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\begin{document}
for asynchronous iterative load balancing}
\author{Raphaël Couturier \and
- Arnaud Giersch \and
- Abderrahmane Sider
+ Arnaud Giersch
}
\institute{R. Couturier \and A. Giersch \at
\email{%
raphael.couturier@univ-fcomte.fr,
arnaud.giersch@univ-fcomte.fr}
- \and
- A. Sider \at
- University of Béjaïa, Béjaïa, Algeria \\
- \email{ar.sider@univ-bejaia.dz}
}
\maketitle
DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous}, propose a
version working with integer load. This work was later generalized by
the same authors in \cite{cedo+cortes+ripoll+al.2007.convergence}.
-{\bf Rajouter des choses ici}.
+\FIXME{Rajouter des choses ici.}
Although the Bertsekas and Tsitsiklis' algorithm describes the condition to
ensure the convergence, there is no indication or strategy to really implement
amount of load. Moreover, when real asynchronous applications are considered,
using asynchronous load balancing algorithms can reduce the execution
times. Most of the times, it is simpler to distinguish load information messages
-from data migration messages. Formers ones allows a node to inform its
+from data migration messages. Former ones allows a node to inform its
neighbors of its current load. These messages are very small, they can be sent
quite often. For example, if an computing iteration takes a significant times
(ranging from seconds to minutes), it is possible to send a new load information
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. 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{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.
$x_3^2(t)$. 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.
-
+%
+\FIXME{Develop: We have the feeling that such a weaker condition
+ exists, because (it's not a proof, but) we have never seen any
+ scenario that is not leading to convergence, even with LB-strategies
+ that are not fulfilling these two conditions.}
\section{Best effort strategy}
\label{Best-effort}
\end{equation*}
\end{enumerate}
+\FIXME{describe parameter $k$}
+
\section{Other strategies}
\label{Other}
-\textbf{Question} faut-il décrire les stratégies makhoul et simple ?
+\FIXME{Réécrire en angliche.}
-\paragraph{simple} Tentative de respecter simplement les conditions de Bertsekas.
-Parmi les voisins moins chargés que soi, on sélectionne :
-\begin{itemize}
-\item un des moins chargés (vmin) ;
-\item un des plus chargés (vmax),
-\end{itemize}
-puis on équilibre avec vmin en s'assurant que notre charge reste
-toujours supérieure à celle de vmin et à celle de vmax.
+% \FIXME{faut-il décrire les stratégies makhoul et simple ?}
-On envoie donc (avec "self" pour soi-même) :
-\[
- \min\left(\frac{load(self) - load(vmin)}{2}, load(self) - load(vmax)\right)
-\]
+% \paragraph{simple} Tentative de respecter simplement les conditions de Bertsekas.
+% Parmi les voisins moins chargés que soi, on sélectionne :
+% \begin{itemize}
+% \item un des moins chargés (vmin) ;
+% \item un des plus chargés (vmax),
+% \end{itemize}
+% puis on équilibre avec vmin en s'assurant que notre charge reste
+% toujours supérieure à celle de vmin et à celle de vmax.
+
+% On envoie donc (avec "self" pour soi-même) :
+% \[
+% \min\left(\frac{load(self) - load(vmin)}{2}, load(self) - load(vmax)\right)
+% \]
\paragraph{makhoul} Ordonne les voisins du moins chargé au plus chargé
puis calcule les différences de charge entre soi-même et chacun des
Doing this, we can expect a faster convergence since nodes have a faster
information of the load they will receive, so they can take in into account.
-\textbf{Question} Est ce qu'on donne l'algo avec virtual load?
+\FIXME{Est ce qu'on donne l'algo avec virtual load?}
+
+\FIXME{describe integer mode}
\section{Simulations}
\label{Simulations}
\paragraph{Load-balancing thread} The load-balancing thread is in
charge of running the load-balancing algorithm, and exchange the
-control messages. It iteratively runs the following operations:
+control messages. As shown in Algorithm~\ref{algo.lb}, it iteratively
+runs the following operations:
\begin{itemize}
\item get the control messages that were received from the neighbors;
\item run the load-balancing algorithm;
\paragraph{}
For the sake of simplicity, a few details were voluntary omitted from
these descriptions. For an exhaustive presentation, we refer to the
-actual code that was used for the experiments, and which is
-available at \textbf{FIXME URL}.
+actual source code that was used for the experiments%
+\footnote{As mentioned before, our simulator relies on the SimGrid
+ framework~\cite{casanova+legrand+quinson.2008.simgrid}. For the
+ experiments, we used a pre-release of SimGrid 3.7 (Git commit
+ 67d62fca5bdee96f590c942b50021cdde5ce0c07, available from
+ \url{https://gforge.inria.fr/scm/?group_id=12})}, and which is
+available at
+\url{http://info.iut-bm.univ-fcomte.fr/staff/giersch/software/loba.tar.gz}.
-\textbf{FIXME: ajouter des détails sur la gestion de la charge virtuelle ?}
+\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?}
\subsection{Experimental contexts}
\label{Contexts}
-\textbf{FIXME once the experimentation is done!}
+In order to assess the performances of our algorithms, we ran our
+simulator with various parameters, and extracted several metrics, that
+we will describe in this section. Overall, the experiments represent
+more than 240 hours of computing time.
+
+\paragraph{Load balancing strategies}
+
+We ran the experiments with the \emph{Best effort}, and with the \emph{Makhoul}
+strategies. \emph{Best effort} was tested with parameter $k = 1$, $k = 2$, and
+$k = 4$. Secondly, each strategy was run in its two variants: with, and without
+the management of \emph{virtual load}. Finally, we tested each configuration
+with \emph{real}, and with \emph{integer} load.
+This gives us as many as 32 different strategies.
+
+\paragraph{Configurations}
+\begin{description}
+\item[\textbf{platforms}] homogeneous (cluster); heterogeneous (subset
+ of Grid5000)
+\item[\textbf{platform size}] platforms with 16, 64, 256, and 1024 nodes
+\item[\textbf{topologies}] line; torus; hypercube
+\item[\textbf{initial load distribution}] initially on a only node;
+ initially on all nodes
+\item[\textbf{comp/comm ratio}] $10/1$, $1/1$, $1/10$
+\end{description}
+
+\paragraph{Metrics}
+
\begin{description}
-\item[platforms] homogeneous ; heterogeneous generated with the SIMULACRUM tool~\cite{QUINSON:2010:INRIA-00502839:1}
-\item[topologies]
-\item[algorithms]
-\item[etc.]
+\item[\textbf{average idle time}]
+\item[\textbf{average convergence date}]
+\item[\textbf{maximum convergence date}]
+\item[\textbf{data transfer amount}] relative to the total data amount
\end{description}
\subsection{Validation of our approaches}
\section{Conclusion and perspectives}
+\begin{acknowledgements}
+ Computations have been performed on the supercomputer facilities of
+ the Mésocentre de calcul de Franche-Comté.
+\end{acknowledgements}
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