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+
\begin{document}
\title{Energy Consumption Reduction for Message Passing Iterative Applications in Heterogeneous Architecture Using DVFS}
Besides platform improvements, there are many software and hardware techniques
to lower the energy consumption of these platforms, such as scheduling, DVFS,
-... DVFS is a widely used process to reduce the energy consumption of a
+\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
\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 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
-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 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 varied
-objectives such as energy reduction or performance. They also could be adapted
-to the execution's environment and the type of the application such as
-sequential, parallel or distributed architecture, homogeneous or heterogeneous
-platform, synchronous or asynchronous application, ...
+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
+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
+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
+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
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+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:
\end{itemize}
-For the first type of platform, the computing intensive parallel tasks are executed on the GPUs and the rest are executed
-on the CPUs. Luley et al.
-~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a heterogeneous
-cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal was to maximize the
-energy efficiency of the platform during computation by maximizing the number of FLOPS per watt generated.
-In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et al. developed a scheduling
-algorithm that distributes workloads proportional to the computing power of the nodes which could be a GPU or a CPU. All the tasks must be completed at the same time.
-In~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, 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.
+For the first type of platform, the computing intensive parallel tasks are
+executed on the GPUs and the rest are executed on the CPUs. Luley et
+al.~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a
+heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
+goal was to maximize the energy efficiency of the platform during computation by
+maximizing the number of FLOPS per watt generated.
+In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
+al. developed a scheduling algorithm that distributes workloads proportional to
+the computing power of the nodes which could be a GPU or a CPU. All the tasks
+must be completed at the same time. In~\cite{Rong_Effects.of.DVFS.on.K20.GPU},
+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 al.~\cite{Naveen_Power.Efficient.Resource.Scaling}
-developed a method that minimizes the value of $energy\cdot delay^2$ (the delay is the sum of slack times that happen during synchronous communications) by dynamically assigning new frequencies to the CPUs of the heterogeneous cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed
-an algorithm that divides the executed tasks into two types: the critical and
-non critical tasks. The algorithm scales down the frequency of non critical tasks proportionally to their slack and communication times while limiting the performance degradation percentage to less than 10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed
- a heterogeneous cluster composed of two types
-of Intel and AMD processors. They use a gradient method to predict the impact of DVFS operations on performance.
-In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks},
- the best frequencies for a specified heterogeneous cluster are selected offline using some
-heuristic. Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic programming approach to
-minimize the power consumption of heterogeneous servers while respecting given time constraints. This approach
-had considerable overhead.
-In contrast to the above described papers, this paper presents the following contributions :
+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
+al.~\cite{Naveen_Power.Efficient.Resource.Scaling} developed a method that
+minimizes the value of $\mathit{energy}\times \mathit{delay}^2$ (the delay is
+the sum of slack times that happen during synchronous communications) by
+dynamically assigning new frequencies to the CPUs of the heterogeneous
+cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling}
+proposed an algorithm that divides the executed tasks into two types: the
+critical and non critical tasks. The algorithm scales down the frequency of non
+critical tasks proportionally to their slack and communication times while
+limiting the performance degradation percentage to less than
+10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
+heterogeneous cluster composed of two types of Intel and AMD processors. They
+use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
+\cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
+frequencies for a specified heterogeneous cluster are selected offline using
+some heuristic. Chen et
+al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
+programming approach to minimize the power consumption of heterogeneous servers
+while respecting given time constraints. This approach had considerable
+overhead. In contrast to the above described papers, this paper presents the
+following contributions :
\begin{enumerate}
\item two new energy and performance models for message passing iterative 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.
power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
have the same network bandwidth and latency.
-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
-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.
+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
+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
as in (\ref{eq:s}).
\begin{equation}
\label{eq:s}
- S = \frac{F_\textit{max}}{F_\textit{new}}
+ 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
vector of scaling factors can be predicted using (\ref{eq:perf}).
\begin{equation}
\label{eq:perf}
- \textit T_\textit{new} =
- \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) + MinTcm
+ \Tnew = \max_{i=1,2,\dots,N} ({\TcpOld[i]} \cdot S_{i}) + \MinTcm
\end{equation}
Where:
\begin{equation}
\label{eq:perf2}
- MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
+ \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
+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
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 $C_L$, the supply voltage $V$ and
+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 C_L \cdot V^2 \cdot F
+ \Pd = \alpha \cdot \CL \cdot V^2 \cdot F
\end{equation}
-The static power $Ps$ captures the leakage power as follows:
+The static power $\Ps$ captures the leakage power as follows:
\begin{equation}
\label{eq:ps}
- Ps = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
+ \Ps = V \cdot \Ntrans \cdot \Kdesign \cdot \Ileak
\end{equation}
-where V is the supply voltage, $N_{trans}$ is the number of transistors,
-$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
+where V is the supply voltage, $\Ntrans$ is the number of transistors,
+$\Kdesign$ is a design dependent parameter and $\Ileak$ 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} = Pd \cdot Tcp + Ps \cdot T
+ \Eind = \Pd \cdot \Tcp + \Ps \cdot T
\end{equation}
-where $T$ is the execution time of the program, $Tcp$ is the computation
-time and $Tcp \le T$. $Tcp$ may be equal to $T$ if there is no
+where $T$ is the execution time of the program, $\Tcp$ is the computation
+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}.
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
-$F_{new}$ from (\ref{eq:s}) can be calculated as follows:
+$\Fnew$ from (\ref{eq:s}) can be calculated as follows:
\begin{equation}
\label{eq:fnew}
- F_\textit{new} = S^{-1} \cdot F_\textit{max}
+ \Fnew = S^{-1} \cdot \Fmax
\end{equation}
-Replacing $F_{new}$ 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}
- {P}_\textit{dNew} = \alpha \cdot C_L \cdot V^2 \cdot F_{new} = \alpha \cdot C_L \cdot \beta^2 \cdot F_{new}^3 \\
- {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \cdot S^{-3}
+ \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 $ {P}_\textit{dNew}$ and $P_{dOld}$ 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
and is given by the following equation:
\begin{equation}
\label{eq:Edyn}
- E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{dOld} \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 energy of a processor after scaling its frequency is computed as follows:
\begin{equation}
\label{eq:Estatic}
- E_\textit{s} = Ps \cdot (Tcp \cdot S + Tcm)
+ \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 $Ps_{i}$
+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
+$\Tcp[i]$ 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
+$\Tcm[i]$ 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
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))}
+ 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
maximum frequency for all nodes) as follows:
\begin{multline}
\label{eq:pnorm}
- P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}}\\
- {} = \frac{ \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) +MinTcm}
- {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
+ \Pnorm = \frac{\Tnew}{\Told}\\
+ {} = \frac{ \max_{i=1,2,\dots,N} (\Tcp[i] \cdot S_{i}) +\MinTcm}
+ {\max_{i=1,2,\dots,N}{(\Tcp[i]+\Tcm[i])}}
\end{multline}
while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
\begin{multline}
\label{eq:enorm}
- E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
- {} = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{New})}}{\sum_{i=1}^{N}{( Pd_i \cdot Tcp_i)} +
- \sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}}
+ \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}
-Where $E_\textit{Reduced}$ and $E_\textit{Original}$ are computed using (\ref{eq:energy}) and
- $T_{New}$ and $T_{Old}$ are computed as in (\ref{eq:pnorm}).
+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
normalized execution time is inverted which gives the normalized performance equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
- P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
- = \frac{\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
- { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm}
+ \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}
\end{multline}
\caption{The energy and performance relation}
\end{figure}
-Then, the objective function can be modeled in order to find the maximum distance
-between the energy curve (\ref{eq:enorm}) and the performance
-curve (\ref{eq:pnorm_inv}) over all available sets of scaling factors. This
-represents the minimum energy consumption with minimum execution time (maximum
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then the objective
-function has the following form:
+Then, the objective function can be modeled in order to find the maximum
+distance between the energy curve (\ref{eq:enorm}) and the performance curve
+(\ref{eq:pnorm_inv}) over all available sets of scaling factors. This
+represents the minimum energy consumption with minimum execution time (maximum
+performance) at the same time, see Figure~\ref{fig:r1} or
+Figure~\ref{fig:r2}. Then the objective function has the following form:
\begin{equation}
\label{eq:max}
- Max Dist =
- \max_{i=1,\dots F, j=1,\dots,N}
- (\overbrace{P_\textit{Norm}(S_{ij})}^{\text{Maximize}} -
- \overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
+ \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.
\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:
+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}
+ \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:
+and the computation scaling factor $\Scp[i]$ as follows:
\begin{equation}
\label{eq:Fint}
- F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
+ 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
-colored in blue 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
+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 colored in blue 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}).
% \footnotesize
\Require ~
\begin{description}
- \item[$Tcp_i$] array of all computation times for all nodes during one iteration and with highest frequency.
- \item[$Tcm_i$] array of all communication times for all nodes during one iteration and with highest frequency.
- \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[{$\Tcp[i]$}] array of all computation times for all nodes during one iteration and with highest frequency.
+ \item[{$\Tcm[i]$}] array of all communication times for all nodes during one iteration and with highest frequency.
+ \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.
\end{description}
- \Ensure $Sopt_1,Sopt_2 \dots, Sopt_N$ is a vector of optimal scaling factors
+ \Ensure $\Sopt[1],\Sopt[2] \dots, \Sopt[N]$ is a vector of optimal scaling factors
- \State $ Scp_i \gets \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i} $
- \State $F_{i} \gets \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}$
+ \State $\Scp[i] \gets \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]} $
+ \State $F_{i} \gets \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\cdots,N}$
\State Round the computed initial frequencies $F_i$ to the closest one available in each node.
\If{(not the first frequency)}
- \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
+ \State $F_i \gets F_i+\Fdiff[i],~i=1,\dots,N.$
\EndIf
- \State $T_\textit{Old} \gets max_{~i=1,\dots,N } (Tcp_i+Tcm_i)$
- \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
- \State $Sopt_{i} \gets 1,~i=1,\dots,N. $
- \State $Dist \gets 0 $
+ \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 $\Eoriginal \gets \sum_{i=1}^{N}{( \Pd[i] \cdot \Tcp[i] + \Ps[i] \cdot \Told)}$
+ \State $\Sopt[i] \gets 1,~i=1,\dots,N. $
+ \State $\Dist \gets 0 $
\While {(all nodes not reach their minimum frequency)}
\If{(not the last freq. \textbf{and} not the slowest node)}
- \State $F_i \gets F_i - Fdiff_i,~i=1,\dots,N.$
- \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,\dots,N.$
+ \State $F_i \gets F_i - \Fdiff[i],~i=1,\dots,N.$
+ \State $S_i \gets \frac{\Fmax[i]}{F_i},~i=1,\dots,N.$
\EndIf
- \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm $
- \State $E_\textit{Reduced} \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} + $ \hspace*{43 mm}
- $\sum_{i=1}^{N} {(Ps_i \cdot T_{New})} $
- \State $ P_\textit{Norm} \gets \frac{T_\textit{Old}}{T_\textit{New}}$
- \State $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
+ \State $\Tnew \gets \max_{i=1,\dots,N} (\Tcp[i] \cdot S_{i}) + \MinTcm $
+% \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot \Tcp[i])} + \sum_{i=1}^{N} {(\Ps[i] \cdot \rlap{\Tnew)}} $
+ \State $\Ereduced \gets \sum_{i=1}^{N}{(S_i^{-2} \cdot \Pd[i] \cdot \Tcp[i] + \Ps[i] \cdot \rlap{\Tnew)}} $
+ \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+ \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
\If{$(\Pnorm - \Enorm > \Dist)$}
- \State $Sopt_{i} \gets S_{i},~i=1,\dots,N. $
+ \State $\Sopt[i] \gets S_{i},~i=1,\dots,N. $
\State $\Dist \gets \Pnorm - \Enorm$
\EndIf
\EndWhile
- \State Return $Sopt_1,Sopt_2,\dots,Sopt_N$
+ \State Return $\Sopt[1],\Sopt[2],\dots,\Sopt[N]$
\end{algorithmic}
\caption{frequency scaling factors selection algorithm}
\label{HSA}
\If {$(k=1)$}
\State Gather all times of computation and\newline\hspace*{3em}%
communication from each node.
- \State Call algorithm \ref{HSA}.
+ \State Call Algorithm \ref{HSA}.
\State Compute the new frequencies from the\newline\hspace*{3em}%
returned optimal scaling factors.
\State Set the new frequencies to nodes.
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 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 applied to different NAS benchmarks classes with
-different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical
-and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time:
-for a heterogeneous 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 $O(F\cdot (N \cdot4) )$, 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.
+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
+applied to different NAS benchmarks classes with different number of nodes. The
+solutions returned by the brute force algorithm and the proposed algorithm were
+identical and the proposed algorithm was on average 10 times faster than the
+brute force algorithm. It has a small execution time: for a heterogeneous
+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.
\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. 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
node because just one process was executed per node. The heterogeneous platform
was composed of four types of nodes. Each type of nodes had different
characteristics such as the maximum CPU frequency, the number of available
-frequencies and the computational power, see Table \ref{table:platform}. The
+frequencies and the computational power, see Table~\ref{table:platform}. The
characteristics of these different types of nodes are inspired from the
specifications of real Intel processors. The heterogeneous platform had up to
144 nodes and had nodes from the four types in equal proportions, for example if
\caption{Running NAS benchmarks on 4 nodes }
% title of Table
\centering
- \begin{tabular}{|*{7}{l|}}
+ \begin{tabular}{|*{7}{r|}}
\hline
+ \hspace{-2.2084pt}%
Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
\caption{Running NAS benchmarks on 8 and 9 nodes }
% title of Table
\centering
- \begin{tabular}{|*{7}{l|}}
+ \begin{tabular}{|*{7}{r|}}
\hline
+ \hspace{-2.2084pt}%
Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
\caption{Running NAS benchmarks on 16 nodes }
% title of Table
\centering
- \begin{tabular}{|*{7}{l|}}
+ \begin{tabular}{|*{7}{r|}}
\hline
+ \hspace{-2.2084pt}%
Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
\caption{Running NAS benchmarks on 32 and 36 nodes }
% title of Table
\centering
- \begin{tabular}{|*{7}{l|}}
+ \begin{tabular}{|*{7}{r|}}
\hline
+ \hspace{-2.2084pt}%
Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
\caption{Running NAS benchmarks on 64 nodes }
% title of Table
\centering
- \begin{tabular}{|*{7}{l|}}
+ \begin{tabular}{|*{7}{r|}}
\hline
+ \hspace{-2.2084pt}%
Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
\caption{Running NAS benchmarks on 128 and 144 nodes }
% title of Table
\centering
- \begin{tabular}{|*{7}{l|}}
+ \begin{tabular}{|*{7}{r|}}
\hline
+ \hspace{-2.2084pt}%
Program & Execution & Energy & Energy & Performance & Distance \\
name & time/s & consumption/J & saving\% & degradation\% & \\
\hline
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_4n},
\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
+\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
nodes. On the other hand, more communications are required when increasing the
number of nodes so the static energy increases linearly according to the
communication time and the dynamic power is less relevant in the overall energy
-consumption. Therefore, reducing the frequency with algorithm~(\ref{HSA}) is
+consumption. Therefore, reducing the frequency with Algorithm~\ref{HSA} is
less effective in reducing the overall energy savings. It 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
\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
+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
The NAS parallel benchmarks were executed again over processors that follow the
new power scenarios. The class C of each benchmark was run over 8 or 9 nodes
-and the results are presented in Tables \ref{table:res_s1} and
+and the results are presented in Tables~\ref{table:res_s1} and
\ref{table:res_s2}. These tables show that the energy saving percentage of the
70\%-30\% scenario is smaller for all benchmarks compared to the energy saving
of the 90\%-10\% scenario. Indeed, in the latter more dynamic power is consumed
limit the increase of the execution time and thus limiting the effect of the
consumed static energy.
-Both new power scenarios are compared to the old one in figure
-(\ref{fig:sen_comp}). It shows the average of the performance degradation, the
+Both new power scenarios are compared to the old one in
+Figure~\ref{fig:sen_comp}. It shows the average of the performance degradation, the
energy saving and the distances for all NAS benchmarks of class C running on 8
or 9 nodes. The comparison shows that the energy saving ratio is proportional
to the dynamic power ratio: it is increased when applying the 90\%-10\% scenario
scenario). Since the proposed algorithm optimizes the energy consumption when
using a higher ratio for dynamic power the algorithm selects bigger frequency
scaling factors that result in more energy saving but less performance, for
-example see Figure (\ref{fig:scales_comp}). The opposite happens when using a
+example see Figure~\ref{fig:scales_comp}. The opposite happens when using a
higher ratio for static power, the algorithm proportionally selects smaller
scaling values which result in less energy saving but also less performance
degradation.
\caption{The results of the 70\%-30\% power scenario}
% title of Table
\centering
- \begin{tabular}{|*{6}{l|}}
+ \begin{tabular}{|*{6}{r|}}
\hline
Program & Energy & Energy & Performance & Distance \\
name & consumption/J & saving\% & degradation\% & \\
\caption{The results of the 90\%-10\% power scenario}
% title of Table
\centering
- \begin{tabular}{|*{6}{l|}}
+ \begin{tabular}{|*{6}{r|}}
\hline
Program & Energy & Energy & Performance & Distance \\
name & consumption/J & saving\% & degradation\% & \\
\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}
+\end{table}
\begin{figure}[!t]
\centering
\caption{The comparison of the three power scenarios}
\end{figure}
-
+\begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
+ \caption{Trade-off comparison for NAS benchmarks class C}
+ \label{fig:compare_EDP}
+\end{figure}
\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 green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Energy\cdot Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
-To fairly compare both algorithms, the same energy and execution time models, equations (\ref{eq:energy}) and (\ref{eq:fnew}), were used for both algorithms to predict the energy consumption and the execution times. Also Spiliopoulos et al. algorithm was adapted to start the search from the
-initial frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm is an exhaustive search algorithm that minimizes the EDP and has the initial frequencies values as an upper bound.
-
-Both algorithms were applied to the parallel NAS benchmarks to compare their efficiency. Table \ref{table:compare_EDP} presents the results of comparing the execution times and the energy consumption for both versions of the NAS benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous nodes. The results show that our algorithm provides better energy savings than Spiliopoulos et al. algorithm,
-on average it results in 29.76\% energy saving while their algorithm returns just 25.75\%. The average of performance degradation percentage is approximately the same for both algorithms, about 4\%.
+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
+green governor that regularly applies an online frequency selecting algorithm to
+reduce the energy consumed by a multicore architecture without degrading much
+its performance. The algorithm selects the frequencies that minimize the energy
+and delay products, $\mathit{EDP}=\mathit{energy}\times \mathit{delay}$ using
+the predicted overall energy consumption and execution time delay for each
+frequency. To fairly compare both algorithms, the same energy and execution
+time models, equations (\ref{eq:energy}) and (\ref{eq:fnew}), were used for both
+algorithms to predict the energy consumption and the execution times. Also
+Spiliopoulos et al. algorithm was adapted to start the search from the initial
+frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm
+is an exhaustive search algorithm that minimizes the EDP and has the initial
+frequencies values as an upper bound.
+
+Both algorithms were applied to the parallel NAS benchmarks to compare their
+efficiency. Table~\ref{table:compare_EDP} presents the results of comparing the
+execution times and the energy consumption for both versions of the NAS
+benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
+nodes. The results show that our algorithm provides better energy savings than
+Spiliopoulos et al. algorithm, on average it results in 29.76\% energy saving
+while their algorithm returns just 25.75\%. The average of performance
+degradation percentage is approximately the same for both algorithms, about 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}),
+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.
-
-
-\begin{table}[!t]
- \caption{Comparing the proposed algorithm}
- \centering
-\begin{tabular}{|l|l|l|l|l|l|l|l|}
-\hline
-\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Program \\ name\end{tabular}}} & \multicolumn{2}{l|}{Energy saving \%} & \multicolumn{2}{l|}{Perf. degradation \%} & \multicolumn{2}{l|}{Distance} \\ \cline{3-8}
-\multicolumn{2}{|l|}{} & EDP & MaxDist & EDP & MaxDist & EDP & MaxDist \\ \hline
-\multicolumn{2}{|l|}{CG} & 27.58 & 31.25 & 5.82 & 7.12 & 21.76 & 24.13 \\ \hline
-\multicolumn{2}{|l|}{MG} & 29.49 & 33.78 & 3.74 & 6.41 & 25.75 & 27.37 \\ \hline
-\multicolumn{2}{|l|}{LU} & 19.55 & 28.33 & 0.0 & 0.01 & 19.55 & 28.22 \\ \hline
-\multicolumn{2}{|l|}{EP} & 28.40 & 27.04 & 4.29 & 0.49 & 24.11 & 26.55 \\ \hline
-\multicolumn{2}{|l|}{BT} & 27.68 & 32.32 & 6.45 & 7.87 & 21.23 & 24.43 \\ \hline
-\multicolumn{2}{|l|}{SP} & 20.52 & 24.73 & 5.21 & 2.78 & 15.31 & 21.95 \\ \hline
-\multicolumn{2}{|l|}{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]
- \centering
- \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
- \caption{Trade-off comparison for NAS benchmarks class C}
- \label{fig:compare_EDP}
-\end{figure}
-
-
\section{Conclusion}
\label{sec.concl}
In this paper, a new online frequency selecting algorithm has been presented. It
% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
% LocalWords: CMOS EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex GPU
-% LocalWords: de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
-% LocalWords: Scp Fmax Fdiff SimGrid GFlops Xeon EP BT GPUs CPUs AMD
+% LocalWords: de badri muslim MPI SimGrid GFlops Xeon EP BT GPUs CPUs AMD
% LocalWords: Spiliopoulos scalability
