-\title{Optimizing Energy Consumption with DVFS for Message \\
- Passing Applications \textcolor{blue}{with iterations} on \\
- Grid Architectures}
+\title{Optimizing the Energy Consumption \\
+of Message Passing Applications with Iterations \\
+Executed over Grids}
In this paper, a new online frequency selecting algorithm for grids, composed of heterogeneous clusters, is presented.
It selects the frequencies and tries to give the best
trade-off between energy saving and performance degradation, for each node
- computing the message passing application \textcolor{blue}{with iterations}.
+ computing the message passing application with iterations.
The algorithm has a small
overhead and works without training or profiling. It uses a new energy model
- for message passing applications \textcolor{blue}{with iterations} running on a grid.
+ for message passing applications with iterations running on a grid.
The proposed algorithm is evaluated on a real grid, the Grid'5000 platform, while
running the NAS parallel benchmarks. The experiments on 16 nodes, distributed on three clusters, show that it reduces on average the
energy consumption by \np[\%]{30} while the performance is on average only degraded
50.32 kilowatts.
Besides platform improvements, there are many software and hardware techniques
-to lower the energy consumption of these platforms, such as DVFS, scheduling \textcolor{blue}{and other techniques}.
+to lower the energy consumption of these platforms, such as DVFS, scheduling and other techniques.
DVFS is a widely used process to reduce the energy consumption of a
processor by lowering its frequency
\cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
trade-off between the energy reduction and performance degradation ratio. In
\cite{Our_first_paper} and \cite{pdsec2015}, a frequency selecting algorithm
was proposed to reduce the energy consumption of message passing
-applications \textcolor{blue}{with iterations} running over homogeneous and heterogeneous clusters respectively.
+applications with iterations running over homogeneous and heterogeneous clusters respectively.
The results of the experiments showed significant energy consumption
reductions. All the experimental results were conducted over the SimGrid
simulator \cite{SimGrid}, which offers easy tools to describe homogeneous and heterogeneous platforms, and to simulate the execution of message passing parallel
applications over them.
-In this paper, a new frequency selecting algorithm, adapted to grid platforms
-composed of heterogeneous clusters, is presented. It is applied to the NAS
+
+This paper presents the following contributions :
+\begin{enumerate}
+\item two new energy and performance models for message passing
+ synchronous applications with iterations running over a heterogeneous grid platform. Both models
+ take into account communications and slack times. The models can predict the
+ required energy and the execution time of the application.
+
+\item a new online frequency selecting algorithm for heterogeneous grid
+ platforms. The algorithm has a very small overhead and does not need any
+ training nor profiling. It uses a new optimization function which
+ simultaneously maximizes the performance and minimizes the energy consumption
+ of a message passing synchronous application with iterations. The algorithm was applied to the NAS
parallel benchmarks and evaluated over a real testbed, the Grid'5000 platform
-\cite{grid5000}. It selects for a grid platform running a message passing
- application \textcolor{blue}{with iterations} the vector of frequencies that simultaneously tries to
-offer the maximum energy reduction and minimum performance degradation
-ratios. The algorithm has a very small overhead, works online and does not need
-any training or profiling.
+\cite{grid5000}.
+
+\end{enumerate}
+
This paper is organized as follows: Section~\ref{sec.relwork} presents some
sequential, parallel or distributed architecture, homogeneous or heterogeneous
platform, synchronous or asynchronous application.
-In this paper, we are interested in reducing energy for message passing
- synchronous applications \textcolor{blue}{with iterations} running over heterogeneous grid platforms. Some
+In this paper, we are interested in reducing the energy consumption of message passing
+ synchronous applications with iterations running over heterogeneous grid platforms. Some
works have already been done for such platforms and they can be classified into
two types of heterogeneous platforms:
\begin{itemize}
al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
programming approach to minimize the power consumption of heterogeneous servers
while respecting given time constraints. This approach had considerable
-overhead. In contrast to the above described papers, this paper presents the
-following contributions :
-\begin{enumerate}
-\item two new energy and performance models for message passing
- synchronous applications \textcolor{blue}{with iterations} running over a heterogeneous grid platform. Both models
- take into account communication and slack times. The models can predict the
- required energy and the execution time of the application.
-
-\item a new online frequency selecting algorithm for heterogeneous grid
- platforms. The algorithm has a very small overhead and does not need any
- training nor profiling. It uses a new optimization function which
- simultaneously maximizes the performance and minimizes the energy consumption
- of a message passing synchronous application \textcolor{blue}{with iterations}.
-
-\end{enumerate}
+overhead.
\label{sec.exe}
\subsection{The execution time of message passing distributed
- applications \textcolor{blue}{with iterations} on a heterogeneous platform}
+ applications with iterations on a heterogeneous platform}
In this paper, we are interested in reducing the energy consumption of message
-passing distributed synchronous applications \textcolor{blue}{with iterations} running over
+passing distributed synchronous applications with iterations running over
heterogeneous grid platforms. A heterogeneous grid platform could be defined as a collection of
heterogeneous computing clusters interconnected via a long distance network which has lower bandwidth
and higher latency than the local networks of the clusters. Each computing cluster in the grid is composed of homogeneous nodes that are connected together via high speed network. Therefore, each cluster has different characteristics such as computing power (FLOPS), energy consumption, CPU's frequency range, network bandwidth and latency.
-The overall execution time of a distributed synchronous application \textcolor{blue}{with iterations}
+The overall execution time of a distributed synchronous application with iterations running
over a heterogeneous grid consists of the sum of the computation time and
the communication time for every iteration on a node.
-\textcolor{blue}{However, nodes from distinct clusters in a grid have different computing powers, thus
-while executing message passing \textcolor{blue}{with iterations} synchronous applications, fast nodes
+However, nodes from distinct clusters in a grid have different computing powers, thus
+while the application, 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. }
+periods are called idle or slack times.
Therefore, the
overall execution time of the program is the execution time of the slowest task
-which has the highest computation time and no slack time. \textcolor{blue}{For example, in Figure \ref{fig:heter} the task 1 is the slower task which has no slack time (not waits for the other nodes) and it is only has the communication times.}
+which has the highest computation time and almost no slack time. For example, in Figure \ref{fig:heter}, task 1 is the slower task and it does not have to wait for the other nodes to communicate with them because they all finish their computations before it.
\begin{figure}[!t]
\centering
\label{eq:s}
S = \frac{\Fmax}{\Fnew}
\end{equation}
-\textcolor{blue}{Where $\Fmax$ is the maximum frequency before applying DVFS and $\Fnew$ is the new frequency after applying DVFS.}
+where $\Fmax$ is the maximum frequency before applying any DVFS and $\Fnew$ is the new frequency after applying DVFS.
+
The execution time of a compute bound sequential program is linearly
proportional to the frequency scaling factor $S$. On the other hand, message
passing distributed applications consist of two parts: computation and
of these clusters, they may get different scaling factors represented by a scaling vector:
$(S_{11}, S_{12},\dots, S_{NM_i})$ where $S_{ij}$ is the scaling factor of processor $j$ in cluster $i$ . To
be able to predict the execution time of message passing synchronous
-applications \textcolor{blue}{with iterations} running over a heterogeneous grid, for different vectors of
+applications with iterations running over a heterogeneous grid, for different vectors of
scaling factors, the communication time and the computation time for all the
tasks must be measured during the first iteration before applying any DVFS
operation. Then the execution time for one iteration of the application with any
\begin{equation}
\label{eq:perf}
\Tnew = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} \cdot S_{ij})
- +\mathop{\min_{j=1,\dots,M_i}} (\Tcm[hj])
+ +\mathop{\min_{j=1,\dots,M_h}} (\Tcm[hj])
\end{equation}
%
where $N$ is the number of clusters in the grid, $M_i$ is the number of nodes in
cluster $i$, $\TcpOld[ij]$ is the computation time of processor $j$ in the cluster $i$
and $\Tcm[hj]$ is the communication time of processor $j$ in the cluster $h$ during the
first iteration. The execution time for one iteration is equal to the sum of the maximum computation time for all nodes with the new scaling factors
-and \textcolor{blue}{the communication time of the slower node without slack time during one iteration.
-The slower node $h$ is the node that gives maximum execution time in all clusters befor scaling its frequency.}
+and the communication time of the slowest node without slack time during one iteration.
+ The slowest node $h$ is the node which takes the maximum execution time to execute an iteration before scaling down its frequency.
It means that only the communication time without any slack time is taken into account.
-Therefore, the execution time of the application \textcolor{blue}{with iterations} is equal to
+Therefore, the execution time of the application is equal to
the execution time of one iteration as in Equation (\ref{eq:perf}) multiplied by the
number of iterations of that application.
-This prediction model is developed from the model to predict the execution time
-of message passing distributed applications for homogeneous and heterogeneous clusters
-~\cite{Our_first_paper,pdsec2015}. \textcolor{blue}{where the homogeneous cluster predication model was used one scaling factor denoted as $S$, because all the nodes in the cluster have the same computing powers. Whereas, in heterogeneous cluster prediction model all the nodes have different scales and the scaling factors have denoted as one dimensional vector $(S_1, S_2, \dots, S_N)$. The execution time prediction model for a grid Equation \ref{eq:perf} defines a two dimensional array of scales
-$(S_{11}, S_{12},\dots, S_{NM_i})$}. This model is used in the method to optimize both the energy consumption and the performance of iterative methods, which is presented in the following sections.
+This model is an adaptation of the one developed in ~\cite{Our_first_paper} which predicts the execution time
+of message passing applications with iterations running on homogeneous clusters.
+In a homogeneous cluster only one scaling factor denoted as $S$ was used because all the nodes in the cluster have the same computing power.
+In a heterogeneous cluster, each node may have a different scaling factor denoted as $(S_i)$ where $i$ is the index of the node. In a grid, each node in each cluster may have a scaling factor. The whole set of scaling factors of all the computing nodes in the grid is denoted by a two dimensional array of scales
+$(S_{11}, S_{12},\dots, S_{NM_i})$ where $N$ is the number of used clusters and $M_i$ is the number of nodes in cluster $i$.
+
+The execution time model, Equation \ref{eq:perf}, is used in the algorithm presented in section \ref{sec.optim}. The latter selects the scaling factors that optimize both the energy consumption and the performance of message passing applications with iterations running on grids.
\subsection{Energy model for heterogeneous grid platform}
In the considered heterogeneous grid platform, each node $j$ in cluster $i$ may have
different dynamic and static powers from the nodes of the other clusters,
-noted as $\Pd[ij]$ and $\Ps[ij]$ respectively. \textcolor{blue}{Therefore, even if the distributed
-message passing application \textcolor{blue}{with iterations} is load balanced, the computation time of each CPU $j$
-in cluster $i$ noted $\Tcp[ij]$ may be slightly different due to the delay caused by the scheduler of the operating system}. Therefore, different frequency scaling factors may be
+noted as $\Pd[ij]$ and $\Ps[ij]$ respectively. Moreover, even if the distributed
+message passing application with iterations is load balanced, the computation time of each CPU $j$ in cluster $i$
+ noted $\Tcp[ij]$ may be slightly different due to the delay caused by the scheduler of the operating system. Therefore, different frequency scaling factors may be
computed in order to decrease the overall energy consumption of the application
and reduce the slack times. The communication time of a processor $j$ in cluster $i$ is noted as
$\Tcm[ij]$ and could contain slack times when communicating with slower nodes,
frequency scaling factor and the dynamic power of each node as in
(\ref{eq:Edyn}), the static energy is computed as the sum of the execution time
of one iteration multiplied by the static power of each processor.
-\textcolor{blue}{ The CPU during the communication times consumes only the static power. While
-in the computation times, it consumes both the dynamic and the static power refer to \cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.}
+ The CPU during the communication times consumes only the static power. While
+in the computation times, it consumes both the dynamic and the static powers, for more information refer to \cite{Freeh_Exploring.the.Energy.Time.Tradeoff}.
The overall energy consumption of a message passing distributed application executed over a
heterogeneous grid platform during one iteration is the summation of all dynamic and
static energies for $M_i$ processors in $N$ clusters. It is computed as follows:
factors $(S_{11}, S_{12},\dots, S_{NM_i})$ 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}. The overall energy consumption
-for the application \textcolor{blue}{with iterations} can be measured by measuring the energy
+for a synchronous application with iterations can be measured by measuring the energy
consumption for one iteration as in (\ref{eq:energy}) multiplied by the number
of iterations of that application.
frequency scaling factors for a homogeneous and a heterogeneous cluster respectively, were proposed.
Both methods selects the frequencies that gives the best trade-off between
energy consumption reduction and performance for message passing
- synchronous applications \textcolor{blue}{with iterations}. In this work we
-are interested in grids that are composed of heterogeneous clusters, \textcolor{blue}{where} the nodes
-have different characteristics such as dynamic power, static power, computation power,
+ synchronous applications with iterations. In this work we
+are interested in grids that are composed of heterogeneous clusters. The nodes from distinct clusters may have
+ different characteristics such as dynamic power, static power, computation power,
frequencies range, network latency and bandwidth.
Due to the heterogeneity of the processors, a vector of scaling factors should be selected
and it must give the best trade-off between energy consumption and performance.
\end{equation}
%
where $Tnew$ is computed as in (\ref{eq:perf}) and $Told$ is computed as in (\ref{eq:told}).
-\textcolor{blue}{
+
\begin{equation}
\label{eq:told}
\Told = \mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\TcpOld[ij]} )
- +\mathop{\min_{j=1,\dots,M_i}} (\Tcm[hj])
+ +\mathop{\min_{j=1,\dots,M_h}} (\Tcm[hj])
\end{equation}
-}
+
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:
\Require ~
\begin{description}
\item [{$N$}] number of clusters in the grid.
- \item [{$M$}] number of nodes in each cluster.
+ \item [{$M_i$}] number of nodes in each cluster.
\item[{$\Tcp[ij]$}] array of all computation times for all nodes during one iteration and with the highest frequency.
\item[{$\Tcm[ij]$}] array of all communication times for all nodes during one iteration and with the highest frequency.
\item[{$\Fmax[ij]$}] array of the maximum frequencies for all nodes.
In this section, the scaling factors selection algorithm for grids, Algorithm~\ref{HSA},
-is presented. It selects the vector of the frequency
+is presented. It selects the vector of frequency
scaling factors that gives the best trade-off between minimizing the
energy consumption and maximizing the performance of a message passing
-synchronous application \textcolor{blue}{with iterations} executed on a grid. It works
-online during the execution time of the message passing program \textcolor{blue}{with iterations}. It
+synchronous application with iterations executed on a grid. It works
+online during the execution time of the application. 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
executed after the first iteration and returns a vector of optimal frequency
program applies DVFS operations to change the frequencies of the CPUs according
to the computed scaling factors. This algorithm is called just once during the
execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
-scaling algorithm is called in the MPI program \textcolor{blue}{with iterations}.
+scaling algorithm is called in the application.
\begin{figure}[!t]
\centering
\end{equation}
Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
algorithm computes the initial frequencies for all nodes as a ratio between the
-maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+maximum frequency of node and its computed scaling factor as
follows:
\begin{equation}
\label{eq:Fint}
\subsection{Grid'5000 architecture and power consumption}
\label{sec.grid5000}
-Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed all over metropolitan France and Luxembourg. All the sites are connected together via a special long distance network called RENATER,
+Grid'5000~\cite{grid5000} is a large-scale testbed that consists of ten sites distributed all over metropolitan France and Luxembourg. All the sites are connected together via a special long distance network called RENATER,
which is the French National Telecommunication Network for Technology.
Each site of the grid is composed of a few heterogeneous
computing clusters and each cluster contains many homogeneous nodes. In total,
Two types of local networks are used, Ethernet or Infiniband networks which have different characteristics in terms of bandwidth and latency.
Since Grid'5000 is dedicated to testing, contrary to production grids it allows a user to deploy its own customized operating system on all the booked nodes. The user could have root rights and thus apply DVFS operations while executing a distributed application. Moreover, the Grid'5000 testbed provides at some sites a power measurement tool to capture
-the power consumption for each node in those sites. The measured power is the overall consumed power by all the components of a node at a given instant, such as CPU, hard drive, main-board and memory. For more details refer to
+the power consumption for each node in those sites. The measured power is the overall consumed power by all the components of a node at a given instant. For more details refer to
\cite{Energy_measurement}. In order to correctly measure the CPU power of one core in a node $j$,
firstly, the power consumed by the node while being idle at instant $y$, noted as $\Pidle[jy]$, was measured. Then, the power was measured while running a single thread benchmark with no communication (no idle time) over the same node with its CPU scaled to the maximum available frequency. The latter power measured at time $x$ with maximum frequency for one core of node $j$ is noted $\Pmax[jx]$. The difference between the two measured power consumptions represents the
dynamic power consumption of that core with the maximum frequency, see Figure~\ref{fig:power_cons}.
The energy model and the scaling factors selection algorithm were applied to the NAS parallel benchmarks v3.3 \cite{NAS.Parallel.Benchmarks} and evaluated over Grid'5000.
-The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. \textcolor{blue}{These benchmarks are message passing applications with iterations compute
-the same block of operations several times, starting from the initial solution until reaching
-the acceptable approximation of the exact solution.}
+The benchmark suite contains seven applications: CG, MG, EP, LU, BT, SP and FT. These benchmarks are considered as message passing applications with iterations because the same block of operations is executed many times.
These applications have different computations and communications ratios and strategies which make them good testbed applications to evaluate the proposed algorithm and energy model.
-The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In this work, class D was used for all benchmarks in all the experiments presented in the next sections.
+The benchmarks have seven different classes, S, W, A, B, C, D and E, that represent the size of the problem that the method solves. In the next sections, the class D was used for all the benchmarks in all the experiments.
\subsection{The experimental results of the scaling algorithm}
\label{sec.res}
In this section, the results of the application of the scaling factors selection algorithm \ref{HSA}
-to the NAS parallel benchmarks are presented. \textcolor{blue}{Each experiment of this section and next sections has been executed many times and the results presented in the figures are the average values of many execution.}
-
-As mentioned previously, the experiments
+to the NAS parallel benchmarks are presented. Each experiment has been executed many times and the results presented in the figures are the average values of many executions. As mentioned previously, the experiments
were conducted over two sites of Grid'5000, Lyon and Nancy sites.
Two scenarios were considered while selecting the clusters from these two sites :
\begin{itemize}
The main reason
for using these two scenarios is to evaluate the influence of long distance communications (higher latency) on the performance of the
scaling factors selection algorithm. Indeed, in the first scenario the computations to communications ratio
-is very low due to the higher communication times which reduce the effect of DVFS operations.
+is very low due to the higher communication times which reduces the effect of the DVFS operations.
The NAS parallel benchmarks are executed over
16 and 32 nodes for each scenario. The number of participating computing nodes from each cluster
& Graphene & Nancy & 5 \\ \cline{2-4}
& Griffon & Nancy & 6 \\
\hline
-\multirow{3}{*}{Tow sites / 32 nodes} & Taurus & Lyon & 10 \\ \cline{2-4}
+\multirow{3}{*}{Two sites / 32 nodes} & Taurus & Lyon & 10 \\ \cline{2-4}
& Graphene & Nancy & 10 \\ \cline{2-4}
& Griffon &Nancy & 12 \\
\hline
using the proposed frequency selection algorithm is measured
using the equation of the reduced energy consumption, Equation~\ref{eq:energy}. This model uses the measured dynamic power showed in Table~\ref{table:grid5000}
and the static
-power is assumed to be equal to 20\% of the dynamic power \textcolor{blue}{as in \cite{Rauber_Analytical.Modeling.for.Energy}}. The execution
+power is assumed to be equal to 20\% of the dynamic power as in \cite{Rauber_Analytical.Modeling.for.Energy}. The execution
time is measured for all the benchmarks over these different scenarios.
The energy consumptions and the execution times for all the benchmarks are
is exponentially related to the CPU's frequency value. On the other hand, the increase in the number of computing nodes can
increase the communication times and thus produces less energy saving depending on the
benchmarks being executed. The results of benchmarks CG, MG, BT and FT show more
-energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes. LU and SP consume more energy with 16 nodes than 32 in one site because their computations to communications ratio is not affected by the increase of the number of local communications.
+energy saving percentage in the one site scenario when executed over 16 nodes than over 32 nodes. LU and SP consume more energy with 16 nodes than 32 nodes on one site because their computations to communications ratio is not affected by the increase of the number of local communications.
\begin{figure*}[!h]
\centering
Figure \ref{fig:per_d} presents the performance degradation percentages for all benchmarks over the two scenarios.
The performance degradation percentage for the benchmarks running on two sites with
16 or 32 nodes is on average equal to 8.3\% or 4.7\% respectively.
-For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes without significantly degrading their performance because the communication times are higher with 32 nodes which results in smaller computations to communications ratio. On the other hand, the performance degradation percentage for the benchmarks running on one site with
+For this scenario, the proposed scaling algorithm selects smaller frequencies for the executions with 32 nodes without significantly degrading their performance because the communication times are high with 32 nodes which results in smaller computations to communications ratio. On the other hand, the performance degradation percentage for the benchmarks running on one site with
16 or 32 nodes is on average equal to 3.2\% and 10.6\% respectively. In contrary to the two sites scenario, when the number of computing nodes is increased in the one site scenario, the performance degradation percentage is increased. Therefore, doubling the number of computing
nodes when the communications occur in high speed network does not decrease the computations to
communication ratio.
presented in Figures~\ref{fig:edp-eng}, \ref{fig:edp-perf} and \ref{fig:edp-dist} respectively.
As shown in these figures, the proposed frequencies selection algorithm, Maxdist, outperforms the EDP algorithm in terms of energy consumption reduction and performance for all of the benchmarks executed over the two scenarios.
-The proposed algorithm gives better results than the EDP method because it
-maximizes the energy saving and the performance at the same time.
+The proposed algorithm gives better results than the EDP method because the former selects the set of frequencies that
+gives the best tradeoff between energy saving and performance.
Moreover, the proposed scaling algorithm gives the same weight for these two metrics.
Whereas, the EDP algorithm gives sometimes negative trade-off values for some benchmarks in the two sites scenarios.
These negative trade-off values mean that the performance degradation percentage is higher than the energy saving percentage.
The algorithm selects the best vector of
frequencies that maximizes the trade-off distance
between the predicted energy consumption and the predicted execution time of the distributed
- applications \textcolor{blue}{with iterations} running over a heterogeneous grid. A new energy model
+ applications with iterations running over a heterogeneous grid. A new energy model
is used by the proposed algorithm to predict the energy consumption
-of the distributed message passing application \textcolor{blue}{with iterations} running over a grid architecture.
+of the application.
To evaluate the proposed method on a real heterogeneous grid platform, it was applied on the
NAS parallel benchmarks and the class D instance was executed over the Grid'5000 testbed platform.
The experiments executed on 16 nodes, distributed over three clusters, showed that the algorithm on average reduces by 30\% the energy consumption
the well known energy and delay product as an objective function. The comparison results showed
that the proposed algorithm outperforms the latter by selecting a vector of frequencies that gives a better trade-off between energy consumption reduction and performance.
-In the near future, \textcolor{blue}{we will adapt the proposed algorithm to take the variability between some iterations in two steps. In the first step, the algorithm selects the best frequencies at the end of the first iterations and apply them to the system. In the second step, after some iterations (e.g. 5 iterations) the algorithm recomputes the frequencies depending on the average of the communication and computation times for all previous iterations. It will change the frequency of each node if the new frequency is different from the old one. Otherwise, it keeps the old frequency.}
-Also, we would like to develop a similar method that is adapted to
-asynchronous applications \textcolor{blue}{with iterations} where iterations are not synchronized and communications are overlapped with computations.
+In the near future, we will adapt the proposed algorithm to take into consideration the variability between some iterations. For example, the proposed algorithm can be executed twice: after the first iteration the frequencies are scaled down according to the execution times measured in the first iteration, then after a fixed number of iterations, the frequencies are adjusted according to the execution times measured during the fixed number of iterations. If the computing power of the system is constantly changing, it would be interesting to implement a mechanism that detects this change and adjusts the frequencies according to the variability of the system. We would like also to develop a similar method that is adapted to
+asynchronous applications with iterations where iterations are not synchronized and communications are overlapped with computations.
The development of such a method might require a new energy model because the
number of iterations is not known in advance and depends on
-the global convergence of the iterative system.
+the global convergence of the iterative system. Finally, it would be interesting to evaluate the scalability of the proposed algorithm by running it on large platforms composed of many thousands of cores. The scalability of the algorithm can be improved by distributing it in a hierarchical manner where a leader is chosen for each cluster or a group of nodes to compute their scaled frequencies and by using asynchronous messages to exchange the the data measured at the first iteration.
