From cb26699a9edd21c00b43b97baa061f04e71bef32 Mon Sep 17 00:00:00 2001 From: afanfakh Date: Mon, 17 Mar 2014 09:42:56 +0100 Subject: [PATCH] Last Update --- compare_class_A.pdf | Bin 27055 -> 27057 bytes paper.tex | 439 ++++++++++++++++++++++---------------------- 2 files changed, 216 insertions(+), 223 deletions(-) diff --git a/compare_class_A.pdf b/compare_class_A.pdf index a8697e02a18934b3f0e05b03c9d9ecae9e56f983..8e7f8b90ed9515cd5efb231fcebaa3a3fb80fb39 100644 GIT binary patch delta 2100 zcmaiwdojAFJ@+hy#y4$WkK^P8F9%#X-ih=|0Yaa=Pb*G43*ZQ9+z3Prym zxv$H#y}87o5!#I2WFoBCrJ61-k-RRsv_h}i_v|?@f9!M4bIx!4tKslfVcM@vnmJU;bpY0Jl!SF_k;%aRT{+GO7Q_fqL=pGTFKmY)!x zS3S$}Guef71Y_f3@w+UAv2=nwNL|3O}n^{kahZxa+2zKrZ)11KV$6GUy_DkLoAMeD)tp@TvzAR6%V-pgr4p&w5h*4lP<8RUgkk%l zUiEUflvdMbH7|NlG}irI)G6j{U|5-IF059$*`}oy6G|z}oh+0EaLis=X>|lUpOGc` z?5=S*oA#`BEzm)b^_q zSJkE|9nljE+dG;kd&hTFsLHp(+}`|Omvv+2HY}Ylczag^b3-SdqrG9gPTK^zCHnWq z^~puc6-QpTUp}zOJO0Hl*)?|_uB#f?#`}znCQQXVTlb_dF}vw24@@IT1;rGs@A9K5yAyaSK9t%crbH^VM9%l6?p2_3R)e*P0 z`dq^gDQ8wsOm20SZvga8^t6elQEl+ZZd)Gxs!XE42i0=QJbb%h`2{~I-=W7N7SePg zx!S(`x?C;Ea|&Pm#KCksH)`-O6sm2KR!R=_?~T*HE(LKz^H1 zOxW>W8$l-9UHt9B$V`Kc)86zt=p?VW@xBEmSd@97JHZB2$+w$)j8mOb)3P&Z{Xw-e5mzn`Ef}mz*TPk! z?W?bsq==0}f3VjM7K5Shr2EZHiqX7`5v8$y0VdU=hj(8V!s8^D^lg#b?xh5~SQq2H zO)tl82^nn`Mv6RN4$9G?@*NNKVw6iZtLRAejoNmF%1``yL*%Zruyd-s=c-p}E_eSt zkv15%(uSZ^{ z;PQ_C&Yd|N?6Dn#4u+~zyjLuR?v3LwvYe`u4m^sWrR0?s%;@wP-g(_6#6-&7WTj5I zs(Izbr}QQQDrdyF`_(FPhqBJ}bdgE%Ar~W$%(NuqID>$c^3vi{hWUE-qus*T)OG$} z1iFO{ZLCnH#cG3S;Zo{rl7lvpoejfGust^6*LyO-YZZFq(& z!H$4GXFGrEgst-3?#<5fhi()3(n@ze;YVL{KAY$j6hD0;SC0T%Yd2h!P# zbZ8Ywrc$WnwHmZ~37JZR)_P$Yu-Z!_!~Y*f001DB0jvgBG;sBGAPoRlF9BB`^S?{} zJ_MxEfwf*bZ4CzjH1f7p2|X3zl+8)QLDHrOV delta 2095 zcmaiwc{J2}AIF;tnQYfwkBO&@8zsri?>FgUE(NEttg zL}j}-DKS&W9=fuWm@F|I>s4ec(d|Lq$GxX>&y7Ex?_ZzuIyeeS{YL=62#2TT7^G4F*}RV|jc!M~?JU8Xfq?)webHH-k6 zcYam5HB%ibFQ!Uo7gPy1UTIGqEwbut&r)hDJ{AG@YdiDk9UF2yEoQ_woC!frI z_d`YCjI#XP^76>oyO0&KRFq$0dn{)LYkc)*)`2S>%DT|yySv+(7!zZ0#!ru1`3CNk zxy6B>XXZp>A1mDSP)yq$T2^cZe`e25D=&TC-+g8~&dyvR!B|NeX5zZbYfsX5Td)T{ zd))ptOAE%!74I2N??d4pKK>9*G__uyjQ=1&`e*s`wPJBiFq+wGME_ z{HsbQSi#ZZc0$&*e812S)l({ytWZu#z%A>fbR{M6e*bY@i8;6_ew$=8!*+h)*J2?| zxFELteL*rACCvR%KP>QNv&HFzGu$3M04`J((y3GaN1vx&;+?LqX?d;@_%`m%jNR~d zOX-S0%m{hb=`-4;XHa$Re(`zlptyQFTpn&G1@ISl2TYFZXgymzyg`5Oex6ze&-$>Y zmA!Oxrg4v@ue3KLD%#aTt=T{#)AJD4#+@l@xhUvszKA_6(oI8t$-=spzvI6; zy)Q4wPwG|GwOmz`rr|Fgn3p}IiDm{Lw8?;^Y3*n2_dH}S5J%na;%l6YM3J+8^MXGt z2H(C+mY0YPs-5%h^{9W@8LqZ9OE;cx<6iB2&MGh@n=Oop&^Nnemviz?afLAeYvIMW ziPQx81H+rbge zV4L@(72}4&nUOA4H|K+s%p#VcL9neN!_LXOXQK5G+HaoLkY;MtEGKWn{R_m+Gc43J z7vfVw&AkTEt36)rO>ljcH%G_QF4{tjPJ1e?v=)~|*@YT?eDi7P^4z@%oj(%rs@Y@5 z;>~8w@-d&&zt5B`7A!R4JOHfT_%E#anLdTids_>+xH*(LWR4Pw|@QI_1wkeMCv;bRg2Ys`?G46FX zO`99V2=gUx$HJl&w7kf!Z_rDp?56_ero-WdDJ885M3bynqPE+?F{!r*#hru6`D6Xh zT6nc-uVbQ{U`EU14TkR(hsWcaF^fyLtrwjGFf+9pCxFe_$jG2nac=mglRK@?zE!R$ zxV=&~rc|@AS~xdk7IOOG=y};E(Xx?(O>Zw0oOSxRB+uOFc(pB>gsrwPb<7F{g4r1>^?@z~GQ_sfZ=^pO<+Ukjwo3#@w>dUGFr;s6; z9Vi0wMaEc^YIH;wxv#zjne6O(GE?M3o96Q|bvw7gbku4Z9bGaOni({Wf2OX5=Eeui>;^f&5jckSJ7S z#uG$XR3TF8r7K#Z2BMH-K6oUZo{yLx2i0h+rX8LG;()Ie5eJY&iN`%QZUI3EAd|=t z@}6siq9FVHcEbc5;m?N)!lC~vK!^sdWdr~st)Bah2LJ#GT+hhFb&N=c)-ej@YcKrw zVgMu%DQiVC1X0&B>e>SVGKI1l(6w5? jlRzee$wr%+qQm%Jq5MddP+u+@BoiqBT345C%|ZVUG0R`) diff --git a/paper.tex b/paper.tex index 962a22b..5e80312 100644 --- a/paper.tex +++ b/paper.tex @@ -24,9 +24,9 @@ \author{% \IEEEauthorblockN{% - Ahmed Badri, Jean-Claude Charr, - Raphaël Couturier and + Raphaël Couturier, + Ahmed Fanfakh and Arnaud Giersch } \IEEEauthorblockA{% @@ -41,7 +41,16 @@ Complete affiliation, add an email address, etc.} \begin{abstract} - \AG{complete the abstract\dots} +The important technique for energy reduction of parallel systems is CPU frequency +scaling. This operation used by many researchers to reduce energy consumption in many +ways. Frequency scaling operation also has big impact on the performance. In some cases, +the performance degradation ratio is bigger than energy saving ratio when the frequency scaled +to down level. Therefore, the trade offs between the energy and performance becomes more +important topic when using this technique. In this paper we developed an algorithm that +select the frequency scaling factor for both energy and performance simultaneously. +This algorithm takes into account the communication times when selecting the frequency scaling +factor. It is works online without training or profiling to have very small overhead. +The algorithm has better energy-performance trade offs compared to other methods. \end{abstract} \section{Introduction} @@ -72,12 +81,12 @@ of the performance. It is important to notice that lowering the frequency to minimum value does not always give the most efficient execution due to energy leakage. The best scaling factor might be chosen during execution (online) or during a pre-execution phase. In this paper we emphasize to develop an -algorithm that selects the optimal frequency scaling factor that takes into -consideration simultaneously the energy consumption and the performance. The +algorithm that selects a frequency scaling factor that simultaneously takes into +consideration the energy consumption and the performance. The main objective of HPC systems is to run the application with less execution -time. Therefore, our algorithm selects the optimal scaling factor online with +time. Therefore, our algorithm selects the scaling factor online with very small footprint. The proposed algorithm takes into account the -communication times of the MPI programs to choose the scaling factor. This +communication times of the MPI program to choose the scaling factor. This 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 @@ -85,16 +94,17 @@ this algorithm to seven MPI benchmarks. These MPI programs are the NAS parallel 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. -\AG{Add citation for Rauber's methods. Moreover, Rauber was not alone to to this work (use ``Rauber et al.'', or ``Rauber and Gudula'', or \dots)} +proposed algorithm with Rauber and R\"{u}nger methods~\cite{3}. The comparison's results show that our algorithm gives better energy-time trade off. -% -\AG{Correctly reword the following}% -In Section~\ref{sec.relwork} we present works from other -authors. Then, in Sections~\ref{sec.ptasks} and~\ref{sec.energy}, we -introduce our model. [\dots] Finally, we conclude in -Section~\ref{sec.concl}. + +This paper is organized as follows: Section~\ref{sec.relwork} presents the works from other authors. +Section~\ref{sec.ptasks} shows the execution of parallel tasks and sources of idle times. Section~\ref{sec.energy} resumes the +energy model of homogenous platform. Section~\ref{sec.mpip} evaluates the performance of MPI program. +Section~\ref{sec.verif} verifies the performance prediction model. Section~\ref{sec.compet} presents +the energy-performance trade offs objective function. Section~\ref{sec.optim} demonstrates the proposed +energy-performance algorithm. Section~\ref{sec.expe} presents the results of our experiments. +Section~\ref{sec.compare} shows the comparison results. Finally, we conclude in Section~\ref{sec.concl}. \section{Related Works} \label{sec.relwork} @@ -102,56 +112,51 @@ Section~\ref{sec.concl}. \AG{Consider introducing the models (sec.~\ref{sec.ptasks}, maybe~\ref{sec.energy}) before related works} -In the this section some heuristics, to compute the scaling factor, are -presented and classified in two parts : offline and online methods. +In the this section some heuristics to compute the scaling factor are +presented and classified in two parts: offline and online methods. \subsection{The offline DVFS orientations} The DVFS offline methods are static and are not executed during the runtime of the program. Some approaches used heuristics to select the best DVFS state -during the compilation phases as an example in Azevedo et al.~\cite{40}. He used -intra-task algorithm -\AG{what is an ``intra-task algorithm''?} -to choose the DVFS setting when there are dependency points +during the compilation phases as for example in Azevedo et al.~\cite{40}. They use +dynamic voltage scaling (DVS) algorithm to choose the DVS 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 -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 -algorithm for the same goal. The offline study that shown the DVFS impact on the -communication time of the MPI program is~\cite{17}, Freeh et al. show that these -times not changed when the frequency is scaled down. +algorithm to do that. Their goal is to save energy with time limits. Another +approach gathers and stores the runtime information for each DVFS state, then +selects the suitable DVFS offline to optimize energy-time +trade offs. As an example Rountree et al.~\cite{8}, use liner programming +algorithm, while in~\cite{38,34}, Cochran et al. use multi logistic regression +algorithm for the same goal. The offline study that shows the DVFS impact on the +communication time of the MPI program is~\cite{17}, where Freeh et al. show that these +times do not change when the frequency is scaled down. \subsection{The online DVFS orientations} -The objective of these works is to dynamically compute and set the frequency of +The objective of online DVFS orientations works is to dynamically compute and set the frequency of the CPU during the runtime of the program for saving energy. Estimating and -predicting approaches for the energy-time trade offs developed by +predicting approaches for the energy-time trade offs are developed by Kimura, Peraza, Yu-Liang et al. ~\cite{11,2,31}. These works select the best DVFS setting depending on the slack times. These times happen when the processors have to wait for data from other processors to compute their task. For example, during the synchronous -communication time that take place in the MPI programs, the processors are -idle. The optimal DVFS can be selected using the learning methods. Therefore, in +communications that take place in MPI programs, some processors are +idle. The optimal DVFS can be selected using learning methods. Therefore, in Dhiman, Hao Shen et al. ~\cite{39,19} used machine learning to converge to the suitable DVFS -configuration. Their learning algorithms have big time to converge when the -number of available frequencies is high. Also, the communication time of the MPI -program used online for saving energy as in~\cite{1}, Lim et al. developed an +configuration. Their learning algorithms take big time to converge when the +number of available frequencies is high. Also, the communication sections of the MPI +program can be used to save energy. 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}, -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 -for maintaining performance. In this paper we compare our algorithm with -Rauber's model~\cite{3}, because his model can be used for any number of -concurrent tasks for homogeneous platform and this is the same direction of this -paper. However, the primary contributions of this paper are: +of analytical modeling used for choosing the optimal frequency as inRauber and R\"{u}nger~\cite{3}. they +developed an analytical mathematical model to determine the +optimal frequency scaling factor for any number of concurrent tasks. They set the slowest task to maximum frequency for maintaining performance. In this paper we compare our algorithm with +Rauber and R\"{u}nger model~\cite{3}, because their model can be used for any number of +concurrent tasks for homogeneous platforms. The primary contributions of this paper are: \begin{enumerate} -\item Selecting the optimal frequency scaling factor for energy and performance - simultaneously. While taking into account the communication time. -\item Adapting our scale factor to taking into account the imbalanced tasks. +\item Selecting the frequency scaling factor for simultaneously optimizing energy and performance, + while taking into account the communication time. +\item Adapting our scaling factor to take into account the imbalanced tasks. \item The execution time of our algorithm is very small when compared to other methods (e.g.,~\cite{19}). \item The proposed algorithm works online without profiling or training as @@ -161,64 +166,61 @@ paper. However, the primary contributions of this paper are: \section{Parallel Tasks Execution on Homogeneous Platform} \label{sec.ptasks} -A homogeneous cluster consists of identical nodes in terms of the hardware and -the software. Each node has its own memory and at least one processor which can +A homogeneous cluster consists of identical nodes in terms of hardware and software. +Each node has its own memory and at least one processor which can be a multi-core. The nodes are connected via a high bandwidth network. Tasks executed on this model can be either synchronous or asynchronous. In this paper we consider execution of the synchronous tasks on distributed homogeneous -platform. These tasks can exchange the data via synchronous memory passing. +platform. These tasks can exchange the data via synchronous message passing. \begin{figure*}[t] \centering - \subfloat[Sync. Imbalanced Communications]{\includegraphics[scale=0.67]{synch_tasks}\label{fig:h1}} + \subfloat[Sync. Imbalanced Communications]{\includegraphics[scale=0.67]{commtasks}\label{fig:h1}} \subfloat[Sync. Imbalanced Computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}} \caption{Parallel Tasks on Homogeneous Platform} \label{fig:homo} \end{figure*} -\AG{On fig.~\ref{fig:h1}, how can there be a synchronization point without communications just before ?\\ -Use ``Sync.'' to abbreviate ``Synchronization''} Therefore, the execution time of a task consists of the computation time and the communication time. Moreover, the synchronous communications between tasks can -lead to idle time while tasks wait at the synchronous point for others tasks to -finish their communications see figure~(\ref{fig:h1}). Another source for idle -times is the imbalanced computations. This happen when processing different -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 : +lead to idle time while tasks wait at the synchronization barrier for other tasks to +finish their communications (see figure~(\ref{fig:h1})). The imbalanced communications happen when nodes have to send/receive different amount of data or each node is communicates with different number of nodes. Another source for idle times is the imbalanced computations. This happen when processing different +amounts of data on each processor (see figure~(\ref{fig:h2})). In +this case the fastest tasks have to wait at the synchronization barrier for the +slowest tasks to finish their job. In both cases the overall execution time +of the program is the execution time of the slowest task as: \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 $i$. +where $T_i$ is the execution time of task $i$. \section{Energy Model for Homogeneous Platform} \label{sec.energy} -The energy consumption by the processor consists of two powers metric: the +The energy consumption by the processor consists of two power metrics: 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~\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} - \textit P_{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f + P_\textit{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} - \textit P_{static} = V \cdot N \cdot K_{design} \cdot I_{leak} + P_\textit{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 +of the dynamic and the static power multiplied by the execution time for example see~\cite{36,15}. \begin{equation} \label{eq:eind} - \textit E_{ind} = ( P_{dyn} + P_{static} ) \cdot T + E_\textit{ind} = ( P_\textit{dyn} + P_\textit{static} ) \cdot T \end{equation} -The dynamic voltage and frequency scaling (DVFS) is a process that allowed in +The dynamic voltage and frequency scaling (DVFS) is a process that is allowed in modern processors to reduce the dynamic power by scaling down the voltage and frequency. Its main objective is to reduce the overall energy consumption~\cite{37}. The operational frequency \emph f depends linearly on the @@ -229,29 +231,27 @@ 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}} + S = \frac{F_\textit{max}}{F_\textit{new}} \end{equation} The value of the scale $S$ is greater than 1 when changing the frequency to -any new frequency value (\emph {P-state}) in governor. -\AG{Explain what's a governor} -It is equal to 1 when the -frequency are set to the maximum frequency. The energy consumption model for -parallel homogeneous platform is depending on the scaling factor \emph S. This -factor reduces quadratically the dynamic power. Also, this factor increases the +any new frequency value~(\emph {P-state}) in governor, the CPU governor is an interface +driver supplied by the operating system kernel (e.g. Linux) to lowering core's frequency. +The scaling factor is equal to 1 when the frequency set is to the maximum frequency. +The energy consumption model for parallel homogeneous platform depends on the scaling factor \emph S. This factor reduces quadratically the dynamic power. Also, this factor increases the static energy linearly because the execution time is increased~\cite{36}. The -energy model, depending on the frequency scaling factor, of homogeneous platform -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}). +energy model depending on the frequency scaling factor for homogeneous platform +for any number of concurrent tasks was developed by Rauber and R\"{u}nger~\cite{3}. This model +considers the two power metrics for measuring the energy of the parallel tasks as +in EQ~(\ref{eq:energy}): \begin{equation} \label{eq:energy} - E = P_{dyn} \cdot S_1^{-2} \cdot + E = P_\textit{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 + P_\textit{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 +where \emph N is the number of parallel nodes, $T_1 $ is the time of the slowest 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 @@ -263,51 +263,50 @@ the time value $T_i$ depends on the new frequency value as in EQ~(\ref{eq:si}). \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} + = \frac{F_\textit{max}}{F_\textit{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 -for homogeneous platform that we work on in this paper. 2-we are compare our -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 +where $F$ is the number of available frequencies. In this paper we depend on +Rauber and R\"{u}nger energy model EQ~(\ref{eq:energy}) for two reasons: (1)-this model is used +for homogeneous platform that we work on in this paper. 2-we compare our +algorithm with Rauber and R\"{u}nger scaling model. Rauber and R\"{u}nger scaling factor that reduce + energy consumption derived from the EQ~(\ref{eq:energy}). They take 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} - \textit S_{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_{dyn}}{P_{static}} \cdot + S_\textit{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) } \end{equation} \section{Performance Evaluation of MPI Programs} \label{sec.mpip} -The performance (execution time) of the parallel MPI applications are depends on +The performance (execution time) of parallel MPI applications depend on the time of the slowest task as in figure~(\ref{fig:homo}). Normally the execution time of the parallel programs are proportional to the operational -frequency. Therefore, any DVFS operation for the energy reduction increase the +frequency. Therefore, any DVFS operation for the energy reduction increases 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 +energy is 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 -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 +to EQ~(\ref{eq:s}), the value of the scale $S$ has inverse relation with +new frequency value ($S \propto \frac{1}{F_{new}}$). Also when decreasing the +frequency value, the execution time increases. Then the new frequency value has +inverse relation with time ($F_{new} \propto \frac{1}{T}$). This leads to the frequency scaling factor $S$ proportional linearly with execution time ($S \propto T$). Large scale MPI applications such as NAS benchmarks have considerable amount of communications embedded in these programs. During the -communication process the processor remain idle until the communication has +communication process the processors remain idle until the communication has finished. For that reason any change in the frequency has no impact on the time of communication but it has obvious impact on the time of -computation~\cite{17}. We are made many tests on real cluster to prove that the +computation~\cite{17}. We have made many tests on a real cluster to prove that the frequency scaling factor \emph S has a linear relation with computation time -only also see~\cite{41}. To predict the execution time of MPI program, firstly -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}). +only. To predict the execution time of MPI program, the communication time and +the computation time for the slower task must be first precisely specified. Secondly, +these times are used to predict 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} - \textit T_{new} = T_{\textit{Max Comp Old}} \cdot S + T_{\textit{Max Comm Old}} + \textit T_\textit{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 @@ -364,26 +363,19 @@ between the consumed energy with scaled frequency and the consumed energy without scaled frequency : \begin{multline} \label{eq:enorm} - \textit E_{Norm} = \frac{\textit E_{Reduced}}{\textit E_{Original}} \\ - {} = \frac{ P_{dyn} \cdot S_i^{-2} \cdot + E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\ + {} = \frac{P_\textit{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 } + P_\textit{static} \cdot T_1 \cdot S_i \cdot N }{ + P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) + + P_\textit{static} \cdot T_1 \cdot N } \end{multline} -\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} -\textit P_{Norm} = \frac{\textit T_{New}}{\textit T_{Old}} - = \frac{T_{\textit{Max Comp Old}} \cdot S + - T_{\textit{Max Comm Old}}}{\textit T_{Old}} + P_\textit{Norm} = \frac{T_\textit{New}}{T_\textit{Old}} + = \frac{T_\textit{Max Comp Old} \cdot S + + T_\textit{Max Comm Old}}{ T_\textit{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 @@ -402,17 +394,17 @@ have the same direction. Therefore, we inverse the equation of normalize performance as follows : \begin{equation} \label{eq:pnorm_en} -\textit P^{-1}_{Norm} = \frac{\textit T_{Old}}{\textit T_{New}} - = \frac{\textit T_{Old}}{T_{\textit{Max Comp Old}} \cdot S + - T_{\textit{Max Comm Old}}} + P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}} + = \frac{ T_\textit{Old}}{T_\textit{Max Comp Old} \cdot S + + T_\textit{Max Comm Old}} \end{equation} \begin{figure*} \centering \subfloat[Converted Relation.]{% - \includegraphics[width=.33\textwidth]{file.eps}\label{fig:r1}}% + \includegraphics[width=.4\textwidth]{file.eps}\label{fig:r1}}% \qquad% \subfloat[Real Relation.]{% - \includegraphics[width=.33\textwidth]{file3.eps}\label{fig:r2}} + \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}} \label{fig:rel} \caption{The Energy and Performance Relation} \end{figure*} @@ -424,15 +416,15 @@ in the same time, see figure~(\ref{fig:r1}). Then our objective function has the following form: \begin{equation} \label{eq:max} - \textit{MaxDist} = \max (\overbrace{\textit P^{-1}_{Norm}}^{\text{Maximize}} - - \overbrace{\textit E_{Norm}}^{\text{Minimize}} ) + \textit{MaxDist} = \max (\overbrace{P^{-1}_\textit{Norm}}^{\text{Maximize}} - + \overbrace{E_\textit{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 -objective of this paper and we choose Rauber's model as an example with two +objective of this paper and we choose Rauber and R\"{u}nger model as an example with two reasons that mentioned before. \section{Optimal Scaling Factor for Performance and Energy} @@ -441,7 +433,7 @@ reasons that mentioned before. In the previous section we described the objective function that satisfy our goal in discovering optimal scaling factor for both performance and energy at the same time. Therefore, we develop an energy to performance scaling algorithm -(EPSA). This algorithm is simple and has a direct way to calculate the optimal +($EPSA$). This algorithm is simple and has a direct way to calculate the optimal scaling factor for both energy and performance at the same time. \begin{algorithm}[tp] \caption{EPSA} @@ -455,10 +447,11 @@ scaling factor for both energy and performance at the same time. \For {$i=1$ to $P_{states} $} \State - Calculate the new frequency as $F_{new}=F_{new} - F_{diff} $ \State - Calculate the scale factor $S$ as in EQ~(\ref{eq:s}). - \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\par + \State - Calculate all available scales $S_i$ depend on $S$ as\par\hspace{1 pt} in EQ~(\ref{eq:si}). + \State - Select the maximum scale factor $S_1$ from the set\par\hspace{1 pt} of scales $S_i$. + \State - Calculate the normalize energy $E_{Norm}=E_{R}/E_{O}$ + \par\hspace{1 pt} as in EQ~(\ref{eq:enorm}). + \State - Calculate the normalize inverse of performance\par\hspace{1 pt} $P_{NormInv}=T_{old}/T_{new}$ as in EQ~(\ref{eq:pnorm_en}). \If{ $(P_{NormInv}-E_{Norm} > Dist$) } \State $S_{optimal} = S$ @@ -480,17 +473,17 @@ system. The algorithm is called just one time during the execution of the program. The DVFS algorithm~(\ref{dvfs}) 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}} +%\AG{Use the same format as for Algorithm~\ref{$EPSA$}} \begin{algorithm}[tp] \caption{DVFS} \label{dvfs} - \begin{algorithmic} + \begin{algorithmic}[1] \For {$J:=1$ to $Some-Iterations \; $} \State -Computations Section. \State -Communications Section. \If {$(J==1)$} - \State -Gather all times of computation and communication from\par each node. + \State -Gather all times of computation and\par\hspace{13 pt} communication from each node. \State -Call EPSA with these times. \State -Calculate the new frequency from optimal scale. \State -Set the new frequency to the system. @@ -505,7 +498,7 @@ 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} \cdot T_i}{S_{optimal} \cdot T_{max}} + F_i = \frac{F_\textit{max} \cdot T_i}{S_\textit{optimal} \cdot T_\textit{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 @@ -578,7 +571,7 @@ same time over all available scales. \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} |} + \begin{tabular}{ | l | l | l |l | r |} \hline Program & Optimal & Energy & Performance&Energy-Perf.\\ Name & Scaling Factor& Saving \%&Degradation \% &Distance \\ \hline @@ -605,146 +598,146 @@ EPSA to selects smaller scaling factor values (inducing smaller energy savings). \section{Comparing Results} \label{sec.compare} -In this section, we compare our EPSA algorithm results with Rauber's +In this section, we compare our EPSA algorithm results with Rauber and R\"{u}nger methods~\cite{3}. He had two scenarios, the first is to reduce energy to optimal level without considering the performance as in EQ~(\ref{eq:sopt}). We refer to -this scenario as $Rauber_{E}$. The second scenario is similar to the first +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 possible. We refer to this -scenario as $Rauber_{E-P}$. The comparison is made in tables~(\ref{table:compare +scenario as $R_{E-P}$. The comparison is made in tables~(\ref{table:compare Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These -tables show the results of our EPSA and Rauber's two scenarios for all the NAS +tables show the results of our EPSA and Rauber and R\"{u}nger scenarios for all the NAS benchmarks programs for classes A,B and C. -\begin{table*}[p] +\begin{table}[p] \caption{Comparing Results for The NAS Class A} % title of Table \centering - \begin{tabular}{ | l | l | l |l | l | l| } + \begin{tabular}{ | l | l | l |l | l | r| } \hline Method&Program&Factor& Energy& Performance &Energy-Perf.\\ - name &name&value& Saving \%&Degradation \% &Distance + Name &Name&Value& Saving \%&Degradation \% &Distance \\ \hline % \rowcolor[gray]{0.85} - EPSA&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline - $Rauber_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline - $Rauber_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline + $EPSA$&CG & 1.56 &37.02 & 13.88 & 23.14\\ \hline + $R_{E-P}$&CG &2.14 &42.77 & 25.27 & 17.50\\ \hline + $R_{E}$&CG &2.14 &42.77&26.46&16.31\\ \hline - EPSA&MG & 1.47 &27.66&16.82&10.84\\ \hline - $Rauber_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline - $Rauber_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline + $EPSA$&MG & 1.47 &27.66&16.82&10.84\\ \hline + $R_{E-P}$&MG &2.14&34.45&31.84&2.61\\ \hline + $R_{E}$&MG &2.14&34.48&33.65&0.80 \\ \hline - EPSA&EP &1.19 &25.32&20.79&4.53\\ \hline - $Rauber_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline - $Rauber_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline + $EPSA$&EP &1.19 &25.32&20.79&4.53\\ \hline + $R_{E-P}$&EP&2.05&41.45&55.67&-14.22\\ \hline + $R_{E}$&EP&2.05&42.09&57.59&-15.50\\ \hline - EPSA&LU&1.56& 39.55 &19.38& 20.17\\ \hline - $Rauber_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline - $Rauber_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline + $EPSA$&LU&1.56& 39.55 &19.38& 20.17\\ \hline + $R_{E-P}$&LU&2.14&45.62&27.00&18.62 \\ \hline + $R_{E}$&LU&2.14&45.66&33.01&12.65\\ \hline - EPSA&BT&1.31& 29.60&20.53&9.07 \\ \hline - $Rauber_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline - $Rauber_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline + $EPSA$&BT&1.31& 29.60&20.53&9.07 \\ \hline + $R_{E-P}$&BT&2.10&45.53&49.63&-4.10\\ \hline + $R_{E}$&BT&2.10&43.93&52.86&-8.93\\ \hline - EPSA&SP&1.38& 33.51&15.65&17.86 \\ \hline - $Rauber_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline - $Rauber_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline + $EPSA$&SP&1.38& 33.51&15.65&17.86 \\ \hline + $R_{E-P}$&SP&2.11&45.62&42.52&3.10\\ \hline + $R_{E}$&SP&2.11&45.78&43.09&2.69\\ \hline - EPSA&FT&1.25&25.00&10.80&14.20 \\ \hline - $Rauber_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline - $Rauber_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline + $EPSA$&FT&1.25&25.00&10.80&14.20 \\ \hline + $R_{E-P}$&FT&2.10&39.29&34.30&4.99 \\ \hline + $R_{E}$&FT&2.10&37.56&38.21&-0.65\\ \hline \end{tabular} \label{table:compare Class A} % is used to refer this table in the text -\end{table*} -\begin{table*}[p] +\end{table} +\begin{table}[p] \caption{Comparing Results for The NAS Class B} % title of Table \centering - \begin{tabular}{ | l | l | l |l | l |l| } + \begin{tabular}{ | l | l | l |l | l |r| } \hline Method&Program&Factor& Energy& Performance &Energy-Perf.\\ - name &name&value& Saving \%&Degradation \% &Distance + Name &Name&Value& Saving \%&Degradation \% &Distance \\ \hline % \rowcolor[gray]{0.85} - EPSA&CG & 1.66 &39.23&16.63&22.60 \\ \hline - $Rauber_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline - $Rauber_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline + $EPSA$&CG & 1.66 &39.23&16.63&22.60 \\ \hline + $R_{E-P}$&CG &2.15 &45.34&27.60&17.74\\ \hline + $R_{E}$&CG &2.15 &45.34&28.88&16.46\\ \hline - EPSA&MG & 1.47 &34.98&18.35&16.63\\ \hline - $Rauber_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline - $Rauber_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline + $EPSA$ &MG & 1.47 &34.98&18.35&16.63\\ \hline + $R_{E-P}$&MG &2.14&43.55&36.42&7.13 \\ \hline + $R_{E}$&MG &2.14&43.56&37.07&6.49 \\ \hline - EPSA&EP &1.08 &20.29&17.15&3.14 \\ \hline - $Rauber_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline - $Rauber_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline + $EPSA$&EP &1.08 &20.29&17.15&3.14 \\ \hline + $R_{E-P}$&EP&2.00&42.38&56.88&-14.50\\ \hline + $R_{E}$&EP&2.00&39.73&59.94&-20.21\\ \hline - EPSA&LU&1.47&38.57&21.34&17.23 \\ \hline - $Rauber_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline - $Rauber_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline + $EPSA$&LU&1.47&38.57&21.34&17.23 \\ \hline + $R_{E-P}$&LU&2.10&43.62&36.51&7.11 \\ \hline + $R_{E}$&LU&2.10&43.61&38.54&5.07 \\ \hline - EPSA&BT&1.31& 29.59&20.88&8.71\\ \hline - $Rauber_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline - $Rauber_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline + $EPSA$&BT&1.31& 29.59&20.88&8.71\\ \hline + $R_{E-P}$&BT&2.10&44.53&53.05&-8.52\\ \hline + $R_{E}$&BT&2.10&42.93&52.80&-9.87\\ \hline - EPSA&SP&1.38&33.44&19.24&14.20 \\ \hline - $Rauber_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline - $Rauber_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline + $EPSA$&SP&1.38&33.44&19.24&14.20 \\ \hline + $R_{E-P}$&SP&2.15&45.69&43.20&2.49\\ \hline + $R_{E}$&SP&2.15&45.41&44.47&0.94\\ \hline - EPSA&FT&1.38&34.40&14.57&19.83 \\ \hline - $Rauber_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline - $Rauber_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline + $EPSA$&FT&1.38&34.40&14.57&19.83 \\ \hline + $R_{E-P}$&FT&2.13&42.98&37.35&5.63 \\ \hline + $R_{E}$&FT&2.13&43.04&37.90&5.14\\ \hline \end{tabular} \label{table:compare Class B} % is used to refer this table in the text -\end{table*} +\end{table} -\begin{table*}[p] +\begin{table}[p] \caption{Comparing Results for The NAS Class C} % title of Table \centering - \begin{tabular}{ | l | l | l |l | l |l| } + \begin{tabular}{ | l | l | l |l | l |r| } \hline Method&Program&Factor& Energy& Performance &Energy-Perf.\\ - name &name&value& Saving \%&Degradation \% &Distance + Name &Name&Value& Saving \%&Degradation \% &Distance \\ \hline % \rowcolor[gray]{0.85} - EPSA&CG & 1.56 &39.23&14.88&24.35 \\ \hline - $Rauber_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline - $Rauber_{E}$&CG &2.15 &45.36&26.70&18.66\\ \hline + $EPSA$&CG & 1.56 &39.23&14.88&24.35 \\ \hline + $R_{E-P}$&CG &2.15 &45.36&25.89&19.47\\ \hline + $R_{E}$&CG &2.15 &45.36&26.70&18.66\\ \hline - EPSA&MG & 1.47 &34.97&21.69&13.27\\ \hline - $Rauber_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline - $Rauber_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline + $EPSA$&MG & 1.47 &34.97&21.69&13.27\\ \hline + $R_{E-P}$&MG &2.15&43.65&40.45&3.20 \\ \hline + $R_{E}$&MG &2.15&43.64&41.38&2.26 \\ \hline - EPSA&EP &1.04 &22.14&20.73&1.41 \\ \hline - $Rauber_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline - $Rauber_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline + $EPSA$&EP &1.04 &22.14&20.73&1.41 \\ \hline + $R_{E-P}$&EP&1.92&39.40&56.33&-16.93\\ \hline + $R_{E}$&EP&1.92&38.10&56.35&-18.25\\ \hline - EPSA&LU&1.38&35.83&22.49&13.34 \\ \hline - $Rauber_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline - $Rauber_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline + $EPSA$&LU&1.38&35.83&22.49&13.34 \\ \hline + $R_{E-P}$&LU&2.15&44.97&41.00&3.97 \\ \hline + $R_{E}$&LU&2.15&44.97&41.80&3.17 \\ \hline - EPSA&BT&1.31& 29.60&21.28&8.32\\ \hline - $Rauber_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline - $Rauber_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline + $EPSA$&BT&1.31& 29.60&21.28&8.32\\ \hline + $R_{E-P}$&BT&2.13&45.60&49.84&-4.24\\ \hline + $R_{E}$&BT&2.13&44.90&55.16&-10.26\\ \hline - EPSA&SP&1.38&33.48&21.35&12.12\\ \hline - $Rauber_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline - $Rauber_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline + $EPSA$&SP&1.38&33.48&21.35&12.12\\ \hline + $R_{E-P}$&SP&2.10&45.69&43.60&2.09\\ \hline + $R_{E}$&SP&2.10&45.75&44.10&1.65\\ \hline - EPSA&FT&1.47&34.72&19.00&15.72 \\ \hline - $Rauber_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline - $Rauber_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline + $EPSA$&FT&1.47&34.72&19.00&15.72 \\ \hline + $R_{E-P}$&FT&2.04&39.40&37.10&2.30\\ \hline + $R_{E}$&FT&2.04&39.35&37.70&1.65\\ \hline \end{tabular} \label{table:compare Class C} % is used to refer this table in the text -\end{table*} +\end{table} As shown in these tables our scaling factor is not optimal for energy saving such as Rauber's scaling factor EQ~(\ref{eq:sopt}), but it is optimal for both -the energy and the performance simultaneously. Our EPSA optimal scaling factors +the energy and the performance simultaneously. Our $EPSA$ optimal scaling factors has better simultaneous optimization for both the energy and the performance -compared to Rauber's energy-performance method ($Rauber_{E-P}$). Also, in -($Rauber_{E-P}$) method when setting the frequency to maximum value for the +compared to Rauber and R\"{u}nger energy-performance method ($R_{E-P}$). Also, in +($R_{E-P}$) method when setting the frequency to maximum value for the slower task lead to a small improvement of the performance. Also the results show that this method keep or improve energy saving. Because of the energy consumption decrease when the execution time decreased while the frequency value @@ -753,8 +746,8 @@ increased. Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving percent and the performance degradation percent. Therefore, this means it is the same resultant of our objective function EQ~(\ref{eq:max}). Our algorithm always -gives positive energy to performance trade offs while Rauber's method -($Rauber_{E-P}$) gives in some time negative trade offs such as in BT and +gives positive energy to performance trade offs while Rauber and R\"{u}nger method +($R_{E-P}$) gives in some time negative trade offs such as in BT and EP. The positive trade offs with highest values lead to maximum energy savings concatenating with less performance degradation and this the objective of this paper. While the negative trade offs refers to improving energy saving (or may @@ -765,7 +758,7 @@ than the first. \includegraphics[width=.33\textwidth]{compare_class_A.pdf} \includegraphics[width=.33\textwidth]{compare_class_B.pdf} \includegraphics[width=.33\textwidth]{compare_class_c.pdf} - \caption{Comparing Our EPSA with Rauber's Methods} + \caption{Comparing Our EPSA with Rauber and R\"{u}nger Methods} \label{fig:compare} \end{figure} @@ -798,4 +791,4 @@ Mésocentre de calcul de Franche-Comté. %%% End: % LocalWords: Badri Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber -% LocalWords: CMOS EQ EPSA Franche Comté Tflop +% LocalWords: CMOS EQ $$EPSA$$ Franche Comté Tflop -- 2.39.5