\usepackage[english]{babel}
\usepackage{algpseudocode}
\usepackage{graphicx}
+\usepackage{algorithm}
\usepackage{subfig}
\usepackage{amsmath}
\newcommand{\TmaxCompOld}{\Xsub{T}{Max Comp Old}}
\newcommand{\Tmax}{\Xsub{T}{max}}
\newcommand{\Tnew}{\Xsub{T}{New}}
-\newcommand{\Told}{\Xsub{T}{Old}}
-
-\begin{document}
+\newcommand{\Told}{\Xsub{T}{Old}}
+\begin{document}
\title{Energy Consumption Reduction in heterogeneous architecture using DVFS}
-
\section{The performance and energy consumption measurements on heterogeneous architecture}
\label{sec.exe}
% paper in homogeneous clusters}
\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 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, \dots{} but they all
+have the same network bandwidth and latency.
\begin{figure}[t]
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}
+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:
+$(S_1, S_2,\dots, S_N)$ 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{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
+ \max_{i=1,2,\dots,N} (TcpOld_{i} \cdot S_{i}) + MinTcm
+\end{equation}
+where $TcpOld_i$ is the computation time of processor $i$ during the first iteration and $MinTcm$ is the communication time of the slowest processor from the first iteration. The model computes the maximum computation time
with scaling factor from each node added to the communication time of the slowest node, it means only the
- communication time without any slack time.
+ communication time without any slack time. Therefore, we can consider the execution time of the iterative application is equal to the execution time of one iteration as in EQ(\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.
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
+ E_\textit{ind} = P_\textit{d} \cdot Tcp + P_\textit{s} \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
$F_{new}$ from EQ(\ref{eq:s}) as follow:
\begin{equation}
\label{eq:fnew}
- F_\textit{new} = S^{-1} . F_\textit{max}
+ F_\textit{new} = S^{-1} \cdot F_\textit{max}
\end{equation}
Replacing $F_{new}$ in EQ(\ref{eq:pd}) as in EQ(\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}
+ {} = \alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dOld} \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.
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:
\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}
+ E_\textit{dNew} = P_{dOld} \cdot S^{-3} \cdot (Tcp \cdot S)= S^{-2}\cdot P_{dOld} \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:
-
\begin{equation}
\label{eq:Estatic}
- E_\textit{s} = P_\textit{s} \cdot (T_{cp} \cdot S + T_{cm})
+ E_\textit{s} = P_\textit{s} \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 platform, each processor $i$ might have different dynamic and static powers, noted as $Pd_{i}$ and $Ps_{i}$ respectively. Therefore, even if the distributed message passing iterative application is load balanced, the computation time of each CPU $i$ noted $Tcp_{i}$ might be different and different frequency scaling factors might be computed in order to decrease the overall energy consumption of the application and reduce the slack times. The communication time of a processor $i$ is noted as $Tcm_{i}$ 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 during one iteration is the summation of all dynamic and static energies for each processor. It is computed as follows:
\begin{multline}
\label{eq:energy}
- E = \sum_{i=1}^{N} {(S_i^{-2} \cdot 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}))}
+ 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 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}.
+
+Reducing the frequencies of the processors according to the vector of
+scaling factors $(S_1, S_2,\dots, S_N)$ may degrade the performance of the
+application and thus, increase the static energy because the execution time is
+increased~\cite{36}. We can measure the overall energy consumption for the iterative
+application by measuring the energy consumption for one iteration as in EQ(\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:
+
+Using the lowest frequency for each processor does not necessarily gives 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 and thus the execution time might be drastically increased during which 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) in order to reduce the overall energy consumption and not significantly increase the execution time. In our previous work~\cite{45}, we proposed a method that selects the optimal
+frequency scaling factor for a homogeneous cluster executing a message passing iterative synchronous application while giving the best trade-off
+ between the energy consumption and the performance for such applications. In this work we are interested in
+heterogeneous clusters as described above. Due to the heterogeneity of the processors, not one but 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 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 computing the ratio between the new execution time (after scaling down the frequencies of some processors) and the initial one (with maximum frequency for all nodes,) as follows:
\begin{multline}
\label{eq:pnorm}
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}}}
+ {} = \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}
-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:
+In the same way, we normalize the energy by computing the ratio between the consumed energy while scaling down the frequency and the consumed energy with maximum frequency for all nodes:
\begin{multline}
\label{eq:enorm}
E_\textit{Norm} = \frac{E_\textit{Reduced}}{E_\textit{Original}} \\
- = \frac{ \sum_{i=1}^{N}{(S_i^{-2} \cdot Pd_i \cdot Tcp_i)} +
+ {} = \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:
-
+Where $T_{New}$ and $T_{Old}$ are computed as in EQ(\ref{eq:pnorm}).
+
+ While the main
+goal is to optimize the energy and execution time at the same time, the normalized energy and execution time curves are not in the same direction. According
+to the equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), 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.
+
+Many researchers used different strategies to solve this nonlinear problem for example
+in~\cite{19,42}, their methods add big overheads to the algorithm to select the
+suitable frequency. In this paper we present a method to find the optimal
+set of frequency scaling factors to simultaneously optimize both energy and execution time
+ without adding a big overhead. \textbf{put the last two phrases in the related work section}
+
+
+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 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}) +\min_{i=1,2,\dots,N} {Tcm_{i}}}
+ { \max_{i=1,2,\dots,N} (Tcp_{i} \cdot S_{i}) + MinTcm}
\end{multline}
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
+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 our objective
function has the following form:
-\begin{multline}
+\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}
+\end{equation}
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}.
+work with any energy model or any power values for each node (static and dynamic powers).
+However, the most energy reduction gain can be achieved when the energy curve has a convex form as shown in~\cite{15,3,19}.
-\section{The heterogeneous scaling algorithm }
+\section{The scaling factors selection algorithm for heterogeneous platforms }
\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}
- \caption{Selecting the initial frequencies}
- \label{fig:st_freq}
-\end{figure}
+
+In this section we propose algorithm~\ref{HSA}) which selects the frequency scaling factors vector that gives the best trade-off between minimizing the energy consumption and maximizing the performance of a message passing synchronous iterative application executed on a heterogeneous platform.
+IT works online during the execution time of the iterative message passing program. It uses information gathered during the first iteration such as the computation time and the communication time in one iteration for each node. The algorithm is executed after the first iteration and returns a vector of optimal frequency scaling factors that satisfies the objective function EQ(\ref{eq:max}). The program apply 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.
-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:
+The nodes in a heterogeneous platform have different computing powers, thus while executing message passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). These periods are called idle or slack times.
+Our algorithm takes into account this problem and tries to reduce these slack times when selecting the frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase the execution times of fast nodes and minimize the differences between the computation times of fast and slow nodes. The value of the initial frequency scaling factor for each node is inversely proportional to its computation time that was gathered from the first iteration. These initial frequency scaling factors are computed as a ratio between the computation time of the slowest node and the computation time of the node $i$ as follows:
\begin{equation}
\label{eq:Scp}
Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
\end{equation}
-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:
+Using the initial frequency scaling factors computed in 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 follows:
\begin{equation}
\label{eq:Fint}
F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
\end{equation}
-\begin{figure}[tp]
+If the computed initial frequency for a node is not available in the gears of that node, the computed initial frequency 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 colored in blue in figure (\ref{fig:st_freq}). This set of frequencies can be considered as a higher bound for the search space of the optimal set of frequencies because selecting frequency scaling factors higher than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the frequency selecting factors algorithm starts its search method from these initial frequencies and takes a downward search direction. The algorithm iterates on all left frequencies, from the higher bound until all nodes reach their minimum frequencies, to compute their overall energy consumption and performance, and select the optimal frequency scaling factors vector. At each iteration the algorithm determines the slowest node according to EQ(\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 EQ(\ref{eq:max}).
+
+
+
+
+
+This algorithm has a small
+execution time: for a heterogeneous cluster composed of four different types of
+nodes having the characteristics presented in table~(\ref{table:platform}), it
+takes \np[ms]{0.04} on average for 4 nodes and \np[ms]{0.15} on average for 144
+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 to selects the best vector of frequency scaling factors that give the results of the next section. \textbf{put the lst paragraph in experiments}
+
+
+
+
+
+
+\begin{algorithm}
\begin{algorithmic}[1]
% \footnotesize
\Require ~
\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 $Sopt_1, \dots ,Sopt_N$ is a set 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 Round the computed initial frequencies $F_i$ to the closest one available in each node.
\If{(not the first frequency)}
- \State $F_i \gets F_i+Fdiff_i,~i=1,...,N.$
+ \State $F_i \gets F_i+Fdiff_i,~i=1,\dots,N.$
\EndIf
- \State $T_\textit{Old} \gets max_{~i=1,...,N } (Tcp_i+Tcm_i)$
+ \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 $Dist \gets 0$
- \State $Sopt_{i} \gets 1,~i=1,...,N. $
+ \State $Sopt_{i} \gets 1,~i=1,\dots,N. $
\While {(all nodes not reach their minimum frequency)}
\If{(not the last freq. \textbf{and} not the slowest node)}
- \State $F_i \gets F_i - Fdiff_i,~i=1,...,N.$
- \State $S_i \gets \frac{Fmax_i}{F_i},~i=1,...,N.$
+ \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}) + min_\textit{~i=1,\dots,N}(Tcm_i) $
+ \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 $E_\textit{Norm}\gets \frac{E_\textit{Reduced}}{E_\textit{Original}}$
\If{$(\Pnorm - \Enorm > \Dist)$}
- \State $Sopt_{i} \gets S_{i},~i=1,...,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{Heterogeneous scaling 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}}
\end{algorithmic}
\caption{DVFS algorithm}
\label{dvfs}
-\end{figure}
+\end{algorithm}
\section{Experimental results}
\label{sec.expe}
-The experiments of this work are executed on the simulator Simgrid/SMPI
-v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We configure the
+The experiments of this work are executed on the simulator SimGrid/SMPI
+v3.10~\cite{casanova+giersch+legrand+al.2014.versatile}. We are 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
+heterogeneous cluster has four different types of nodes. Each node in the 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
+(\ref{table:platform}). These different types of processing nodes are 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
+benchmarks that used. We are use the same number from each type of nodes when we
+run the iterative MPI programs, for example if we are execute the program on 8 node, there
+are 2 nodes from each type participating in the computation. 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
+and static power values proportional to their computing power (FLOPS), 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
+80\% for dynamic power and 20\% for static power of the total power
+consumption of a CPU, 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}
\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
-The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.
+The proposed algorithm was applied to seven MPI programs of the NAS benchmarks (EP, CG, MG, FT, BT, LU and SP) NPB v3.3
\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}.
+In this experiments we are interested to run 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 iterative 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 are used under the same assumption used by \cite{45,3}, we are used a percentage of 80\% for dynamic power and 20\% for static of the total power consumption of a CPU. The heterogeneous nodes in table (\ref{table:platform}) have different simulated computing power (FLOPS), ranked from the node of type 1 with smaller computing power (FLOPS) to the highest computing power (FLOPS) for node of type 4. Therefore, the power values are used proportionally increased from nodes of type 1 to nodes of type 4 that with highest computing power. Then, we are used an assumption that the power consumption is increased linearly when the computing power (FLOPS) of the processor is increased, see \cite{48}.
\begin{table}[htb]
\caption{Running NAS benchmarks on 4 nodes }
\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.
+The results of applying the proposed scaling algorithm to the 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 are show the experimental 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. Also the distance is decreased by the same direction of the energy saving. This because when we are run the iterative MPI programs on a big number of nodes the communications times is increased, so the static energy is increased linearly to these times. The tables also show that the performance degradation percent still approximately the same ratio or decreased in a very small present when the number of computing nodes is increased. This is gives a good prove that the proposed algorithm keeping the performance degradation as mush as possible is the same.
\begin{figure}
\centering
- \subfloat[Balanced nodes type scenario]{%
+ \subfloat[CG, MG, LU and FT benchmarks]{%
\includegraphics[width=.23185\textwidth]{fig/avg_eq}\label{fig:avg_eq}}%
\quad%
- \subfloat[Imbalanced nodes type scenario]{%
+ \subfloat[BT and SP benchmarks]{%
\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}
+ \caption{The average of energy and performance for all NAS benchmarks running with difference number of nodes}
\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}).
+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,
+\dots{}) of nodes. The other benchmarks such as BT and SP executed on 2 to a
+power of (1, 2, 4, 9, \dots{}) of nodes. We are take the average of energy
+saving, performance degradation and distances for all results of NAS
+benchmarks. The average of values 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 because the communication times is increased as mentioned
+before. Thus, the average of distances (our objective function) is decreased
+linearly with energy saving while keeping the average of performance degradation approximately is
+the same. In BT and SP benchmarks, the average of the 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 power consumption scenarios}
+
+The results of the previous section are obtained using a percentage of 80\% for
+dynamic power and 20\% for static power of the total power consumption of a CPU. In this
+section we are change these ratio by using two others power scenarios. Because is
+interested to measure the ability of the proposed algorithm when these power ratios are changed.
+In fact, we are used 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 these as:
+70\%-20\%, 80\%-20\% and 90\%-10\% scenario respectively. The results of these scenarios
+running the 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}
\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.
+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 these results, the energy saving ratio is increased when using a higher percentage for dynamic power (e.g. 90\%-10\% scenario), due to increase in dynamic energy. 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 is optimizes the static energy consumption that is always related to the 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.
+The precision of the proposed algorithm mainly depends on the execution time prediction model EQ(\ref{eq:perf}) and the energy model EQ(\ref{eq:energy}). The energy model is significantly depends on the execution time model, that the static energy is related linearly. So, our work is depends mainly on execution time model. To verifying thid model, we are compare the predicted execution time with the real execution time (Simgrid time) values that gathered offline from 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 equal to 0.03 for all the NAS benchmarks. The second verification that we are made is for the proposed scaling algorithm to prove its ability to selects the best vector of the 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 available scaling factors of the other nodes, all possible solutions. This version of the algorithm is applied to different NAS benchmarks classes with different number of nodes. The results from the expanded algorithms and the proposed algorithm are identical. While the proposed algorithm is runs by 10 times faster on average compare to the expanded algorithm.
\section{Conclusion}
\label{sec.concl}
%%% ispell-local-dictionary: "american"
%%% End:
-% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
+% 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: de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
+% LocalWords: Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
