\ead{arnaud.giersch@femto-st.fr}
\address{FEMTO-ST, University of Franche-Comté\\
- 19 avenue de Maréchal Juin, BP 527, 90016 Belfort cedex , France\\
- % Tel.: +123-45-678910\\
- % Fax: +123-45-678910\\
-}
+ 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France}
\cortext[cor]{Corresponding author.}
network performance and the message size, the time of the reception of the
message also varies.
-In the following of this paper, Section~\ref{BT algo} describes the Bertsekas
-and Tsitsiklis' asynchronous load balancing algorithm. Moreover, we present a
-possible problem in the convergence conditions. Section~\ref{Best-effort}
-presents the best effort strategy which provides an efficient way to reduce the
-execution times. This strategy will be compared with other ones, presented in
-Section~\ref{Other}. In Section~\ref{Virtual load}, the virtual load mechanism
-is proposed. Simulations allowed to show that both our approaches are valid
-using a quite realistic model detailed in Section~\ref{Simulations}. Finally we
-give a conclusion and some perspectives to this work.
+In the following of this paper, Section~\ref{sec.bt-algo} describes the
+Bertsekas and Tsitsiklis' asynchronous load balancing algorithm. Moreover, we
+present a possible problem in the convergence conditions.
+Section~\ref{sec.besteffort} presents the best effort strategy which provides an
+efficient way to reduce the execution times. This strategy will be compared
+with other ones, presented in Section~\ref{sec.other}. In
+Section~\ref{sec.virtual-load}, the virtual load mechanism is proposed.
+Simulations allowed to show that both our approaches are valid using a quite
+realistic model detailed in Section~\ref{sec.simulations}. Finally we give a
+conclusion and some perspectives to this work.
\section{Bertsekas and Tsitsiklis' asynchronous load balancing algorithm}
-\label{BT algo}
+\label{sec.bt-algo}
In order prove the convergence of asynchronous iterative load balancing
Bertsekas and Tsitsiklis proposed a model
the amount of load of processor $i$ at time $t+1$ is given by:
\begin{equation}
x_i(t+1)=x_i(t)-\sum_{j\in V(i)} s_{ij}(t) + \sum_{j\in V(i)} r_{ji}(t)
-\label{eq:ping-pong}
+\label{eq.ping-pong}
\end{equation}
x_3(t)=99.99\\
x_3^2(t)=99.99\\
\end{eqnarray*}
-In this case, processor $2$ can either sends load to processor $1$ or processor
-$3$. If it sends load to processor $1$ it will not satisfy condition
-(\ref{eq:ping-pong}) because after the sending it will be less loaded that
+In this case, processor $2$ can either sends load to processor $1$ or processor
+$3$. If it sends load to processor $1$ it will not satisfy condition
+(\ref{eq.ping-pong}) because after the sending it will be less loaded that
$x_3^2(t)$. So we consider that the \emph{ping-pong} condition is probably to
strong. Currently, we did not try to make another convergence proof without this
condition or with a weaker condition.
algorithm.
\section{Best effort strategy}
-\label{Best-effort}
+\label{sec.besteffort}
In this section we describe a new load-balancing strategy that we call
\emph{best effort}. First, we explain the general idea behind this strategy,
Concretely, once $s_{ij}$ has been evaluated as before, it is simply divided by
some configurable factor. That's what we named the ``parameter $k$'' in
-Section~\ref{Results}. The amount of data to send is then $s_{ij}(t) = (\bar{x}
-- x^i_j(t))/k$.
-\FIXME[check that it's still named $k$ in Sec.~\ref{Results}]{}
+Section~\ref{sec.results}. The amount of data to send is then $s_{ij}(t) =
+(\bar{x} - x^i_j(t))/k$.
+\FIXME[check that it's still named $k$ in Sec.~\ref{sec.results}]{}
\section{Other strategies}
-\label{Other}
+\label{sec.other}
Another load balancing strategy, working under the same conditions, was
previously developed by Bahi, Giersch, and Makhoul in
\section{Virtual load}
-\label{Virtual load}
+\label{sec.virtual-load}
In this section, we present the concept of \emph{virtual load}. In order to
use this concept, load balancing messages must be sent using two different kinds
\FIXME{describe integer mode}
\section{Simulations}
-\label{Simulations}
+\label{sec.simulations}
In order to test and validate our approaches, we wrote a simulator
using the SimGrid
characteristics of the running platform, etc. Then several metrics
are issued that permit to compare the strategies.
-The simulation model is detailed in the next section (\ref{Sim
- model}), and the experimental contexts are described in
-section~\ref{Contexts}. Then the results of the simulations are
-presented in section~\ref{Results}.
+The simulation model is detailed in the next section (\ref{sec.model}), and the
+experimental contexts are described in section~\ref{sec.exp-context}. Then the
+results of the simulations are presented in section~\ref{sec.results}.
\subsection{Simulation model}
-\label{Sim model}
+\label{sec.model}
In the simulation model the processors exchange messages which are of
two kinds. First, there are \emph{control messages} which only carry
\end{algorithm}
\paragraph{}\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?
-par ex, donner l'idée générale de l'implémentation. l'idée générale est déja décrite en section~\ref{Virtual load}}
+ par ex, donner l'idée générale de l'implémentation. l'idée générale est déja
+ décrite en section~\ref{sec.virtual-load}}
\subsection{Experimental contexts}
-\label{Contexts}
+\label{sec.exp-context}
In order to assess the performances of our algorithms, we ran our
simulator with various parameters, and extracted several metrics, that
time.
\subsubsection{Metrics}
+\label{sec.metrics}
In order to evaluate and compare the different load balancing strategies we had
to define several metrics. Our goal, when choosing these metrics, was to have
\subsection{Experimental results}
-\label{Results}
+\label{sec.results}
In this section, the results for the different simulations will be presented,
and we'll try to explain our observations.
This suggests that the relative performances of the different strategies are not
influenced by the characteristics of the physical platform. The differences in
the convergence times can be explained by the fact that on the grid platforms,
-distant sites are interconnected by links of smaller bandwith.
+distant sites are interconnected by links of smaller bandwidth.
Therefore, in the following, we'll only discuss the results for the grid
platforms.
when the load to balance is initially randomly distributed over all nodes.
On both figures, the computation/communication cost ratio is $10/1$ on the left
-column, and $1/10$ on the right column. With a computatio/communication cost
+column, and $1/10$ on the right column. With a computation/communication cost
ratio of $1/1$ the results are just between these two extrema, and definitely
don't give additional information, so we chose not to show them here.
are given for the process topology being, from top to bottom, a line, a torus or
an hypercube.
-\FIXME{explain how to read the graphs}
-
-each bar -> times for an algorithm
-recall the different times
-no bar -> not run or did not converge in allocated time
-
-repeated for the different platform sizes.
+Finally, on the graphs, the vertical bars show the measured times for each of
+the algorithms. These measured times are, from bottom to top, the average idle
+time, the average convergence date, and the maximum convergence date (see
+Section~\ref{sec.metrics}). The measurements are repeated for the different
+platform sizes. Some bars are missing, specially for large platforms. This is
+either because the algorithm did not reach the convergence state in the
+allocated time, or because we simply decided not to run it.
\FIXME{donner les premières conclusions, annoncer le plan de la suite}
+\FIXME{comparer be/makhoul -> be tient la route (parler du cas réel uniquement)}
\subsubsection{With the virtual load extension}
+\FIXME{valider l'extension virtual load -> c'est 'achement bien}
+
\subsubsection{The $k$ parameter}
-\subsubsection{With an initial random repartition, and larger platforms}
+\FIXME{proposer le -k -> ça peut aider dans certains cas}
+
+\subsubsection{With an initial random distribution, and larger platforms}
+
+\FIXME{dire quoi ici ?}
\subsubsection{With integer load}
+\FIXME{conclure avec la version entière -> on n'a pas l'effet d'escalier !}
+
\FIXME{what about the amount of data?}
-\begin{itshape}
-\FIXME{remove that part}
-Dans cet ordre:
-...
-- comparer be/makhoul -> be tient la route
- -> en réel uniquement
-- valider l'extension virtual load -> c'est 'achement bien
-- proposer le -k -> ça peut aider dans certains cas
-- conclure avec la version entière -> on n'a pas l'effet d'escalier !
-Q: comment inclure les types/tailles de platesformes ?
-Q: comment faire des moyennes ?
-Q: comment introduire les distrib 1/N ?
-...
-
-On constate quoi (vérifier avec les chiffres)?
+\FIXME{On constate quoi (vérifier avec les chiffres)?
\begin{itemize}
\item cluster ou grid, entier ou réel, ne font pas de grosses différences
-
\item bookkeeping? améliore souvent les choses, parfois au prix d'un retard au démarrage
-
\item makhoul? se fait battre sur les grosses plateformes
-
\item taille de plateforme?
-
\item ratio comp/comm?
-
\item option $k$? peut-être intéressant sur des plateformes fortement interconnectées (hypercube)
-
\item volume de comm? souvent, besteffort/plain en fait plus. pourquoi?
-
\item répartition initiale de la charge ?
-
\item integer mode sur topo. line n'a jamais fini en plain? vérifier si ce n'est
pas à cause de l'effet d'escalier que bk est capable de gommer.
-
-\end{itemize}
+\end{itemize}}
% On veut montrer quoi ? :
% Prendre un réseau hétérogène et rendre processeur homogène
% Taille : 10 100 très gros
-\end{itshape}
\section{Conclusion and perspectives}
\FIXME{conclude!}
-\section*{Acknowledgements}
+\section*{Acknowledgments}
Computations have been performed on the supercomputer facilities of the
Mésocentre de calcul de Franche-Comté.
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-% LocalWords: ik isend irecv Cortés et al chan ctrl fifo Makhoul GFlop xml pre
-% LocalWords: FEMTO Makhoul's fca bdee cdde Contassot Vivier underlaid
+% LocalWords: Raphaël Couturier Arnaud Giersch Franche ij Bertsekas Tsitsiklis
+% LocalWords: SimGrid DASUD Comté asynchronism ji ik isend irecv Cortés et al
+% LocalWords: chan ctrl fifo Makhoul GFlop xml pre FEMTO Makhoul's fca bdee
+% LocalWords: cdde Contassot Vivier underlaid du de Maréchal Juin cedex calcul
+% LocalWords: biblio
