X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy.git/blobdiff_plain/c25c567d8c7358554605d785e62623b9fc7cda8e..e10a2fae800c41166633eabe0cff9e7815befda7:/paper.tex?ds=inline diff --git a/paper.tex b/paper.tex index da3fb81..e62908a 100644 --- a/paper.tex +++ b/paper.tex @@ -4,7 +4,7 @@ \usepackage[utf8]{inputenc} \usepackage[english]{babel} \usepackage{algpseudocode} -\usepackage{graphicx,graphics} +\usepackage{graphicx} \usepackage{subfig} \usepackage{amsmath} @@ -40,6 +40,7 @@ \newcommand{\Pdyn}{\Xsub{P}{dyn}} \newcommand{\PnormInv}{\Xsub{P}{NormInv}} \newcommand{\Pnorm}{\Xsub{P}{Norm}} +\newcommand{\Tnorm}{\Xsub{T}{Norm}} \newcommand{\Pstates}{\Xsub{P}{states}} \newcommand{\Pstatic}{\Xsub{P}{static}} \newcommand{\Sopt}{\Xsub{S}{opt}} @@ -310,8 +311,8 @@ EQ~\eqref{eq:sopt}. \section{Performance evaluation of MPI programs} \label{sec.mpip} -The performance (execution time) of parallel synchronous MPI applications depends -on the time of the slowest task. If there is no +The execution time of a parallel synchronous iterative application is +equal to the execution time of the slowest task. If there is no communication and the application is not data bounded, the execution time of a parallel program is linearly proportional to the operational frequency and any DVFS operation for energy reduction increases the execution time of the parallel @@ -336,12 +337,13 @@ for each processor as presented in the next section. \section{Performance and energy reduction trade-off} \label{sec.compet} -This section presents our approach for choosing the optimal scaling factor. -This factor gives maximum energy reduction while taking into account the execution -times for both computation and communication. The relation between the performance -and the energy is nonlinear and complex. Thus, unlike the relation between the performance and the scaling factor, the relation of energy with the scaling factor 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 energy by calculating the ratio between -the consumed energy with scaled frequency and the consumed energy without scaled +This section presents our method for choosing the optimal scaling factor that +gives the best tradeoff between energy reduction and performance. This method +takes into account the execution times for both computation and communication to +compute the scaling factor. Since the energy consumption and the performance +are not measured using the same metric, a normalized value of both measurements +can be used to compare them. The normalized energy is the ratio between the +consumed energy with scaled frequency and the consumed energy without scaled frequency: \begin{multline} \label{eq:enorm} @@ -352,32 +354,17 @@ frequency: \Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + \Pstatic \cdot T_1 \cdot N } \end{multline} -In the same way we can normalize the performance as follows: +In the same way, the normalized execution time of a program is computed as follows: \begin{equation} \label{eq:pnorm} - \Pnorm = \frac{\Tnew}{\Told} + \Tnorm = \frac{\Tnew}{\Told} = \frac{\TmaxCompOld \cdot S + \TmaxCommOld}{ \TmaxCompOld + \TmaxCommOld} \end{equation} -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 performance curves are not at the same direction see -Figure~\ref{fig:rel}\subref{fig:r2}. While the main goal is to optimize the -energy and performance in the same time. According to the -equations~\eqref{eq:enorm} and~\eqref{eq:pnorm}, the scaling factor $S$ reduce -both the energy and the performance simultaneously. But the main objective is -to produce maximum energy reduction with minimum performance 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 present a method to find the -optimal scaling factor $S$ to optimize both energy and performance -simultaneously without adding a big overhead. Our solution for this problem is -to make the optimization process for energy and performance follow the same -direction. Therefore, we inverse the equation of the normalized performance as -follows: +The relation between the execution time and the consumed energy of a program is nonlinear and complex. In consequences, the relation between the consumed energy and the scaling factor is also nonlinear, for more details refer to~\cite{17}. Therefore, the resulting normalized energy consumption curve and execution time curve, for different scaling factors, do not have the same direction see Figure~\ref{fig:rel}\subref{fig:r2}. To tackle this problem and optimize both terms, we inverse the equation of the normalized execution time as follows: \begin{equation} \label{eq:pnorm_en} - \Pnorm^{-1} = \frac{ \Told}{ \Tnew} + \Pnorm = \frac{ \Told}{ \Tnew} = \frac{\TmaxCompOld + \TmaxCommOld}{\TmaxCompOld \cdot S + \TmaxCommOld} @@ -392,7 +379,7 @@ follows: \label{fig:rel} \end{figure} Then, we can model our objective function as finding the maximum distance -between the energy curve EQ~\eqref{eq:enorm} and the inverse of performance +between the energy curve EQ~\eqref{eq:enorm} and the inverse of the execution time (performance) curve EQ~\eqref{eq:pnorm_en} over all available scaling factors. This represents the minimum energy consumption with minimum execution time (better performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}. Then @@ -400,7 +387,7 @@ our objective function has the following form: \begin{equation} \label{eq:max} \MaxDist = \max_{j=1,2,\dots,F} - (\overbrace{\Pnorm^{-1}(S_j)}^{\text{Maximize}} - + (\overbrace{\Pnorm(S_j)}^{\text{Maximize}} - \overbrace{\Enorm(S_j)}^{\text{Minimize}} ) \end{equation} where $F$ is the number of available frequencies. Then we can select the optimal @@ -443,10 +430,10 @@ the objective function described above. \Pdyn \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + \Pstatic \cdot T_1 \cdot N }$ - \State $\PnormInv \gets \Told / \Tnew$ - \If{$(\PnormInv - \Enorm > \Dist)$} + \State $\Pnorm \gets \Told / \Tnew$ + \If{$(\Pnorm - \Enorm > \Dist)$} \State $\Sopt \gets S$ - \State $\Dist \gets \PnormInv - \Enorm$ + \State $\Dist \gets \Pnorm - \Enorm$ \EndIf \EndFor \State Return $\Sopt$ @@ -524,9 +511,9 @@ Each node in the cluster has 18 frequency values from \np[GHz]{2.5} to \np[MHz]{800} with \np[MHz]{100} difference between each two successive frequencies. The nodes are connected via an ethernet network with 1Gbit/s bandwidth. -\subsection{Performance prediction verification} +\subsection{Execution time prediction verification} -In this section we evaluate the precision of our performance prediction method +In this section we evaluate the precision of our execution time prediction method based on EQ~\eqref{eq:tnew} by applying it to the NAS benchmarks. The NAS programs are executed with the class B option to compare the real execution time with the predicted execution time. Each program runs offline with all available @@ -566,14 +553,14 @@ and Rünger in~\cite{3}. The results showed that the algorithm selected different scaling factors for each program depending on the communication features of the program as in the plots from Figure~\ref{fig:nas}. These plots illustrate that there are different distances between the normalized energy and -the normalized inverted performance curves, because there are different +the normalized inverted execution time curves, because there are different communication features for each benchmark. When there are little or no -communications, the inverted performance curve is very close to the energy +communications, the inverted execution time curve is very close to the energy curve. Then the distance between the two curves is very small. This leads to small energy savings. The opposite happens when there are a lot of communication, the distance between the two curves is big. This leads to more energy savings (e.g. CG and FT), see Table~\ref{table:compareC}. All discovered -frequency scaling factors optimize both the energy and the performance +frequency scaling factors optimize both the energy and the execution time simultaneously for all NAS benchmarks. In Table~\ref{table:compareC}, we record all optimal scaling factors results for each benchmark running class C. These scaling factors give the maximum energy saving percentage and the minimum @@ -604,7 +591,7 @@ savings). In this section, we compare our scaling factor selection method with Rauber and Rünger methods~\cite{3}. They had two scenarios, the first is to reduce energy -to the optimal level without considering the performance as in +to the optimal level without considering the execution time as in EQ~\eqref{eq:sopt}. We refer to this scenario as $R_{E}$. The second scenario is similar to the first except setting the slower task to the maximum frequency (when the scale $S=1$) to keep the performance from degradation as mush as @@ -684,10 +671,10 @@ In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed application running on a homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy -consumption and the performance of the parallel application at every available +consumption and the execution time of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best trade-off between energy reduction and performance which is the maximum distance between -the energy and the inverted performance curves. To evaluate this method, we +the energy and the inverted execution time curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber and Rünger methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber and Rünger's methods in terms of energy-performance @@ -714,6 +701,7 @@ supporting his work. % the document is modified later %\IEEEtriggeratref{15} +\newpage \bibliographystyle{IEEEtran} \bibliography{IEEEabrv,my_reference} \end{document}