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\begin{document}
-\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
+\title{Energy Consumption Reduction with DVFS for \\
+ Message Passing Iterative Applications on \\
+ Heterogeneous Architectures}
\author{%
\IEEEauthorblockN{%
Arnaud Giersch
}
\IEEEauthorblockA{%
- FEMTO-ST Institute\\
- University of Franche-Comté\\
+ FEMTO-ST Institute, University of Franche-Comté\\
IUT de Belfort-Montbéliard,
19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
% Telephone: \mbox{+33 3 84 58 77 86}, % Raphaël
\begin{abstract}
+
\end{abstract}
\section{Introduction}
\label{sec.intro}
+
\section{Related works}
\label{sec.relwork}
-
-
-
\section{The performance and energy consumption measurements on heterogeneous architecture}
\label{sec.exe}
-% \JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'',
-% can be deleted if we need space, we can just say we are interested in this
-% paper in homogeneous clusters}
+\subsection{The execution time of message passing distributed iterative
+ applications on a heterogeneous platform}
-\subsection{The execution time of message passing distributed iterative applications on a heterogeneous platform}
-In this paper, we are interested in reducing the energy consumption of message passing distributed iterative synchronous applications running over heterogeneous platforms. We define a heterogeneous platform as a collection of heterogeneous computing nodes interconnected via a high speed homogeneous network. Therefore, each node has different characteristics such as computing power (FLOPS), energy consumption, cpu's frequency range, ... but they all have the same network bandwidth and latency.
+In this paper, we are interested in reducing the energy consumption of message
+passing distributed iterative synchronous applications running over
+heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
+heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth
+and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
-
-\begin{figure}[t]
+\begin{figure}[!t]
\centering
- \includegraphics[scale=0.6]{fig/commtasks}
+ \includegraphics[scale=0.6]{fig/commtasks}
\caption{Parallel tasks on a heterogeneous platform}
\label{fig:heter}
\end{figure}
- The overall execution time of a distributed iterative synchronous application over a heterogeneous platform consists of the sum of the computation time and the communication time for every iteration on a node. However, due to the heterogeneous computation power of the computing nodes, slack times might occur 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 have the highest computation time and no slack time.
-
-Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in modern processors, that reduces the energy consumption
-of a CPU by scaling down its voltage and frequency. Since DVFS lowers the frequency of a CPU and consequently its computing power, the execution time of a program running over that scaled down processor might increase, especially if the program is compute bound. The frequency reduction process can be expressed by the scaling factor S which is the ratio between the maximum and the new frequency of a CPU as in EQ (\ref{eq:s}).
+The overall execution time of a distributed iterative synchronous application
+over a heterogeneous grid 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 clusters, slack times may occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:heter}). Therefore, the
+overall execution time of the program is the execution time of the slowest task
+which has the highest computation time and no slack time.
+
+Dynamic Voltage and Frequency Scaling (DVFS) is a process, implemented in
+modern processors, that reduces the energy consumption of a CPU by scaling
+down its voltage and frequency. Since DVFS lowers the frequency of a CPU
+and consequently its computing power, the execution time of a program running
+over that scaled down processor may increase, especially if the program is
+compute bound. The frequency reduction process can be expressed by the scaling
+factor S which is the ratio between the maximum and the new frequency of a CPU
+as in (\ref{eq:s}).
\begin{equation}
\label{eq:s}
- S = \frac{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 distributed applications consist of two parts: computation and communication. The execution time of the computation part is linearly proportional to the frequency scaling factor $S$ but the communication time is not affected by the scaling factor because the processors involved remain idle during the communications~\cite{17}. The communication time for a task is the summation of periods of time that begin with an MPI call for sending or receiving a message till the message is synchronously sent or received.
-
-Since in a heterogeneous platform, each node has different characteristics, especially different frequency gears, when applying DVFS operations on these nodes, they may get different scaling factors represented by a scaling vector: $(S_1, S_2,..., S_N)$ where $S_i$ is the scaling factor of processor $i$. To be able to predict the execution time of message passing synchronous iterative applications running over a heterogeneous platform, for different vectors of scaling factors, the communication time and the computation time for all the
- tasks must be measured during the first iteration before applying any DVFS operation. Then the execution time for one iteration of the application with any vector of scaling factors can be predicted using EQ (\ref{eq:perf}).
-
-
-
-\begin{multline}
+The execution time of a compute bound sequential program is linearly
+proportional to the frequency scaling factor $S$. On the other hand, message
+passing distributed applications consist of two parts: computation and
+communication. The execution time of the computation part is linearly
+proportional to the frequency scaling factor $S$ but the communication time is
+not affected by the scaling factor because the processors involved remain idle
+during the communications~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}. The
+communication time for a task is the summation of periods of time that begin
+with an MPI call for sending or receiving a message until the message is
+synchronously sent or received.
+
+Since in a heterogeneous grid each cluster has different characteristics,
+especially different frequency gears, when applying DVFS operations on the nodes
+of these clusters, they may get different scaling factors represented by a scaling vector:
+$(S_{11}, S_{12},\dots, S_{NM})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
+be able to predict the execution time of message passing synchronous iterative
+applications running over a heterogeneous grid, for different vectors of
+scaling factors, the communication time and the computation time for all the
+tasks must be measured during the first iteration before applying any DVFS
+operation. Then the execution time for one iteration of the application with any
+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}) + TcmOld_{j}
-\end{multline}
-where $TcpOld_i$ is the computation time of processor $i$ during the first iteration and $TcmOld_j$ is the communication time of the slowest processor $j$.
- 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.
+ \Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\TcpOld[ij]} \cdot S_{ij})
+ +\mathop{\min_{j=1,\dots,M}} (\Tcm[hj])
+\end{equation}
+
+where $N$ is the number of clusters in the grid, $M$ is the number of nodes in
+each cluster, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$
+and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during 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 in the slowest cluster $h$.
+It means only the communication time without any slack time is taken into account.
+Therefore, the execution time of the iterative application is equal to
+the execution time of one iteration as in (\ref{eq:perf}) multiplied by the
+number of iterations of that application.
-This prediction model is based on our model for predicting the execution time of message passing distributed applications for homogeneous architectures~\cite{45}. The execution time prediction model is used in our method for optimizing both energy consumption and performance of iterative methods, which is presented in the following sections.
+This prediction model is developed from the model to predict the execution time
+of message passing distributed applications for homogeneous and heterogeneous clusters
+~\cite{Our_first_paper,pdsec2015}. The execution time prediction model is
+used in the method to optimize both the energy consumption and the performance
+of iterative methods, which is presented in the following sections.
\subsection{Energy model for heterogeneous platform}
-Many researchers~\cite{9,3,15,26} 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 $P_{d}$ is related to the switching
-activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
-operational frequency $F$, as shown in EQ(\ref{eq:pd}).
+Many researchers~\cite{Malkowski_energy.efficient.high.performance.computing,
+ Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling,
+ Rizvandi_Some.Observations.on.Optimal.Frequency} divide the power consumed by
+a processor into two power metrics: the static and the dynamic power. While the
+first one is consumed as long as the computing unit is turned on, the latter is
+only consumed during computation times. The dynamic power $\Pd$ is related to
+the switching activity $\alpha$, load capacitance $\CL$, the supply voltage $V$
+and operational frequency $F$, as shown in (\ref{eq:pd}).
\begin{equation}
\label{eq:pd}
- P_\textit{d} = \alpha \cdot C_L \cdot V^2 \cdot F
+ \Pd = \alpha \cdot \CL \cdot V^2 \cdot F
\end{equation}
-The static power $P_{s}$ captures the leakage power as follows:
+The static power $\Ps$ captures the leakage power as follows:
\begin{equation}
\label{eq:ps}
- P_\textit{s} = 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
-technology-dependent parameter. The energy consumed by an individual processor
+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} = P_\textit{d} \cdot T_{cp} + P_\textit{s} \cdot T
+ \Eind = \Pd \cdot \Tcp + \Ps \cdot T
\end{equation}
-where $T$ is the execution time of the program, $T_{cp}$ is the computation
-time and $T_{cp} \leq T$. $T_{cp}$ 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{37}. The operational frequency $F$
-depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot F$ with some
-constant $\beta$. This equation is used to study the change of the dynamic
-voltage with respect to various frequency values in~\cite{3}. The reduction
-process of the frequency can be expressed by the scaling factor $S$ which is the
-ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
-The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. we can calculate the new frequency
-$F_{new}$ from EQ(\ref{eq:s}) as follow:
+The main objective of DVFS operation is to reduce the overall energy
+consumption~\cite{Le_DVFS.Laws.of.Diminishing.Returns}. The operational
+frequency $F$ depends linearly on the supply voltage $V$, i.e., $V = \beta \cdot
+F$ with some constant $\beta$.~This equation is used to study the change of the
+dynamic voltage with respect to various frequency values
+in~\cite{Rauber_Analytical.Modeling.for.Energy}. The reduction process of the
+frequency can be expressed by the scaling factor $S$ which is the ratio between
+the maximum and the new frequency as in (\ref{eq:s}). The CPU governors are
+power schemes supplied by the operating system's kernel to lower a core's
+frequency. The new frequency $\Fnew$ from (\ref{eq:s}) can be calculated as
+follows:
\begin{equation}
\label{eq:fnew}
- F_\textit{new} = S^{-1} . F_\textit{max}
+ \Fnew = S^{-1} \cdot \Fmax
\end{equation}
-Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\ref{eq:fnew}) gives the following equation for dynamic
-power consumption:
+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 \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 new frequency and the maximum frequency respectively.
-
-According to EQ(\ref{eq:pdnew}) the dynamic power is reduced by a factor of $S^{-3}$ when
-reducing the frequency by a factor of $S$~\cite{3}. Since the FLOPS of a CPU is proportional to the frequency of a CPU, the computation time is increased proportionally to $S$. The new dynamic energy is the dynamic power multiplied by the new time of computation and is given by the following equation:
+where $\PdNew$ and $\PdOld$ are the dynamic power consumed with the
+new frequency and the maximum frequency respectively.
+
+According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
+$S^{-3}$ when reducing the frequency by a factor of
+$S$~\cite{Rauber_Analytical.Modeling.for.Energy}. Since the FLOPS of a CPU is
+proportional to the frequency of a CPU, the computation time is increased
+proportionally to $S$. The new dynamic energy is the dynamic power multiplied
+by the new time of computation and is given by the following equation:
\begin{equation}
\label{eq:Edyn}
- E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (T_{cp} \cdot S)= S^{-2}\cdot P_{dOld} \cdot T_{cp}
+ \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{3,46}, we assume that the static power of a processor is constant during idle and computation periods, and for all its available frequencies.
-The static energy is the static power multiplied by the execution time of the program. According to the execution time model in EQ(\ref{eq:perf}),
-the execution time of the program is the summation of the computation and the communication times. The computation time is linearly related
-to the frequency scaling factor, while this scaling factor does not affect the communication time. The static energy
-of a processor after scaling its frequency is computed as follows:
-
+The static power is related to the power leakage of the CPU and is consumed
+during computation and even when idle. As
+in~\cite{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
+the static power of a processor is considered as constant during idle and
+computation periods, and for all its available frequencies. The static energy
+is the static power multiplied by the execution time of the program. According
+to the execution time model in (\ref{eq:perf}), the execution time of the
+program is the sum of the computation and the communication times. The
+computation time is linearly related to the frequency scaling factor, while this
+scaling factor does not affect the communication time. The static energy of a
+processor after scaling its frequency is computed as follows:
\begin{equation}
\label{eq:Estatic}
- E_\textit{s} = P_\textit{s} \cdot (T_{cp} \cdot S + T_{cm})
+ \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 $P_{di}$ and $P_{si}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $T_{cpi}$ might be different and different frequency scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $T_{cmi}$ and could contain slack times if it is 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 EQ(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time of each processor multiplied by its static power. The overall energy consumption of a message passing distributed application executed over a heterogeneous platform is the summation of all dynamic and static energies for each processor. It is computed as follows:
+In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
+different dynamic and static powers from the nodes of the other clusters,
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively. Therefore, even if the distributed
+message passing iterative application is load balanced, the computation time of each CPU $j$
+in cluster $i$ noted $\Tcp[ij]$ may be different and different frequency scaling factors may be
+computed in order to decrease the overall energy consumption of the application
+and reduce slack times. The communication time of a processor $j$ in cluster $i$ is noted as
+$\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
+see Figure~\ref{fig:heter}. Therefore, all nodes do not have equal
+communication times. While the dynamic energy is computed according to the
+frequency scaling factor and the dynamic power of each node as in
+(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
+of one iteration multiplied by the static power of each processor. The overall
+energy consumption of a message passing distributed application executed over a
+heterogeneous grid platform during one iteration is the summation of all dynamic and
+static energies for $M$ processors in $N$ clusters. It is computed as follows:
\begin{multline}
\label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot P_{di} \cdot T_{cpi})} +\\
- {}\sum_{i=1}^{N} {(P_{si} \cdot (\max_{i=1,2,\dots,N} (T_{cpi} \cdot S_{i}) +}
- {}\min_{i=1,2,\dots,N} {T_{cmi}))}
- \end{multline}
-
-Reducing the the frequencies of the processors according to the vector of scaling factors $(S_1, S_2,..., S_N)$ may degrade the performance of the application and thus,
-increase the static energy because the execution time is increased~\cite{36}.
+ E = \sum_{i=1}^{N} \sum_{i=1}^{M} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} +
+ \sum_{i=1}^{N} \sum_{j=1}^{M} (\Ps[ij] \cdot {} \\
+ (\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}({\Tcp[ij]} \cdot S_{ij})
+ +\mathop{\min_{j=1,\dots M}} (\Tcm[hj]) ))
+\end{multline}
+
+Reducing the frequencies of the processors according to the vector of scaling
+factors $(S_{11}, S_{12},\dots, S_{NM})$ may degrade the performance of the application
+and thus, increase the static energy because the execution time is
+increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption
+for the iterative application can be measured by measuring the energy
+consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
+of iterations of that application.
\section{Optimization of both energy consumption and performance}
\label{sec.compet}
-Applying DVFS to lower level not surly reducing the energy consumption to minimum level. Also, a big scaling for the frequency produces high performance degradation percent. Moreover, by considering the drastically increase in execution time of parallel program, the static energy is related to this time
-and it also increased by the same ratio. Thus, the opportunity for gaining more energy reduction is restricted. For that choosing frequency scaling factors is very important process to taking into account both energy and performance. In our previous work~\cite{45}, we are proposed a method that selects the optimal frequency scaling factor for an homogeneous cluster, depending on the trade-off relation between the energy and performance. In this work we have an heterogeneous cluster, at each node there is different scaling factors, so our goal is to selects the optimal set of frequency scaling factors, $Sopt_1,Sopt_2,...,Sopt_N$, that gives the best trade-off between energy consumption and performance. The relation between the energy and the execution time is complex and nonlinear, Thus, unlike the relation between the performance and the scaling factor, the relation of the energy with the frequency scaling factors is nonlinear, for more details refer to~\cite{17}. Moreover, they are not measured using the same metric. To solve this problem, we normalize the execution time by calculating the ratio between the new execution time (the scaled execution time) and the old one as follow:
-\begin{multline}
+
+Using the lowest frequency for each processor does not necessarily give the most
+energy efficient execution of an application. Indeed, even though the dynamic
+power is reduced while scaling down the frequency of a processor, its
+computation power is proportionally decreased. Hence, the execution time might
+be drastically increased and during that time, dynamic and static powers are
+being consumed. Therefore, it might cancel any gains achieved by scaling down
+the frequency of all nodes to the minimum and the overall energy consumption of
+the application might not be the optimal one. It is not trivial to select the
+appropriate frequency scaling factor for each processor while considering the
+characteristics of each processor (computation power, range of frequencies,
+dynamic and static powers) and the task executed (computation/communication
+ratio). The aim being to reduce the overall energy consumption and to avoid
+increasing significantly the execution time. In our previous
+work~\cite{Our_first_paper,pdsec2015}, we proposed a method that selects the optimal
+frequency scaling factor for a homogeneous and heterogeneous clusters executing a message passing
+iterative synchronous application while giving the best trade-off between the
+energy consumption and the performance for such applications. In this work we
+are interested in heterogeneous grid as described above. Due to the
+heterogeneity of the processors, a vector of scaling factors should be selected
+and it must give the best trade-off between energy consumption and performance.
+
+The relation between the energy consumption and the execution time for an
+application is complex and nonlinear, Thus, unlike the relation between the
+execution time and the scaling factor, the relation between the energy and the
+frequency scaling factors is nonlinear, for more details refer
+to~\cite{Freeh_Exploring.the.Energy.Time.Tradeoff}. Moreover, these relations
+are not measured using the same metric. To solve this problem, the execution
+time is normalized by computing the ratio between the new execution time (after
+scaling down the frequencies of some processors) and the initial one (with
+maximum frequency for all nodes) as follows:
+\begin{equation}
\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}) +\min_{i=1,2,\dots,N} {Tcm_{i}}}
- {\max_{i=1,2,\dots,N}{(Tcp_i+Tcm_i)}}
-\end{multline}
+ \Pnorm = \frac{\Tnew}{\Told}
+\end{equation}
-By the same way, we are normalize the energy by calculating the ratio between the consumed energy with scaled frequency and the consumed energy without scaled frequency:
-\begin{multline}
+Where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told})
+\begin{equation}
+ \label{eq:told}
+ \Told = \mathop{\max_{i=1,2,\dots,N}}_{j=1,2,\dots,M} (\Tcp[ij]+\Tcm[ij])
+\end{equation}
+In the same way, the energy is normalized by computing the ratio between the
+consumed energy while scaling down the frequency and the consumed energy with
+maximum frequency for all nodes:
+\begin{equation}
\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})}}
-\end{multline}
-Where $T_{New}$ and $T_{Old}$ is computed as in EQ(\ref{eq:pnorm}). The second problem
-is that the optimization operation for both energy and performance is not in the same direction.
-In other words, the normalized energy and the normalized execution time curves are not at the same direction.
-While the main goal is to optimize the energy and execution time in the same time. According to the
-equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the set of frequency scaling factors $S_1,S_2,...,S_N$ reduce both the energy and the
-execution time simultaneously. But the main objective is to produce maximum energy
-reduction with minimum execution time reduction. Many researchers used different
-strategies to solve this nonlinear problem for example see~\cite{19,42}, their
-methods add big overheads to the algorithm to select the suitable frequency.
-In this paper we are present a method to find the optimal set of frequency scaling factors to optimize both energy and execution time simultaneously
-without adding a big overhead. Our solution for this problem is to make the optimization process
-for energy and execution time follow the same direction. Therefore, we inverse the equation of the normalized
-execution time, the normalized performance, as follows:
+ \Enorm = \frac{\Ereduced}{\Eoriginal}
+\end{equation}
-\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}) +\min_{i=1,2,\dots,N} {Tcm_{i}}}
-\end{multline}
+Where $\Ereduced$ is computed using (\ref{eq:energy}) and $\Eoriginal$ is
+computed as in ().
+\textcolor{red}{A reference is missing}
+\begin{equation}
+ \label{eq:eorginal}
+ \Eoriginal = \sum_{i=1}^{N} \sum_{j=1}^{M} ( \Pd[ij] \cdot \Tcp[ij]) +
+ \mathop{\sum_{i=1}^{N}} \sum_{j=1}^{M} (\Ps[ij] \cdot \Told)
+\end{equation}
-\begin{figure}
+While the main goal is to optimize the energy and execution time at the same
+time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way.
+According to the equations~(\ref{eq:pnorm}) and (\ref{eq:enorm}), the
+vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduce both the energy
+and the execution time simultaneously. But the main objective is to produce
+maximum energy reduction with minimum execution time reduction.
+
+This problem can be solved by making the optimization process for energy and
+execution time follow the same evolution according to the vector of scaling factors
+$(S_{11}, S_{12},\dots, S_{NM})$. Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance
+equation, as follows:
+\begin{equation}
+ \label{eq:pnorm_inv}
+ \Pnorm = \frac{\Told}{\Tnew}
+\end{equation}
+
+\begin{figure}[!t]
\centering
- \subfloat[Homogeneous platform]{%
- \includegraphics[width=.22\textwidth]{fig/homo}\label{fig:r1}}%
- \qquad%
- \subfloat[Heterogeneous platform]{%
- \includegraphics[width=.22\textwidth]{fig/heter}\label{fig:r2}}
+ \subfloat[Homogeneous cluster]{%
+ \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
+
+ \subfloat[Heterogeneous grid]{%
+ \includegraphics[width=.33\textwidth]{fig/heter}\label{fig:r2}}
\label{fig:rel}
\caption{The energy and performance relation}
\end{figure}
-Then, we can model our objective function as finding the maximum distance
-between the energy curve EQ~(\ref{eq:enorm}) and the performance
-curve EQ~(\ref{eq:pnorm_inv}) over all available sets of scaling factors. This
-represents the minimum energy consumption with minimum execution time (better
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}) . Then our objective
-function has the following form:
-\begin{multline}
+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}} )
-\end{multline}
-where $N$ is the number of nodes and $F$ is the number of available frequencies for each nodes.
-Then we can select the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}). Our objective function can
-work with any energy model or energy values stored in a data file.
-Moreover, this function works in optimal way when the energy curve has a convex
-form over the available frequency scaling factors as shown in~\cite{15,3,19}.
-
-\section{The heterogeneous scaling algorithm }
+ \MaxDist =
+\mathop{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}}_{k=1,\dots,F}
+ (\overbrace{\Pnorm(S_{ijk})}^{\text{Maximize}} -
+ \overbrace{\Enorm(S_{ijk})}^{\text{Minimize}} )
+\end{equation}
+where $N$ is the number of clusters, $M$ is the number of nodes in each cluster and
+$F$ is the number of available frequencies for each node. Then, the optimal set
+of scaling factors that satisfies (\ref{eq:max}) can be selected.
+The objective function can work with any energy model or any power
+values for each node (static and dynamic powers). However, the most important
+energy reduction gain can be achieved when the energy curve has a convex form as shown
+in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
+
+\section{The scaling factors selection algorithm for grids }
\label{sec.optim}
-In this section we proposed an heterogeneous scaling algorithm, (figure~\ref{HSA}), that selects the optimal set of scaling factors from each node.
-The algorithm is numerates the suitable range of available scaling factors for each node in the heterogeneous cluster, returns a set of optimal frequency scaling factors for each node. Using heterogeneous cluster is produces different workloads for each node. Therefore, the fastest nodes waiting at the barrier for the slowest nodes to finish there work as in figure (\ref{fig:heter}). Our algorithm takes into account these imbalanced workloads when is starts to search for selecting the best scaling factors. So, the algorithm is selecting the initial frequencies values for each node proportional to the times of computations that gathered from the first iteration. As an example in figure (\ref{fig:st_freq}), the algorithm don't test the first frequencies of the fastest nodes until it converge their frequencies to the frequency of the slowest node. If the algorithm is starts test changing the frequency of the slowest nodes from beginning, we are loosing performance and then not selecting the best tradeoff (the distance). This case will be similar to the homogeneous cluster when all nodes scales their frequencies together from the beginning. In this case there is a small distance between energy and performance curves, for example see the figure(\ref{fig:r1}). Then the algorithm searching for optimal frequency scaling factor from the selected frequencies until the last available ones.
-\begin{figure}[t]
- \centering
- \includegraphics[scale=0.5]{fig/start_freq.pdf}
- \caption{Selecting the initial frequencies}
- \label{fig:st_freq}
-\end{figure}
-
-To compute the initial frequencies, the algorithm firstly needs to compute the computation scaling factors $Scp_i$ for each node. Each one of these factors represent a ratio between the computation time of the slowest node and the computation time of the node $i$ as follow:
-\begin{equation}
- \label{eq:Scp}
- Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
-\end{equation}
-Depending on the initial computation scaling factors EQ(\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 follow:
-\begin{equation}
- \label{eq:Fint}
- F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
-\end{equation}
-\begin{figure}[tp]
+\begin{algorithm}
\begin{algorithmic}[1]
% \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 [{$N$}] number of clusters in the grid.
+ \item [{$M$}] number of nodes in each cluster.
+ \item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
+ \item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
+ \item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
+ \item[{$\Pd[ij]$}] array of the dynamic powers for all nodes.
+ \item[{$\Ps[ij]$}] array of the static powers for all nodes.
+ \item[{$\Fdiff[ij]$}] array of the differences between two successive frequencies for all nodes.
\end{description}
- \Ensure $Sopt_1, \dots ,Sopt_N$ is a set of optimal scaling factors
+ \Ensure $\Sopt[11],\Sopt[12] \dots, \Sopt[NM_i]$, a vector of scaling factors that gives the optimal tradeoff between energy consumption and execution time
- \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.
+ \State $\Scp[ij] \gets \frac{\max_{i=1,2,\dots,N}(\max_{j=1,2,\dots,M_i}(\Tcp[ij]))}{\Tcp[ij]} $
+ \State $F_{ij} \gets \frac{\Fmax[ij]}{\Scp[i]},~{i=1,2,\cdots,N},~{j=1,2,\dots,M_i}.$
+ \State Round the computed initial frequencies $F_i$ to the closest available frequency for each node.
\If{(not the first frequency)}
- \State $F_i \gets F_i+Fdiff_i,~i=1,...,N.$
- \EndIf
- \State $T_\textit{Old} \gets max_{~i=1,...,N } (Tcp_i+Tcm_i)$
- \State $E_\textit{Original} \gets \sum_{i=1}^{N}{( Pd_i \cdot Tcp_i)} +\sum_{i=1}^{N} {(Ps_i \cdot T_{Old})}$
- \State $Dist \gets 0$
- \State $Sopt_{i} \gets 1,~i=1,...,N. $
- \While {(all nodes not reach their minimum frequency)}
+ \State $F_{ij} \gets F_{ij}+\Fdiff[ij],~i=1,\dots,N,~{j=1,\dots,M_i}.$
+ \EndIf
+ \State $\Told \gets $ computed as in equations (\ref{eq:told}).
+ \State $\Eoriginal \gets $ computed as in equations (\ref{eq:eorginal}) .
+ \State $\Sopt[ij] \gets 1,~i=1,\dots,N,~{j=1,\dots,M_i}. $
+ \State $\Dist \gets 0 $
+ \While {(all nodes have not reached their minimum \newline\hspace*{2.5em} frequency \textbf{or} $\Pnorm - \Enorm < 0 $)}
\If{(not the last freq. \textbf{and} not the slowest node)}
- \State $F_i \gets F_i - Fdiff_i,~i=1,...,N.$
- \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,...,N.$
+ \State $F_{ij} \gets F_{ij} - \Fdiff[ij],~{i=1,\dots,N},~{j=1,\dots,M_i}$.
+ \State $S_{ij} \gets \frac{\Fmax[ij]}{F_{ij}},~{i=1,\dots,N},~{j=1,\dots,M_i}.$
\EndIf
- \State $T_{New} \gets max_\textit{~i=1,\dots,N} (Tcp_{i} \cdot S_{i}) + min_\textit{~i=1,\dots,N}(Tcm_i) $
- \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 $ computed as in equations (\ref{eq:perf}).
+ \State $\Ereduced \gets $ computed as in equations (\ref{eq:energy}).
+ \State $\Pnorm \gets \frac{\Told}{\Tnew}$
+ \State $\Enorm\gets \frac{\Ereduced}{\Eoriginal}$
\If{$(\Pnorm - \Enorm > \Dist)$}
- \State $Sopt_{i} \gets S_{i},~i=1,...,N. $
+ \State $\Sopt[ij] \gets S_{ij},~i=1,\dots,N,~j=1,\dots,M_i. $
\State $\Dist \gets \Pnorm - \Enorm$
\EndIf
\EndWhile
- \State Return $Sopt_1,Sopt_2, \dots ,Sopt_N$
+ \State Return $\Sopt[11],\Sopt[12],\dots,\Sopt[NM_i]$
\end{algorithmic}
- \caption{Heterogeneous scaling algorithm}
+ \caption{Scaling factors selection algorithm}
\label{HSA}
-\end{figure}
-When the initial frequencies are computed the algorithm numerates all available scaling factors starting from these frequencies until all nodes reach their
-minimum frequencies. At each iteration the algorithm remains the frequency of the slowest node without change and scaling the frequency of the other nodes. This is gives better performance and energy tradeoff.
-The proposed algorithm works online during the execution time of the MPI
-program. Its returns a set of optimal frequency scaling factors $Sopt_i$ depending on the objective function EQ(\ref{eq:max}). The program changes the new frequencies of the CPUs according to the computed scaling factors. This algorithm has a small execution time:
-for an heterogeneous cluster composed of four different types of nodes having the characteristics presented in
-table~(\ref{table:platform}), it takes \np[ms]{0.04} on average for 4 nodes and
-\np[ms]{0.1} on average for 128 nodes. 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 on average from 12 to 20 iterations for all the NAS benchmark on class C to selects the best set of frequency scaling factors. Its called just once during the execution of the program. The DVFS figure~(\ref{dvfs}) shows where and when the algorithm is
-called in the MPI program.
-\begin{figure}[tp]
+\end{algorithm}
+
+\begin{algorithm}
\begin{algorithmic}[1]
% \footnotesize
\For {$k=1$ to \textit{some iterations}}
\If {$(k=1)$}
\State Gather all times of computation and\newline\hspace*{3em}%
communication from each node.
- \State Call algorithm from Figure~\ref{HSA} with these times.
+ \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.
\end{algorithmic}
\caption{DVFS algorithm}
\label{dvfs}
+\end{algorithm}
+
+\subsection{The algorithm details}
+
+\textcolor{red}{Delete the subsection if there's only one.}
+
+In this section, the scaling factors selection algorithm for grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency
+scaling factors 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 grid. It works
+online during the execution time of the iterative message passing program. It
+uses information gathered during the first iteration such as the computation
+time and the communication time in one iteration for each node. The algorithm is
+executed after the first iteration and returns a vector of optimal frequency
+scaling factors that satisfies the objective function (\ref{eq:max}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors. This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
+scaling algorithm is called in the iterative MPI program.
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.45]{fig/init_freq}
+ \caption{Selecting the initial frequencies}
+ \label{fig:st_freq}
\end{figure}
+Nodes from distinct clusters in a grid 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 vector of the frequency
+scaling factors. 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[ij] = \frac{ \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M}(\Tcp[ij])} {\Tcp[ij]}
+\end{equation}
+Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
+algorithm computes the initial frequencies for all nodes as a ratio between the
+maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+follows:
+\begin{equation}
+ \label{eq:Fint}
+ F_{ij} = \frac{\Fmax[ij]}{\Scp[ij]},~{i=1,2,\dots,N},~{j=1,\dots,M}
+\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 powers in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors. The resulting new
+frequencies are highlighted in Figure~\ref{fig:st_freq}. This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting higher frequencies
+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 until reaching the nodes' minimum frequencies or lower bounds. A node's frequency is considered its lower bound if the computed distance between the energy and performance at this frequency is less than zero.
+A negative distance means that the performance degradation ratio is higher than the energy saving ratio.
+In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances.
+
+Therefore, the algorithm iterates on all remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall
+energy consumption and performance and selects the optimal vector of the frequency scaling
+factors. At each iteration the algorithm determines the slowest node
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+while it lowers the frequency of all other nodes by one gear. The new overall
+energy consumption and execution time are computed according to the new scaling
+factors. The optimal set of frequency scaling factors is the set that gives the
+highest distance according to the objective function (\ref{eq:max}).
+
+Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
+consumed energy for an application running on a homogeneous cluster and a
+ grid platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous cluster the search for the optimal scaling
+factor should start from the maximum frequency because the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand, in
+the grid platform the performance is maintained at the beginning of the
+plot even if the frequencies of the faster nodes decrease until the computing
+power of scaled down nodes are lower than the slowest node. In other words,
+until they reach the higher bound. It can also be noticed that the higher the
+difference between the faster nodes and the slower nodes is, the bigger the
+maximum distance between the energy curve and the performance curve is, which results in bigger energy savings.
+
+
\section{Experimental results}
\label{sec.expe}
-The experiments of this work are executed on the simulator Simgrid/SMPI v3.10. We configure the simulator to use a heterogeneous cluster
-with one core per node. The proposed heterogeneous cluster has four different types of nodes. Each node in cluster has different characteristics
-such as the maximum frequency speed, the number of available frequencies and dynamic and static powers values, see table (\ref{table:platform}). These different types of processing nodes simulate some real Intel processors. The maximum number of nodes that supported by the cluster is 144 nodes according to characteristics of some MPI programs of the NAS benchmarks that used. We are use the same number from each type of nodes when running the MPI programs, for example if we execute the program on 8 node, there are 2 nodes from each type participating in the computing. The dynamic and static power values is different from one type to other. Each node has a dynamic and static power values proportional to their performance/GFlops, for more details see the Intel data sheets in \cite{47}. Each node has a percentage of 80\% for dynamic power and 20\% for static power from the hole power consumption, the same assumption is made in \cite{45,3}. These nodes are connected via an ethernet network with 1 Gbit/s bandwidth.
-\begin{table}[htb]
- \caption{Heterogeneous nodes characteristics}
- % title of Table
- \centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Node & Similar & Max & Min & Diff. & Dynamic & Static \\
- type & to & Freq. GHz & Freq. GHz & Freq GHz & power & power \\
- \hline
- 1 & core-i3 & 2.5 & 1.2 & 0.1 & 20~w &4~w \\
- & 2100T & & & & & \\
- \hline
- 2 & Xeon & 2.66 & 1.6 & 0.133 & 25~w &5~w \\
- & 7542 & & & & & \\
- \hline
- 3 & core-i5 & 2.9 & 1.2 & 0.1 & 30~w &6~w \\
- & 3470s & & & & & \\
- \hline
- 4 & core-i7 & 3.4 & 1.6 & 0.133 & 35~w &7~w \\
- & 2600s & & & & & \\
- \hline
- \end{tabular}
- \label{table:platform}
-\end{table}
+While in~\cite{mpi-energy2} the energy model and the scaling factors selection algorithm were applied to a heterogeneous cluster and evaluated over the SimGrid simulator~\cite{SimGrid.org},
+in this paper real experiments were conducted over the grid'5000 platform.
+
+\subsection{Grid'5000 architature and power consumption}
+\label{sec.grid5000}
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed over all metropolitan France and Luxembourg. All the sites are connected together via a special long distance network called RENATER,
+which is the French National Telecommunication Network for Technology.
+Each site of the grid is composed of few heterogeneous
+computing clusters and each cluster contains many homogeneous nodes. In total,
+ grid'5000 has about one thousand heterogeneous nodes and eight thousand cores. In each site,
+the clusters and their nodes are connected via high speed local area networks.
+Two types of local networks are used, Ethernet or Infiniband networks which have different characteristics in terms of bandwidth and latency.
+
+Since grid'5000 is dedicated for testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the grid'5000 testbed provides at some sites a power measurement tool to capture
+the power consumption for each node in those sites. The measured power is the overall consumed power by by all the components of a node at a given instant, such as CPU, hard drive, main-board, memory, ... For more details refer to
+\cite{Energy_measurement}. To just measure the CPU power of one core in a node $j$,
+ firstly, the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $P\max[jx]$. The difference between the two measured power consumption represents the
+dynamic power consumption of that core with the maximum frequency, see figure(\ref{fig:power_cons}).
+
+\textcolor{red}{why maximum and minimum, change peak in the equation and the figure}
+
+The dynamic power $\Pd[j]$ is computed as in equation (\ref{eq:pdyn})
+\begin{equation}
+ \label{eq:pdyn}
+ \Pd[j] = \max_{x=\beta_1,\dots \beta_2} (P\max[jx]) - \min_{y=\Theta_1,\dots \Theta_2} (\Pidle[jy])
+\end{equation}
-
-%\subsection{Performance prediction verification}
+where $\Pd[j]$ is the dynamic power consumption for one core of node $j$,
+$\lbrace \beta_1,\beta_2 \rbrace$ is the time interval for the measured peak power values,
+$\lbrace\Theta_1,\Theta_2\rbrace$ is the time interval for the measured idle power values.
+Therefore, the dynamic power of one core is computed as the difference between the maximum
+measured value in peak powers vector and the minimum measured value in the idle powers vector.
+On the other hand, the static power consumption by one core is a part of the measured idle power consumption of the node. Since in grid'5000 there is no way to measure precisely the consumed static power and in~\cite{Our_first_paper,pdsec2015,Rauber_Analytical.Modeling.for.Energy} it was assumed that the static power represents a ratio of the dynamic power, the value of the static power is assumed as np[\%]{20} of dynamic power consumption of the core.
-\subsection{The experimental results of the scaling algorithm}
-\label{sec.res}
+In the experiments presented in the following sections, two sites of grid'5000 were used, Lyon and Nancy sites. These two sites have in total seven different clusters as in figure (\ref{fig:grid5000}).
-The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.
-\cite{44}, which were run with three classes (A, B and C).
-In this experiments we are focus on running of the class C, the biggest class compared to A and B, on different number of
-nodes, from 4 to 128 or 144 nodes according to the type of the MPI program. Depending on the proposed energy consumption model EQ(\ref{eq:energy}),
- we are measure the energy consumption for all NAS MPI programs. The dynamic and static power values used under the same assumption used by \cite{45,3}. We used a percentage of 80\% for dynamic power from all power consumption of the CPU and 20\% for static power. The heterogeneous nodes in table (\ref{table:platform}) have different simulated performance, ranked from the node of type 1 with smaller performance/GFlops to the highest performance/GFlops for node 4. Therefore, the power values used proportionally increased from node 1 to node 4 that with highest performance. Then we used an assumption that the power consumption increases linearly with the performance/GFlops of the processor see \cite{48}.
-
-\begin{table}[htb]
- \caption{Running NAS benchmarks on 4 nodes }
- % title of Table
- \centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Method & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
- \hline
- CG & 64.64 & 3560.39 &34.16 &6.72 &27.44 \\
- \hline
- MG & 18.89 & 1074.87 &35.37 &4.34 &31.03 \\
- \hline
- EP &79.73 &5521.04 &26.83 &3.04 &23.79 \\
- \hline
- LU &308.65 &21126.00 &34.00 &6.16 &27.84 \\
- \hline
- BT &360.12 &21505.55 &35.36 &8.49 &26.87 \\
- \hline
- SP &234.24 &13572.16 &35.22 &5.70 &29.52 \\
- \hline
- FT &81.58 &4151.48 &35.58 &0.99 &34.59 \\
-\hline
- \end{tabular}
- \label{table:res_4n}
-\end{table}
+Four clusters from the two sites were selected in the experiments: one cluster from
+Lyon's site, Taurus cluster, and three clusters from Nancy's site, Graphene,
+Griffon and Graphite. Each one of these clusters has homogeneous nodes inside, while nodes from different clusters are heterogeneous in many aspects such as: computing power, power consumption, available
+frequency ranges and local network features: the bandwidth and the latency. Table \ref{table:grid5000} shows
+the details characteristics of these four clusters. Moreover, the dynamic powers were computed using the equation (\ref{eq:pdyn}) for all the nodes in the
+selected clusters and are presented in table \ref{table:grid5000}.
-\begin{table}[htb]
- \caption{Running NAS benchmarks on 8 and 9 nodes }
- % title of Table
- \centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Method & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
- \hline
- CG &36.11 &3263.49 &31.25 &7.12 &24.13 \\
- \hline
- MG &8.99 &953.39 &33.78 &6.41 &27.37 \\
- \hline
- EP &40.39 &5652.81 &27.04 &0.49 &26.55 \\
- \hline
- LU &218.79 &36149.77 &28.23 &0.01 &28.22 \\
- \hline
- BT &166.89 &23207.42 &32.32 &7.89 &24.43 \\
- \hline
- SP &104.73 &18414.62 &24.73 &2.78 &21.95 \\
- \hline
- FT &51.10 &4913.26 &31.02 &2.54 &28.48 \\
-\hline
- \end{tabular}
- \label{table:res_8n}
-\end{table}
-\begin{table}[htb]
- \caption{Running NAS benchmarks on 16 nodes }
- % title of Table
- \centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Method & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
- \hline
- CG &31.74 &4373.90 &26.29 &9.57 &16.72 \\
- \hline
- MG &5.71 &1076.19 &32.49 &6.05 &26.44 \\
- \hline
- EP &20.11 &5638.49 &26.85 &0.56 &26.29 \\
- \hline
- LU &144.13 &42529.06 &28.80 &6.56 &22.24 \\
- \hline
- BT &97.29 &22813.86 &34.95 &5.80 &29.15 \\
- \hline
- SP &66.49 &20821.67 &22.49 &3.82 &18.67 \\
- \hline
- FT &37.01 &5505.60 &31.59 &6.48 &25.11 \\
-\hline
- \end{tabular}
- \label{table:res_16n}
-\end{table}
-\begin{table}[htb]
- \caption{Running NAS benchmarks on 32 and 36 nodes }
- % title of Table
- \centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Method & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
- \hline
- CG &32.35 &6704.21 &16.15 &5.30 &10.85 \\
- \hline
- MG &4.30 &1355.58 &28.93 &8.85 &20.08 \\
- \hline
- EP &9.96 &5519.68 &26.98 &0.02 &26.96 \\
- \hline
- LU &99.93 &67463.43 &23.60 &2.45 &21.15 \\
- \hline
- BT &48.61 &23796.97 &34.62 &5.83 &28.79 \\
- \hline
- SP &46.01 &27007.43 &22.72 &3.45 &19.27 \\
- \hline
- FT &28.06 &7142.69 &23.09 &2.90 &20.19 \\
-\hline
- \end{tabular}
- \label{table:res_32n}
-\end{table}
-\begin{table}[htb]
- \caption{Running NAS benchmarks on 64 nodes }
- % title of Table
+\begin{figure}[!t]
\centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Method & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
- \hline
- CG &46.65 &17521.83 &8.13 &1.68 &6.45 \\
- \hline
- MG &3.27 &1534.70 &29.27 &14.35 &14.92 \\
- \hline
- EP &5.05 &5471.1084 &27.12 &3.11 &24.01 \\
- \hline
- LU &73.92 &101339.16 &21.96 &3.67 &18.29 \\
- \hline
- BT &39.99 &27166.71 &32.02 &12.28 &19.74 \\
- \hline
- SP &52.00 &49099.28 &24.84 &0.03 &24.81 \\
- \hline
- FT &25.97 &10416.82 &20.15 &4.87 &15.28 \\
-\hline
- \end{tabular}
- \label{table:res_64n}
-\end{table}
+ \includegraphics[scale=1]{fig/grid5000}
+ \caption{The selected two sites of grid'5000}
+ \label{fig:grid5000}
+\end{figure}
+
+
+The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over grid'5000.
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, the class D was used for all benchmarks in all the experiments presented in the next sections.
-\begin{table}[htb]
- \caption{Running NAS benchmarks on 128 and 144 nodes }
- % title of Table
- \centering
- \begin{tabular}{|*{7}{l|}}
- \hline
- Method & Execution & Energy & Energy & Performance & Distance \\
- name & time/s & consumption/J & saving\% & degradation\% & \\
- \hline
- CG &56.92 &41163.36 &4.00 &1.10 &2.90 \\
- \hline
- MG &3.55 &2843.33 &18.77 &10.38 &8.39 \\
- \hline
- EP &2.67 &5669.66 &27.09 &0.03 &27.06 \\
- \hline
- LU &51.23 &144471.90 &16.67 &2.36 &14.31 \\
- \hline
- BT &37.96 &44243.82 &23.18 &1.28 &21.90 \\
- \hline
- SP &64.53 &115409.71 &26.72 &0.05 &26.67 \\
- \hline
- FT &25.51 &18808.72 &12.85 &2.84 &10.01 \\
-\hline
- \end{tabular}
- \label{table:res_128n}
-\end{table}
-The results of applying the proposed scaling algorithm to NAS benchmarks is demonstrated 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}). These tables show the results for running the NAS benchmarks on different number of nodes. In general the energy saving percent is decreased when the number of the computing nodes is increased. While the distance is decreased by the same direction. This because when we are run the MPI programs on a big number of nodes the communications is biggest than the computations, so the static energy is increased linearly to these times. The tables also show that performance degradation percent still approximately the same or decreased a little when the number of computing nodes is increased. This gives good a prove that the proposed algorithm keeping as mush as possible the performance degradation.
-\begin{figure}
+\begin{figure}[!t]
\centering
- \subfloat[Balanced nodes type scenario]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
- \quad%
- \subfloat[Imbalanced nodes type scenario]{%
- \includegraphics[width=.23185\textwidth]{fig/avg_neq}\label{fig:avg_neq}}
- \label{fig:avg}
- \caption{The average of energy and performance for all Nas benchmarks running with difference number of nodes}
+ \includegraphics[scale=0.6]{fig/power_consumption.pdf}
+ \caption{The power consumption by one core from Taurus cluster}
+ \label{fig:power_cons}
\end{figure}
-In the NAS benchmarks there are some programs executed on different number of nodes. The benchmarks CG, MG, LU and FT executed on 2 to a power of (1, 2, 4, 8, ...) of nodes. The other benchmarks such as BT and SP executed on 2 to a power of (1, 2, 4, 9, ...) of nodes. We are take the average of energy saving, performance degradation and distances for all results of NAS benchmarks. The average of these three objectives are plotted to the number of nodes as in plots (\ref{fig:avg_eq} and \ref{fig:avg_neq}). In CG, MG, LU, and FT benchmarks the average of energy saving is decreased when the number of nodes is increased due to the increasing in the communication times as mentioned before. Thus, the average of distances (our objective function) is decreased linearly with energy saving while keeping the average of performance degradation the same. In BT and SP benchmarks, the average of energy saving is not decreased significantly compare to other benchmarks when the number of nodes is increased. Nevertheless, the average of performance degradation approximately still the same ratio. This difference is depends on the characteristics of the benchmarks such as the computation to communication ratio that has.
-\subsection{The results for different powers scenarios}
-The results of the previous section are obtained using a percentage of 80\% for dynamic power and 20\% for static power of total power consumption. In this section we are change these ratio by using two others scenarios. Because is interested to measure the ability of the proposed algorithm to changes it behaviour when these power ratios are changed. In fact, we are use two different scenarios for dynamic and static power ratios in addition to the previous scenario in section (\ref{sec.res}). Therefore, we have three different scenarios for three different dynamic and static power ratios refer to as: 70\%-20\%, 80\%-20\% and 90\%-10\% scenario. The results of these scenarios running NAS benchmarks class C on 8 or 9 nodes are place in the tables (\ref{table:res_s1} and \ref{table:res_s2}).
- \begin{table}[htb]
- \caption{The results of 70\%-30\% powers scenario}
+
+\begin{table}[!t]
+ \caption{CPUs characteristics of the selected clusters}
% title of Table
\centering
- \begin{tabular}{|*{6}{l|}}
+ \begin{tabular}{|*{7}{c|}}
\hline
- Method & Energy & Energy & Performance & Distance \\
- name & consumption/J & saving\% & degradation\% & \\
+ Cluster & CPU & Max & Min & Diff. & no. of cores & dynamic power \\
+ Name & model & Freq. & Freq. & Freq. & per CPU & of one core \\
+ & & GHz & GHz & GHz & & \\
\hline
- CG &4144.21 &22.42 &7.72 &14.70 \\
- \hline
- MG &1133.23 &24.50 &5.34 &19.16 \\
- \hline
- EP &6170.30 &16.19 &0.02 &16.17 \\
- \hline
- LU &39477.28 &20.43 &0.07 &20.36 \\
+ Taurus & Intel & 2.3 & 1.2 & 0.1 & 6 & \np[W]{35} \\
+ & Xeon & & & & & \\
+ & E5-2630 & & & & & \\
\hline
- BT &26169.55 &25.34 &6.62 &18.71 \\
- \hline
- SP &19620.09 &19.32 &3.66 &15.66 \\
- \hline
- FT &6094.07 &23.17 &0.36 &22.81 \\
-\hline
- \end{tabular}
- \label{table:res_s1}
-\end{table}
-
-
-
-\begin{table}[htb]
- \caption{The results of 90\%-10\% powers scenario}
- % title of Table
- \centering
- \begin{tabular}{|*{6}{l|}}
+ Graphene & Intel & 2.53 & 1.2 & 0.133 & 4 & \np[W]{23} \\
+ & Xeon & & & & & \\
+ & X3440 & & & & & \\
\hline
- Method & Energy & Energy & Performance & Distance \\
- name & consumption/J & saving\% & degradation\% & \\
+ Griffon & Intel & 2.5 & 2 & 0.5 & 4 & \np[W]{46} \\
+ & Xeon & & & & & \\
+ & L5420 & & & & & \\
\hline
- CG &2812.38 &36.36 &6.80 &29.56 \\
- \hline
- MG &825.427 &38.35 &6.41 &31.94 \\
- \hline
- EP &5281.62 &35.02 &2.68 &32.34 \\
- \hline
- LU &31611.28 &39.15 &3.51 &35.64 \\
+ Graphite & Intel & 2 & 1.2 & 0.1 & 8 & \np[W]{35} \\
+ & Xeon & & & & & \\
+ & E5-2650 & & & & & \\
\hline
- BT &21296.46 &36.70 &6.60 &30.10 \\
- \hline
- SP &15183.42 &35.19 &11.76 &23.43 \\
- \hline
- FT &3856.54 &40.80 &5.67 &35.13 \\
-\hline
\end{tabular}
- \label{table:res_s2}
-\end{table}
+ \label{table:grid5000}
+\end{table}
-\begin{figure}
- \centering
- \subfloat[Comparison the average of the results on 8 nodes]{%
- \includegraphics[width=.22\textwidth]{fig/sen_comp.pdf}\label{fig:sen_comp}}%
- \quad%
- \subfloat[Comparison the selected frequency scaling factors for 8 nodes]{%
- \includegraphics[width=.24\textwidth]{fig/three_scenarios.pdf}\label{fig:scales_comp}}
- \label{fig:avg}
- \caption{The comparison of the three power scenarios}
-\end{figure}
-To compare the results of these three powers scenarios, we are take the average of the performance degradation, the energy saving and the distances for all NAS benchmarks running on 8 or 9 nodes of class C, as in figure (\ref{fig:sen_comp}). Thus, according to the average of the results, the energy saving ratio is increased when using the a higher percentage for dynamic power (e.g. 90\%-10\% scenario). While the average of energy saving is decreased in 70\%-30\% scenario.
-Because the static energy consumption is increase. Moreover, the average of distances is more related to energy saving changes. 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). The raison behind these relations, that the proposed algorithm optimize both energy consumption and performance in the same time. Therefore, when using a higher ratio for dynamic power the algorithm selecting bigger frequency scaling factors values, more energy saving versus more performance degradation, for example see the figure (\ref{fig:scales_comp}). The inverse happen when using a higher ratio for static power, the algorithm proportionally selects a smaller scaling values, less energy saving versus less performance degradation. This is because the
-algorithm also keeps as much as possible the static energy consumption that is always related to execution time.
-\subsection{The verifications of the proposed method}
-\label{sec.verif}
-The precision of the proposed algorithm mainly depends on the execution time prediction model and the energy model. The energy model is significantly depends on the execution time model EQ(\ref{eq:perf}), that the energy static related linearly. So, we are compare the predicted execution time with the real execution time (Simgrid time) values that gathered offline for the NAS benchmarks class B executed on 8 or 9 nodes. The execution time model can predicts
-the real execution time by maximum normalized error 0.03 of all NAS benchmarks. The second verification that we are make is for the scaling algorithm to prove its ability to selects the best set of frequency scaling factors. Therefore, we are expand the algorithm to test at each iteration the frequency scaling factor of the slowest node with the all scaling factors available of the other nodes. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the two algorithms is identical. While the proposed algorithm is runs faster, 10 times faster on average than the expanded algorithm.
+\subsection{The experimental results of the scaling algorithm}
+\label{sec.res}
+
+\subsection{The experimental results of multi-cores clusters}
+\label{sec.res}
+
+\subsection{The results for different power consumption scenarios}
+\label{sec.compare}
+
+
+
+
+\subsection{The comparison of the proposed scaling algorithm }
+\label{sec.compare_EDP}
+
+
\section{Conclusion}
\label{sec.concl}
+
\section*{Acknowledgment}
+This work has been partially supported by the Labex ACTION project (contract
+``ANR-11-LABX-01-01''). Computations have been performed on the supercomputer
+facilities of the Mésocentre de calcul de Franche-Comté. As a PhD student,
+Mr. Ahmed Fanfakh, would like to thank the University of Babylon (Iraq) for
+supporting his work.
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-% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
+% 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 SimGrid GFlops Xeon EP BT GPUs CPUs AMD
+% LocalWords: Spiliopoulos scalability
