\begin{table}[!t]
- \caption{Comparing the proposed algorithm}
- \centering
-\begin{tabular}{|*{7}{r|}}
-\hline
-Program & \multicolumn{2}{c|}{Energy saving \%} & \multicolumn{2}{c|}{Perf. degradation \%} & \multicolumn{2}{c|}{Distance} \\ \cline{2-7}
-name & EDP & MaxDist & EDP & MaxDist & EDP & MaxDist \\ \hline
-CG & 27.58 & 31.25 & 5.82 & 7.12 & 21.76 & 24.13 \\ \hline
-MG & 29.49 & 33.78 & 3.74 & 6.41 & 25.75 & 27.37 \\ \hline
-LU & 19.55 & 28.33 & 0.0 & 0.01 & 19.55 & 28.22 \\ \hline
-EP & 28.40 & 27.04 & 4.29 & 0.49 & 24.11 & 26.55 \\ \hline
-BT & 27.68 & 32.32 & 6.45 & 7.87 & 21.23 & 24.43 \\ \hline
-SP & 20.52 & 24.73 & 5.21 & 2.78 & 15.31 & 21.95 \\ \hline
-FT & 27.03 & 31.02 & 2.75 & 2.54 & 24.28 & 28.48 \\ \hline
-
-\end{tabular}
-\label{table:compare_EDP}
+ \caption{Comparing the proposed algorithm}
+ \centering
+ \begin{tabular}{|*{7}{r|}}
+ \hline
+ Program & \multicolumn{2}{c|}{Energy saving \%}
+ & \multicolumn{2}{c|}{Perf. degradation \%}
+ & \multicolumn{2}{c|}{Distance} \\
+ \cline{2-7}
+ name & EDP & MaxDist & EDP & MaxDist & EDP & MaxDist \\
+ \hline
+ CG & 27.58 & 31.25 & 5.82 & 7.12 & 21.76 & 24.13 \\
+ \hline
+ MG & 29.49 & 33.78 & 3.74 & 6.41 & 25.75 & 27.37 \\
+ \hline
+ LU & 19.55 & 28.33 & 0.00 & 0.01 & 19.55 & 28.22 \\
+ \hline
+ EP & 28.40 & 27.04 & 4.29 & 0.49 & 24.11 & 26.55 \\
+ \hline
+ BT & 27.68 & 32.32 & 6.45 & 7.87 & 21.23 & 24.43 \\
+ \hline
+ SP & 20.52 & 24.73 & 5.21 & 2.78 & 15.31 & 21.95 \\
+ \hline
+ FT & 27.03 & 31.02 & 2.75 & 2.54 & 24.28 & 28.48 \\
+ \hline
+ \end{tabular}
+ \label{table:compare_EDP}
\end{table}
\begin{figure}[!t]
\includegraphics[width=.33\textwidth]{fig/three_scenarios}\label{fig:scales_comp}}
\label{fig:comp}
\caption{The comparison of the three power scenarios}
-\end{figure}
+\end{figure}
\begin{figure}[!t]
\centering
\subsection{The comparison of the proposed scaling algorithm }
\label{sec.compare_EDP}
+
In this section, the scaling factors selection algorithm, called MaxDist, is
compared to Spiliopoulos et al. algorithm
\cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP. They developed a
performance degradation percentage is approximately the same for both
algorithms, about \np[\%]{4}.
-
For all benchmarks, our algorithm outperforms Spiliopoulos et al. algorithm in
terms of energy and performance trade-off, see Figure~\ref{fig:compare_EDP},
because it maximizes the distance between the energy saving and the performance
degradation values while giving the same weight for both metrics.
-
\section{Conclusion}
-\label{sec.concl}
+\label{sec.concl}
+
In this paper, a new online frequency selecting algorithm has been presented. It
selects the best possible vector of frequency scaling factors that gives the
maximum distance (optimal trade-off) between the predicted energy and the
iterative applications running over heterogeneous platforms. To evaluate the
proposed method, it was applied on the NAS parallel benchmarks and executed over
a heterogeneous platform simulated by SimGrid. The results of the experiments
-showed that the algorithm reduces up to \np[\%]{35} the energy consumption of a
+showed that the algorithm reduces up to \np[\%]{34} the energy consumption of a
message passing iterative method while limiting the degradation of the
performance. The algorithm also selects different scaling factors according to
the percentage of the computing and communication times, and according to the
\section*{Acknowledgment}
-This work has been partially supported by the Labex
-ACTION project (contract “ANR-11-LABX-01-01”). As a PhD student,
-Mr. Ahmed Fanfakh, would like to thank the University of
-Babylon (Iraq) for supporting his work.
-
+This work has been partially 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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-
+
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