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
+ 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.
\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_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:
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
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 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}
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
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.
