Section~\ref{sec.expe} presents the results of applying the algorithm on the NAS parallel benchmarks and executing them
on a heterogeneous platform. It also shows the results of running three
different power scenarios and comparing them.
-Finally, we conclude in Section~\ref{sec.concl} with a summary and some future works.
+Finally, in Section~\ref{sec.concl} the paper is ended with a summary and some future works.
\section{Related works}
\label{sec.relwork}
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 platforms. A heterogeneous platform is defined 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
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 \subsection{The verifications of the proposed method}
-\label{sec.verif}
+\label{sec.verif.method}
The precision of the proposed algorithm mainly depends on the execution time prediction model defined in
EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}).
The energy model is also significantly dependent on the execution time model because the static energy is
of iterations and $N$ is the number of computing nodes. The algorithm needs from 12 to 20 iterations to select the best
vector of frequency scaling factors that gives the results of the sections (\ref{sec.res}) and (\ref{sec.compare}).
slowest node, it means only the communication time without any slack time.
-Therefore, we can consider the execution time of the iterative application is
+Therefore, 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 developed from our model for predicting the execution time of
+This prediction model is developed from the model for predicting the execution time of
message passing distributed applications for homogeneous architectures~\cite{Our_first_paper}.
-The execution time prediction model is used in our method for optimizing both
+The execution time prediction model is used in the method for optimizing both
energy consumption and performance of iterative methods, which is presented in the
following sections.
process of the frequency can be expressed by the scaling factor $S$ which is the
ratio between the maximum and the new frequency as in EQ(\ref{eq:s}).
The CPU governors are power schemes supplied by the operating
-system's kernel to lower a core's frequency. we can calculate the new frequency
-$F_{new}$ from EQ(\ref{eq:s}) as follow:
+system's kernel to lower a core's frequency. The new frequency
+$F_{new}$ from EQ(\ref{eq:s}) can be calculated as follows:
\begin{equation}
\label{eq:fnew}
F_\textit{new} = S^{-1} \cdot F_\textit{max}
\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{Rauber_Analytical.Modeling.for.Energy,Zhuo_Energy.efficient.Dynamic.Task.Scheduling},
-we assume that the static power of a processor is constant
+ the static power of a processor is considered as 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
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{Kim_Leakage.Current.Moore.Law}. 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})
+increased~\cite{Kim_Leakage.Current.Moore.Law}. The overall energy consumption for the iterative
+application can be measured by measuring the energy consumption for one iteration as in EQ(\ref{eq:energy})
multiplied by the number of iterations of that application.
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{Freeh_Exploring.the.Energy.Time.Tradeoff}.
-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
+Moreover, they are not measured using the same metric. To solve this problem, the
+execution time is normalized 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}
\end{multline}
-In the same way, we normalize the energy by computing the ratio between the consumed energy
+In the same way, the energy is normalized 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}
-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:
+This problem can be solved by making the optimization process for energy and
+execution time follow the same direction. Therefore, the equation of the
+normalized execution time is inverted which gives the normalized performance equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
P_\textit{Norm} = \frac{T_\textit{Old}}{T_\textit{New}}\\
\caption{The energy and performance relation}
\end{figure}
-Then, we can model our objective function as finding the maximum distance
+Then, the objective function can be modeled 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 (maximum
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then our objective
+performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then the objective
function has the following form:
\begin{equation}
\label{eq:max}
\overbrace{E_\textit{Norm}(S_{ij})}^{\text{Minimize}} )
\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 any power values for each node
+Then, the optimal set of scaling factors that satisfies EQ~(\ref{eq:max}) can be selected.
+The objective function can 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{Zhuo_Energy.efficient.Dynamic.Task.Scheduling,Rauber_Analytical.Modeling.for.Energy,Hao_Learning.based.DVFS}.
\label{sec.optim}
\subsection{The algorithm details}
-In this section we propose algorithm~(\ref{HSA}) which selects the frequency scaling factors
+In this section algorithm~(\ref{HSA}) is presented. It 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.
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
+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 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
\end{algorithm}
\subsection{The verifications of the proposed algorithm}
-\label{sec.verif}
+\label{sec.verif.algo}
The precision of the proposed algorithm mainly depends on the execution time prediction model defined in
EQ(\ref{eq:perf}) and the energy model computed by EQ(\ref{eq:energy}).
The energy model is also significantly dependent on the execution time model because the static energy is
\section{Conclusion}
\label{sec.concl}
-In this paper, we have presented a new online selecting frequency scaling factors algorithm
-that selects the best possible vector of frequency scaling factors for a heterogeneous platform.
+In this paper, a new online frequency selecting algorithm have been presented. It selects the best possible vector of frequency scaling factors for a heterogeneous platform.
This vector gives the maximum distance (optimal tradeoff) between the predicted energy and
the predicted performance curves. In addition, we developed a new energy model for measuring
and predicting the energy of distributed iterative applications running over heterogeneous
