From c0e417f74debcd978ae5229e414981fc3c45a19a Mon Sep 17 00:00:00 2001 From: jean-claude Date: Mon, 21 Sep 2015 14:46:01 +0200 Subject: [PATCH] I have corrected section 5 --- Heter_paper.tex | 47 +++++++++++++++++++++++------------------------ 1 file changed, 23 insertions(+), 24 deletions(-) diff --git a/Heter_paper.tex b/Heter_paper.tex index afad54b..0cc9720 100644 --- a/Heter_paper.tex +++ b/Heter_paper.tex @@ -407,7 +407,7 @@ values for each node (static and dynamic powers). However, the most important energy reduction gain can be achieved when the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}. -\section{The scaling factors selection algorithm for heterogeneous grid platforms } +\section{The scaling factors selection algorithm for grids } \label{sec.optim} \begin{algorithm} @@ -479,10 +479,11 @@ in~\cite{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modelin \subsection{The algorithm details} \textcolor{red}{Delete the subsection if there's only one.} -In this section, Algorithm~\ref{HSA} is presented. It selects the frequency -scaling factors vector that gives the best trade-off between minimizing the + +In this section, the scaling factors selection algorithm for grids, algorithm~\ref{HSA}, is presented. It selects the vector of the frequency +scaling factors that gives the best trade-off between minimizing the energy consumption and maximizing the performance of a message passing -synchronous iterative application executed on a heterogeneous grid platform. It works +synchronous iterative application executed on a grid. It works online during the execution time of the iterative message passing program. It uses information gathered during the first iteration such as the computation time and the communication time in one iteration for each node. The algorithm is @@ -500,13 +501,13 @@ scaling algorithm is called in the iterative MPI program. \label{fig:st_freq} \end{figure} -The nodes in a heterogeneous grid have different computing powers, thus +Nodes from distinct clusters in a grid have different computing powers, thus while executing message passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their computations before being able to synchronously communicate with them as in Figure~\ref{fig:heter}. These periods are called idle or slack times. The algorithm takes into account this -problem and tries to reduce these slack times when selecting the frequency -scaling factors vector. At first, it selects initial frequency scaling factors +problem and tries to reduce these slack times when selecting the vector of the frequency +scaling factors. At first, it selects initial frequency scaling factors that increase the execution times of fast nodes and minimize the differences between the computation times of fast and slow nodes. The value of the initial frequency scaling factor for each node is inversely proportional to its @@ -527,25 +528,24 @@ follows: \end{equation} If the computed initial frequency for a node is not available in the gears of that node, it is replaced by the nearest available frequency. In -Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in +Figure~\ref{fig:st_freq}, the nodes are sorted by their computing powers in ascending order and the frequencies of the faster nodes are scaled down according to the computed initial frequency scaling factors. The resulting new frequencies are highlighted in Figure~\ref{fig:st_freq}. This set of frequencies can be considered as a higher bound for the search space of the -optimal vector of frequencies because selecting scaling factors higher +optimal vector of frequencies because selecting higher frequencies than the higher bound will not improve the performance of the application and it will increase its overall energy consumption. Therefore the algorithm that selects the frequency scaling factors starts the search method from these initial frequencies and takes a downward search direction toward lower -frequencies or reaching to the lower bound. The lower bound is used to stop -the algorithm search process when the new computed distance between the energy and performance is less than zero. -The new negative distance is mean that the performance degradation ratio is higher than energy saving ratio. -Therefore, the algorithm must stop the iterations before reaching to the end of the search space, the minimum frequencies, -because the all the coming new distances are negative values. -The algorithm iterates on all remaining frequencies, from the higher -bound until all nodes reach their minimum frequencies or to the lower bound, 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 +frequencies until reaching the nodes' minimum frequencies or lower bounds. A node's frequency is considered its lower bound if the computed distance between the energy and performance at this frequency is less than zero. +A negative distance means that the performance degradation ratio is higher than the energy saving ratio. +In this situation, the algorithm must stop the downward search because it has reached the lower bound and it is useless to test the lower frequencies. Indeed, they will all give worse distances. + +Therefore, the algorithm iterates on all remaining frequencies, from the higher +bound until all nodes reach their minimum frequencies or their lower bounds, to compute the overall +energy consumption and performance and selects the optimal vector of the frequency scaling +factors. At each iteration the algorithm determines the slowest node according to the equation (\ref{eq:perf}) and keeps its frequency unchanged, while it lowers the frequency of all other nodes by one gear. The new overall energy consumption and execution time are computed according to the new scaling @@ -553,18 +553,17 @@ factors. The optimal set of frequency scaling factors is the set that gives the highest distance according to the objective function (\ref{eq:max}). Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and -consumed energy for an application running on a homogeneous platform and a -heterogeneous grid platform respectively while increasing the scaling factors. It can -be noticed that in a homogeneous platform the search for the optimal scaling +consumed energy for an application running on a homogeneous cluster and a + grid platform respectively while increasing the scaling factors. It can +be noticed that in a homogeneous cluster the search for the optimal scaling factor should start from the maximum frequency because the performance and the consumed energy decrease from the beginning of the plot. On the other hand, in -the heterogeneous grid platform the performance is maintained at the beginning of the +the grid platform the performance is maintained at the beginning of the plot even if the frequencies of the faster nodes decrease until the computing power of scaled down nodes are lower than the slowest node. In other words, until they reach the higher bound. It can also be noticed that the higher the difference between the faster nodes and the slower nodes is, the bigger the -maximum distance between the energy curve and the performance curve is while the -scaling factors are varying which results in bigger energy savings. +maximum distance between the energy curve and the performance curve is, which results in bigger energy savings. \section{Experimental results} -- 2.39.5