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 in cluster $h$ 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:
E = \sum_{i=1}^{N} \sum_{i=1}^{M_i} {(S_{ij}^{-2} \cdot \Pd[ij] \cdot \Tcp[ij])} +
\sum_{i=1}^{N} \sum_{j=1}^{M_i} (\Ps[ij] \cdot {} \\
(\mathop{\max_{i=1,\dots N}}_{j=1,\dots,M_i}({\Tcp[ij]} \cdot S_{ij})
- +\mathop{\min_{j=1,\dots M_i}} (\Tcm[hj]) ))
+ +\mathop{\min_{j=1,\dots M_h}} (\Tcm[hj]) ))
\end{multline}
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:
While the main goal is to optimize the energy and execution time at the same
time, the normalized energy and execution time curves do not evolve (increase/decrease) in the same way.
According to (\ref{eq:pnorm}) and (\ref{eq:enorm}), the
-vector of frequency scaling factors $S_1,S_2,\dots,S_N$ reduces both the energy
+vector of frequency scaling factors $S_{11},S_{12},\dots,S_{NM_i}$ reduces both the energy
and the execution time, but the main objective is to produce
maximum energy reduction with minimum execution time reduction.
This problem can be solved by making the optimization process for energy and
execution time follow the same evolution according to the vector of scaling factors
-$(S_{11}, S_{12},\dots, S_{NM})$. Therefore, the equation of the
+$(S_{11}, S_{12},\dots, S_{NM_i})$. Therefore, the equation of the
normalized execution time is inverted which gives the normalized performance
equation, as follows:
\begin{equation}
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
communication ratio.
The performance degradation percentage of the EP benchmark after applying the scaling factors selection algorithm is the highest in comparison to
-the other benchmarks. Indeed, in the EP benchmark, there are no communication and slack times and its
+the other benchmarks. Indeed, in the EP benchmark, there are no communication and no slack times and its
performance degradation percentage only depends on the frequencies values selected by the algorithm for the computing nodes.
The rest of the benchmarks showed different performance degradation percentages which decrease
when the communication times increase and vice versa.
The execution times for most of the NAS benchmarks are higher over the multi-core per node scenario
than over the single core per node scenario. Indeed,
- the communication times are higher in the one site multi-core scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.
+ the communication times are higher in the multi-core scenario than in the latter scenario because all the cores of a node share the same node network link which can be saturated when running communication bound applications. Moreover, the cores of a node share the memory bus which can be also saturated and become a bottleneck.
Moreover, the energy consumptions of the NAS benchmarks are lower over the
one core scenario than over the multi-core scenario because
the first scenario had less execution time than the latter which results in less static energy being consumed.
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.
\section*{Acknowledgment}
This work has been partially supported by the Labex ACTION project (contract
-``ANR-11-LABX-01-01''). Computations have been performed on the Grid'5000 platform. As a PhD student,
+``ANR-11-LABX-01-01''). Computations have been performed on the Grid'5000
+platform and on the mésocentre of Franche-Comté. As a PhD student,
Mr. Ahmed Fanfakh, would like to thank the University of Babylon (Iraq) for
supporting his work.
