X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy.git/blobdiff_plain/f2f3e27d9d560225f78b031375f3d02f1948f9a4..9cf953b2bfe1f9fdfbf76e564f4df0ad6f64b4ed:/paper.tex diff --git a/paper.tex b/paper.tex index 72cf122..29f175b 100644 --- a/paper.tex +++ b/paper.tex @@ -9,6 +9,7 @@ \usepackage{subfig} \usepackage{listings} \usepackage{colortbl} +\usepackage{amsmath} % \usepackage{sectsty} % \usepackage{titlesec} % \usepackage{secdot} @@ -16,12 +17,21 @@ %\usepackage[font=scriptsize,labelfont=bf]{caption} \usepackage{lmodern} +\usepackage{todonotes} +\newcommand{\AG}[2][inline]{\todo[color=green!50,#1]{\sffamily\small\textbf{AG:} #2}} + \begin{document} \title{Optimal Dynamic Frequency Scaling for Energy-Performance of Parallel MPI Programs} \author{A. Badri \and J.-C. Charr \and R. Couturier \and A. Giersch} \maketitle +\AG{``Optimal'' is a bit pretentious in the title} + +\begin{abstract} + \AG{FIXME} +\end{abstract} + \section{Introduction} The need for computing power is still increasing and it is not expected to slow @@ -59,11 +69,11 @@ algorithm has ability to predict both energy consumption and execution time over all available scaling factors. The prediction achieved depends on some computing time information, gathered at the beginning of the runtime. We apply this algorithm to seven MPI benchmarks. These MPI programs are the NAS parallel -penchmarks (NPB v3.3) developed by NASA~\cite{44}. Our experiments are executed -using the simulator Simgrid/SMPI v3.10~\cite{45} over an homogeneous distributed -memory architecture. Furthermore, we compare the proposed algorithm with -Rauber's methods. The comparison's results show that our algorithm gives better -energy-time trade off. +benchmarks (NPB v3.3) developed by NASA~\cite{44}. Our experiments are executed +using the simulator SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183} +over an homogeneous distributed memory architecture. Furthermore, we compare the +proposed algorithm with Rauber's methods. The comparison's results show that our +algorithm gives better energy-time trade off. \section{Related Works} @@ -78,7 +88,7 @@ during the compilation phases as an example in Azevedo et al.~\cite{40}. He used intra-task algorithm to choose the DVFS setting when there are dependency points between tasks. While in~\cite{29}, Xie et al. used breadth-first search algorithm to do that. Their goal is saving energy with time limits. Another -approaches gathers and stores the runtime information for each DVFS state , then +approaches gathers and stores the runtime information for each DVFS state, then used their methods offline to select the suitable DVFS that optimize energy-time trade offs. As an example~\cite{8}, Rountree et al. used liner programming algorithm, while in~\cite{38,34}, Cochran et al. used multi logistic regression @@ -103,7 +113,7 @@ program used online for saving energy as in~\cite{1}, Lim et al. developed an algorithm that detects the communication sections and changes the frequency during these sections only. This approach changes the frequency many times because an iteration may contain more than one communication section. The domain -of analytical modeling used for choosing the optimal frequency as in ~\cite{3}, +of analytical modeling used for choosing the optimal frequency as in~\cite{3}, Rauber et al. developed an analytical mathematical model for determining the optimal frequency scaling factor for any number of concurrent tasks, without considering communication times. They set the slowest task to maximum frequency @@ -145,8 +155,9 @@ amounts of data on each processor as an example see figure~(\ref{fig:h2}). In this case the fastest tasks have to wait at the synchronous barrier for the slowest tasks to finish their job. In both two cases the overall execution time of the program is the execution time of the slowest task as : -\begin{equation} \label{eq:T1} - Program Time=MAX_{i=1,2,..,N} (T_i) \hfill +\begin{equation} + \label{eq:T1} + \textit{Program Time} = \max_{i=1,2,\dots,N} T_i \end{equation} where $T_i$ is the execution time of process $i$. @@ -154,24 +165,27 @@ where $T_i$ is the execution time of process $i$. The energy consumption by the processor consists of two powers metric: the dynamic and the static power. This general power formulation is used by many -researchers see ~\cite{9,3,15,26}. The dynamic power of the CMOS processors +researchers see~\cite{9,3,15,26}. The dynamic power of the CMOS processors $P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and operational frequency $f$ respectively as follow : -\begin{equation} \label{eq:pd} - \displaystyle P_{dyn} = \alpha . C_L . V^2 . f +\begin{equation} + \label{eq:pd} + P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f \end{equation} The static power $P_{static}$ captures the leakage power consumption as well as the power consumption of peripheral devices like the I/O subsystem. -\begin{equation} \label{eq:ps} - \displaystyle P_{static} = V . N . K_{design} . I_{leak} +\begin{equation} + \label{eq:ps} + P_{static} = V \cdot N \cdot K_{design} \cdot I_{leak} \end{equation} where V is the supply voltage, N is the number of transistors, $K_{design}$ is a design dependent parameter and $I_{leak}$ is a technology-dependent parameter. Energy consumed by an individual processor $E_{ind}$ is the summation of the dynamic and the static power multiply by the execution time for example see~\cite{36,15} . -\begin{equation} \label{eq:eind} - \displaystyle E_{ind} = (P_{dyn} + P_{static} ) . T +\begin{equation} + \label{eq:eind} + E_{ind} = ( P_{dyn} + P_{static} ) \cdot T \end{equation} The dynamic voltage and frequency scaling (DVFS) is a process that allowed in modern processors to reduce the dynamic power by scaling down the voltage and @@ -182,8 +196,9 @@ 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 are expressed by scaling factor \emph S. The scale \emph S is the ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}). -\begin{equation} \label{eq:s} - S=\:\frac{F_{max}}{F_{new}} \hfill \newline +\begin{equation} + \label{eq:s} + S = \frac{F_{max}}{F_{new}} \end{equation} The value of the scale \emph S is grater than 1 when changing the frequency to any new frequency value(\emph {P-state}) in governor. It is equal to 1 when the @@ -196,8 +211,11 @@ for any number of concurrent tasks develops by Rauber~\cite{3}. This model consider the two powers metric for measuring the energy of the parallel tasks as in EQ~(\ref{eq:energy}). -\begin{equation} \label{eq:energy} - E= \displaystyle \;P_{dyn}\,.\,S_1^{-2}\;.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_1\;\,.\,N +\begin{equation} + \label{eq:energy} + E = P_{dyn} \cdot S_1^{-2} \cdot + \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) + + P_{static} \cdot T_1 \cdot S_1 \cdot N \hfill \end{equation} Where \emph N is the number of parallel nodes, $T_1 $ is the time of the slower @@ -205,11 +223,14 @@ task, $T_i$ is the time of the task $i$ and $S_1$ is the maximum scaling factor for the slower task. The scaling factor $S_1$, as in EQ~(\ref{eq:s1}), selects from the set of scales values $S_i$. Each of these scales are proportional to the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}). -\begin{equation} \label{eq:s1} - S_1=MAX_{i=1,2,..,F} (S_i) \hfill +\begin{equation} + \label{eq:s1} + S_1 = \max_{i=1,2,\dots,F} S_i \end{equation} -\begin{equation} \label{eq:si} - S_i=\:S\: .\:(\frac{T_1}{T_i})=\: (\frac{F_{max}}{F_{new}}).(\frac{T_1}{T_i}) \hfill +\begin{equation} + \label{eq:si} + S_i = S \cdot \frac{T_1}{T_i} + = \frac{F_{max}}{F_{new}} \cdot \frac{T_1}{T_i} \end{equation} Where $F$ is the number of available frequencies. In this paper we depend on Rauber's energy model EQ~(\ref{eq:energy}) for two reasons : 1-this model used @@ -218,10 +239,11 @@ algorithm with Rauber's scaling model. Rauber's optimal scaling factor for optimal energy reduction derived from the EQ~(\ref{eq:energy}). He takes the derivation for this equation (to be minimized) and set it to zero to produce the scaling factor as in EQ~(\ref{eq:sopt}). -\begin{equation} \label{eq:sopt} - S_{opt}= {\sqrt [3~]{\frac{2}{n} \frac{P_{dyn}}{P_{static}} \Big(1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^3}\Big) }} \hfill +\begin{equation} + \label{eq:sopt} + S_{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_{dyn}}{P_{static}} \cdot + \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) } \end{equation} -%[\Big 3] \section{Performance Evaluation of MPI Programs} @@ -232,7 +254,7 @@ frequency. Therefore, any DVFS operation for the energy reduction increase the execution time of the parallel program. As shown in EQ~(\ref{eq:energy}) the energy affected by the scaling factor $S$. This factor also has a great impact on the performance. When scaling down the frequency to the new value according -to EQ(~\ref{eq:s}) lead to the value of the scale $S$ has inverse relation with +to EQ~(\ref{eq:s}) lead to the value of the scale $S$ has inverse relation with new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decrease the frequency value, the execution time increase. Then the new frequency value has inverse relation with time ($F_{new} \propto \frac{1}{T}$). This lead to the @@ -249,16 +271,16 @@ must be precisely specifying communication time and the computation time for the slower task. Secondly, we use these times for predicting the execution time for any MPI program as a function of the new scaling factor as in the EQ~(\ref{eq:tnew}). -\begin{equation} \label{eq:tnew} - \displaystyle T_{new}= T_{Max \:Comp \:Old} \; . \:S \;+ \;T_{Max\: Comm\: Old} - \hfill +\begin{equation} + \label{eq:tnew} + T_{new} = T_{\textit{Max Comp Old}} \cdot S + T_{\textit{Max Comm Old}} \end{equation} The above equation shows that the scaling factor \emph S has linear relation with the computation time without affecting the communication time. The communication time consists of the beginning times which an MPI calls for sending or receiving till the message is synchronously sent or received. In this paper we predict the execution time of the program for any new scaling factor -value. Depending on this prediction we can produce our energy-performace scaling +value. Depending on this prediction we can produce our energy-performance scaling method as we will show in the coming sections. In the next section we make an investigation study for the EQ~(\ref{eq:tnew}). @@ -271,7 +293,9 @@ real execution time with the predicted execution time. Each program runs offline with all available scaling factors on 8 or 9 nodes to produce real execution time values. These scaling factors are computed by dividing the maximum frequency by the new one see EQ~(\ref{eq:s}). In all tests, we use the simulator -Simgrid/SMPI v3.10 to run the NAS programs. +SimGrid/SMPI v3.10 to run the NAS programs. +\AG{Fig.~\ref{fig:pred} is hard to read when printed in black and white, + especially the ``Normalize Real Perf.'' curve.} \begin{figure}[width=\textwidth,height=\textheight,keepaspectratio] \centering \includegraphics[scale=0.60]{cg_per.eps} @@ -288,7 +312,7 @@ frequencies. For more details on the characteristics of the platform refer to table~(\ref{table:platform}). This lead to 18 run states for each program. We use seven MPI programs of the NAS parallel benchmarks : CG, MG, EP, FT, BT, LU and SP. The average normalized errors between the predicted execution time and -the real time (Simgrid time) for all programs is between 0.0032 to 0.0133. AS an +the real time (SimGrid time) for all programs is between 0.0032 to 0.0133. AS an example, we are present the execution times of the NAS benchmarks as in the figure~(\ref{fig:pred}). @@ -303,12 +327,28 @@ is not straightforward. Moreover, they are not measured using the same metric. For solving this problem, we normalize the energy by calculating the ratio between the consumed energy with scaled frequency and the consumed energy without scaled frequency : -\begin{equation} \label{eq:enorm} - E_{Norm}=\displaystyle\frac{E_{Reduced}}{E_{Orginal}}= \frac{\displaystyle \;P_{dyn}\,.\,S_i^{-2}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,.\,S_i\;\,.\,N }{\displaystyle \;P_{dyn}\,.\,(T_1+\sum\limits_{i=2}^{N}\frac{T_i^3}{T_1^2})+\;P_{static}\,.\,T_1\,\,.\,N } +\begin{equation} + \label{eq:enorm} + E_{Norm} = \frac{E_{Reduced}}{E_{Original}} + = \frac{ P_{dyn} \cdot S_i^{-2} \cdot + \left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + + P_{static} \cdot T_1 \cdot S_i \cdot N }{ + P_{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + + P_{static} \cdot T_1 \cdot N } \end{equation} +\AG{Use \texttt{\textbackslash{}text\{xxx\}} or + \texttt{\textbackslash{}textit\{xxx\}} for all subscripted words in equations + (e.g. \mbox{\texttt{E\_\{\textbackslash{}text\{Norm\}\}}}). + + Don't hesitate to define new commands : + \mbox{\texttt{\textbackslash{}newcommand\{\textbackslash{}ENorm\}\{E\_\{\textbackslash{}text\{Norm\}\}\}}} +} By the same way we can normalize the performance as follows : -\begin{equation} \label{eq:pnorm} - P_{Norm}=\displaystyle \frac{T_{New}}{T_{Old}}=\frac{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}}{T_{Old}} \;\; +\begin{equation} + \label{eq:pnorm} + P_{Norm} = \frac{T_{New}}{T_{Old}} + = \frac{T_{\textit{Max Comp Old}} \cdot S + + T_{\textit{Max Comm Old}}}{T_{Old}} \end{equation} The second problem is the optimization operation for both energy and performance is not in the same direction. In other words, the normalized energy and the @@ -325,8 +365,11 @@ optimize both energy and performance simultaneously without adding big overheads. Our solution for this problem is to make the optimization process have the same direction. Therefore, we inverse the equation of normalize performance as follows : -\begin{equation} \label{eq:pnorm_en} - \displaystyle P^{-1}_{Norm}= \frac{T_{Old}}{T_{New}}=\frac{T_{Old}}{T_{Max \:Comp \:Old} \;. \:S \;+ \;T_{Max\: Comm\: Old}} +\begin{equation} + \label{eq:pnorm_en} + P^{-1}_{Norm} = \frac{T_{Old}}{T_{New}} + = \frac{T_{Old}}{T_{\textit{Max Comp Old}} \cdot S + + T_{\textit{Max Comm Old}}} \end{equation} \begin{figure} \centering @@ -341,14 +384,16 @@ curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors. This represent the minimum energy consumption with minimum execution time (better performance) in the same time, see figure~(\ref{fig:r1}). Then our objective function has the following form: -\begin{equation} \label{eq:max} - \displaystyle MaxDist = Max \;(\;\overbrace{P^{-1}_{Norm}}^{Maximize}\; -\; \overbrace{E_{Norm}}^{Minimize} \;) +\begin{equation} + \label{eq:max} + \textit{MaxDist} = \max (\overbrace{P^{-1}_{Norm}}^{\text{Maximize}} - + \overbrace{E_{Norm}}^{\text{Minimize}} ) \end{equation} Then we can select the optimal scaling factor that satisfy the EQ~(\ref{eq:max}). Our objective function can works with any energy model or static power values stored in a data file. Moreover, this function works in optimal way when the energy function has a convex form with frequency scaling -factor as shown in ~\cite{15,3,19}. Energy measurement model is not the +factor as shown in~\cite{15,3,19}. Energy measurement model is not the objective of this paper and we choose Rauber's model as an example with two reasons that mentioned before. @@ -374,15 +419,14 @@ scaling factor for both energy and performance at the same time. \State - Calculate all available scales $S_i$ depend on $S$ as in EQ~(\ref{eq:si}). \State - Select the maximum scale factor $S_1$ from the set of scales $S_i$. \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ as in EQ~(\ref{eq:enorm}). - \State - Calculate the normalize inverse of performance $P_{NormInv}=T_{old}/T_{new}$ - - as in EQ~(\ref{eq:pnorm_en}). - \If{ $(P_{NormInv}-E_{Norm}$ $>$ $Dist$) } - \State $S_{optimal}=S$ + \State - Calculate the normalize inverse of performance\par + $P_{NormInv}=T_{old}/T_{new}$ as in EQ~(\ref{eq:pnorm_en}). + \If{ $(P_{NormInv}-E_{Norm} > Dist$) } + \State $S_{optimal} = S$ \State $Dist = P_{NormInv} - E_{Norm}$ \EndIf \EndFor - \State $ Return \; \; (S_{optimal})$ + \State Return $S_{optimal}$ \end{algorithmic} \end{algorithm} The proposed EPSA algorithm works online during the execution time of the MPI @@ -397,6 +441,7 @@ system. The algorithm is called just one time during the execution of the program. The following example shows where and when the EPSA algorithm is called in the MPI program : \begin{minipage}{\textwidth} +\AG{Use the same format as for Algorithm~\ref{EPSA}} \begin{lstlisting}[frame=tb] FOR J:=1 to Some_iterations Do -Computations Section. @@ -415,33 +460,35 @@ After obtaining the optimal scale factor from the EPSA algorithm. The program calculates the new frequency $F_i$ for each task proportionally to its time value $T_i$. By substitution of the EQ~(\ref{eq:s}) in the EQ~(\ref{eq:si}), we can calculate the new frequency $F_i$ as follows : -\begin{equation} \label{eq:fi} - F_i=\frac{F_{max} \; . \;T_i}{S_{optimal} \; . \;T_{max}} \hfill +\begin{equation} + \label{eq:fi} + F_i = \frac{F_{max} \cdot T_i}{S_{optimal} \cdot T_{max}} \end{equation} According to this equation all the nodes may have the same frequency value if they have balanced workloads. Otherwise, they take different frequencies when have imbalanced workloads. Then EQ~(\ref{eq:fi}) works in adaptive way to change -the freguency according to the nodes workloads. +the frequency according to the nodes workloads. \section{Experimental Results} -The proposed ESPA algorithm was applied to seven MPI programs of the NAS -benchmarks (EP ,CG , MG ,FT , BT, LU and SP). We work on three classes (A, B and +The proposed EPSA algorithm was applied to seven MPI programs of the NAS +benchmarks (EP, CG, MG, FT, BT, LU and SP). We work on three classes (A, B and C) for each program. Each program runs on specific number of processors proportional to the size of the class. Each class represents the problem size ascending from the class A to C. Additionally, depending on some speed up points for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes -respectively. Our experiments are executed on the simulator Simgrid/SMPI +respectively. Our experiments are executed on the simulator SimGrid/SMPI v3.10. We design a platform file that simulates a cluster with one core per node. This cluster is a homogeneous architecture with distributed memory. The -detailed characteristics of our platform file are shown in -thetable~(\ref{table:platform}). Each node in the cluster has 18 frequency -values from 2.5 GHz to 800 MHz with 100 MHz difference between each two -successive frequencies. +detailed characteristics of our platform file are shown in the +table~(\ref{table:platform}). Each node in the cluster has 18 frequency values +from 2.5 GHz to 800 MHz with 100 MHz difference between each two successive +frequencies. \begin{table}[ht] \caption{Platform File Parameters} % title of Table \centering + \AG{Use e.g. $5\times 10^{-7}$ instead of 5E-7} \begin{tabular}{ | l | l | l |l | l |l |l | p{2cm} |} \hline Max & Min & Backbone & Backbone&Link &Link& Sharing \\ @@ -453,7 +500,7 @@ successive frequencies. \end{table} Depending on the EQ~(\ref{eq:energy}), we measure the energy consumption for all the NAS MPI programs while assuming the power dynamic is equal to 20W and the -power static is equal to 4W for all experiments. We run the proposed ESPA +power static is equal to 4W for all experiments. We run the proposed EPSA algorithm for all these programs. The results showed that the algorithm selected different scaling factors for each program depending on the communication features of the program as in the figure~(\ref{fig:nas}). This figure shows that @@ -485,6 +532,9 @@ same time over all available scales. \caption{Optimal Scaling Factors Results} % title of Table \centering + \AG{Use the same number of decimals for all numbers in a column, + and vertically align the numbers along the decimal points. + The same for all the following tables.} \begin{tabular}{ | l | l | l |l | l | p{2cm} |} \hline Program & Optimal & Energy & Performance&Energy-Perf.\\ @@ -676,6 +726,7 @@ than the first. \label{fig:compare} \end{figure} +\AG{\texttt{bibtex} gives many errors, please correct them} \bibliographystyle{plain} \bibliography{my_reference} \end{document} @@ -686,3 +737,6 @@ than the first. %%% fill-column: 80 %%% ispell-local-dictionary: "american" %%% End: + +% LocalWords: Badri Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber +% LocalWords: CMOS EQ EPSA