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-\newcommand{\Dist}{\textit{Dist}}
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+\newcommand{\fxheight}[1]{\ifx#1\relax\relax\else\rule{0pt}{1.52ex}#1\fi}
+
+\newcommand{\CL}{\Xsub{C}{L}}
+\newcommand{\Dist}{\mathit{Dist}}
+\newcommand{\EdNew}{\Xsub{E}{dNew}}
\newcommand{\Eind}{\Xsub{E}{ind}}
\newcommand{\Enorm}{\Xsub{E}{Norm}}
\newcommand{\Eoriginal}{\Xsub{E}{Original}}
\newcommand{\Ereduced}{\Xsub{E}{Reduced}}
-\newcommand{\Fdiff}{\Xsub{F}{diff}}
-\newcommand{\Fmax}{\Xsub{F}{max}}
+\newcommand{\Es}{\Xsub{E}{S}}
+\newcommand{\Fdiff}[1][]{\Xsub{F}{diff}_{\!#1}}
+\newcommand{\Fmax}[1][]{\Xsub{F}{max}_{\fxheight{#1}}}
\newcommand{\Fnew}{\Xsub{F}{new}}
\newcommand{\Ileak}{\Xsub{I}{leak}}
\newcommand{\Kdesign}{\Xsub{K}{design}}
-\newcommand{\MaxDist}{\textit{Max Dist}}
+\newcommand{\MaxDist}{\mathit{Max}\Dist}
+\newcommand{\MinTcm}{\mathit{Min}\Tcm}
\newcommand{\Ntrans}{\Xsub{N}{trans}}
-\newcommand{\Pdyn}{\Xsub{P}{dyn}}
-\newcommand{\PnormInv}{\Xsub{P}{NormInv}}
+\newcommand{\Pd}[1][]{\Xsub{P}{d}_{\fxheight{#1}}}
+\newcommand{\PdNew}{\Xsub{P}{dNew}}
+\newcommand{\PdOld}{\Xsub{P}{dOld}}
\newcommand{\Pnorm}{\Xsub{P}{Norm}}
-\newcommand{\Tnorm}{\Xsub{T}{Norm}}
-\newcommand{\Pstates}{\Xsub{P}{states}}
-\newcommand{\Pstatic}{\Xsub{P}{static}}
-\newcommand{\Sopt}{\Xsub{S}{opt}}
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-\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
-\newcommand{\Tmax}{\Xsub{T}{max}}
+\newcommand{\Ps}[1][]{\Xsub{P}{s}_{\fxheight{#1}}}
+\newcommand{\Scp}[1][]{\Xsub{S}{cp}_{#1}}
+\newcommand{\Sopt}[1][]{\Xsub{S}{opt}_{#1}}
+\newcommand{\Tcm}[1][]{\Xsub{T}{cm}_{\fxheight{#1}}}
+\newcommand{\Tcp}[1][]{\Xsub{T}{cp}_{#1}}
+\newcommand{\TcpOld}[1][]{\Xsub{T}{cpOld}_{#1}}
\newcommand{\Tnew}{\Xsub{T}{New}}
\newcommand{\Told}{\Xsub{T}{Old}}
+
\begin{document}
\title{Energy Consumption Reduction for Message Passing Iterative Applications in Heterogeneous Architecture Using DVFS}
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 frequencies selecting algorithm for heterogeneous
-platforms is presented. It selects the frequency which 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
+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 35\% while limiting the performance degradation as much as
-possible. Finally, the algorithm is compared to an existing method, the
+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.
\end{abstract}
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
+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
+performance degradation percentage to less than \np[\%]{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
+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
+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.
+\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.
-\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.
+\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}
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
+$\Tcm[i]$ 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
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}
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.
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 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
% \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}
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 $F$
+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
+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
-experiments were executed on the simulator SimGrid/SMPI which offers easy tools
+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
+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
-characteristics of these different types of nodes are inspired from the
-specifications of real Intel processors. The heterogeneous platform had up to
+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
+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
a benchmark was executed on 8 nodes, 2 nodes from each type were used. Since the
-constructors of CPUs do not specify the dynamic and the static power of their
-CPUs, for each type of node they were chosen proportionally to its computing
-power (FLOPS). In the initial heterogeneous platform, while computing with
-highest frequency, each node consumed an amount of power proportional to its
-computing power (which corresponds to 80\% of its dynamic power and the
-remaining 20\% to the static power), the same assumption was made in
-\cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}. Finally, These
+constructors of CPUs do not specify the dynamic and the static power of their
+CPUs, for each type of node they were chosen proportionally to its computing
+power (FLOPS). In the initial heterogeneous platform, while computing with
+highest frequency, each node consumed an amount of power proportional to its
+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.
\caption{Heterogeneous nodes characteristics}
% title of Table
\centering
- \begin{tabular}{|*{7}{l|}}
+ \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 & 2.5 & 1.2 & 0.1 & 20~W &4~W \\
+ 1 &40 & 2.50 & 1.20 & 0.100 & \np[W]{20} &\np[W]{4} \\
\hline
- 2 &50 & 2.66 & 1.6 & 0.133 & 25~W &5~W \\
+ 2 &50 & 2.66 & 1.60 & 0.133 & \np[W]{25} &\np[W]{5} \\
\hline
- 3 &60 & 2.9 & 1.2 & 0.1 & 30~W &6~W \\
+ 3 &60 & 2.90 & 1.20 & 0.100 & \np[W]{30} &\np[W]{6} \\
\hline
- 4 &70 & 3.4 & 1.6 & 0.133 & 35~W &7~W \\
+ 4 &70 & 3.40 & 1.60 & 0.133 & \np[W]{35} &\np[W]{7} \\
\hline
\end{tabular}
\end{tabular}
\label{table:res_128n}
\end{table}
-The overall energy consumption was computed for each instance according to the
-energy consumption model (\ref{eq:energy}), with and without applying the
+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},
-\ref{table:res_8n}, \ref{table:res_16n}, \ref{table:res_32n},
+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
-values from many experiments for energy savings and performance degradation.
+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 35\%) 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. 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 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
-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
-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 significantly increased when
-the number of computing nodes is increased because the computation times are
-small when compared to the communication times.
+on different number of nodes. The experiments show that the algorithm
+significantly reduces the energy consumption (up to \np[\%]{35}) 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.
+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
+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 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 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
+significantly increased when the number of computing nodes is increased because
+the computation times are small when compared to the communication times.
\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 80\% composed of dynamic
-power and of 20\% of static power. In this section, these ratios are changed and
-two new power scenarios are considered in order to evaluate how the proposed
-algorithm adapts itself according to the static and dynamic power values. The
-two new power scenarios are the following:
+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
+are changed and two new power scenarios are considered in order to evaluate how
+the proposed algorithm adapts itself according to the static and dynamic power
+values. The two new power scenarios are the following:
\begin{itemize}
-\item 70\% of dynamic power and 30\% of static power
-\item 90\% of dynamic power and 10\% of static power
+\item \np[\%]{70} of dynamic power and \np[\%]{30} of static power
+\item \np[\%]{90} of dynamic power and \np[\%]{10} of static power
\end{itemize}
-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
-\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
-when nodes are running on their maximum frequencies, thus, scaling down the
-frequency of the nodes results in higher energy savings than in the 70\%-30\%
-scenario. On the other hand, the performance degradation percentage is smaller
-in the 70\%-30\% scenario compared to the 90\%-10\% scenario. This is due to the
-higher static power percentage in the first scenario which makes it more
-relevant in the overall consumed energy. Indeed, the static energy is related
-to the execution time and if the performance is degraded the amount of consumed
-static energy directly increases. Therefore, the proposed algorithm does not
-really significantly scale down much the frequencies of the nodes in order to
-limit the increase of the execution time and thus limiting the effect of the
+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
+\ref{table:res_s2}. These tables show that the energy saving percentage of the
+\np[\%]{70}-\np[\%]{30} scenario is smaller for all benchmarks compared to the
+energy saving of the \np[\%]{90}-\np[\%]{10} scenario. Indeed, in the latter
+more dynamic power is consumed when nodes are running on their maximum
+frequencies, thus, scaling down the frequency of the nodes results in higher
+energy savings than in the \np[\%]{70}-\np[\%]{30} scenario. On the other hand,
+the performance degradation percentage is smaller in the \np[\%]{70}-\np[\%]{30}
+scenario compared to the \np[\%]{90}-\np[\%]{10} scenario. This is due to the
+higher static power percentage in the first scenario which makes it more
+relevant in the overall consumed energy. Indeed, the static energy is related
+to the execution time and if the performance is degraded the amount of consumed
+static energy directly increases. Therefore, the proposed algorithm does not
+really significantly scale down much the frequencies of the nodes in order to
+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
-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
-because at maximum frequency the dynamic energy is the most relevant in the
-overall consumed energy and can be reduced by lowering the frequency of some
-processors. On the other hand, the energy saving decreases when the 70\%-30\%
-scenario is used because the dynamic energy is less relevant in the overall
-consumed energy and lowering the frequency does not return big energy savings.
-Moreover, the average of the performance degradation is decreased when using a
-higher ratio for static power (e.g. 70\%-30\% scenario and 80\%-20\%
-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
-higher ratio for static power, the algorithm proportionally selects smaller
-scaling values which result in less energy saving but also less performance
+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
+\np[\%]{90}-\np[\%]{10} scenario because at maximum frequency the dynamic energy
+is the most relevant in the overall consumed energy and can be reduced by
+lowering the frequency of some processors. On the other hand, the energy saving
+decreases when the \np[\%]{70}-\np[\%]{30} scenario is used because the dynamic
+energy is less relevant in the overall consumed energy and lowering the
+frequency does not return big energy savings. Moreover, the average of the
+performance degradation is decreased when using a higher ratio for static power
+(e.g. \np[\%]{70}-\np[\%]{30} scenario and \np[\%]{80}-\np[\%]{20}
+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
+higher ratio for static power, the algorithm proportionally selects smaller
+scaling values which result in less energy saving but also less performance
degradation.
\begin{table}[!t]
- \caption{The results of the 70\%-30\% power scenario}
+ \caption{The results of the \np[\%]{70}-\np[\%]{30} power scenario}
% title of Table
\centering
\begin{tabular}{|*{6}{r|}}
\begin{table}[!t]
- \caption{The results of the 90\%-10\% power scenario}
+ \caption{The results of the \np[\%]{90}-\np[\%]{10} power scenario}
% title of Table
\centering
\begin{tabular}{|*{6}{r|}}
\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.
+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\%.
+Spiliopoulos et al. algorithm, on average it results in \np[\%]{29.76} energy
+saving while their algorithm returns just \np[\%]{25.75}. The average of
+performance degradation percentage is approximately the same for both
+algorithms, about \np[\%]{4}.
For all benchmarks, our algorithm outperforms Spiliopoulos et al. algorithm in
degradation values while giving the same weight for both metrics.
-
-
-\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
- \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
-selects the best possible vector of frequency scaling factors that gives the
-maximum distance (optimal trade-off) between the predicted energy and the
+selects the best possible vector of frequency scaling factors that gives the
+maximum distance (optimal trade-off) between the predicted energy and the
predicted performance curves for a heterogeneous platform. This algorithm uses a
-new energy model for measuring and predicting the energy of distributed
-iterative applications running over heterogeneous platforms. To evaluate the
+new energy model for measuring and predicting the energy of distributed
+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 35\% 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 values of the static
-and dynamic powers of the CPUs. Finally, the algorithm was compared to
-Spiliopoulos et al. algorithm and the results showed that it outperforms their
-algorithm in terms of energy-time trade-off.
+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
+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
+values of the static and dynamic powers of the CPUs. Finally, the algorithm was
+compared to Spiliopoulos et al. algorithm and the results showed that it
+outperforms their algorithm in terms of energy-time trade-off.
In the near future, this method will be applied to real heterogeneous platforms
to evaluate its performance in a real study case. It would also be interesting
% 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
